CONTRACTTHEORY

CONTRACTTHEORY

PatrickBoltonand MathiasDewatripont

The MITPressCambridge,MassachusettsLondon,England

\302\251 2005 by the Massachusetts Institute of Technology.

All rights reserved. No part of this book may bereproduced in any form by any electronicor mechanical means (including photocopying, recording, or information storage andretrieval) without permission in writing from the publisher.

This book was set in Times Ten by SNP Best-set Typesetter Ltd., Hong Kong, and wasprinted and bound in the United States of America.

Library of Congress Cataloging-in-Publication Data

Bolton, Patrick, 1957-Contract theory / Patrick Bolton and Mathias Dewatripont.

p. cm.Includes bibliographical references and index.ISBN 0-262-02576-01.Contracts\342\200\224Methodology. I. Dewatripont, M. (Mathias) II.Title.

K840.B65 2004346.02'01\342\200\224dc22

2004055902

Contents

Preface xv

1 Introduction 11.1 Optimal Employment Contractswithout Uncertainty, Hidden

Information, or HiddenActions 41.2 Optimal Contractsunder Uncertainty 7

1.2.1Pure Insurance 81.2.2Optimal Employment Contractsunder Uncertainty 11

1.3 Information and Incentives 141.3.1AdverseSelection 151.3.2MoralHazard 20

1.4 Optimal Contracting with Multilateral AsymmetricInformation 251.4.1Auctions and Tradeunder Multilateral Private

Information 261.4.2MoralHazard in Teams,Tournaments, and

Organizations 271.5 TheDynamics of IncentiveContracting 30

1.5.1Dynamic AdverseSelection 311.5.2Dynamic MoralHazard 34

1.6 IncompleteContracts 361.6.1Ownershipand Employment 371.6.2IncompleteContractsand Implementation Theory 391.6.3BilateralContractsand Multilateral Exchange 40

1.7 Summing Up 42

Part I STATICBILATERALCONTRACTING 45

2 HiddenInformation, Screening 472.1 The Simple EconomicsofAdverseSelection 47

2.1.1First-BestOutcome:PerfectPriceDiscrimination 482.1.2AdverseSelection,LinearPricing, and Simple

Two-PartTariffs 492.1.3Second-BestOutcome:OptimalNonlinear Pricing 52

2.2 Applications 572.2.1CreditRationing 572.2.2Optimal IncomeTaxation 62

vi Contents

2.2.3Implicit Labor Contracts 672.2.4 Regulation 74

2.3 MoreThan TwoTypes 77

2.3.1Finite Number ofTypes 77

2.3.2Random Contracts 812.3.3A Continuum ofTypes 82

2.4 Summary 932.5 LiteratureNotes 96

HiddenInformation, Signaling 993.1 Spence'sModelof Educationas a Signal 100

3.1.1Refinements 1073.2 Applications 112

3.2.1CorporateFinancing and Investment Decisionsunder Asymmetric Information 112

3.2.2Signaling Changesin CashFlow through DividendPayments 120

3.3 Summary and LiteratureNotes 125

HiddenAction, MoralHazard 1294.1 TwoPerformanceOutcomes 130

4.1.1First-BestversusSecond-BestContracts 1304.1.2The SecondBestwith BilateralRisk Neutrality and

ResourceConstraints for the Agent 1324.1.3SimpleApplications 1334.1.4BilateralRisk Aversion 1354.1.5TheValue of Information 136

4.2 LinearContracts,Normally DistributedPerformance,and

Exponential Utility 1374.3 The Suboptimality of LinearContractsin the Classical

Model 1394.4 GeneralCase:TheFirst-OrderApproach 142

4.4.1Characterizingthe SecondBest 1424.4.2 When is the First-OrderApproachValid? 148

4.5 Grossmanand Hart'sApproachto the Principal-AgentProblem 152

4.6 Applications 157

vii Contents

4.6.1ManagerialIncentiveSchemes 1574.6.2The Optimality of DebtFinancing under Moral

Hazardand Limited Liability 1624.7 Summary 1684.8 LiteratureNotes 169Disclosureof Private CertifiableInformation 1715.1 Voluntary DisclosureofVerifiable Information 172

5.1.1Private, UncertifiableInformation 1725.1.2Private, CertifiableInformation 1735.1.3TooMuch Disclosure 1745.1.4Unraveling and the Full DisclosureTheorem 1755.1.5Generalizingthe Full DisclosureTheorem 176

5.2 Voluntary Nondisclosureand Mandatory-DisclosureLaws 1785.2.1Two Examplesof No Disclosureor Partial

Voluntary Disclosure 1795.2.2 Incentivesfor Information Acquisition and the

Role of Mandatory-DisclosureLaws 1805.2.3 No Voluntary DisclosureWhen the Informed

Party CanBeEither a Buyer or a Seller 1865.3 CostlyDisclosureand DebtFinancing 1905.4 Summary and LiteratureNotes 197Multidimensional IncentiveProblems 1996.1 AdverseSelectionwith Multidimensional Types 199

6.1.1An ExampleWhere Bundling IsProfitable 2006.1.2When IsBundling Optimal?A LocalAnalysis 2016.1.3Optimal Bundling: A GlobalAnalysis in the 2 x2

Model 2046.1.4GlobalAnalysis for the GeneralModel 212

6.2 MoralHazardwith Multiple Tasks 2166.2.1Multiple Tasksand Effort Substitution 2186.2.2Conflicting Tasksand Advocacy 223

6.3 An ExampleCombining MoralHazardand AdverseSelection 2286.3.1Optimal Contractwith MoralHazardOnly 2306.3.2Optimal Contractwith AdverseSelectionOnly 231

viii Contents

6.3.3Optimal Saleswith Both AdverseSelectionandMoralHazard 231

6.4 Summary and LiteratureNotes 233

Partn STATICMULTILATERALCONTRACTING 237

7 Multilateral Asymmetric Information: BilateralTrading andAuctions 2397.1 Introduction 2397.2 BilateralTrading 243

7.2.1 TheTwo-TypeCase 2437.2.2 Continuum ofTypes 250

7.3 Auctions with PerfectlyKnown Values 2617.3.1 OptimalEfficient Auctions with IndependentValues 2627.3.2 Optimal Auctions with IndependentValues 2657.3.3 Standard Auctions with IndependentValues 2677.3.4 Optimal Independent-ValueAuctions with a

Continuum ofTypes:The RevenueEquivalenceTheorem 271

7.3.5 Optimal Auctions with CorrelatedValues 2767.3.6 The Roleof Risk Aversion 2787.3.7 The Role ofAsymmetrically DistributedValuations 280

7.4 Auctions with Imperfectly Known Common Values 2827.4.1 TheWinner's Curse 2837.4.2 StandardAuctions with Imperfectly Known

CommonValues in the 2 x2 Model 2857.4.3 Optimal Auctions with Imperfectly Known

Common Values 2887.5 Summary 2907.6 LiteratureNotes 2927.7 Appendix: Breakdownof RevenueEquivalencein a 2 x3

Example 294

8 Multiagent MoralHazardand Collusion 2978.1 MoralHazard in Teamsand Tournaments 299

8.1.1UnobservableIndividual Outputs:The Needfor aBudget Breaker 301

ix Contents

8.1.2UnobservableIndividual Outputs:Using OutputObservationsto Implement the First Best 305

8.1.3ObservableIndividual Outputs 3118.1.4Tournaments 316

8.2 Cooperationor Competitionamong Agents 3268.2.1Incentivesto Helpin Multiagent Situations 3268.2.2Cooperationand Collusionamong Agents 331

8.3 Supervisionand Collusion 3388.3.1Collusionwith Hard Information 3388.3.2Application: Auditing 342

8.4 Hierarchies 3518.5 Summary 3608.6 LiteratureNotes 362

Part IDREPEATEDBILATERALCONTRACTING 365

9 Dynamic AdverseSelection 3679.1 Dynamic AdverseSelectionwith FixedTypes 367

9.1.1CoasianDynamics 3699.1.2Insuranceand Renegotiation 3799.1.3SoftBudgetConstraints 3849.1.4Regulation 388

9.2 RepeatedAdverseSelection:Changing Types 3969.2.1Banking and Liquidity Transformation 3979.2.2Optimal Contracting with Two Independent

Shocks 4029.2.3Second-BestRisk Sharing betweenInfinitely

LivedAgents 4089.3 Summary and LiteratureNotes 415

10 Dynamic MoralHazard 41910.1TheTwo-PeriodProblem 420

10.1.1NoAccessto Credit 42210.1.2MonitoredSavings 42610.1.3Free Savings and Asymmetric Information 429

10.2TheT-periodProblem:SimpleContractsand the Gainsfrom Enduring Relations 431

x Contents

10.3

10.4

10.5

10.610.7

10.2.1RepeatedOutput10.2.2RepeatedActions10.2.3RepeatedActions and Output10.2.4Infinitely RepeatedActions, Output, and

ConsumptionMoralHazardand Renegotiation10.3.1RenegotiationWhen Effort IsNot Observedby

the Principal10.3.2RenegotiationWhen Effort IsObservedby the

PrincipalBilateralRelationalContracts10.4.1MoralHazard10.4.2AdverseSelection10.4.3ExtensionsImplicit Incentivesand CareerConcerns10.5.1The Single-TaskCase10.5.2The Multitask Case10.5.3TheTrade-OffbetweenTalentRisk and Incentives

under CareerConcernsSummaryLiteratureNotes

433434435

447450

450

456461468468470470473475

481483484

Part IV INCOMPLETECONTRACTS 487

11 IncompleteContractsand Institution Design 48911.1Introduction: IncompleteContractsand the Employment

Relation 48911.1.1The Employment Relation 49011.1.2A Theoryof the Employment RelationBasedon

ExPostOpportunism 49111.2Ownershipand the Property-Rights Theoryof the Firm 498

11.2.1A GeneralFramework with ComplementaryInvestments 500

11.2.2A Framework with Substitutable Investments 51511.3Financial Structure and Control 521

11.3.1Wealth Constraints and Contingent AllocationsofControl 523

Contents

11.3.2Wealth Constraints and Optimal DebtContractswhen EntrepreneursCanDivert CashFlow 534

11.4Summary 54911.5LiteratureNotes 551

Foundations of Contracting with Unverifiable Information 55312.1Introduction 55312.2Nash and Subgame-PerfectImplementation 555

12.2.1Nash Implementation: Maskin'sTheorem 55512.2.2Subgame-PerfectImplementation 558

12.3TheHoldupProblem 56012.3.1SpecificPerformanceContractsand Renegotiation

Design 56312.3.2OptionContractsand Contracting at Will 56612.3.3DirectExternalities 57012.3.4Complexity 572

12.4ExPostUnverifiable Actions 57812.4.1Financial Contracting 57912.4.2Formal and Real Authority 585

12.5ExPostUnverifiable Payoffs 58812.5.1The Spot-ContractingMode 59112.5.2The Employment Relationand Efficient Authority 594

12.6Summary and LiteratureNotes 597

Markets and Contracts 60113.1(Static)AdverseSelection:Market Breakdownand

ExistenceProblems 60113.1.1The Caseof a SingleContract 60213.1.2The Caseof Multiple Contracts 604

13.2Contractsas a Barrier to Entry 60613.3Competitionwith BilateralNonexclusiveContractsin the

Presenceof Externalities 60913.3.1The Simultaneous Offer Game 61413.3.2The SequentialOfferGame 62313.3.3TheBidding Game:CommonAgency and Menu

Auctions 62813.4Principal-Agent Pairs 630

xii Contents

13.5Competitionas an Incentive Scheme 63613.6Summary and LiteratureNotes 641

APPENDIX 645

14 Exercises 647

References 687Author Index 709

SubjectIndex 715

To Our Families

Preface

Contracttheory, information economics,incentive theory, and organizationtheory have beenhighly successfuland activeresearchareas in economics,finance, management, and corporate law for more than three decades.Anumber of founding contributors have beenrewardedwith the Nobelprizein economicsfor their contributions in this generalarea,including RonaldCoase,HerbertSimon, William Vickrey, JamesMirrlees,GeorgeAkerlof,

JosephStiglitz, and MichaelSpence.There is now a vast literature relatingto contracttheory in leadingeconomics,finance, and law journals, and yeta relatively small number ofcorenotions and findings have found their way

into textbooks.The most recent graduate textbooks in microeconomics1devote a few chapters to basicnotions in incentive and information

economics like adverseselection,moral hazard, and mechanism design,but this

material servesonly as an introduction to theseenormoustopics.The goalof this bookis to providea synthesis of this huge area by

highlighting the common themes and methodologiesthat unite this field.Thebookcan serveboth as a complementary text for a graduate or advancedundergraduate coursein microeconomicsand for a graduate coursein

contract theory. Although we aim to provide very broad coverageof theresearchliterature, it is impossibleto do justiceto all the interestingarticles and all subfields that have emergedoverthe past30years.As a remedyagainst the most obvious gapsand omissions,we make a limited attempt to

providea short guide to the literature at the end of each chapter.Even if

this bookleavesout largeportionsof the literature, it still contains far toomuch material for evena full-semestercoursein contracttheory. Ourintention was to give instructors somediscretionoverwhich chapterstoemphasize and to leaveit to the students to do the background reading.

The bookalso presentsmethodologicalresults in the key applicationareaswherethey have beendeveloped,be they in laboreconomics,organization theory, corporatefinance,or industrial organization, for example.Inthis way, the bookcanalsoserveasa referencesourcefor researchersinterested in the very many applicationsof contract theory in economics.Thephilosophy of the bookis to stressapplicationsrather than generaltheorems, while providing a simplified yet self-containedtreatment of the keymodelsand methodologiesin the literature.

We owean immeasurableintellectual debt to our advisers,OliverHart,Andreu Mas-Colell,Eric Maskin, John Moore, and Jean Tirole.Their

1.See,for example, the books by Kreps (1990)and Mas-Colell,Whinston, and Green (1995).

xvi Preface

influence is visible on almost every page of this book.And although wehave not had the good fortune to have them as our advisers,theintellectual influence of Bengt Holmstrom, Jean-JacquesLaffont, Paul Milgrom,JamesMirrlees,and RogerMyersonhasbeenjust as important. Theinspiration and support of our coauthors,Philippe Aghion, ChristopherHarris,Ian Jewitt, Bruno Jullien, Patrick Legros,SteveMatthews, Patrick Rey,Alisa

Roell, Gerard Roland, Howard Rosenthal, David Scharfstein,Ernst-Ludwig von Thadden,MichaelWhinston, and Chenggang Xu, has beeninvaluable. In particular, PhilippeAghion and Patrick Rey have playedamajor role throughout the long gestationperiodof this book.

Over the years,every chapter of the bookhas beentested in theclassroom. We thank our students at ECARES(Universite LibredeBruxelles),Tilburg, Princeton,MIT,Helsinki, and the summer schoolsin Oberweseland Gerzenseefor their comments.We also thank Philippe Aghion andOliverHart for using our manuscript in their contract theory coursesatHarvard and for their feedback.We are grateful to Kenneth Ayotte, EstelleCantillon, Antonio Estache, Antoine Faure-Grimaud,Denis Gromb,ChristopherHennessy,Andrei Hagiu, JacquesLawarree,JoelShapiro,JeanTirole,and three anonymous MITPressreadersfor comments and advice.We are particularly grateful to KathleenHurley, Diana Prout, and EllenSklar for all their help in preparingthe manuscript. We are alsoenormouslygrateful to our editors,Terry Vaughn and John Covell,for their continuing

support and for making sure that we bring this project to completion.

Introduction

Economicsis often definedas a field that aims to understand the processby which scarceresourcesare allocatedto their most efficient uses,andmarkets are generally seenas playing a central role in this process.But,more fundamentally, the simple activity of exchangeof goodsand services,whether on organizedexchangesor outsidea market setting, is the basicfirst step in any productionor allocationof resources.For a long timeeconomic theory has beenable to analyze formally only very basicexchangeactivities like the barter of two different commoditiesbetween two

individuals at a given placeand point in time.Mostmicroeconomicstextbooks1begin with an analysis of this basicsituation, representingit in the classic\"Edgeworth box.\" A slightly more involved exchangesituation that canalsoberepresentedin an Edgeworth boxis betweentwo individuals trading atdifferent points in time. Simplelending, investment, or futures contractscanbe characterizedin this way. However,such a simple reinterpretationalreadyraisesnew issues,like the possibility of default or nondelivery bythe other party in the future.

Until the 1940sor 1950sonly situations of simple exchangeof goodsand serviceswere amenable to formal analysis.More complexexchangeactivities like the allocationand sharing of risk began to be analyzedformally only with the introduction of the idea of \"state-contingent\"commodities by Arrow (1964)and Debreu(1959)and the formulation of atheory of \"choiceunder uncertainty\" by von Neumann and Morgenstern(1944)and others.Thenotion of exchangeofstate-contingentcommoditiesgave a precisemeaning to the exchangeand allocationof risk. Preferenceorderings over lotteries provided a formal representation of attitudes

toward risk and preferencesfor risk taking. Theseconceptualinnovations

are the foundations of modern theories of investment under risk and

portfoliochoice.In the late 1960sand 1970syet another conceptualbreakthrough took

placewith the introduction of \"private information\" and \"hidden actions\"

in contractual settings.The notions of \"incentive compatibility\" andincentives for \"truth telling\" providedthe basicunderpinnings for the theory ofincentives and the economicsof information. Theyalsoprovidedthe first

formal tools for a theory of the firm, corporatefinance, and, moregenerally,

a theory of economicinstitutions.

1.See,for example,Part 4of the celebrated book by Mas-Colell,Whinston, and Green (1995).

2 Introduction

Finally, much of the existing theory of long-term or dynamic contractingwas developedin the 1980sand 1990s.Contractrenegotiation,relationalcontracts,and incompletecontractsprovidedthe first toolsfor an analysisof \"ownership\" and \"control rights.\" Thesenotions, in turn, completethefoundations for a full-fledged theory of the firm and organizations.

Thereare by now many excellentfinance and economicstextbookscovering the theory of investment under risk, insurance, and riskdiversification. As this is alreadywell-exploredterritory, we shall not provide any

systematic coverageof theseideas.In contrast,to date there are only a fewbookscovering the theory of incentives, information, and economicinstitutions, which is generally referredto in short ascontracttheory.2 Therehasbeensuch a largeresearchoutput on these topicsin the last 30years that

it is an impossibletask to give a comprehensivesynthesis of all the ideasand methodsof contracttheory in a single book.Nevertheless,our aim isto be as wide ranging as possibleto give a senseof the richnessof the

theory\342\200\224its core ideasand methodology\342\200\224as well as its numerous possibleapplicationsin virtually all fields of economics.

Thus, in this bookwe attempt to coverall the major topicsin contracttheory that are taught in most graduate courses.Part Istarts with basicideasin incentive and information theory like screening,signaling, and moralhazard.Part IIcoversthe lesswell trodden material of multilateral

contracting with private information or hidden actions.In this part we providean introduction to auction theory, bilateraltradeunder private information,and the theory of internal organization of firms. Part IIIdealswith long-term contractswith private information or hidden actions.Finally, Part IVcoversincompletecontracts,the theory of ownership and control,and

contracting with externalities.Exercisesare collectedin a specificchapter atthe end of the book.

There is obviously too much material in this bookfor any one-semestercoursein contracttheory. Rather than imposeour own preferencesand ourown pet topics,we thought that it would be better to coverall the mainthemesof contracttheory and let instructors pick and choosewhich partsto coverin depth and which ones to leaveto the students to read.

Consistentwith our goalofproviding broadcoverageof the field,we haveaimedfor a style ofexpositionthat favors simplicity overgenerality orrigor.

2.Seein particular the textbooks by Salanie (1997)and Laffont and Martimort (2002).

3 Introduction

Our primary goal is to illustrate the core ideas,the main methodsin their

simplest self-containedform, and the wide applicability of the centralnotions of contract theory. More often than not, researcharticlesin

contract theory are hard to penetrate evenfor a well-trained reader.We have

goneto considerablelengths to make the centralideasand methodsin thesearticlesaccessible.Inevitably, we have been led to sacrificegenerality toachievegreatereaseofunderstanding. Ourhope is that oncethe main ideashave beenassimilatedthe interestedreaderwill find it easierto read the

original articles.In the remainderof this chapterwe providea briefoverview of the main

ideasand topicsthat are coveredin the bookby consideringa singleconcrete situation involving an employer and an employee.Dependingon the

topicwe are interestedin we shall take the employerto bea manager hiringa worker, or afarmer hiring asharecropper,or evena company ownerhiringa manager. Throughout the bookwe discussmany other applications,andthis briefoverview should not be taken to be the leading applicationofcontract theory. Before we proceedwith a brief descriptionof the multiplefacetsof this contracting problem,it is useful to beginby delineating theboundariesof the framework and stating the main assumptions that apply

throughout this bdok.The benchmark contracting situation that we shall consider in this

bookis one between two parties who operate in a market economywith

a wen-functioning legal system. Under such a system, any contract the

parties decideto write will be enforcedperfectlyby a court, provided,ofcourse, that it doesnot contravene any existing laws. We shall assumethroughout most of the bookthat the contracting parties do not needto

worry about whether the courts are able or willing to enforce the termsof the contractprecisely.Judgesare perfectlyrational individuals, whoseonly concernis to stick as closelyas possibleto the agreed terms of thecontract.The penaltiesfor breachingthe contractwill be assumedto besufficiently severe that no contracting party will ever consider the

possibility of not honoring the contract.We shall step outsidethis framework

only occasionallyto consider, for example, the case of self-enforcingcontracts.

Thus, throughout this bookwe shall assumeaway most of the problemslegalscholars,lawyers, and judgesare concernedwith in practiceandconcentrate only on the economicaspectsof the contract.We shall beprimarily

interested in determining what contractual clausesrational economic

4 Introduction

individuals are willing to sign and what types of transactions they are willing

to undertake.If the transaction is a simple exchangeofgoodsor servicesfor money, we

shall beinterestedin the terms of the transaction.What is the priceperunit

the parties shall agree on? Doesthe contractspecifyrebates?Are therepenalty clausesfor late delivery?If so,what form do they take?And soon.Alternatively, if the transaction is an insurancecontract,we shall beinterested in detennining how the terms vary with the underlying risk, with therisk aversionofthe parties,orwith the private information the insureeor theinsurer might have about the exactnature of the risk. We beginby briefly

reviewing the simplest possiblecontractual situation an employer and

employeemight face:a situation involving only two parties,transacting only

once,and facing nouncertainty and no private information orhidden actions.

1.1Optimal Employment Contractswithout Uncertainty, HiddenInformation,or HiddenActions

Considerthe following standard bilateralcontracting problembetweenan

employerand employee:the employeehas an initial endowment of time,which she can keepfor herself of sell to the employeras labor services,becausethe employercan make productive use of the employee'stime.

Specifically,we can assume therefore that the parties'utility functions

dependboth on the allocationof employeetime and on their purchasing

power.Let us denotethe employer'sutility function as U(l, t) where/ is the

quantity of employeetime the employerhas acquiredand t denotes the

quantity of \"money\"\342\200\224or equivalently the \"output\" that this money canbuy3\342\200\224that he has at his disposal.Similarly, employeeutility is u(l, t), where/ is the quantity of time the employeehas kept for herselfand t is the

quantityof money that she has at her disposal.

Supposethat the initial endowment of the individuals is (lu i{)= (0,1)for the employer (hereafter individual 1) and (l2, i2) = (1> 0) for the

employee(hereafterindividual 2).That is,without any trade,the employergets no employeetime but is assumedto have all the money, while the

employeehas all of her time for herselfbut has no money.

3. Indeed, the utility of money here reflects the utility derived from the consumption of acomposite good that can bepurchased with money.

5 Introduction

Both individuals could decidenot to trade, in which casethey would

each achievea utility levelof U = U(0,1)and u = u(l,0),respectively.If,however,both utility functions are strictly increasingin both arguments and

strictly concave,then both individuals may be able to increasetheir jointpayoff by exchanging labor services/ for money/output. What will be theoutcomeof their contractualnegotiations?That is,how many hours ofworkwill the employeebe willing to offer and what (hourly) wagewill she bepaid?

As in most economicstexts, we shall assumethroughout this bookthat

contracting partiesare rational individuals who aim to achievethe highest

possiblepayoff. The joint surplus maximization problemfor bothindividuals can be representedas follows.If we denote by /,- the amount ofemployeetime actually consumedand by tt the amount of output consumedby eachparty i =1,2 after trade, then the partieswill solvethe following

optimization problem:

mzxU(!i,ti)+nu(!T,t2) (1-1)u,u

subjectto aggregateresourceconstraints:

k+h=k+h=land h+t2 =?i+?2=1Hereji canreflectboth the individuals' respectivereservationutility levels,U and u, and their relativebargaining strengths.

When both utility functions are strictly increasing and concave,themaximum is completelycharacterizedby the first-orderconditions

Ul+(iui=0=Ut+(Jict (1.2)which imply

Ui _ Ui

Ut ut

SeeFigure 1.1,whereindifferencecurvesare drawn.

In otherwords,joint surplus maximization is achievedwhen the marginalrates of substitution betweenmoney and leisurefor both individuals areequalized.

Thereare gains from trade initially if

Ui iii

Ut ut

6 Introduction

Figure 1.1Classical Edgeworth Box

Howthesegains are sharedbetweenthe two individuals is determinedby

/i.The employeegets a higher share of the surplus the higher /i is.Thehighest possibleutility that the employeecan get is given by the solutionto the following optimization problem:

maxu(l2, t2) subjectto \302\2437(1 -l2,l-t2)>U

Similarly, the highest payoff the employercan get is given by the solutionto

max\302\243/(/i,/i) subjectto u(l-ll,l-ti)>uThese extremeproblemscan be interpreted as simple bargaining gameswhereoneparty has all the bargaining powerand makesa take-it-or-leave-it offer to the other party. Note, however,that by increasingu in the

employer'sconstrainedmaximization problem or U in the employee'sproblemonecanreducethe surplus that either individual gets.Thus a givendivision of the surplus canbe parameterizedby either jj., U, or u,

dependingon how the joint surplus maximization problemis formulated.

7 Introduction

Throughout this bookwe shall representoptimal contracting outcomesas solutions to constrainedoptimization problemslike the two precedingproblems.We thus take as starting point the Coasetheorem(1960),that is,the efficient contracting perspective,as long as informational problemsarenot present.4Although this representationseemsquite natural, it isimportant to highlight that behind it lie two implicit simplifying assumptions.First,the final contractthe partiesend up signing is independentof the

bargaining processleading up to the signature of the contract.In reality it is likelythat most contractsthat we seepartly reflectprior negotiations and eachparty's negotiating skills.But, if the main determinants of contractsare the

parties' objectives,technologicalconstraints, and outsideoptions, then it isnot unreasonable to abstract from the potentially complexbargaining

gamesthey might be playing. At least as a first approach,this simplifying

assumption appearsto be reasonable.Second,as we have alreadymentioned,the other relevant dimension of

the contracting problem that is generally suppressedin the precedingformal characterizationis the enforcementof the contract.Without legalinstitutions to enforcecontractsmany gains from tradeare left unexploitedby rational individuals becauseone or both fear that the other will fail tocarry out the agreedtransaction.In the absenceof courtsor other modesof enforcement,a transaction betweentwo or morepartiescan take placeonly if the exchangeof goodsor servicesis simultaneous. Otherwise,the

party who is supposedto executeher trade last will simply walk away. In

practice,achieving perfectsimultaneity is almost impossible,so that

important gains from trademay remain unexploitedin the absenceof an efficientenforcementmechanism.

1.2Optimal Contractsunder Uncertainty

Thereis more to employment contractsthan the simple characterizationinthe previoussection.One important dimension in reality is the extent towhich employeesare insured against economicdownturns. In most

developedeconomiesemployeesare at leastpartially protectedagainst the risk

of unemployment. Most existing unemployment insurance schemesare

4. As we shall detail throughout this book, informational problems will act asconstraints onthe set of allocations that contracts can achieve.

Introduction

nationwide insurance arrangements, funded by employerand employeecontributions, and guaranteeing a minimum fraction of a laid-offemployee'spay over a imnimum time horizon (ranging from one year toseveralyearswith a sliding scale).A fundamental economicquestionconcerning these insurance schemesis how much \"business-cycle\" and other\"firm-specific\" risk should be absorbedby employersand how much by

employees.Should employerstake on all the risk, and if so,why? Onetheory, dating back to Knight (1921)and formalizedmore recentlybyKihlstrom and Laffont (1979)and Kanbur (1979),holdsthat employers(or\"entrepreneurs\") should take on all the risk and fully insure employees.Thereason is that entrepreneursare natural \"risk lovers\" and are bestable toabsorbthe risk that \"risk-averse\" employeesdo not want to take.

To be able to analyze this question of optimal risk allocationformallyone must enrich the framework of section1.1by introducing uncertainty.At one level this extensionis extremely simple.All it takes is theintroduction of the notions of a state of nature, a state space,and a state-contingent commodity. Arrow (1964)and Debreu(1959)were the first to

explorethis extension.Theydefine a state of nature as any possiblefuture

event that might affect an individual's utility. Thestatespaceis then simplythe set of all possiblefuture events, and a state-contingentcommodity is agood that is redefinedto be a new commodity in every different state ofnature. For example,a given number of hours of work is a different

commodity in the middle of an economicboom than in a recession.The difficulty is not in defining all thesenotions.The important

conceptual leap is rather to supposethat rational individuals are able to form acompletedescriptionof all possiblefuture eventsand, moreover,that allhave the samedescriptionof the state space.Once this commondescription is determined,the basiccontracting problemcan be representedlikethe precedingone,although the interpretation of the contractwill bedifferent. More precisely,it is possibleto represent a simple insurancecontract,which specifiestrades between the employerand employeein

different statesofnature, in an Edgeworth box.Beforedoing so,let usconsider a pure insurance problemwithout production.

Pure Insurance

Considerthe simplest possiblesetting with uncertainty. Assume that thereare only two possiblefuture statesof nature, 6L and 6H.Tobe concrete,let6Lrepresentan adverseoutput shock,or a \"recession,\"and 6Ha goodoutput

9 Introduction

realization,or a \"boom.\" For simplicity, we disregardtime endowments.Then the state of nature influences only the value of output eachindividual hasas endowment. Specifically,assumethe following respectiveendowments for each individual in eachstate:

(tw Jil)=(2,1), for individual 1(hH,hi)=(2,1), for individual 2

Thevariablefy thereforedenotesthe endowment of individual i in state ofnature dr Note that in a \"recession\"aggregateoutput\342\200\2242\342\200\224is

lowerthan in

a boom\342\200\2244.

Beforethe stateofnature is realizedeachindividual has preferencesoverconsumption bundles (tL, tH) representedby the utility functions V(tL, tH)

for the employerand v(tL, tH) for the employee.If the two individuals donot exchangeany contingent commodities,their

ex ante utility (beforethe state of nature is realized)is V = V(2,1)and v

= v(2, 1).But they can also increase their ex ante utility by coinsuring

against the economicrisk. Note,however, that someaggregaterisk isuninsurable: the two individuals can do nothing to smooth the differencein

aggregateendowments between the two states. Nevertheless,they canincreasetheir ex ante utility by poolingtheir risks.

As before, the efficient amount of coinsuranceis obtained when thefinal allocationsof eachcontingent commodity {{hL, t2{), (tm, ?2h)} are suchthat

VtL +livtL =0=VtH +/ivtH (1.3)

which implies

SeeFigure 1.2,whereindifferencecurvesare drawn.

It should be clearby now that the analysis of pure exchangeunder

certainty can be transposedentirely to the casewith uncertainty once oneenlargesthe commodity spaceto include contingent commodities.

However,to obtain a full characterizationof the optimal contracting

problem under uncertainty one needs to put more structure on this

framework. Indeed,two important elementsare hidden in the precedingcharacterizationof the optimal insurance contract:one is a descriptionof

10 Introduction

Slope=-

Figure 1.2Optimal Coinsurance

Allocation under optimalinsurance

-Initial

endowment: equalshares of output

-*-L

ex post utility once the state of nature has beenrealized,and the other isthe probability of eachstate occurring.

The first completeframework of decisionmaking under uncertainty,which explicitly specifiesthe probability distribution overstatesand the expost utility in eachstate,is due to von Neumann and Morgenstern(1944).It is this framework that is usedin most contracting applications.Interestingly,

eventhough there is by now a largeliterature exploring a wide rangeof alternative modelsof individual choiceand behaviorunder uncertainty,there have beenrelatively few explorationsof the implications for optimalcontracting of alternative modelsof behaviorunder uncertainty.

In the setup consideredby von Neumann and Morgenstern,individual

expost utility functions are respectivelyU(t) and u(f) for the employerandemployee,whereboth functions are increasingin t. If we callp,- e (0,1)the

probability of occurrenceof any particular state of nature dh the ex anteutility function is simply definedas the expectationover ex post utilityoutcomes:

V(f1L, t1H)=pLU(t1L)+pHU(t1H)

Introduction

and

v(t2L, t2H) =pLu{t2L) +pHu{t2H)

Theeasiestway of thinking about the probability distribution \\p,) is simplyas an objectivedistribution that is known by both individuals. But it is alsopossibleto think of {p;} as a subjectivebelief that is common to bothindividuals. In most contracting applicationsit is assumedthat all partiessharea common priorbeliefand that differencesin (posterior)probability beliefsamong the parties only reflect differencesin information. Although this

basic assumption is rarely motivated, it generally reflectsthe somewhat

vague idea that all individuals are born with the same\"view of the world\"

and that their beliefsdiffer only if they have had different life experiences.Recently, however, there have beensomeattempts to exploretheimplications for optimal contracting of fundamental differencesin beliefsamongcontracting parties.

It is instructive to consider the optimal insurance conditions (i.3)when the individuals' ex ante utility function is assumedto be the Von

Neumann-Morgensternutility function that we have specified.In that

casethe marginal rate of substitution between commodities1and 2 is

given by

Vu. _ Pl UVu.)VtH Ph U'(t1H)

As this expressionmakesclear,the marginal rate of substitution betweenthe two contingent commoditiesvarieswith the probability distribution.

Moreover,the marginal rate of substitution is constant along the 45\302\260 line,where t1L = t1H.

Optimal Employment Contractsunder Uncertainty

Using the framework of von Neumann and Morgenstern,let us comebackto the contracting problemof section1.1with two goods,leisure / and aconsumption good t, which canbe readily extendedto include uncertaintyas follows:

Let (liL, hO and (hit, hH) represent the two different state-contingenttime/output bundles of the employer,and (l2L, t2L) and (l2H, t2H) the two

different state-contingenttime/output bundles of the employee.Also let

(li}, Uj) denote their respectiveinitial endowments, (i =1,2;;'=L,H).Then

12 Introduction

the optimal insurance contractsignedby the two individuals canberepresented as the solution to the optimal contracting problem:

max[pLU(l1L, t1L)+pHU(l1H,t1H)]

subjectto

pMhL,hd +Phu(12h , hH) ^ u (1.4)and

k, +l2, < kj +h, for j=L,H

hj +12,< h, +t2j for ] =L,Hwhere

u =pLu(l2L, t2L)+pHu(l2H, t2H)

One important advantage of the von Neumann and Morgensternformulation is that an individual's attitude toward risk canbe easilycharacterizedby the curvature of the ex post utility function. Thus, if both

\302\243/(\342\200\242)

and u(-)are strictly concave,then both individuals are risk averseand want to sharerisk, whereasif both

\302\243/(\342\200\242)

and u(-) are strictly convex,then bothindividuals are risk loving and want to trade gambleswith eachother.

For now, supposethat both individuals are risk averse,so that their expost utility functions are strictly concave.Then the contract-maximizingjoint surplus is fully characterizedby the first-orderconditions:

Ui\\hjihj) _ ui\\hj,hi) /-i r\\

Ut(lij,tij) ut(l2ht2j)

1 ln lj constant across0/s (1-6)ui \\hi j h;)

ut\\l2l,hj)constant across9/s (1-7)

Condition (1.5)is the familiar condition for efficient trade ex post.Thismeansthat ex ante efficiencyis achievedif and only if the contractis alsoexpostefficient.We shall seethat when incentive considerationsenter into

the contracting problemthere is usually a conflict betweenex ante and expostefficiency

13 Introduction

Conditions(1.6)and (1.7)are conditionsof optimal coinsurance.Condition (1.7)is sometimesreferred to as the Borchrule (1962):optimalcoinsurance requiresthe equalizationof the ratio ofmarginal utilities ofmoneyacrossstatesof nature.

A risk-neutral individual has a constant marginal utility of money.Thus,if one of the two individuals is risk neutral and the other individual is risk

averse,the Borch rule says that optimal insurance requires that the risk-averseindividual must alsohave a constant marginal utility ofmoney acrossstatesof nature. In other words, the risk-averseindividual must getperfectinsurance.This is exactlythe solution that intuition would suggest.

To summarize, optimal contracting under uncertainty would resultin perfect insurance of the employeeagainst economicrisk only if the

employeris risk neutral. In general,however,when both employerand

employeeare risk averse,they will optimally sharebusinessrisk. Thus the

simple Knightian idea that entrepreneursperfectly insure employeesis

likely to hold only under specialassumptions about risk preferencesofentrepreneurs.An individual's attitude toward risk is driven in part by initial

wealth holdings. Thus it is generallyacceptedthat individuals' absoluteriskaversiontends to decreasewith wealth. If extremely wealthy individuals areapproximately risk neutral and poor individuals are risk averse,then onespecialcasewherethe Knightian theory would bea goodapproximation iswhen wealth inequalities are extremeand a few very wealthy entrepreneursprovidenearly perfectjobsecurity to a massofpooremployees.

It should be clearfrom this brief overview of optimal contracting under

uncertainty that the presumption of rational behaviorand perfectenforceability

of contractsis lessplausiblein environments with uncertainty than

in situations without uncertainty. In many contracting situations in practiceit is possiblethat the contracting partieswill beunable to agreeon acomplete descriptionof the state spaceand that, as a consequence,insurancecontractswill be incomplete.The rationality requirements imposedon the

contracting parties and the enforcementabilities assumedof the courtsshould bekept in mind as caveatsfor the theory of contracting when facedwith very complexactual contractualsituations wherethe partiesmay havelimited abilities to describepossiblefuture events and the courts havelimited knowledgeto be able to effectively stick to the original intentions

of the contracting parties.Another important simplifying assumption to bear in mind is that it is

presumed that each party knows exactly the intentions of the other

14 Introduction

contracting parties.However,in practicethe motives behind an individual's

willingness to contractare not always known perfectly.As a consequence,suspicionabout ulterior motives often may lead to breakdownofcontracting.

These considerationsare the subject of much of this bookand arebriefly reviewedin the next sectionsof this chapter.

1.3Information and Incentives

The precedingdiscussionhighlights that even in the best possiblecontracting environments, where comprehensiveinsurance contractscan bewritten, it is unlikely that employeeswill beperfectlyinsured againstbusiness risks.Thereasonis simply that the equilibrium priceofsuch insurancewould be too high if employerswerealsoaverseto risk.

Another important reasonemployeesare likely to get only limitedinsurance is that they needto have adequate incentives to work. If the output

producedby employeestends to be higher when employeesexertthemselves more, or if the likelihoodof a negative output shock is lower if

employeesare more dedicatedor focusedon their work, then economicefficiency requires that they receive a higher compensationwhen their

(output) performanceis better.Indeed,if their pay is independentofperformance and if their job security is not affectedby their performance,why

should they put any effort into their work? This is a well-understoodidea.Even in the egalitarian economicsystem of the former SovietUnion the

provision of incentives was a generally recognizedeconomicissue,and overthe years many ingenious schemeswere proposedand implemented toaddressthe problemofworker incentives and alsofactory managers'incentives. What was lesswell understood,however,was the trade-offbetweenincentives and insurance.How far should employeeinsurance be scaledbackto make way for adequatework incentives?Howcouldadequateworkincentives bestructured while preservingjob security as much aspossible?Theseremainedopenand hotly debatedquestionsoverthe successivefive-year plans.

Much of Part Iof this bookwill be devotedto a formal analysis of this

question.Twogeneraltypes of incentive problemshave beendistinguished.One is the hidden-information problem and the other the hidden-actionproblem.The first problemrefers to a situation where the employeemayhave private information about her inability or unwillingness to take on

Introduction

certaintasks.That is,the information about somerelevant characteristicsofthe employee(her distastefor certain tasks, her levelof competence)arehidden from her employer.The secondproblemrefers to situations wherethe employercannot seewhat the employeedoes\342\200\224whether she works ornot, how hard she works, how carefulshe is, and soon. In these situations

it is the employee'sactionsthat are hidden from the employer.Problemsof hidden information are often referred to as adverse

selection, and problemsofhidden actionsasmoral hazard.In practice,ofcourse,most incentive problems combine elements of both moral hazard andadverseselection.Also, the theoreticaldistinction betweena hidden-actionand a hidden-information problemcan sometimesbe artificial.

Nevertheless,it is useful to distinguish betweenthese two types of incentive

problems, in part becausethe methodologythat has beendevelopedto analyzetheseproblemsis quite different in eachcase.

AdverseSelection

Chapters2 and 3 provide a first introduction to optimal contractswith

hidden information. These chapters examineoptimal bilateral contractswhen one of the contracting parties has private information. Chapter 2explorescontracting situations wherethe party making the contractoffersis the uninformed party. Thesesituations are often referredto asscreeningproblems,sincethe uninformed party must attempt to screenthe different

piecesof information the informed party has.Chapter3 considerstheopposite situation wherethe informed party makesthe contractoffers.Thesesituations fall under the generaldescriptiveheading ofsignaling problems,asthe party making the offermay attempt to signal to the otherparty what it

knows through the type of contractit offersor other actions.The introduction of hidden information is a substantial breakfrom the

contracting problemswe have alreadyconsidered.Now the underlying

contracting situation requiresspecificationof the private information one ofthe partiesmight have and the beliefsof the other party concerningthat

information in addition to preferences,outsideoptions, initial endowments,a state space,and a probability distribution overstatesof nature.

In the contextof employment contractsthe type of information that mayoften beprivate to the employeeat the time ofcontracting is herbasicskill,

productivity, or training. In practice,employerstry to overcomethis

informational asymmetry by hiring only employeeswith some training or only

high schooland collegegraduates.

16 Introduction

In a pathbreaking analysis Spence(1973,1974)has shown how educationcan be a signal of intrinsic skill or productivity. The basicidea behind his

analysis is that more-ableemployeeshave a lowerdisutility of educationand therefore are more willing to educate themselves than less-ableemployees.Prospectiveemployersunderstand this and thereforeare willing

to pay educatedworkersmore even if educationperse doesnot add anyvalue.We review Spence'smodel and other contracting settings with

signaling in Chapter3.Another way for employersto improve their pool of applicants is to

commit to pay greater than market-clearingwages.This tends to attract

better applicants,who generally have betterjob opportunities and aremorelikely to do well in interviews. Sucha policy naturally gives risetoequilibrium unemployment, as Weiss (1980)has shown. Thus, as Akerlof, in his1970article,and Stiglitz, in many subsequentwritings, had anticipated,the

presenceof private information about employeecharacteristicscanpotentially explain at a microeconomiclevel why equilibrium unemploymentand other forms of market inefficienciescanarise.With the introduction ofasymmetric information in contracting problemseconomistshave at lastfound plausibleexplanationsfor observedmarket inefficienciesthat had

long eluded them in simpler settings of contracting under completeinformation.

Or at least they thought so.Understandably, given the importance of thebasiceconomicissue,much of the subsequentresearchon contracting under

asymmetric information has tested the robustnessof the basicpredictionsof market inefficienciesand somewhat deflatedearly expectationsabout ageneraltheory of market inefficiencies.

A first fundamental question to be tackledwas, Just how efficient cancontracting under asymmetric information be?The answer to this questionturns out to besurprisingly elegantand powerful. It is generally referredtoas the revelation principleand is one of the main notions in contracteconomics. The basic insight behind the revelationprincipleis that todetermine optimal contractsunder asymmetric information it sufficesto consideronly one contract for each type of information that the informed party

might have, but to make sure that each type has an incentive to selectonlythe contractthat is destinedto him/her.

Moreconcretely,consideran employerwho contractswith two possibletypes of employees\342\200\224a \"skilled\" employeeand an \"unskilled\" one\342\200\224and

who doesnot know which is which. The revelationprinciplesaysthat it is

17 Introduction

optimal for the employerto consideroffering only two employmentcontracts\342\200\224one destinedto the skilledemployeeand the other to the unskilledone\342\200\224but to make sure that each contractis incentive compatible.That is,that each type of employeewants to pick only the contractthat is destinedto her.Thus, accordingto the revelationprinciple,the employer'soptimalcontracting problemreducesto a standard contracting problem,but with

additional incentive compatibility constraints.As a way of illustrating a typical contracting problemwith hidden

information, let us simplify the previousproblemwith uncertainty with a simpleform of private information added.Supposefirst that employeetime and

output enter additively in both utility functions: define them as U[ccd(l-/)-t] for the employerand u(6l+ t) for the employee,where

\342\200\242 (1-/) is the employeetime soldto the employer,and / is the time the

employeekeepsfor herself;\342\200\242 t is the monetary/output transfer from the employerto the employee;\342\200\242 a is a positive constant; and\342\200\242 6measuresthe \"unit value of time,\" or the skill levelof the employee.

The variable 6 is thus the state of nature, and we assumeit is learnedprivately by the employee before signing any contract. Specifically, the

employeeknows whether sheis skilled,with a value of time 6H,or unskilled,with a value of time 6L < 6H.The employer,however,knows only that the

probability of facing a skilled employeeis pH.When the employerfacesa skilled employee,the relevant reservation

utility is uH = u(6H), and when he faces an unskilled employee,it is uL =u(6L)-5Assume that the employee'stime is more efficient when soldto the

employer;that is,assumea > 1.Then,if the employercouldalsolearn the

employee'stype, he would simply offer in state 0, a contractwith atransfer tj

= 6, in exchangefor all her work time (that is,1-/,\342\200\242

=1).Suchacontract would maximize production efficiency, and since the employee'sindividual rationality constraint, u(tj) > u(6j), would be binding under this

contract,it would maximize the employer'spayoff.When employee productivity is private information, however, the

employerwould not beableto achievethe samepayoff, for if the employeroffers a wage contract

t,-= 0, in exchangefor 1unit of work time, all

5. Indeed, in state6>;,

the employee's endowment l2]= 6r

18 Introduction

employeetypes would respondby \"pretending to be skilled\" to get the

higher wage 6H-Note that for the employee type to be truly private information it

must also be the casethat the employee'soutput is not observable.If it

were,the employercouldeasilyget around the informational asymmetryby including a \"money-back guarantee\" into the contract should the

employee'soutput fall short of the promisedamount. Assumptions similarto the nonobservability of output are required in these contractingproblems with hidden information. A slightly morerealisticassumption servingthe samepurposeis that the employee'soutput may be random and that

a \"no-slavery\" constraint prevents the employer from punishing the

employeeex post for failing to reach a given output target.If that is thecase,then an inefficient employee can always pretend that she was

\"unlucky.\" Even if this latter assumption is moreappealing,we shall simplyassumehere for expositionalconvenience(asis often done) that output isunobservable.

Under that assumption, the only contractsthat the employercan offerthe employeeare contractsoffering a total payment of t(l) in exchangefor

(1-/) units of work. Although this classof contractsismuch simpler than

most real-worldemployment contracts,finding the optimal contractsin theset of all (nonlinear) functions [t(l)}couldbe a daunting problem.Fortunately, the revelationprincipleoffersa key simplification. It saysthat all the

employerneedsto detennineisa menu of two \"point contracts\":(tL, lL) and

{tH, Ih), where,by convention, (tn /,) is the contractchosenby type j.Thereasonwhy the employerdoesnot needto specifya full (nonlinear)contract t([) is that each type of employeewould pick only one point on thefull schedulet(t) anyway. So the employermight as well pick that point

directly. However,eachpoint has to be incentive compatible.That is, type

6H must prefercontract(tH, lH) over(tL, lL),and type 6L contract(tL, lL)over(tH, Ih)-

Thus the optimal menu of employment contractsunder hiddeninformation canberepresentedas the solution to the optimal contracting problemunder completeinformation:

msLx{pLU[aeL(1-lL)-tL]+pHU[a6H(1-lH) -tH]}(/,,r,)

subjectto

u(iLeL+tL)>u(eL)

19 Introduction

and

u{iHeH+tH)>u{eH)

but with two additional incentive constraints:

u(lHOH+tH)^u(lLeH+tL)

and

u(iLeL+tL)>u(iH9L+tH)

The solution to this constrainedoptimization problem will produce themost efficient contractsunder hidden information. As this problemimmediately reveals,the addition of incentive constraints will in generalresultin lessefficient allocationsthan under completeinformation. In general,optimal contractsunder hidden information will be second-bestcontracts,which do not achieve simultaneously optimal allocative anddistributive efficiency.Much of Chapter2 will be devoted to the analysis of thestructure of incentive constraints and the type of distortions that resultfrom the presenceof hidden information. The generaleconomicprinciplethat this chapter highlights is that hidden information results in a form

of informational monopoly power and allocativeinefficienciessimilar tothose produced by monopolies.In the precedingexample,the employermight chooseto suboptimally employ skilled employees(by setting 1- lH

< 1)to be able to pay unskilled employeesslightly less.6In a nutshell, themain trade-off that is emphasizedin contracting problemswith hiddeninformation is one between informational rent extractionand allocativeefficiency.

6. Oneoption is to have a contract with allocative efficiency, that is, lH = lL = 0 and tH =tL = 6H.This leaves informational rents for the unskilled employee relative to her outsideopportunity. It may therefore be attractive to lower skilled employment (that is, set lH >

0)\342\200\224

at an allocative cost of (a -l)lH\342\200\224in

order to lower tL without violating the incentiveconstraint:

Intuitively, lowering skilled employment allows the employer to lower tH by a significantamount, since the skilled employee has a high opportunity cost of time. Therefore,

\"pretending

to beskilled\" becomeslessattractive for the unskilled employee (who has a lower

opportunitycost of time), with the result that a lower transfer tL becomescompatible with the

incentive constraint. The trade-off between allocative efficiency and rent extraction will bedetailed in the next chapter.

Introduction

If the presenceof hidden information may give rise to allocativeinefficiencies such as unemployment, it doesnot follow that public intervention

is warranted to improve market outcomes.Indeed,the incentive constraintsfacedby employersare alsolikely to be facedby plannersor publicauthorities. It is worth recallinghere that the centrally plannedeconomyof theSovietUnion was notoriousfor its overmanning problems.It may not havehad any official unemployment, but it certainly had huge problemsofunderemployment. Chapters7 and 13will discussat length the extent towhich market outcomesunder hidden information may be first- or second-best efficient and when the \"market mechanism\" may be dominated bysomebetter institutional arrangement.

Our discussionhas focusedon a situation where the employeehas aninformational advantage over the employer.But, in practice,it is often the

employerthat has more information about the value of the employee'swork. Chapter 2 also exploresseveral settings where employershave

private information about demandand the value ofoutput. As we highlight,these settings are perhaps more likely to give rise to unemployment.Indeed,layoffs can be seenas a way for employersto crediblyconveytotheir employeesthat the economicenvironment of their firm hasworsenedto the extent that pay cuts may beneededfor the firm to survive.

MoralHazard

Chapter 4 introducesand discussesthe other major classof contracting

problemsunder asymmetric information: hidden actions.In contrast tomost hidden information problems,contracting situations with hiddenactions involve informational asymmetries arising after the signing of acontract.In these problemsthe agent (employee)is not askedto choosefrom a menu of contracts,but rather from a menu of action-rewardpairs.

Contracting problemswith hidden actionsinvolve a fundamental

incentive problem that has long beenreferred to in the insurance industry asmoral hazard:when an insureegets financial or other coverageagainst abad event from an insurer she is likely to belesscarefulin trying to avoidthe bad outcomeagainst which she is insured.This behavioralresponsetobetter insurance arisesin almost all insurancesituations, whether in life,health, fire, flood,theft, or automobile insurance.When a persongetsbetterprotectionagainst a bad outcome,shewill rationally invest fewerresourcesin trying to avoid it. One of the first and most striking empiricalstudiesof

21 Introduction

moral hazard is that of Peltzman (1975),who has documentedhow theintroduction of laws compellingdrivers to wear seat beltshas resultedin

higher averagedriving speedsand a greaterincidenceof accidents(involving,

in particular, pedestrians).How do insurers deal with moral hazard? By charging proportionally

morefor greatercoverage,thus inducing the insureeto trade off thebenefits of better insurance against the incentive costof a greaterincidenceofbad outcomes.

Incentive problems like moral hazard are also prevalent in

employment relations.As is now widely understood,if an employee'spay and

job tenure are shieldedagainst the risk of bad earnings, then shewill worklessin trying to avoid theseoutcomes.Moralhazard on the jobwas one ofthe first important new economicissuesthat Sovietplannershad to contendwith. If they were to abolishunemployment and implement equaltreatment of workers, how could they also ensure that workerswould work

diligently? As they reluctantly found out, there was unfortunately nomiraclesolution. For a time ideologicalfervor and emulation of modelworkersseemedto work, but soonmajor and widespreadmotivationproblems arose in an economicsystem foundedon the separationof pay from

performance.Employerstypically respondto moral hazard on the job by rewarding

goodperformance(through bonus payments, piecerates,efficiencywages,stock options, and the like) and/or punishing bad performance(through

layoffs).As with insurancecompanies,employersmust trade off thebenefits of better insurance (in terms of loweraveragepay) against the costsinlowereffort provision by employees.The most spectacularform ofincentive pay seennowadays is the compensationof CEOsin the United States.Arguably, the basictheory of contracting with hidden actionsdiscussedinChapter 4 providesthe main theoreticalunderpinnings for the types ofexecutivecompensationpackagesseentoday.According to the theory, evenrisk-averseCEOsshould receivesignificant profit- and stock-performance-basedcompensationif their (hidden)actionshave a major impact on thefirm's performance.7

While it is easy to graspat an intuitive level that there is a basictradeoff betweeninsurance and incentivesin most employment relations, it is

7. Actual CEOcompensation packages have also shown someserious limitations, which the

theory has addressed too; more on this topic will follow.

22 Introduction

lesseasyto seehow the contractshould bestructured to besttradeoff effort

provision and insurance.Formally, to introduce hidden actions into the precedingemployment

problemwith uncertainty, supposethat the amount of time (1-/) workedby the employeeis private information (a hidden action).Suppose,inaddition, that the employeechoosesthe action(1-/) beforethe state ofnature

6j is realizedand that this actioninfluencesthe probability of the state ofnature:when the employeechoosesaction(1-/), output for the employeris simply 9H with a probability function pH[l-/],increasingin 1-/ (and 9L

with a probability function pL[l-/] =1-pH[l-/]).8Theusualinterpretation here is that (1-/) standsfor \"effort,\" and moreeffort produceshigher

expectedoutput, at cost1-/ for the employee,say.9 However,it is notguaranteed to bring about higher output, sincethe bad state of nature 6L maystill occur.Note that if output wereto increasedeterministically with effortthen the unobservability of effort would not matter becausethe agent'shidden effort supply couldbe perfectlyinferred from the observationofoutput.

Sinceeffort (1-/) is not observable,the agent canbecompensatedonlyon the basisof realizedoutput Q-r

The employeris thus restrictedto

offeringa compensationcontractt(6}) to the employee.Also the employermust

now take into accountthe fact that (1-/) will bechosenby the employeeto maximize her own expectedpayoff under the output-contingent

compensation schemet(6j). In otherwords, the employercannow make only abestguessthat the effort levelchosenby the employeeis the outcomeofthe employee'sown optimization problem:

(l-l)esLrgmsix{pL[l-l]u[t(eL)+l]+pH[l-l]u[t(6H)+l]}I

Therefore,when the employerchoosesthe optimal compensationcontract[t(6j)}to maximize his expectedutility, he must make sure that it is in the

employee'sbestinterest to supply the right levelof effort (1-/).In otherwords, the employernow solvesthe following maximization problem:

msix{pL[l-l]U[eL-t(eL)]+pH[l-l]U[9H-t(dH)]}

8.Where pff[-] (respectively pL[-]) is an increasing (respectively, decreasing) function of(1-0-9. For simplicity, we assume here that the opportunity cost of time for the employee isindependent of the state of nature.

23 Introduction

subjectto

pL[l-l]u[t(6L)+l]+pH[l-l]u[t(6H)+l]>u=u(l) (IR)and

(l-Qeaigmax{pdl-lMt{eL)+l]+pH[l-l]u[t(eH)+rft (IC)I

As in contracting problemswith hidden information, when the actionsupplied by the employeeisnot observablethe employermust take into

consideration not only the employee'sindividual rationality constraint but alsoher incentive constraint.

Determining the solution to the employerproblemwith both constraintsis not a trivial matter in general.Chapter4providesan extensivediscussionof the two main approaches toward characterizingthe solution to this

problem.For now we shall simply point to the main underlying idea that anefficient trade-offbetweeninsuranceand incentives involves rewarding the

employeemost for output outcomesthat are most likely to arisewhen sheputs in the requiredlevelofeffort and punishing her the most for outcomesthat are most likely to occurwhen sheshirks.Theapplicationof this

principlecangive rise to quite complexcompensationcontractsin general,often

morecomplexthan what we seein reality. There is one situation, however,wherethe solution to this problemis extremely simple:when the employeeis risk neutral. In that caseit is efficient to have the employeetake on all the

output risk so as to maximize her incentivesfor effort provision.That is,when the employeeis risk neutral, she should fully insure the employer.

One reason why this simple theory may predict unrealistically

complex incentive schemesis that in most situations with hidden actionstheincentive problem may be multifaceted.CEOs,for example,can takeactionsthat increaseprofits, but they canalsomanipulate earnings, or \"run

down\" assetsin an effort to boostcurrent earnings at the expenseof future

profits. Theycan alsoundertake high-expected-returnbut high-riskinvestments. It hasbeensuggestedthat when shareholdersor any otheremployerthus face a multidimensional incentive problem, then it may beappropriate

to respondwith both less \"high powered\"and simpler incentiveschemes.We exploretheseideasboth in Chapter6,which discusseshiddenaction problemswith multiple tasks, and in Chapter10,which considerslong-term incentive contracting problemswhere the employeetakesrepeated hidden actions.

24 Introduction

Chapter6 alsoconsidersmultidimensional hidden information problemsaswell asproblemscombining both hidden information and hidden actions.All these problemsraise new analytical issuesof their own and produceinteresting new insights. We providean extensivetreatment of someof themost important contracting problemsunder multidimensional asymmetricinformation in the researchliterature.

Part I of our book also discussescontracting situations with anintermediate form of asymmetric information, situations where the informed

party can credibly discloseher information if she wishes to do so.Thesesituations, which are consideredin Chapter 5, are mostly relevant for

accounting regulation and for the designof mandatory disclosurerules,which are quite pervasivein the financial industry. Besidestheir obvious

practicalrelevance,thesecontractualsituations are alsoofinterestbecausethey dealwith a very simple incentive problem,whether to discloseor hiderelevant information (while forging information is not an availableoption).Becauseof this simplicity, the contractualproblemsconsideredin Chapter5 offeran easyintroduction to the generaltopicof contracting underasymmetric information. One of the main ideasemerging from the analysis ofcontracting problemswith private but verifiable information is that

incentives for voluntary disclosurecan be very powerful. The basiclogic,which

is sometimesreferred to as the \"unraveling result,\" is that any sellerof agoodor service(e.g.,an employee)hasevery incentive to revealgoodinformation about herself,such as high test scoresor a strong curriculum vitae.

Employersunderstand this fact and expectemployeesto be forthcoming.If an employeeis not, the employerassumesthe worst. It isfor this reasonthat employeeshave incentivesto voluntarily discloseall but the worst

pieceof private verifiable information. This logicis so powerful that it isdifficult to seewhy there should bemandatory disclosurelaws. Chapter5discussesthe main limits of the unraveling result and explainswhen

mandatorydisclosurelaws might bewarranted.

Finally, it is worth stressingthat although our leadingexamplein this

introduction is the employment relation, each chapter contains severalother classicapplications,whether in corporatefinance, industrial

organization, regulation, public finance,or the theory of the firm. Besideshelpingthe readersto acquaint themselveswith the core conceptsof the theory,theseapplicationsare alsomeant to highlight the richnessand broadrelevance of the basictheory of contracting under private information.

25 Introduction

1.4Optimal Contracting with Multilateral Asymmetric Information

The contracting situations we have discussedsofar involve only one-sidedprivate information or one-sidedhidden actions.In practice,however,thereare many situations whereseveralcontracting partiesmay possessrelevant

private information or be calledto take hidden actions.A first basicquestion ofinterestthen iswhether and how the theory ofcontracting with

onesided private information extends to multilateral settings. Part IIof this

bookisdevotedto this question.It comprisestwo chapters.Chapter7 dealswith multilateral private information and Chapter8 with multilateral hidden

actions.Besidesthe obvioustechnicaland methodologicalinterest in

analyzing these more general contractual settings, fundamental economicissuesrelating to the constrainedefficiency of contractual outcomes,therole of competition, and the theory of the firm are alsodealtwith in thesechapters.

While the generalmethodologyand most of the core ideasdiscussedinPart I extend to the generalcaseof multilateral asymmetric information,there is one fundamental difference.In the one-sidedprivate information

casethe contractdesignproblemreducesto a problemof controlling theinformed party's response,while in the multilateral situation the

contracting problembecomesone of controlling the strategicbehaviorof severalparties interacting with each other.That is, the contract designproblembecomesone of designing a gamewith incomplete information.

One of the main new difficulties then ispredicting how the gamewill beplayed.Thebestway of dealing with this issueis in fact to designthecontract in such a way that eachplayerhas a unique dominant strategy. Thenthe outcomeof the game is easy to predict,sincein essenceall strategicinteractionshave then beenremoved.Unfortunately, however, contractswhereeachparty hasa unique dominant strategy aregenerally not efficient.Indeed,in a major result which builds on Arrow's (1963)impossibility

theorem,Gibbard (1973)and Satterthwaite (1975)have shown that it is

impossiblein generalto attain the full-information efficient outcomewhen

there are more than two possibleallocationsto choosefrom and when the

contracting parties'domain ofpreferencesisunrestricted (that is,when theset ofpossibletypes of eachcontracting party is very diverse).Rather than

stick to predictablebut inefficient contracts,it may then generally bedesirable to agree on contractswhere the outcomeis lesspredictablebut on

Introduction

average more efficient (that is, contracts where each party's responsedependson what the other contracting partiesare expectedto do).From atheorist'sperspectivethis isa mixed blessingbecausethe proposedefficientcontracts(or \"mechanisms,\" as they are often referredto in multilateral

settings) may besomewhat fragile and may not always work in practiceas the

theory predicts.

Auctions and Tradeunder Multilateral Private Information

Perhapsthe most important and widely studiedproblemofcontracting with

multilateral hidden information is the design of auctions with multiplebidders,each with his or her own private information about the value ofthe objectsthat are put up for auction.10Accordingly, Chapter7 devotesconsiderablespaceto a discussionof the main ideasand derivation of keyresults in auction theory, such as the revenue equivalencetheorem or the

winner's curse.The first result establishesthat a number of standardauctions yield the sameexpectedrevenueto the sellerwhen biddersare riskneutral and their valuations for the object are independently and

identicallydistributed. The secondidea refers to the inevitable disappointment

of the winner in an auction wherebiddersvalue the objectin a similar way

but have different prior information about its worth: when shelearns that

shewon shealsofinds out that her information ledher to be overoptimisticabout the value of the object.

In recentyearstherehasbeenan explosionof researchin auction theory

partly becauseof its relevanceto auction designin a number of important

practicalcases.Coveringthis researchwould requirea separatebook,and

Chapter7 can serveonly as an introduction to the subject.Auction designwith multiple informed biddersis by no means the only

example of contracting with multilateral hidden information. Another

leadingexample,which isextensively discussedin Chapter7, is trade in

situations whereeachparty hasprivate information about how much it valuesthe goodor the exchange.

A major economicprinciple emerging from the analysis of contractingwith one-sidedhidden information is the trade-offbetweenallocativeefficiency and extractionof informational rents.If the bargaining power lies

10.Despite its relative fragility, the theory of contracting with multilateral hiddeninformation has proved to beof considerable practical relevance, as for example in the design ofspectrum and wireless telephone license auctions (seefor example Klemperer, 2002).

Introduction

with the uninformed party, as we have assumed,then that party attemptsto appropriatesomeof the informational rents of the informed party at the

expenseof allocativeefficiency.But note that if the informed party (e.g.,the employeein our example)has all the bargaining powerand makesthecontractoffer,then the contracting outcomeis always efficient. So,if the

overriding objectiveis to achievea Pareto efficient outcome(with, say, nounemployment), then there appearsto be a simple solution when there is

only one-sidedhidden information: simply give all the bargaining powertothe informed party.

In practice,however,besidesthe difficulty in identifying who theinformed party is,there isalsothe obvious problemthat generally all partiesto the contractwill have somerelevant private information. Therefore,thenatural contracting setting in which to posethe question of the efficiencyof tradeunder asymmetric information and how it varieswith the

bargaining powerof the different parties isone of multilateral asymmetricinformation. A fundamental insight highlighted in Chapter7 is that the mainconstraint on efficient trade is not so much eliciting the parties'privateinformation as ensuring their participation.Efficient trade can (almost)always beachievedif the parties'participation isobtainedbeforethey learntheir information, while it cannot be achievedif participation is decidedwhen they alreadyknow their type.

Applying this insight to our labor contracting example,the analysis in

Chapter7 indicatesthat labormarket inefficiencieslike unemployment areto be expectedin an otherwisefrictionlesslabor market when employershave market power and employeesprivate information about their

productivity, or when there is two-sidedasymmetric information. It must bestressed,however, that policyintervention that is not basedon anyinformation superiorto that availableto the contracting partieswill not beableto reduceor eliminate these inefficiencies.But labor market policiesthat

try to intensify competitive bidding for jobsor for employeesshould lowerinefficienciescausedby hidden information.

MoralHazard in Teams,Tournaments, and Organizations

Contracting situations whereseveralparties take hidden actionsare oftenencounteredin firms and other organizations. It is for this reason that the

leading applicationof contracting problems involving multisided moralhazard isoften seento be the internal organization of firms and othereconomic institutions. Someprominent economictheorists of the firm like

28 Introduction

Alchian and Demsetz(1972)or Jensen and Meckling (1976)go as far asarguing that the resolution of moral-hazard-in-teamsproblems (whereseveralagentstake complementary hidden actions)is the raisond'etreofa firm. They contendthat the roleofa firm's ownerormanager is to monitor

employeesand make sure that they take efficient actionsthat are hiddento others.Hence,the analysis ofcontracting problemswith multisided moralhazard is important if only as an indirect vehiclefor understandingeconomic organizations and firms.

Accordingly, Chapter8 coversmultiagent moral hazard situations with aparticular focuson firms and their internal organization. To illustrate someof the key insights and findings coveredin this chapter,considerthesituation where our employernow contractswith two employees,A and B,eachsupplying a (hidden) \"effort\" (1- lA) and (1-lB).A key distinction in

contracting problemswith multisided moral hazard concernsthe measureofperformance:Iseach employee'sperformancemeasuredseparately,or isthere a single aggregatemeasureof both employees'contributions?

In the former case,when the output of eachemployeeis observableandis given by, say, 6A]with probability pAj and 6Bj with probability pBj, for ;'=L,H, the employer'sproblemis similar to the single-agentmoral hazardproblemdescribedearlier,with the new featurethat now the employercanalsobasecompensationon eachemployee'srelative performance:

9a, ~6b,

An important classof incentive contracting situations in which agentsarerewardedon the basisof how well they did relative to othersis rank-ordertournaments. Many sports contests are of this form, and promotionsofemployeesup the corporateladdercanalsobeseenas a particular form oftournament.

Thus, in our employment problemwith two employeesand observableindividual outputs, the employermay be able to providebetter incentiveswith lessrisk exposureto the two employeesby basing compensationonhow well they perform relative to each other.Thispossibility can be seenas one reason why firms like to provide incentives to their employeesthrough promotion schemes,appointing only the better employeesto

higher paying and more rewarding jobs.The reasonwhy relativeperformance evaluation improves incentivesis that when employeesare exposedto the same exogenousshocksaffecting their performance(changesin

demand for their output or quality of input supplies,say), it ispossibleto

29 Introduction

shieldthem against theserisks by filtering out the common shockfrom their

performancemeasure.Toseehow this works, think that the probability pA,

(iesp.,pBl)dependsnot only on individual effort (1-lA) [resp.,(1-lB)]but

also on a random variable that affectsboth agents.In this case,it makessenseto link an employee'scompensationpositively to her own

performance but negatively to the other employee'sperformance.Chapter 8discusses extensively how to make the best use of relative performancemeasuresin generalproblemswith multisided moral hazard.

It is worth noting here that as compelling and plausibleas the caseforrelative performancemay be,many criticalcommentatorson CEOcompensation in the UnitedStateshave pointedto the absenceof such relative

performance evaluation for CEOs.For example,Bebchuk, Fried, andWalker (2002)have criticizedCEOcompensationcontractsin the UnitedStatesfor not optimally correctingcompensationby filtering out commonstockmarket shocksthrough indexing.Theyargue that this is a majordeviation from optimal incentive contracting and is evidenceofa failure in

corporate governancein most large U.S.companies.Others,however, haverationalizedthe absenceof explicit indexing as an optimal way of getting

managersto do their own hedging when this is cheaper,or as an optimalresponseto competitive pressuresin the market for CEOs(seeGarvey andMilbourn, 2003;Jin, 2002).

Let us now turn to the secondcase,whereobservableoutput isa singleaggregatemeasuregiven by

with the probability of higher realizationsthat dependspositively on eachemployee'seffort. Then the employer faces a moral-hazard-in-teamsproblem.Indeed,the amount of time workedby eitherof the two

employees is a public goodbecause,by raising joint output, it benefits bothemployees. As is easy to understand, in such situations a major difficulty for the

employeris to prevent free riding by oneemployeeon the otheremployee'swork.

Alchian and Demsetz(1972)proposedthat free riding of employeescanbepreventedthrough monitoring by the employer.That is, the employer'smain role in their view is one of supervising employeesand making surethat they all work. They alsoargue that the employershould be the

residual claimant on the firm's revenuesand that employeesshould bepaidfixed

wagesto make sure that the employerhas the right incentives to monitor.

30 Introduction

When monitoring is too costly or imperfect,however,then employeesalsoneedto bemotivated through compensationbasedon aggregateperformance. An important insight of Holmstrom (1982),which we discussinChapter8,is that optimal provision of incentives by giving sharesofaggregate output to employeesrequiresbudget breaking in general.That is, thesum of the sharesof the team membersshould not always add up to one.The residualshould then besoldto a third party, which canbe thought ofas outsideshareholders.

When the number of employeesto be monitored is large,it is notreasonable to think that a single employeris able to effectively monitor all

employees.Multiple supervisorsare then required,and someonewill haveto monitor the monitors. If the number of supervisorsis itself large,then

multiple monitors of supervisorswill beneeded.And soon.Thus, by

specifying the spanof controlof any supervisor(the number of employeesthat

canreasonablybemonitoredby a single supervisor)and the lossof controlas more tiers of supervisorsare added(intuitively, there will be an overallreductionin efficiencyof supervision of bottom-layeremployeesas morelayersare addedbetweenthe top and the bottom of the hierarchy), onecandevelopsimultaneously a simple theory of the optimal firm size and the

optimal hierarchicalinternal organization of the firm. Again, Chapter 8

gives an extensivetreatment of this theory of organizations.One of the reasonswhy there may be a lossof controlas more

supervisorytiers areaddedis that midlevel supervisorsmay attempt to colludewith

their employeesagainst top management or the firm's owners.Recentcorporate scandalsin the United Stateshave painfully remindedinvestors ofthe risk of collusionbetweenauditors and the agents they are meant tomonitor. Theseexamplesvividly draw attention to the importanceofconsidering the possibility of collusionin multiagent contracting situations.

Chapter8 providesan extensivediscussionof someof the main modelsofoptimal contracting with collusion.It emphasizesin particular the idea that

beyond incentive and participation constraints, optimal multilateral

contracts are alsoconstrainedby \"no-collusionconstraints.\"

1.5The Dynamics of IncentiveContracting

In practice,many if not most contracting relationsare repeatedor longterm. Yet the theory we developin the first two parts of the bookdealsonly

Introduction

with static or one-shotcontracting situations. In Part IIIwe providesystematic coverageof long-term incentive contracting, mostly in a bilateralcontracting framework. In Chapter9 we discussdynamic adverseselectionand in Chapter10dynamic moral hazard.

Methodologically,there is no significant changein analyzing optimal mul-

tiperiodcontractsas long as the contracting partiescan commit to a singlecomprehensivelong-term contractat the initial negotiation stage.As wehave alreadynoted in the contextof intertemporal coinsurancecontracting

problems,when full commitment is feasible the long-term contract canessentiallybereducedto a slightly more complexstatic contractinvolvingtrade of a slightly richerbasketof state-contingentcommodities,services,and transfers. What this conclusionimplies in particular for contractingunder hidden information is that the revelationprinciple still appliesunderfull commitment.

However,if the contracting partiesare allowedto renegotiate the initial

contractas time unfolds and new information arrives, then new conceptualissuesneedto be addressedand the basicmethodology of optimal static

contracting must be adapted.Mainly, incentive constraints must then bereplacedby tighter renegotiation-proofnessconstraints.

A number of new fundamental economicissuesarise when the partiesare involved in a long-term contractual relation.How is privateinformation revealedover time? How is the constrainedefficiencyof contractualoutcomesaffectedby repeatedinteractions?How doesthe possibility ofrenegotiationlimit the efficiencyof the overalllong-term contract?Towhat

extent can reputation serveas a more informal enforcementvehiclethat isan alternative to courts?We discussthese and other issuesextensively in

this third part of the book.

Dynamic AdverseSelection

There are two canonical long-term contracting problems with hiddeninformation: one where the informed party's type doesnot changeovertime and the other where a new type is drawn every period.In the first

problem the main new conceptual issue to be addressedrelates tolearning and the gradual reductionof the informed party's informational

advantage over time. The secondclassof problemsis conceptuallymuchcloserto a static contracting problem, as the information asymmetrybetween the two contracting parties remains stationary. The main noveleconomic question in this class of problems concerns the trade-off

32 Introduction

between within-period and intertemporal insurance or allocativeefficiency.

Toseeone important implication of learning of the informed party's typeover time, consider again our employment-contracting problem with

private information of the employee'sproductivity. That is,supposethat the

employeecan supply labor (1-I) to produceoutput ctdH(l - l) when sheis skilled and output adL(l- t) when she is unskilled at opportunity cost(1-l)6j(;'=H,L),whereher productivity and opportunity costof laborarehidden information. And suppose,as before,that the employerknows onlythe probability of facing a skilled employee,pH.As we have seen,in a staticcontracting situation the employer would pursue an optimal trade-offbetweenrent extractionand allocativeinefficiency with a menu ofemployment contracts.Thecontractfor the skilled employeewould specifyan

inefficiently low level of employment, and the contract for the unskilled

employeewould leave her an informational rent relative to her outsideopportunity.

Now, considera twice-repeatedrelation with spot contracting in eachperiod.In this situation the menu ofcontractsthat we have describedwould

no longerbe feasiblein the first period:indeed,if in the first period the

type-;'employeechoosesoption (/,-,tj), shewill have identified herselfto the

employer.In the secondperiodthe employerwould then know her outsideopportunity, and would in particular not leave any informational rent

anymore to the unskilled employee.Therefore, unless the employercommits not to respondin this way, the unskilled employeewill bereluctant to revealher type by separating.Consequentlymorepoolingof typesis to beexpectedin early stagesof the contracting relation.

Note that this commitment issueis a very generalone that arisesin manydifferent contexts.It was known for example to analysts of the Sovietsystem as the ratchet effect(seeWeitzman, 1976),which denotesthebehavior of centralplannersthat dynamically increasefirm performancetargetswhen they realizethat they are facing very productive firms.

Under full commitment to a comprehensivelong-term contract the

precedingproblemsdisappear.But full commitment will not be feasibleif the contracting parties are allowed to sign long-term contracts but

cannot commit not to renegotiatethem in the future if they identify Pareto-improving opportunities.Indeed,in our employment example,thecontracting parties will always want to renegotiate the optimal long-termcontract,as this contractspecifiesan inefficiently low labor supply for the

33 Introduction

skilled employee.Oncethe high skill of the employeeis revealed,there aregains from trade to renegotiating the contract to a higher levelof laborsupply. But if this renegotiationis anticipated,then the unskilled employeewill again want to pretend to be skilled.In general, then, the optimalrenegotiation-proofcontract will differentiate the types lessin the earlystagesof the relation,and the hidden information about the employee'stype will only gradually be revealed to the employer.Chapter9 providesan extensivediscussionof the dynamics of contracting under adverseselection. It alsoillustrates the relevanceoftheseideaswith severalapplications.

One general lessonemerging from our analysis in this chapter is that

there are no gains from enduring relationshipswhen the type of theinformed party is fixed.Indeed,the bestthe contracting parties can hopeto achieve is to repeat the optimal static contract. In contrast, when

the informed party's type changesover time, there are substantial gainsfrom repeatingthe relationship.While it is stretching our imagination tothink that an employee'sintrinsic productivity may changerandomly overtime, it is much more plausible to think of the hidden type as an un-

observableincomeshock and to think of the contracting problemas aninsurance problem with unobservableincome shocks.Indeed,the first

formal model of this problem by Townsend(1982)considersexactly this

application.The starting point of this analysis is that there canbeno gains from

contracting at all in a one-shotcontracting relationbecausethe informed partywill always claim to have had a low incomerealizationin order to receivean insurancecompensation.But even in a twice-repeatedcontractingrelation there canbe substantial gains from insurance contracting.The reasonis that in a relation that is repeatedtwice or more,greaterinsurance againstincome shockswithin the first periodcan be traded off against betterintertemporal allocationof consumption. In very concreteterms anindividual who getsa low-incomeshockin the first periodcanborrow againsther future incometo smooth consumption. Vice versa,an individual who

getsa high incomein the first periodcansavesomeof this incometoward

future consumption. The key insight of Townsendand the subsequentliterature on this problemis that borrowing and lending in a competitivedebt market providesinefficiently low insurance.The optimal long-termincentive-compatiblecontract would provide more within-periodinsurance against low-incomeshocks.As we highlight in Chapter 9, this

insight is particularly relevant for understanding the role of banks and their

Introduction

greater ability in providing liquidity (that is, within-period insurance)than financial markets.

Dynamic MoralHazard

Dynamic contracting problemswith moral hazardhave a similar structure

to dynamic adverseselectionproblemswhere the type of the informed

party is drawn randomly every period.As with thesecontracting problems,there are gains from enduring relations,and the optimal long-term contractinducesa similar distortion in intertemporal consumption allocationsrelative to what would obtain under repeated-spot-incentivecontracting and

simple borrowing and lending. That is, an optimal long-term employmentcontractwith hidden actionsby the employeewill induceher to consumerelatively more in earlierperiods.If the employeewere free to save anyamount of her first-periodincomeat competitive market rates, then shewould chooseto savemore than is optimal under a long-term incentive

contract, which would directly controlher savings. The broad intuition for this

generalresult is that by inducing the employeeto consumemore in earlyperiodsthe employercankeepher \"hungry\" in subsequentperiodsand thus

doesnot need to raise her level of compensationto maintain the sameincentives.

Chapter10beginswith a thorough analysis of a generaltwice-repeatedcontracting problemwith moral hazard and of the generalresult we justmentioned.It then proceedswith a detailed discussionof the different

effectsin play in a repeatedrelationwith moral hazard and identifies two

important sourcesof gains and one important source of lossesfrom an

enduring relation.A first positive effect is that repetitionof the relationmakes the employeelessaverse to risk, since she can engage in \"self-

insurance\" and offset a bad output shock in one period by borrowing

against future income.A secondpotentialpositive effectcomesfrom betterinformation about the employee'schoice of action obtained from

repeated output observations.Offsetting these two positiveeffects,however,is a negative effect,which comesfrom the greater flexibility affordedthe

employeeto act in responseto dynamic incentives.In an enduring relationshe can slackoff following a goodperformancerun or make up for poorperformancein one periodby working extrahard the next period.

It is this ability to modulate her effort supply in responseto good orbad output changes that drives a striking insight due to Holmstromand Milgrom (1987)concerningthe shapeof the optimal long-term incen-

35 Introduction

tive contract.Onewould think that when the relationbetweenan employerand employeeis enduring, the complexity of the employment contractwould grow with the length of the relation,somuch sothat the optimalcontract predicted by incentive theory in any realisticsetup would becomehopelesslycomplex.One might then fear that this extreme complexitycouldeasilydefeatthe practicaluseof the theory.Holmstrom and Milgrom,however, argue that as the employee'sset of possibleactionsgrows with

the length of the relation, the set of incentive constraints that constrictthe shape of the optimal contract becomesso large that the incentive-compatiblelong-term contractendsup taking a simple linear form in final

accumulatedoutput. In short, under an enduring relationthe optimal long-term contractgains in simplicity. This observation,which tends to accordwell with the relative simplicity of actual employment contracts, is,however, theoreticallyvalid only under somespecificconditions onpreferences and technology.

Another important simplification that is availableunder fairly generalconditions is that the incentive effectsunder an optimal long-termincentive contract may be replicablewith a sequenceof spot contracts.Thisobservationmay be of particular relevancefor evaluating long-term CEOcompensationcontracts.A common practiceis to let CEOsexercisetheir

stockoptionsand selltheir equity stake early but to \"reload\" their stockoptionsto providecontinuing incentives to CEOs.This hasbeenviewed asan inefficient practiceby some commentators(e.g.,Bebchuk,Fried, andWalker, 2002),but it may alsobeseenas consistentwith the idea ofreplication of the efficient long-term contractthrough a sequenceof short-termcontracts.

Explicit long-term employment contractsmay alsotake a simple form in

practice becausein an ongoing employment relation efficiency may beattained by providing a combination of explicitand implicit incentives. Theexplicitly written part of the contractmay then appearto besimple becauseit is supplementedby sophisticatedimplicit incentives.Looselyspeaking,the term \"implicit incentives\" refers to notions like reputation building,careerconcerns,informal rewards,and quid proquos.In reality many long-term employment relations do not provide a completespecificationofemployerobligations and employeeduties.Instead they are sustained by

implicit rulesand incentives.Relianceon such incompleteexplicit contractsmay often bea way of economizingon contract-drafting costs.Accordingly,

Chapter10providesan extensivetreatment ofso-calledrelational contracts,

36 Introduction

which combineboth explicit and implicit incentives, and of the implicitincentives derivedfrom \"market\" perceptionsabout employee\"talent\" andtheir implications, for example,in terms of outsideoffers.

Thechapteralsodealswith renegotiation.As with dynamic adverseselection, the possibility of renegotiationunderrnines efficient incentiveprovision. Oncea risk-averseemployeehas taken her action,there is no pointin further exposingher to unnecessaryoutput risk, and gains fromrenegotiation open up by letting the employerprovidebetter insuranceto the

employee.But if such renegotiationis anticipated,then the employeewill

have lowerincentives to put in effort.Thisissueis particularly relevant forCEOcompensationwherethe \"action\" to betaken by the CEOmay bethe

implementation of a new investment projector a new businessplan.Oncethe projecthasbeenundertaken, there is no point in exposingthe CEOtofurther risk, and it may be efficient to let her sellat leastpart ofher equitystake.This is indeed the predictionof optimal incentive contracting with

renegotiation,as we explain in Chapter10.

1.6IncompleteContracts

Our discussionofexplicit and implicit incentives alreadyalludesto the factthat most long-term contractsin practiceare incomplete,in that they donotdealexplicitly with all possiblecontingenciesand leavemany decisionsandtransactions to bedeterminedlater.It is easyto understand intuitively why

this is the case.Mostpeoplefind it hard to think through even relativelysimple dynamic decisionproblemsand prefer to leavemany decisionstobe settledat a later stagewhen they becomemorepressing.If this is true

for dynamic decisionproblems,then this must be the casea fortiori for

dynamic contracting problems,wherethe partiesmust jointly think throughand agreeon future transactions or leavethem to bedeterminedat a laterstage.

Our formulation of optimal contracting problemsin the first three partsof the bookabstractsfrom all these issues.There are no contract-drafting

costs,there are no limits on contractenforcement,and partiesare able to

instantly determinecomplexoptimal long-term contracts.This is clearlyadrasticalbeit convenient simplification. In the fourth and final part of thebookwe depart from this simple framework and explorethe implicationsof contractual incompleteness.

Introduction

As in Part III,Part IV is concernedwith long-term contracts.But thefocusis different. When contractsare incomplete,some transactions anddecisionsmust bedeterminedby the contracting partiesat somelaterstage.The questionthen arises,Who makesthesedecisions?The principal focusof this part of the bookwill be to addressthis question.The form of the

(incomplete)long-term contractwill beprespecifiedexogenouslyin at leastsomedimensions,and the optimizing variableswill bemainly the allocationamong contracting parties of ownership titles, control rights, discretion,authority, decision-makingrules, and soon.

Hence,the formulation of the basic incompletecontracting probleminvolves a major methodologicalchange.Indeed,to emphasizethis changewe considermostly problemsinvolving little or no asymmetric information

at the contracting stage.Thispart of the bookalsoinvolves a fundamental

substantive change:In the first threeparts the focuswas exclusivelyonmonetary rewardsfor the provision of incentives.In contrast,in Part IV thefocuswill be on the incentive effectsof controland ownership protections.In otherwords, this part emphasizesother institutional factorsbesidesmonetary remuneration in the provision of incentives.In a nutshell, the

incomplete contracting approach offers a vehicle to explorethe analysis ofeconomicinstitutions and organizations systematically.

Ownership and Employment

In Chapter11we begin our treatment of incompletecontractsby

assumingthat an inability to describecertaineventsaccuratelybeforethe fact is

the principal reason why contracts are incomplete.We shall, however,assumethat after the fact theseeventsare easilydescribedand their

implications fully understood.It has beena matter of debatehow muchlimitations on language are a constraint for drafting fully comprehensivecontractsboth in theory and in practice.We providean extensivediscussion of this debate in Chapter12.

In Chapter11we specifyexogenouslywhich eventsthe contractcannotbe basedon and focuson the implications of contractual incompletenessfor institution design.We shall be interestedprimarily in the role of two

ubiquitous institutions of market economies,ownership rights andemployment relations.

The first formal modelof an incompletecontracting problemby Simon

(1951)dealswith a fundamental aspectof the employment relation we havenot hitherto considered:the authority relation betweenthe employerand

38 Introduction

employee.Sofar we have describedan employment contractlike any othercontractfor the provision of an explicit serviceor \"output.\" But in realitymost employment contractsdefineonly in very broad terms the duties ofthe employeeand leaveto the discretionof the employerthe future

determination of the specifictasks of the employee.In short, employmentcontracts are highly incompletecontractswhere the employeeagreesto putherselfunder the (limited) authority of the employer.Employmentcontracts thus specifya different mode of transaction from the negotiationmode prevalent in spot markets. It is for this reason that Simon, Coase,Williamson, and others have singled out the employment relation as the

archetypal form of an economicinstitution that is different from market

exchange.Simon views the choicebetweenthe two modesof transaction as a

comparison betweentwo long-term contracts:a \"salescontract,\" in which theserviceto beprovidedis preciselyspecifiedin a contract,and an\"employment contract,\" in which the serviceis left to the discretionof the buyer

(employer)within some contractually specifiedlimits. The employmentcontractis preferredwhen the buyer is highly uncertain at the time ofcontracting about which servicehe prefersand when the seller(employee)iscloseto indifferent betweenthe different tasks the employercan choosefrom. Chapter11discussesthe strengths and limitations of Simon'stheoryand provides an extensivetreatment of a \"modernized\" version of his

theory that allows for ex ante relation-specificinvestments and ex postrenegotiation. Chapter 12further builds on Simon'stheory by explicitly

modeling \"orders\" or \"commands\" given by the employer,to which the

employeerespondsby either \"quitting\" or \"executing\" the order.Thenotion that the presenceof relation-specificinvestments createsthe

needfor modesofexchangeother than trade in spotmarkets hasbeenarticulated and emphasizedforcibly in Williamson's writings (1975,1979,1985).In a pathbreaking article that builds on his insights, Grossmanand Hart(1986)developeda simple theory and modelof ownership rights basedonthe notion of residualrights of control.They definea firm as a collectionofassetsownedby a common owner, who has residualrights of controloverthe use of the assets.A key new notion in their articleis that ownershipservesas a protectionagainst future holdups by other trading partnersandthus may give strongerincentives for ex ante relation-specificinvestments.That is, the owner of an assethas a bargaining chip in future negotiationsover trades not specifiedin the initial incompletecontracts.He can sell

Introduction

accessto the productive assetto future trading partnerswho needthe assetfor production.The owner can thus protect the returns from ex ante

relation-specificinvestments.

Building on these notions, Grossmanand Hart are able to provide asimple formal theory of the costsand benefits of integration and theboundaries of the firm. Chapter11providesan extensivetreatment of this theory.The advantage of integration is that the bargaining positionof the ownerof the newly integrated firm is strengthened.This strongerpositionmayinducehim to invest more.The drawback, however, is that the previousowner'sbargaining positionis weakened.Thisagent may thereforeinvestless.Dependingon the relative sizeof these costsand benefits ofintegration, Grossmanand Hart are able to determine when it is optimal to

integrate or not. They are thus able to articulate for the first time a simpleand rigorous theory of the boundariesof the firm, which has beenfurther

elaboratedby Hart and Moore(1990)and synthesized by Hart (1995).

IncompleteContractsand Implementation Theory

While this theory of the firm has improved our understanding of the specialroleof ownership, a major theoreticalissueremains only partially resolved,at leastin its initial versions.As Maskin andTirole(1999a)have pointedout,there is a basiclogicaltensionin the theory. On the one hand, contracting

partiesare assumedto beable to fully anticipatethe consequencesof then-

current actionsfor the future, like the potential for holdups or theprotections given by ownership.And yet, on the otherhand, they arealsoassumedto beunable to limit expectedfuture abuseby trading partnerswith explicitcontractual clauses.All they cando to improve their future negotiatingposition is to trade a very standardizedcontract:ownership titles.

We discussthe delicate theoreticalissuesrelating to this basic logicaltension in Chapter12.Thischapterbeginsby coveringthe theory of Nash

implementation (Maskin, 1977)and its subsequent developments.This

theory dealswith issuesof contractdesignin situations wherean event isdifficult to describeex ante (or identify by a third party) but easilyrecognized by all contracting parties ex post.It exploitsthe idea that contractscanbemadecontingent on such eventsby relying on reportsby the partieson which event occurred.A striking general result of implementationtheory is that by designing suitable revelation gamesfor the contracting

parties it is often possibleto achievethe sameoutcomesas with fully

contingent contracts.

Introduction

Thus the challengefor the theory of incompletecontracts,which relieson the distinction between observability and nonverifiability (or non-describability) of an event, is to explainwhy the contracting partiesdo not

attempt to contractaround this constraint by designing sophisticatedrevelation games(or Maskin schemesas they are commonly called).This is not

necessarilyan abstruse theoretical issue,as Chapter 12illustrates with

examplesof plausibleoptionlike contractsthat achieveefficiency in

contracting problemsinvolving relation-specificinvestments and holdupproblems. As is readily seen,these contracts can be interpreted as simplerevelationgames.

Oneway the challengehas beentaken up is to argue that Maskin schemeshave limited power in improving efficiency over simple incompletecontracts in complexcontracting environments wherethere may be manydifferent statesofnature or potentially many different servicesor goodsto betraded (seeSegal,1999a).Chapter12providesan extensivediscussionofthese arguments. It also exploresanother foundation for the theory ofauthority, based on actions that are both ex ante and ex post non-contractable.If one assumes that one can contract on who controlsthese actions,one can derivepredictionsabout the optimal allocationofauthority within organizations,either in a one-shotor in a repeatedcontext.

BilateralContractsand Multilateral Exchange

Finally, Chapter 13dealswith another common form of contractual

incompleteness: the limited participation of all concernedparties in a singlecomprehensivemultilateral contract.Employment contracts,for example,are generally bilateral contractsbetween an employerand an employeeeven in situations where the employeeworks together in a team with

otheremployees,or in situations wherethe employeris involved in a wholenexus of contracts with suppliers,clients, lenders,and other providersof capital.When (incomplete)bilateral contracts are written in suchmultilateral contractsettings, any bilateralcontractmay imposean

externality on the other parties.The equilibrium outcomeof the contracting

game may then be inefficient. Thus a central focusof this chapter is thecharacterizationof situations wherebilateralcontracting results in efficientoutcomes.

An important distinction that is drawn in the literature is whether

the bilateral contract is exclusive or nonexclusive\342\200\224that is, whether the

41 Introduction

employeecansign only an exclusivecontractwith oneemployerorwhether

she cansign up with severalemployersfor severalpart-time jobs.Interestingly,

most employment contractsare exclusive.But this is not always thecasefor other contracts.For example,for health insurance it is generallypossiblefor the insureeto acquiresupplementary insurance.Similarly, with

creditcarddebt or other loans,a borrowercanbuild up debton severaldifferent cardsor take out loansfrom severaldifferent lenders.Exactly why

exclusivity is requiredfor sometypesof contractsbut not othersinvolvingexternalitieshas not beenfully explored.Intuitively, one should expecttoseeexclusivecontractswhen the externality is potentially largeand when

exclusivity is easy to enforce.Whether exclusivity is enforced or not,however, one should expectinefficient equilibrium outcomesto obtain in

general in bilateralcontracting games,sincebilateral contractsalone areinsufficient to fully internalize all externalitiesacrossall affectedparties.This is a centraltheme in the common agency literature, which studiesmultiple

bilateral incentive contracting betweena single agent and severalprincipals (see,for example,Bernheim and Whinston, 1985,1986a,1986b).Chapter13providesan extensivetreatment of this important contracting

problem.Becauseof the presenceof a potential externality, an obvious concernis

whether the bilateral contracting game has a well-definedequilibriumoutcome.An early focusof the contracting literature has indeedbeenthe

potential nonexistenceof equilibrium in such contracting games.In alandmark article Rothschild and Stiglitz (1976)have thus shown that if two

insurers compete for exclusivebilateral contracts with one or severalinsureeswho have private information about their likelihoodof facing anadverseshock(oraccident),then a well-definedequilibrium outcomeof the

contracting game may not exist.The reason is that each insurer has anincentive to respondto the contract offers of the other insurer by onlycreamskimming the good risks and leaving the high-risk insureestocontract with their rival. Chapter13discussesthis striking result and the vastliterature it has spawned.

Other important topicsare touchedon besidesthese two broad themes,such as the strategicvalue ofcontracting in duopoly or barrier-to-entrysettings, or the impact of product-marketcompetitionon the sizeof agencyproblems.But it is fair to say that Chapter13doesnot attempt to providea systematic treatment of the existing literature on bilateral contracting

42 Introduction

with competition (whether static or dynamic, with adverseselection,moralhazard, or both) simply becausethe existing literature that toucheson this

topicis at this point both toovast and toodisconnectedto beableto providea systematic and comprehensivetreatment in only one chapter.11

1.7 Summing Up

The analysis of optimal contracting in this bookhighlights the fact that

common contract forms and institutions that we take for granted, like

employment contractsand ownership rights, are sophisticatedand multi-

faceted \"institutions.\" For a long time economistshave beenable to giveonly an oversimplified analysis of these contract forms, which ignoreduncertainty, asymmetric information, incentives, and controlissues.As this

introductory chaptermakesclear,however,a basiceconomicrelationlikethe employment relationhas to deal with thesevarious facets.

Thegoalof this bookis thereforeto explain how existing contracttheoryallows one to incorporatethesefeatures,not just in employment relationsbut also in many other applications.In fact, we have chosento illustrate

contract-theoreticanalysesin many different contexts,typically choosingthe applicationthat has beenthe most influential in economicsfor theparticular generalproblemunder consideration.

The bookgives an overview of the main conceptualbreakthroughs ofcontracttheory, while alsopointing out its current limitations. As contracttheory has grown to becomea largefield,we have beenforcedto limit our

coverageby making difficult choiceson what to leaveout. While we have

given carefulconsiderationto what material to cover,our choicesinevitablyalsoreflectour limited knowledgeof the field and our own personalpreferences. As the readermay alreadyhave noted,our biaseshave likely beenin the directionof overemphasizing economicideas,insights, and simplicityovergenerality.

11.For example, we do not cover here the growing literature on contracting in a \"generalequilibrium\" setting, for example, the impact of credit rationing on macroeconomicfluctuations (seeBernanke and Gertler, 1989,and Kiyotaki and Moore, 1997)or income distribution

(seeBanerjee and Newman, 1991,1993,and Aghion and Bolton, 1997).Nor do we cover moretraditional general equilibrium analysis with moral hazard or adverse selection (seePrescottand Townsend, 1984,and Guesnerie, 1992).Indeed, providing a self-contained treatment ofthese various topics would require dealing with many technical issues that go beyond contracttheory.

43 Introduction

The final chapterof the bookcontainsa set of exercises,which servetwo

purposes.First and foremost,theseexerciseshelpthe readerto mastersomeof the basicanalytical techniquesto solveoptimal contracting problemsandto developa deeperunderstanding of the most important arguments. Thesecondpurposeof this chapteris to coversomeclassicarticlesthat we havenot had the spaceto coverin the main chapters.For this reason it may beworth leafing through this chaptereven if the readerdoesnot intend to try

to solveany of the problems.

STATICBILATERALCONTRACTING

The first part of this bookdealswith the classicaltheory of incentives and

information in a static bilateral contracting framework. The fundamental

conceptualinnovation in the contracting problemsconsideredhere relativeto the classicaldecisionproblemsin microeconomicstextbooksis that oneof the contracting parties, the principal, is now controlling the decisionproblemof another party, the agent. That is, the principal'soptimizationproblemhasanother optimization problemembeddedin its constraints: the

agent's optimization problem.The first part of this bookprovidesasystematic expositionof the generalmethods that have beendevelopedto

analyze this type of nestedoptimization problem.The first two chaptersare concernedwith the generalproblemof

contracting under hidden information. Chapter2 considersoptimal contractsdesignedby the uninformed party. This isgenerally referredto asthe

screening problem.Chapter 3 turns the table and considersoptimal contractsofferedby the informed party. Thisinvolves a signaling problemand is

generally referred to as the informed principalproblem.Chapter4 dealswith

the contracting probleminvolving hidden actions.Thisis generally known

as the moral hazardproblem.Much of the material in thesethreechapterscan also be found in advancedmicroeconomicstextbooks,such as Mas-Cole11,Whinston, and Green(1995)and in the contracttheory textbooksofSalanie(1997)and Laffont and Martimort (2002).Chapter5,on the otherhand, coversthe generalproblemof disclosureof verifiable privateinformation and introducesmaterial that is not found in other texts.This chapterdealswith a subclassof contracting problemsunder hidden information,which is relevant in particular to auditing and securitiesregulations. Thenext chapter (Chapter6) coversricher contracting problemsthat involvemultidimensional private information or hidden actions,and alsoacombination of hidden information and hidden actions.Again, most of thematerial in this chapteris not coveredin depth in other textbooks.

HiddenInformation, Screening

In this chapterwe focuson the basicstatic adverseselectionproblem,with

one principal facing one agent who has private information on her \"type,\"that is,her preferencesor her intrinsic productivity. Thisproblemwas first

formally analyzed by Mirrlees(1971).We first explain how to solvesuch

problemswhen the agent canbeofonly two types,acasethat alreadyallowsus to obtain most of the key insights from adverseselectionmodels.We dosoby looking at the problemof nonlinear pricing by a monopolisticsellerwho facesa buyer with unknown valuation for his product.

We then move on to other applications,still in the casewhere theinformed party canbe of only two types:credit rationing, optimal incometaxation, implicit labor contracts,and regulation. Thisis only a partial list

ofeconomicissueswhereadverseselectionmatters. Nevertheless,theseareall important economicapplicationsthat have madea lasting impressiononthe economicsprofession.For eachof them we underline both the economicinsights and the specificitiesfrom the point of view of contracttheory.

In the last part of the chapterwe extend the analysis to more than two

types, returning to monopoly pricing. We especiallyemphasizethecontinuum case,which is easierto handle.Thisextensionallowsus to stresswhich

results from the two-type caseare general and which ones are not. Themethodswe presentwill alsobe helpful in tackling multiagent contexts,inparticular in Chapter7.

2.1The SimpleEconomicsofAdverseSelection

Adverseselectionnaturally arisesin the following context,analyzed first byMussaand Rosen (1978),and subsequently by Maskin and Riley (1984a):Considera transaction betweena buyer and a seller,wherethe sellerdoesnot know perfectlyhow much the buyer is willing to pay for a good.Suppose,in addition, that the sellersets the terms of the contract.Thebuyer'spreferencesare representedby the utility function

u(q,T,6)=fp(x,e)dx-Twhereq is the number of units purchased,T isthe total amount paid to theseller,and P(x,9) is the inversedemandcurve of a buyer with preferencecharacteristics6.Throughout this sectionwe shall considerthe following

specialand convenient functional form for the buyer'spreferences:

u(q,T,6)=0v(q)-T

Hidden Information, Screening

wherev(0) = 0, v'(q) > 0, and v\"(q) < 0 for all q.The characteristics9 areprivate information to the buyer.The sellerknows only the distribution of0,F(9).

Assuming that the seller'sunit productioncostsare given by c > 0, his

profit from selling q units against a sum of money T isgiven by

n =T-cqThe question of interest here is, What is the best,that is, the profit-

maximizing, pair (T,q) that the sellerwill be able to induce the buyer tochoose?The answer to this questionwill dependon the information thesellerhason the buyer'spreferences.We treat in this sectionthe casewherethere are only two types ofbuyers:6e[6L,6H], with 6H > 0L.Theconsumeris of type 6L with probability j3 6 [0,1]and of type 6H with probability

(1-j3).Theprobability j3 can alsobe interpretedas the proportionofconsumers of type 6L.

First-BestOutcome:PerfectPrice Discrimination

Tobeginwith, supposethat the selleris perfectlyinformed about the buyer'scharacteristics.The sellercan then treat each type of buyer separatelyandofferher a type-specificcontract,that is, (T,-,qi) for type 6t (i = H,L).Thesellerwill try to maximize his profits subject to inducing the buyer to

accept the proposedcontract.Assume the buyer obtains a payoff of u if

shedoesnot take the seller'soffer.In this case,the sellerwill solve

max Ti -cqiT\342\200\236qt

subjectto

We can call this constraint the participation, or individual-rationality,constraint of the buyer. The solution to this problemwill be the contract(qt, Ti) such that

eiv\\ql) =c

and

9lv(qi) =fl+u

Intuitively, without adverseselection,the sellerfinds it optimal to maximizetotal surplus by having the buyer selecta quantity such that marginal utility

Static Bilateral Contracting

equalsmarginal cost,and then setting the payment soas to appropriatethefull surplus and leaveno rent to the buyer aboveu. Note that in a market

context,u couldbe endogenized,but here we shall treat it as exogenousand normalize it to 0.

Without adverseselection,the total profit of the selleris thus

P(TL-cqL)+(l-P)(TH-cqH)and the optimal contractmaximizes this profit subjectto the participationconstraints for the two types of buyer. Note that it canbe implemented by

type-specifictwo-part tariffs, where the buyer is allowedto buy as much asshewants of the goodat unit pricec providedshepays a type-specificfixedfee equal to 0,-v(#i)-cqt.

The idea that, without adverseselection,the optimal contractwill

maximize total surplus while participation constraints will detenninethe way in

which it is sharedis a very generalone.1Thisceasesto be true in thepresence of adverseselection.

AdverseSelection,LinearPricing, and SimpleTwo-PartTariffs

If the sellercannotobservethe type of the buyer anymore, he has to offerthe samecontractto everybody.The contractset is potentially large,sinceit consistsof the set of functions T(q).We first lookat two simple contractsof this kind.

2.1.2.1LinearPricing

The simplest contractconsistsin traditional linear pricing, which is asituation wherethe seller'scontractspecifiesonly a priceP.Giventhis contractthe buyer choosesq to maximize

6[V(q)-Pq, wherei =L,HFrom the first-orderconditions

e<v\\q) =Pwe can derivethe demandfunctions of each type:2

1.This idea also requires that surplus befreely transferable across individuals, which will notbethe caseif someindividuals face financial resource constraints.2.The assumed concavity of v(.) ensures that there is a unique solution to the first-orderconditions.

50 Hidden Information, Screening

qi=Di(P)The buyer'snet surplus cannow be written as follows:

Si(P)=eiv[Dl(P)]-PDi(P)Let

D(P)=PDL(P)+0.-P)DH(P)S(P)=PSL(P)+(X-P)SH(P)

With linear pricing the seller'sproblemis the familiar monopoly pricing

problem,where the sellerchoosesP to solve

max(P-c)D(P)p

and the monopoly price is given by

D(P)Pm=c-D'{P)In this solution we have both positive rents for the buyers [S(P)> 0] and

inefficiently low consumption, that is, Ojy'iq) = P > c, since the sellercanmake profits only by setting a pricein excessofmarginal costand

\302\243>'(\342\200\242)

< 0.Note that, dependingon the valuesof /J, 6L, and 6H,it may beoptimal for thesellerto serve only the 9H buyers.We shall, however,proceedunder the

assumption that it is in the interestof the sellerto serveboth markets.Can the sellerdo better by moving away from linearpricing? Hewill be

able to do so only if buyers cannot make arbitrage profits by trading in asecondarymarket: if arbitrage is costless,only linear pricing is possible,becausebuyers would buy at the minimum averagepriceand then resellinthe secondarymarket if they do not want to consumeeverything they

bought.

2.1.2.2SingleTwo-PartTariff

In this subsectionwe shall work with the interpretation that there is onlyone buyer and that j3 is a probability measure.Underthis interpretationthere are no arbitrage opportunities open to the buyer. Therefore,a singletwo-part tariff (Z, P),where P is the unit price and Z the fixed fee,will

improve upon linearpricing for the seller.Note first that for any given priceP,the minimum fixed fee the sellerwill set is given by Z = SL(P).(This is

51 Static Bilateral Contracting

the maximum fee a buyer of type 9L is willing to pay.)A type-0H buyer will

always decideto purchasea positive quantity ofq when chargeda two-parttariff T(q)= SL(P)+ Pq,since 6H > 6L.If the sellerdecidesto serveboth

types of customersand therefore setsZ = SL(P),he also choosesP tomaximize

maxSL(P)+(P-c)D(P)p

The solution for Punder this arrangement is given by

S'L(P)+D(P)+(P-c)D'(P)=0

which implies

D(P)+Sj(P)P=cD\\P)

Now, by the envelopetheorem,S'L(P)= -DL(P),so that D(P)+ S'L(P)is

strictly positive;in addition, D'(P)< 0, so that P > c.Thus, if the sellerdecidesto serveboth types ofcustomers[andthereforesetsZ =SL(P)],thefirst-best outcome cannot be achieved and underconsumption remainsrelative to the first-best outcome.3Another conclusionto be drawn fromthis simple analysis is that an optimal single two-part tariff contract is

always preferredby the sellerto an optimal linear pricing contract [sincethe seller.canalways raise his profits by setting Z = SL(Pm)].We can alsoobservethe following: If Pm, Pd, and Pc,respectively,denote the monopolyprice,the marginal price in an optimal single two-part tariff, and the (first-bestefficient) competitive price,then Pm> Pd> Pc= c.To seethis point,note that a small reductionin pricefrom Pm has a second-order(negative)effecton monopoly profits (Pm-c)D(Pm),by definition of Pm.But it has afirst-order(positive)effect on consumersurplus, which increasesby anamount proportionalto the reduction in price.The first-order(positive)effectdominates, and, therefore,the selleris better off lowering the pricefrom Pm when he can extract the buyer's surplus with the fixed fee Z =SL(Pm). Similarly, a small increasein price from Pchas a first-order(positive)

effecton (Pc-c)D(Pc),but a second-order(negative) effecton S(PC),by definition of Pc.

3. If he decides to set an even higher fixed feeand to price the type-0^ buyer out of the market,he doesnot achieve the first-best outcome either; either way, the first-best solution cannot beattained under a single two-part tariff contract.

Hidden Information, Screening

SlopeP-

B'h.0H-type indifference curve

y bh

Figure 2.1Two-Part Tariff Solution

fy-type indifference curve

%

An important feature of the optimal single two-part tariff solution isthat the 0/rtype buyer strictly prefersthe allocationBH to BL,as illustrated

in Figure 2.1.As the figure alsoshows,by setting up moregeneralcontractsC= [q,T(q)],the sellercan do strictly better by offering the sameallocation to the 0L-type buyer, but offering, for example,someother allocationB'H *BH to the 0fl-type buyer. Notice that at B'H the sellergets a highertransfer T(q)for the sameconsumption (this is particular to the example).Also, the 6H buyer is indifferent betweenBL and B'H.These observationsnaturally raise the question of the form of optimal nonlinear pricingcontract.

Second-BestOutcome:Optimal Nonlinear Pricing

In this subsectionwe show that the sellercangenerally do better by

offeringmore general nonlinear pricesthan a single two-part tariff. In the

processwe outline the basicmethodology of solving for optimal contractswhen the buyer'stype is unknown. Sincethe sellerdoesnot observethe

type of the buyer, he is forced to offer her a set of choicesindependentof her type. Without loss of generality, this set can be describedas

53 Static Bilateral Contracting

[q,T(q)];that is, the buyer facesa schedulefrom which she will pick theoutcomethat maximizes her payoff. The problemof the selleris thereforeto solve

maxP[T(qL)-cqL]+0.-P)[T(qH)-cqH]T(q)

subjectto

qx=argmax 6iV(q) -T(q) for i =L,H

i

and

0^(4,-)-7%)>0for i =L,HThe first two constraints are the incentive-compatibility (IC)constraints,while the last two are participation or individual-rationality constraints

(IR).Thisproblemlooksnontrivial to solve,sinceit involves optimizationover a scheduleT(q) under constraints that themselvesinvolveoptimization problems.Such adverse selectionproblemscan, however, be easilysolvedstep-by-stepas follows:

Step1:Apply the revelation principle.From Chapter1we canrecallthat without lossofgenerality we canrestricteachscheduleT(q)to the pair of optimal choicesmadeby the two types ofbuyers [[T(qL),qL]and [T(qH),qH]}; this restrictionalso simplifies greatlythe incentive constraints.Specifically,if we define T(qt) = Tt for i = L,H,then the problemcanbe rewritten as

maxj3CTL -cqL)+0.-P)(TH-cqH)T\342\200\236qt

subjectto

6Hv(qH)-TH>eHv(qL)-TL (ICH)

eLv(qL)-TL>dLv{qH)-TH (ICL)

6Hv(qH)-TH>0 (IRH)

eLv(qL)-TL>0 (IRL)

The sellerthus faces four constraints, two incentive constraints [(/C,-)means that the type-fy buyer should prefer her own allocationto theallocationof the other type of buyer] and two participation constraints

54 Hidden Information, Screening

[(IRi) means that the allocationthat buyer of type 0,choosesgives her anonnegative payoff].Step1has thus alreadygreatly simplified the problem.We cannow try to eliniinate someof theseconstraints.

Step2:Observethat the participation constraint of the \"high\" type will notbind at the optimum.

Indeed(IRH)will be satisfiedautomatically becauseof (IRL)and (ICH):

eHv(qH)-TH>eHv(qL)-TL>eLv(qL)-TL>0where the inequality in the middle comesfrom the fact that 6H > 6L.

Step3:Solvethe relaxedproblemwithout the incentive constraint that is

satisfiedat the first-best optimum.

The strategy now is to relax the problemby deletingone incentiveconstraint, solvethe relaxedproblem,and then checkthat it doessatisfy this

omitted incentive constraint. In order to choosewhich constraint to omit,considerthe first-best problem.It involves efficient consumption and zerorents for both types of buyers, that is, 6iV'(qi) = c and Q,y(q) = Tr Thisoutcomeis not incentive compatible,becausethe 6H buyer will prefer tochoose(qL, tL) rather than her own first-best allocation:while this

inefficiently restrictsher consumption, it allows her to enjoy a strictly positivesurplus equalto (6H-6L)qL, rather than zerorents.Instead,type 6L will notfind it attractive to raiseher consumption to the levelqH: doing sowould

involve paying an amount TH that exhauststhe surplus of type 6H and would

thereforeimply a negative payoff for type 6L,who hasa lowervaluation forthis consumption. In step3,we thus chooseto omit constraint (JCL).Notethat the fact that only one incentive constraint will bind at the optimumis driven by the Spence-Mirrleessingle-crossingcondition, which can bewritten as

d[\"

du/dql~de[~du/dTjThis condition means that the marginal utility of consumption (relativetothat of money, which is here constant)riseswith 9.Consequently,optimalconsumption will have to risewith 6.

Step4:Observethat the two remaining constraints of the relaxedproblemwill bind at the optimum.

Rememberthat we now look at the problem

55 Static Bilateral Contracting

maxj3CrL -CqL)+(X-pXTH -cqH)T,,qi

subjectto

eHv(qH)-TH>eHv(qL)-TL (ICH)

6Lv(qL)-TL>0 (IRL)In this problem,constraint (ICH)will bind at the optimum; otherwise,thesellercanraise TH until it doesbind: this stepleavesconstraint (IRL)unaffected while improving the maximand. And constraint (IRL)will alsobind;

otherwise,the sellercanraise TLuntil it doesbind: this step in fact relaxesconstraint (ICH)while improving the maximand [it is here that havingomitted (ICL)matters, since a rise in TL couldbe problematicfor this

constraint].

Step5:Eliminate TLand TH from the maximand using the two binding

constraints, perform the unconstrained optimization, and then checkthat

(ICL)is indeedsatisfied.

Substituting for the valuesof TLand TH in the seller'sobjectivefunction,we obtain the following unconstrainedoptimization problem:

msixl3[dLv(qL)-cqL]+(l-l3)[dHv(qH)-cqH-(eH-eL)v(qL)]iL,qn

The first term in bracketsis the full surplus generatedby the purchasesof type 6L, which the sellerappropriatesentirely becausethat type is left

with zerorents.Instead,the secondterm in bracketsis the full surplusgenerated by the purchasesof type 9H minus her informational rent (6H -0l)v(<?l),which comesfrom the fact that she can \"mimic\" the behaviorofthe other type.Thisinformational rent increaseswith qL.

The following first-orderconditionscharacterizethe unique interiorsolution (q*, q%) to the relaxedprogram, if this solution exists:4

eHv'(q%)=c

Mtf)= n-peH--K]>cI P eL J

4. If the denominator of the second expression is not positive, then the optimal solutioninvolves q*=0,while the other consumption remains determined by the first-order condition.

56 Hidden Information, Screening

This interior solution implies q\\ < q%. One can then immediately verifythat the omitted constraints are satisfiedat the optimum (q* Tf,i = L,H)given that (ICH)binds. Indeed,

0Hv(ql)-T%=dHv(q*L)-tf (ICH)

togetherwith 6L < 6H and q*<q%,implies

eLv{q*H)-T%<GLv{qt)-TZ (ICL)We have therefore characterizedthe actual optimum. Two basic

economic conclusionsemergefrom this analysis:

1.The second-bestoptimal consumption for type 6H is the same as thefirst-best optimal consumption (qH), but that of type 9L is lower.Thus

only the consumption of one of the two types is distortedin the second-bestsolution.

2.The type-0Lbuyer obtains a surplus ofzero,while the other type obtainsa strictly positive \"informational\" rent.

Thesetwo conclusionsare closelyrelated to eachother:the consumptiondistortion for type 9L is the result of the seller'sattempt to reducetheinformational rent of type 6H.Sincea buyer of type 6H is moreeagerto consume,the sellercanreducethat type'sincentive to mimic type 6L by cutting down

on the consumption offeredto type 6L.By reducing type Oh'sincentives tomimic type 6L, the sellercan reduce the informational rent of (or,equiva-lently, chargea higher price to) type 6H. Lookingat the first-orderconditions for qf indicatesthat the sizeof the distortion, qL-q*, is increasingin

the potential sizeof the informational rent of type 6H\342\200\224asmeasuredby the

difference(6H- 6L)\342\200\224and decreasingin /J.For /J and {dH-6L) largeenoughthe denominator becomesnegative.In that casethe sellerhits the constraint

qL>0.As the latter part of the chapter will show, what will remain true with

more than two types is the inefficiently low consumption relative to thefirst best (exceptfor the highest type: we will keep \"efficiency at the

top\") and the fact that the buyer will enjoy positiveinformational rents

(exceptfor the lowesttype).Beforedoing this extension,letus turn to otherapplications.

57 Static Bilateral Contracting

2.2Applications

2.2.1CreditRationing

Adverseselectionarisesnaturally in financial markets. Indeed,a lenderusually knows lessabout the risk-return characteristicsof a project than

the borrower.In other words, a lender is in the samepositionas a buyerof a secondhandcar:5it doesnot know perfectly the \"quality\" of the

projectit invests in. Becauseof this informational asymmetry, inefficienciesin the allocationof investment funds to projectsmay arise.As in the caseof secondhandcars, these inefficienciesmay take the form that \"goodquality\" projects remain \"unsold\" or are denied credit.This type ofinefficiency is generally referred to as \"credit rationing.\" There is now anextensiveliterature on credit rationing. The main early contributions areJaffeeand Modigliani (1969),Jaffeeand Russell(1976),Stiglitz and Weiss

(1981),Bester(1985),and De Meza and Webb (1987).We shall illustrate

the main ideaswith a simple examplewhere borrowerscan be of two

different types.Considera population of risk-neutral borrowerswho eachown a project

that requiresan initial outlay of/=1and yieldsa random return X, whereXe [R,0}.Letpe [0,1]denote the probability that X=R.Borrowershaveno wealth and must obtain investment funds from an outsidesource.Aborrower canbe of two different types i =s,r, wheres stands for \"safe\" and rfor \"risky.\" Theborrowerof type i has a projectwith return characteristics(Pi, R).We shall make the following assumptions:

Al: PiRt =m, with m > 1A2: ps>pr and Rs < Rr

Thus both types ofborrowershave projectswith the sameexpectedreturn,but the risk characteristicsof the projectsdiffer. In general,project typesmay differ both in risk and return characteristics.It turns out that the earlyliterature mainly emphasizesdifferencesin risk characteristics.

A bank can offer to finance the initial outlay in exchangefor a future

repayment. Assume for simplicity that there is a single bank and excess

5.Akerlof (1970),in a pioneering contribution, has analyzed the role of adverse selection inmarkets by focusing in particular on used cars. SeeChapter 13on the role of adverseselection in markets more generally.

58 Hidden Information, Screening

demand for funds: the bank has a total amount a < 1of funds, and it facesa unit massofborrowers.Theproportionof \"safe\" borrowersis /J (and that

of \"risky\" borrowersis 1-/J).Assume, moreover,that a > max {/J, 1-/J},so that availablefunds are sufficient to avoid crowding out either type ofborrowercompletely.

What type of lending contractshould the bank offera borrowerin this

situation? Undersymmetric information and no contractual restrictions,thebank would lend all of a and would be indifferent about financing eithertype of borrower:it would specifya repayment \302\243>;

for borrowerof type i,with Dt =Rt, which it would obtain with probability pv

What about adverseselection?The early literature on credit rationingdoesnot allow for much contractual flexibility: it considersonly contractswhere the bank specifiesa fixed repayment, D, in exchangefor the initial

outlay 1=1.Assume that type-z borrowersapply for funds if and only if

D < Rt. If D> Rs, only \"risky\" borrowerswill considerapplying. In this case,setting D = Rr is optimal and gives the bank a profit of

(1-jSXm-l) (2.1)Instead, if D < Rs, both types of borrowerswill apply. Assuming each

applicant has an equal chance of being financed, the bank then finds it

optimal to set D = Rs and obtains

a\\fi(.m -1)+(1-P)(prRs -D] (2.2)This secondoutcomeallows the bank to use its funds fully. However,it

is now earning lessthan m on \"risky\" borrowers,sincethey repay only Rs,with the relatively low probability pr. Whether expression(2.1)or (2.2)ishigher will dependon parametervalues.Ceterisparibus, expression(2.2)will behigher than expression(2.1)when 1-/J is small enough, or whenpris closeenough to ps.When this is the case,onecansay that creditrationingarises:some \"risky\" borrowerscannot get credit;they would be ready to

accepta higher D,sincethey obtain strictly positivesurplus when financed;however,the bank finds it optimal not to raise D,becauseit would then

losethe \"safe\" borrowers.This story begsthe following question,however:Can't the bank dobetter

by offerring more sophisticatedcontracts?Let us keepthe restrictiontodebt contracts,or fixed-repaymentcontracts (more on this topic will

follow), but assumethat the bank can offer contractsthat differ both in

59 Static Bilateral Contracting

terms of repayment and in terms of probability of obtaining the scarcefunds. Specifically,call

(x\342\200\236 \302\243>;)a contract that offersfinancing with

probability xi and repayment \302\243),-.Thebank then setsits contractsto solvethe

following problem:

max[jfo,(p,Z),-1)+(X-P)xriprDr-1)]x\342\200\236Dt

subjectto

0<x;<lfor all i =s,r (2.3)Di<Ri for all i =s,r (2.4)

xjiiRi-DitexjPiiRi-Dj)for all i,j=s,r (2.5)

pxs+(l-p)xr<cc (2.6)whereexpression(2.3)is a set of feasibility constraints; expressions(2.4)and (2.5)are the individual rationality and incentive compatibility

constraints; and expression(2.6)is the resourceconstraint given by the bank'savailability of funds.

SinceDt <R\342\200\236

one can easilyseethat the binding individual rationalityconstraint is

DS<RS

while the binding incentive constraint is

xr(Rr-Dr)>xs(Rr-Rs) (2.7)The binding incentive constraint implies that extracting all the rents on

risky borrowers(Dr= Rr) requiresexcluding safeonesfrom borrowing (xs=0).Instead,giving both types equalaccessto credit(xr =xs)implies equalrepayments when successful(Dr = Rs).These two outcomesare the onesdiscussedearlier.Can the bank do better? To answer this question, notethat the binding incentive constraint (2.7)canbe rewritten as

Dr=Rr-^(Rr-Rs)Xr

so that the precedingproblembecomes

max{pxs(psRs-l)+0-p)[xr(prRr-l)-xsPr(Rr-Rs)]}Xs,Xr

60 Hidden Information, Screening

subjectto

0<xs<xr<lPxs+(1-P)xr<a

SincepsRs =prRr = m, this problemleadsto xr =1:there is no distortion

at the top, that is,no rationing of risky borrowers.As for safeones,we havexs=0 whenever

P(psRs-l)'-0.-P)pr(Rr-Rs)<0while xs= [a-(1-P)xr]/potherwise.

Intuitively, in this solution, the bank trades off the rents extractedfrom

\"risky\" borrowerswith the ability to lendall its funds. When \"safe\"

borrowers arenumerous enough, it doesnot make senseto excludethem by setting

xs=0.As a result, risky borrowersearn rents.However,it is possibleto dobetter than financing everybodywith a repayment Rs:one induces\"risky\"

borrowersto choosea higher repayment by giving them preferentialaccessto funds. Thebank endsup going \"all the way\" when setting xr =1.

Note that creditrationing has disappearedhere:whether or not \"risky\"

borrowersearnrents (that is,whether xs> 0 or xs =0),they are not rationed.The only oneswho are not fully funded are \"safe\" borrowers,who are in

fact indifferent about being funded, since they never earn rents when

financed.Credit rationing reappears,however,when assumption Al is

replacedby

A3: psRs>1, but prRr <1while assumption A2 is maintained. In this case,the bank would like to turn

away \"risky\" borrowersbut is unable to do so.Consequently, either both

types of projects are funded and there is cross-subsidizationbetweentypes\342\200\224this outcomeoccurswhenever[fips + (1-P)pr]Rs ^ 1\342\200\224or neither

project is funded and there is \"financial collapse,\"a severeform of creditrationing. This happenswhen \\fips + (1-p)pr]Rs < l.6

Theprecedingexampleleadsto severalremarks:

1.The example illustrates that even though asymmetric information is

likely to be a pervasivephenomenonin financial markets, it doesnot in

6. Note that the lender would never set a contract with D>Rs,since then only \"risky\"projects (with a negative net present value) would befunded.

61 Static Bilateral Contracting

itself inevitably give riseto credit rationing. It may bepossibleto achieveallocativeefficiency despiteasymmetric information. Moreover,even if

there are inefficienciesresulting from asymmetric information, they maynot take the form of credit rationing. Negative-net-present-valueprojectsmay be funded.

2.When assumptions Al and A2 hold, credit market inefficienciesariseonly as a result of exogenousrestrictionsimposedon the allowableset ofcontracts.To seethis point, supposethat the lender can set an arbitrary

repayment schedulecontingent on realizedreturns: Dsif X= Rs and Dr if

X = Rr. Then it suffices to set\302\243>,

= Rh and there is no longera need to try

to discriminate betweenborrowertypes, sinceex post eachborrowertypeendsup paying the relevant contingent repayment. Thus in this examplethe

potential inefficienciesarising from asymmetric information can beovercome if the lendercan freelydecidehow to structure the investmentcontract. This approachworks very straightforwardly herebecausethe expostreturns are different for the two types. It should bestressed,however, that

evenin moregeneralexamplesit is more efficient to discriminate betweenborrower types by offering general-return-contingentrepayments. Whensuch contractsare allowed,the incidenceof creditrationing is much lower.Thereadermay wonderat this point why loancontractsin practicedo not

specifygeneral-return-contingentrepayments if they aresomuch moreefficient. We shall return to this important questionin Chapter11.3.When assumptions A3 and A2 hold instead,then inefficienciesmay ariseeven if a general-return-contingentrepayment schedulecan be specified.In this casethe lenderdoesnot want to finance projectsof type r.It canattempt to discouragetype-rapplicantsby setting Dr = Rr. However,evenwith such a high repayment, type-r applicantsmay still want to apply.

Suppose,for example,that the borrowersget positive utility just by

undertaking the project;then type-rborrowerswould want to apply for fundingeven if their expectedmonetary return (net of repayments) from

undertaking the projectis zero.The maximum expectedreturn for the lenderin

that caseis given by

PpsRs+(l-P)prRr-lWhen j3 is sufficiently large, this return is positive and there is cross-subsidization.However,when /J is small the expectedreturn is negative andthere is financial collapse.

64 Hidden Information, Screening

Under adverseselection,the following set of incentive constraints mustbe imposedon the government's optimization problem:

eLeL-tL -yr(eL)>9HeH-tH \" v(^p-) (2-10)

eHeH-tH-Wh )> eLeL-tL- [^J (2.ii)Indeed,when the government cannot identify eachproductivity type, it canonly offereverybody an income-contingenttax, where tt has to be paid if

incomeqt = Qfii is produced.This would allow an individual with

productivity 6j to pay t; by producing qt at the cost of an effort level dfije,.As

always, the incentive constraints ensure that this behavioris not attractivefor either type of individual.

When u(-) is concave,the completeinformation optimum is such that

qL~tL-V(eL) =qH-tH-V(eH)

This allocation, however, violates incentive constraint (2.11):high-

productivity individuals would then prefer to choose(qL, tL) instead of(<?h, tH)? Therefore,condition (2.11)must be binding in a second-bestoptimum, and the same is of course true of the budget constraint (2.8).Using these two constraints to eliroinate the tax levelsfrom the maximandand taking the first-orderconditionswith respectto eH and eLthen yields

V'(eH)=6H (2.12)and

yf'(eL)=eL-(x-pyr (2.13)

wherey= {u'L-Uh)IWl + (1-P)uk]-

That is, /representsthe differencebetweenthe marginal utilities of low-and high-ability individuals as a percentageof averagemarginal utilities.

Condition (2.12)tells us, as is by now familiar, that the second-bestallocation for type 6H is efficient (efficiencyat the top).Condition(2.13)tells

9.Note that this conclusion is robust to changes in the utility function: it would remain true,for example,if effort appeared additively outside the utility function {e.g.,if we assume apayoffu[q ~t]~yr(e), with y(-) being linear or convex in effort).

65 Static Bilateral Contracting

us that type 9L underprovideseffort.Indeed,first-best efficiency requiresthat y/(eL)= dL. But since6L < 9H, we have 0<yr'(eL)0--eL/eH)<yr\\eL)-QlI^hW'^l^lIQh)-Moreover,for any strictly increasing,concaveutility

function u(-),y is strictly positive,since then u'L>u'H> 0.Therefore,thesecondterm of the right-hand side(RHS)of condition (2.13)is strictly

positive, resulting in underprovision of effort.The reasonwhy it is second-bestefficient to underprovide effort here is that a lowereLlimits the welfaredifferencebetweenhigh- and low-productivity individuals, which is givenby

[8HeH -tH-Y(eH)]-[6LeL-tL -y(eL)]=y^eJ-y^^J (2.14)

when the incentive constraint (2.11)is binding. This brings about a first-

ordergain in the utilitarian welfarefunction that exceedsthe second-orderlossfrom a reduction in productivity of the low type. A reduction in eLbrings about a reductionin inequality ofwelfarebecausehigh-productivityindividuals have the option to produceoutput qL while saving a proportion(Oh-0l)/0hof the effort eLthat low-productivity individuals have to exert.

How can we reinterpretconditions(2.12),(2.13),and (2.14)in terms ofexisting featuresof the incometax code?One can think of governmentsfirst setting marginal tax ratesas well as uniform tax rebates,while

individuals respondby choosingeffort.In this perspective,\342\200\242 Equation (2.12)implies that the marginal tax rate at output qH = QHeH

should bezero,in order to induceefficient effort.\342\200\242 By contrast,the marginal tax rate at output qL = dLeL should bepositive,and equalto

where/and eLare the second-bestvalues;indeed,this marginal tax rate isneededto induce the (inefficiently low) choice of eL given by equation(2.13).\342\200\242 In between these two output levels,the marginal tax rate should bepositive and such that the additional tax payment coming from high-

productivity individuals will be sufficient to reduce the ex post utility

difference betweenthe two types to that given by condition(2.11)[recallthat,

66 Hidden Information, Screening

from budget constraint (2.8),all marginal tax revenuesare returned toindividuals, in the form of uniform tax rebates].

Thisframework thus seemsto pleadfor positive marginal tax ratesat lowincomelevels,and for zero marginal tax rates at high incomelevels.Thisresult appearsparadoxical,sincewe started from a government objectivethat was tilted toward redistribution. In fact, there is of coursepositiveredistribution here, through the uniform tax rebate.And the positivemarginal tax rate at low incomesis there to achievehigher redistribution, asexplainedearlier.The theory in fact offersa rationalization for the

widespread practiceof making various welfarebenefitsconditional on havinglow incomes:this results in very high effectivemarginal tax rates at lowincomelevels.As for the zero-marginal-tax-rate-at-the-topresult, it shouldnot be overemphasized:If we allowedfor a continuum of types (as in

section2.3),this zero marginal tax rate would appear only at the verymaximum of the incomedistribution.

We closethis subsectionby analyzing the effectof changesin thegovernment's information structure in this setup.Following Maskin and Riley(1985),we couldindeedask the following question:how would tax levelsbe affected if, instead of observing \"income\" q = d-e, the governmentobservedindividual \"input,\" that is,effort (or,equivalently, hours ofwork)?Would this new information structure result in more or lessinequality, andmore or lessallocativedistortion?

In comparisonwith the precedinganalysis, only the incentive constraints

(2.10)and (2.11)would be affected.Theywould become

eLeL-tL-yr(eL)> 6LeH-tH-yr(eH) (2.16)dHeH -tH-Wn)>QneL-tL-WL) (2.17)since individuals now face effort-contingent taxes, instead of output-

contingent taxes.As before,the problemis to prevent high-abilityindividuals from mimicking the choiceof low-ability individuals. Consequently,condition (2.17)must be binding at the optimum, and the inequality in

welfarebetweenthe two types is given by

[9HeH-tH-y/(eH)]-[9LeL-tL-y/(eL)]=(9H-eL)eL (2.18)Thus, for a given eL,welfareinequality is higher in equation(2.18)than

in equation (2.14)whenever 6H > yr\\eL). This is always the caseat the

Static Bilateral Contracting

optimum, since9H> 9L>y/(e*).Intuitively, an effort-contingent tax makesit more attractive for the high type to mimic the behaviorof the low type,sincea given effort levelallows the high type to enjoy comparatively moreoutput than the low type, thanks to his higher productivity. Instead,in an

income-contingentscheme,mimicking the low type really meanssticking toa low income.It is true that this allows the high type to saveon effort cost,but with a convexeffort cost,the benefit of this lowereffort is limited.

Solving the optimum tax problem with equation (2.17)instead ofequation(2.11),onecanshow that efficient effort results again for the high

type. Instead,the effort of the low type is further distorteddownward, in

comparisonwith the income tax case:when welfareinequality is higher,the marginal benefit from reducing inequality at a given allocativecost isalso higher. In this setting, effort monitoring is thus inferior to incomemonitoring, sinceit leadsboth to more inequality and to more allocativeinefficiency.

This last discussionis especiallyrelevant when the principal is able tomake decisionsabout which information structure to put in place.It would

apply naturally, for example,in the contextof the internal organization ofa firm, where decisionshave to bemade about which information systemsto set up for monitoring purposes.On this subject,seethe generalanalysisof Maskin and Riley (1985)on the superiority of output- overinput-monitoring

schemes.

Implicit Labor Contracts

The optimal labor contracting approachhas beenusedsincethe 1970stotry to understand labor-marketfluctuations, particularly perceivedwagerigidities and employment variability over the businesscycle.It is not our

purposehere to discussthe debatesamong macroeconomistssurroundingthe stylized facts,but simply to illustrate how the contracting paradigm mayor may not be consistentwith perceivedfacts.

2.2.3.1Contracting without AdverseSelection

The first wave of labor contractmodels(seeAzariadis, 1975;Baily, 1974;Gordon,1974)abstractfrom adverseselectionto focussolelyon risk

sharing. They are basedon the idea that employment relationships typically

involve set-upcosts(e.g.,specifictraining) so that firms and workershavean interestin organizing their future relationsthrough a long-term contract

68 Hidden Information, Screening

that has to be contingent on random future events.Becauseof imperfectinsurance markets, moreover,parties to the contractare risk averse.Thisobservationis especiallytrue ofworkers, whosehuman capitalis much lesseasily diversifiable than financial capital. Consequently,labor contractsserveboth to setproductiondecisionsand to allocaterisk efficiently.

Formally, considera firm that can produceoutput q using a technologyq = 6Q(L),where 9 is a random productivity parameter,L is the levelofemployment, and Q(-)is an increasingconcavefunction. The firm canhirefrom a pool of identicalworkerswhoseutility is u[w + G(l- \302\243)],

wherew

is the wagelevel,J is the individual's potentialworking time, I is effectiveindividual labor time, and G is a positive constant.We thus assumehereperfect substitutability betweenmoney and leisure(more on this topic to

follow).Let us normalize I to 1and assumeindivisibilities to be such that

Ican only take the values 0 or 1.In contrast to the analysis of this chapterso far, contracting now takes

place at the ex ante stage, that is, before the realizationof 6 is known.

Assume symmetric information at this stage:all partiesexpect6e{9L,6H]and expectProb[0= 0J = #,with /JL + /JH =1.Moreover,thesebeliefsarecommon knowledge.Theproblemhas two stages:

Stage1(exante stage):Hiring and contracting decisions.Stage 2 (expost stage):6t is learned by all parties,and the contract isexecuted.

CallL the stockofworkershired in stage1and o,Lthe employment levelgiven 9t in stage2,sothat (1-o,)Lcorrespondsto the workersthat are laidoff.A contractdetermines,for each

6\342\200\236the valuesof a,, and of

w\342\200\236

and wsi,

the wage and severancepay levels.Assume that workers have market

opportunities that amount to u in utility terms, and that the firm has an

increasingand concaveutility function V(-) of profits. The firm then solves

Max Tpy[etQ(flctL)-wtatL-wsl (1-at)L]Wi.at.L i

subjectto

X A [cCiu(wi )+(l-al)u(wsl +G)]>ui

and the feasibility constraints

a,<1 for i =L,H

69 Static Bilateral Contracting

Denotingfirm profits in eachstate 0,by

n, =diQia.L)-WidiL -wsl (1-a,)Lthe first-orderconditionswith respectto w, and wsi, respectively,yield, fori = L,H,

-plV,(Ul)aiL+Xplaiu'(wi) =0 (2.19)

-ptV'(n,)(1-at )L+Xpi (1-a,)u\\wsl +G)=0 (2.20)Whenever workersare risk averse {u\" < 0),equations(2.19)and (2.20)imply wt = wsl + G:for a given total monetary payment to workersit makesno sensefor the firm to have individual workersbear risk over who isretained and who is dismissed.In fact, given that they have idential

productivity and outsideopportunities, it is optimal to give them eachexactlyu in expectedterms ex ante.

In this model, therefore,there is no involuntary unemployment, sinceworkers are indifferent between being laid off and staying on the job.What about wage rigidity acrossthe businesscycle? Stretching

interpretation, state 6H couldbe describedas a \"boom\" and state 6L (<6H) as a\"recession.\"The extent of wage rigidity is determinedby the followingcondition:

u\\wL) _ V'(TlL)

u\\wH) V'(IIh)This is simply the Borchrule, which requires that wL = wH if the firm is

risk neutral {V\" =0).Thismodel thus predictscompletewage \"rigidity,\" in

the specialcasewhere firms are risk neutral. But is risk neutrality a goodassumption? Diversifiability of financial capitaldoessuggest that V is lessconcavethan u, and that wagesshould be more stable than profits.

However, macroeconomicshocksare by definition hard to diversify, so that risk

neutrality of the firm is too extremean assumption in this case.Still, what are the consequencesof the predictedwage \"rigidity\" across

states of nature? Doesit lead to excessiveemployment fluctuations, as in

the usual spot-marketview of macroeconomictextbooks?To assessthis

issue,considerthe first-orderconditionwith respectto a,,for i =L,H.Jia,is strictly between0 and 1,we have

PtV'$ii)[eiQ'(atL)-(wl-wsl)]L+m^M-u'(wsl+G)]=0

70 Hidden Information, Screening

But sincewt =wsi +G, this is equivalent to 9iQ\\(XiL) = G.In otherwords,inefficient underemployment neveroccurs(note also that we always haveaL< aH=1:indeed,it only makessenseto hire workers that will at leastbe retained in the good state of nature). Thus, as in other applications,allocativeefficiencyobtains under symmetric information.

What is somewhat paradoxicalabout the precedingformulation is that

employment is not only ex ante efficient,but alsoprofit maximizing expost,given that wt =wsi +G.It is thus not evennecessaryto seta,exante,becauseit would beself-enforcinggiven wt and wsi.

Finally, what canwe say about riiring decisions?Thefirst-orderconditionwith respectto L is not particularly illuminating. It simply saysthat riiringdecisionsare such that expectedmarginal cost of labor (given the outsideopportunity u) is set to be equal to expectedmarginal productivity.

Tosum up, the modelpredictswagesthat are stablerelative to profits (atleastwhen the firm is risk neutral), but this stability doesnot comeat the

expenseof allocativeefficiency, nor does it induce job rationing (sinceworkersare indifferent ex post betweenbeingretainedor being laid off).That is, this theory cannot offer a rationale for either inefficient

underemployment or involuntary unemployment.Howrobust are theseinsights? It turns out that changing workers'utility

functions matters for their comparativeexpost treatment. Forexample,onecoulddrop the perfectsubstitutability betweenmoney and leisure,and takea utility function u(w, 1-

\302\243),

where leisure is a normal good.Take,for

example,the following Cobb-Douglasformulation:

u(w, 1-1)=logw+logCK+1-1)whereK is a positive constant.The individual rationality constraint now

becomes

^PM(\\ogwi+\\ogIO+(X-oii)(\\ogwsi+\\o^K+l))\\>ui

and the first-orderconditionswith respectto w, and wsi yield

Wi =Wsi

Although allocativeefficiency obtains, the relation betweenemployedand laid-offworkers has now changed:one can now speakof involuntary

employment! Exante, it is Pareto optimal to equate the marginal utility ofmoney acrossworkers, and here this meanssetting w, = wsi. Thisfeatureofinvoluntary employment is in fact present for any utility function where

71 Static Bilateral Contracting

leisureis a normal good (seeAzariadis and Stiglitz, 1983).Note also that

the contractcannot leave the firm the freedomto setemployment expostanymore: with the severancepay equal to the wage, the firm has anincentive to chooseoci as high as possible,whatever ft>

2.2.3.2Contracting with AdverseSelection

A secondwave of labor contractmodels(seeAzariadis, 1983;Chari,1983;Greenand Kami, 1983;and Grossmanand Hart, 1983)have addedadverseselectionto the previousframework. In earliersections,adverseselectionintroduceda trade-offbetween rent extractionand allocativeefficiency.In this subsection,this will translate into a trade-offbetween insuranceand allocativeefficiency,becausecontracting takes place at the ex ante

stageand not at the interim stage.In the previoussubsection,the optimalcontractwas efficient and reproducedthe spot-marketallocation[that is,we had dQ'iOiL)= G wheneverlayoffs occurred,that is,whenever a, < 1].Assume now instead that contractexecutiontakes place at a time when

only the firm knows the value of ft As a result of this information

asymmetry,incentive constraints must be imposedon the optimal contracting

problem.Tokeepthings simple,assumeperfectsubstitution betweenmoneyand leisureand set1=1.Moreover,assumeL =1,and interpret <% now asthe number of hours worked by each individual worker, while w, isindividual compensation.10We thus have the utility function u[wi +(1-Ok)G\\

Under theseassumptions, the firm's optimization problembecomes

MaxXA-m-fi(a/)-w,]

subjectto

cCi<l i =L,HXA\"[wi+(l-a,)G]>Hi =L,H

i

O.Qia,)-Wi> OiQifx,)-Wj i, j=L,HAs before,this problemcan be solvedby first looking at the incentive

constraint that is violatedin the first-bestoutcome.An interesting special

10.We thus abstract away here from the issue of potential unequal treatment among workers.

72 Hidden Information, Screening

caseoccurswhen the firm is risk neutral, for then it is optimal to fully insureworkers and set

wL+(l-aL)G=wH +(l-aH)Gso that the incentive constraints reduce to

OiQioci)-Wi>eaty)-(a,-at)G-wt

But these conditionsare automatically satisfiedwhen a, is ex postefficient: intuitively, by setting a wagelevel that risesby G times theemployment level<Xi, the optimal contractinducesfirms to internalize the value ofleisurethat workerslosewhen they have to work, and to chooseex postefficient employment levels.

On the one hand, under firm risk neutrality, adverseselectiondoesnot

changethe optimal contracthere.On the other hand, when the firm is risk

averse,it is efficient to reducewL relativeto wH, in order to sharerisk with

workers.This action,however, createsan incentive problemin that the firm

may want to claim falsely to be in state 6L in order to reduceits wage cost.This problemariseswhenever V(-) is sufficiently concaverelative to u(-).

Tomake the point, considerthe extremecasewherethe firm is risk averse(V\" < 0) but the worker is risk neutral [u\" =0,or u(x) =x].In this case,therelevant incentive constraint is

&hQ(och)-wh=9HQ(aL)-wL

Sincethe individual rationality constraint is alsobinding at the optimum,we have a maximization problemwith two equality constraints. The first-

order conditionswith respectto w, and a, are then given by

-PlV'(nL)+Kfa +h =0 (2.21)

-PhV\\YIh) +KPh -h=0 (2.22)

PLV'(nL)eLQ'(aL)-KhG-heHQ'{ocL)>o (2.23)

pH V'QIh)eHQ\\aH)-Xa$HG +XbeHQ\\aH) > 0 (2.24)whereXa and A6 are the Lagrangemultipliers for the individual-rationalityand incentive constraints, respectively.11

11.Remember that, by the feasibility constraints, a,<l.

73 Static Bilateral Contracting

Conditions(2.22)and (2.24)yield

9HQ'{aH)>G (2.25)while conditions(2.21)and (2.23)yield

9LQ'{aL)>G+^^^-Q\\aL) (2.26)

Moreover,these two conditionsare binding whenever workers are fully

employed,that is,whenevera,=1.Finally, conditions(2.21)and (2.22)imply

nH>nL (2.27)sinceA6> 0.

The intuition for these results is as follows:Becausethe incentiveconstraint binds, risk sharing is imperfect,and underemployment occurs,but

only in the bad stateof nature 9L [seeequations(2.25)and (2.26)].We thus

obtain again the result of no distortion at the top.Note that while theseresults have beenobtainedunder worker risk neutrality, allowing for a little

concavity for u(-) would keeptheseresults unchanged, providedthe firm is

(sufficiently) risk averse.This modelthus generatesunderemployment in bad statesof nature, but

not wagestability relative to profits: underemployment relieson thesufficient concavity of VQ relative to u(-).Moreover,interpreting 9= 9L as arecessionis difficult, since6has to beprivate information to the firm (see,however,Grossman,Hart, and Maskin (1983),for a macroeconomicextension of this framework that is immune to this criticism).Finally, the

underemployment result is itself dependent on the particular choiceof utility

functions: assumeinsteadfirm risk neutrality (V\" =0) and a Cobb-Douglasutility function for workers[u(-)= logw + log(K+1-a)].In this case,thefirst-best allocation(aL< aH and wL = wH) violatesthe incentive constraint

of the firm when it is in the bad state of nature 9L:sincetotal payments toworkersdo not dependon the realizationof 9but employment is higher in

the goodstate 9H, announcing 9= 0L is never incentive compatible.Solvingthe modelwith the incentive constraint

0lQ(ccl)-wl=9LQ(aH)-wH

then yields expost efficient employment when 6 = 9L and possibleoveremployment when 9= 9H, thus overturning the previouspredictionofunderemployment.

Hidden Information, Screening

In conclusion,the optimal contracting approach, by stressing theinsurance role of labor contracts,has offered an interesting motive for

wagestability. However,it has failedto deliverrobust predictionsoninvoluntary unemployment or underemployment. From this perspective,moralhazard modelsbasedon \"efficiency wages\" have beenmoresuccessful(seeChapter4).

Regulation

Regulation of natural monopoliesis another area whereadverseselectionideashave beenemphasized,becausepublic regulatorsareoften at aninformational disadvantage with respectto the regulated utility or natural

monopoly.Baron and Myerson(1982)have written a pioneeringcontribution in the field, deriving the optimal output-contingent subsidy schemefora regulator facing a firm with a privately known productivity parameterthat

belongsto a continuous interval.12 Later,Laffont and Tirole(1986)pointedout that in reality subsidy schemesare also cost contingent and built amodel where accounting costsare observable,but reflectboth \"intrinsic

productivity\" and an \"effort to cut costs.\"Interestingly, this frameworkcombinesboth an adverseselectionand a moral hazard (hidden action)element.In this subsection,we study the simplest possibleversionof theLaffont-Tirolemodel,which is the cornerstoneof the \"new economicsofregulation\" (seetheir 1993book).

The basiceconomicproblem involves a regulator concernedwith

protecting consumerwelfareand attempting to force a natural monopoly to

chargecompetitive prices.The main difficulty for the regulator is that hedoesnot have full or adequateknowledgeof the firm's intrinsic coststructure. More formally, considera natural monopoly with an exogenouscostparameter 6 e {6L,6H}.DefineAd = 6H- 6L.The firm's costof producingthe goodis c = 9-e,wheree stands for \"effort.\" Expending effort has costy(e)= [max{0,e}]2/2,which is increasingand convexin e.Assume that the

regulator wants the good to be producedagainst the lowest possiblepayment P=s +c (wheres is a \"subsidy,\" or payment to the firm in excessof accounting costc).Minimizing the total payment to the firm canbejustified on the grounds that it has to be financed out of distortionary taxa-

12.This paper has been very influential in terms of dealing with adverse selection with morethan two types (seesection 2.3).

75 Static Bilateral Contracting

tion, which a benevolentregulator should be anxious to minimize. Thepayoff of the firm is P-c- \\ff(e)

=s- y/\"(e). If the regulator observesthevalue of the costparameter 9, it tries to achieve

min s+c=s+9-esubjectto the individual-rationality constraint

j-[max{0,e}f/2>0This approachyields an effort levele* =1and a subsidy s* =0.5.One caninterpret this outcome as the regulator offering the firm a fixed total

payment P,which inducesthe firm to minimize c+ ys(e),and thus to choosee*=1.This is what the regulation literature callsa \"price-capscheme\";the

oppositeof price caps is a \"cost-plus\" scheme,where the firm receivesaconstant subsidy s irrespectiveof its actual cost(such a schemewould resultin zero effort, as is easy to see).Between these two extreme regulatory

regimes,we have cost-sharingagreements,where the incentive schemeinducesmoreeffort the closerit is to the pricecap.With adverseselection,pricecapsmay besuboptimal becausethey fail to optimally extracttheregulated firm's informational rent (seeLaffont and Tirole,1993,fordiscussions of regulation issues).

Indeed,assume that the regulator cannot observe the cost parameterof the firm but has prior beliefas follows:Pr(0= 9L) = /J. Call (sL,cL) thecontract that will be chosen by type 9L (who then expendseffort eL =9L-cL),and call (sH,cH) the contractthat will be chosenby type 9H (who

then expendseffort eH=9H-cH).Theregulatorchoosesthesecontractssoas to minimize fi(sL + cL)+ (1-P)(sH+ cH),or, equivalently (sincethe &sare exogenous),to solve

mm{p(sL-eL)+(1-P)(sH-eH)}

subjectto participation and incentive constraints

5L-[max{0,eL}]2/2>0^-[max{0,eH}]2/2>0sL-[max{0,eL}fll>sH-[max{0,eH-A9}fjlsH -[max{0,eH}f/2>sL-[max{0,eL+A9}fjl

76 Hidden Information, Screening

The first two inequalities are participation constraints, and the next two areincentive constraints. Achieving cH implies effort level eH for type 6H but

effort leveleH-Ad for type 9L, and similarly for costcL.We shall be interested in avoiding corner solutions.Accordingly, we

assumethat eH=6H-cH>Ad, so that 6L-cH>0.This is a condition

involving equilibrium effort, and we shall expressit directly in terms ofexogenous parametersof the model.

The firstrbest outcomehas the same effort level for both types (sinceboth effort cost functions and marginal productivities of effort areidentical)

and thereforethe samelevelof subsidy, but a higher actual costfor theinefficient type.Theincentive problemin this first-best outcomearisesfromthe fact that the efficient type wants to mimic the inefficient type, to collectthe samesubsidy while expendingonly effort e*-Ad, and achieving actualcostcH.As a result, the relevant incentive constraint is that of the efficient

type, while the relevant participation constraint is that of the inefficient

type, or

SL-el/2=SH-(eH-A9)2/2

sH ~e2H/2=0

Rewriting the optimization problemusing these two equalitiesyields

mm{p(el/2-eL+[e2H/2-(eH-A9)2/2])+(l-P)(e2H/2-eHj\\which implies as first-orderconditions

eL=land

We thus have again the by-now-familiar \"ex post allocativeefficiency atthe top\" and underprovision of effort for the inefficient type, since this

underprovision reducesthe rent of the efficient type [which equalse#/2-(eH-A0)2/2]. Becauseof the increasingmarginal cost of effort, a loweractual cost cH benefits the efficient type lessthan the inefficient type in

terms of effort costsavings. The incentive to lowereH increasesin the costparameter differential and in the probability of facing the efficient type,whoserents the regulator is trying to extract.

77 Static Bilateral Contracting

In terms of actual implementation, the optimal regulation thereforeinvolves offering a menu that inducesthe regulatedfirm with costparameter 9L to choosea price-capscheme(with a constant pricePL= sL+ cL=$l+Ql~eL,where 6L and eLare the solutions computedpreviously) andthe regulatedfirm with costparameter 6H to choosea cost-sharingarrangement (with a lowerpriceplus a shareofexpostcoststhat would make efforteH optimal and leaveit zero rents).As appealingas this solution appearstobe,it is unfortunately still far from beingappliedsystematically in practice(seeArmstrong, Cowan,and Vickers, 1994).

2.3MoreThanTwoTypes

We now return to the generalone-buyer/one-sellerspecificationof section2.1and, following Maskin and Riley (1984a),discussthe extensionsof thebasicframework to situations wherebuyers canbeofmore than two types.We shall considerin turn the generalformulation of the problemfor n > 3

types and for a continuum of types.This latter formulation, first analyzedin the contextof regulation by Baron and Myerson(1982),is by far themore tractableone.

2.3.1Finite Number ofTypes

Recallthat the buyer has a utility function

u(q,T,9l)=eiv(q)-TBut supposenow that there are at least three different preferencetypes:

@n > @n-l > \342\200\242\342\200\242->6iwith n > 3.Call /J, the proportionof buyers of type 6t in the population.Let

{(q\342\200\236 T);i = 1,...,n) be a menu of contractsofferedby the seller.Then,by the revelationprinciple,the seller'sproblemis to choose

{(q\342\200\236 Tt);i = 1,...,n) from among all feasiblemenus of contracts to solve the

program

max{(*,r,)} Z\"=1(^-C^)A-I subjecttofor all; eAq^-Ti^o.forall i, j dAqJ-Tt >e^q^-T,

78 Hidden Information, Screening

Just as in the two-type case,among all participation constraints only theone concerningtype 9X will bind; the other oneswill automatically hold

given that

0iv(?i)-7:>M?i)-71>M?i)-71>OThe main difficulty in solving this program is to reduce the number of

incentive constraints to a moretractableset ofconstraints [in this program,there are n(n -1)incentive constraints].This reductioncanbe achievedifthe buyer's utility function satisfiesthe Spence-Mirrleessingle-crossingcondition:

dddu/dqdu/dT.

>0

With our chosenfunctional form for the buyer'sutility function, 9y(q)-T, this condition is satisfied,as can be readily checked.This observationleadsus to our first step in solving the problem:

Step1:Thesingle-crossingcondition implies monotonicity and the

sufficiency of \"local\" incentive constraints.

Suinming the incentive constraints for types dt & dp that is

GAq^-Ti^eAq^-Tjand

OjViq^-Tj^djviqJ-T,we have

Sincev'(q)> 0, this equationimplies that an incentive-compatiblecontractmust be such that q, > q, whenever0, > dr That is, consumption must bemonotonically increasingin 6when the single-crossingcondition holds.

It is this important implication of the single-crossingproperty that

enablesus to considerablyreduce the set of incentive constraints. To seewhy monotonicity ofconsumption reducesthe setofrelevant incentive

constraints, considerthe three types d{_i < 0,< 6i+1,and considerthe followingincentive constraints, which we can call the localdownward incentive

constraints, or LDICs:

79 Static Bilateral Contracting

Oi+iv(ql+i )-Tl+i>ei+iv(qt)-Tt

and

diVi.qJ-TiZeAqi-J-'Z-iThis secondconstraint, togetherwith q, > q^u implies that

Oi+iviq,)-Tt>eMv{qi-i)-T,_i

which in turn implies that the downward incentive constraint for type 6M

and allocation(qt_i, T^i)alsoholds:

0i+iv(g,+i)-TM > 0Mv{q^i)-7}_i

Therefore,if for each type 6t, the incentive constraint with respectto type

0,_i holds\342\200\224in otherwords, the LDICis satisfied\342\200\224then all other downward

incentive constraints (for 0, relative to lower 0's)are also satisfiedif the

monotonicity condition q, > q^ holds.We are thus able to reducethe set ofdownward incentive constraints to the setof LDICsand the monotonicitycondition qt > q^.One can easilyshow that the sameis true for the setofupward incentive constraints (that is, for 0,relative to upper 0's).

We can thus replace the setof incentive constraints by the setof localincentive constraints and the monotonicity condition on consumption.Thenext questionis whether the setof incentive constraints canbereducedstill

further.

Step2:Togetherwith the monotonicity of consumption, the relevant setof incentive constraints is the setof LDICs,which will bind at the

optimum.

Just as in the two-type analysis of section2.1,we can start by omitting theset of local upward incentive constraints (LUICs)and focus solely on

monotonicity ofconsumption togetherwith the setofLDICs.Thenit is easyto show that the optimum will imply that all LDICsare binding. Indeed,supposethat an LDICis not binding for some type 0i; that is,

M9,)-7]>M9i-i)-3rI_iIn this case,the sellercan adapt his scheduleby raising all 7}'sfor ;*> i

by the same positive amount so as to make the preceding constraint

binding. This method will leaveunaffectedall other LDICswhile

improvingthe maximand.

80 Hidden Information, Screening

In turn, the fact that all LDICsare binding, togetherwith the monoto-nicity of consumption, leadsall LUICsto besatisfied.Indeed,

GiViqt)-7] =9i vfo-i)-7]_i

imphes

0,-iv(g,)-7] < 0i-iv(?,_i)-7;_i

sinceqt_i <qh Therefore,if the single-crossingcondition is verified, only theLDICconstraints are binding when the monotonicity condition q^ < qt

holds,so that the seller'sproblemreducesto

max{(g\342\200\236r,)} X\"=1(71-c#,)Asubjectto

Mgi)-7i=0for all i 6lv(qt)-Tt=0,-vfo-i)-rM.and qt > q} where0; > 0;

Step3:Solving the reducedprogram.The standard procedure for solving this program is first to solve therelaxedproblemwithout the monotonicity condition and then to checkwhether the solution to this relaxed problem satisfiesthe monotonicitycondition.

Proceedingas outlined, considerthe Lagrangian

^ =\302\243{[7;- -cqtfi+X,[0lv(ql)-eiv(qi.l)-Ti+Ti_1 +pi[d1v(qi)-T1]i=l

The Lagrangemultiplier associatedwith the LDICfor type 6, is thus A*,

while ji is the multiplier associatedwith the participation constraint for type

0i.The first-orderconditionsare,for 1< i < n,

\342\200\224 =KQiV'iqd-Ai+10i+1v'(g,)=cfadqt

-r=r =Pi-X,+Xl+i=0

and, for i =n,

Static Bilateral Contracting

-\342\200\224 =hnenv'(qn) =cj3noqn

Thus,for i =n, we have 6nv'(qn) =c.In otherwords,consumption is efficientfor i = n. However,for i < n, we have d,y'(q)> c. In other words, all typesother than n underconsumein equilibrium.

Theseare the generalizationsto n types of the results establishedfor two

types. If onewants to further characterizethe optimal menu of contracts,itis more convenient to move to a specificationwherethere is a continuum

of types.Before doing so,we briefly take up an important issue that our

simple setting has enabledus to sidestepso far.

Random Contracts

Sofar, we have restrictedattention to deterrninistic contracts.Thisrestriction involves no lossof generality if the seller'soptimization program isconcave.In general,however,the incentive constraints are such that theconstraint setfacedby the selleris nonconvex.In thesesituations the sellermay beableto do strictly better by offering random contractsto the buyer.A random contractis such that the buyer facesa lottery insteadof buyinga fixed allocation:

Wi) ={[qiO,,a),Wt, a)];A(dt, a)=Probability of a given 0;}We shall considera simple examplewherestochasticcontractsstrictlydominate deterrninistic contracts:Let [q(9t),T(d)\\ i =1,2,bean optimaldeterministic contract.Assume a utility function u(q, T, 6) that is concavein qand such that the buyer of type Oi is more risk aversethan the buyer of typedi (wherefy > 6i).Sinceboth types ofbuyers are risk averse,they are willing

to pay morefor ql than for any random q\\ with mean qt. In particular, if type

&!is indifferent between (T*i, q{),where Tx is fixed, and (7\\, q{),we musthave Ti<ThIn otherwords, by introducing a random scheme,the selleriscertain to losemoney on type di, so the only way this can be beneficialisif he can chargetype fy more.By assumption, type &i is more risk averse,so that she strictly prefers (7\\, qi) to (Ti,q{).The sellercan thereforefind

S> 0 such that type 62 (weakly) prefers (T2+ S,q2) to (7\\, qx). If type 02 is

sufficiently more risk averse than type &y, the seller'sgain of d outweighs

Hidden Information, Screening

the lossof T\\-

T\\. Instead,if type 02 is lessrisk aversethan type &i, then

the random contract doesnot dominate the optimal deterministiccontract.13 This intuition is indeed correct,as Maskin and Riley (1984a)haveshown. It stemsfrom the fact that the relevant incentive constraints are thedownward constraints; there is therefore no point in offering type 02 arandom allocation,sinceit involves an efficiencylossfor this type without

any gain in terms of rent extractionon the other type, who is alreadyat herreservationutility level.

An exampleof a random contractin the real world is an economy-classairline ticket, which comeswith somany restrictionsrelative to a business-classticket that it effectively imposessignificant additional risk on thetraveler. It is in part becauseof such restrictionsthat businesstravelers, who

often feel they cannot afford to take such risks, are preparedto paysubstantially more for a business-classticket.

A Continuum ofTypes

Supposenow that 0 doesnot take a finite number of valuesanymore but

is distributed accordingto the density /(0) [with c.d.f.F(9)]on an interval

[0,0].Thanks to the revelationprinciple,the seller'sproblemwith acontinuum of types canbewritten as follows:

maxg(9),r(9)\302\243

[T(6)-cq(6)]f(0)d6, subjectto

(IR) 6v[q(6)]-T(6)>0 for all 0e[0,0].(IC) 6v[q(6)]-T(0)> 6v[q{6)]-T{\302\247) for all 0,0e[0,0]

Note first that we canreplacethe participation constraints (IR) by

(IR') 0v[^(0)]-T(0)>O

given that all (IC)'shold.Note also that the sellermay want to excludesome types from consumption; this canbe formally representedby setting

q(9)= T(0)=0 for the relevant types.It is convenient to decomposethe seller'sprobleminto an

implementation problem [which functions q(6) are incentive compatible?]and an

13.In fact, one can show that the optimal contract is deterministic in our simple casewhereu(q,T,8) = 8v(q) -T.

83 Static Bilateral Contracting

optimization problem[among all implementable q{9)functions, which oneis the bestfor the seller?].2.3.3.1The Implementation Problem

With a continuum of types it is evenmore urgent to get a tractablesetofcontraints than in the problemwith a finite setof types n.As we shall show,

however, the basiclogicthat ledus to concludethat all incentive constraintswould hold if (1)consumption q(9)is monotonically increasingin 9and (2)all localdownward incentive constraints are binding alsoappliesin the casewherethere is a continuum of types.

More formally, we shall show that if the buyer'sutility function satisfiesthe single-crossingcondition

dddu/dq >0du/dT.

as it doesunder our assumedfunctional form

u(q,T,6)=6v(q)-Tthen the setof incentive constraints in the seller'soptimization problemis

equivalent to the following two setsof contraints:

Monotonicity:

dd

Local incentive compatibility:

Ov\\q(0)]^r=T\\d) for all 9e[9,9]d9

Toseethis point, suppose,first, that all incentive constraints are satisfied.Then,assuming for now that the consumption q{9)and transfer T{9)schedules are differentiable, it must be the casethat the following first- andsecond-orderconditions for the buyer'soptimization problemare satisfiedat 9= 9:

9vMo)]^-r(e)=o focda

84 Hidden Information, Screening

and

'dq(eY\\ a ,r (XY]d2q(9)

dv\"[q{e\\~leJ +6vUo)]-^~-T,,(e)<o socThus the first-orderconditionsof the buyer's optimization problemare

the same as the local incentive compatibility constraints earlier. If wefurther differentiate the localincentive compatibility condition with respectto 6we obtain

ev\"[^ei^fiW<^+M<K0]^-nB)-obut from the buyer'sSOC,this equationimplies that

da

or,sincev'[q(9)\\ > 0, that

dd

Suppose,next, that both the monotonicity and the local incentive

compatibility conditionshold.Then it must be the casethat all the buyer'sincentive compatibility conditionshold. To seethis result, supposebycontradictionthat for at leastone type 6 the buyer'sincentive constraint isviolated:

ev[q(e)]-T(9)<ev[q{0)]-T{d)for at leastone 6^9.Or, integrating,

By assumption we have dq(x)/dx> 0,and if 6 > 6,we have

0v'[q(x)]< xv'[q(.x)]

Therefore,the localincentive constraint implies that

f \\ev'[q(x)] *l-r(x)]dx< 0Je L dx J

85 Static Bilateral Contracting

a contradiction.Finally, if 6 < 6, the same logic leads us to a similarcontradiction.This result establishesthe equivalencebetween the mono-tonicity condition togetherwith the localincentive constraint and the full

setof the buyer'sincentive constraints.

2.3.3.2The Optimization Problem

Theseller'sproblemcan thereforebe written as

max \\e[T(e)-cq(d)]f(0)d6

subjectto

6y[q(i)]-T(\302\243)>0 (2.28)

dq{0)dd

>0 (2.29)

TX9)=9v\\qm^r (2.30)da

The standard procedure for solving this program is first to ignore the

monotonicity constraint and solvethe relaxedproblemwith only equations(2.28)and (2.30).This relaxed problem is reasonablystraightforward tosolvegiven our simplified utility function for the buyer.

Toderivethe optimal quantity function, we shall follow a procedurefirst

introducedby Mirrlees(1971),which has becomestandard.Define

W(9) =6v[q(6)]-T(9)=max{6v[q{0)]-T{0)}

By the envelopetheorem,we obtain

dW(9) dW(9)-dW-=-dT=v[qmor, integrating,

W(e) =j\\[q(x)]dx+W(e)

At the optimum the participation constraint of the lowesttype is binding,so that W(9) =0 and

W(e) =fv[q(x)]dx

86 Hidden Information, Screening

Since

T(9)=9v[q(9)]-W(6)

we can rewrite the seller'sprofits as

n =\302\243 [ev[q(9)]

-[\302\243 v[q(x)]dx]

-cq(9)]f(9)d9

or,after integration by parts,14

k =\\l ({9v[q(9)]-cq(9)}f(9)-v[q(9)][l-F(9)])d9

Themaximization of n with respectto the scheduleq{-)requiresthat theterm under the integral be maximized with respectto q{9)for all 9.Thuswe have

9v'[q{9)]=c+^-^-v'[q(9)] (2.31)j\\P)

or

[e-^Ym-cFrom this equation, we can immediately make two useful

observations:

1.Since first-best efficiency requires 9v'[q(9)] = c, there isunderconsumption for all types 9 < 9.

14.Remember that

jV=[\"v]|-jVv

Here, let V =/(0)and u =J V[q(x)]dx sothat

jeg (jegv[q{x)]dx)f{e)d8=[jegv[q{x)]dxf(0)J-

\302\243

v[q(e)]F(e)de

which is equal to

fgV[q(e)][i-F(e)]de

87 Static Bilateral Contracting

2.We can obtain a simple expressionfor the price-costmargin: Let T'(6)= P[q(0)].Then from equation(2.30)in the seller'soptimization problemwe have P[q(6)]= Qv\\q(d)\\ Substituting in equation(2.31),we get

P-c_l-F(9)P

\"

0/(0)It finally remains to checkthat the optimal solution definedby equations

(2.31)and (2.30)satisfiesthe monotonicity constraint (2.29).In generalwhether condition (2.29)is satisfiedor not dependson the form ofbuyer'sutility function and/or on the form of the density function/(0).A sufficient

condition for the monotonicity constraint to be satisfiedthat is commonlyencounteredin the literature is that the hazard rate,

h{6)-l-F(9)is increasingin 0.15

It is straightforward to verify that if the hazard rate is increasingin 0,then condition (2.29)is verified for the solution given by equations(2.31)and (2.30).Indeed,letting

g(e)='e-l~F{0)'/(0)

the first-orderconditionscanbe rewritten as

g(ey[q(9)]=c

Differentiating this equationwith respectto 0 then yields

dq _ gW[g(0)]d6 v\"te(0)]g(0)

Since v(-) is concaveand g(6)> 0 for all 0, we note that dq/d6>0ifg'(9)> 0.A sufficient condition for g'(9)> 0 is then that l/h(9)is

decreasingin 0.

15.In words, the hazard rate is the conditional probability that the consumer's typebelongs to the interval [6,6+ dd], given that her type is known to belong to the interval

[e,6].

88 Hidden Information, Screening

Thehazard rate h{9)is nondecreasingin 0for the uniform, normal,exponential, and other frequently useddistributions; therefore,the precedingresults derived without imposing the monotonicity constraint (2.29)arequite general.Thehazard rate,however,decreaseswith 0if the density f(9)decreasestoo rapidly with 9,in otherwords, if higher 0'sbecomerelativelymuch lesslikely. In this case,the solution given by equations(2.31)and

(2.30)may violate condition (2.29).When the monotonicity constraint isviolated,the solution that we obtainedmust bemodified soas to \"iron out\"

the nonmonotonicity in q(9).For the sakeof completeness,we now turn tothis case.2.3.3.3Bunching and Ironing

Callthe solution to the problemwithout the monotonicity constraint (2.29)q*(9).Sowe have

fl 1-F(g)l\342\200\224fWiv

Assume that dq*(9)/d9< 0 for some 9 e [9,9],as in Figure 2.2.

q(Q)i

Figure 2.2Violation of the Monotonicity Constraint

89 Static Bilateral Contracting

This could be the casewhen the hazard rate h(9) is not everywhereincreasingin 9.Thenthe sellermust choosethe optimal q(6)[which we will

callq(6)]to maximize the constrainedproblem

max7r Je ev[q(6)]-cq(e)-v[q(6)]h(6)

f{9)dd

subjectto

dg(6)dd

\342\226\240>0

Assume that the objectivefunction is strictly concavein q(9) and that

the unconstrained problem is such that dq*(6)/ddchangessign only afinite number of times. Then the following \"ironing\" procedure can beapplied to solve for the optimal nonlinear contract:Rewrite the seller'sproblemas

ev[q(e)]-cq(d)-v[q(0)]h(6) .

f{B)ddmax7r= fq(8) J6

subjectto

The Hamiltonian for this program is then

v[q(6)]H(6,q,^X)= 6v[q(e)]-cq(6)-h(6)

m+mmAnd by Pontryagin's maximum principle,the necessaryconditions for an

optimum [q(6),p.(9)]are given by

1.H[9,q(9),fL(6),A(0)] > H[6,q(6), (6),X(6)]2.Exceptat points of discontinuity of q(6),we have

dX{9)dd d-m)Am]-c.m (2.32)

3.The transversality conditionsA(0) = X(d) =0 are satisfied.

90 Hidden Information, Screening

Theseconditions are alsosufficient if H[6,q(9),fJ.(9),A(0)] is a concavefunction of q.16

Integrating equation(2.32),we canwrite

m=-t e-w>}'[m]-c.f{d)dd

Using the transversality conditions,we then have

1\\'\\q(0)]-<O=A(0) =A(0)=-J e-~~)v'[q(e)]-c\\f(e)de

Next, the first condition requires that n(9) maximize H(6,q, n, X) subjectto n(0) > 0.This requirement implies that X{6) < 0 or

\302\243l(\302\260-w^[q(e)]-c\\fWe>o

Whenever X(d) < 0 we must then have

Thus we get the following complementary slacknesscondition:

dq{0)dd J\302\243 L1

6--T7TTV[g(0)]-c\\f(6)d6 =0h{d))'

for all 6e[\302\243 9].

It follows from this condition that if q(6) is strictly increasingoversomeinterval, then it must coincidewith q*(6).Toseethis conclusion,notethat

_,.. dq(6) dHd)m=-dW>0^m=0^~dT=\302\260^ie-w}'[m]-c=0

But this is preciselythe condition that definesq*(6).It therefore onlyremains to determinethe intervals over which q(6) is constant.ConsiderFigure 2.3.To the left of 0i and to the right of &z, we have

16.SeeKamien and Schwartz (1991),pp. 202,205.

91 Static Bilateral Contracting

<7(6);q{QH

Bunching

Figure 2.3Bunching and Ironing Solution

A(0) =O and jt(0) =_dq(9) dq*(9)dd dd >0

And for any 0between dx and 02, we have

A(0)<O and n{0)=Q

By continuity of A(0), we must have X(9X) = X(&z) =0,so that

f e-m}'[m]-'=o

In addition, at 9X and 0iwe must have g*(0i)=qr*(0l2).Thisfollowsfrom the

continuity of q(9).Thus we have two equationswith two unknowns,

allowingus to determinethe values of 6X and 02.An interval [0l5 02] overwhich

<2(0)is constant is known as a bunching interval.To gain some intuition for the procedure,considerthe density function

depictedin Figure 2.4.Thisdensity doesnot have a monotonicallyincreasing

hazard rate. In this example,there is little likelihoodthat the buyer'stype liesbetweendx and 02.Now supposethat the sellerofferssomestrictly

92 Hidden Information, Screening

f{Q)i

Figure 2.4Violation of the Monotone Hazard Rate Condition

(7(0)1

Figure 2.5Improvement over Strictly Monotonic Consumption

93 Static Bilateral Contracting

increasingschedule q*(9).Remember that consumer surplus W{6) =0v[q(6)]-T[q(6)]canbe rewritten as

W(6) =fgv[q(x)]dx

so that the utility of a buyer of type 6increasesat a rate that increaseswith

q(6).Now the sellercanreduce the rent to all types 6> 9X by specifying ascheduleq(6) that is not strictly increasing,as in Figure 2.5.

This rent reductioninvolves a cost,namely, that all types 6e [&i, 0J will

pay a lowertransfer than the total transfer that couldbeextractedfrom themwithout violating the incentive constraints. But there is alsoa benefit, sincethe incentive constraints for the types 6 > Oi are relaxed,so that a highertransfer canbeextractedfrom them. Thebenefit will outweigh the costif thelikelihoodthat the buyer'stype falls between dx and $2 is sufficiently low.

2.4 Summary

Thebasicscreeningmodeldescribedin this chapterhas beenhugelyinfluential in economics.We have detailedthe way in which it canbesolvedand its

main contract-theoreticinsights, aswell asa number ofkey applications.Wecansummarize the main results in terms ofpurecontracttheory asfollows:

\342\200\242 The two-type caseprovidesa useful paradigm for the screeningproblem,sincemany of its insights carry over to the casewith more than two types.\342\200\242 When it comesto solving the screeningproblem,it is useful to start fromthe benchmark problemwithout adverseselection,which involves

maximizing the expectedpayoff of the principal subjectto an individual

rationalityconstraint for each type ofagent.At the optimum, allocativeefficiency

is then achieved,becausethe principal can treat each type of agentseparately and offera type-specific\"package.\"\342\200\242 In the presenceof adverseselection,however,the principal has to offerall types ofagentsthe samemenu ofoptions.Hehas to anticipate that eachtype of agent will chooseher favorite option.Without lossof generality, hecan restrict the menu to the setof optionsactually chosenby at leastonetype of agent.This latter observation,known as the revelationprinciple,reducesthe program of the principal to the maximization of his expectedpayoff subjectto an individual-rationality constraint and a setof incentiveconstraints for each type of agent.

94 Hidden Information, Screening

\342\200\242 In the nonlinear pricing problemwith two types, one can moreover(1)disregardthe individual rationality constraint of the high-valuation type, aswell as the incentive constraint of the low-valuation type; (2) observethat

a reductionof the consumption of the low-valuation type lowerstheinformational rent of the high-valuation type. The optimal contractthen tradesoff optimally the allocativeinefficiency of the low-valuation type with theinformational rent concededto the high-valuation type. In contrast,thereisno allocativeinefficiency for the high-valuation type and no rent for thelow-valuation type, sinceno buyer wants to pretend that an objectis worth

more to her than it really is.\342\200\242 In otherapplications,the technicalresolutionmethod is broadly the sameas in the nonlinear pricing problem.While the structure of incentiveconstraints is similar acrossapplications,the same is not always true forindividual rationality constraints:(1)for example,in the labor-contractapplication,contracting takesplaceex ante, before the firm learnsits type,so that there is a single individual-rationality constraint for the firm; (2) in

the optimal-income-taxapplication,the government is ableto forceparticipation

of the agent; its utilitarian objective function, together with its

budget constraint, inducesa specifictrade-offbetweenallocativeefficiencyand the distribution of rents acrossindividuals; (3) finally, in the credit-rationing and regulation problems,we have the sametrade-offbetweenrentextraction and allocative

efficiency\342\200\224and the same type of individual-

rationality constraints\342\200\224as in the nonlinear pricing problemof section2.1.\342\200\242 For generalizationsto more than two types, the continuous-type caseisoften easiestto analyze, becausethe setof incentive constraints can oftenbereplacedby a simple differential equationplus a monotonicity conditionfor the allocationsof each type of agent (itself implied by the \"Spence-Mirrlees\" monotonicity condition on the agent utility with respectto hertype).The problemwith a continuum of types revealsthat what is robustin the two-type caseis the existenceof positiveinformational rents and ofallocativeinefficiency (indeed,these two featureshappen for all types in

the interior of the interval of types, so that they are \"probability-one

events\.\342\200\242 Undernatural restrictionson the distribution of types (e.g.,a monotonehazard rate),we have \"full separation\"; that is, any two different types endup choosingdifferent points in the menu of optionsofferedby the princi-

95 Static Bilateral Contracting

pal,implying that the monotonicity condition is strictly satisfied.Thepoint-wisechoiceof optionsin the contractis the outcomeof the same type oftrade-offas in the two-type casebetween allocativeefficiency and rentextraction.\342\200\242 In somecases,the distribution of types doesnot lead to full

separation\342\200\224

for example,when there are intermediate types that the principalconsiders to be of low prior probability, as in the caseof a multimodal typedistribution. There would then be an incentive for the principal to havesevereallocativeinefficiency for thesetypes, in order to reducethe rents ofadjacenttypes.But this incentive conflictswith the monotonicity conditionmentionedearlier.In this case,a procedureof \"bunching and ironing\" hasbeenoutlined to solve for the optimal contract,relying on Pontryagin'smaximum principle.Themonotonicity condition then binds for someintervals of types where bunching occurs.Theseintervals are derivedby tradingoff allocativeefficiency and rent extractionwith respectto the bunchinginterval as a whole.

Beyondthese contract-theoreticinsights, there are important economiclessonsto be obtainedfrom the screeningparadigm:

\342\200\242 The analysis of nonlinear pricing has rationalizedquantity discounts,which increaseallocativeefficiencyrelative to linearpricing.\342\200\242 Adverseselectionin financial markets can rationalize the phenomenonof equilibrium credit rationing, even though this result is sensitive to theset of financial contractsthat is allowed(seeChapter11in particular on

optimal financial contractdesign).\342\200\242 Redistributive concernsby a government that has imperfectinformation

about individual types can rationalizedistortionary incometaxation, that

is,positive marginal incometax rates and thereforeinefficiently low laborsupply, especiallyby low-productivity individuals, in order to reduce theincomegap betweenlessand more productive individuals.

\342\200\242 A desireby the government not to abandontoo many rents to regulatedfirms can rationalize deviations from \"price cap regulation\" in favor ofpartial coverageof costs,at the expenseof efficient cost cutting byregulated firms.\342\200\242

Finally, the screeningparadigm hasbeenusedto investigate the effectofoptimal labor contracting on equilibrium levelsof employment. Although

96 Hidden Information, Screening

labor contractsclearlyhave a (partial) insurancerole, the theory revealsthat the introduction of adverseselectiondoesnot producerobustpredictions on deviations from efficient employment.

2.5 LiteratureNotes

The material coveredin this chapter is \"classical\" and has by now

generated a huge literature. In this section,we stressonly pathbreakingcontributions that have beenat the origin of this literature, surveys of its various

branches,and selectedkey recentadvancesthat gobeyondthe scopeofourbook.

The pioneeringarticle developingthe formal approach to the basicadverse selectionproblem is Mirrlees (1971).The revelation principle,which also applies in a multi-agent context, has appeared in various

contributions, e.g.,Gibbard (1973),Greenand Laffont (1977),Myerson(1979),and Dasgupta,Hammond, and Maskin (1979).The solution to the

problem with a continuum of types owes much to Baron and Myerson(1982).Seealso(1)Guesnerieand Laffont (1984)for a generalcharacterization of \"necessaryand sufficient\" conditionsfor the existenceof asecond-bestoptimum in the one-dimensionalcaseand (2) Jullien (2000)and Rochet and Stole (2002) for explorationsof type-dependentandrandom individual rationality constraints.

Somekey applicationsinvolving adverseselectionand screeningare the

following:

\342\200\242 Mussa and Rosen (1978)on nonlinear pricing, and Maskin and Riley(1984a),on which section2.1is based.For an in-depthexplorationofnonlinear pricing, the bookby Wilson (1993)is alsoan excellentsource.\342\200\242 A centralreferenceon credit rationing is Stiglitz and Weiss (1981).Seealso the surveys by Besterand Hellwig (1987),Bester(1992),and Harrisand Raviv (1992),which coversfinancial contracting more generally.\342\200\242 On optimal taxation, beyondthe classiccontribution by Mirrlees(1971),seealso the surveys by Mirrlees(1986)and Tuomala(1990).Recentcontributions not coveredhere that further operationalizethe Mirrleesframework are Roberts (2000)and Saez(2001),for example.\342\200\242 On implicit labor contracts under adverse selection,our discussionisbasedon the papersby Azariadis (1983),Chari (1983),Greenand Kahn

97 Static Bilateral Contracting

(1983),and Grossmanand Hart (1983).Seealso the summaries providedby Azariadis and Stiglitz (1983)and Hart (1983a).\342\200\242

Finally, for regulation under adverseselection,beyondthe initial

contribution by Baron and Myerson(1982),we should mention the paperbyLaffont and Tlrole(1986)which allows for the observability of costsandwhich has led to a vast researchoutput culminating in their 1993book.Seealsothe survey by Caillaud,Guesnerie,Rey, and Tirole(1988).

HiddenInformation, Signaling

In Chapter2 we considereda contracting problemwherethe party makingthe contractoffer (the principal) is attempting to reducethe informational

rent of the other party (the agent).In this chapterwe considertheopposite casewhere the principal hasprivate information and may convey someof that information to the agent either through the form of the contractofferor through observableactionsprior to the contracting phase.This typeof contracting problem under asymmetric information is commonlyreferredto as a signaling or informed principalproblem.

The classicexampleof a signaling problemis the modelof educationasa signal by Spence(1973,1974).We shall baseour expositionof the generalprinciplesof contracting by an informed principal on that model.

The basic setup consideredby Spenceis a competitive labor marketwhere firms donot know perfectlythe productivity of the workers they hire.In the absenceof any information about worker productivity, the

competitive wage reflectsonly expectedproductivity, so that low-productivityworkers are overpaid and high-productivity workersunderpaid.In this

situation the high-productivity workershave an incentive to try to reveal(or signal) their productivity to the firms.

Spenceconsideredthe idea that educationprior to entering the labormarket may actas a signal of future productivity. In a nutshell, his ideawasthat education might be less difficult or costly for high-productivityworkers.They could therefore distinguish themselvesby acquiring moreeducation.It is important to emphasizethat Spencedid not argue that

education per se would raise productivity, nor did he argue that educationwould reveal ability through test scores.His point was rather that higheducationwould signal high productivity becauseit would be too costly for

low-productivity types to acquirehigh education.Spence'sideais the first exampleofaprecontractualsignaling activity. For

later referenceit is useful to point out that the form of signaling consideredby Spenceis onewherethe informed principal (the worker) takesan actionto convey information prior to contracting. As we shall see,this is a form ofsignaling that is different from signaling in the contracting phasethrough theform of the proposedcontract.An exampleof the latter type of signalingwould be the saleofsharesin a company by the original owner:by selling alargeror smallerfraction of the firm's sharesthe owner may signal what heknows about the value of the company (seeLelandand Pyle,1977).

The informed-principal problemraisesnew conceptualdifficulties. Thereasonis that an informed principal'saction,by conveying new information

100 Hidden Information, Signaling

to the agent, changesthe agent'sbeliefsabout the type of principal he is

facing.Therefore,to determineequilibrium actions,one needstounderstand the processby which the agent'sbeliefsare affectedby theseactions.

The approachinitiated by Spence,subsequently refined by Kreps andWilson (1982)and others,hasbeento beginby specifying the agent'spriorbeliefsabout the principal'stype, as well as the agent'sbeliefsabout what

action each type of principal will take, then to determineeach principaltype'sactiongiven the agent'sbeliefs,and finally to define an equilibriumoutcomeas a situation where the agent'sbeliefsabout what actioneachtype of principal takesare correct(that is, the actionbelievedto bechosenby a given type is indeedthe actionchosenby that type) and where, giventhe agent'supdatedbeliefs(following the action),each type of principal is

acting optimally.

Unfortunately, the additional conceptualdifficulty is not just tounderstand this complexprocessof revision ofbeliefsfollowing an actionby the

principal.Indeed,oneof the unresolvedproblemswith this approachis that

the equilibrium outcomecannot beperfectlypredicted,sincethere are toomany degreesof freedomin choosingthe agent'sprior beliefsabout eachprincipal type'saction.Thisissueis addressedin subsequentwork by Choand Kreps(1987)and Maskin and Tirole(1992)in particular.

In this chapter,we detail the original Spencemodelas well as its

subsequent theoreticaldevelopments.We then turn to severalfinanceapplications that illustrate the generalrelevanceof signaling models.

3.1Spence'sModelof Educationas a Signal

We shall consideronly the simplest versionofSpence'smodelwith two

productivity levels:a worker'sproductivity canbe either rH or rL, with rH > rL> 0.Let fy e [0,1]be the firm's prior belief that r = rt. Workers are willing

to work at any wage w > 0, and firms are willing to hire any worker at awage lessthan the worker'sexpectedproductivity.

A worker of type i =L,Hcan get e yearsof educationat costc(e)= dtebeforeentering the labormarket.1Thekey assumption here is that 9H < 6L,

or, in words, that the marginal cost of education is lower for high-

1.This is a model of voluntary education acquisition; e= 0 thus means that the worker doesnot acquire any education beyond the rriiriimum legal requirement.

w,r *\342\226\240

Figure 3.1Single-CrossingCondition

productivity workers.Underthis assumption the indifferencecurvesof ahigh-productivity and low-productivity worker can crossonly once, asindicatedin Figure 3.1.

Note that it is assumedhere that educationdoesnot affectfuture

productivity and is therefore purely \"wasteful\": its only purposeis to allow

highly productive workers to distinguish themselves.This extreme,andunrealistic, assumption caneasilyberelaxed.Its advantage here is to allowus to identify very clearlythe signaling role of education.

In his original article,Spenceconsidersa competitive labor market,whereworkers determinetheir levelof educationanticipating that when

they enter the labor market the equilibrium wage is set equal to the

expectedproductivity of the worker conditional on observededucation.Weshall postponeany discussionof equilibrium determination in acompetitive market until Part IV, as the analysis of competition in the presenceofasymmetric information raisessubtle new issues.For now, we shall consideronly the problemof educationas a signal in a bilateralcontracting settingwith one firm and one worker.

102 Hidden Information, Signaling

In a first stage the worker chooseseducation,and in a secondstage the

wage is determined through bargaining. To stick as closelyto Spence'soriginal analysis, we assumethat in the bargaining phasethe worker has allthe bargaining power.2

Suppose,to beginwith, that the worker'sproductivity is perfectlyobservable. Then it should be obvious that eachworker type choosesa levelofeducation et = 0.Indeed,since the worker'sproductivity is known, the

highest offerthe firm is willing to acceptis wt = r, regardlessof the levelofeducationof the worker.

Now supposethat productivity is not observable.Thenthe first-bestsolution eL= eH = 0 and w* =

rx can no longerbe an equilibrium outcome.Toproceedfurther with our analysis of contracting under asymmetricinformation we needto specifypreciselythe gameplayedby the worker andfirm, as well as the notion of equilibrium.

In the first stageof the game the worker choosesa levelof education,possiblyrandomly, to maximize his expectedreturn. Let pt(e) denotethe probability that the worker/principal of type i chooseseducationlevele.In the secondstageof the game the outcomeis entirely driven by how

the agent'sbeliefshave beenaffectedby the observationof the principal'seducation level. Let j3(6i\\e) denote the agent's revised beliefs about

productivity upon observing e.Then the equilibrium wage in the secondstageis given by

wie)=P(eH\\e)rH+P(eL\\e)rL

whereP(6L\\e) =1-P(6H\\e). This is the maximum wagethe firm is willing to

pay given its updatedbeliefs.As we hinted at earlier,the key difficulty in this gameis determining the

evolution of the agent'sbeliefs.Imposing the minimum consistencyrequirements on the agent'sconditional beliefsin equilibrium leadsto thedefinition of a so-calledperfectBayesianequilibrium:

definition A perfectBayesianequilibrium (PBE)is a set of (possiblymixed) strategies{p,(e)}for the principal'stypes and conditional beliefs/J(0jle)for the agent such that

2.Note that in the opposite case,where the employer has all the bargaining power, theequilibrium wage is w =0no matter what the worker's productivity is. In this case,it is not in theworker's interest to acquire costly education soas to signal his productivity.

103 Static Bilateral Contracting

1.All educationlevelsobservedwith positiveprobability in equilibriummust maximize workers'expectedpayoff: that is, for all e* such that p,(e*)> 0,we have

e*eargmax{j3(0H|e)rH+P{0L\\e)rL-eie}e

2.Firms' posteriorbeliefsconditionalon equilibrium educationlevelsmust

satisfy Bayes'rule:

whenever pt(e)> 0 for at least one type.3. Posteriorbeliefsare otherwisenot restricted:if pt(e)=0 for i =L,H(sothat 2\302\243=iAPt(e)

=0,and Bayes'rule givesno predictionfor posteriorbeliefs),then j3(0,le)can take any value in [0,1].4. Firms pay workerstheir expectedproductivity:

w(e)=P(6H\\e)rH+P(eL\\e)rL

When a PBEis taken to be the outcomeof this signaling game,essentially

the only restrictionthat is imposedis that in equilibrium the agent'sbeliefsare consistentwith the agent'sknowledgeof the optimizingbehavior of the principal.

To solve for a PBE one typically proceedsas follows. Using one'sbasicunderstanding ofhow the gameworks, oneguessesconditional beliefsj3(0,le)for the agent.Thenonedeterminesthe principal'sbestresponsept(e)given these beliefs.Finally one checkswhether the beliefs /J(0,le) areconsistentwith the principal's optimizing behavior.In signaling gamesthe difficulty is usually not to find a PBE.Rather, the problemis that thereexist too many PBEs.

Using our intuitive understanding ofSpence'smodel,wecaneasilyguessthe following PBE.If the observededucationlevelis high, it is likely that

the worker has high productivity. Indeed,evenif a low-productivity workercouldobtain a wagew = rH by acquiring education,he would not bewilling

to choosea levelof educationabovee,where e is given by rH - 6Le= rL.

Thus a candidatefor a PBEis to setp(6H\\e) =1for all e > e and j3(0Hle) =0 for all e< e.If the principal optimizesagainst thesebeliefs,then he choosesno educationwhen he has low productivity [pz,(e)=0 for all e > 0] and he

104 Hidden Information, Signaling

choosesthe levelof educatione when he has high productivity. It is easytoseethat with this bestresponseby the principal, the agent'sbeliefsareconsistent in equilibrium.

ThePBEwe have identified capturesin a simpleand stark way Spence'sidea that the levelofeducationcanactas a signal ofproductivity. Thehigh-

productivity worker distinguishes himself by acquiring a levelof educatione > e and obtains a high wage,rH, while the low-productivity worker doesnot acquireany educationand getsa low wage,rL.

However,the intuitively plausiblePBEwe have identified is by no meansthe only one,sothat the effectivenessofeducationasa signal is by nomeansguaranteed.There are many PBEs,which can be classifiedinto threedifferent categories:1.SeparatingPBEssuch as the one we have identified, where the signalchosenby the principal identifies the principal'stype exactly.2.PoolingPBEs,wherethe observedsignal revealsno additionalinformation about the principal'stype.3.SemiseparatingPBEs,where some but not all information about the

principal'stype is obtainedfrom the observationof the signal.

We focushere on giving a precisedefinition and characterizationof thesetofseparatingand poolingequilibria, while commenting briefly on semi-separatingPBEswheneverappropriate (indeed,this last type ofequilibrium is not very important in this economiccontext,as will be explainedlater).3

definition A separatingequilibrium is a PBEwhereeachtype ofprincipal

choosesa different signal in equilibrium: eH *eLso that P(9H\\eH) -1,P(6i)eL)=1,and w, = r,.

It is straightforward to checkthat the setofseparatingequilibrium levelsof educationis given by

Ss=\\(eH,eL)\\eL =0andeHe

We illustrate one separating equilibrium in Figure 3.2.Intuitively,since the low-productivity type is getting the worst possiblewage, he has

3.For more on signaling models,seegame theory texts,such as Fudenberg and Tirole (1991).

rH-rL rH-rL9r 9H 1

105 Static Bilateral Contracting

U, = r,

w,r n

= r,-vue.

Figure 3.2Separating Equilibrium

no incentive to acquireeducation:he can obtain this wagewith noeducation at all. Instead, the high-productivity type is getting the highest

possiblewagerH. Incentivecompatibility requiresrH -6LeH< rL; otherwise,the low-productivity type would prefer to acquire educationlevel eH in

order to get wage rH. Similarly, we need rH - QueH > rL; otherwise,the

high-productivity type would preferno educationand a wageof rL. Beyondthese restrictions, it is easy to find out-of-equilibrium beliefsthat supportany eH in Ss.For example,as in Figure 3.2,P(0H\\e) =1whenevere > eH, and

P(6H\\e) =0 otherwise;under thesebeliefs,only 0 and eH caneverbeoptimalworker choices.

definition A poolingequilibrium is a PBEwhereeach type of principalchoosesthe samesignal in equilibrium: eH=eLsothat P(6H\\eH) =

j3H, j3(0LleL)=

j3L, and w(eH) =w(eL)= j3LrL +pHrH = f.Again, it is straightforward to verify that the setof poolingequilibrium

levelsof educationis

106 Hidden Information, Signaling

r=PLrL + pHrH,

\342\200\242H- <-vH*p

Figure 3.3Pooling Equilibrium

Sp=keH,eL)\\eL=eH=epaadepe0, pLfL+pHrH-rLeT

A pooling equilibrium is illustrated in Figure 3.3.Intuitively, Sp isdeterminedby the fact that the worker couldobtain a wageof at least rL

without any education.This option is particularly attractive for the low-productivity type. Incentivecompatibility thus requiresf - 6Lep > rL. It isthen easy to find out-of-equilibrium beliefsthat support any ep in Sp.For

example,as in Figure 3.3,fi(9H\\e) =fiH and fi(6L\\e)

=/JL whenevere > ep, and

P(6H\\e) =0 otherwise.Underthesebeliefs,only 0 and ep caneverbeoptimalworker choices.

Finally, note that therealsoexistsasetofsemiseparatingequilibria, whereat leastone type ofprincipal is mixing betweentwo signals, one ofwhich isalsochosenwith positive probability by the other type ofprincipal.Thekeyrequirement here (asstressedin part 1of the definition ofaPBE)is that, for

mixing to beoptimal, the type ofprincipal who is mixing has to beindifferent betweenthe two signals that are playedwith positive probability.

107 Static Bilateral Contracting

3.1.1Refinements

Thefact that there may besomany different equilibria suggestsat the veryleast that the theory of contracting with an informed principal is

incomplete. Indeed,in his original work Spencesuggestedthat additionalconsiderations,such as social custom or conventions, are relevant in

detennining how a particular equilibrium situation may come about.Spencedid not venture into the difficult terrain of socialcustoms and wasrather agnosticas to what type of signaling equilibrium might besupportedby socialcustom.

Subsequently, a largebody of researchin gametheory has attempted to

developcriteriabasedon theoreticalconsiderationsalone for selectingaparticular subsetofequilibria.4Thebasicideabehind this researchprogramis to enrich the specificationof the game by introducing restrictions onthe set of allowableout-of-equilibrium beliefsusing the observationthat

some types of players are more likely to choosesome deviations than

others.

3.1.1.1Choand Kreps'Intuitive Criterion

Themost popularrefinement usedin signaling gamesis the so-calledCho-Kreps intuitive criterion (seeCho and Kreps,1987).The basicobservationthis selectioncriterionbuilds on is that most deviations from equilibrium

play canneverbein the interestofsomeprincipal types.Beliefsconditionalon out-of-equilibrium actions,therefore,ought to reflectthe fact that theseactionsare more likely to be chosenby some types than others.In otherwords, beliefsconditionalon out-of-equilibrium actionsmust berestrictedto reflect the fact that only some types are ever likely to choosetheseactions.

Formally, the restrictionimposedby Cho and Kreps on beliefsconditional on out-of-equilibrium actionsin our simple exampleis the following:

definition Cho-Krepsintuitive criterion:Let m? =wf (e,)-0,e,denotethe

equilibrium payoff of type i.Then,j3(0;-le)=0,for e*(e,-;e}),whenever rH -d,e< u* and rH -dfi > wf (i =L,H;i ^ ;').

The Cho-Krepsintuitive criterion is essentiallya stability requirementabout out-of-equilibrium beliefs:it saysthat when a deviation is dominated

4. See,for example, Fudenberg and Tirole (1991).

108 Hidden Information, Signaling

for one type of playerbut not the other one, this deviation should not beattributed to the playerfor which it is dominated.By dominated, onemeansthat the playeris getting a worsepayoff than his equilibrium payoff for anybelief of the uninformed party following the deviation.Here,the mostfavorablebelieffollowing a deviation is fi(9H\\e) =1,leading to a wageof rH,so that a deviation is dominated for type 6>;-

but not 6i if and only if rH -6te> uf and rH - d,e< u*. In this case,the Cho-Krepscriterionsays that the

out-of-equilibrium beliefshould be j3(0;le)=0.Any equilibrium that doesnot satisfy this criterionshould bediscarded.

Applying this test to poolingequilibria, one is led to discardall of them.Indeed,at any poolingequilibrium, the indifferencecurvesof the two typesmust crossas shown in Figure 3.4.But then the 9H type can always find aprofitable deviation, by, say, increasinghis educationlevelto ed(seeFigure3.4).At ed the firm is readyto accepta wageofw(ed) =rH, sinceat that wagethe deviation canbeprofitable only for a 6H type, while it is dominated forthe other type. Using an argument similar to the one for pooling equilibria,it caneasilybeshown that all semiseparatingequilibria canbediscarded.

Figure 3.4Cho-Kreps Unstable Pooling Equilibrium

Static Bilateral Contracting

U, = r,

Figure 3.5Least-Cost Separating Equilibrium

Next, applying the Cho-Kreps intuitive criterion to all separatingequilibria, one eliminates all but one equilibrium, the so-called\"least-cost\" separatingequilibrium in which eL= 0 and eH ={rH -rL)/dL.Figure3.5detailsthis equilibrium. The Cho-Krepsintuitive criterionthus selectsaunique pure-strategyequilibrium, which is given by the \"least-cost\"

separating equilibrium.This extensionof the original theory, which placesgreater demandsof

rationality on the formation of beliefsthan doesSpence'stheory, makesastrong and unambiguous prediction:when signals such as educationareavailableto allow higher ability workers to distinguish themselvesfrom

averageworkers, these signals will be correctlyunderstoodby employersand, consequently, will be usedby the higher ability workersto separatethemselvesout.

As plausibleas the Cho-Krepsintuitive criterion may be,it doesseemto predict implausible equilibrium outcomesin some situations. Note in

particular that the \"least-cost\" separatingequilibrium is the same for all

110 Hidden Information, Signaling

valuesof$> 0; i = L,H.Supposenow that j3L, the prior probablity that aworker of type 6L is present,is arbitrarily small (/JL= 5 ->0).In that case,it seemsan excessivecost to pay to incur an educationcost of c(eH;6H) =QJjh-ri)l9Ljust to be able to raise the wageby a commensurately smallamount [Aw = rH -(1-5)rH -5rL = 5{rH - rL) ->0].Indeed,in that casethe poolingequilibrium where no educationcostsare incurred seemsamoreplausibleoutcome,sinceit Pareto-dominatesthe Cho-Krepsequilibrium.5 Moreover,note that without adverseselectionat all, that is, j3L

= 0,no educationis chosenin equilibrium, with the result that it is the poolingequilibrium without educationthat is the limit of this complete-informationcase,and not the Cho-Krepsequilibrium.

This particular argument should serve as a useful warning not to relytoo blindly on selectioncriteriasuch as Cho and Kreps'intuitive criterionto single out particular PBEs.6It is fair to say nevertheless that the

Cho-Krepscriterionhas acquiredpreeminencein the many applicationsofthe signaling approach.

3.1.1.2Maskin and Tirole'sInformed-PrincipalProblem

Interestingly, the problem of multiplicity of PBEsis reduced when the

timing of the gameis changedso as to let the principal offer the agent acontingent contractbefore the choiceof the signal.This is one importantlessonto bedrawn from Maskin and Tirole(1992).Toseethis, considerthelinear modelofeducationspecifiedpreviously, and invert the stagesofcontracting and educationchoice.That is,now the worker offershis employer

5. In all equilibria, firms earn zero profits and are therefore indifferent. As for the

low-productivity worker, his preferred equilibrium is the pooling one without education,where he is subsidized by the high-productivity worker without having to incur educationcosts. Finally, for fiL low enough, the education level required to separate from the low-

productivity worker is too expensive for the high-productivity worker in comparisonwith the pooling wage (but not in comparison with the low-productivity wage, ofcourse).6. Some game theorists cast further doubt on such selection criteria that do not\"endogenize\" out-of-equilibrium strategies: while it makes sense not to attribute adeviation to a type for whom the deviation is dominated, why attribute it to the other type,since by construction of the PBE the deviation gives the player a lower payoff than hisequilibrium payoff, given the original beliefs about out-of-equilibrium strategies? It wouldtherefore be better to build a theory of equilibrium selection that would embody a theory ofout-of-equilibrium strategies. For more on these issues,see,for example,Fudenberg and Tirole(1991).

Ill Static Bilateral Contracting

a contractbeforeundertaking education.This contractthen specifiesa wageschedulecontingent on the levelof educationchosenby the worker after

signing the contract.Let [w(e)}denotethe contingent wageschedulespecifiedby the contract.

Thereare two different casesto consider:

1.r =/3HrH +/3LrL<rH-9H(rH-rL)/6L.In this casea high-productivityworker is better off in the \"least-cost\" separatingequilibrium than in theefficient pooling equilibrium.

2. f>rH-9H(rH-rL)/dL.Here,on the contrary, the high-productivityworker is better off in the efficient poolingequilibrium.

In the first case, the least-cost separating equilibrium is \"interim

efficient,\" in that there is no other incentive-compatiblecontract that

managesto strictly improve the equilibrium payoff of the firm or of atleast one type of worker without causing another payoff to deteriorate.In this case,the unique equilibrium contract to be signedis the one that

specifies the same wage schedule as in the \"least-cost\" separatingequilibrium:

rH for e> \342\226\240

[rL otherwise J

Indeed,note first that the firm is willing to accept this contract. If it

does, the high-productivity worker then choosesan education leveleH =(rH -rL)/&L and the low-productivity worker, being indifferent

betweeneH and eL=0,chooseseL=0.In eithercasethe firm breakseven,so that it is willing to accept this contractirrespectiveof whether it

originates from a low- or a high-productivity worker.Second,it is easyto verifythat in the casewhere r <rH -6H(rH -rL)/dLthe high-productivity workercannot do better than offering this contract,nor can the low-productivityworker.More precisely,the high-productivity worker strictly prefers this

contractover any contractresulting in poolingor any contractwith morecostly separation.As for the low-productivity worker, he cannot offer abetter incentive-compatiblecontractthat would be acceptableto the firm.

In this case,therefore,changing the timing of the gameleadsto a uniquePBE.

112 Hidden Information, Signaling

In the alternative casehowever,that is, when r>rH-6H(rH-rL)/dL,the least-costseparatingequilibrium can be improved upon in a Paretosense.In that case,Maskin and Tiroleshow that the equilibrium setof this

game consistsin the incentive-compatibleallocationsthat weakly Pareto-dominate the least-costseparatingequilibrium. This multiplicity ofequilibria brings us backto the lack of restrictionsimposedon out-of-equilibriumbeliefsby the PBEconcept.

Changing the timing of the gamesothat a contracting stageprecedesthe

stagewhen workers choosetheir educationlevelsis thus particularlysignificant in situations where the least-costseparatingequilibrium is interim

efficient, sinceit is then selectedas a unique PBE,and this without havingto appealto restrictionson out-of-equilibrium beliefs.

3.2Applications

3.2.1CorporateFinancing and Investment Decisionsunder AsymmetricInformation

One of the founding blocks of modern corporate finance is the

ModigHani-Miller theorem (1958),which states that capital structure (inother words, the form and source of financing) is irrelevant for firms'investment decisionswhen there are no tax distortions, transactionscosts,agency problems,or asymmetries of information. In a frictionlesscapital market firms maximize profits when they decideto undertakeinvestment projects if and only if the net present value (NPV) of the

project is positive.Considerationsother than cash-flowand investment

costs,such as dividend policy and debt-equityratios, are irrelevant forinvestment decisions.The basic reasoning behind this theorem is the

following: sinceshareholderscan buy and sell financial assetsat will, theycandesignthe payout stream from the firm that bestsuits their needs.Thefirm has no advantage over investors in designing a particular payoutstream.

In Chapter2 we showedhow in the presenceof adverseselectionsomefirms may beunable to fund their positiveNPV projectswith debt.Here,we apply the signaling paradigm to show how the choiceof firm financing\342\200\224

equity or debt\342\200\224may affect a firm's investment policy when the firm hasbetter information about investment returns than outside investors, thus

leading to a departurefrom Modigliani-Miller irrelevance.

113 Static Bilateral Contracting

Much of the following discussionis basedon the articleby Myers and

Majluf (1984).Theseauthors attempt to explain the following stylized fact:new equity issueson averageresult in a drop in the stockpriceof the issuer.Thisis somewhat paradoxical,sincea new issuerevealsto the market that

the firm has new investment opportunities available.If theseare positive-NPV projects,the stockpriceshould go up rather than down.

Myers and Majluf show, however,that if the firm hasprivate information

about the value of its assetsand new investment opportunities, then the

negative-stock-pricereactionfollowing the announcement of a new equityissuecanbeexplainedasa rational responseby lesswell informed investors.

When the firm has private information, the stockissuemay actasa signalof firm value and thus conveyinformation to the market. In other words,the firm facesa signaling situation, where the signal is the firm's decisionto raisenew capital.

We shall considera generalizationof the simple exampleintroducedby

Myers and Majluf that incorporatesuncertainty. In this examplea firm run

by a risk-neutral manager has assetsin placewhosevalue canbe0 or 1in

the future, and a new investment projectwhosegrossvalue can alsobe 0or 1;the start-up costof the new projectis 0.5.

Denote by y and 77, the respectivesuccessprobabilitiesfor the assetsinplaceand the new projectin state of nature i (with outcomesuncorrectedconditional on theseprobabilities).Supposethat there are only two statesof nature, i-=G,B,eachoccurring with probability 0.5.In state G,the goodstate,we have yG > yB and T7G

> T]B.We introduceprivate information by

assuming that the managerobservesthe true stateof nature beforemakingthe investment decision,but that outside investors do not observe the

underlying state the firm is in.Assume that the firm is initially fully ownedby its manager.Then,if she

had 0.5in cash,shewould start the investment projectin state i if and onlyif

r\\i>0.5.Suppose,however, that the new investment projecthas to befully

funded externally by a risk-neutral investor.

3.2.1.1Case1:T]GkT]Bk0.5In Case1 it is always efficient to undertake the new project. But weshall seethat the manager may be led to make inefficient investmentdecisions becauseofher informational advantage overthe market. Letus beginby specifying the financial contractusedto raise new funds. To this end,denote by r, the ex post repayment when the grossvalue of the firm V is

114 Hidden Information, Signaling

respectivelyj = 1,2.Sincethe ex post value of the asset in placeand thenew projectcan be either0 or 1,the total ex post value of the firm canbeeither 2 (when both assetin place and new projectare worth 1),1(whenonly one of the two assetshas expost value of 1),or 0 (when neither assethasa value of1).Notealsothat when the expostvalue is 0 the firm cannotmake any repayment. Therefore,we need to specifyonly rx and r2 todescribea generalinvestment contract.

Equity Financing

We begin by consideringthe signaling gamethe firm plays when it isconstrained to raising capitalin the form of a new standard equity issue.Thisamounts to

r2 =2rx

Under this constraint the firm facesa very simple signaling problemwith

only two actions:\"issue new equity and invest in the new project\"and \"do

not issue.\"

In state B the firm always wants to issueand invest, since,at worst,investors correctlybelievethat the state of nature is bad and are willing toinvest 0.5in the firm in exchangefor an equity stake of at least

0.5Yb+tIb

of the sharesof the firm. Indeed,with such a stakenew investors obtain an

expectedreturn of (yB +rjB )[0.5/(yB+T]B)]=0.5,equalto their initial

investment. In a competitive capitalmarket with a discount rate ofzero they arewilling to acceptsuch an offerfrom the firm. Thisofferleavesthe firm with

a payoff equal to

(.Yb+tiJi :)=7b+i1b-0.5

which is, by assumption, higher than yB, the payoff the firm obtains when

not undertaking the project.If, in equilibrium, the firm were to issueshares in both states of nature,

investors would not be able to infer the state of nature from the firm'saction.Unableto observethe state,they would then value the firm's assetsat 0.5(7G+ rjG) +0.5(yB+ r\\B) and ask for an equity stake of

115 Static Bilateral Contracting

0.5

o.5X(r.-+*7.)

in return for an initial investment of 0.5.Under such an offer the firm isovervaluedby investors in state B and undervalued in state G.

In state G the firm then obtains the following expectedpayoff if it goesaheadwith the investment:

( \\XyG+r]G)-(yB+r}B)

(Yg+tIg) 1-(Yi+rii)

=7g+77g-0-5-0.5w\"'\"/\"\" ,D/ (3.1)

The last term representsthe subsidy from the firm to investors in stateG, resulting from the undervaluation of the firm in that state.This subsidyis sometimesreferred to as a \"dilution cost\" associatedwith a new equityissue.It should beclearfrom this expressionthat if this subsidy is too high,the manager'spayoff from issuing equity will be lowerthan yG, the payoffwhen she doesnot undertake the investment project.Thus,whenever

r,G-0.S<0.5(ro+^)-(r\302\260+

\"\302\260)

ZiiYi +li)I

the firm issuesequity only in the bad state. In that case,the firm'sactionperfectlyrevealsthe firm's information, and we have a separatingequilibrium.

Thereare two important observationsto bedrawn from this equilibriumoutcome:

1.The value of the firm is lower than it would have been,had it had the

necessarycash to fund the investment projectbeforehand,sinceit fails toundertake a positive-NPV projectin the goodstate of nature.

2.When the firm announcesa new equity issue,investors learn not onlythat it has a new investment project availablebut also that the firm isin state B;as a result, firm value dropsfrom 0.5yG + 0.5(yB+ T]B

-0.5)toyB + t]b -0.5.Thus in this separatingequilibrium there is a negative stock-price reaction to the announcement of a new equity issue.It is worth

emphasizing that despitethe expecteddrop in stockpriceit is in the firm'sinterest to go aheadwith the issue.

116 Hidden Information, Signaling

When instead

XyG+riG)-(rB+TiB)%-0.5> 0.5-X(7<+77<)

then a poolingequilibrium existswherethe firm issuesequity in both states.However,it is not the only equilibrium if the following condition alsoholds:

(7g+%(i--^-)<7g (3.2)

When these two conditionshold simultaneously, then both the poolingand the separatingequilibria exist.Hereinvestors' beliefs can be self-fulfilling: the firm in state G issuesequity if and only if the market thinks

that it does.If the market believesthat the firm in state G issuesequity,it is ready to give the firm more favorableterms, which in turn makes it

more attractive for the firm to issueequity.

Finally, if

(7g+^(i--^-Wg (3.3)v 7b+T]bJ

then only the poolingequilibrium exists.Note that this condition alwaysholds when yG

=yB (no asymmetric information on the value of existing

assets)or,equivalently, when the new projectis funded entirely externally,as an independent firm. In such cases,all positive-NPV projects areundertaken.

Note finally that all the equilibria we have describedsatisfy the Cho-Kreps intuitive criterion.This is trivially the casefor the separatingequi-Ubrium, whereboth signals areusedwith positiveprobability in equilibriumand no out-of-equilibrium beliefsneedbe specified.But it is also trivially

satisfiedfor the poolingequilibrium, for which deviating meansnot issuing

equity, so that out-of-equilibrium beliefsare irrelevant.

DebtFinancing

We have seenthat whenever

(7g+T7g)-(7b+^b)77G-0.5<0.5-Xfr<-+77<)

117 Static Bilateral Contracting

there is such a high dilution cost involved in issuing new equity in state Gthat the firm prefers to forgo a profitable investment opportunity. Thisresult raisesthe questionof whether under these conditions alternativemodesof financing, such as debt, are availablethat would allow the firm

to reduce the cost of outside financing enough to go aheadwith the newinvestment.

To this end,we can expressthe differencebetweenequity and debt in

terms of rh the expostrepayment when the grossvalue of the firm is,respectively;

=1,2.We defineda standard equity contractas onewherer2 =2rx.In contrast, a standard (risky) debt contractwith facevalue D is such that

ri =min{\302\243>, 1}and r2 =

min{\302\243>, 2}so that r2 < 2rx wheneverD < 2.Thus,under debt financing, the firm repaysrelatively more when the ex postfirm value is 1than under equity

financing. Indeed,it is a feature of debt repayment to be concavein firm value,while equity repayment here is linear in firm value (and becomesconvexin firm value when debt has alsobeenissued,sincedebt has priority overequity).

Becauseof these differencesin repayment functions it is likely that thedilution costsassociatedwith thesetwo instruments will bedifferent. Inparticular, it seemsplausiblethat debt financing would reducethe subsidy fromthe firm to the market in state G if the mispricing problemis lessseverewhen the value of the firm is 1rather than 2, that is,whenever

Prob(y=ll\302\243) Prob(y=

2l\302\243)

Prob(V =1|G)>

Prob(V =2|G)

or

YBlX-nB) +(\\-yB)T]B > YbIb (3<4)7G(l-%)+(l-rG)% YolG

an inequality that is always satisfiedunder our assumption that Yg^-Yb and

t]g ^ t]b-We now show that debt financing reducesthe subsidy to the market in

state G if and only if condition (3.4)holds.To seethis, considerthe localdeviation from an equity contract,where instead of repaying (ri, r2) =(ru 2ri) the firm agreesto repay [rx + 5, 2rx -e(6)],where5 > 0 and e(5)ischosensoas to keepthe expectedrepayment constant:

118 Hidden Information, Signaling

wherek is a constant.If the subsidy under the deviation is lowerthan underthe equity contract, then clearly debt financing involves lower dilution

costs.The subsidy, S,in state G under a contract [ri + <5, 2rx -e(<5)]is given by

the differencebetween the true value of the repayment stream [ri + <5,

2rx -e(6)]and the investor'svaluation of this stream:

S=(r1+5)[yG0\342\200\224T]G) +(l-yG)T]G]+[2r1-\302\243(5)]yGT]G-

o.5XrIa-^J+a-rO^Vi+5)+[o.5Xr^l(2r1-e)Now considerthe change in the subsidy, S,when <5 is increasedbut the

expectedrepayment is kept constant:

-77=7g (1-r\\a) +(1-7g)1g-7gIg\342\200\224

do d\302\243

n W1 , [o-5l,r\302\253(i-'7,)+(i-r.)'7,]=7G0--riG)+Q--7G)riG-7GTlG rn gV -,[0.51,7,77,]

Rearranging and simphfying, it is easyto checkthat dS/dd<0 if and onlyif condition (3.4)holds.Thus,under our assumptions, debt finance would bebetter than equity finance.But, debt finance, in turn, is dominated byinternal finance,since,as we pointedout, internal finance doesnot involve any\"dilution\" cost.Theseobservationsare consistentwith the \"pecking-ordertheory of finance\" put forward by Myers (1977),which suggests that

retained earnings are the best source of finance, followedby debt, and

finally, equity, which is the least efficient form of finance.Strictly speaking,equity would neverbeusedby the firm in the narrow contextofour model.Thisfinding hasledsomecriticsto argue that the theory proposedby Myersand Majluf is not empirically plausiblebecauseit leadsto the conclusionthat firms neverissueequity. However,it is not difficult to extendthe modelto introducebankruptcy costsand thus to obtain someequity financing in

equilibrium in order to reduce the firm's exposureto thesecosts.Perhapsa more important criticism is that this peckingorder suggestedby Myersdoesnot hold for all relevant parametervalues.If, for example,condition

119 Static Bilateral Contracting

(3.4)doesnot hold, then equity is the preferredinstrument overdebt,andthe peckingorderis changed.Furthermore, the theory consideredherepredicts that in generalneither equity nor debt is the most suitable instrument,as we shall illustrate.

OptimalExternalFinancing

While debt is a better way of financing the firm externally than equity, it isnot necessarilyoptimal.Theoptimal form ofexternalfinance is the one that

minimizes the differencein value of repayment acrossstatesofnature. Thisfinding may mean that the repayment should not be monotonicin therevenueof the firm. Consider,for example,the casewhere1=

r\\G > r\\B=0.5

and yG =0.5> yB =0.In this case,Prob(V=1)is 0.5in both statesofnature.

Consequently, a repayment stream such that rx = 1and r2 = 0 satisfiestheinvestor's participation constraint and eliroinates the subsidy to newinvestors in state G.

While the exampleis extreme in terms of allowing for a repaymentburden that is identicalacrossstates of nature, it illustrates in a stark waythe criterionthat dictatesthe attractivenessoffinancing modesin this setup.Note,however, that one unattractive featureof this contractis that the firm

could make money here by borrowing secretly to artifically boost its

revenue and thereby reduce its total repayment. As will be discussedinChapter4 (in a moral hazardcontext),Innes (1990)has analyzed optimalfinancial contracting under \"monotonicity constraints,\" which requirerepayments to be weakly increasingin the revenue of the firm, and hasderivedconditionsunder which debt then becomesan optimal contract.

3.2.1.2Case2:J]G >0.5> J]B

We closeour discussionof this model with a few remarks about the casewhere the efficient outcomeis separatingand has the firm undertake thenew investment project only in the good state of nature. If this were an

equiUbrium, investment would take placeif and only if the new projecthad

positive NPV, sincethe market would correctlyperceivethe stateofnature

the firm is in. However,in this situation, the market would subsidizethefirm if it were in state B and decidedto invest. For example,under equity

finance, the firm would then obtain a payoff of

(7b+t]b)\\ 1 =7b +77B-0.5+0.5V Yg+tIgJ Yg+tIg

Hidden Information, Signaling

The last term of this expressionis the subsidy from the market to thefirm. If it is largeenough, the manager'spayoff will be higher than yB, andan efficient separatingequilibrium will not existunder equity financing. A

separating equilibrium may, however,exist under alternative financingmodes\342\200\224for sure under self-financing, but possiblyalsounder debtfinancing. Indeed,as in Case1,debt finance is preferableto equity finance here,for an identicalreason:financing instruments are better the more stableisthe repayment burden acrossstatesof nature.

Signaling Changesin CashFlow through DividendPayments

TheModigliani-Miller theoremstatesnot only that corporatecapitalstructure is irrelevant for firms' investment decisions,but also that changesin

dividend policy do not have any effecton firm value (when capitalmarketsare competitive, there are no tax distortions, and the firm has noinformational advantage overoutsideinvestors).Toseethe basiclogicbehind the

Modigh'am-Miller theorem (1958)more clearly, consider the following

highly simplified formal argument that underliestheir result.Taketwo firms

that are identicalin every respect,exceptfor their dividend policy.Tokeepthings assimple aspossible,we shall reducea firm to a cashflow and adividend policy.Supposethat the two firms have the samecashflow, C[= C'2= C,but a different dividend policy,d[ *d\\ for t =0,1,2...,\302\260\302\260. Let p > 0denote the constant market interest rate, ra[ the number of new sharesissued(or bought) by firm i = 1,2 at date t, and P\\ the market price of ashare in firm i =1,2at date t. Then the net present value of each firm atdate t =0 canbewritten as

d[+m[Pl'(1+p)' .

In words, the current net value of firm i = 1,2 is simply the discountedvalue of dividend payments and sharerepurchases(or new issues).In otherwords, it is the discountedvalue of all net payments from the firm to theshareholders.It would seemfrom this formula that the current value ofboth firms must differ becausethey have different dividend policies.As

Modigliani and Miller have pointed out, however, this assumption is notcorrect,since we have failed to take into account the basic accountingidentity:

C^df+mlP;

121 Static Bilateral Contracting

which states that in any period,revenue(C-)must beequalto expenditure(d\\ +mlPl), as an accounting construction.Oncewe take this identity into

account,it is obvious that the two firms' current values cannot be different

if they have the samecashflows.

As compelling as the ModigUani-Miller logic is, in reality dividend

policy doesmatter, and firm value is significantly affectedby changesin

dividend policy. For example, it has been widely documented that afirm's sharepricedropssubstantially following the announcement of adividend cut. Similarly, firms that have neverpaidany dividends seetheir stockpriceincreasesignificantly (on average)following the announcement of achange in policy to positivedividend payouts (seeAsquith and Mullins,

1983).The reason why dividend policymatters is still not well understood.

Perhapsthe bestavailableexplanationis that, in a world wheremanagershave better information than investors about cash flow, dividend policyservesas a signal of cashflow, and changesin dividend policy as a signal ofchangesin cashflow. The first modelof signaling through dividends is dueto Bhattacharya (1979).We shall consideran extremely simple exampleadapted from his model.This examplehighlights the common structure

betweenthe signaling problemsconsideredby Bhattacharya and by Myersand Majluf. It alsointroducesa moresatisfactory objectivefunction for the

manager.Considera firm run by a managerwho owns an equity stake a > 0 in

the firm. This firm generatesa cash flow in periodst = 1and t = 2 that

can take two possibleexpectedvalues, ye {yG, yB], where yG > yB. As in themodel of Myers and Majluf consideredearlier,we shall supposethat in

eachperiodthere is a random (i.i.d.)draw of cashflows and that realizedcash flows can take only the value zero or one.The variable y(i = G, B)thus denotesthe probability that realizedcash flow in any period is equalto one.At date t = 0 the manager learns the true value of y but othershareholdersremain uninformed. Shareholders'prior beliefsare given by

j3=Pr(7= 7G).The managerof the firm is risk neutral but has uncertain liquidity

preferences; that is, she doesnot know in advance when she will need toconsume.We model this problemas in the celebratedpaperby Diamondand Dybvig (1983)(seeChapter 9 for further discussionof this issue).Specifically,we assumethat the manager'sstate-contingentutility function

takes the following simple form:

122 Hidden Information, Signaling

. [cx with probability p ]u{ci,c2)=\\ \\

lc2 with probability 1-p)

where,ct(t = 1,2) denotesconsumption in periodt, and p is the ex ante

probability that the managerwants to consumein periodt=l.Theidea isthe following: at date t =0 the managerdoesnot yet know when shewants

to consume;shelearnsher preferredconsumption date only at the

beginningofperiodt= 1.If it turns out that shewants to consumein periodt=l,

then sheneedsto sellher stake a in the firm. Thus at date t =0 the managercaresboth about the final value of her stake at date t = 2 and the marketvalue ofher stakeat date t=l.

The precisetiming of events in period t = 1 is as follows:(1) thefirm's cash flow for that period is realized and observed by both

manager and shareholders;(2) the firm borrowswhatever is neededtomeet its dividend payments; (3) the managerlearnsher consumptionpreferences; (4) the manager decideswhether or not to liquidate her stake in

the firm.We shall considerthe decisionfacedby a managerofa firm with expected

cashflow perperiod of yG. In a world of symmetric information the valueof the firm at date t = 0 would be 2yG (assuming a discount rate of zero),but in the absenceof any information identifying the type of the firm, themarket would value this firm only at 2\\fiyG + (1-P)Yb\\-

We shall now show that the managercouldraisethe market value of thefirm by announcing a dividend payout d such that 1> d > 0 at date t =1.Inotherwords, the announceddividend payout canactasa signal of the firm'scashflow. As usual, the promiseddividend payout works as a signal if and

only if it is too costlyfor a firm with low cashflow, yB, to commit to such apayout stream.Thecostsfrom commiting to a dividend payout d ariseherefrom the costof borrowing that must be incurred when realizedcashflow

in period1falls short of the promiseddividend payment.Let 8 > 0 denote the unit deadweightcost of borrowing,7 and suppose

that the market's posteriorbelief about the firm's cash-flowtypeconditional on an announceddividend payment d is given by

7. Note that the presence of a strictly positive deadweight cost of borrowing is essential forthe theory. This cost might be a debt-mispricing cost or a debt-collection cost, or simply amonitoring cost.Whatever interpretation one takes of this cost, it is worth emphasizing thatthe signaling theory works only if there is also an imperfection in credit markets.

123 Static Bilateral Contracting

10 for d < d)

It is then too costly for a managerof a low-cash-flowfirm to commit to adividend payout d if and only if

a2yB > cc[2yG-d(l-yB )<5] (3.5)The left-hand side(LHS)of this inequality is the manager'spayoff if

she commits to no dividend payout. In that case,the market identifiesher firm as a low-cash-flowfirm and correctlyvalues her stake at a2yB.TheRHSis the manager'spayoff if she commits to a payout of d.In that case,the market is fooled to identify her firm as a high-cash-flow firm at datet=l.Given that the market is then led to overvaluethe firm, it is in the

manager'sinterest to liquidate her stake at date t = 1,whether she wants

to consumein that period or not. Also, given that the manager sellsherstake after the realizationof cash flow (and, therefore,after the firm's

outstanding debt has beendetermined),her expectedex ante payoff fromthat strategy is

yB(a2yG)+(l-yB)a{2yG-d5)=a[2yG-d(l-yB)<5]

Similarly, it is worth commiting to a dividend of d for a manager of ahigh-cash-flow firm if and only if

cc[2yG-d(l-yG)5]>cc[p2yB+(l-p)2yG]The LHSof this inequality is the manager'sexpectedpayoff if she

commits to a dividend of d and thus inducesthe market to correctlyvalue thefirm at 2yG minus the expecteddeadweight cost of borrowing d(l- yG)5.The payoff on the RHScan be understoodas follows:on the one hand,if the managermakesno dividend payment, the firm is identified as a low-cash-flowfirm at datet=l,and the manageris forcedto sellher stakebelowthe true value whenevershe has a liquidity need(with probability p);onthe other hand, if she doesnot have any liquidity need,she canhold on toher stake until date t = 2 and realize the full value of her stake without

incurring any deadweight costof borrowing.Rearranging and simplifying these two inequalities, we obtain

2p(yG-yB)> 2(yG-yB)(l-yG)5

~ ~(l-yB)5

124 Hidden Information, Signaling

Thus, as long as 5 is largeenough (sothat d < l),8and

then it is possibleto find a separatingequilibrium, wherea high-cash-flowfirm commits to a dividend payment of d > 0,mcurring an expecteddeadweight costof borrowing of (1-yG)5d, and a low-cash-flowfirm would notcommit to any dividend payment. Intuitively, a high p helps the existenceof a separatingequilibrium, sinceit raisesthe costof not committing to adividend payment for the high-cash-flow firm; and the sameis true for anincreasein yG or a decreasein yB, which amplify the differencebetweenthetwo types and thereforethe differencebetweentheir expecteddeadweightcostof borrowing.

Even though the signaling theory of dividends, as outlined in this simpleexample,is rather plausibleand is also consistentwith U.S.data (seeBernheim and Wantz, 1995),it is not entirely satisfactory. Onemajordifficulty

with the theory is that its results are sensitiveto small, seeminglyinnocuous changes in specifictiming assumptions. For instance,if the

manager could liquidate her stake any time after the announcement of dbut before the realizationof cashflows at date t=l,then it would be easyto seethat dividends couldno longeract as crediblesignals of cash flow.

Indeed,in that casecondition (3.5)becomes

a2yB>a[2yG-d{l-yG)S\\

However,we alsohave

a[2yG-d(X-yG)8]>a\\p2yB +(l-p)2yG]

so that it is impossibleto find a dividend payment that would be chosenonly by a high-cash-flow firm. Moregenerally, a centraldifficulty with the

theory outlined here is that a dividend policy is in essencea \"promise to

pay\" and not an \"obligation to pay.\" The signaling theory outlined hereassumesthat when a firm has announceda future dividend of d, then it

always sticks to its commitment ex post.But it is not obvious that it is in

8.The condition that d < 1is stronger than necessary.It is possible to allow for higher valuesof d, but then our expressions for the manager's payoff would be changed. Also, if d is toolarge, the policy is not credible because the firm would not be able to borrow the full amountneeded in caseof cash-flow shortfall.

125 Static Bilateral Contracting

the firm's interestnot to renegeon its promise.Toestablishthat it is in thefirm's interest to stick to its word, the theory needsto beenrichedby

bringingin reputation considerations.

3.3 Summary and LiteratureNotes

As the analysis in this chapter highlights, signaling modelsinvolve moresophisticatedgame-theoreticarguments than screeningmodels.In the latterclassof modelsthe principal simply solvesan optimization problemconstrained by a set of incentive constraints. That is, the revelationprincipleand the fact that the uninformed party movesfirst allow for a simpletransformation of the bilateralcontracting probleminto a decisionproblem.

In contrast, in signaling modelsit is the informed party who movesfirst.

This fact enrichesthe set of equilibrium outcomesbecausemanyconditional beliefsof the uninformed party canbeself-fulfilling. Themultiplicityof equilibria is due to the fact that Bayes'rule providesno restrictionsonbeliefsfor zero-probabilitycontractoffers\"off the equilirium path.\"

In an attempt to providesharperpredictionson likely equilibriumoutcomes in signaling games,a large literature on equilibrium refinementshas emergedfollowing Spence'scontribution. The basicapproachof this

literature is to introduce some noise into the contracting game, which

guaranteesthat all \"off the equilibrium\" paths are reachedwith positiveprobability, so that beliefscan be tied down everywhereusing Bayes'rule

(seevan Damme,1983,for an extensivetreatment of equilibrium

refinements).Themost popularrefinement is the one by Choand Kreps(1987).

It is basedon the following heuristic argument: when a deviation fromequilibrium behavior is observed that is dominated for some types of theinformed party but not others,then this deviation should not be attributed

to the types for which it is dominated.Application of this refinement cutsdown the equilibrium set, often to a unique refined equilibrium. For

example,in the original modelby Spenceit uniquely selectsthe \"least-costseparatingequilibrium.\" As we have shown, although this refinement hasconsiderableintuitive appeal,it may selecta Pareto-inefficientequilibrium.In contrast, when oneallowsthe uninformed party to restrict the setofcontracts from which the informed party can chooseprior to the signaling

action,as in Maskin and Tirole(1992),then the least-costseparatingequilibrium is the unique perfect Bayesianequilibrium only when it is also

126 Hidden Information, Signaling

Paretoefficient.When it isnot, all incentive-compatiblepayoffs that Pareto-dominate the least-costseparatingpayoffs may be supportedas perfectBayesianequilibria.

The influence of signaling modelsin economicsis considerableand on

par with the screeningmodels consideredin Chapter 2.Remarkably,signaling ideashave also influenced subjectsas far afield as evolutionary

biology. An extremely successfulapplication of signaling theory, for

example,has been to explain the striking yet unwieldy shape of malepeacocktails!A widely acceptedexplanation nowadays is that peacocktails

signal strength, for only the fittest peacockswould be able to survive with

such a handicap (seeGrafen,1990;Zahavi, 1975).In economics,signaling ideas have found prominent applications,

for example,in labor and education,corporate finance, and industrial

organization:

\342\200\242 Labor:Following the pioneeringwork of Spence(1973,1974),educationhas beenseenasnot only a human-capital-enhancing activity but alsoas aselectiondevice(see,e.g.,Weiss, 1983,for a model combining both

elements). Someof the subsequenttheoreticalliterature on educationas asignal has beenconcernedwith robustnessissuesof the basictheory

proposed by Spence.For example,Noldekeand van Damme (1990)have

exploredthe issueof the value of educationas a signal when prospectivejobapplicants cannot commit to a particular duration of education.Otheraspectsof labor markets involving elementsof signaling have also beenexplored.It has beenargued,for instance,that a spell of unemploymentmay bea way of signaling worker productivity, for only productive workerscan afford to stay without a jobfor long (seeMa and Weiss, 1993).\342\200\242 Corporate Finance:Besidesthe classiccontributions of Bhattacharya(1979)and Myersand Majluf (1984)discussedin this chapter,otherimportant examplesof signaling behaviorby corporationsin financial marketshave beenexplored.Tojust name the first applicationofsignaling in finance,Lelandand Pyle (1977)show how a risk-averseowner-managercan signalthe underlying quality of the firm it is floating in an initial public offering

(IPO)by retaining a substantial undiversified stake in the firm. Thisis aneffectivesignal becauseowner-managersof low-quality firms facegreatercostsof retaining largestakesin their firm. Each of theseclassiccontributions has spawnedits own literature testing the robustnessof the proposedtheory. For example,Brennan and Kraus (1987),Constantinides and

127 Static Bilateral Contracting

Grundy (1989),and Goswami,Noe, and Rebello (1995),among severalother studies, explorethe robustnessof the \"pecking-ordertheory\" ofcorporate finance implied by the analysis ofMyersand Majluf. Similarly, Johnand Williams (1985)and Bernheim (1991),among other studies, extendBhattacharya'ssignaling theory ofdividends to include,among other things,dividend taxation. Finally, Welch (1989)and Allen and Faulhaber (1989),among others,have extendedLelandand Pyle'stheory of signaling in IPOs.\342\200\242 Industrial Organization: Limit (or predatory) pricing, introductory

pricing, and advertising are threeprominent examplesofsignaling in

industrial organization. Theclassiccontribution by Milgrom and Roberts(1982)has put new life in an old discreditedtheory of entry prevention throughlow pricing by an incumbent monopolist.The authors show how low pricesmay bea signal of low costor low demand.Theirarticlehas spawneda largeliterature exploring various extensionsand robustnessof the theory (seeTirole,1988,for a discussionof this literature).As for advertising, Milgromand Roberts(1986a)show how a firm selling a high-quality goodcansignalthe quality of its product through wasteful advertising expenditures.Likethe peacock'stail, only a firm sellinghigh-quality goodscanbear the burdenof such expenditures.In an important later contribution BagwellandRiordan (1991)show how introductory pricing can act as a signal in a way

similar to advertising (seeBagwell,2001,for a survey of the substantial

literature on pricing and advertising as signals of quality that has beenspawnedby theseearly contributions).

Other examplesof signaling in industrial organization that are moredirectly linked to contracting includeAghion and Bolton (1987),Aghionand Hermalin (1990),and Spier(1992),among others.Thesecontributions

take the form of the contract to be the signal. Aghion and Bolton,for

example,show how short-term or incompletecontracts offered by aninformed sellerto buyers can signal a low probability of future entry byrival sellers.Similarly, Spiershows how complexcompletecontractoffers,like prenuptial agreements,may neverbemadein equilibrium, for they maysignal a tough-minded streak in the party making the proposaland thus

scare off the other contracting party. Aghion and Hermalin explorehow

mandatory legalrules, such as bankruptcy laws, can improve on incompletecontractual outcomesthat result from a signaling game at the contractnegotiation stage.

HiddenAction, MoralHazard

In this chapterwe analyze the following contracting problembetweenaprincipal and an agent:the principal hires the agent to perform a task; the

agent choosesher \"effort intensity\" a, which affects\"performance\" q.Theprincipal caresonly about performance.But effort is costly to the agent,and the principal has to compensatethe agent for incurring thesecosts.Ifeffort is unobservable,the bestthe principal can do is to relatecompensation to performance.Thiscompensationschemewill typically entail a loss,sinceperformanceis only a noisy signal of effort.

This classof principal-agentproblems with moral hazard has beenwidely usedas a representationof various standard economicrelations.Among the most well-known applicationsare the theory ofinsuranceunder\"moral hazard\" (Arrow, 1970,and Spenceand Zeckhauser,1971,provideearly analysesof this problem);the theory of the managerial firm (Alchianand Demsetz,1972;Jensenand Meckling,1976;Grossmanand Hart, 1982);optimal sharecroppingcontractsbetweenlandlordsand tenants (Stiglitz,

1974;Newbery and Stiglitz, 1979);efficiency wage theories (Shapiroand

Stiglitz, 1984);and theoriesof accounting (seeDemskiand Kreps,1982,fora survey).Thereare ofcoursemany otherapplications,and for eachof them

specificprincipal-agentmodelshave beenconsidered.Thebasicmoral hazard problemhasa fairly simple structure, yet general

conclusionshave beendifficult to obtain. As yet, the characterizationofoptimal contractsin the contextof moral hazardis still somewhat limited.

Very few generalresultscan be obtainedabout the form of optimalcontracts. However,this limitation hasnot preventedapplicationsthat usethis

paradigm from flourishing, as the short list in the precedingparagraphalreadyindicates.Typically, applicationshave put more structure on themoral hazard problemunder consideration,thus enabling a sharpercharacterization of the optimal incentive contract.Thischapterbeginsby

outlining the simplest possiblemodelof a principal-agentrelation,with onlytwo possibleperformanceoutcomes.Thismodelhas beenvery popularin

applications.Another simple model,which we considernext, is that of anormally distributed performancemeasuretogetherwith constant absoluterisk-aversepreferencesfor the agent and linear incentive contracts.Following the analysis of these two specialmodels,we turn to a moregeneralanalysis, which highlights the centraldifficulty in deriving robust predictionson the form of optimal incentive contractswith moral hazard.Todo so,webuild in particular on the classicalcontributions by Mirrlees(1974,1975,1976),Holmstrom (1979),and Grossmanand Hart (1983a).We then turn

130 Hidden Action, Moral Hazard

to two applications,relating to managerial incentive schemesand the opti-mality of debt as a financial instrument, respectively.

4.1Two PerformanceOutcomes

Supposefor now that performance,or output q, can take only two values:q e {0,1}.When-g = 1the agent'sperformanceis a \"success,\" and when q=0 it is a \"failure.\" Theprobability ofsuccessisgiven by Pr(g= IIa) =p(a),which is strictly increasingand concavein a.Assume that p(0)=0, p(\302\260\302\260)

=1,and p'(0)> 1.Theprincipal'sutility function is given by

V(q -w)whereV(')> 0 and V'(.')-0-The agent'sutility function is1

u(w) -\\j/(a)

whereu'(-)> 0, u\"{-) < 0, i^'(')> 0\302\273

and\302\245\"(')

-0-We can make theconvenient simplifying assumption that \\\\f{a)

=a,which doesnot involve muchlossof generality in this specialmodel.

4.1.1First-Bestversus Second-BestContracts

When the agent'schoiceof actionis observableand verifiable, the agent'scompensationcanbemade contingent on actionchoice.The optimalcompensation contract is then the solution to the following maximization

problem:2

max p(a)V (1-wx)+[1-p(a)]V(-w0)

subjectto

p(a)u(wl) +[1-p(a)]u(wQ)-a>uwhereu is the agent'soutsideoption.Without lossof generality wecansetu = 0.Denotingby X the Lagrangemultiplier for the agent's individual-

1.It is standard in the literature to assume that the agent's utility is separable in income andeffort. This assumption conveniently eliminates any considerations of income effects on the

marginal cost of effort. In the general model without wealth constraints it guarantees that theagent's individual rationality constraint is always binding at the optimum.

131 Static Bilateral Contracting

rationality constraint, the first-orderconditionswith respectto wi and w0

yield the following optimal coinsurance,or so-calledBorch rule, betweenthe principal and agent (seeBorch,1962):

Vd-Wi)= V'(-w0)u'(wi) u(w0)

The first-ordercondition with respectto effort is

p\\a)[V{l -wx)-V(-w0)]+Xp/(a)[u(w1) -u(w0)]-X =0

which, togetherwith the Borchrule, detenninesthe optimal actiona.

Example 1:Risk-Neutral Principal [V(x) = x] The optimum entails full

insurance of the agent, with a constant wagew* and an effort levela* suchthat

1u(w*) =a* and p'(a*)= \342\200\224\342\200\224.\342\200\224-

u (w*)

That is, marginal productivity of effort is equated with its marginal cost(from the perspectiveof the principal, who has to compensatethe agent forher costa*).

Example 2:Risk-Neutral Agent \\u(x) =x] The optimum entails full

insurance of the principal, with

wf -wJ =1 and p'(a*)=1Onceagain,marginal productivity ofeffort is equatedwith its marginal costfor the principal.

When the agent'schoiceofactionis unobservable,the compensationcontract cannot bemadecontingent on actionchoice.Thenthe agent'soutput-

2.Alternatively, the maximization problem can beformulated as

max{p(a)u(w-i) + [1-p(a)]u(w0)-a}

subject to

p(a)V(l -Wl)+[1 -p(a)]V(-w0) >V

By varying V or u it is possible to allow for any division of surplus between the principal and

agent. This is auseful shortcut that allows us to separate the analysis of the form of the optimalcontract from the bargaining game between principal and agent.

Hidden Action, Moral Hazard

contingent compensationinducesher to choosean actionto maximize herpayoff:

maxp(a)u(wi) +[1-p(a)]u(w0)-aa

The second-bestcontractis then obtainedas a solution to the following

problem:

max p(a)V(V- wi) +[1-p(a)]V(-w0)a,w,

subjectto

p(a)u(wi) +[1-p(a)]u(w0)-a>0and

aeaigmaxp(a)u(wi) +[l-p(a)]u(wQ)-a (IC)a

Thefirst-ordercondition of the agent'soptimization problemisgiven by

Jp/(a)[\"(wi)-w(w0)]=l (4.1)Givenour assumptions onp(-)and u(-),there is a unique solution to this

equationfor any compensationcontract(w0; w{).We can thereforereplacethe agent's incentive constraint (IC)by the solution to equation(4.1).Ingeneral,replacingthe agent'sincentive constraint by the first-orderconditions of the agent's optimization problem will involve a strict relaxationof the principal'sproblem,as we shall see.However,in this specialtwo-outcomecasethe principal'sconstrainedoptimization problem remains

unchanged following this substitution. Thissubstitution simplifies theanalysis enormously, as will becomeclear subsequently. We begin by analyzingthe second-bestproblem in two classicalcases:in the first case,principaland agent are risk neutral, but the agent facesa resourceconstraint; in thesecondcase,at leastone of the contracting partiesis risk averse.

The SecondBestwith BilateralRisk Neutrality and ResourceConstraints

Agent

When the agent isrisk neutral, sothat u(x) =x, first-best optimality requiresthatp'(a*)=l.Thefirst-ordercondition of the agent'soptimization problemthen alsobecomesp\\a){wx-w0) = 1,so that the first-best actioncouldbe

Static Bilateral Contracting

implemented with w\\-

w% = 1.This solution can be interpreted as an

upfront \"sale\" of the output to the agent for a price-w%> 0.If w0 =0 and w1-w0=l,the agent would obtain an expectedpayoff equal

to

p(a*)-a*which is strictly positive, sincep\"(a) is strictly negative and p'(a*)= 1.Bycontrast,the principal would obtain a zero payoff, sincehe is selling the

output upfront at a zeroprice.A risk-neutral principal facedwith theconstraint w0> 0 would thereforechoosew0 =0 and, facedwith Wi =l/p'(a)inthe second-bestproblem,would choosea to solvethe following problem:

maxp(a)(l-wi)a

subjectto

p'(a)wi =1Solving this problemyields

p(a)=l \342\200\224

[p\\a)f

As can be readily checked,the solution to this equation is smaller than

#*.Thisresult is intuitive, sinceinducing more effort provision by the agenthere requiresgiving her more surplus.

Simple Applications

Theresultsjust presentedcanbeinterpretedin severalways. For example,one can think of the agent as a manager of a firm and the principal aninvestor in this firm. Then wx is interpretedas \"inside\" equity and (1-w{)as \"outside\" equity (Jensenand Meckling, 1976);or, still thinking of the

agent as a manager and wx as inside equity, (1-w{) can be thought of asthe outstanding debt of the firm (Myers, 1977);an alternativeinterpretation is that the agent is an agricultural laborerunder a sharecroppingcontract (Stiglitz, 1974).Under all these interpretations a lower wx reducesincentives to work, and it can evenbecomeperversefor the principal,generating a form of \"Laffer curve\" effect.Thiscan mean, for example,that

134 Hidden Action, Moral Hazard

reducing the facevalue of debt can increaseits real value by reducing thedebt burden \"overhanging\" the agent (Myers,1977).

The standard agencyproblemassumesthat actionsareunobservablebut

output is observable.But for somecontracting problemsin practiceevenperformancemay bedifficult to observeor describe.Tocapturethesetypesof situations some applicationsinvolving moral hazard assumethat

contracting on q is too costly but actionscan be observedat a cost through

monitoring. One important exampleof suchapplicationsis the efficiencywagemodel(Shapiroand Stiglitz, 1984).In this modelthe focusis on tryingto induceeffort through monitoring. Assume, for example,that effort canbe verified perfectly at a monitoring expenseM. The full monitoring

optimum then solves

maxp(a)-w-M

subjectto

w-a>0and

w>0

which yields w* = a* and p'(a*)=1.But supposenow that the principal is able to verify the agent'saction

with probability 0.5without spending M. If the agent is found shirking, it is

optimal to give her the lowestpossiblecompensation.In this case,with alimited wealth constraint, w > 0, the variablew isset equalto 0.If the

principaldecidesnot to spendM, his problemis

max p(a)-w

subjectto

w-a>0.5wTheLHSof the incentive constraint is the agent'spayoff if shechoosesthe

prescribedactiona.TheRHSis the agent'smaximum payoff if shedecidesto shirk. In that caseit is bestnot to exert any effort and gamble on the

possibility that the agent will not becaught. Now the principal has to givethe agent a compensationthat is twice her effort level.In other words, the

principal gives the agent rents that she would losewhen caught shirking.

Static Bilateral Contracting

Having to concederents onceagain lowersthe principal'sdesireto induceeffort, sothat optimal effort is lowerthan a*.Finally, the choiceofwhether

or not to monitor dependson the sizeof M.

BilateralRisk Aversion

Let us now return to our simple theoreticalexample.In the absenceof riskaversionon the part of the agent and no wealth constraints, the first bestcan be achievedby letting the agent \"buy\" the output from the principal.In contrast,if only the agent is risk averse,the first-best solution requiresaconstant wage, independent of performance.Of course,this completelyeliminates effort incentives if effort is not observable.Optimal risk sharingunder bilateral risk aversiondoesnot provide for full insurance of the

agent, but risk aversionon the part of the agent still preventsfirst-bestoutcomes under moral hazard in this case.Indeed,the principal then solves

ma.xp(a)V (1-wi) +[1-p(a)]V(-w0)a,w,

subjectto

p(a)u(w1) +[l-p(a)]fi(wQ)-a>0 (IR)

and

p'(a)[\"(wi)-K(w0)]=l (IC)

Letting X and fi denote the respectiveLagrangemultipliers of the (IR)and (IC)constraints and taking derivatives with respectto w0 and wx yields

V'\302\260--Wl)-X\\p'{a)

\302\253Vi) p(a)

and

V'(-w0)^. p\\a)

\302\253Vo) l-p(a)When fi = 0,we obtain the Borchrule.However,at the optimum \\i > 0

under quite generalconditions,aswe shall show, so that optimal insuranceis distorted:the agent gets a larger (smaller)share of the surplus relativeto the Borchrule in caseof high (low) performance.Specifically,in orderto induce effort, the agent is rewarded (punished) for outcomeswhosefrequency rises(falls) with effort. In our two-outcomesetting, this result is

Hidden Action, Moral Hazard

particularly simple:q =1is rewarded,and q = 0 is punished.Paradoxically,the principal perfectlypredictsthe effort levelassociated

with any incentive contract and realizes that the reward or punishment

correspondsonly to good or bad luck. Nevertheless,the incentive schemeis neededto induceeffort provision.

The Value of Information

Assume now that the contractcan be made contingent not only on q but

also on another variables e {0,1}.This variablemay be independentofeffort (think of the state of the businesscycle)or may dependon it (thinkofconsumersatisfaction indices,if the agent is a salesperson).However,thevariables doesnot enter directly into the agent'sor the principal'sobjective functions. Specifically,assumePr(g = i, s =j I a) =pi,(a).The principalcan now offerthe agent a compensationlevelwi}, to solve

i i

max XX Pv (fl)^'-wn)a'w\" i=0;=0

subjectto

i i

XX#;(fl)\"K)^ (IR)i=0 ;'=0

Xi>>)\302\253K)=i (ic)i=0 ;=0

Again, letting X and fi denote the respectiveLagrangemultipliers of the

(IR) and (IC)constraints, and taking derivatives with respectto wi;-, oneobtains the following first-orderconditionswith respectto wi;-:

V'd-Wij) pljia)\342\200\224 =X +

fj,\342\200\224\342\200\224

u'(wij) pti(a)

Hencethe variables dropsout of the incentive scheme(i.e.,w;;-= wt for

i =0,1)if for all a

\342\200\224tt=

\342\200\224TT

for* =0,lPmW Pawwherea denotesthe second-bestactionchoice.

Integrating theseconditionsyields

137 Static Bilateral Contracting

Pio (a)=kipa (a) for i =0,1where the

fc\302\243'sare positiveconstants.The condition for s to beabsentfrom

the agent'sincentive schemeis thus that, for eachq, changing effort yieldsthe samechangein relativedensity whatever s may be.When this issatisfied, one saysthat q is a \"sufficient statistic\" for (q,s) with respectto a, orthat s isnot \"informative\" about a given q.

When q is not a sufficient statistic for a, taking s into accountimproves the principal'spayoff by allowing for a more precisesignal abouteffort, and thus a more favorabletrade-offbetweeneffort provision andinsurance.

The sufficient statistic result is due to Holmstrom (1979)(seealsoShavell,1979a,1979b;and Harris and Raviv, 1979).It extendsto generalsettings, as will be shown later on in this chapterand in Chapters6 and 8.

4.2 LinearContracts,Normally DistributedPerformance,and ExponentialUtility

Next to the two-performance-outcomecase,another widely usedspecialcaseinvolves linear contracts,normally distributed performance,andexponential utility. Performanceis assumedto be equalto effort plus noise:q =a + e,wheree is normally distributed with zero mean and variancea2.Theprincipal isassumedto berisk neutral.Theagent has constant absoluterisk-averse (CARA) risk preferencesrepresentedby the following negativeexponentialutility function:

u(w,a) =-e-T][w-^a)]

wherew is the amount of monetary compensationand 77 > 0 is the agent'scoefficientof absoluterisk aversion(77

=-u\"lu').Note that in contrast with

the earlierformulation, effort costhere ismeasuredin monetary units. For

simplicity the cost-of-effortfunction is assumedto bequadratic: \\\\f{a)=\\ca2.

Supposethat the principal and agent can write only linear contractsofthe form

w =t+sqwhere t is the fixed compensationleveland s is the variable,performance-

138 Hidden Action, Moral Hazard

related componentof compensation.The principal'sproblem is then tosolve

max E(q-w)a,t,s

subjectto

E{-e-n[w-^a)])>u(w) (IR)

and

a earg max JE(-e-\"[^(fl)]) (IC)a

whereu(w) is the reservationutility levelof the agent and w denotestheminimum acceptablecertainmonetary equivalent of the agent'scompensation contract.

Maximizing expectedutility is equivalent to maximizing

=(_e-\"('+--ica2))\302\243(e-^)

Moreover,when a random variable e is normally distributed with zeromeanand variancea2,we have,3 for any y

\302\243(e^)=erV/2

Therefore,maximizing expectedutility is equivalent to maximizing

_eMt+S\"-2Ca2-2sZ,j2) =_g-U*(\302\253)

3. Indeed, E(en) is then equal to

i JL i [<?-mJ-)_\302\261_r%r\302\253g i^de=-TL=ri ** de-Jim J~\302\260 -12na J-

-llna J~\302\260 -Jlna J~\302\260

This last expression is equal to ey \302\260 n since

L=ri~^~dElira J\342\200\224

\342\226\240Jlita

is the area under a normal distribution with mean ycr2 and variance a2, that is,1.

139 Static Bilateral Contracting

wherew(a) is the certainty equivalent compensationof the agent, which isthus equalto her expectedcompensationnet ofher effort costand of a risk

premium, which, for a given s, is increasingin the coefficientof riskaversion 7] and in the varianceof output a2.It is alsoincreasingin s,sincethe

higher the s, the more the agent bearsthe risk associatedto q.Thus the exponentialform of the expectedutility function makes it

possibleto obtain a closed-formsolution for the certainty equivalent wealth

w(a) here.An explicit solution is generally not obtainablefor otherfunctional forms.

The optimization problemof the agent is therefore:

a earg maxw(fl) = t +sa\342\200\224 ca2\342\200\224s2o21 1which yieldsthe simple result a = s/c.This equationgives us the agent'seffort for any performanceincentive s.Knowing that a =sic,the principalsolves

s ( s2max\342\200\224 t +\342\200\224

t,s C V C

subjectto

S2 C S2 77 , , _t + \342\200\224-s2o2 =w

c 2 c2 2

Thisderivation yields

1s= -1+riccr

Effort and the variable compensationcomponentthus go down when c(cost of effort), J] (degreeof risk aversion),and o2(randomnessofperformance) go up, a result that is intuitive.

4.3 The Suboptimality of LinearContractsin the ClassicalModel

The previouscaseis attractive in that it leadsto a simple and intuitive

closed-formsolution under standard assumptions: normally distributed

performanceand a CARA utility function for the agent.Unfortunately,

however,linear contractsare far from optimal in this setting.Toseethis point, considerfirst a simple casewhereperformanceis such

that q =a +e and wherethe support of e is given by [-k,+k],where \302\260o > k> 0.Supposefor simplicity that e isuniformly distributed on this interval.

Through her choiceof action,the agent can then move the support of q.Thisseems,at first glance,to bea reasonabledescriptionof the agent'stechnology. Unfortunately, under this specification,the agency problemdisappears altogether:the first bestcan always beachieved.Calla* the first-bestaction,and w* the first-best transfer. With boundedsupport, the principalcan rule out certainperformancerealizations,providedthe agent choosesa*.Thus by punishing the agent very severelyfor performanceoutcomesoutsideof [a*-k, a* +k] the principal can give the right incentives to the

agent.At the sametime, the principal canperfectlyinsure the agent, sincewhen the latter choosesa*, she has a constant transfer w* irrespectiveofthe performancerealizationsin [a*-k,a*+k].

One responseto this observationis to assume that e has a fixed orunbounded support and thus avoid having perfectlyinformative signalsabout the agent'sactionchoice.As Mirrlees(1975)has shown, however,performance signals may be arbitrarily informative even when e hasunbounded support.This is the case,for example,when e is normallydistributed.

Mirrleesconsidersthe same setting as before:

q =a+e, where e ~N(0,a2)

Now, with a normal density we have

f(q\\a) da crIn other words, the likelihoodratio fjfcan.take any value betweenplus

and minus infinity. That is, the principal canget an almost exactestimate ofa at the tails of the distribution. Mirrleesshows that this information canbeusedto approximate the first bestarbitrarily closely.That is, the principalcanchooseq such that, for all q < q, the transfer to the agent w(q) is verylow (a form of extremepunishment), but for q > q the transfer is fixed at

w(q) = w* + 8, slightly higher than the first-best wage level.Under such acompensationschemethe agent facesa negligible risk of getting punishedwhen shechoosesa*,sothat her (IR) constraint issatisfied if sheisoffereda fixed wage w* + 8,with 8positive but arbitrarily small.

141 Static Bilateral Contracting

Toseethis point, let Kbe the (very low) transfer the agent receiveswhen

failing to achieveoutput of at least q.For any arbitrary q the value of Kcanbe set soas to inducethe agent to choosethe first-best levelof efforta*, that is,

j^u(K)fa(q,a*)dq+^u(yv*)fa(q,a*)dq=yf\\a*)

where \\ff(a) is the (convex)cost-of-effortfunction. This incentive schemeviolatesthe agent's(IR) constraint only by the positive amount

Jl[u(w*)-u(K)]f(q,a*)dq

However,we know that for any arbitrarily largeM, there existsa q low

enough such that

ftt<-\"f(q\\a)

for all q < q.Therefore,the shortfall to meet the (IR) constraint is lessthan

^\\r[u(w*)-u(K)]fa(q\\a*)dq

which, using the incentive constraint, isequal to

Finally, as

fa(q\\a*)=(^:p-f(q\\a*)

this expressionis given (and finite) for a given M and tends to zero for M

large enough.As a result, the first bestcan be approximatedby settingextremepunishments that are almost surely avoidableprovidedthat the

agent choosesthe optimal actiona*.Unboundedpunishments areofcoursecrucialfor the argument here.It is paradoxicalthat they are usedagainst arisk-averseagent\342\200\224and they are:the first bestcanbeapproximatedbut not

142 Hidden Action, Moral Hazard

achieved;a constant w* generatesno effort whatsoever!But the key insightis that, as the sizeof the punishment grows, its relative occurrencefalls veryfast.

While the result of this subsectioncasts a shadowover the linear\342\200\224

CARA\342\200\224normal model,Holmstrom and Milgrom (1987)have identified

conditionsunder which linear contracts are optimal. Beyond assumingCARA preferences,they considera dynamic modelwhere effort is chosenin continuous time by the agent.TheHolmstrom-Milgrom result, discussedin Chapter10,providesfoundations, under admittedly specificassumptions,to this user-friendly model discussedin the previoussubsection,and has

partly contributed to its popularity.

4.4 GeneralCase:The First-OrderApproach

4.4.1Characterizingthe SecondBest

As the previoussubsectionhashighlighted, linear incentive schemes,while

easy to analyze, are,however,not optimal in general.Following Mirrlees(1974,1975,1976)and Holmstrom (1979)in particular, we now turn to thecharacterizationof generalnonlinear incentive schemesin a frameworkwhereperformancemeasuresare describedin broad terms as q = Q(6,a),where6e 0issomerandom variable representingthe state of nature, anda e A c !Rdenotesthe agent'seffort or action.The principal may be riskaverseand has a utility function given by

V(q-w)

whereV'(-)> 0 and V'\\-) < O.Theagent is risk averseand alsoincurs privateeffort costs.Herutility function takes the generalseparableform

u(w) -iff (a)

where u'{-)> 0, u\"{-) < 0, \\ff'(-) > 0, and!//\342\200\242\"(\342\226\240)

> 0.4As in the two-outcomecase,it is convenient to formulate the optimal contracting problemby

specifying a probability distribution function overoutput that is conditional onthe agent'sactionchoice.One advantage of this formulation is that it givesvery intuitive necessaryconditionsfor an optimal contract.Supposethat

performanceis a random variable q e [q,q] with cumulative distribution

4. It is standard in the literature to assume that the agent's utility is separable into incomeand effort.

143 Static Bilateral Contracting

function F(q I a),wherea denotesthe agent'sactionchoice.Let theconditional density of q bef(q I a).Then the principal'sproblemcanbe written

as

max \\9V[q-w(q)]f(q\\a)dq

subjectto

f u[w(q)]f(q|a)dq-yr(a) > u (IR)J\302\243

and

a eargmaxj [ u[w(q)]f(q \\ a)dq-yia) \\ (IC)aeA lJ\302\243

J

As in the casewith only two output outcomes,it is tempting to replacethe (IC)constraint by the first- and second-orderconditionsof the agent'sproblem:

P u[w(q)]fa (q|a)dq=\\f/'(a) (ICa)

J\302\243

f uMqWm (q I a)dq-y\"(a) < 0 (ICb)

In fact, what many authors do is to leaveasidethe second-ordercondition (ICb)and replace(IC)by condition (ICa)only. Proceedingin this way

and replacingthe (IC)constraint by the agent'sfirst-ordercondition, weobtain the Lagrangean

L=f Mq-w(q)]f(q|a)+X[u{w{q))f{q \\ a)-\302\245(a)

-u]1+ll[u(w(q))fa (q\\a)-y/\\a)fidq

Differentiating with respectto w(q) insidethe integral sign, we obtain the

by-now-familiar first-order conditions for the principal's problem.An

optimal incentive compensationschemew(q) is such that for all q

Vlq-w(q)]= fM*l (42)uWq)]

*f(q\\a)

K )

144 Hidden Action, Moral Hazard

Theseconditionsreduceto Boren'srule for optimal risk sharing if ji =0.However,intuition suggeststhat generally fi = 0 cannot be an optimumwhen a is not observedby the principal.Indeed,as a way of inducing the

agent to put in higher effort, onewould expectthe principal to pay the agentmore for higher g'sthan would be optimal for pure risk-sharing reasons.Implicit in this intuition, however,is the idea that higher effort tendsto raiseoutput. Thisassumption is true, however,only if one assumesat least that

the conditional distribution function over output satisfiesfirst-orderstochastic dominance:Fa(q I a) < 0 with a strict inequality for somevalue of q.Thiscondition meansthat, for all q e [q, q],there is always a weakly (andat times strictly) lowerprobability that q < q when effort is higher (seeFigure 4.1for an illustration). Note that in the two-outcomecase,first-orderstochasticdominanceis implied by the assumption that effort raisesthe

probability of success.

F(qla) a

Figure 4.1First-Order Stochastic Dominance

145 Static Bilateral Contracting

With this additional assumption on F(-) and our other assumptionson y/(a) we can show that fi > 0 if the agent is risk averse(if the agent isrisk neutral, shecanbemaderesidualclaimant and moral hazard is not anissue,sincewe donot have resourceconstraints here).Todo so,proceedbycontradictionand assumeinstead that \\i < 0.Take the derivative of the

Lagrangian with respectto a:

f Mq~w(q)]fa (q\\a) +X[u(w(q))fa (q\\a)-y'(a)])l+n[u(w(q))faa (q\\a)-y\"(a)]}dq=0

Using the first- and second-orderconditionsof the agent'soptimizationproblem(ICa)and (ICb),we note that fi < 0 isequivalent to

\\qV[q-w(q)]fa(q\\a)dq<0 (4.3)lCallthe first-best wage schedulewFB(q).By Borchrule, wFB(q) and q -

wFB(q) are (weakly) monotonically increasingin q.When \\i < 0,by (4.2),thesecond-bestwage will beweakly belowwFB(q) for output levelsq such that

fa(q I a) > 0,and it will beabovewFB(q) for output levelsq such that fa(q I a)< 0.Consequently,we have

\\qV[q-w(q)]fa(q\\a)dq>\\qV[q-wFB(q)]fa(q\\a)dq (4.4)1 1However, since Fa(q,a)=Fa(q,a)=0 for all a, integration by parts

implies

\\qV[q-wFB(q)]fM\\a)dq =-\\qVf[q-wFB(q)][l-wFB(q)]Fa(q\\a)dq(4.5)\302\243 \302\243

Moreover,differentiating the optimality condition (4.2)with fi = 0yields

' ( n V\"[q-wFB{q)\\WFBW

Xu\"[wFB(q)] +V\"[q-wFB(q)]

146 Hidden Action, Moral Hazard

When the agent is risk averse(that is, u\" < 0),this implies 1> w'(q)> 0(which simply means that a risk averseagent should not be fully residualclaimant in an optimal risk-sharing contract).SinceV'[q -w(q)]> 0, first-

orderstochasticdominanceimplies that (4.5)isstrictly positive and, in turn,that conditions(4.3)to (4.5)are incompatible.This contradictionprovesthat fi must be positive at the second-bestoutcome.Thisalsomeans that

the principal would like to raise the effort of the agent in the second-bestcontract.

When fi > 0, the first-orderconditionsof the principal'sproblemwith

respectto w(q) summarize the trade-offbetween risk sharing andincentives. Tobetter interpret theseconditionsit is helpful to considerthe specialcasewherethe agent canchooseonly among two levelsof effort, aH and aL(aH > aL).Then,assuming that it is optimal to elicitaH, the first-orderconditions are rewritten as

ttqWYf(q\\aH).

If the principal is risk neutral, this equationsimplifies to

u'[w(q)]

Thisequationtellsus that w(q) ishigher if f{q\\aL)j'f{q\\aH)< 1:the agentgetsa higher transfer for an output performancethat is more likely underaH. Vice versa,if f(q\\aL)/'f(q\\aH)>l,shegetsa lowertransfer than under

optimal risk sharing. The agent is thus punished for outcomesthat revisebeliefsabout aH down, while sheisrewardedfor outcomesthat revisebeliefsup. As Hart and Holmstrom (1987,page80)have put it:

The agency problem is not an inference problem in a strict statistical sense;conceptually, the principal is not inferring anything about the agent's action from q

V'[q-w(q)]u'\\w{q)\\

=h +fl

1-f(q\\aL)f(q\\aH).

147 Static Bilateral Contracting

becausehe already knows what action is being implemented. Yet, the optimalsharing rule reflects preciselythe pricing of inference.

Becausethe transfer function w(q) is directly related to the likelihoodratio, it will be difficult to derive strong properties about this function

without making strong assumptions about F{q I a).Thus, w(q) isnotmonotone in generalevenunder first-orderstochasticdominance,Fa(q I a) < 0.Toseethis fact considerthe following example.

Example:Nonmonotone Transfer Function Supposethat there are onlythree possibleperformancerealizations(qL< qM< <?h), and that the agenthas two possibleeffort levels{aL< aH).The conditional densitiesf(q I aH)and j\\q I aL) are given in the following table:

KQl\\o) f(qM\\a) f(qH\\a)

aL 0.5 0.5 0.0aH 0.4 0.1 0.5

Here,the second-besttransfer function is such that

w(qH)>w(qL)>w(qM)

The point is that when q = qL it is almost as likely that the agent choseaHasaL.Therefore,the principal doesnot want to punish the agent too muchfor low performancerealizations.

One should expectto have a monotone transfer function only if lower

performanceobservationsindicatea lowereffort choiceby the agent.Thisintuitive requirement correspondsto the statistical assumption of amonotone likelihood ratio property (MLRP).When f(q\\a) satisfiesMLRP,a high-

performancerealizationisgoodnews about the agent'schoiceof effort (seeMilgrom, 1981a).Toseethis point, considerthe necessaryconditionsfor an

optimal compensationin the two-actioncase,when the principal is riskneutral:

Hidden Action, Moral Hazard

u'[w(q)]=h +fl \\_f(q\\aL)

f(q\\aH).

We know that u'(-)> 0,so that

f(q\\aL)'dw n d\342\200\224 >0<&\342\200\224

dq dq .f(q\\aH).<0

This last inequahty is preciselythe MLRP condition.In the continuous-actioncase,the MLRPcondition takes the following form:

d_

dqfa(q\\ay

f(q\\a)>0

To get a monotonic transfer function, we must therefore make strong

assumptions about f(q I a).Toobtain optimal linear incentive schemes,evenstronger assumptions are required.This is unfortunate, as optimal linearincentive schemesare relatively straightforward to characterize.Also, in

practice such schemesare commonly observed(as in sharecropping,for

example).However,one also observesnonlinear incentive schemeslikestockoptionsfor CEOsor incentive contractsfor fund managers.

A final comment on the monotonicity of the transfer function: Perhapsa goodreasonwhy the function w(q) must bemonotonicis that the agentmay beableto costlesslyreduceperformance,that is,\"burn output\". In the

precedingexample,he would then loweroutput from qM to qL wheneverthe outcomeis qM.

When Isthe First-OrderApproachValid?

4.4.2.1An Examplewherethe First-OrderApproachIsnot Valid

We pointedout earlier that in generalone cannotsubstitute the (IC)constraint with the agent'sfirst-ordercondition (ICa).Thefollowing exampledue to Mirrlees(1975)providesan illustration of what cangowrong.5 Mir-rleesconsidersthe following principal-agentexample:

5, This example is admittedly abstract, but this is the only one to our knowledge that addressesthis technical issue.

149 Static Bilateral Contracting

Theprincipal'sobjectiveis to maximize

-(x-1)2-(z-2)2with respectto z.

The agent, however, choosesx to maximize her objective:

u(x,z)=ze-{x+1)2+eAx-1)2

For any z, the first-ordercondition of the agent'smaximization problemis

z(x+l)e-{x+lf+(x-l)e-(*-1)2=0

or

z =~ e1+x

Now the reader can checkthat for z between0.344and 2.903there arethree valuesofx that solvethis equation,one of which is the optimal valuefor the agent.Toseewhich is the optimal value, observethat

u(z, x)-u(z,-x)=-{z-l)(e4x-l)e-{x+1)2

Thus, for z > 1the maximum of u occursfor negative x.When z < 1,themaximum of u occursfor positive x.In eithercase,this observationidentifies the optimal value ofx.When z = 1,u is maximized by setting x =0.957or x =-0.957.

By sketching indifferencecurvesfor the principal

(x-l)2+(z-2)2=K

in a diagram, one finds that the solution of the maximization problemofthe principal is x =0.957and z =1.

Thissolution is not obtainedif one treats the problemas a conventionalmaximization problemwith the agent'sfirst-ordercondition asa constraint.One then obtains insteadthe following first-orderconditionsto the

principal's problem:

(2-z)+(l-x) =0

Solution of the

principal-agent

problem

y 0.957Agent's optimal response

0.364 r

V

Principal's indifference curves

Wrong solution selectedbythe first-order approach

1.99 2.903Other first-order solutions

Agent's optimal response

-0.957

Figure 4.2Solution of \"Relaxed Problem\" under the First-Order Approach

where

dz_j4(l-x*)-2dx { (x+x)2

f2(l-x2)-l

Sothat

2z(2-z)= (l-*2)(l+x)(2x2-l)

There are three solutions x to this equation. One of those, defined byz =1.99and x =0.895,achievesthe highest value for the maximand

-{x-lf-{z-lfbut this solution doesnot maximize u(x, z).This solution is only a localmaximum.6

6. The second solution is ineligible on all possible grounds: it is a local minimum of u(x, z)-The third solution is a global maximum of u(x, z)but doesnot maximize the principal's payoff.

151 Static Bilateral Contracting

Graphically the problem pointed out by Mirrleescan be representedas seenin Figure 4.2.The points on the bold curve representthe solutionsto the agent's maximization problem.These points are globaloptima ofthe agent's problem.The point (x = 0.957;z = 1) is the optimum of the

principal-agentproblem.As the figure illustrates, by replacingthe agent's(IC)constraint by only the first-orderconditionsof the agent'sproblem(ICa),we are in fact relaxing someconstraints in the principal'soptimization problem.As a result, we may identify outcomesthat are actually notattainable by the principal (in this case,the point x =0.895;z =1.99).

4.4.2.2A Sufficient Condition for the First-OrderApproachto BeValid

If the solution to the agent'sfirst-ordercondition is unique and the agent'soptimization problem is concave,then it is legitimate to substitute the

agent'sfirst-ordercondition for the agent's(IC).Rogerson(1985a)gives sufficient conditionsthat validate this

substitution: if MLRP,togetherwith a convexity of the distribution function

condition (CDFC)holds, then the first-orderapproach is valid. The CDFCrequiresthat the distribution function F(q\\a) beconvexin a:

F[q\\fr +0.-Qa']ZCF(q\\a)+0.-QF(q,a')for all a,a'e A, and

\302\243

e [0,1].The two conditions essentiallyguarantee that the agent's optimization

problemis concaveso that the first-orderconditionsfully identify globaloptima for the agent.The following heuristic argument shows why, underMLRPand CDFC,the agent'sfirst-orderconditionsare necessaryandsufficient. Supposethat the optimal transfer function w(q) is differentiablealmost everywhere.7 The agent'sproblemis to maximize

f u[w(q)]f(q|a)dq-y/(.a)

with respectto a e A.

7. Note that since w(q) is endogenously determined, there is no reason, a priori, for w(q) tobedifferentiable.

152 Hidden Action, Moral Hazard

Integrating by parts, we can rewrite the agent's objectivefunction asfollows:

u[w(q)]-Hu'[w(q)\\w\\q)F(q\\a)dq -y/(a)1Differentiating this expressiontwice with respectto a, we obtain

-P u'[w(q)]w\\q)Faa {q\\ a)dq-y/-\"(a)1Then,by MLRP [w'(q)> 0] and CDFC[Faa(q I a) > 0], the secondderivative with respectto a is always negative.

Unfortunately, CDFCand MLRPtogetherarevery restrictive conditions.For instance,noneof the well-known distribution functions satisfy bothconditions simultaneously. Jewitt (1988)has identified conditionssomewhatweakerthan CDFCand MLRPby making strongerassumptions about theform of the agent'sutility function.

Notice,however,that the requirementthat the agent'sfirst-orderconditions benecessaryand sufficient is too strong. All we needis that thesubstitution of (IC)by its first-ordercondition yields necessaryconditionsforthe principal'sproblem.These considerationssuggestthat an alternative

approachto the problemthat doesnot put stringent restrictionson theconditional distribution function F(q\\a) is useful. Suchan approachhas beendevelopedby Grossmanand Hart (1983a).

4.5 Grossmanand Hart'sApproachto the Principal-Agent Problem

This approachrelieson the basicassumption that there are only a finite

number of possibleoutput outcomes.Thus, supposethat there are only N

possibleq{.Q< qi < q2 <...< qN, and let pt(a) denote the probability ofoutcomeq{ given actionchoicea.The agent'sactionsetA is taken to be acompactsubsetof !R\". Tokeepthe analysis simple we take the principal tobe risk neutral: V(-) =q-w.The agent'sobjectivefunction, however, nowtakes the moregeneralform

u(w, a)=<j)(a)u[w(q)]-iff (a)

153 Static Bilateral Contracting

Underthis specification,the agent'spreferencesover incomelotteriesareindependentofher choiceofaction.8Thisutility function contains asspecialcasesthe multiplicatively separableutility function [y(fl) =0 for all a] andthe additively separable utility function [<j>(a)

= 1for all a].Moreover,u[w(g)]is such that u(-) is continuous, strictly increasing,and concaveonthe open interval (w, +\302\260\302\260)

and

limw(w) =-oow\342\200\224\302\273>v

The principal'sproblem is to choosea e A and wt = w(qt) e (w, +\302\260\302\260)

tomaximize

V =^pi(a)(qi-wi)i=i

subjectto the agent's(IR) constraint

iV

^ {<j>(a)u(Wi)-y/(a)}pi (a)> u

i=i

In addition, if a is unobservablethe principal alsofacesthe (IC)constraint

X {(t>(.a)u(wi)-

yr(a)}pi (a)>^ {^(a)w(w\302\243-)-

yr(a)}pi (a)i=i i=i

for all a e A.

Suppose,to begin with, that a is observableto the principal.Sincethe

agent is risk averseand the principal risk neutral, the (first-best) optimalcontractinsures the agent perfectly.The optimal transfer is determinedby

maintaining the agent on her individual rationality constraint. Then the

principal can implement the first bestby setting

8. This is the most general representation of the agent's preferences such that the agent'sparticipation (or IR) constraint is binding under an optimal contract. Under a more generalrepresentation, such that the agent's cost of effort depends on the agent's income or exposure torisk, it may beoptimal for the principal to leave a monetary rent to the agent soas to lowerthe cost of effort and induce a higher effort choice. In that casethe IR constraint wouldnot bebinding. As a consequence the optimal incentive contract would besubstantially moredifficult to characterize.

154 Hidden Action, Moral Hazard

w =u 1 u+y/(a)

and choosinga to maximize

'u+\\ff(a)'V =^Jpl(a)qi-u!<f>(a) J

Now supposethat a is not observableto the principal.The innovation in

this approachis to divide the principal'sprobleminto two stages:first

determine the minimum transfers to the agent that implement any given actiona, then maximize overactions:

Stage1:Implementation Thisstageinvolves solving the following problem

min y.pl(a)wi(wi,- >\302\273w) l=1

subjectto

XPi(a^iaMwi)-y/(a)} > u (IR)

XPi(a^iaMwi)-iff (a)}> JTPi{a){^{a)u{wi)-yr(a)} for all a eA (IC)

This program is solvedfor any actiona e A. Noticethat we have a finite

number of constraints so that the Kuhn-Tucker theorem can be appliedhere.But the program aswe have written it is not necessarilyconcave.Thetrick is then to make a simple transformation and take ut =u(wi) rather than

wt as the principal'scontrolvariables.Definealsoh = u~l and ut =w(w\302\243)

sothat Wt =h(u$.

Thisapproachrequiresan additional technicalassumption. Let

U ={u|u(w)=u for somew e{w,+ \302\260\302\260)}

Thenassumethat

[u+\\f/(a)]

Ha)eU for all a eA

155 Static Bilateral Contracting

In words,for every effort a, there existsa wagethat meetstheparticipation constraint of the agent for that effort.

Then the transformed program is

N

min ^y,pl(a)h(ul)(m, mn) i=1

subjectto

N

XPiWWafc -yr(a)}> u (IR)

iV N

XPi(aK0(fl)\".-iff (a)}>XPi(a){p{a)ui-yf(a)} for all a eA (IC)i=i

We now have linear constraints and a convexobjective,sothat the Kuhn-

Tuckerconditions are necessaryand sufficient.

Let u =(ui,...,uN) and define

C(a)=inf < XPi(fl)Kui) \\

u implements a

Note that for somea e A there may existno u that implements a.Forsuch actions,let C(a)= +\302\260\302\260.

Grossmanand Hart show that whenp\302\243(a)> 0 for i =1,...,N, thereexists

a solution (u%,..., u%) to the precedingminimization problem,so that thecost function C(a) is well defined.As the Mirrleesexamplehighlights, apotential difficulty here is that somecomponentsofu may besuch that h(ui)is unbounded even if C(a)remains bounded.If this is the case,a solutionto the maximization problem may not exist.One can rule this problemout by assuming that pt(a) > 0, so that C(a)alsobecomesinfinitely largewhen h(ui) is unbounded for some iii's.An infinitely largeC(a)cannot beoptimal, so that one can then safely assume that all relevant h(u) areboundedand that the constraint setis compact.Theexistenceof a solutionin the set

X_Pi(fl)Mwi) In implements a

156 Hidden Action, Moral Hazard

then follows by Weierstrass'stheorem.

Stage2:Optimization The secondstepis to choosea e A to solve

max] Yp, (\302\253)$,\342\226\240-C(a)

Once the cost function C(a)has been determined,this is astraightforward step.

We have now completed our outline of the approach proposedbyGrossmanand Hart to solvethe principal-agentproblem.Notice that the

only assumption on the distribution function over output they make toobtain an optimal contractis that p^a)> 0.The remainderof their articleestablishes results obtained in previous formulations (in this moregeneralsetting) and alsosomenew results.Particulary noteworthy are the

following new results:First, as alreadydiscussed,the (IR) constraint is binding if the agent's

utility function is additively or multiplicatively separableand

limu(w) = -\302\260\302\260

H>-\302\273>V

Second,even if MLRPholds,the incentive schememay not be mono-tonic.This result is due to the possibleindifferenceof the agent betweenseveralactionsat an optimum (seeexample1in Grossmanand Hart).Toobtain monotonicity, one needsan evenstrongerspanning condition (seeGrossmanand Hart).

Beyondthese results, the most that can besaidabout optimal incentive

schemes,when no restrictionsare put on pl(a),exceptfor pi{a)> 0, is that

the transfers wu ...,wN cannot be decreasingeverywherein performance,nor increasingeverywherefasterthan performance.That is,while the Borchrule requiresthat

W: -W;_i\342\200\2360<\342\200\224

\342\200\224<1

qi-qi-iin this generalcasethe most that canbesaidis that we do not have

wiZw1\302\261<Q

everywhere,nor

157 Static Bilateral Contracting

qi-qt-i

everywhere!Theseare rather weak predictions.Just as sharp characterizationsof the

optimal incentive schemeare not obtainablein general,simplecomparative staticsresults are unavailable. For instance,a reductionin the agent'scostofeffort canmake the principal worseoff, as this may make it easierforthe agent to take someundesirableactions(seeGrossmanand Hart for an

example).Finally, it should comeasno surprisethat sharp characterizationsof the agent'ssecond-bestactionare alsonot obtainable.The second-bestactioncanbehigher or lowerthan the first-best action.Thereis no generalreason that hidden actionsnecessarilyresult in underprovision of effort.

4.6 Applications

As the introductory and concluding sectionsof this chapterstress,there isa paradox in moral hazard theory: While the model is very relevant for

many real-worldcontexts,the above discussionindicatesthat it deliversvery few generalpredictions.Only when significant simplifying assumptionsare introducedcanwe obtain cleanand intuitive results.This explainswhy

applicationsin the area take these simplifying assumptions on board.Thissectiondetailstwo such applications.

4.6.1ManagerialIncentive Schemes

The conflict of interest betweenshareholdersand managersof a firm is aclassicexampleof a principal-agentproblem.Indeed,managementcompensation packagesareperhapsthe best-known examplesof incentivecontracts. The design of executive compensationschemes has become asophisticatedart, somuch so that most companiesnowadays hire theservices of consultants specializedin managerial compensationissuesto designthe remuneration packagesof their executives.It is thereforenatural to askwhat lessonscan be drawn from principal-agenttheory for the designofmanagerial compensationcontractsand whether current practiceisconsistent with the predictionsof the theory.

In most casesa manager'scompensationpackagein a listed companycomprisesa salary, a bonus related to the firm's profits in the current year,and stock options (or other related forms of compensationbasedonthe firm's share price).The overallpackage also includesvarious other

158 Hidden Action, Moral Hazard

benefits, such as pension rights and severancepay (often describedas\"golden parachutes\.9") In other words, a manager's remuneration canbroadly bedivided into a \"safe\" transfer (the wage),a short-term incentive

component(the bonus),and a long-term incentive component(the stockoption).

At first sight, the overallstructure of this packageis difficult to relate tothe optimal contractsconsideredin this chapter.This difficulty should notcomeasa surprisegiven that the problemofproviding adequateincentivesto managers is much richer than the stylized principal-agentproblemswe have consideredin this chapter. First, the relationship between amanager and shareholdersis long-term.Second,the manager can often

manipulate performancemeasuressuch as profits. Third, and perhapsmost

importantly, the managerial incentive problemdoesnot just boil down to

eliciting moreeffort from the manager.It alsoinvolves issuesof risk taking,efficient cost cutting, adequatepayout provisions, empirebuilding, hubris,and so on. If anything, what is surprising is the relative simplicity ofobservedmanagerial compensationpackagesgiven the complexity of theincentive problem.

Another important differencebetweenpracticeand theory is the

contracting protocol. In the abstract problem we have consideredin this

chapter,a single principal makesa take-it-or-leave-itoffer to the agent.Inpractice,however, it is the manager (or a compensationcommittee, often

appointedby the manager) who draws the compensationpackageand getsit approvedby the directors (who are, for all practicalpurposes,on the

manager'spayroll).It comesas no surprise then that managersare oftengenerouslyrewardedevenwhen their company is doing poorly.A key focusof corporategovernanceregulation is preciselyon this contractualprotocol. A number of SECor other securitiesregulations are aimedat

reducingthe risk of capture of the remuneration committee and the board by

management. Despitethese regulations, many commentatorshave arguedthat managerial compensationhas more to do with rent extractionthan

with providing incentives to CEOs(see,for example,Bebchuk,Fried,andWalker, 2002).

9. In somecasesthe problem is not somuch to insure the manager against job loss as to tiethe manager to the firm. Then the compensation package might include a \"golden handcuff'(which is essentially a payment from the manager to the firm if the manager leaves the firmbefore the expiration of his contract) instead of a \"golden parachute.\" Another, more exotic,type of clause that is sometimes observed is a \"golden coffin\" (which is essentially an excep-tionaly generous life insurance for executives).

159 Static Bilateral Contracting

With all thesedifferencesbetweentheory and practiceit would seemthat

the simple principal-agenttheory developedin this chapterhas little to sayabout the problemof managerial compensation.We shall show, however,that a slightly reformulated versionof the generalmodelconsideredin this

chapter can yield the general structure of the managerial compensationpackagewe have describedand provide insights as to how the packageshould vary with the environment the manageroperatesin.

Considera single manager who contractswith a single representativeshareholder(say, the head of the remuneration committee).The managertakeshidden actionsaeA, which affectboth current profits q and the stockpriceP.Takeprofits q to benormally distributed with mean a and varianceoj:q =a+

\302\243q

where eq~ Af(0, <jq).Similarly, take the stockprice to be normally

distributed with meana and variance0$:

P=a+epwhereep is a normally distributed random variable:ep

~ Af(0, dp).In general,the stockprice variable behavesdifferently from the profit

variable becausestockpricesincorporateinformation other than that

contained in profit reports.For example,the stockpricemay reflectinformation stockanalysts have acquireddirectly about the manager'sactions(seeHolmstrom andTirole,1993,for a moredetailedmodelofmanagerialcompensation that illustrates how analyst's information getsreflectedin stockprices).Even if they behavedifferently, however,they are likely to becorrelated random variables.Thus, let oqP denote the covarianceof q and P.

Take the manager to have constant absoluterisk-averse(CARA)preferences, and representher preferenceswith the negative exponentialutility

function

u{w,a)=-e-T,[w-^a)]

where, asbefore,y/(a) = l/2ca2.Theprincipal, however,is assumedto beriskneutral.10

10.This assumption is justified as an approximation if we think of the principal as arepresentative shareholder with a well-diversified portfolio.

160 Hidden Action, Moral Hazard

As in section4.2,restrictattention to linear incentive packages.This is acommon (though not innocuous) simplification in the economicsoraccounting literature on managerial compensationcontracts.Supposethat

the managergetsa compensationpackagew of the form

w =t +sq+fP

wheret denotesthe salary (independentof performance),s the fraction ofprofits the managerobtains as a bonus, and /the manager'sshareof equity

capital.11We shall think of the owner as a \"buy and hold\" investor, who caresonly

about the firm's profit performance.That is, the owner is not concernedabout sellinghis sharesin the secondarymarket at a goodpriceandtherefore is not directly concernedabout stockprice.Thus the owner'sproblemis

'maxfl>M jE[q-w]subjectto

fl\342\202\254argmaxa E Aw-cf)

and

_Aw-<t)\\E -e

where,as before,w denotesthe manager'sreservationwage.As we have shown in section4.2,we can rewrite the principal'sand

agent's objectivesin terms of their certainty-equivalent wealth and thus

obtain the following simple reformulation of the principal'sproblem:

'maXa^^/O--s-f)a -1subjectto

a \342\202\254 argmaxa \\ (s+f)a +1-\342\200\224ca2-\342\200\224T][s2o2q

+2sfoqP+f2o2P ] \\

and 11 _(s+f)a+1--ca2--ri[s2a2+2sfaqP+f2a2P ]> w

11.Note that by restricting attention to linear incentive schemes we rule out (nonlinear) stockoption plans.

Static Bilateral Contracting

As in section 4.2,we can solve this problem using the first-orderapproach.Accordingly, we substitute the incentive constraint with the first-

order conditionsfor the agent'sproblem:

nS+fa=

c

The owner'sproblemis then simplified to

s+fmaxttStf0\342\200\224s-f)

1c' subjectto

Substituting for the value of t in the individual-rationality constraint and

maximizing with respectto s and /, one then obtains the following solutionfor the optimal linearcompensationpackage:

<Tq+2<TqP+<Tp 1+T]CQ.

<Jq +2<JqP +<J2p 1+T]CQ.

where

o\\ +2oqp+op

Theseexpressionsdeterminehow the manager'scompensationpackagevariesas a function of the underlying environment the firm operates in.

Specifically,the compensationpackagecan be directly tied to thestochastic structure of the firm's cash flow and stockprice.To seehow the

compensation packagevaries with the volatility of cash flow or stock price,considerfirst the specialcasewhereeq and ePare independently distributed.

In that case,oqP =0,and the expressionsfor s* and f* reduceto

162 Hidden Action, Moral Hazard

It is easy to seefrom these expressionsthat as the firm's stock pricebecomesmorevolatile (dp increases),the manager'sshareholding decreases(/* is smaller),but the share of profit increases(s* increases).Similarly, if

profit becomesmorevolatile, the manager'sstockparticipation goesup, but

the profit participation is reduced.Another interesting observationis that

the manager doesnot becomea 100%equity holderof the firm evenwhen

she is almost risk neutral (as 77 \342\200\224> 0).However,she doesbecomethe soleresidualclaimant sinces* +f* \342\200\224> 1as 77 \342\200\224> 0.

Considernext the specialcasewhere eP= eq +\302\243,

where\302\243

is an

independently normally distributed random variable:\302\243

~ N(Q,a}).In that casethe stockprice is a more noisy signal of the manager'seffort than profits.In other words, profits are then a sufficient statistic, and the result due toHolmstrom (1979)suggeststhat there should be no stockparticipation in

the manager'scompensationpackage.Underthis stochasticstructure, dp =a\\ + o\\ and oqP =

o\\, so that /*= 0 and s*=l/(l+rjcGq), as the generaltheory would predict.

An important assumption in the agencytheory of executivecompensation that we have outlined is that the stockmarket is informationally

efficient, so that the stock price is an unbiased estimate of the firm'sfundamental value and the CEO'sactionchoicea.When one allowsfor the

possibility of speculativebubblesin stockmarkets, then the optimalcompensation contractmay put more weight on short-term stock priceperformance than on long-term fundamental value, as Bolton,Scheinkman,and Xiong (2003)have shown.

4.6.2The Optimality of DebtFinancing under MoralHazardand Limited

Liability

In Chapter3 we discussedwhy debt financing can be a cheapersourceoffunding for firms than equity or any other form of financing when the firm

has private information about the expectedvalue of its assetsandinvestments. Sinceunder a debt contractthe repayment to investors varieslesswith performancethan under any other financial contract,this repaymentstream is likely to be lessunderpricedby uninformed investors than the

payment stream under any other form of financing.

163 Static Bilateral Contracting

We now illustrate that when (costlyand hidden)managerial/entrepreneurial effort raisesthe return on investment, then the most incentive-efficient form of outside financing of the entrepreneur's project underlimited liability may be someform of debt financing. A specialcaseof this

result has first beenestablishedby Jensenand Meckling (1976),who have

pointedout that when the investment projectcanbefinanced entirely with

\"safe debt\"\342\200\224that is,debt that is repaidin full for sure\342\200\224then a risk-neutral

entrepreneurwill have first-bestincentivesto provideeffort.Thereasonissimply that in this casethe entrepreneurgetsto appropriatethe entire

marginal return from her effort.Innes(1990)has shown that the incentive

efficiency of debt financing extendsto situations where debt is \"risky\"\342\200\224that

is, debt is not always repaid in full\342\200\224if the setof feasibleinvestmentcontracts is such that the repayment to investors is always (weakly) increasingin the return on investment.

Innes'stheory, along with Myers and Majluf's and severalothers wediscussin later chapters,provideseveralcompelling explanationsfor why

most small firms and most householdsraiseoutsidefunds primarily in theform of debt.Until recentlydebt financing was the only form of outsidefinancing availableto firms and households.And even though it is possiblenowadays to get funding in the form of venture capitalfinancing or equity

participation by an investment fund, debt financing is still the mostprevalent form of outsidefinancing.

Innes'sbasic idea is that, in the presenceof limited liability, when thedownside of an investment is limited both for the entrepreneurand theinvestor, the closestone canget to a situation where the entrepreneuris a\"residual claimant\" is a (risky) debt contract.In other words, a debtcontract providesthe bestincentivesfor effort provision by extracting asmuchaspossiblefrom the entrepreneurunder low performanceand by giving herthe full marginal return from effort provision in high-performancestateswhererevenuesare abovethe facevalue of the debt.

The basicsetup is as follows.Supposethat a risk-neutral entrepreneurcan raise revenuesfrom investment q by increasingeffort a. Specifically,supposethat revenuesare distributed accordingto the conditional density

f(q\\a) and the conditional cumulative distribution F(q\\a). Theentrepreneur's utility function is separablein incomeand effort (asusual) and is

given by v(w, a) = w -iff (a),with y/, y\" > 0.

Supposealsothat the risk-neutral entrepreneurhasno funds and that thefirm's setup cost/ must be funded by a risk-neutral investor in exchange

164 Hidden Action, Moral Hazard

for a revenue-contingentrepayment r(q).We now show that if the firm isa limited liability corporationand if feasible investment contractsmust

specify repayments that are monotonically increasing in q, then the

uniquely optimal investment contract is a risky debt contract taking theform r(q)=D for q > D and r(q)= q for q < D,whereD > 0 is set so that

(discounted)expectedrepayments are equal to the initial investment I.More formally, the key restrictionsimposedon the set of feasible

contracts r(q) are

1.A two-sidedlimited liability constraint 0 < r(q)< q2.A monotonicity constraint 0 < r\\q)

The first restriction meansthat the entrepreneurcannot beaskedto paybackmore than she earns [r(q) < q],and also that the investor cannot beaskedto pay more than I in total (this latter part of the constraint is not

important for the result in the end).The secondrestriction seemslessnatural at first. It can,however,bejustified asfollows:Supposethat 0> r\\q)for a subsetof revenueoutcomesq. In that casethe entrepreneurwould

strictly gain by borrowing money at par from another sourceand thus

marginally boosther profit performance,

d{[q-riq)]-q}=-Aq)>0dq

If this kind ofborrowing cangoon undetected,then any contractsuch that

0 > r\\q) would simply encouragethe entrepreneurto engagein this typeofarbitrage activity. Note that the secondrestriction would alsoberequiredif the principal wereable to costlesslyreduceprofits expost.

Under these restrictions,a striking conclusionemerges:If the densityfunction f{q I a) satisfiesthe monotonelikelihood ratio property (MLRP),an optimal investment contractis a debt contract.Moreover,Innes showsthat under the optimal investment contractthere is always underprovisionof effort relative to the first best,wherethe entrepreneurdoesnot needtoraise any outsidefunding. Let us providethe intuition for this result.

We begin by showing that in the absenceof any monotonicity constraint

0 < r\\q), MLRPimplies that it isoptimal to \"punish\" the entrepreneurfor\"bad\" performance(q < Z) by taking away all the revenuesfrom her \\r(q)

165 Static Bilateral Contracting

= q],and to \"reward\" her for \"good\" performance{q > Z) by leaving herall the revenuesof the investment [r(q)=0].

Formally, the entrepreneur'soptimal contracting problemis to maximizeher expectednet return from the investment, subject to meeting (1)anincentive compatibility constraint, (2) the investor'sbreak-evencondition,and (3) the two-sidedlimited liability constraint. If we substitute theincentive compatibility constraint by the first-order condition of the

entrepreneur'soptimization problemwith respectto a, then we can write

the constrainedoptimization problemas follows:

max T[q-r(q)]f(q\\a)dq -y/(a)-1subjectto

\\l k-r{q)]fa(q I a)dq =y\\a) (IC)

\\qr(q)f(q\\a)dq=I (IR)

0<r(q)<q (LL)As we have explained,in general,substitution of the incentive constraints

with the first-orderconditionsof the agent'sproblemresults in a \"relaxed\"

problem for the principal.However,here the restrictionson the set offeasiblecontractsas well as the MLRP ensure that the substitution ofthe incentive constraints with the first-orderconditionsdoesnot resultin a strictly relaxedproblem.We leaveit to the reader to verify this findingas an exercise.

Letting (i and X denotethe Lagrangemultipliers associatedwith theconstraints (IC)and (IR),respectively,we can write the Lagrangeanof this

problemas

L=JJ [q-r(q)]f(q\\ a)dq-y/(a) +

jt[\302\243

[q-r(q)]fa(q \\ a)dq-y/(tf)]

+l[\\\\{q)f{q\\a)dq-l\\

Rearranging, the Lagrangeancanbe rewritten as

166 Hidden Action, Moral Hazard

\302\243=

\\ r(q) X-li -1Jo *

|_ f(q\\a)

,fa(q\\a)J>l+A*

f(q\\a)l

f(q\\a)dq

f(q |a)dq-Hf(a)-

fJ.\\ff'(a)-XI

Thisformulation of the problemmakesit clearthat the objectiveis linearin r(q)for all q.Therefore,providedthat the constraint (IC)is binding andthereforethat \\i > 0,12the optimal repayment scheduler*(q) is such that

r*(q)=q ifX>l+n

0 ifX<l+fi

fa(q\\a)

f(q\\a)

faiqla)f(q\\a)

We can then concludefrom this analysis that it is optimal to \"reward\"

the entrepreneur for revenue outcomes such that the likelihood ratio

fa (q I a)/f(q|a) is higher than the threshold (X-1)1fi.UnderMLRPthelikelihood ratio fa(q\\a)/f(q\\a) is increasingin q. Therefore,there exists arevenuelevelZ such that it isoptimal to set

r*(q)=0 ifq>Zq ifq<Z

SeeFigure 4.3for an illustration of this contract.Note that this contractis highly nonmonotonic and doesnot resemblethe descriptionof astandard debt contract[rD(q)}such that

rD(q) = D ifq>Dq ifq^D

However,if oneimposesthe additional constraint that r'(q)> 0,it is clearfrom the precedingargument that the constrainedoptimal contracttakesthe form a standard debt contractrD (q)whereD is the lowestvalue that

solvesthe (IR) constraint:

12.Otherwise, the first-best can bereached.

167 Static Bilateral Contracting

r*(q)n

Figure 4.3Optimal Nonmonotonic Contract

f* qf(q |a*)dq +[1-F(D \\ a*)]D=IJo

and a* isgiven by the (IC)constraint

\\\\q-D)fa(q\\a*)dq =w'(a*)

The intuition of Innes'sresult is straightforward. The incentive problemis to try to induce the entrepreneur to supply high effort. Given that

high revenue outcomes are most likely when the entrepreneur workshard, it makessenseto rewardher as much as possiblefor high-revenueperformance.

One important implication of this result is that when externalfinancingis constrainedby limited liability, it will generally not bepossibletomitigate

the debt-overhangproblem,which was highlighted by Myers(1977)and discussedat the beginning of this chapter,by looking for other formsof financing besidesdebt.Indeed,Innes'sresult indicatesthat under quite

168 Hidden Action, Moral Hazard

generalconditionsit is not possibleto get around this problemby

structuring financing differently. Debt is already the financial instrument that

minimizes this problemwhen there is limited liability.

One key feature Innes abstractsfrom is risk aversion.While debtprovides maximum effort incentives, it is not good from the point of view ofinsuring the entrepreneuragainst risk. In Chapter10,we discuss,however,the contribution of Dewatripont,Legros,and Matthews (2003),who showthat Innes'sinsight survives in a dynamic setting wherethe initial financial

contract can be renegotiatedafter the investor has observed the effortchoiceof the entrepreneur.The renegotiatedcontractcan then be reliedupon to provideoptimal insurance,while the role of the initial contractisto maximize effort incentives.For this purpose,starting with debt financingis optimal.

4.7 Summary

Thenotion ofmoral hazardhas for a long time beenconfinedto theinsurance industry and mainly beendiscussedby actuaries.Nowadays,however,this is a widely understoodnotion, which is seenas highly relevant in manydifferent subdisciplinesranging from international finance to economicdevelopment.Although the trade-offbetweenrisk sharing and incentivesis widely recognized,there is unfortunately no tractablegeneralmodelthat

isas widely used.In this chapter,we have highlighted someof theconceptual and mathematical difficulties inherent in a generalcontracting problemwith hidden actions.Despitethese difficulties, several general lessonsemergefrom our analysis.

\342\200\242 When the agent is risk neutral and wealthy, a simple solution to any

contracting problem with moral hazard is to let the agent be a \"residual

claimant\" on the output sheproduces.\342\200\242 When the agent isrisk neutral but has limited wealth, then the incentive

problem is worse the pooreris the agent.Sometimesthe poverty of the

agent may be soextremethat it pays the principal to hand overwealth tothe agent in the form of aid or an efficiencywage.Much of developmentpolicy to very poor countriescanbe rationalizedon thesegrounds.\342\200\242 When the agent is risk averse,then more incentives comeat the costof arisk premium that the principal must pay the agent.Whether the principal

169 Static Bilateral Contracting

should exposethe agent to lessrisk when the agent is more risk averse,however,is generally not clear.Similarly, whether the principal should

optimally elicit higher effort from the agent when the agent's effort costdecreasesisnot generally clear.\342\200\242 Among the main generalpredictionsof the modelis the informativeness

principle,which says that the incentive contract should be basedon allvariables that provide information about the agent's actions.Also, in

general the agent must be exposedto some risk to provide adequateincentives.

\342\200\242 When the distribution of output conditional on the agent'sactionchoicesatisfiesthe monotonelikelihoodratio property (MLRP),then the agent'sremuneration is increasingin her performance(providedthat the agent isnot alsosubjectto a limited liability or wealth constraint).But in generalthere is no reasonto expectthe optimal rewardfunction of the agent to belinear.\342\200\242 Someof the main areas of applicationof the principal-agentframeworkwith hidden actions have been to executive compensation,corporatefinance, and the theory of the firm. But this framework has also beenappliedwidely in international finance, agricultural economics,andmacroeconomics. A leading application in macroeconomicshas been to theincentive problemof centralbankers(seeWalsh, 1995).

4.8 LiteratureNotes

The topicscoveredin this chapterare standard material.Sincereferencesfor key applicationswere mentioned in the introduction, we concentratehere on theoreticalreferences.Arrow's (1970)essayson risk bearinghave

playedan important role in exposingthe economicsprofessionto the ideasof moral hazard and the widespreadproblemsof incentives when actionsarehidden.Thefirst articleto characterizethe first-best optimal risk-sharingrule is due to Borch(1962).Wilson (1968)considersamoregeneralproblemthan Borch,and his articleis the first to formally touch on the problemofoptimal risk sharing with hidden actions.Much of the discussionin Wilson

(1968)and later in Spenceand Zeckhauser(1971)and Ross(1973)isconfined to the questionof identifying situations where the first bestcan beachievedwith hidden actions.The pioneeringarticlesthat first attempt to

170 Hidden Action, Moral Hazard

formulate the second-bestproblemare due to Mirrlees(1974,1975,1976).Later, Holmstrom (1979)(seealso Shavell,1979a,1979band Harris andRaviv, 1979)built on Mirrlees5work to providethe first articulation of theinformativeness principle(seealsoKim, 1995and Jewitt, 1997)on the valueof information systems in moral hazard settings).Sappington (1983)andKahn and Scheinkman (1985)have providedsolutions to the bilateralrisk

neutrality problem with resourceconstraints for the agent. Finally, theresearch effort of formulating a general framework for the second-bestproblemculminates with the modeland approachproposedby Grossmanand Hart (1983a)and the characterizationof sufficient conditionsunderwhich the first-orderapproachis valid by Rogerson(1985a)and the later

generalizationsproposedby Jewitt (1988).Thestatic bilateralcontracting problemwith hidden actionshasofcourse

beensubsequently extendedin many different directions,most ofwhich we

explorein detailin subsequentchapters.Thus we considermultitask

extensions aswell ashybrid modelsinvolving both hidden information and actionin Chapter6.Multiagent and dynamic extensionsare pursued further in

Chapters8 and 10.One important extension,however,that we have notbeenable to cover is the framework consideredby Baker(1992,2000),wherethe contractis signedbefore the agent learnsher true costof effort.Thisassumption not only is realistic,but alsoprovidesa contractingstructure that is fairly tractableand gives riseto sharpercharacterizations.

Disclosureof Private CertifiableInformation

Up to now we have consideredsituations where private information isneither observablenor verifiable.That isto say, we have consideredprivateinformation about such things as individual preferences,tastes,ideas,intentions, quality ofprojects,and effort costs,which cannot bemeasuredobjectively by a third party. But there are other forms of private information,such as an individual's health, the servicing and accidenthistory of a car,potential and actual liabilities of a firm, and earned income,that can becertifiedor authenticated once disclosed.For these types of information

the main problemis to get the party who has the information to discloseit.This is a simpler problemthan the one we have consideredsofar, sincetheinformed party cannot report false information. It can only choosenot to

report somepieceof information it has available.There is a small but nonethelesssignificant literature dealing with the

problemof disclosureof verifiable information. Our review of the mainresults and ideasof this literature is basedon Townsend(1979),Milgrom(1981a),Grossmanand Hart (1980),Grossman(1981),Gale and Hellwig(1985),Okuno-Fujiwara, Postlewaite,and Suzumura (1990),Shavell (1994),and Fishman and Hagerty (1995).

In many situations whereone contracting party may have privateverifiable information, a standard regulatory responseis to introducemandatorydisclosurelaws.Thispracticeis particularly prevalent in financial markets,where firms that issueequity or bonds are required to regularly discloseinformation about their performanceand activities. Also, financial

intermediaries are requiredto discloseinformation about their trading activitiesand pricing decisions.Although at first sight mandatory disclosurelaws mayseemlike a rather sensibleregulatory responseto overcomesomeforms ofinformational asymmetry, we shall seethat someof these laws arecontroversial, at least in financial markets. The reasonis simply that accordingtothe theory outlined in this chapter there are sufficient incentives forcontracting parties to voluntarily discloseinformation. Therefore,it is often

argued,mandatory disclosurelaws are at bestredundant and may imposeunnecessarycostson the contracting parties.We begin this chapterby

considering the incentives for voluntary disclosure.We then identify situations

where there may be inefficient voluntary disclosureand investigate themerits ofmandatory disclosureregulations.We closethe chapterwith adiscussion ofcostly information productionand disclosure.We show that when

there are positivedisclosureor verification costsit is optimal to iriinimize

overalldisclosurecostsand to commit to produce and discloseonly the

172 Disclosureof Private Certifiable Information

most relevant or essentialinformation. We illustrate how this basiclogiccanbe appliedto the designof financial contractsand financial reporting. In

particular, we cover the arguments by Townsend(1979)and Gale and

Hellwig (1985)that debt contractshave the desirableproperty that theyminimize the extent of costly financial reporting.

5.1Voluntary DisclosureofVerifiable Information

Sincemandatory disclosurelaws are particularly prevalent in financial

markets, we shall addressthe problemof disclosureof verifiableinformation in the contextof a finance application.Specifically,we shall considerthe problemof a firm issuing equity on a stockmarket. In practice,securities laws and stockmarkets imposea number of rather stringent disclosurerequirementson any memberfirm. Theserequirementsare often invokedas an important reasonwhy somefirms may prefernot to go public.

Consideragain the model of Myers and Majluf (1984)introduced in

Chapter3.Recallthe basicelementsof their model:\342\200\242 A risk-neutral owner-managerof a firm with assetsin place is

considering issuing equity to finance a new project.\342\200\242 The final value of the assetsin placeis uncertain and may beeither1with

probability /or0 with probability 1-y\342\200\242 Thesetup costof the new projectis 0.5,and its expecteddiscountedgrossvalue is also uncertain; it may be either 1with probability 77 or 0 with

probability 1- 77, wherer\\

> 0.5.\342\200\242 Theowner-managerhasprivate information about the expectedvalue ofassetsin place.Sheknows whether 7= yG or 7=yB, whereyG > yB.

\342\200\242 Outsideinvestors, however,know only that 7= yG with probability 0.5.Sincethey are all risk neutral, they value the assetsin placeat 0.5(7G+ yB).

As a benchmark, we begin by briefly reviewing the equilibrium outcomesin this modelwhen the private information of the owner-manageris notcertifiable.We then show how the equilibrium outcome is drastically

changedwhen information is certifiable.

5.1.1Private UncertifiableInformation

If the owner-manager'sprivate information about 7 is not certifiable,then

we are dealing with a signaling problemsimilar to that analyzed in Chapter

Static Bilateral Contracting

3.Recallthat in this signaling problemthe owner-manageralways wants toissuesharesand invest in the new project if y= yB. Indeed,even if outsideinvestors have the most pessimisticvaluation of the firm's assetsin placeand demand a shareof equity of 0.5/\\yB +77)in return for an investment of0.5(to coverthe setupcostsof the new project)the owner-managergetsapayoff equal to

(yB+ml:\342\200\224]

=7B+77-O.5

which is, by assumption, higher than yB, the payoff the owner-managerwould obtain if she did not undertake the new investment project.

If 7= Yg, however,then the owner-managermay or may not issuenew

equity, dependingon how much outside investors undervalue the firm'sassetsin place.In a poolingequilibrium the owner-managerissuesshareswhether assetsin placehave a high or low expectedvalue. In that casethe

owner-manager'sactionrevealsno information, and outsideinvestors valuethe firm at 0.5(7G+ yB) + (77

-0.5).In a separatingequilibrium the owner-manager issuessharesonly when the expectedvalue of assetsin place islow (7=yB).1

Since7> 0.5,only the pooling equilibrium is efficient. But for someparameterconstellationsonly the separatingequilibrium exists.Moreover,for a wide rangeof parametervaluesboth types of equilibrium may exist,and there is no guarantee that the pooling equilibrium is the natural

outcomeof the signaling game.

Private, CertifiableInformation

If the firm is able to discloseits information about the value of assetsinplaceby certifying the true value of 7, then the equilibrium outcomein this

model is radically different. To seethis, supposethat the firm can get anunderwriter, a rating agency,or an accounting firm to certify the true value

1.Recall that a separating equilibrium exists if and only if

(ra+rj/l-w-^ -W+ rj-0.5-0.5j^\"7* <ycI X.^'+rl); 2,(7.+77)

or if

t7-0.5<0.5x^g/Yl

Disclosureof Private Certifiable Information

of y at a costK > 0.Then,whenever7=Yg, the owner-managerwould want

to discloseher certifiedinformation if outside investors believethat thevalue of assetsin placeis strictly lessthan (Yg +W)lQ+22f)-r].Indeed,following disclosurethe owner-managercan then getfinancing of0.5in return

for a stakeof 0.5/{yg+if) and obtain a payoff of Yg +T]-0.5-K,while if shedoesnot discloseher information shegetsno morethan Yg (under an inefficient

separatingequilibrium) or (yG +rf)[l-(0.5+K)/(Yg +V)] =Yg +V-0-5-K

(under an efficient poolingequilibrium). In other words, by disclosingthevalue of assetsin place,the owner-managercan overcometheundervaluation problemin the Myers and Majluf model.

When 7=7b, however,it is not in the interestof the firm to disclosethat

outsideinvestors may be overvaluing assetsin place,so that no voluntarydisclosureof information takes place in that event.But rational investorscaninfer asmuch information from the firm's equity issuedecisionsas fromthe firm's disclosureactions,and here the firm's decisionnot to discloseitsinformation when 7= Yb inevitably revealsits type.

To summarize, when the firm is able to certify its private information

about assetsin placeat sufficiently low cost,then the unique equilibriumoutcomeis for the firm to disclosegoodinformation, to always invest in thenew project,and to raisenew funding by issuing equity at fair terms.

TooMuch Disclosure

It is tempting to concludefrom the precedingdiscussionthat efficiencyis always improved by making disclosurepossible(say, by introducing anew information-certification technology).However,sincecertification is

costly,it would be more efficient to have a poolingequilibrium. Yet if the

technology were available,the firm might want to use it to prevent apoolingequilibrium. Indeed,a firm with assetsin placeworth Yg might want

to discloseits private information about the value of these assetssoas toobtain better terms for the equity issue.More precisely,as long ascertification costsare lowerthan the subsidy to investors [which is equalto0.5 (yG -7b)I^LX7i+if))]>in a poolingequilibrium, the firm would want to

certify the value of its assetyG.

It is only against a separating-equilibrium situation that disclosureiswelfareenhancing.In such a situation, certification by type Ghas no impacton the payoff of type B,so that it occursif and only if it is efficient.All in

all, then, there is too much disclosurein this model,evenif disclosureisvoluntary. A mandatory disclosurelaw would only make things worsehere.

Static Bilateral Contracting

Unraveling and the Full DisclosureTheorem

One striking conclusionemerging from the analysis of this simple modelisthat (when certification costsare not too high) the firm endsup revealingall private information in equilibrium. In other words, the firm is unable to

exploitits informational advantage when it cancrediblydisclosethis

information. An obvious questionis whether this result extendsto moregeneralsettings.

A central result in the literature on disclosureof information (due toGrossmanand Hart (1980),Grossman(1981),and Milgrom (1981a))isthat when certification is costless(K = 0), there is full disclosureofinformation in equilibrium under very general conditions.In particular,the full-disclosuretheorem holds not just with two types, but with anynumber of types.The basiclogicbehind this full-disclosuretheoremis asfollows.

Supposethat assetsin place can now take N different values, % e I\" ={ji, Yi, \342\226\240\342\200\242

\342\200\242, Yn}, whereY\\

< Yi < \342\226\240\342\226\240\342\226\240< Yn- Sinceoutsideinvestors have onlya probability distribution over all possiblevalues

y\342\200\236

the expectedmarket value y is strictly lessthan yN (unlessthe probability distribution is

degenerate).As a result, type yN has a strict incentive to discloseso as toeliminate the market'sundervaluation of assetsin place.

Hence,if the firm doesnot discloseany information, outsideinvestors areable to infer that assetsin place are worth at most Yn-i and probably lessthan that. This downward revision in market expectationsin turn promptstype Yn-i to disclosealso,so that no disclosureof information leadsoutsideinvestors to believethat assetsin placeare worth only Yn-i- This logiccanberepeatedfor types Yn-i, Yn-3, and soon.

To complete the induction argument, supposethat some type i > 1doesnot disclose.Considerthe highest such type, say, type ; > i. Bydisclosing, type / could obtain funding at actuarially fair terms. By not

disclosing, type ;'is pooledwith at leasttype 1,for whom disclosingis a weaklydominated strategy. But then type ;'gets financing at worsethan actuariallyfair terms, so that type ; is strictly better off disclosingits type.

Note that we have heavily reliedon the assumption that disclosureinvolves no costs(K-0) to obtain this rather striking result.Of course,ifK is very high, then disclosuremay not pay evenfor type N.In that case,thefirm's problemreducesto a signaling gameakin to that analyzed in Chapter3.With intermediatevaluesofK, however,partial disclosurecanobtain.To

Disclosureof Private Certifiable Information

seethis possibility, take,forexample,y e {yu y_, y$) with yx < ji < %,but Ji-Jismall relative to y,

- ji.Then the incentive for type 3 to discloseis much

higher than for type 2.We then obtain disclosurefor all types abovea givenlevel y and no disclosurefor lower types. This simple dichotomy obtainsbecausethe net incentive to discloseincreaseswith y.

As hinted at earlier, the full-disclosuretheorem has implications for

mandatory-disclosurelaws. If all private information is alreadydisclosedvoluntarily, then mandatory-disclosurelaws are at bestsuperfluous.Moreover, they may be harmful to the extent that they may force firms todisclose totally irrelevant information. Perhapsa more important implicationof this result and its underlying logicis that it suggeststhat \"privacy\" laws,such as the right to keepone'shealth history private, may be ineffective if

they do not alsopunish voluntary disclosureof this information. In otherwords, if any healthy employeeis allowedto volunteer the information ofher goodhealth to her employerby providing an updatedmedicalexamination, then employeeswith health problemsget no protectionat all from

any privacy laws.The only way to enforceprivacy in that casewould be toallow individuals to forge health certificatesand to prevent courts from

punishing such forgery, evenif private contractsexplicitly wereto allowforsuch punishments!

Generalizingthe Full-DisclosureTheorem

Howgeneralis the full-disclosuretheorem?We have shown so far that it

holdsfor an arbitrary number of types when disclosureis costlessand when

it is common knowledgethat the firm knows its type y. Would the resultstill hold if, for example,the firm did not know its type perfectlybut knew

only that it belongs to a subsetof all possibletypes? Okuno-Fujiwara,Postlewaite,and Suzumura (1990)have investigated this issue.Theyprovidetwo sufficient conditionsunder which full disclosureobtains, as well asexampleswhereit fails when any of theseconditionsare not satisfied.Theirfull-disclosuretheorem essentiallygeneralizesas much as is possiblethe

underlying unraveling logicoutlined previously. The main interestof then-

result lies in the identification of two conditions,which in fact are almostnecessaryin the sensethat exampleswhere either of the conditionsdoesnot hold give riseto partial or no disclosurein equilibrium.

Let i^y,b) be the manager'sequilibrium payoff when the true value ofthe firm's assetsis y but outsideinvestors have beliefb about this value.

177 Static Bilateral Contracting

Beliefsare (possiblydegenerate)probability distributions overT [so that,for eachb, calling b(yd trie probability that the value of the firm's assetsisYi under beliefb, we have b(Y) + b(Yi) + \342\226\240\342\200\242\342\200\242+ K7w) = 1]-Beliefsmay beinfluenced (through Bayesianupdating) by certifiedstatements that the

manager sends.Thesecertifiedstatementsare assumedto besubsetsof T,and for eachy, the managerhas accessto a setof certifiedstatements that

all include Y- In otherwords,the manager cancertify with moreor lessaccuracy that the value of her assetsis at least as high as % or no greater than

Yi, but shecannot lie and announcethat her true type is in a given subsetwhen it is not. Okuno-Fujiwara, Postlewaite,and Suzumura (1990)definethe following two conditions:

1.For any y, the managercan certify to the market that her type is at leastYi- More formally, the manager can send a certified statement whoseniinimum elementis Y-

2.For any beliefb, considerbeliefb'that first-orderstochasticallydominates b:that is,for all y,

X^(7;)^lMr;)

wherea strict inequality holdsfor at least one yx. Then k(y, b')> n{yu b).Condition 1says that the set of certifiablestatements is rich enough to

allow the manager to prove that the assetsin placeare worth at least their

true value. Condition 2 is a monotonicity condition on beliefs,which saysthat the manager'spayoff is higher when market beliefsaremoreoptimisticabout the value of the assetsof the firm.

We have argued that, following a statement c cT,outsideinvestors useBayes'rule in order to revisetheir beliefsfrom b to b(c).Okuno-Fujiwara,Postlewaite,and Suzumura (1990)alsodefinebeliefsto beskepticalif,following a statement c,updatedbeliefsb(c)put full probability weight on thelowestelementof c.

In this setup,they establishthe following full-disclosuretheorem:Under

conditions 1and 2, the equilibrium involves full revelation of informationand skepticalmarket beliefs.

Thistheoremcan beproved in three steps:

178 Disclosureof Private Certifiable Information

\342\200\242 First, onecanrule out incompletedisclosurein equilibrium (which is itself

incompatiblewith skepticalbeliefs).Indeed,incompletedisclosuremeansthat there exist several types of manager sending the same certifiedstatement c.Considerthe highest type of managerwho is to be pooledwith some other type(s).By condition 1,she could separate from such

type(s)by sending acertifiedstatement with her true type aslowestelementof the statement.She would then be either fully separated or pooledwith higher types than hers.In any case,by condition 2, she would

have strictly raisedher payoff. Thus theredoesnot exist a partial-disclosureequilibrium.\342\200\242 Considernext a full-disclosureequilibrium wherethere existsa report c(which might or might not besent with positiveprobability in equilibrium)for which the associatedequilibrium beliefis not skeptical.In this case,itwould bein the interest of the lowesttype includedin c to deviateand sendthis report:shewould thereby strictly raiseher equilibrium, full-revelation

payoff, becauseshewould have generatedbeliefsthat first-orderstochastically

dominate the (correct)equilibrium belief.Thus there doesnot existan equilibrium without skepticalbeliefs.\342\200\242

Finally, it is clear that it is an equilibrium for each type to senda reportwith one'strue type as lowestelement,and for investors to anticipate this

behavior,that is, to have skepticalbeliefs.

5.2 Voluntary Nondisclosureand Mandatory-DisclosureLaws

If the conditionsof the abovefull-disclosuretheoremalways held in

practice, then mandatory-disclosurelaws would be redundant. We begin this

section by showing that when either or both of the conditionsof their

full-disclosuretheoremdo not hold, then there may be only partialdisclosure in equilibrium. We then proceedby analyzing the manager'sincentivesto acquire costly information in the first place when this information

may be wastedex post through voluntary disclosure.Finally, we closethis

sectionby showing that an implicit assumption behind the full-disclosuretheoremis that the informed party is always on onesideof the transaction.That is, the informed party is always a buyer or always a seller.If,however,the informed party is sometimesa buyer and sometimesa seller,then

Static Bilateral Contracting

full disclosurebreaksdown even when both conditionsof the theoremhold.

TwoExamplesofNo Disclosureor Partial Voluntary Disclosure

Although conditions1and 2 are only sufficient conditions, they appear tobe the right conditions, since the full-disclosureresult easilybreaksdown

in exampleswhereeitherone of them doesnot hold.Toseethis point,consider the following two examples,wherein turn conditions 1and 2 are notsatisfied.

Example 1 In this examplethe manager may not know her true type. Inother words, with positiveprobability the manager may be as uninformed

as outside investors. If there are only two possiblevalues for assetsin

place\342\200\224/is either yG oryB\342\200\224then

there are three types of managersin this

example:thosewho know that y= yB, thosewho know that y= yG, and thosewho do not know anything. We denote that last manager type by 7= yGB.

Thus, let the setof types beT={yB, yBG, yG}, whereyB < yBG < 7G-Thisexample

satisfiescondition 2, since the manager'spayoff is monotonic in y. But it

violatescondition 1.The reason is simply that it is not possibleto proveone'signorance,so that type yBG cannot certify that her type is at least yBG.

As a result, the equilibrium is such that type yG discloses,but type yB is ableto pool with type yBG (only, of course,as long as issuing equity is optimalfor type yBG).

Example2 This exampleis rather contrived within our simplified setting;basedon an examplefrom Okuno-Fujiwara, Postlewaite,and Suzumura

(1990)it conveysits generalidea,but doesnot do full justiceto it. In this

example,disclosureof information exogenouslyreducesthe manager'spayoff so that condition 2 is violated.In Okuno-Fujiwara, Postlewaite,andSuzumura (1990),disclosurereducesthe manager'spayoff becauseof theincreasedcompetition generatedby the news that the firm hasa high valueof assetsin place.2Here,keepingthis story in reducedform, let T= {yB, yG),

with yB < yG, and assumethat if type yG disclosesher type, then her payoff

2. Concretely, consider a pharmaceutical company for which certifying that it has assets in

place that are worth a lot means disclosing results on preliminary testing of a new wonder

drug. In such a case,it is conceivable that, asa result of the disclosure, its future profits arereduced because of the increased future competition by other pharmaceutical companies thatare working in parallel with this company. If competition is expected to besufficiently fierce,the firm may well decide not to disclose its information (see,for example, Green and Scotch-mer, 1995,on the issue of disclosure and patenting with sequential innovations).

180 Disclosureof Private Certifiable Information

dropsfrom yG to yG-A. Disclosureof type yG would, ofcourse,improve the

firm's financing terms and make the new investment profitable toundertake. But if disclosureleadsto an excessivereductionin the value of assetsin place (that is, if A is too large),a type-7G firm may well decidenot todisclose. In a poolingequilibrium situation, disclosingleadsto morefavorablefinancing terms for the type-7G firm if yG

-A > 0.5(yG+ yB). This conditiondoesnot imply that disclosurewill happen,however:if A >

r\\-0.5,the loss

ofvalue on assetsin placeexceedsthe net presentvalue of the investment,so that a type-7G manager prefersnot to discloseher type and not to investin the new projecteven though this decisionmeanspassingup a valuableinvestment opportunity.

Example1identifies an important reason why full disclosuremay notoccurin equilibrium. If there is any doubt about whether the manager is atall informed, then full voluntary disclosuremay not occurbecausemanagers who receivebad information can always claim that they areuninformed. Giventhat voluntary disclosureresults in only partial disclosure,itappearsthat theremay bea role for mandatory-disclosurelaws in this case.Indeed,if mandatory-disclosurelaws are effective,they may bring aboutfull disclosure.There are,however,potentially two objectionsagainst this

line ofargument. First, it isnot clearthat full disclosureis desirable.Supposethat there are strictly positivedisclosurecosts(K> 0);then, as long as type

yBG is willing to invest in the new positive-net-present-valueproject (in asituation where it might be pooledwith type yB), there is nothing to begainedfrom full disclosure.Second,even if mandatory disclosureisdesirable, it may not beeffective.Indeed,if the manager is unable to prove that

sheis ignorant, how can a judgedo so?

5.2.2Incentivesfor Information Acquisition and the Role of Mandatory-DisclosureLaws

So far we have not found very compellingreasons for introducing

mandatory-disclosurelaws.The only possiblejustification wehave found isthat voluntary disclosuremay only be partial when there is someuncertainty about whether the manager is informed or not.We shall pursue this

line further here by asking what the sourceof this uncertainty might be.Specifically,we shall considera stage prior to the information-disclosurestagewherethe manager canendogenouslydeterminehow muchinformation sheacquires.

181 Static Bilateral Contracting

Formally, this approach involves consideringa model combining bothmoral hazard elements (how much information to acquire)and adverseselectionelements(how to act on the basisof the information acquired).Themodelwe shall analyze is a simplified version of Shavell (1994).In theinformation-acquisition stage we allow the manager to randomizeoverbecominginformed about her type or not. If she randomizes, then herchoiceof probability p of acquiring information is private information.We also assumethat information acquisition is costly and involves a costiff

> 0.3To allow for the possibility that the information acquiredby the

manager is sociallyvaluable, we let the manager'stype affect the value ofboth the assetsin placeand a new investment opportunity. Thus we assumethat yG

>yB, and alsothat 77 e {t]g, t]b} with r]G > ^.Theinformation acquired

by the manager then has positive(gross)socialvalue whenever r\\G> 0.5

and T]B < 0.5.We shall considerin turn two cases\342\200\224the first, when theinformation has no socialvalue, and the second,when the information hasstrictly positivesocialvalue.

5.2.2.1When Information Acquisition HasZeroGrossSocialValue

We first considerthe manager'sincentive for information acquisition under

voluntary disclosurewhen r\\G>r]B> 0.5.Toseethe effectsof endogenousinformation acquisition in the first stage,it is important to bear in mind that

in the information-disclosurestage the problemis essentiallyidenticaltoExample1.Themanager has a positiveincentive to discoverher type hereif outsideinvestors believethat sheis likely to beuninformed in theinformation-disclosure stage,for shecanthen hidebehind type Ybg whenever shelearnsthat her true type is Yb- Moreover,if shelearnsthat her type is Yg, shecan surpriseoutside investors by disclosingthis information and obtainbetter financing terms. Thesebenefits must, of course,bebalancedagainstthe information-acquisition cost yr. Also, these benefits are likely to besmallerthe moreoutsideinvestors believethat sheis in fact informed. It thus

appears that there may be some subtle feedbackeffectsbetweenoutsideinvestors'beliefsand the manager'sincentive to acquireinformation.

As before,we supposethat outsideinvestors form their beliefsrationally,and consequentlywe define the outcome of this game to be a perfectBayesian equilibrium (PBE)where the manager's actions are a best

3. In practice, the certification process plays the role both of improving the manager'sinformation and of credibly disclosing to the market information the manager may already possess,so that the acquisition and disclosure of information can also be thought of as a joint action.

182 DisclosureofPrivate Certifiable Information

responseto investors'beliefsand investors form their beliefsusing Bayes'rule given the manager'sbestresponsefunction. We beginby determiningthe manager'sbestresponsefunction. Supposeinvestors believethat the

manager acquires information with probability p.Then, conditional on

getting the information, the manager is of type yG with probability 0.5.Inthis case,she voluntarily disclosesher type, obtains financing at actuariallyfair terms, and invests in the new project.With equalprobability, her typeis yB, in which caseshe doesnot discloseher type, obtains funding at

actuarially favorableterms, and invests in the new project.How favorablethe terms are dependson whether the manager has an

incentive to invest in the new projectwhen she remains uninformed. Onthe one hand, if she decidesto invest even when she is uninformed, then

type yB can hide behind type yBG and obtain better than actuarially fair

terms. If, on the other hand, the manager prefersnot to invest when

uninformed, then type yB can obtain only actuarially fair terms. Indeed,by

seekinginvestment funds and not disclosingher type she actually revealsher type, sincetype yBG doesnot want to raise funds.

Considerfirst the situation wherethe manager invests in the new projecteven when uninformed. Then outsideinvestors'expectationof firm valueconditional on no disclosureis

v _Q-5(1--pXrG+%)+Q-5(rfl+^)p l-0.5pNotice that dVp/dp =025(yB+T]B-yG-r}G)/(l-0.5p)2<0.In words,

the market value of the firm conditionalon no disclosureis decreasingin

the probability that the manager is informed. This result is quiteintuitive: the lesslikely type yBG is, the more weight outside investors put ontype yB when there is no disclosure.Giventhe market'sconditionalexpectation Vp, the firm's costof capitalwhen the managerremains uninformed isgiven by

\342\200\224{0.5(yG+T]G+yB+T]B)}*p

However,when the managerinforms herself,the expectedcostof capitalis

0.5J0.5+\342\200\224(yB+J]B)\\+\\ff

183 Static Bilateral Contracting

Therefore, the manager decidesto become informed (with positiveprobability) if and only if

o,5+M(r,+^H2y<\302\260-*rG+%+r.^)VP VP

There,is therefore,

1.A (pure-strategy)PBE without information acquisition if and onlyif

Ya +riG-rB-riB4(yG+r]G+7B+r]B)

2.A (pure-strategy)PBEwith information acquisition if and only if

^YG+riG-YB-riB4(7B+77B)

3.A mixed-strategy PBEif and only if

v =Tg+t]g_p l+Ayor if and only if

1> _ (l+4y)(yG+Tfc+yB +r]B)-2(yG+J]G)P

(Yg+Vg)^that is, if and only if

Ya+riG-rB-riB < YG+riG-YB-riB4(.Yg+t1g+Yb+t1b) 4(7b+77b)

We first concludethat when the cost of information acquisition y is

high, there is a unique pure-strategyequilibrium with no information

acquisition. When instead the cost of information acquisition falls below(Yg +1g~Yb-1b)/[4(7b+tjb)], the manager may acquire information

with positive probability. There is a unique full-information equilibriumwhen the cost of information acquisition falls below (Yg+1g~Yb -Vb)/[4(7g+Wg +Yb +1b)]-In other words, as soon as the cost of information

184 Disclosureof Private Certifiable Information

acquisition is sufficiently low, the manager has a strict incentive to

acquire information, and there is then too much information acquisitionin equilibrium. Interestingly, when y/ falls below (7g+*7g ~7b~t\\b)I

[4(7g+^g+7b+^7b)Lthe manager's attempt to take advantage of herprivate information fails, and the cost of information acquisition is

completely wasted.Thereasonis that with full information acquisition ex antethere is alsounraveling and full disclosureex post.

To summarize, when rjG> T]B>0.5,equilibria with voluntary disclosuretend to induceexcessiveinformation acquisition.Only when theinformation acquisition cost is high is the social optimum attained.There is,however,a simple way of restoring efficiencyhere.It suffices to adopt amandatory-disclosurelaw, that is, to threaten to punish the manager forfailing to discloseany information she might have acquired.Then the

manager obtains no return from information acquisition.But, as notedbefore,mandatory-disclosurelaws may be ineffective here if it is difficult

to prove that the manager is informed.

5.2.2.2When Information Acquisition HasPositiveGrossSocialValue

Considernow the manager'sincentive for information acquisition under

voluntary disclosurewhen rjG> 0.5> 77B.Two caseshave to bedistinguished:

one where it is efficient to invest when uninformed about the value of 77,

that is, when 0.5(77G+ rjB) > 0.5;the other when it is not worth investing,that is,when 0.5(77G+ r\\B) < 0.5.

In the secondcase,the equilibrium outcome is easy to see:when the

manager is uninformed, she doesnot invest in the new project;therefore,onceinformation hasbeenacquired,type yB cannot\"hide\" behind type yBG.

The net value of information acquisition for the manager, computedexante, is then 0.5(77G-0.5)- yAThis alsohappensto be the net socialvalueof information. Therefore,in the subcasewhere 0.5(77G+ r]B) < 0.5,the

manager has the correctincentives for information acquisition under

voluntary disclosure.Note that in this subcasea mandatory disclosurelaw

would not make any difference.In the former subcase,where0.5(77G+ r]B)> 0.5,the manager'snet value

of information is at most 0.5(0.5-r]B)- yr. the value of finding out that it

is not worth investing if the manager'stype is yB. In this case,the manageralsohas the correctincentives for information acquisition under voluntarydisclosure.A mandatory disclosurelaw here would again make nodifference. But, if the manager prefers to invest when her type is yB, then there

185 Static Bilateral Contracting

may be too much or too little information acquisition relative to the socialoptimum. A type-^manager here invests if and only if

(7b+^)(1--^-WbGiven that r\\B

< 0.5,the manager would not have an incentive to invest if

p =1,sincethen

(7b +Vb )(1-y-J=Yb +Vb-0.5< yB

But the manager may have a strict incentive to invest if, at p =0,

V VqJ YG+lG+rB+riBThereis then a mixed-strategy equilibrium given by

v 0m5r*\302\261H>L

1bor

?7b (7g +%+7b +77b )-(7b +77b )P = -

T7b(7g+%)-0.5(7b+77b)In this mixed-strategy equilibrium the manager acquirestoo little

information relative to the socialoptimum when 0.5(0.5-rjB) > y/, and toomuchwhen 0.5(0.5-rjB) < yr. Thesedistortions canagain becorrectedby

imposinga mandatory-disclosurelaw, sinceunder such a law the mixed-strategy

equilibrium breaksdown, and the type-7B manager strictly prefersnot toinvest, so that the manager's incentives for information acquisition arealigned with socialincentives.

5.2.2.3Conclusion

We have found that, when the managercan acquireinformation at a cost,thereoften tendsto betoomuch (but at times too little) information

acquisition under voluntary disclosure.In this setup,mandatory-disclosurelaws,which threaten to punish managersfor failing to discloseany information

they might have acquired,have beenshown to correctthesedistortions andto lead to sociallyoptimal information acquisition.This finding assumes,

186 DisclosureofPrivate Certifiable Information

however,that such laws are effective,that is, that managerscannot succeedin falsely claiming to beuninformed.

5.2.3No Voluntary DisclosureWhen the Informed Party CanBeEither a Buyeror a Seller

As we show in this section,limited voluntary disclosureof information caneasilyresult when the informed party is not always on the samesideof atransaction, aswe have assumedsofar by letting the informed party alwaysbe a sellerof equity. We shall explorethis insight by consideringinformation disclosureby tradersin financial markets. Financial market regulationssometimesrequiresomeinvestors to disclosetheir trades.Forexample,corporate managersand insidersin the UnitedStates(investorswho own morethan 5 percent of a firm's stock) must disclosetheir trades in the firm'sstock.

We providea simple exampleadaptedfrom Fishman and Hagerty (1995)illustrating how informed traders who may decideto buy or sell a stockhave little or no incentive to voluntarily disclosetheir information or their

trades.Therefore,mandatory-disclosurerulesmay berequiredto make that

information public.We show, however,that theserulesmay not necessarilybenefit uninformed traders.

Considerthe secondarymarket for sharesof the firm discussedearlier,and supposefor simplicity that there are no new investment opportunitiesavailableto the firm. Recallthat the final value pershareof the firm isuncertain and may beeither1,with probability y or0,with probability 1-^There-fore, the current expectedvalue of a share in the secondarymarket,

assuming away discounting, is y Themanager of the firm may have privateinformation about the expectedfinal value ofshares.With probability /Jshelearnswhether the sharesare worth yG > /oryB < y. Uninformed

shareholders, however,know only that y= yG with probability 0.5.Sincethey are allassumedto berisk neutral, they value eachshareat 0.5(yG+ yB)

=yex ante.Supposethat there are two trading dates,t = 1and t =2, and that each

shareholdercan buy or sell only one share per trading date.The examplecan be adapted to allow for trading of a finite number of sharesat eachtrading date.What is important for the exampleis that trades are spreadoverthe two trading dates.Although we do not modelthem explicitly here,there are goodreasonswhy this would be the casein practice:most

importantly,concentrationoftradesin a single trading periodwould result in both

higher pricerisk and moreadverseterms of trade.

187 Static Bilateral Contracting

Suppose,in addition, that trading in the secondarymarket is anonymousand that the sizeof the manager'strades is small relative to the market, sothat she cannot move the price.With probability (1- /J), the manager isuninformed. Sheis then assumedto be equally likely to buy or selloneshare for liquidity or portfolio-rebalancingreasons.When she is informed,however,she sellsone share at date 1when she learns that the final

expectedvalue is yB and buys one share when she learns that the final

expectedvalue is yG. Hernet payoff from trading at date 1(her\"informational rent\") when she is informed is then

0.5(rG-y)+0.5(7-yB) =0.5(rG-yB) =0.5Ay

Therefore,if she doesnot discloseher information or her trade at date1,the manager can expectto get a total payoff of 2/J0.5Ay If she disclosesher trade at date 1,however,the price at which she can trade at date 2will move against her. If she disclosesa \"buy,\" the price of shares in the

secondarymarket will risefrom yto

j37G+(l-j3)7The price moves to this level becauseuninformed traders update their

beliefsthat the underlying expectedvalue of sharesis yG following a \"buy\"

by the informed party. Theyreason that the manager couldhave bought asharebecauseshe was informed (with prior probability /J) or becauseshewas uninformed but repositionedher portfolio[with prior probability (1-P)].The informed manager'spayoff from trading at date 2 is then only

(l-j3)(yG-y)

Similarly, if she disclosesa \"sell,\" the pricewill fall from yto

j3rB+(l-j3)rand her payoff from trading at date 2 is only

(l-j3)(y-yB)

Clearly, an informed manager would prefer not to discloseher date-1trade.When her trade was a \"buy,\" she would lose/J(yG

-y), and when it

was a \"sell,\" she would lose/J(y- yB).If her private information wereverifiable, shewould alsonot want to discloseher information. Often,however,private information isnot verifiable,and only tradesare.It is for this reasonthat regulation concentrateson disclosurerequirements of trades.

188 Disclosureof Private Certifiable Information

What about the disclosureincentives of an uninformed manager?Ironically,

uninformed managershave a weak incentive to disclosetheir tradeshere.Indeed,if their disclosureis taken as news by the market, they standto gain from the resulting price manipulation. Suppose,for example,that

an uninformed manager disclosesa date-1\"buy.\"This couldresult in a priceincrease,which the manager knows to be unjustified. Shecould then gainfrom this pricemove by selling one share at date2.Sheknows the share'sexpectedvalue is % but shewould be able to sellit for a higher price.Ofcourse,in a perfectBayesianequilibrium this couldnot be the case:sinceonly uninformed managers would disclose,disclosurewould simply beignoredby the market.

Thereare two reasonswhy informed managerscannot beinducedto

voluntarily disclosetheir trades here.First, and foremost,the market herecannot punish nondisclosure,sinceit doesnot know the nature of theinformation. If the market interprets nondisclosureas the managerwithholdingbad news, then the manager can profit by buying shares in the secondarymarket, and vice versa by selling shares.The secondreason why the

manager doesnot discloseinformation voluntarily is that her first tradedoesnot move the market. Supposethat trading in the secondarymarketwas not anonymous; then any party trading with the manager at date 1would only acceptto sella shareat price j3yG +(1-j3)y and to buy a sharefrom the manager at price (5yB +(1-/3)y. If the manager had no choicebut

to trade at thoseprices,shewould be indifferent betweendisclosingor not

disclosingher first trade.If there is no voluntary disclosureof trades by the informed manager,

can a casebe made for mandatory disclosure?Note first that mandatorydisclosurecan be justified here only as a way of protectinguninformedtraders.Indeed,in this simple example,trading under asymmetricinformation doesnot induceany distortions. It affectsonly the distribution ofvalueacrossshareholders.In a richersetup,however,one can imagine that sometraders' informational advantage might adversely affect liquidity andinvestment. Disclosurepolicy might then be directedat rninimizing suchinvestment or liquidity distortions. It is conceivablethat this might requireprotecting uninformed investors.

Interestingly, even if the objectiveof protectinguninformed investors istaken for granted, there is not always a casefor introducing mandatory-disclosurerules in this simple example.That is, there may be situations

where a mandatory-disclosurerule actually benefits the manager at the

189 Static Bilateral Contracting

expenseof other shareholders.How can such situations arise?We haveshown that an informed manager is always worseoff as a result of thedisclosure ofher date-1trade.Therefore,if there is a gain to beobtainedfroma mandatory-disclosurerule, it would have to befor uninformed managers.As we have hinted, uninformed managersactually gain under mandatorydisclosure.Theyobtain an informational advantage becausethey know that

the pricemove following the disclosureof their trade was not warranted.

Following disclosureof a \"buy,\" uninformed managers obtain a net

expectedgain from a date-2\"sell\" of

Piya-r)and following disclosureof a \"sell,\" they obtain a net expectedgain by

buying a shareat date 2 of

P(Y-Yb)

Thus, on average,an uninformed manager gains (P/2)(yG-yB)undermandatory disclosure,but an informed manager can expect to lose(P/2)(yG-7B).Thenet gain from mandatory disclosureis therefore

^Yg-Jb)+ (!-\302\243) ^(Yg-Yb)

In otherwords,if the manageris lesslikely to beinformed than uninformed

(that is, j3 < V2), it is she,and not uninformed investors, who gains from amandatory-disclosurerule.

This examplebrings out the closeconnectionbetweendisclosureandmarket manipulation. Heremandatory disclosuremay facilitate market

manipulation by increasingthe credibility of the disclosedinformation. Ifdisclosurewas voluntary, rational uninformed participants would simply

ignore the information, since they understand that only an uninformed

manager would have an incentive to disclose.4Let us end this subsectionby stressing that Fishman and Hagerty's

example is yet another illustration of the subtle effectsof mandatory-disclosurerules.Theiranalysis and Shavell'sactually do not make a strongcasefor mandatory-disclosurerules.A more recent articleby Admati and

4. Interestingly, for someparameter values, one can check that mandatory disclosure couldactually lead to an extreme form of market manipulation here, where the informed managerwould begin by selling when observing yG soas to beable to buy shares back on the cheap.

190 Disclosureof Private Certifiable Information

Pfleiderer(2000)makes a strongercase.They assume that disclosureofinformation involves transaction costsand that there is an informational

externality on otherfirms. In otherwords,disclosureis partly a public good.Sinceindividual firms do not capturethe full socialinformational value ofdisclosureand since they must pay the full cost, there will generally beunderprovision of information in equilibrium. To improve the productionof socially valuable information, Admati and Pfleiderer argue that

mandatory-disclosurerulesmay berequired.

5.3 CostlyDisclosureand DebtFinancing

We closethis chapterwith a treatment of an influential researchline onfinancial engineering, the costly state verification (CSV)approach.TheCSVperspectivestarts from the premisethat when disclosure(or verification)of a firm's performance(profits or earnings) is costly,the designoffinancial contractsmay be driven to a largeextent by the objectiveofminimizing

disclosure(or audit) costs.A striking and central result of the CSVapproachto financial engineering is that it is generally optimal to committo a partial, state-contingentdisclosurerule. What is more,under someadmittedly strong conditions,the optimal rule is implemented by a standarddebt contractfor which there is no disclosureof the debtor'sperformanceas long as debtsare honored,but there is full disclosureor verification in

the event of default. Viewed from the CSVperspective,the main function

of bankruptcy institutions is to establisha clear inventory of all assetsandliabilities and to assessthe net value of the firm.

The CSVapproachto financial contracting considersa slightly different

disclosureproblemthan we have seenso far in this chapter.The informed

party now has to make a decisionwhether to certify its information to beable to discloseit credibly.

The financial contracting problemconsideredby Townsend(1979)andGale and Hellwig (1985)involves two risk-neutral agents, an entrepreneurwith an investment projectbut no investment funds, and a financier with

unlimited funds. In its simplest form the contracting problemis as follows:The fixed investment requiresa setupcostofI> 0 at t =0 and generates

random cashflows at t = 1of n e [0,-h\302\273),with density function f(n).The

entrepreneurobservesrealizedcashflows and cancredibly disclosen to theinvestor only by incurring a certification cost K > 0.The contract-design

191 Static Bilateral Contracting

problemis to specifyin advancewhich cash-flowrealizationsshould becertified and which not.

It is easyto seethat absentany certification the entrepreneurwill neverbe able to raise funding for the project.Indeed,the rational investor then

anticipatesthat at t = 1the entrepreneurwill always pretend that n = 0 toavoid making any repayments to the investor.

One way, of course,of guaranteeing financing is to always verify or audit

the realizedreturn of the project.Securitiesregulations mandating theperiodic disclosureof firm performancecan be seenas instancesof suchsystematic verification of performance.Interestingly, the CSV perspectivehighlights the potential inefficiency of such regulations. Indeed,we shall

explain that such systematic disclosurerules tend to imposeexcessivedisclosure costs.

The set of contractsfrom which the contracting parties can chooseisdescribedas follows:In generalterms, the contractwill specifywhether anaudit should take place following the realizationof cash flows and what

fraction of realizedcash flows the entrepreneur should pay back to theinvestor. More formally, we can reduce the set of relevant contractsby

appealingto the revelation principle(seeChapter2).Thus, following the realizationof n, the entrepreneur(truthfully) reveals

n, and the contractcanspecifythe probability p(n)e [0,1]of certifying (orauditing) cash flows. When there is no certification, the contractcan only

specifya repayment r(n).But, when there is certification of cashflows thecontractcan specifya repayment contingent on both the announcedcashflow it and the (true) cashflow certifiedby the audit n, r(n, 7t). Applicationof the revelationprinciplewill, of course,ensure that in equilibrium the

entrepreneuralways truthfully reports cash flows: it = n, for all n.Nevertheless, the contractwill specifya different repayment when it ^ rcasa threat

(which will neverbeexercisedin equilibrium) to provideincentives to the

entrepreneurto tell the truth. It is easyto seethat, sincethis threat is notexercisedin equilibrium, it is efficient to providemaximum incentives totell the truth by maximizing the punishment for lying and setting r{it, n) =n whenever it ^ n.5 Undersuch a threat there is no benefit from falselyreporting a cash flow that triggers an audit for sure. Consequently, such

5.Since the entrepreneur has no resources to start with, it is impossible to induce arepayment in excessof %.

192 Disclosureof Private Certifiable Information

cash-flowrealizationsare always truthful, and we candenotethe repaymentfor audited cashflows simply as rjjz).

As we alreadyalludedto, a centralresult of the CSVapproachtofinancial contracting is that standard debt contractsmay be optimal financial

contracts.This result only obtains,however,under fairly strong assumptions.One critical assumption, in particular, is that financial contractscanspecifyonly deterministic certification policies:

assumption: For all n, p(n)e (0,1}.In other words, commitment to random audits is not feasible.We now

show how under this assumption a standard debt contract is an optimalfinancial contract.

Assuming that the projecthas a positive net present value, the optimalcontracting problemreducesto a problemof minimizing expectedaudit

costs,subjectto meeting the entrepreneur'sincentive constraints for truth

telling, and the financier'sparticipation constraint:

p(iz)f(iz)diz,\342\200\236\342\200\236.v..,\342\200\236av\342\200\236,

opin

subjectto:

(1)the investor'sindividual-rationality constraint:

(2) a set of incentive constraints:

ra(ni)<r(n2) for all tz\\ *k2 such thatp(^)=1a.ndp(n;2) =0

r(7ii) =r(n2) =r for all K\\*n2 such thatp(^)=0=p(n2)

and

(3) a set of limited wealth constraints:

ra (k)<k for all n such that p(n) =1r(n)<n for all n such that p(n) =0

The incentive constraints have a particularly simple structure. For anytwo cash-flowrealizationsthat do not requirecertification, the repaymentto the financier has to be the same.Obviously,if this werenot the case,the

193 Static Bilateral Contracting

rfa); ra{n) \302\253

45\302\260 line

p(iz) = 0No audit

p(7t) = 0No audit

Figure 5.1An Incentive-Compatible Repayment Schedule

entrepreneurwould want to lieabout realizedcashflows and announcetheone with the lower repayment. Similarly, a cash-flowrealization%y that

requirescertification should not entail a higher repayment ra{n{) than the

repayment for a cash-flowrealizationJh that doesnot involve certification

r{na). If that were the case,then the entrepreneurwould want to lie aboutthe realizationof tz\\ and thus make a smallerrepayment. The set ofincentive and limited wealth constraints thus imposessweepinglimits on the setof feasiblecontracts.As Figure 5.1illustrates, all feasiblecontractshave thetwo featuresthat repayments outsidethe certification subsetmust beindependent of realizedcash flows, and that repayments in the certificationsubsetmust be lessthan thoseoutsidethe set.

Basically,the only dimensions along which the financial contractcanbevaried are (1)the certification subsetand (2) the sizeof the repayment in

the certificationsubset.Characterizingthe optimal contractcanbeachievedthanks to the following observations.First, it is easy to seethat anyfeasible contractmust include the cash-flowrealizationn=0 in the audit subset.If this werenot the case,then the entrepreneurcouldalways claim to havea cash flow of zero, thus avoiding an audit as well as any repayments tothe financier.Second,it is alsostraightforward to seethat any contractthat

Disclosureof Private Certifiable Information

minimizes expectedaudit costsmust be such that for any cash-flowrealization n in the audit subset,ra(n) = min{^, r}.This result can be seenasfollows.Supposefirst that r < n.Then incentive compatibility precludesany

ra(n) > r, as we have alreadynoted.But a contractwith ra(n) < r would

involve inefficiently high audit costs.Indeed,it would bepossibleto raiserjjz) to r without violating any incentive or wealth constraints. Doingsowould relax the financier'sparticipation constraint by generating higher

expectedrepayments and thus would make it possibleto slightly reducetheaudit subset(and thereforeexpectedaudit costs)without violating the

participation constraint. Supposenow that n < r.Then any contractsuch that

ra(n) < n would be inefficient for the samereason.Figure 5.2modifies the

repayment scheduleof Figure 5.1to take advantage of theseobservations.The third and final observationis that any contractwith a disconnected

audit subset[0,tc\\ u [n, n\\ (with n < n) as depictedin Figure 5.3would beinefficient, sincethen an obvious improvement is availableby shifting to aconnectedsubsetwith the sameprobability mass (seeFigure 5.4).Suchashift would make it possibleto raise r, and thereforealsoto raiseexpectedaudit repayments ra(n). Once again, the higher expectedrepayments thus

generated would relax the participation constraint and enable a small

saving of expectedaudit costs.

r(7t); ra(7t) m

45\302\260 line

P(tz) = 0 p(n) = 0

Figure 53.Incentive-Compatible Repayment Schedule with ra (n) =min(?i;, r]

195 Static Bilateral Contracting

/\342\226\240(\302\253); ra(7c) I

45\302\260 line

Audit Audit

Figure 5.3Inefficient Audit Region

r(7t); ra(7t)45\302\260 line

Figure 5.4Efficient Audits

196 DisclosureofPrivate Certifiable Information

Piecingall theseobservationstogether,we are able to concludethat the

uniquely optimal financial contract,which minimizes expectedaudit costs,is such that (1)there is a single connectedaudit region IIa = [0,n\\, with

7r < oo; (2) over this regionaudit repayments are ra(n) = tt, and (3) for cashflows n> n, which arenot audited,the repayment is r =Jz.Theunique cutoff

% is given by the solution to the participation constraint

T (k -K)f{n)dn+[1-F(k)]r =IJO

and expectedaudit costsare F(7i)K.In otherwords, the optimal financial contractis equivalent to a standard

debt contract with facevalue It, which alsogives the creditorthe right to all

\"liquidation proceeds\"n in the event of default. Underthis interpretationK is a fixed bankruptcy cost that must be borne to be able to obtain any

liquidation proceeds.That standard debt contracts have the property that they minimize

expectedcertification or disclosurecostsis an insightful and importantobservation.It shedslight on anotherdesirableproperty of debt contracts,and it highlights the potential costsofmandatory-disclosureregulations that

requireequaldisclosureofperformancewhether goodor bad.Interestingly,the CSVapproachhighlights the benefitsofmorefine-tuned disclosureregulations, which would callfor more disclosurewhen performanceis bad.

It is important to bear in mind, however,that the optimality of standarddebtcontractsin a CSVframework only obtains under strong assumptions.As Gale and Hellwig (1985)have shown, standard debt contractsare nolongeroptimal when the entrepreneur is risk averse(seeQuestion19in

Chapter14).Similarly, when random audit policiesare introduced,standarddebt is no longeroptimal (seeMookherjeeand Png, 1989).

Another important weaknessof the CSVapproachis that standard debtcontractsareoptimal only if the partiescancommit to the audit policyspecified in the contractat t =0.If they cannot commit to the audit policy,theywould not want to carry out the audit calledfor by the contractex post,sinceto do sowould be wasteful. Indeed,if the entrepreneurexpectstheaudit to take placeas specifiedin the contract,then shewill always reportcash-flow realizationstruthfully. But if truthful reporting is anticipated,there is no point in carrying out the audit. Interestingly, the needfor acommitment devicecouldprovidea justification for mandatory-disclosurerules,

197 Static Bilateral Contracting

sincesuch rules typically cannot be renegotiatedby individual contracting

parties.Finally, the result that standard debt contractsare optimal financial

contracts doesnot extendto slightly richersettings, wherethe firm may investin more than one project,where the investment extendsover more than

oneperiod,or in the presenceofmultiple financiers (seeGaleand Hellwig,1989;Chang, 1990;Webb, 1992;Winton,1995).

5.4 Summary and LiteratureNotes

The private-but-certifiable-information paradigm is very useful andrelevant for a wide rangeof applications,including health, innovation, auditing,and financial market trading. Underthis variation of the adverseselectionparadigm detailedin Chapter2, the informed party is not allowedto claimto beof a type different from her true type: her only choiceis betweenfull

disclosure,partial disclosure,and no disclosureof the truth. Such aparadigm has generateda key result, known as the \"full-disclosure theorem,\"first establishedby Grossman and Hart (1980),Milgrom (1981a),andGrossman (1981),and subsequently generalized by Okuno-Fujiwara,Postlewaite,and Suzumura (1990).This result highlights that full disclosureobtains in equilibrium when the following four conditionsare satisfied:(1)it is common knowledgethat the informed party is indeedinformed abouther type: (2) the type spaceis one-dimensional,and types can be ranked

monotonically in terms of payoffs of the informed party; (3) the informed

party is able to senda certifiedmessageproving that her type is not below(resp.above) her true type; and (4) disclosureis costless.An important

implication of this result is that when the conditionsfor full disclosurearemet, one should not expect\"privacy laws\" to bevery effective.Even if

individuals have the right not to discloseparts of their personalhistory or torefuseundergoing a test, if they are likely to be drawn into disclosingthis

information voluntarily, then these privacy protectionsdo not have muchbite.To strengthen these laws one might also have to let individuals getaway with forging certificates.Otherwise\"good types\" will want to

voluntarily certify their types, thus inducing \"worse types\" to do soaswell.When all conditionsfor full disclosureare not met, voluntary disclosure

is not going to be sufficient in generalto generateall relevant publicinformation. Mandatory-disclosurerules may then be desirable.Only a few

198 DisclosureofPrivate Certifiable Information

formal analysesof the role of mandatory-disclosurerulesas a supplementto deficient voluntary disclosureexist.Shavell (1994)and Fishman and

Hagerty (1995)identify conditionswhen mandatory-disclosurerules maybe desirablegiven that voluntary disclosureis incomplete.Interestingly,both analysespoint out that when mandatory-disclosurelaws are desirablethey alsotend to be difficult to enforce.One reasonis that mandatorydisclosure is generally desirablein situations where the informed party canpretend to be ignorant. But then it is much more difficult to prove that oneis uninformed (or,for example,that one doesnot possessa weapon).

The last sectionof the chapterhas detailed the costly state verification

approachto optimal financial contracting.Thisapproachfocuseson a one-shot financing problem of an entrepreneur who privately observes thereturn of the projectwhile the financier observesit only at a positive cost.Townsend(1979)and Galeand Hellwig (1985)have shown that the optimalfinancing mechanism, that is, the mechanism that minimizes expectedverification costs,is a standard debt contract, under the following assumptions:(1) only deterministic contracts are possible;(2) both parties are risk

neutral; and (3)the partiescancommit in advanceto undertaking the costlyverification policy.Thesepapersthereforeprovidean alternativejustification to the optimality of debt contractsto the moral-hazard-basedone dueto Innes (1990),discussedin Chapter4.

MultidimensionalIncentive Problems

Mosteconomiccontracting problemscannot beeasilyreducedto pureone-dimensional adverse-selectionorpureone-dimensionalmoral-hazardproblems. Either they involve multidimensional characteristicsor multiple tasks,or they combineboth adverse-selectionand moral-hazardfeatures.Thischapterillustrates how the purecasesofadverseselectionand moral hazardcanbeextendedto incorporatemultidimensional aspectsand how hidden-action and hidden-information considerationscan be combinedwithin asingle framework. Theseextensionsdo not involve any fundamental

reformulation of the general theory of contracting under asymmetricinformation. As will becomeclear,most extentionscanbeanalyzed by adapting themethods outlined in previouschapters.

There are, obviously, many different ways of introducingmultidimensional incentives into the standard frameworks. Indeed,many different

extensionshave alreadybeenexploredin the literature\342\200\224far too numerousto be all coveredin this chapter.We shall exploreonly three of the most

important extensions.Thefirst introducesmultidimensional aspectsinto thestandard screeningproblemand the secondinto the standard moral-hazardproblem.Thethird combinesadverseselectionand moral hazard in a simpleway.

6.1AdverseSelectionwith Multidimensional Types

The classicsituation of adverseselectionwith multidimensional types isthe multiproduct monopoly problem.This is the problemof any sellerwith

some market power who sellsat least two different goods.An extremeexampleof a multiproduct monopoly problem is that faced by a largesupermarket or department store.A supermarket sellsseveral thousand

different items. It can offer quantity discountson any one of these itemsand specialdealson any bundle of them. Sothe optimal nonlinear pricing

problemof a largesupermarket is potentially very complex.One approachto this problemmight be to simply considereachgoodin

isolationand determinea nonlinear price for that good separately.Thiswould involve only a minor conceptualextensionof the basicsingle-productmonopoly problem.But the multiproduct monopolistmay fail to maximize

profits by taking this approach.Indeed,it is easy to construct exampleswhere he would gain by bundling different goodstogether (and sellthebundle at a different pricefrom the sum of the componentparts).We shall

6

200 Multidimensional Incentive Problems

provide one example later. This bundling option cannot be analyzednaturally as a one-dimensionalscreeningproblem;a full analysis requiresa multidimensional extensionof the basictheory consideredin Chapter2.Becausesuch an extensioncanbe technically rather involved, we beginby consideringthe simplest bundling problem,with two goods,and with

one unit of eachgoodonly. We then proceedto analyze the multiunit case.Thebundling problemof a multiproduct monopoly was first analyzed by

Adams arid Yellen(1976).Theyconsidera monopolistselling two different

items, good1and good2.Thebuyer'sreservationvalues for eachof the two

goodsare given by (v!,v2) > 0,and the costsofproductionof eachgoodarecx > 0 and c2> 0.The sellerdoesnot know the true reservationvalues andhaspriorbeliefsrepresentedby the cumulative distribution function F(vu v2).

Note that this formulation excludesany complementaritiesinconsumption. If such complementaritieswere present, there would be a natural

reason for bundling the two productstogether.What is more interestingand surprising is that even in the absenceof any complementaritiesthesellermay gain by bundling the two products.

The sellerhas three different pricing strategies:(1) sell each goodseparately;(2) offer the two goodsonly as a bundle; and (3) sell eitherseparatelyor bundled.We begin our discussionwith an example,due toAdams and Yellen (1976),illustrating how the sellercanbenefit by sellingthe two goodseitherbundled or separately.

6.1.1An ExampleWhere Bundling IsProfitable

In this examplect = 0, and the buyer may have four different

characteristics,

eachwith equal probability; that is, the reservationprices(v1;v2)maytake four different values given in the following table:

State ofNature Vi v2 Prob

A 90

B 80

C 40

D 10

It is easy to verify that, when selling each good separately,the sellermaximizes expectedprofits by setting individual prices,PX =P2=80.In this

case,each good is soldwith probability A-, and expectedprofits are 80.

10 i4

40 i4

80 i4

90 i4

When offering the bundle only (\"pure bundling\,") the sellermaximizes

profits by setting the priceof the bundle, Pb,at 100,at which it is soldwith

probability 1,thereby improving upon \"no bundling.\" Finally, selling both

separatelyand bundled, a strategy Adams and Yellen refer to as \"mixed

bundling,\" the optimal pricesare Pi =P2=90,Pb =120.Thisis the optimalstrategy, sinceexpectedprofits are then 105:the bundle is soldwith

probability |,and eachgood is soldindividually with probability ^.Having establishedthat bundling may beprofitable, the next question is,

Howbroad is the rangeof caseswherebundling dominates?On the basisof this example,as well as others,Adams and Yellen concludethat mixed

bundling tends to dominate selling each good separatelywhenever \"the

correlationcoefficientlinking an individual's valuation of one good to his

valuation of the other goodis not strongly positive.\" A moregeneralcharacterization is given by McAfee,McMillan, and Whinston (1989),hereafterMMW.

When Is Bundling Optimal?A LocalAnalysis

Following MMW, we considergeneralenvironments wherethe cumulativedistribution i7(v1,v2)representingthe prior beliefsof the selleriscontinuously

differentiable with support [vi,vi] x [v2,v2].Also, we define G;(v;lv;)[and giiy^Vj)] as the associatedconditionalcumulative distribution (anddensity) functions, and H^vi) [and /z\302\243(v,)]

as the associatedmarginaldistribution (and density) functions for i =1,2.Following MMW, we candistinguish

betweenthe \"home bundling\" case,where the buyer can \"construct

her own bundle\" by buying each goodseparately,and the casewhere thesellercan prevent the buyer from purchasing more than one goodseparately. In the first casethe sellerfaces the constraint that Pb <

P\302\261+ P2.

We assumehere that home bundling is possibleand let (Pf, P2) be the

monopoly pricesunder separate sales.When would the introduction ofbundling raise profits above those obtained with separate monopolypricing? A sufficient condition for bundling to dominate is easilyobtainedby asking whether a policyof introducing a bundle at price Pb = P*+ P*and selling good 2 at the slightly higher price P2= P* + e (e> 0) raisesthe seller'sexpectedprofits. Undersuch a bundling policy,\342\200\242 All consumertypes with vi > Pf and v2 < P*would consumegood1only.\342\200\242 All consumertypes with v2 > P*+ e and vi < Pf -e would consumegood2 only.

202 Multidimensional Incentive Problems

Po+ e 1U

Figure 6.1Mixed Bundling

\342\200\242 All consumertypes with vx + v2> P% + P*, v2 > P*, and vx > P% - epurchasethe bundle (seeFigure 6.1).

Notice that the introduction of a bundle at price Pb = P% + P\\ and theincreasein price for good 2 to P2 = P* + \302\243 affectneither the purchasingdecisionsof consumertypes with vx > P*nor the price they endup paying.

Any gain from introducing bundling must thereforebe obtainedfrom theconsumertypes with vx < P*.Theseller'sprofit from theseconsumertypesas a function of e is

* _

K(e)=(pt+\302\243-c2)j^'\302\243\\jV^J(v1,v2)dv2\\dv1

+(pt+Pt-Cl-ci)QJ^+pf_vif(yl1v1)dv1]dVl

For e small, this equationcanbe rewritten as

- r *K(e)=(pt +

\302\243-c2)\\vl \\

1

f(yi,v2)dvi

+(Pf -Ci)Jf*^[J^*/(vi,v2)dv2] dVi

(6.1)

dv7

203 Static Bilateral Contracting

Intuitively, raising e above 0 has two effects:(1)the usual monopolytrade-offbetween price and quantity, but limited to those buyers who

considerbuying good 2 or no good at all, becausevx < Pf; and (2) anincreasein the sale of good 1,bundled with good2, for those buyers forwhom Vi is \"just below\" Pf while v2 > Pf. The secondeffectis positive, sothat bundling is attractive for the seller,providedthe first effectis not toonegative.

It is thus a sufficient condition for bundling to be profitable that thederivative of 7t(e) with respectto ebepositiveat e=0, that is, that

+{P*i-c1)fef(p*1,v2)dv2>0

which canbe rewritten as

J^{[l-G2(pf|v1)]-(P*-C2)g2(p*|v1)}^1(v1)dv1+(pr-Cl)[l-G2(p?|Pf)]^(pt)>0PtThesecondterm in this inequality correspondsto the areaabedin Figure

6.2.It representsthe additional profit the monopolist cangeneratethrough

bundling from thoseconsumerswho without bundling would buy only good2 but with bundling are encouragedto buy both goods(effect2).Thisterm is always positive.The sign of the first term, which representsthe

monopoly trade-offfor buyers for whom Vi < Pf (effect1),is ambiguous. Itturns out, however, that, if the reservationvalues are independentlydistributed, this first term is of secondorder,becausePf is alsothe monopolyprice for buyers for whom Vi < Pf. It therefore vanishes, and bundling

strictly dominates.To seethis result, note that, in this case,the precedinginequality simplifies to

H\\pX)ll-H1{pi]]-{p%-c1)h1{pl)}+(p*1-c1)[l-H2(p*2)]h1(pt)>0

Since Pf is the monopoly price under separate sales,we must have

[1-tf2(Pf)]-(Pf -c2)/z2(Pf) =0, so that the inequality reducesto

204 Multidimensional Incentive Problems

v2 A

Po+8

Figure 6.2Additonal Salesthrough Bundling

(pr-Cl)[i-7f2(p!)]^(pt)>o (6.2)This condition always holdsprovided that there is a positive measureofreservationvalues abovecost.

In sum, bundling is optimal for the multiproduct monopoly for alarge range of cases.In particular, it is optimal in the case wherethe demands for the two goodsby the consumer are unrelated (orindependent).

6.1.3Optimal Bundling: A GlobalAnalysis in the 2x2Model

From MMW's analysis we wereable to identify sufficient conditionsunderwhich bundling dominatesno bundling locally.Following Armstrong andRochet (1999),we now providea globalanalysis of the optimal bundling

problem in the specialcasewhere there are only two possiblevaluations

for eachgood:Vi e {vf;vf} with Ai = vf- v[> 0, and v2 e {v^v?} with A2 =v?-V2> 0.Thereare then only four possibletypes of consumers,(vL,vL),(vL,vH), (vH,vL), and (vH,vH). Theseller'sprior probability distribution overthe buyers' types is Pr(v,-,v;) = #,;i = L,H;j = L,H.We shall make theinnocuous simplifying assumption that fiLH

=j3HL

= j3.

205 Static Bilateral Contracting

The seller'sproblemhere is to offer a menu of contracts{7^-,x\\, xl) tomaximize his net expectedprofit, where Tq denotes the payment and x\\

(resp.x$) the probability of getting good 1(resp.good 2) for buyer type

(v,-,v;).The seller'sexpectedprofit is then

n =XA/ft -*ici ~x2ci)y

and the buyer'snet surplus (or informational rent) is

R^^vi+xH-TtjA more convenient expressionfor the seller'sexpectedprofit is

K =S(x)-JJj3ijRi,o

where

S(x)=JJl5ij[x!(v[-c1)+4{vi-c2)]y

denotesthe expectedsocialsurplus.The constraints facedby the sellerare,as usual, incentive-compatibility

(IC)and individual-rationality (IR) constraints. As in the one-dimensionalproblem,the individual-rationality constraint RLL > 0and the incentivecompatibility constraints imply that the other individual-rationality constraintsare satisfied.As for the setof incentive constraints, it canbewritten as

R(j> Rkl +Xlkl{v{ -v{)+x?{v{-v{) for all ij and kl

As in the one-dimensionalproblem, incentive compatibility impliesmonotonicity conditionsfor the saleprobabilitiesx\\ and x%

LL1 \342\200\224 xl 5 Xi d.X\\ , X2 ^ Xl , X2 ^.X2y1Xl

In words, incentive compatibility requires that for either good the

probability of getting it is (weakly) increasingwith the buyer'svaluation ofthat good.If these conditionsdo not hold, truthful revelationof typescannot be hopedfor.

In the one-dimensionalproblem,whether we have two or moretypes, allincentive constraints except the downward adjacentonescan be ignoredwhen the buyer's preferenceshave the single-crossingproperty and the

monotonicity conditionshold (seeChapter 2).Unfortunately, this useful

206 Multidimensional Incentive Problems

property doesnot extendto the multidimensional problem.Nevertheless,weshall beginby characterizing the solution to the seller's\"relaxedproblem,\"whereonly the downward incentive constraints,the monotonicity conditions,and the individual-rationality constraint for type LLare imposed.We then

identify situations wherethe solution to the relaxedproblemis the solutionto the seller'sproblem.Finally, we characterizethe solution to the seller'sproblemin situations wherethe relaxedproblemis not valid.

In our two-dimensionalproblem,the downward incentive constraints are

#hl>All+*iLLAi

Rlh>Rll+*2llA2

RHH > tmlx{Rlh +x^AuRhl +x?LA2, RLL +xtLAx +x2LLA2}

Given that the individual rationality constraint binds at the optimum forthe relaxedproblem(RLL =0),theseincentive constraints canbe simplifiedto

RHL>XlLLAi

RLH>x2LLA2

RHH >xtLAl+x2LLA2 +max{{x1LH -xtL)Ax,{x?L-x2LL)A2,0}

Theseller'srelaxedproblemthus takes the form

subjectto

Rhl =%i Ai

RLH=x2LLA2

RHH =xtLAi +xkLA2 +max{(xtH -xtL)Ax,{x?L-*2LL)A2,0}

XlHH>xtH, x?L>xtL,x?H>x?L,xkH>xkL

It is immediately apparent from this problem that the optimal contractfor type (vH,vH) takes the form: XiH = x%H = 1,since lowering theseprobabilitiesof getting the goodswould not relaxany incentive constraint.In otherwords, there is no distortion of the consumption allocationfor the

highest type, just as in the one-dimensionalproblem.As we shall see,while

207 Static Bilateral Contracting

this result is true generally in the one-dimensionalproblem,it holdsfor therelaxedproblemonly in the multidimensional case.

To characterizethe optimal contractsfor the other types, there aredifferent casesto consider,dependingon which of the downward incentiveconstraints for type (vH,vH) is binding. One casecorrespondsto a situation

with no bundling and the others to (partial) bundling:

1.No bundling: The case with no bundling is that where x\\L > x\\H,

X2L > X2L, so that the relevant incentive constraint for type (vH,vH) is

RHH=xtLA1+x2LLA2

This is a situation with no bundling in that the probability of getting objecti is not increasingin the valuation for object;(xfL>x^H).As our previousdiscussionindicates,bundling the two objectsaffectsthe buyer'schoiceofconsumption only when she has a low valuation for one objectand a highvaluation for the other.In this casethe buyer might want to buy only the

high-value object if given a choice,but bundling forcesher to buy both

objects.Thus (partial) bundling is formally equivalent to raising the

probability of selling objecti when the value for object;is higher.Substituting for the RJs in the seller'sobjectivefunction, the relaxed

problemin the caseof no bundling reducesto

max{^}ZyA/[*?(vi-Ci)+4M-C2)]-Ph[xiLLAi+*2llA2]

subjectto

x1HH>x1HL, x?H>xkH, x?L>xtL,xkH>xkL,and

xtL>xtH,xkL>x?Lwherej3H

=j3HH + j3 denotesthe marginal probability that the buyer'svalue

for (either)one of the objectsis high.

2. Bundling: When the relevant downward constraint for type (vH,vH) iseither

Rhh =xtLAi +xhLA2 +(x?L-xkL)A2

or

Rhh =xtLA, +xkLA2 +(XlLH -x1LL)A1

208 Multidimensional Incentive Problems

there is someform of bundling, sincethen eitherx2L> x2L or x\\H > x[Lorboth, so that the probability of getting good i is increasingin the valuation

for good;'.Substituting for the R1' in the seller'sobjectivefunction, weobtain the reducedproblem:

max{^,4}Z0^[4(vi-cMxM-c2)]-pH[xtLAi+*2LLA2]

-pHHmax{(;t?L-x$L)A2,(xtH-xtL)A,}subjectto

x1HH>x1HL, x?H>x2LH, x?L>xtL,xkH>xkL,and

xtL<xtH,xkL<xkH

6.1.3.1Optimal Contractin the Symmetric Model

Toproceedfurther, it is helpful to beginby imposing symmetry: Ax = A2 =A, Ci= c2=c,and v[ = v2 = v' for i = L,H.The optimal contractis then alsosymmetric and simplifies to [xHH,Xth, xHL, x111),wherex1*11 =xj?H, Xth =XkL

for k = 1,2, while xHL = x?L= x2H (the probability of getting a good forwhich onehas announceda high valuation while onehas announceda lowvaluation for the other good),and similarly x^H =xtH =x2L(theprobability

of getting a good for which one has announceda low valuation while

one has announceda high valuation for the other good).In this case,theincentive constraint

RHH =xtLAi +xhLA2 +msLx{(XlLH-xtL)A,, {x?L-x2LL)A2,0}

becomes

RHH =2xLLA +maxlU -xLL)A,0}so that the relaxedproblemcanbe rewritten as

max{;cV} 2[(pHHxHH +fixHL){vH -c)+(j3xLH +pLLxLL)(vL-c)]-[2pxLL+j3HH{2xLL+m<ix{{xLH-xLL),0}]A

subjectto

xHH>xLH, xHL>xLL

Note that, in this program, it is suboptimal to setxLH < xLL: if this wereto happen,the coefficientofx111in the seller'sobjectivefunction would be

209 Static Bilateral Contracting

positive and would lead to xLH =1,a contradiction.Therelaxedproblemisthus

maxH2[(pHHxHH +fixHL)(yH -c)+(PxLH +pLLxLL)(vL-c)]-[(2p+pHH)xLL +pHHxLH]A

subjectto

What is more,when the condition xHL > xLH is satisfied,it canbe shown

that the solution to the relaxedsymmetric problemis actually the solutionto the seller'soptimal contracting problem.1To find this solution, all weneedto determine is the sign of the coefficientsof the x1':

\342\200\242 As is intuitive, the coefficientsof x11\" and xHL always have a positive sign,so that it is optimal for the sellerto setx\"\" = xHL =1.\342\200\242 Thecoefficientofx111is 2/J(vL-c)-/W^; therefore,if

-\302\243-\302\253-\302\243- (6.3)

then it is optimal for the sellerto minimize xLH and to setx1\" =xLL. If the

oppositeinequality holds, then the sellersetsxLH =x?11=1.\342\200\242 The coefficientof Xth is 2/JLL(vL

-c)-(2/3+ /W)A; therefore,the sellerwants to minimize Xth =xLH if

^ 2pLL _ pLL

vL-c 2p+pHH l-pLLNote that this inequality is implied by inequality (6.3)whenever

hL <4~ (6.4)1-/3

1.This result can beshown asfollows: first, the fact that, in the relaxed problem, four

downward incentive constraints (for types HL and LHwith respect to type LL,and for type HHwith respect to types HLand LH) are binding plus monotonicity {x?H >xhH, x?L >

y^L~) impliesthat the five upward incentive constraints are satisfied. Second, the fact that yPL > x11* ensuresthat the last two constraints, involving types HLand LH, are also satisfied given that thedownward and upward constraints are.

210 Multidimensional Incentive Problems

We thus have five cases:(i) when conditions(6.3)and (6.4)hold, it is optimalfor the sellerto set xLL =x111=0.If, on the other hand, (ii)

2-\302\243->-A->2-^-Phh vl-c 1-Pu.

holds,then the sellersetsXth =0 and x1\" =jc\342\204\242*

= 1,while if (iii)

Phh 1-Pll vl-cholds,then xLL =x111=x\"\" =1.Finally, when condition (6.4)doesnot hold,then (iv) the seller'ssolution is given by xLL =xLH =x\"11 =1if

P vL-cand (v) Xth = x1-11 =0 otherwise.

Tosummarize, the key result in the symmetric modelis the optimality forthe sellerto have pure bundling, where xLL = 0 and x11*= xHL = x\"\" = 1,when the buyer'svaluations are not too positively correlatedand when theinformational rents are sufficiently high that it pays the sellerto excludetype (vL,vL)but not sohigh that it is bestto serveall other types. Indeed,the condition on valuations is condition (6.4)or, equivalently

(remembering

that ft = pa +P and that pHH +2p+pLL= 1),

PllPhh-P2 PhPlPh 1-0

TheLHSof this inequality is the correlationcoefficientlinking the buyer'svaluations for the two items. This inequality is always satisfiedwhen thevaluations are independent,sincethe RHSis always positive.This analysisthus confirms Adams and Yellen'sintuition that bundling is an efficient

screeningdevicewhenever \"the correlationcoefficientlinking anindividual's valuation ofonegoodto his valuation of the othergoodis not strongly

positive.\"Theothercasesthat we have detailed,whereeitherxLL =x1 =xHL =x\"11

=1or Xth =x1\" =0 and x\"1* =x\"\" =1,unfortunately are ambiguous, as theycan be interpreted either as independentsalesor as bundling. This is alimitation of the specialsetting with only two possiblevaluations for eachitem. The advantage of this simple formulation, however,is that it high-

211 Static Bilateral Contracting

lights exactly when and how the methodology for one-dimensionalscreeningproblemsextendsto multidimensional problems.

6.1.3.2Optimal Contractin the Asymmetric Model

In the asymmetric problemwhere,without lossof generality, Ax > A2, thesolution is similar to that in the symmetric problemwhenever onlydownward incentive constraints are binding at the optimum. The only slight

analytical complicationis that now three different regions[dependingonwhich downward incentive constraint for type (vH,vH) is binding] must beconsidered.Also, an interesting conceptual differenceis that now the

optimal menu of contractsmay involve random allocations.To seethis

possibility, considerthe examplewhere there are only two equally likely

types of buyers, buyer A with valuations (v!,v2) = (1,2)and buyer B with

fvi,v2) = (3,1).Supposemoreover that the seller'scostsare cx = c2= 0.CallingP( the priceof the bundle chosenin equilibrium by buyer type i, theseller'soptimum is the result of

max 0.5PA+0.5PB

such that

xf +2xf-PA>0and

3x?+x$ -PB>max{0,3xf+x?-PA]

Indeed,it can be checkedthat the relevant constraints concernparticipation by buyer type A as well as both participation and incentive

compatibilityfor buyer type B.The result is that xf =*f = 1(no distortion at the

top), as well asxi=1(sincebuyer A's valuation for good2 is higher than

buyer 5's,lowering xibelow its efficient level only hurts incentive

compatibility). Theseller'sproblemthus becomes

max 0.5PA+0.5PB

such that

xf +2-PA=0and

4-PB =max{0,3xf+1-PA }

212 Multidimensional Incentive Problems

or

max 0.5{jcf*+2}+0.5{4-max[0,2jcf* -1]}It is then optimal for the sellerto set 0 =2xi-1or x\\ =0.5.Tosummarize,the optimum implies giving both items for sure to type B and giving object2 for surebut object1only with probability 0.5to type A.

In general, in the asymmetric problem, other incentive constraintsbesidesdownward incentive constraints may bebinding at the optimum. Inthat casethe relaxedproblemconsideredsofar is no longerrelevant.Moreprecisely,when the buyer'svaluations for the two items are negativelycorrelated and when Ai is sufficiently larger than A2, then someupwardincentive constraints may be binding. In that casethe optimal contract maydistort the consumption allocationof the highest type (yH,vH). It thus

appearsthat eventhe most robust result of the one-dimensionalproblem\342\200\224

no distortion at the top\342\200\224does not survive in the multidimensional problem.However,evenif the main conclusionsof the one-dimensionalproblemdonot extend in general to the multidimensional setting, the main

methodological principlesdevelopedfor the one-dimensionalproblemcarry overto the moregeneralsetting.

6.1.4GlobalAnalysis for the GeneralModel

The analysis for the 2x2model seemsto suggest that outside thesymmetric model few general insights emerge about the multidimensional

screeningproblem.It would thus seemthat evenlesscanbesaidin any

generality about the generalmodelwith more than two items and a continuum

of valuations. There is ground for optimism, however,for at least threereasons.First, a completecharacterizationof the generalproblemhasbeenobtainedby Armstrong (1996)and Rochet and Chone (1998).Second,asshown by Armstrong (1996),in the two-dimensional problemwith acontinuum of valuations a generalinsight emergesconcerningthe optimalityof exclusionof the lowestvaluation types.Third, as Armstrong (1999)hasshown, when the number of items is largeand the valuations for each itemare independently distributed, the optimal contractcan be approximatedby a simple two-part tariff. We shall not attempt to providethecharacterization of the generalproblem.We shall limit ourselveshere to illustratingthe latter two points.

213 Static Bilateral Contracting

6.1.4.1The Optimality of Exclusion

To understand this result derived by Armstrong (1996),consider the

problemwith two items where v, is uniformly distributed on the interval

[6,6+ 1],i = 1,2,and supposethat c,= 0.Let P, denote the price of item i

and Pb the price of the bundle.Sincethe problemis symmetric, considerthe symmetric contract[Pi= P2= P,Pb}-The localanalysis of MMW

suggests that when valuations are independent,somebundling is optimal, sothat 2P > Pb.We shall solvehere for the optimal pricesP* and P* and showthat in the two-dimensional problem it is always optimal to excludethelowestvaluations, while in the one-dimensionalcaseall consumertypes areservedwhen 6 > 1.

Undermixed bundling, 6+1> P> 6 and Pb>P+6.Thenall types with

v2 >Pand Vi<Pb-Pconsumeonly item 2;region62in Figure 6.3representsall thosetypes.Similarly, all types with vx >P and v2<Pb-Pconsumeonlyitem 1;regionQx in Figure 6.3representsall thosetypes.All types in regionQbpurchasethe bundle and have valuations Vi +v2 > Pb.Finally, all types in

region G0are excluded.In this case,the seller'sexpectedrevenue from

selling individual items is2P(6+1-P)(Pb-P-6),and his expectedrevenue

Figure 6.3Exclusion under Mixed Bundling

214 Multidimensional Incentive Problems

from the bundle is Pb(6+l-P)(6+l+P-Pb)+\302\261Pb(2P-Pb)(26 +2-Pb),sothat his profit is given by

2P(e+l-P)(Pb-P-9)(e+l-P)(6+l+P-Pb)+^(2P-Pb)(26+2-Pb)

Under pure bundling P > Pb - 6, and for 2d + 1> Pb > 2d the seller'sexpectedprofits are given by

l-|(P&-20)2Theseller'sexpectedprofit function is thus piecewisecubicin (P,Pb)and

nonconcave.Solving the seller'sunconstrainedmaximization problem,oneobtains the result that pure bundling is optimal when 6 > 0 and thesolution is given by

p&=i(40+V402 +6) with P >Pb-d

The fraction of types excludedfrom consumption is then

18v ;This fraction is decreasingin 6,but it is strictly positive for all 6> 1.Incontrast, in the one-dimensionalproblemthere is no exclusionfor all 6>l.Theintuition for this result is that by lowering Pb by e the sellerlosesrevenueefrom all thosetypes that werealreadypreparedto purchasethe bundle and

gainsPb -efrom at most afraction s2ofnew consumers.Thus,at Pb=6,whereno types are excluded,a small increasein pricehas a negligible effectondemand but a first-ordereffecton revenues.It is for this reason that someexclusionis always optimal in the two-dimensional problem.In the one-dimensional problem,in contrast, the fraction ofnew consumersis ofordere.6.1.4.2Approximating the Optimal Contractwith a Two-PartTariff

Extending the 2x2modelin a different direction,Armstrong (1999)showsthat when the dimensionality of the seller'sproblembecomeslarge,the

optimal contractcanbe approximatedby a simple two-part tariff.

Considerthe seller'sproblemwhen he canproducen > 2 items at a costc;> 0 for item i. Thebuyer'sutility function is now

^ViXi-T1=1

where xt e [0,1]is the probability of getting good i, v = (yu...,v\342\200\236)

isthe buyer's type, and T is the payment from the buyer to the seller.LetG(y) denote the seller'sprior distribution over buyer types. The seller'soptimal contracting problemthen is to choosea payment scheduleT(x)=T(xi,...,xn) to solve

max E{T[;t(v)]-^Cix]

subjectto

xiy) earg max*, ^T\"=i ViXt-T(x)

2^v,jc((v)-T[jc(v)]>0For some distributions of buyer types G(v)\342\200\224in particular, when the

valuations for the different items v, are independently distributed\342\200\224it is

possibleto approximate the optimal contractwith a two-part tariff P(x)=F+ Ii%iCiXi when n \342\200\224> \302\260\302\260.

Toseethis possibility, let S(v)denotethe first-best socialsurplus obtainedwhen the sellerand the type-v buyer trade with eachother,and let \\i ando2 denote the mean and varianceof S(v).The seller'sfirst-best expectedprofit is then equalto fi, and his second-bestexpectedprofits arenecessarily

such that 7^B < ji.If the selleroffersa two-part tariff P(x)=F+ XILiQa:;, then a type-v buyer

acceptsthis contractif and only if S(v)>F.Conditionalon the buyeraccepting

the contract,the sellerobtains a profit ofF.Let tP denotethe maximum

profit from the two-part tariff obtainedby optimizing overF.If the sellersetsF= (1-E)fi, then

np > (1-e)liPr{5(v)> (1-s)ii}>(l-e)A\302\243(l-Pr{]5(v)-At|^^})

But from Chebyshev'sinequality,

Pr{|S(v)-Ai|><5}<-^0

so that

^-o^-evh1 e 2 2

or2/3'

A*.L ^A^

for e =[2(<72/^2)] (while /cisa positive constant).Consequently,

so that the differencein profit between the optimal second-bestcontractand the optimal two-part tariff P(x)is decreasingin a/fi and vanishes when

a/fi tendsto 0.If the valuations for the individual items are independently distributed,

and if we callfi, and o;the mean and varianceof ^(v,),the first-best socialsurplus on item i for buyer type v, we have

n n

a*=Xa*\302\253and ff2=Zff?

[=1 [=i

Substituting in the precedinginequality, one then obtains

where o=max^a;}and \\i =min^}.Letting n ->

\302\260\302\260,

one observesthat the optimal two-part tariff contractP(x)is approximately optimal when the sellerhas many items for sale(like a department store) and the buyer's valuations for these items areindependently distributed. This is a rather striking result indicatingthat in complexenvironments simple contracts may be approximatelyoptimal, a theme that will be illustrated again in Part IV, this time in thecontextof dynamic incompletecontracting. Note finally that this result is

by no means dependent on the independenceassumption: Armstrong

(1999)shows that a similar result holdswhen valuations are correlatedandwhen the sellercan offera menu of two-part tariffs.

6.2MoralHazardwith Multiple Tasks

The classicalmoral hazardproblemanalyzedin Chapter4 capturesafundamental aspect of contracting under uncertainty and hidden actions.It

217 Static Bilateral Contracting

highlights the basic trade-offbetween risk sharing and incentives, and it

reveals the importanceof the likelihoodratio in estimating the agent'sactionchoiceand, thus, deterniining the shapeof the optimal contract.Yet

beyondtheseinsights the generaltheory haslimited practicalinterest unlessmore structure is imposedon the problemunder consideration.Also, the

generalproblemis too abstractto be able to capturethe main featuresofan actual bilateralcontracting problemwith moral hazard.Specifically,inmost real-worldcontracting problems,the agent'sactionset is considerablyricherthan the theory describes,and the variablesthe contractcanbeconditioned on aremuch harder to specifypreciselyin a contractor to observe.

Tomention just oneexample,a CEO'sjobinvolves many dimensions andcannot adequatelybe reducedto a simple problemof effort choice.CEOsdo not make only investment, production,and pricing decisions.They arealso responsiblefor hiring other managersand employees,and for

determining the overallstrategy of the firm and the managerial structure insidethe firm. They lay out long-term financial objectivesand mergerandacquisition plans, and they continually interact with the largest clients andinvestors.Tocapturethe CEO'scompensationproblem,the classicalprincipal-agent problemneedsto be modified in a number of dimensions.At

the very leasther actionset must include the range of different tasks he is

responsiblefor, performancemeasuresmust bemultidimensional, and thetime dimension must be incorporatedinto the basicproblem.In this

subsection we shall mainly be concernedwith the problemof multiple tasksand objectives.The time dimension isexploredsystematically in Part III.

When an agent is responsiblefor several tasks, the basiccontracting

problem can no longer be reducedto a simple trade-offbetween risk

sharing and incentives.Now the principal must alsoworry about how

incentives to undertake one task affectthe agent'sincentives to undertake othertasks.When a company'sremuneration committee decidesto tie the CEO'scompensationto the firm's profit and stock price, it must considerhow

such a compensationpackageaffectsthe CEO'svarious tasks that are not

directly or immediately relatedto profit or stockprice,such asemploymentand promotion decisions;mergers,acquisitions,and other forms ofrestructurings; and soon.

Thesametypes ofconsiderationsarisein otherjobcontexts.For example,with the hotly debated issueof incentive pay for teachers,and the

proposal of linking teachers'pay to pupil performancein those standardized(multiple-choice)tests that are usedfor collegeadmissions.The advantage

218 Multidimensional Incentive Problems

of thesetestsis that they providean independentmeasureofperformance;their disadvantage is that they measure only part of what teachers aremeant to teach to their pupils. If the rest (e.g.,ability to write, to argue,to philosophize,to behaveproperlyin the world) is not measured,there isa definite risk that incentive pay basedon standardizedtestswould induceteachersto focusexcessivelyon these tests and to neglectthe othercomponents of education.

At first sight it may appearthat, asa result of enriching the agent'sactionset, the principal-agentproblemis more intractable, and even lesscan besaid about this problem in any generality than about the single-taskproblem.This is actually not the case.A moment's thought reveals that

in fact the classicprincipal-agentproblem consideredin Chapter 4 is ageneralmultitask problem,when the agent'sactionis definedto bea givenprobability distribution over outcomesrather than effort.Then,providingincentives to achievea particular outcomemay alsobeseenas affecting the

agent'sincentives to obtain otheroutcomes.Seenfrom this perspective,themain weaknessof the classicproblemis its excessivegenerality, and, indeed,the multitask principal-agentproblemsconsideredin the literature provideresults beyondthe classicalonespreciselybecausethey imposemorestructure on the classicalproblem.Thus the multitask principal-agentproblemconsideredin Holmstrom and Milgrom (1991)focuses on the linear-contract,normal-noise,CARA-preferencesmodeldiscussedin Chapter4,and makesassumptions about the interconnectionof the agent'sdifferent

tasks in order to obtain predictionsabout high- or low-poweredincentive

schemes,job design,and various other employment practicesin

organizations. As we shall see,this approachis basedon the competitionbetweentasks that arisesfrom the effort-costfunction.

6.2.1Multiple Tasksand Effort Substitution

The basicsetup consideredby Holmstrom and Milgrom is as follows:Theagent undertakesat least two tasks a, (i =1,...,n\\ n > 2),each taskproducing somemeasurableoutput q\342\200\236

whereq, = a, + e,.The randomcomponent of output \302\243t

is assumedto benormally distributed with meanzeroandvariance-covariancematrix

S=(ol

\\0\\n

\\

Onl\"

olj

219 Static Bilateral Contracting

where o]denotesthe varianceof\302\247,

and<7V

is the covarianceof e;and\302\243,.

The agent has constant absoluterisk aversion,and her preferencesarerepresentedby the utility function

u(yv,a) =-e-n[w-v{au-an)]

where 77 is the coefficientof absolute risk aversion (77= -u7u'),and

!//(#!,...,an) is the agent'sprivate costofchoosingactions(#i,...,an).Theprincipal is risk neutral and offerslinear incentive contractsto the agent.

Within this setup,Holmstrom and Milgrom explorehow the incentivecontractvarieswith the shapeof the costfunction y^a^...,an), in

particular how the choiceof actionlevelfor one task affectsthe marginal costofundertaking other tasks.Theyalsoask how the incentive contractbasedonthe subsetof observableoutputs qt is affectedby the nonobservability ofthe outputs ofother tasks.This somewhat abstractline of questioning turns

out to be pertinent for many employment situations in organizations, asthe following examplestudied by Anderson and Schmittlein (1984)andAnderson(1985)illustrates.

The subjectof these authors' study is employment contractsand jobdescriptionsof salespeoplein the electronic-componentsindustry. Themultiple

activities of an individual salespersonin this industry canbe classifiedinto three categoriesof tasks:(1)selling the productsof a manufacturer; (2)finding new customers;and (3)possibly selling the productsof othermanufacturers. In principle, thesetasks couldbeundertaken as well byindependent salespeopleasby employeesof a given manufacturer, but Anderson andSchmittlein find that in this industry most salespeopleare employeesrather

than independentcontractors.This observationis explainedby Holmstromand Milgrom as the efficient contractual outcomeof a multitask principal-agent problemwherethe different tasks of the agent are substitutes and the

outputs of someof the activities are not observable(or measurable).Moreprecisely,let a representsellingactivitiesfor the principal,a2finding

new customers,and a3sellingonbehalfofanothermanufacturer. Supposeinaddition that the principal canobserveonly qlt the volume ofsalesof its own

components.Then,Holmstrom and Milgrom argue that if the principal is

mostly concernedabout the volume ofsalesqx, it may beefficient to preventthe salesmanfrom selling for other manufacturers. This purpose can beachievedby hiring the salespersonas an employeerather than an outsidecontractorwho is freeto choosehow to organizehis worktime.They explain:

220 Multidimensional Incentive Problems

Incentives for a task can be provided in two ways: either the task itself can berewarded or the marginal opportunity costfor a task canbe lowered by removingor reducing the incentives on competing tasks. Constraints are substitutes for

performance incentives and are extensively used when it is hard to assessthe

performance of the agent. (Holmstrom and Milgrom, 1991,p.27)

Thus, by imposing more structure on the classicalprincipal-agentproblem,the scopeof incentive theory canbebroadenedto include aspectsof incentive, design(suchasjobdescription)other than simply the designofcompensationcontracts.We shall illustrate the core idea in Holmstromand Milgrom's work by consideringthe simplest possiblesetting wherethe agent can undertake only two tasks (n = 2) whose outputs areindependently distributed (<712

= 0) and where the cost function takes the

simple quadratic form y/\"(tfi, a2)=\\{c\\a\\ +c2a2)+5a1a2,with 0 < 5< VciC2.When 5=0,the two efforts are technologicallyindependent,while when

8 = VciC2, they are \"perfect substitutes,\" with no economiesof scopebetweenthem. Whenever <5>0,raising effort onone task raisesthe marginalcostofeffort on the other task, the so-calledeffort substitution problem.

With two observableoutputs, the principal offersan incentive contract:

w =t +s1ql+s2q2

to the agent.Under this contractthe agent'scertainty equivalentcompensation is given by (seeChapter4 for a definition and derivation)

77 1t +s1a1+s2a2--{slo\\+s2ol)--{c1al+c2a2)-5a1a2

Theagent choosesat to maximize this certainty equivalent compensation.Thefirst-orderconditionsof the agent'sproblemequatethe marginal return

from effort and its marginal costofeffort:

Si =Cjfl, +daj

for i =1,2,j=2,1.As in Chapter4, this problemis simplified by the factthat the agent doesnot control the riskinessof her compensation,only its

mean.With effort substitution, however, one has to take into accountthefact that the marginal cost of effort on one task dependson the effortexertedon the other task. The unique solution to the agent'sproblemis

given by

Sic.--5sia. = ,2

CiCj -O

221 Static Bilateral Contracting

In the following discussionwe assumeaway cornersolutions and focusoninterior solutions,at > 0,which implies in particular <5< ^clc2:under perfectsubstitutability, the agent would perform only a single task, the one with

the higher return, sinceshe facesa single marginal costof effort,Theprincipal'sproblemis then to choose{t;Si,s2}to solve

max\302\253i(l

-Ji)+a2(l-s2)-1subjectto

s1c2-5s2 s2c1-ds1*= cc S2> a*=cc 52 (IC)C\\C2

\342\200\224 O C2C\\\342\200\224 O

77 1 _t +s1a1+s2a2--{s2o2+s2o2)--(c1a\\+c2a2)-5a1a2>w (IR)

At the optimum, the (IR) constraint is binding, so that we cansubstitute

for t and (ax, a2) to obtain the unconstrainedproblem:

naxfSlC2~3S2+^ \"&1)-%(stt+slol)

n,n \\ C\\C2-0 ) 2maxlC\\C2

1 (S\\C2-6si\\ 1 f s2c1-8s1^\\ (S]_c2-5s2\\(s2Ci-8sx2C\\Clc2-52J2C2L2Cl-<52J \\Clc2-52Ac2Cl-52

Thefirst-orderconditionsfor this unconstrained problemyield

_ c2-5+5s2c2+Wi(cic2-52)

and

_ cx-8+5sx

c1+j]ol{c1c2-52)so that the solution is given by

* l+(c2-5)f]ols\\ =\342\226\240

l+f]c2ol+ricxG2 +ri2<j2<J2(cic2 -52)

and

* l+(Ci-<5)77<7?j _l+J]C2ol+T]CXC>1 +772<7i2<7|(CiC2

-<52)

222 Multidimensional Incentive Problems

Onceagain,let us focushereon interior solutions for the sfs.It isinstructive to comparethis solution to the one obtainedwhen the two tasks aretechnologicallyindependentand 5 = 0.In that case,the first-orderconditions for the agent'sproblemreduceto at =Si/ch and the optimal incentivecontractfor each task is the sameas in the single-taskproblemconsideredin Chapter4,

J' \342\200\224

-i 21+77C;07

Thus, in the degeneratecaseof technologicalindependencebetweentasks,optimal incentives for each task canbe determineddirectly from thestandard trade-offbetweenrisk sharing and incentives.This is no longer thecasewhen tasks are substitutes.

Using the formulas for sf we can indeeddetermine how the optimal(linear) incentive contractvarieswith the quality of output measuresforeach task. Assume, for example,that the measurability of task 2 worsens,that is, of increases.Then,as is intuitive, s* goesdown. But what is moreinteresting is that s^alsogoesdown. Indeed,the sign of the partialderivative of s* with respectto 02 is the sign of <5t7[77<7i(<5

-Ci) - l],2 which is

negative sinceit has the sign oppositeto that of s*, which is positive.Thekey observationof this multitask approach is thus the complementaritybetweenthe sfs in the presenceof effort-substitution problems.

Holmstrom and Milgrom push this logicto the limit and point out that, when

the principal essentiallycaresabout the task with unobservableoutput (taski with d\\ \342\200\224> 00),then it may bebestto give no incentives at all to the agent.Following Williamson, they definesuch a situation asonewith \"low-poweredincentives.\" This is easiestto understand if we think of at =0 asanormalization rather than a \"true cornersolution.\" It isindeedoften natural to expectthe agent, if shelikesherjob at all, to exertsomeeffort evenin the absenceof any financial incentive (the effort choiceof the agent will thus equatehermarginal nonfinancial benefit with her marginal cost).In those situations,becauseof the effort-substitution problem,any incentivesbasedon theobservableoutput q-,may actually becounterproductive,sinceit will inducethe agent to divert effort away from the task i valued by the principal.

2. Indeed, starting from the expression of s*,the sign of its partial derivative with respect toa\\ is the sign of (c2- <5)77[1 + r\\cra\\ + r\\cxa\\ +

\342\226\240rfa\\a\\{cxc2

- 82)]- [1+ (c2- 8)r]al][r]c2+772oi(c1c2

-<52)], which is equal to 5t)[t]g\\(5- c{)-1].

223 Static Bilateral Contracting

Another observationfrom this approachconcernsthe casewhere the

principal may not be able to control all the dimensions of the incentivescheme.In the precedinganalysis we assumedthat both sf'swerecontrolledby the principal.In reality, this may not be the case:for example,task 2 mayrefer to \"outside work\" that the agent couldengagein.Then,clearly,if themarket starts valuing such work more highly, so that s*rises,the agent will

divert effort away from task l.Toavoid this outcome,the principal will haveto eitherraises*or prevent the agent from engaging in outsidework (whilecompensatingher soas to keepsatisfying her participation constraint).

6.2.2Conflicting Tasksand Advocacy

In the Holmstrom-Milgrom model that we just discussed,multitask

problems arisebecauseof the effort-substitution problem,that is, the fact that

raising effort on one task raisesthe marginal cost of effort on the othertasks.In the absenceof such an effect,we are back to the moral hazard

problemof Chapter4,whereincentiveson each task canbe derivedindependently if individual outputs are independently distributed. This is nottrue anymore if there is a directconflict between tasks. Think, for example,3about an agent/salespersonwho has to try to sell two productsmadeby agiven principal/manufacturer, products1and 2, which are imperfectsubstitutes. For simplicity, assumethat the levelof saleson producti is qt e {0,1},and the g/sare distributed independently with

Pr(g, =1)=a+pat-

yat for i =1,2,/ =2,1Effort to promoteproduct i is denotedby a, e {0,1}.One unit ofpromotion effort on any given productcostsan agent y> 0.Effort raisesthe

probability ofsaleon the given productby p,but this comespartly at the expenseof lowering the saleprobability for the otherproductby y We assumethat

expendingeffort to promoteboth productsis efficient in that p-y> yr.

We have explicitly ruled out here the effort-substitution problemstressedby Holmstrom and Milgrom, by assuming that the marginal cost ofpromotion effort a, isindependentof the levelof a,-.Thedirectconflict betweentasks, however, can make it expensiveto induce ax = a2 = 1by a singleagent.Indeed,assumeagents are risk neutral but resourceconstrained;namely, their payoff equals their wage w minus their effort cost, and

wages cannot be negative.If both products have to be promoted by

3.This example is based on Dewatripont and Tirole (2003).

224 Multidimensional Incentive Problems

a single agent, the principal can offer her a wage that is contingent onboth g/s,that is,a schedulewiy. Giventhe symmetry of the problem,we haveWio = Woi,which we denoteby wv Moreover,the principal clearlywants toset w00equalto zero.Defining (p = a+p-% the problemof the principal isthus

min (p2wn +2<p(l-(p)wx

such that

(p2wn +2<p(l-<p)wi -2\\ff>

(<P+7)(<P-p)wn +[(<p+y)+(<p-p)-2(<p+y)((p-p)]wi -yand

<p2wn +2<p(l-<p)wi -2y>a2wu +2a(l-a)wi

These two incentive constraints state respectivelythat the agent has to

prefer exerting two efforts to exerting one or zero.It turns out that the

optimal incentive schemehas w\\ = 0,4and that the binding incentiveconstraint is the secondone,5which reducesto

(p2wn-2yr>a2wn

4. Assume this is not the case,Then the LHS of the two incentive constraints is unchangedif we make the following change in the compensation schedule:

<pAw, = Awn <02(1-9)

Straightforward calculations show that this relaxes the RHS of both incentive constraints, andtherefore allows us to improve the maximand by lowering expected compensation. This canbedone until we reach w\\ = 0.5.When Wj = 0,and using the fact that cp

= a + p-y, the two constraints can berewritten as

p +y +ap-ay-pY^-J\342\200\224

Wll

and

p2 + y2 + 2ap-lay-2py > \342\200\224\342\200\224

Wll

Since p2+ y2 > 0,the second constraint implies the first.

225 Static Bilateral Contracting

At the optimum, we thereforehave

2w

This leavesthe agent with a rent equal to c?wn or

2\\ff

a2(p2-a2

Let us comparethis outcometo the casewhere the principal hires two

agentsand askseach of them to promote one product.Assume that the

principal offerseach agent an incentive schemeand, if they both accept,they play a Nash equilibrium in effort choices.Once again, each agent'sincentive schemecan be contingent on both g/s.We can concentrateon asingle agent, and by symmetry the other agent will receivethe sameincentive scheme.Clearly,the agent'swage will bepositive only when she issuccessful in selling her own product.Onceagain, we thus have to determinewn and w10.The cost-minimizing incentive schemethat inducesax =a2= 1is given by

min cp2wn +<p(l-cp)w10WW Ml

such that

cp2wn +<p(l-

cp)ww -yf>(<p-p)[((p+y)wn +(l-(p-y)ww]

Sinceeachagent undertakesonly one task, there is only a single incentiveconstraint. Not exerting effort given that the other agent exerts effortlowersone'ssuccessprobability from (p to (p-p and raisesthe otheragent'ssuccessprobability from (p to (p + y The result is that the optimal incentiveschemehas Wi0 > 0 but wn =0:6intuitively, the failure of the other agent is

6.Assume this is not the case.Then the LHS of the incentive constraint is unchanged if wemake the following change in the compensation schedule:

Awn =\342\200\224Awio

<09

This relaxes the RHS of the incentive constraint, and therefore allows us to improve themaximand by lowering expected compensation. This can bedone until we reach wn =0.

226 Multidimensional Incentive Problems

rewardedbecauseit is itself an indication that the agent has exertedhigheffort. The optimum is therefore

\302\245

ww =(pO\342\200\224(p)-((p-p)Q.-(p-y)

Thisleaveseachagent with a rent equal to

(<p-p)(l-<p-y)<p(X-<p)-(<p-p)(X-<p-y)

\302\245

In orderto determine whether it isbetter to hire oneor two agents,wehaveto comparethe total rents they obtain. Rememberingthat (p= ct+p-% therents left to one and two agentsare,respectively,

a1\302\245

(cc+p-y) -aand

(a-y)Q--cc-p)(a+p-7XI-a-p+y) -(a-7XI-cc-p)

2\\ff

Straightforward computations indicate that hiring a single agent is thebetter solution if and only if

2(\302\253

+p-7)2 (q+p-yXl-q-p+y)a2 (a-y)(l-a-p)

The LHSof this condition decreasesin 7 which measuresthe intensity ofconflict between tasks. Its RHSis increasingin y? For 7=0,the LHSis

7. Indeed, the sign of the derivative of the RHS is the sign of

{a-y){l-a-p)[2(a+p-y)-l]+{a+ p-y)[l-a-p+ y](l-a-p)

or the sign of

2(a-y)(a+p-y)+ p[l-a-p+ y]

which is positive.

227 Static Bilateral Contracting

bigger,sohiring a single agent is optimal.8 However,for /tending to a, theRHStends to

+\302\253>,and hiring two agentsis optimal.

To sum up, we have considereda setting where, without any conflict

between tasks, multitasking is beneficialfor the principal, who prefers tohire a single agent to undertake both tasks rather than hiring one agent pertask. However,introducing conflicts between tasks can changethe resultbecause,intuitively, agents take away rents from one another by exertingeffort. In the limiting casewhere /tendsto a,the agent obtains no

compensation whatsoeverwithout exerting effort if the other agent does.Thiseffect is not present with a single agent, who can coordinateher effortchoices(it is for this reason that the binding incentive constraint is the onewhere he considersnot exerting any effort at all).

Directconflictsbetweentasks have beenconsideredby DewatripontandTirole(1999)in a modelwherea principal has to hire one or two agentstosearchfor information about the prosand cons of a decisionunderconsideration by the principal.They assumethat agentscannot be rewardeddirectly on the basisof the amount of information generatedbut only onthe decisiontaken, which is itself positively related to the amount of\"positive\" information and negatively related to the amount of \"negative\"

information. Therefore,there is a directconflict betweenthe task \"searching for

positive information\" and the task \"searching for negative information,\"and thus it may be optimal for the principal to hire two agents, one beingaskedto look for positive information and the other one for negativeinformation, and to endow them with incentive schemeswhere they are\"advocates,\" respectively,for and against the decisionthat is beingconsidered. A natural exampleof this setting is the judicial system, whosegoalisto have decisionsmade by judges\342\200\224convicting a defendant, awarding

damagesto a plaintiff\342\200\224based on the bestpossibleinformation. Searchingfor this information is costly,and it is difficult to designincentive schemesbasedon the amount of information generated.However,decision-basedrewards are common, whether they are explicit \"contingency fees\" orreputation-baseddelayedincentives(sincecourt outcomesare easierto

8. Indeed, the condition then amounts to

2{a+pf? a+p[ 1

a2 awhich is true.

228 Multidimensional Incentive Problems

rememberthan detailsabout the information providedduring the trial). Inthis setting, just asbefore,onecanshow that splitting the tasks betweentwo

agentsand making them advocatesof the defenseand prosecutionis the

cost-minimizing way to reach informed judicial decisions.

6.3 An ExampleCombining MoralHazard and AdverseSelection

Mostcontractual situations involve elementsofboth adverseselectionandmoral hazard.Often,however,one of these two incentive problemsstandsout as the key problem,so that the otherproblemcanbe ignoredas a first

approximation. In somecasesit is not possibleto decidea priori which ofthe two incentive problemsis most important, or to disentangle the moralhazard from the adverseselectiondimension.Many of these caseshavebeen explored in the literature. One of the earliest models combiningadverseselectionwith moral hazard is Laffont and Tirole(1986),who

consider the problemof regulating a monopoly using cost observations.Themonopoly may have a more or lessefficient productiontechnology (theadverseselectiondimension) and may put in more or lesseffort in

reducingcost (the moral hazard dimension).Several other models with this

multidimensional incentive structure appliedto other contextshave beendevelopedsince(see,for example,the bookby Laffont and Tirole,1993).

Recall that, in Chapter 2, we considereda simplified version of theLaffont-Tiroleoriginal setting, wherethe effort choiceledto adeterministic cost realization.As a result, the key dimension of the analysis wasadverseselection,eventhough moral hazard was alsopresent.Herewe shall

extendthe analysis to random performance,as in Chapter4, and we shall

detail the respectiverolesof moral hazard and adverseselectionand the

implications of their simultaneous presence.Todo so,we considerthe problemof selling a firm or any other

productive asset to a new owner-manager.Thisis an obvious exampleinvolvingboth adverseselectionand moral hazard, sincethe new owner may bemoreor lessableand shemay put in more or lesseffort in running the firm. Aswe shall see,the prescriptionson the optimal salescontractfor this problemdependin an essentialway on whether only the adverse-selectionor themoral-hazardaspectorboth are explicitly taken into account.Although the

problemwe consideris highly stylized, it is relevant for detennining how

to structure corporateacquisition dealsand privatization transactions.

229 Static Bilateral Contracting

The examplewe consideris basedon a model consideredby Bolton,Pivetta, and Roland (1997),but this problem has beenexploredmorecompletelyby Rhodes-Kropfand Viswanathan (2000)and Lewis and

Sappington (2000;seealso McAfee and McMillan, 1987b;Riley, 1985;Riordan and Sappington, 1987a).Considerthe problemof a risk-neutralsellerof a firm transacting with a risk-neutral buyer. Thebuyer cangenerate an uncertain revenuestreamby running this firm. Supposethat thereare only two possiblerevenuerealizations:X e {0,R}.The buyer may bemoreor lessableat running this firm, and buyer ability here translates into

a higher or lowerprobability of getting the high revenueoutcomeR. Inaddition, the buyer canraisethe probability ofgetting R by working harder.

Let 6 denote ability, and supposethat 6 e [6L, 6H} with 6L < 6H.The sellerdoesnot know the buyer's type, and his prior is that the buyerhas a high ability 6H with probability /J and a low ability 6L with probabil-ity(l-j3).

Letedenoteeffort here,and supposethat abuyer with ability 6who supplieseffort egenerateshigh revenuesR with probability 6eat a private costof9

\\ff(e) =-e2

Supposefor simplicity that the sellerisdeterminedto sellbecausehe isunable to generateany revenuewith the firm. The seller'sproblemthen isto offer a\" menu of contracts

(/\342\200\242, rt) to maximize his expectedreturn fromthe sale,where t( denotes an up-front cash payment for the firm and r{ arepayment to bepaidfrom the future revenuesgeneratedby the new buyer.Assume that the firm is a limited liability corporationso that X > r,\342\200\242> 0.Inthat casethere is no lossof generality in consideringcontractsof the form

(tt, r(), where r, is paid only if X =R.Thebuyer'spayoff under such a contractthen takes the form

O^R-ri-tt-^e2,i =L,H (6.5)

Heroptimal choiceof effort under a contract(tt, r,) is thus

e^OtiR-rdlc (6.6)

9. To ensure that we have well-defined probabilities, take cto belarge enough that the buyerwould never want to choose a level of effort e such that Q^e >1.

230 Multidimensional Incentive Problems

and her maximum payoff under the contractis

2c

As onemight expect,the buyer'soptimal choiceofeffort is independentoftt but is decreasingin rt.

The seller'sprogram then is

rri^itlMtH+e2H(R-rH)rHlc]+(l-P)[tL+e2L(R-rL)rLlc]}

subjectto

\342\200\224MR-r,)]2-h >

\342\200\224[em-r.)f-tj for all;*i, and all i =L,H2c 2c

1-mR-rtf-ttZO,i =L,H2c

6.3.1Optimal Contractwith MoralHazard Only

Supposethat the sellerisableto observethe buyer'sability sothat the only

remaining incentive problemis moral hazard.Thenthe seller'sproblemcanbe treated separatelyfor eachbuyer type i and reducesto

maxfc^fr+Q}(R-ri)r,/c}subjectto

-^-[OtiR-r,)]1-hZO,i =L,H2c

Sincethe participation constraint is binding, the problembecomes

rnax{-\302\261-[Ol(R-ri)]1+-0?(R-n)r\\

n 12c C J

or equivalently

max^-ft2^2-^2)n 2c

Thisproblemhas a simple solution:rH = rL = 0 and tl =9?R2/(2c).Toavoid the moral hazard problemthe sellershould sellthe firm for cashonly

Static Bilateral Contracting

and not keepany financial participation in it. This simple yet striking resultliesbehind many privatization proposalssuggesting that the state shouldselloff 100%of state-ownedcompaniesin order to minimize managerialincentive problems.The only reasonwhy a sellermight want to keepsomefinancial participation in the pure moral-hazardcaseis that the buyer maybe financially constrainedand may not have all the cashavailableup-front

(or the buyer may be risk averse).

Optimal Contractwith AdverseSelectionOnly

Supposenow that the buyer'seffort levelis fixed at somelevele but that

the sellercannot observethe buyer'stalent. Theseller'sprogram then is to

max(tj A) {P(tH +6HerH)+(1-p)(tL +6LerL)}

subjectto

0,-eGR-ri)-tt>ete{R-rt)-1,for all;*1,and all i =L,H

dieOl-ri-ti-lpZO;i =L,H

This problemalsohasa simple solution:rH = rL =R and tt =-ce2/2.Thiscontractminimizes the buyer's informational rent. In fact, it extractsallof it without inducing any distortions, Intuitively, in the absenceofmoral hazard, the best incentive-compatibleway to extract rents fromthe buyer is to ask for 100%of the future return (and thereforeno cashup-front).

Optimal Saleswith Both AdverseSelectionand MoralHazard

Thesimplicity of the precedingsolutions is ofcoursedriven by the extremenature of the setup.It appearsthat neither extremeformulation is an

adequate representationof the basicproblemat hand and that it is necessaryto allow for both types of incentive problemsto have a plausibledescription of assetsalesin practice.

Not surprisingly, the optimal menu of contracts when both types ofincentive problemsare present is some combination of the two extremesolutions that we have highlighted. Solving the problemof the sellercanbe done by relying on the pure adverse-selectionmethodology detailedin Chapter 2.Specifically,one can first note that only the individual-

rationality constraint of the low buyer type and the incentive-compatibilityconstraint of the high buyer type will bebinding. Indeed,(1)when the low

232 Multidimensional Incentive Problems

type earnsnonnegative rents, sowill the high type, who can always mimicthe low type; and (2) in the symmetric-information (that is, pure moral-hazard)optimum, the sellermanagesto leavethe high type with no rents,but this outcomeis what would induceher to mimic the low type.Therefore, the sellerhas to solve

msuLltlMtH+e2H(R-rH)rH/c]+0.-P)[tL+0l(R-rL)rL]/c}

subjectto

\342\226\240\302\243-[0h(R

~rH)f ~tH=^-[eH(R-rL)f-tL2c 2c

~WL(R-rL)f-tL=02c

Using the two binding constraints to eliminate tH and tL from the maxi-mand, we obtain the usual efficiency-at-the-topcondition rH = 0 (asin the

puremoral-hazardcase).However,raising rL allows the sellerto reducetheinformational rent enjoyedby the high buyer type, which is

j-(ei-el)(R-rLfThe first-ordercondition with respectto rL involves the usual trade-offbetweensurplus extractionfrom the low buyer type and informational rentconcessionto the high buyer type, and leadsto

P(e2H-el)RrL p(e2H-ol)+a-P)elwhich isbiggerthan O.Theoptimal menu ofcontractsis thus such that thereis no effort-supply distortion for the high-ability buyer becauseshe is a100%residualclaimant. But there is a downward effort distortion for the

low-ability buyer that servesthe purposeofreducing the informational rentof the high-ability buyer. Theextentof the distortion, measuredby the sizeof rL, dependson the sizeof the ability differential (6$- 6l) and on theseller'sprior j3:Themore confident the selleris that he facesa high buyertype, the largeris his stake rL and the larger is the up-front payment tH.

233 Static Bilateral Contracting

6.4 Summary and LiteratureNotes

This chapter has first consideredthe multidimensional adverseselectionparadigm, whoseanalysis was started by Adams and Yellen (1976)in thecontextof optimal selling strategiesby a monopoly. A recurring theme in

the literature hasbeento establishconditionsunder which mixed bundling

(the simultaneous separateand joint sale of multiple products)is optimal.As shown by McAfee,McMillan, and Whinston (1989),mixed bundlingis optimal when consumer valuations for the different goodsare not

strongly positively correlated,thereby extending Adams and Yellen'soriginal insight.

The subsequentliterature illustrates the difficulty of generalizing the

analysis of the one-dimensionalsetting consideredin Chapter 2 (seeArmstrong, 1996,and Rochet and Chone,1998,for generalmodels,Armstrong and Rochet,1999,for a complete solution of a simple case,andRochet and Stole,2003,for an overview).These studieshave generatedseveralnew insights. In particular, Armstrong (1996)has establishedthe

prevalenceof exclusionin optimal contracts:In the presenceofmultidimensional uncertainty, it typically becomesattractive to excludebuyerswith the lowestvaluation for both products,becausethe lossthe sellerincurs by excluding them is typically of secondorderrelative to the gain hethereby makes in terms of rent extraction on the other buyer types. Asecondinsight, illustrated in Armstrong (1999),concernsthe optimality ofsimple contracts:A monopolistselling many items and facing consumerswith independently distributed valuations can approximatethe optimalcontractwith a simple two-part tariff.

We then turned our attention to multidimensional moral hazard.Thecoreprinciple stressedhere is the desirability of keepinga balancebetweenincentives acrosstasks to avoid a form of \"task arbitrage\" by the agent that

results in sometasks being neglected(when higher effort on one task raisesthe marginal costof effort on the other task).Consequently, incentives ona given task should be reducedwhen a competingtask becomesharder tomonitor. In extreme cases,providing no incentives at all may be the

optimum! An alternative route is to prevent the agent from undertakingthe competing task altogether.

Our treatment of these issueshas been based on Holmstrom and

Milgrom (1991).Additional referencesincludethe following:

234 Multidimensional Incentive Problems

\342\200\242 Laffont and Tirole(1991)and Hart, Shleifer,and Vishny (1997)analyzethe trade-offbetweenincentives to cut costsand incentives to raiseproductquality.\342\200\242 Laffont and Tirole (1988b)and Holmstrom and Tirole (1993)discussincentives to improve short-term and long-term profits.\342\200\242 Martimort (1996),Dixit (1996),and Bernheim and Whinston (1998a)analyze multiprincipal models(aswe do in Chapter 13)and discussthebenefitsof exclusivity, that is, of preventing the agent from dealing with

more than oneprincipal.

Multitask problemsalso arise in the absenceof an effort-substitution

problemwhen tasks are directly in conflict. In such a case,eliciting efforton both tasks from a single agent is more expensivefor the principalbecausesuccesson the secondtask directly undermines performanceon thefirst task.This considerationleadsto the optimality ofhiring one agent pertask and making them advocatesfor the task they are askedto perform,rather than having them internalize the overallobjectivefunction of the

principal.Thisquestionhas beeninvestigated by Dewatripontand Tirole(1999),who analyze the optimality of organizing the judicial system (that

is, the \"pursuit of justice\") as a competitionbetween prosecutionanddefenselawyers (seealsoMilgrom and Roberts,1986b;Shin, 1998).Another

exampleofconflicting tasks concernsexante and expostcontrolof\"illegal\"

behavior:Uncovering illegality ex post indicatesa failure to have stoppedit ex ante.Endowing the two tasks to separateagentsthen avoids\"regulatory cover-ups,\" as discussedby Boot and Thakor(1993)and Dewatripontand Tirole(1994b).

Note finally that combining tasks can at times be good for incentives,by reducing adverseselectionproblems:see,for example,Riordan and

Sappington (1987b)(and more generally Lewisand Sappington, 1989,on

\"countervailing incentives\.We concludedthe chapterwith an illustration of a modelthat combines

adverse selectionand moral hazard.Various models share this feature,including Laffont and Tirole(1986)and McAfeeand McMillan (1987b)inthe contextof a problemof regulation of a natural monopoly; and Riley(1985),Riordanand Sappington (1987a),Lewisand Sappington (1997),andCaillaud,Guesnerie,and Rey (1992)in an agencysetting. Cremer andKhalil (1992)alsoconsiderthe relatedproblemofan agent gathering inf or-

235 Static Bilateral Contracting

mation (at a cost)about the state of nature before signing a contractwith

the principal.We have reliedhere on an applicationexploredin Bolton,Pivetta, and

Roland(1997)on the sale of a firm to a new risk-neutral owner-manager,which has beenexploredindependently and more fully by Rhodes-Kropfand Viswanathan (2000)and Lewisand Sappington (2000).In this

application,under pure moral hazard, the optimal strategy is to sellthe firm to the

manager for a fixed price in order to inducesubsequentefficientmanagerial effort.Underpure adverseselection,the sellershould insteadask for100%of the future revenueof the firm, soas to leaveno informational rentto the manager.In contrast to these two extreme solutions, when bothinformation problemsare present,the optimal contractresemblesthecontracts analyzed in Chapter2, with efficient effort for one type and under-provision of effort for the other type.

STATICMULTILATERALCONTRACTING

In this secondpart we considerstatic optimal contracting problemsbetweenn > 3 parties,or betweenone principal and two or more agents.The keyconceptualdifferencebetween the bilateral contracting problemsconsidered in the first part and the problemsconsideredin this part is that

the principal'scontract-designproblemis now no longeroneof simply

controlling a single agent's decisionproblem, but is a much more complexproblem of designing a game involving the strategicbehaviorof severalagentsinteracting with each other.

When it comesto game (or mechanism) design,one is confrontedwith amajor new theoreticalissue:predicting how the game will beplayedby the

agents.Thus,a centralquestionwe shall have to addressin this part is what

are likely or reasonableoutcomesof the game among agentsdefinedby thecontract.This is, of course,the central question that game theoristshavebeenconcernedwith ever since the field of game theory was foundedbyvon Neumann and Morgenstern(1944).Given that game theoristshave

proposeddifferent approachesand equilibrium notions it should comeasno surprisethat contracttheorists(often the samepeoplewearingdifferent hats) have alsoconsideredseveraldifferent equilibrium notions for the

gamesspecifiedby a multilateral contract.In many ways the most satisfactory approachis to designthe gamesin

such a way that each player has a unique dominant strategy. Then theoutcomeof the gameis easy to predict.It is the unique dominant strategy

equilibrium. Whenever it is possibleto achievethe efficient outcomewith

such games,this is the most desirableway to proceed.But when it is not

possibleto achievefirst-best efficiencywith contractswhere eachparty hasa unique dominant strategy, it may be desirableto considera largerclassof contractswhere the contracting parties are playing a game with a lessclear-cutbut hopefully more efficient outcome.1

In otherwords,it may bein the principal'sbestinterestto acceptthat thecontractcreates a game situation with some inevitable uncertainty as tohow the agentswill play the game,if the most plausibleoutcomeof play is

1.Situations where it is not possible to achieve first-best efficiency with contracts where eachplayer has a well-defined dominant strategy are by no means uncommon. Oneof the mainresults in dominant strategy implementation due to Gibbard (1973)and Satterthwaite (1975)is an impossibility result akin to Arrow's impossibility theorem establishing that dominant

strategy contracts cannot in general achieve first-best efficiency when there are at least threeoutcomes and when the players' domain of preferences is unrestricted.

238 Static Multilateral Contracting

more efficient.Now, although there is always someuncertainty as to how

the partiesplay the game, it is not implausible that they will attempt to playsomebestresponse.Thereforeit couldbe arguedthat a plausibleoutcomeof play by the agentsmay be someNash or (with incompleteinformation)Bayesianequilibrium of the game inducedby the contract.

Most of the literature on contractand mechanism designtheory takesthis route and assumesthat the set of outcomesof the game among agentsis the setofNash or Bayesianequilibria.As our purposein this bookisnotto offera crashcoursein game and mechanism designtheory, we shall

consider only optimal multilateral contractswith this equilibrium notion.2

The first chapter in this secondpart (Chapter7) considersmultilateral

contracting problems under hidden information. Perhaps the leadingexampleof such contracting problemsis an auction involving a seller(theprincipal) and multiple buyers (the agents)with private information abouttheir valuation of the goodsfor sale.The secondchapter (Chapter8)considers multilateral contracting problemsunder hidden actions.A classicalexample of such a contracting problem is the moral hazard in teams

problem.More generally, this paradigm allows us to addressmanyquestions pertaining to the internal designof organizations.

2.For an overview of mechanism design under private information and other equilibriumnotions we refer the interested reader to Mas-Colell,Whinston, and Green (1995,chap. 23),Fudenberg and Tirole (1991,chap. 7), and Palfrey (1992).

Multilateral Asymmetric Information: BilateralTradingand Auctions

7.1 Introduction

In this chapter we consider optimal contracts when several contracting

parties have private information. One central questionwe shall beconcerned with is how trade betweena buyer and a selleris affectedbyasymmetric information about both the buyer'svalue and the seller'scost.Weshall alsoask how a seller'soptimal nonlinear pricing policy is affectedby

competitionbetweenseveralbuyers with unknown value.We shalldemonstrate that the generalmethodology and most of the core ideasdevelopedin Chapter2 carry over in a straightforward fashion to the generalcaseofmultilateral asymmetric information.

Considern > 2 parties to a contract,each having private information

about some personal characteristic.The game between these agents is

generally referred to as a gameof incompleteinformation. The accepteddefinition ofa game with incompleteinformation sinceHarsanyi (1967-68)is as follows:

definition A game ofincompleteinformation C=(S,0,j3, u) iscomposedof the following:

\342\200\242 Thestrategy spaceS=Sx x...xSn, with one strategy setSt for eachplayeri =1,...,n.\342\200\242 The type space0= 0ix...x 0\342\200\236,

with 0,denoting playerVs set ofpossible types.

\342\200\242 The probability profile ft = fa x...xf}n, where#\342\226\240

= {/^(ftJQ)}denotesplayer fs probability distribution over other players' types, 6L,- = (6U ...,0i_i, 0I+i,...,6n), conditionalon his own type 0;.The #'sare derivedusing

Bayes'rule from /3*(0),the joint distribution over the type space0.\342\200\242 The utility profile u = ux x...xun whereu^s^,st\\ 6L,,d) denotesplayert'svon Neumann-Morgensternutility as a function of the vectorof strategiess and the vectorof types 6.

The structure of C is assumedto be common knowledgeamong all the

playersin the game.That is, it is common knowledgethat the distribution

240 Multilateral Asymmetric Information: Bilateral Trading and Auctions

for 6 is j8* and that # is simply the marginal distribution for 0_, conditionalon 6i.For readersunfamiliar with thesenotions, an extensiveexpositionofgameswith incompleteinformation canbe found in Fudenbergand Tirole(1991),Binmore (1992),or Myerson(1991).

The analogueof Nash equilibrium for gamesof incompleteinformation

is the notion of Bayesianequilibrium definedas follows:

definition A Bayesianequilibrium of a game with incompleteinformation Cisaprofile ofbest-responsefunctions <7=(ai,...,<r\342\200\236)

=(ci*,a;)whereGi'.0,\342\200\224> Siis such that

0_, E0-, 0-,E0-,

for all i, all 6, and all j,.In words,as for Nash equilibria in gameswith completeinformation, the

profile of best-responsefunctions is an equilibrium if no agent type gainsby deviating from the prescribedbestresponse,when all other agentsareassumedto play accordingto the prescribedbest-responsefunctions.

Generallygameswith incompleteinformation have at leastoneequilibrium (possiblyin mixed strategies)when strategy setsare compactand

payoff functions are continuous. When the equilibrium is unique, there islittle ambiguity, and the outcomeof the game specifiedby the contractcanbe taken to be that equilibrium. When there aremultiple equilibria, it is lessclear what the outcomeshould be,but a casecan be made for taking themost efficient equilibrium as the natural outcome.1In most applicationsinthis chapterthe optimal contractspecifiesa unique equilibrium, so that wedo not needto confront this delicateissue.Nevertheless,it is important tobear in mind that even if an equilibrium is unique there is no guaranteethat the outcomeof the game is that equilibrium.2

With this word ofcaution we shall proceedand assumethat the outcomeof the game specifiedby the contractis the (unique) Bayesianequilibriumof that game.The contract is then describedformally by (C,a) wherea = a[o*(6)]is the outcomefunction specifying an outcomefor each type

1.On mechanisms that achieve unique equilibria, seePalfrey (1992).2. In an interesting experiment Bull, Schotter, and Weigelt (1987)have indeed found that thevariance in subjects' responses is considerably larger in contractual situations where the partiesare effectively playing a game with incomplete information than in situations where they onlyhave to solve a decision problem.

241 Static Multilateral Contracting

profile 0 e 0 associatedwith the game C.This outcomefunction simply

maps the (unique) Bayesianequilibrium <7* =(cf,...,o%) into an outcomefor the game.What is done here is therefore simply to redefineutilities in

terms of outcomesand types, insteadof strategiesand types:u,[o*(0)l0]=K,(a[o*(0)]l0).

The principal'scontractdesignproblemcan then be describedformallyas follows:LetA denote the set ofall possibleoutcomes,and let V(-):0 \342\200\224>

A denotesomeobjectivefunction of the principal.That is, V{6) denotesanoutcomefavoredby the principal when the profile of types is 0.Thecontract (C,a) is then saidto implement VQ if a[o*(0)]= V(6).Theprincipal'sproblemthen is to achievethe most preferredobjectiveV(-) by optimizingoverthe set of feasiblecontracts(C,a).

Although the generaldescriptionof a game of incompleteinformation

and of a Bayesianequilibrium looksrather daunting, in most applicationsin this bookthis descriptionis both simple and natural. In particular, thoseapplicationswhere agentsare assumedto beof only two possibletypes canbehandledwithout too much difficulty.

In addition, although there is a very largeset of contractsfrom which the

principal can choose,many contractswill in fact be redundant. Just as in

Chapter2, it is possibleto simplify the principal'sproblemby eliminatingmost irrelevant alternatives and restricting attention to a subsetofso-calledrevelationcontractsin which the players'strategy setsare simply their typesetS(=0,and where playerstruthfully announcetheir type in the (unique)Bayesianequilibrium.

A revelationcontract(C/0,a), whereS is replacedby 0 in C,is said to

truthfully implement VQ if a[(f*(9)]= V(0), where o*(0) = [^(0),...,0\342\200\236(0)]

is the Bayesianequilibrium of the revelationgame whereeachplayertruthfully reportshis type.Therevelationprincipleestablishesthat for everycontract (C,a) that implements V(-) an alternative revelationcontract(C/0,a) canbe found that also truthfully implements V(-).

the revelation principle Let (C,a) be a contractthat implements V(.);then a revelationcontract(C/0,a) canbe found that tmthfully implements]/(\342\226\240).

One canprove this result as follows:Let o*(0)be the Bayesianequilibrium of the game C,and define the compositefunction a = aoo*.That is,

242 Multilateral Asymmetric Information: Bilateral Trading and Auctions

d(6)= 4o*(0)]for all 6 e 0.Toprove that (C/0,a) truthfully implements

VQ it suffices to show that o(0)= [0i(0),...,6n(6)]is a Bayesianequilibrium of (C/0,a).By construction we have

X A-(M0;)\";M<7*(0)]|e}>X j8i(e-i|0i)\302\253,{4ff*(0),J.]}e-,e9-, 6-,e9-,

But, by definition of a, we have tf[<7*(0)] = d(6) and a[<7*i(0), 5;] =Q.{9[,6-i)for some type announcement

6\302\243^ 0;.Sowe concludethat for all

i =1,...,n we have

X A-(0-;|0;)iaa(e)ie]>X ftCe-iiejkW.e-Jie]0_,e0_, 0-,e0_,

as required,which provesthe result.

Note that, in contrastto the result obtainedin the single-agentproblem,this result doesnot say that there is no lossof generality in restrictingattention to revelationcontracts,since it doesnot establishthat there is

necessarilya unique Bayesian-Nashequilibrium in the revelationcontract(C/0,a) constructedfrom the contract (C,a),even when the contract(C,a) admits only a unique equilibrium. In the transformation processfrom

(C,a) to (C/0,a), the players' strategy setsmay well become smaller,and this may enlarge the equilibrium set (seeDasgupta,Hammond, andMaskin, 1979).In other words, in contrast to the single-agentproblem,the restriction to revelationcontractsmay result in greater uncertainty

concerningthe outcomeof the contract(a largerequilibrium set).Here,wechoose to disregard this problem,which is treated at length in Palfrey(1992).

We shall considerin turn two classicpure exchangeproblems.The first

is a bilateraltradeproblemunder two-sidedasymmetric information (aboutthe seller'scostand the buyer'sreservationvalue).Thesecondis an auction

setting where two (or more) buyers with private information aboutreservation valuescompeteto purchasea goodfrom a seller.As in Chapter2,we beginour analysis of contracting under multilateral asymmetricinformation with a detailedexpositionof the specialcasewhereeachparty has

only two different types.This specialcaseis moretractableand is sufficient

to illustrate most economicinsights. We then proceedto a more advancedanalysis of the more general and elegant problem with a continuum oftypes.

243 Static Multilateral Contracting

7.2 BUateral Trading

7.2.1TheTwo-TypeCase

In Chapter2 we found that when an uninformed party (say, a seller,\"he\makesa take-it-or-leave-itofferto an informed party (a buyer, \"she\,") then

inefficient trade may result.For example,take a buyer with two possiblevaluations for a good:vH > vL > 0,and a sellerwho has a value of zero forthe good.Let /J denote the seller'sprior belief that the buyer has a highvaluation. Then the (risk-neutral) seller'sbestoffer to the buyer is P= vH

whenever j3vH > vL. For such offers,there is (ex post) inefficient tradewith probability (1-/J) wheneverthe buyer'strue value is vL.

If insteadthe informed party (here,the buyer) makesthe contractoffer,then there is always expostefficient trade.Thebuyer simply makesan offerP= 0 which the selleralways accepts.Thus, if the objectiveis to promoteefficient trade,there is a simple solution availablein the caseof unilateral

asymmetric information: simply give all the bargaining powerto informed

parties.This simple examplehighlights one obvious facet of the fundamental

trade-offbetween allocativeefficiency and the distribution of informationalrents:If the bargaining powerlieswith the uninformed party, then that party

attempts to appropriatesomeof the informational rents of the informed

party at the expenseof allocativeefficiency.By implication, to reducetheallocativedistortions associatedwith informational rent extraction, oneshould removebargaining power from uninformed agentswhenever it is

possibleto do so.Stateddifferently, if all bargaining powerof uninformed

agentscansomehowbe removed,oneneedno longerbe concernedaboutallocativeefficiency.

But this solution is not availablewhen there isbilateralasymmetricinformation. An obvious questionthen is whether efficient trade is still

attainable under bilateralasymmetric information if the surplus from trade canbedistributed arbitrarily betweenthe buyer and seller.It turns out that this

questionhas a simple generalanswer.It has beenelegantly put by JosephFarrell (1987)as follows:

Supposethe problem is which of two peopleshould have an indivisible object,a\"seller\" (who originally has it) or a \"buyer.\" The efficient solution is that whoeverin fact values it more should have it, with perhaps somepayment to the other.

244 Multilateral Asymmetric Information: Bilateral Trading and Auctions

(That is, every Pareto-efficient outcome has this form.) The king can easilyachieve this outcome using an incentive-compatible scheme if participation iscompulsory: for example, he can confiscate the item from the \"seller\" and thenauction it off, dividing the revenues equally between the two people.But thissolution is not feasiblewith voluntary trade; the sellermay prefer to keep the

objectrather than to participate and risk having to repurchase (or lose)somethinghe already has.A lump sum payment to the sellercould solve that problem, butthen the buyer (who would have to make the payment) might prefer to withdraw,

(p.120)

As Farrell'sexplanation highlights, the difficulty here isnot somuch

eliciting the parties' private information as ensuring their participation.Weshow in this sectionthat efficient trade can (almost) always be achievedif

the parties' participation is obtainedex ante, before they learn their type,while it cannot be achievedif the parties'participation decisionis madewhen they alreadyknow their type. In other words, when the contracting

partiesalreadyknow their type, the extent to which the surplus from tradecan be distributed arbitrarily between them while mamtaining their

participation is too limited to always obtain efficient trade.Considerthe situation wherethe sellerhas two possiblecostscH > cL> 0

and the buyer's two possiblevaluations are such that vH > cH > vL > cL.3To simplify matters, assumethat valuation and costare independentlydistributed. Let the buyer'sprior beliefsthat the sellerhas low costbe y. Theprincipal'sobjectivehere is to achieveefficient trade. Let x = 1denote\"trade\" and x =0 denote \"no trade.\"

Applying the revelationprinciple, the principal must find pricescontingent

on the buyer'svalue and seller'scostannouncements, P(v,c),such that

efficient trade,that is,x(y,c)=1if and only if v > c,is incentive compatibleand individually rational for both buyer and seller.As we arguedpreviously,the main difficulty here is not to find incentive-compatibleprices:they canbe found becauseefficient tradesare monotonic, that is, the probability oftrading under efficiencyis increasingin the buyer'svaluation and

decreasingin the seller'scost.The problemis rather to find incentive-compatible

pricesthat alsosatisfy both parties' individual-rationality constraints.

3.These are the values for which establishing efficient trade is most difficult. Indeed, supposethat instead we have vH > vL > cH > cL. In that caseefficient trade is guaranteed by fixing aprice P e [cH,vJ.Alternatively, suppose that cH> vH > vL > cL. Then efficient trade isguaranteed by setting a price P e [cL,vL] and letting the seller decide whether he wants to trade atthat price. Similarly, when vH>cH>cL>vL, efficient trade is established by settingP e [vH,cH]and letting the buyer decide whether to trade.

245 Static Multilateral Contracting

Whether efficient trade can be achievedwill dependcrucially on which

individual-rationality constraints are relevant:interim or ex ante IRconstraints. Interim constraints are relevant wheneverthe buyer and the sellerknow their type when signing the contract.Exante constraints are relevantwhen they sign the contractbeforeknowing their exactcostor valuerealization. We begin by showing a general result\342\200\224first establishedin thecontextof efficient public good provision by d'Aspremont and Gerard-Varet (1979)\342\200\224that when ex ante individual-rationality constraints arerelevant, then efficient trade can always be achieveddespitethe doubleasymmetry of information about costsand values.

7.2.1.1Efficient Tradeunder Ex Ante Individual-RationalityConstraints

Supposethat the principal (say, a socialplanner) offersthe buyer and seller,beforeeachonehas learnedhis or her type, the following bilateraltradingcontract:C= (P(v/,c;)= P,,-; x(vhcj) =

Xif, i=L,H;j = L,H),wherexl} = 1if

v; >Cj and xlS

=0 otherwise.Theseller'sincentive-compatibility (IC)and exante individual-rationality (IR) constraints then take the form

(1-P)[Pll-cl]+I3[Phl-cl]>(X-I3)Plh+P[Phh-cl] (7.1)

(1-P)Plh+P[Phh-ch]>(1-P)[Pll-cH]+p[PHL -cH] (7.2)

7[(1-P)Pll+PPhl-cl]+(1-7)[(1-I3)PLh+P(Phh-ch)]>0 (7.3)In theseequationsnote that when the buyer has a low value vL and the

sellerhas a high costcH there is no trade,that is,xLH =0.We thereforerequirethe type-/sellerto prefertruth telling given his type

and given that he expectsthe buyer to tell the truth about her type, and tobreakevengiven that he himself expectsto beof type L with probability y.

Similarly, the buyer's(IC)and ex ante (IR) constraints are

y[vL -Pll]-{1-7)Plh>7[vl-Phl]+d-7)bz.-Phh] (7.4)

y[vH -Phl]+0--7)[vh-Phh]>7[vh-Pll]-(X-Y)Plh (7.5)

p[vH -rPHL-0.-r)PHH]+0--P)[rvL-rPLL-a-r)PLH]>o(7.6)In order to analyze this set of inequalities, let us call P the expected

payment the buyer will have to make to the seller:

P=7[(l-P)PLL+PPHL]+0--y)l0--P)PLH+PPHH]

246 Multilateral Asymmetric Information: Bilateral Trading and Auctions

Then the two (IR) constraints canbe rewritten as

j3vH +(1-j3)yvL >P> ycL +(1-y)j3cH

These constraints thus require the expectedpayment P to divide the

expectedfirst-best surplus, which isclearlypositive, into two positive shares.This requirement implies a condition on the expectedlevel of payments,which can be adjustedwithout any consequenceon incentive constraints,

which dependonly on the differencesof payments acrossrealizationsofcostsand valuations. These incentive constraints can be redefinedfor thesellerand buyer, respectively,as

(1-(3)ch>0.-P)[Pll-Plh]+P[Phl-Phh]>0.-P)cland

{l-y)vH>y[PHL-PLL\\+iX~y)[PHH-PLH]>{l-y)vL

The incentive constraints for the sellercan be interpreted as follows:given truth telling by the buyer, if the sellerannouncesa low cost,tradewill

take placewith probability 1.Instead,if the sellerannouncesa high cost,trade will take placeonly with probability j3 (that is,when the buyer has ahigh valuation). The increasein expectedpayment the sellerwill receiveupon announcing low costshould thus coverthe increasein expectedproduction cost (due to trade taking placewith probability 1insteadof j3) if

the seller'scostis cL,but not if the seller'scostiscH.Theideais exactlythe samefor the buyer: given truth telling by the seller,

if the buyer announcesa high valuation, trade will take placewith

probability 1.Instead,if the buyer announcesa low valuation, trade will take

placeonly with probability y(that is,when the sellerhas low cost).Theincrease in expectedpayment the buyer will incur upon announcing a highvaluation should thus be lowerthan the increasein expectedvalue ofconsumption (due to trade taking placewith probability 1insteadof y) if the

buyer'svaluation is vH, but not if the buyer'svaluation is vL.If constraints (7.1)to (7.6)are satisfied,both parties are happy to

participateand truth telling results, so that we have implemented the expost efficient allocation.The fact that payments exist such that allconstraints are satisfied is easy to see.Indeed,the constraints canbe satisfied

recursively. First of all, note that the incentive constraints are not

incompatible for any party, sincevH > vL and cH > cL.Second,considerany vector

247 Static Multilateral Contracting

(PLL,PLH, PHL, PHH) such that the incentive constraints of the selleraresatisfied. It is easyto seethat any changein (Pll,Plh) that keeps(PLL-PLH)

unchanged will leavethe incentive constraints of the sellerunaffected.Thisobservationprovidesa degreeof freedom that can be usedto satisfy theincentive constraints of the buyer.Finally, we have alreadyobservedthat

any changein (PLL,PLH, PHL, PHH) that involves an adjustment (upward ordownward) of all four payments by the same amount leavesall incentiveconstraints unaffected.This fact providesanother degreeof freedomto setP to satisfy the participation constraints of the parties.This system ofinequalities thereforeadmits a solution, and it is possibleto find incentive-compatibleand individually rational contractsthat achieveefficient trade.

This result holdsvery generally. It holds, in particular, for public-goodsproblemswith an arbitrary number ofagents,eachwith an arbitrary number

of independently distributed types, as d'Aspremont and Gerard-Varet(1979)have shown. But efficient trade canbe achievedevenwhen agents'types are correlated.Thispossibility is straightforward to establishin our

simple example.In fact, correlationof types canmake it easierto achieveefficiency, aswe shall seewhen we considerauctions with correlatedvalues.

A simple generalexplanationbehind this result runs as follows:Assuming

that one of the agentstruthfully revealsher type, the other agent facesdifferent oddsdependingon whether she tellsthe truth or not.Thisdifference in oddscan then be exploitedto construct lotteriesthat have anegative net expectedvalue when the agent lies,but a positive value when the

agent truthfully revealsher type. Then,by ensuring that the ex ante

compound lottery (the lottery overlotteriesfor eachtype) has a positive exantenet expectedvalue, the principal canmake sure that both buyer and sellerare willing to participate.Concretely,here,if for examplethe sellerhas costcH, then he (weakly) prefersthe lottery requiring him to produceat a pricePhh with probability j3, and not to produceand receivea transfer PLH with

probability (1-j3).Similarly, if he has costcL,he prefersthe lottery wherehe obtains (PHL -cL)with probability /J, and (PLL-cL)with probability (1-P).As long as the compoundlottery (composedof the lottery for type cHand the lottery for type cL)has a positive expectedvalue, the selleris willing

to acceptthe contract.And the sameistrue, mutatis mutandis, for the buyer.A helpful economicintuition for this result, which is often given for

problemsof public-goodsprovision under asymmetric information, is that

efficiency requires essentiallythat each agent appropriate the expectedexternality her actionsimposeon other agents.In this way, each agent is

248 Multilateral Asymmetric Information: Bilateral Trading and Auctions

inducedto internalize the whole collectivedecisionproblemand is

effectively maximizing the socialobjective.Sincethe sum of all expectedexternalities is just the socialsurplus, trade can be implemented wheneverit isefficient.

7.2.1.2Inefficient Tradeunder Interim Individual-RationalityConstraints

Now supposethat the buyer and sellerknow their type before signing thecontract.Thenthe exante participation constraints must bereplacedby the

following four interim constraints:

0--P)Pll+PPhl-cl>00-P)Plh+P(Phh-ch)>0Vh-tPhl-(X-Y)Phh>0

rvL-rPLL-a-r)PLH>oIn fact, for somevaluesof /3 and 7 thesefour constraints togetherwith theincentive constraints (7.1),(7.2),(7.4),and (7.5)cannot all simultaneouslyhold.For example,let the probability of the sellerfacing a high-valuation

buyer, /3, tend to 1,then from conditions(7.1)and (7.2)we have PHH \342\200\224 PHL,which we shall denoteP.Therefore,the interim participation constraints ofthe sellerreduceto

P-cL>0P-cH>0In words, when the sellerthinks the buyer is almost surely of type vH,

he knows he will have to producewith probability closeto 1whatever his

type; as a result, the expectedpricePhas to coverhis cost,whether low orhigh.

In addition, when the probability of facing a high-valuation buyer, j3,tends to 1and PHH \342\200\224 PHL = P, by condition (7.5),we have

vH-P>y[vH-PLL\\-{l-y)PLH

which says that, for the buyer, telling the truth about vH and buying the

goodfor sureat pricePshould bebetter than lying and getting vH with

probability 7 (that is, when the sellerhas low cost) and paying the associated

249 Static Multilateral Contracting

expectedprice.This condition togetherwith (3 \342\200\224 1implies that the interim

participation constraints of the buyer reduce to

P<vH

P<yvL+(l-y)vHThis expressionsaysthat the high-valuation buyer cannot bemade to paymore than vH, obviously, but alsonot more than vH -

y(yH- vL): this last

term is the information rent the high-valuation buyer can obtain by

pretending to have a low valuation and trading with probability yat terms that

give the low-valuation buyer zero rents.Collectingall interim participation constraints together,we must have

yvL+(i-y)vH>cHBut if the probability that the sellerhasa low cost,y tendsto 1,this

inequalitydoesnot hold:becausethe buyer believesthat she is almost certainly

facing a low-costseller,it becomesvery attractive for her to pretendto havea low valuation, since in any casethe probability of trade is almost 1.Toprevent the buyer from pretending this, we needP< vL, but this is

incompatible with P> cH, sincewe have assumedcH> vL in order to make the

probleminteresting.In other words, when prior beliefsare such that the sellerthinks that he

is most likely to face a high-value buyer and the buyer thinks that she ismost likely to facea low-costseller,then each one wants to claim a shareof the surplus that is incompatiblewith the other party's claim, and tradebreaksdown.

Thus, when the optimal contract must satisfy both incentive-

compatibility and interim individual-rationality constraints, efficient tradecannot always be obtained under bilateral asymmetric information. Thisresult can be strengthenedby consideringthe same setting with acontinuum of types, as we shall seein section7.2.2.

To summarize, bilateral trade may be inefficient when both buyerand sellerhave private information about their respectivereservationpricesand when this information is available before they engage in

voluntary trade.The reasonwhy trade may breakdown is that eachpartycould attempt to obtain a better deal by misrepresentinghis or hervaluations. Toavoid such misrepresentations,the high-value buyer and thelow-costsellermust get informational rents, but the sum of the rents

may exceedthe surplus from trade in somecontingencies,and with interim

250 Multilateral Asymmetric Information: Bilateral Trading and Auctions

individual-rationality constraints, informational rents cannot be freelytraded acrosscontingencies.

Thepossibility of inefficient tradeunder thesecircumstancesis animportant theoreticalobservationbecauseit providesone explanation for why

rational self-seekingtrading partners leave some gains from trade unex-

ploited.It suggeststhat the Coasetheoremmay breakdown in voluntary

trading situations with multilateral asymmetric information. It alsopointsto the potential value of institutions with coercivepower that can breakinterim participation constraints and secureparticipation at an ex ante

stage.

7.2.2Continuum ofTypes

Most of the analysis of the special2x2model can be straightforwardlyextendedto a modelwith a continuum of types by adapting the

methodologyoutlined in section2.3.In fact,an evensharpercharacterizationof the

main results canbe obtainedin the generalproblem.The material of this section follows closely the model and analysis

of Myersonand Satterthwaite (1983).They considera bilateral trading

problem where a single good is to be traded between two risk-neutral

agents who have private, independently distributed, information abouthow much they value the item. Suppose,without lossof generality, that

agent 1is in possessionof the item and that his privately known valuation

Vi has differentiable strictly positive density fi(v{) on the interval [v.i, vj.Agent 2'sprivately known valuation v2 has differentiable strictly positivedensity/2(v2) on the interval [v2,v2].As in section7.2.1,the interesting caseto consideris that where Hi < v2 < v^ < v2. In that casethere are bothsituations wheretrade is (first-best)efficient and situations wheretrade isinefficient.

7.2.2.1Inefficiency of BilateralTradewith Interim Individual-

Rationality Constraints

This sectionestablishesthat efficient trade cannot be achievedif

contracting takes place at the interim stage. In order to prove this point,we first determine the set of implementable trades that satisfy interim

individual rationality. We then show that efficient trading isnot in this set.

Step1:The Set of Implementable Trades Applying the revelationprinciple, the principal's problem is to choose a revelation contract

251 Static Multilateral Contracting

{x(vi, v2); P(yu v2)l\342\200\224where x(vu v2) denotesthe probability of trade and

P(yu v2) the payment from agent 2 to agent 1contingent on announcements

(yi, v2)\342\200\224which satisfiesincentive-compatibility and individual-rationalityconstraints and maximizes the expectedgains from trade.

Let us start with somedefinitions: In any truth-telling equilibrium, agentl'sexpectednet revenueis given by

P1e(y1)=rP(yuv2)f2(y2)dv2Jv-2

and agent 2'sexpectedpayment by

P2e(v2)=\\V1P(y1,v2)f1(y1)dv1

Also, their respectiveexpectedprobabilitiesof trade are given by

A (vi) = P*0>i,v2)/2(v2)dv2

x{(v2)= fV1x(vi, v2)/i (vi )dVi

Therefore,each agent type'sexpectedpayoff under the contractcan bewritten as

u1(y1)=P1e(v1)-v1x[(y1)

u2 (v2)=v2xi (v2)-P{(v2)

The incentive-compatibility and interim individual-rationality constraintsfor both agentsare then given by

ux (vi) >P{(vi)-vijcf (Vi) for all vx, vx e[vx, vx ] (IC1)

u2 (v2)> v2xl (v2)-P2e (vi) for aU v2, v2 e[v2, i72] (IC2)

Mi (vi) > 0 for all Vi e[v!,vi ] (IR1)u2 (v2)> 0 for all v2 e[v2,v2] (IR2)

How can we reduce this set of constraints to a more manageableconstrained optimization problem?As we learned in Chapter 2, when both

agents'preferencessatisfy the single-crossingcondition

252 Multilateral Asymmetric Information: Bilateral Trading and Auctions

d [ duldx6 1 n\342\200\224 '-. >0dvl du/dPe]as they do here,we ought to expectthe following:

1.Only the individual-rationality constraints for the lowesttypes may bebinding: Ui(v{)> 0 and u2(v2) ^ 0.Notehere that the \"lowest type\" for agent1is Vv Indeed,sinceagent 1is the owner of the item to beginwith, hisvaluation should be interpretedas an opportunity cost,so that his rents will

be lowestwhen his costis highest.2.Each agent's expectedprobability of trade must be monotonicallyincreasingin his type: xf(v2) is (weakly) increasingin v2, and x{(yi) is(weakly) decreasingin v^

3.Given monotonicity, only local incentive-compatibility constraintsmatter, sothat eachagent type'sinformational rent issimply the sum of thelowesttype'srent and the integral ofall inframarginal types'marginal rents:

\302\253i (vi) = \"i (v\"i) + fl x\\ (yi )dyi

and

\302\2532 (V2 ) =\302\2532 (V,)+ P*20^^2

Thiseducatedguessis indeedcorrect,as the following argument shows:From the (IC)constraints one observesthat, sincefor any two Vi > vx wemust have

ux (vi) >Pi (vi)-vixf (vi) =\302\253i (vi)+(vi -Vi)x{(Vi)

and

u1(v1)>P1e(v1)-v1xe1(v1)=u1(v1)+(v1-v1)xe1(v1)

It follows that

(vi -vi )x{(vi) > (vi -Vi)x{(vi)

This inequality then implies that x\\ is weakly decreasingin v^ Next, by

dividing by {yx-v{) and letting Vi tend to vl5 one obtains

\"i(vi) =-4(vi)

253 Static Multilateral Contracting

which, by integrating with respectto vl5 is equivalent to

\"i(vi) =ui(vi)+ f x{0>i)dyxJvi

Note that sincex(yi, v2) \342\202\254 [0,1]one canalsoobservethat Ui(yi) > u^Vi)for all Vi e [vb vi].One easilyseesthat exactly the sameargument canbeappliedto agent 2'sincentive constraints.

Thus, as in the situation with one-sidedasymmetric information, we canreduce all the incentive constraints to essentiallytwo conditionsfor eachagent, a monotonicity condition and the first-ordercondition for eachagent'soptimization problem,which requiresthat the marginal increasein

informational rent [-u[(vi)for agent 1and u2(v2) for agent 2] be equal tothe expectedprobability of trade [x{(v{)for agent 1and x2(v2)for agent 2].Also, at most one individual-rationality constraint will bebinding for eachagent.

The intuition for these results is straightforward. Agent 1(the seller)gains by pretending that he is reluctant to sellsoas to induceagent 2 (thebuyer) to offer a higher price.Thus, other things equal,he would alwaysclaim to have the highest possibleopportunity cost v\"i. Toinducethe sellerto truthfully announcea valuation vx < vi, he must thereforebegiven someinformational rent, and giving him this rent requiresraising the likelihoodof trade when he announcesa lower opportunity cost v^ A similar logicapplies to the buyer, who wants to underestimateher valuation for the

good.Thus any buyer with valuation v2 > y2 must begiven somerent to beinducedto tell the truth, or equivalently the likelihoodof trade must beincreasingin the buyer'sannouncedvaluation.

Step2:Efficient Trading IsNot Implementable Can efficient trading beimplemented?Efficient trading requiresthat trade take placewith

probabilityone whenever Vi < v2 and with probability zero when Vi > v2. Note

first that this requirement doesnot conflictwith monotonicity: the expectedprobability of trade will clearlybe increasingin v2 and decreasingin Vi.

What doescreate a problem,however,is the sharing of rents: given the

requirementsof localincentive compatibility and interim individual

rationality,efficient trading imposesa lowerbound on the expectedrents to be

concededto eachparty: for party 2,evenassuming u2(v2) =0,we must have

u2(y2)= r Fl{y2)dy1Jv-2

254 Multilateral Asymmetric Information: Bilateral Trading and Auctions

And for party 1,evenassuming ux{v-j)= 0, we must have

\302\253i(vi)= \\V1[1-F2(yi)]dy1

Taking the expectationsof theseexpressionswith respectto v2 and vi and

summing them, it may well be that this lowerbound on the expectedrentsto be given to the two partiesmore than exhauststhe total surplus to beshared.And it turns out that indeeda contractimposing the first-best tradeprobabilities[x*(vl5v2)= 1for vx < v2 and x*(vu v2) = 0 for vx > v2] and

satisfying all incentive-compatibility constraints inevitably leadsto a violationof interim individual rationality.

Toseethis result, observefirst that

Pi (vi) =Mi (vi)+vxx{ (vi) =vxx{ +ux (vi)+ fn x{(yx )dyl

P{(y2)=v2xe2 (v2)-u2 (v2)=v2xe2-

Mi (v2)- I\"* x|(y2)dy2

For each expressionPf(v,), taking expectationswith respectto v\342\200\236

integrating the last term by parts, and then substituting for the definitions ofPf(v,) and xei(vi), one obtains, respectively,

f f P(yl,v2)fl(yl)f2(y2)dvldv2

=f J Vi*(Vl, v2)fM)f2(y2)dv1dv2 +ul(vl)+ PPx(vu v2)f1(v1)f2(v2)dv1dv2

=f f v2x(v1,v2)f1(v1)f2(v2)dv1dv2-u2(v2)

-PP*0i,v2)/i(viXl-F2(y2)]dv1dv2

Rearranging the last two expressions,one then obtains

Ui(v1)+u2(v2)=

l-F2(v2Yv2 \342\200\224

/2O2)

'AW/l(Vl).

*0i,V2)/i (Vi )/2(V2 )dVidv2

255 Static Multilateral Contracting

This expectedpayoff must be nonnegative to satisfy both interim

individual-rationality constraints. But substituting x(yu v2) in this

expression with the first-best trading probabilitiesx*(vx, v2), we obtain

\"i(v\"i) +\302\2532(v2)

^2/2^2)+F2(v2)-l]f1(y1)dv1dv2Jv-2 Jv-1

r\\>2 /\342\226\240min{v2,?t}-I [vifi(vi) +Fi(v1)]f2(v2)dv1dv2

= P[V2/2(v2) +F2(y2)-lJFi(v2)dv2 -Pmm{v2Fi(v2),Pi}/2(v2)dv2

where the last term isobtainedafter integration by parts.Note,moreover,that i7i(v2)=1for v2 > vx, which meansin particular that

min{v2i7'i(v2), vx) = v2i71(v2)wheneverv2 < V! and rmn[v2Fi{v2), vx) =^whenever v2 > Vi. Consequently, the precedingexpressioncanbe rewritten as

\"1 (vi)+u2 (v2)=-J [1-F2 (v2)]i?(v2)dv2 + C(v2 -vi )/2(v2)dv2Jv2 Jvl

and, again integrating by parts,

\"1 (vi)+u2(v2) =-J [1-F2 (v2)]i?(v2)dv2-r[i\302\243 (v2)-l]dv2

Finally, sincei7i(v2)= 1for v2 > vl5 this expressionisequal to

=-f[l-JF2(v2)]i=l(v2)dv2

This last expression,however,is strictly negative, sincev2 < Vi.This resultestablishesthat the bilateral trading problemwhereboth buyer and sellerhave private information at the time of trading always results in tradeinefficiencies when eachcontracting party hasa continuous distribution oftypeswith strictly positive density on overlapping supports, as we have assumedhere.

7.2.2.2Second-BestContract

If the first-best outcome is unattainable, the natural next question is,What isthe second-bestoutcome?Remarkably, it turns out that the simpledouble auction first consideredby Chatterjeeand Samuelson(1983)is a

256 Multilateral Asymmetric Information: Bilateral Trading and Auctions

second-bestoptimal contracthere.We now show this result by first

characterizing the second-bestoutcomeand then showing that it can beimplemented with the Chatterjee-Samuelsondoubleauction.

Step1:Characterizing the SecondBest Supposefor simplicity that eachagent'svalue for the item is uniformly and independently distributed onthe interval [0,l].Thesecond-bestex ante optimal outcomemaxirnizes the

expectedsum of utilities of the two parties subject to the incentive-compatibility and interim individual-rationality constraints for eachagent.That is, formally the second-bestoutcomeis a solution to the followingconstrainedoptimization problem:

max f1 f1 f1 f1axJq

ux (vi )dvl +Jq

u2 (v2)dv2=Jq Jq

(v2-Vi)x(yi, v2)dvxdv2

subjectto

Jo Jq(2v2-1-2vi)x(vls v2)dvxdv2 > 0

The constraint is simply the expressionfor the participation constraint

x(vi, v2)/i (n )/2(v2)dvidv2 > 0j-K2fVL

hi hi

'1-F2(y2)

fi(v2) .Vi +

/l(Vl).

when both v,'s are uniformly distributed on the interval [0,1].Recall that when this condition holds, all incentive-compatibility and

interim individual-rationality constraints are satisfied.The constraint

must be binding at the optimum. Indeed,if it is not, then the solution

to the unconstrained problemis simply the first-best rule, x*(vi, v2) = 1if

(y2-

Vi) > 0 and x*(vx,v2) = 0 if (v2-Vi) < 0.But for this rule it is easilyverified that

Jo JQ(2v2-1-2vi )x(yi, v2)dvxdv2 <0

Next, pointwise optimization implies that the optimal x(vi, v2) must takethe generalform

fl \\fO- +2X)v1<Q.+2A)v2-%10 otherwise

257 Static Multilateral Contracting

whereX is the Lagrangemultiplier associatedwith the constraint. Thus theconstraint canbe rewritten as

Li_f 1+2H2v2-l-2Vl)dvidv2=01+2A

After integration, this constraint reducesto the following simple expression:

3A2+2A3-1

6(1+2X)3

so that, at the optimum, X=\\, and the second-besttrading rule isgiven by

10 otherwise J

Inefficient trade thus obtains wheneverv2 - \\< vx < v2. That is, under

symmetric information it would beefficient to trade,but under the second-bestefficient contracttrade doesnot take place.

Step 2:Implementing the SecondBest through a Double Auction

Chatterjeeand Samuelson(1983)considerthe trading game where eachagent simultaneously makesa bid for the item, bt. Mbx<b2, agent 1tradeswith agent 2 at a price P=(bi+b2)/2.Otherwise,no trade takes place.There existsa Bayesianequilibrium of this double-auctiongame whereagent 1and agent 2,respectively,choosestrategies

b1=<j1(y1)= \302\261 +jv1

b2=o2(y2)= \302\261 +iv2

To checkthat this is indeed a Bayesianequilibrium, note that agent l'sand agent 2'sbestresponsefunctions of this double-auctiongameare givenby the solutions to

m^L{^^-Ag2{bi)db2and

max \\v2-

IgiilhWh

Eachparty i is thus trying to maximize his surplus (which isnonzeroonly if

bx < b2)under the assumption that party ;\"'s bid isdistributed accordingto the

density \302\243,(\302\243>,).Denotingthe cumulative density by G;(\302\243>;)

and using Leibniz'srule,oneobtains the first-orderconditionsfor theserespectiveproblems:

^[l-G2(b1)]=(b1-v1)g2(b1)

and

1-G1(b2)=(v2-b2)g1(b2)

In addition, for the candidateequilibrium strategieswe have

G2(W =Pr(b2<W =Pr(v2<|ft1-|j=|ft1-|and thus

giQh)=GiQh)=\\

and similarly

G1(b2)=Pr(b1<b2)=-b2--and thus

Substituting the valuesof g^bj) and G^bj)in the first-orderconditions,one cancheckthat we obtain the bidding functions we startedwith, that is,

1 2k=<7i(Vi) =-+\342\200\224Vi

and

1 2b2=a2(v2)= \342\200\224 +-v2

which are indeedbest-responsefunctions to one another and thus form aBayesianequilibrium. In this equilibrium, trade takesplaceas in the secondbest,that is, if and only if

2 12 1 1\342\200\224Vi+\342\200\224

<\342\200\224 v2 +\342\200\224r or Vi<v2 \342\200\224

3 4 3 12 4

Note,however, that thereare otherequilibria of this game,which arenecessarily inefficient comparedwith the secondbest.Theworst equilibrium isone whereagent 1always bidsbx =1and agent 2 always respondswith b2=0.In that casetradeneveroccurs.Unfortunately, it is not obvious a priorithat the two agentswould naturally chooseto play the most efficient

equilibrium. Therecouldthus be gains to specifying a more complicatedgamewherethe setofBayesianequilibria is smaller (possiblyunique) sothat theoutcomeof the game ismorepredictable.

7.2.2.3Summary and Caveat

A central insight of the theory of optimal contracting under asymmetricinformation is that one should expectallocativeinefficienciesto arise that

are similar to the classicinefficiency associatedwith monopoly pricing.Thenovel lessonof the theory is that monopoly poweris not just derivedfrom monopoly ownership of a scarceresource,but also from theinformational monopoly rent present in most contracting situations with

asymmetric information. We began this chapter by pointing out that the

potential allocativedistortions, arising from the uninformed party's attemptto extract the informed party's informational rent, are reducedor evenentirely eliminated if the bargaining power is shifted to the informed

party. In more general contracting situations, however, where everyparty hasprivate information at the time of contracting, there is no clearlyidentifiable \"uninformed party\" from whom the bargaining power is tobe taken away. In such situations, all parties are \"uninformed\" with

respectto the otherparties'information, soto speak.Oneshould thereforeexpectallocativeinefficienciesto ariseunder optimal contracting that aredriven by eachparty's attempts to extract the other party's informational

rent. Conversely,aswe have illustrated, when neither party hasprivateinformation at the time ofcontracting (and thereforeno informational rent), oneshould expectallocativeefficiencyto obtain, evenif the contracting partiesacquireprivate information at a laterstage,which must betruthfully elicitedto be able to implement the efficient allocation.In other (more abstract)words, allocativeefficienciesare not attributable to the presenceofincentive compatibility constraints perse,but rather to the combination ofincentive compatibility and (interim) participation constraints.

We closethis sectionwith an important caveatto this generalprinciple.The analysis of contracting under private information in Chapters2 and 6

260 Multilateral Asymmetric Information: Bilateral Trading and Auctions

and section7.2has consideredonly asymmetric information about privatevalues.When instead the informational advantage concerns commonvalues, then allocativeinefficienciesgenerally obtain even in simplesituations wherethere is only one informed party. We illustrate this observationwith the following simple example.

Considerthe problemfacedby a buyer (agent1)and a seller(agent 2) ofa painting with the following valuations asa function of the state ofnature 6:

v1(9)=a19-k1v2(6) =a29-k2with

0< ax < a2

0< ki < k2

Thestateof nature might measurehow fashionablethe painting is likelyto be.Supposethat only the sellerobserves9e [9,6],where

a2-axand that the buyer'sprior belief is that 9 is uniformly distributed on theinterval [9,9].Efficiency requiresthat the buyer acquirethe goodwhenever

a2-axHowever,sinceonly the sellerknows the value of 9, it is not possibleto

inducehim to give away the goodonly for high valuesof ft the higher the9, the morereluctant he will be to part with the good.4In fact, the most one

4. Applying the revelation principle, consider the set of contracts [x(ff), P(6)}over whichthe contracting patties can optimize, with x(6) e [0,1]the probability that the saletakesplace contingent on announcement 6 by the seller and P(6)the payment from the buyer tothe seller contingent on announcement 61Incentive compatibility requires that for any two 6and 6we have

M-kx )[i-x(6)]+P(e)>[axe-kx )[i-x(e)]+p(e)

and

(fll0-fc1)[i-*(e)]+p(e)s(a10-fc1)[i-*(0)]+p(0)or, combining these constraints,

261 Static Multilateral Contracting

canachieveis eitherno trade at all or unconditional trade for a given price(providedthis doesnot violate IR constraints),dependingon what

generates a higher expectedsurplus.This simple exampleillustrates how, when there is private information

about common values,quite generally trade will be inefficient evenif thereisonly one-sidedasymmetric information.

7.3 Auctions with PerfectlyKnown Values

In the previoussubsectionwe showedhow in the presenceof bilateralasymmetric information it may not always bepossibleto achieveefficienttrade. In this subsectionwe show how the potential inefficiency resultingfrom bilateralasymmetric information canbe reducedwhen there is

competition among agents.We shall illustrate this general principle in the

simplest possibleexamplewherea sellerfacestwo risk-neutral buyerscompeting to buy one indivisible good(say,a house).As before,eachbuyer hastwo possiblevaluations for the good:

jvH with probability /3, 1

lvL with probability 1-j3Jwhere vH > vL > 0 and /3, denotesthe seller'sprior belief that buyer i =1,2 has a reservationvalue of vH. To keep things as simple as possible,we shall assumehere that /3, = /3 and that each buyer's value is drawn

independently.When the seller(with costnormalizedto 0) setsthe terms of trade and

there is a single buyer, he setsa price equal to vH whenever/3v# > vL, sothat inefficient trade occurswith probability (1-(3).With two buyersefficiency is improved even if the sellersticks to the samepricing policy,sinceinefficient trade now occursonly with probability (1-/3)2.In other words,efficiencyis improved with two buyers just becausethe likelihood of thesellerfacing a high-valuation buyer has increased.To some extent this

improvement is an artifact of the simple structure of our example.Thegeneralreasonwhy efficiencyis improved with two buyers is that by getting

(e-e)[x(e)-x(ej\\>oIt follows that any incentive-compatible contract must besuch that x(6)is (weakly)decreasing

in 6.

Multilateral Asymmetric Information: Bilateral Trading and Auctions

the buyers to competein an auction the sellercan extract their

informational rent with smaller trade distortions.

Optimal Efficient Auctions with IndependentValues

The seller'soptimal auction designproblem in this exampletakes a verysimple form. Sinceboth buyers are identicalex ante, the sellercan,without

lossof generality, restrict attention to symmetric auctions, where both

buyers are treated equally. Applying the revelationprinciple, an optimalauction then specifiesthe following:

\342\200\242 Payments from buyers to the sellercontingent on their announcedreservation values {PHH, PHL, PLH, PLL).\342\200\242 Contingent trades{xHH, xHL, xLH,xLL], wherext] denotesthe probability ofbuyer i getting the goodgiven announcements v; and v;.

Thesecontingent payments and trades must satisfy the usual constraints:

1.Feasibility: 2xHH < 1;2xLL < 1;xHL +Xlh < 1.2.Buyer participation for high- and low-valuation types, respectively:

PixnnVn -Phh)+0-~P)(xHLvH -PHL)>0 (IRH)and

(3(xLHvL -PLH)+0--P)(xLLvL-PLL)>0 (IRL)

3.Incentive compatibility for high- and low-valuation types, respectively:

P(xHHvH-PHH) +(1-fi)(xHLvH

-PHL) >

P(xLHvH -PLH) +(1-P){xLLvH ~Pll) (ICH)and

P(xLHvL -PLh)+0--P)(xllvl-Pll)>P(.xhhvl-Phh)+{1-P){xhlVl-Phl) (ICL)An auction is efficient if and only if the item is allocatedto the highest-

value user.SincevH > vL, this statement means in particular xHL = 1andxlh = 0.Sinceboth buyers are identicalex ante, the sellercan, without

lossof generality, restrict attention to symmetric auctions, that is, caseswhere both buyers are treated equally. Therefore,efficiency also meansXHH = Xll = 1-

263 Static Multilateral Contracting

Here,we restrict attention to efficient auctions and characterizethe onethat maximizes the seller'sexpectedrevenuein this class.We know that, asin Chapter2,constraint (IRH)will hold, given that (IRL)and (ICH)bothhold.Moreover,(ICH)is likely to be binding at the optimum while (ICL)islikely to beslack:indeed,if the sellerknew the type of the buyer, hewould

manage to extract all of her expectedsurplus. This possibility, however,would inducea high-valuation buyer to understateher valuation in orderto limit this payment. Thus let us proceedunder the assumption that only

(IRL)and (ICH)bind. Observingthat all parties are risk neutral, we canalso expressall payoffs in terms of the expectedpayments P\302\243

= j3PLH +(1-P)PLLand PeH = fiPHH+ (1-P)PHL-Therefore,the efficient auction that

maximizes the seller'sexpectedrevenuesolvesthe following constrainedoptimization program:

max2[pP^+a-P)P\302\243]

subjectto

PjvH+(l-/i)vH-Pi>(1-|3)ivH-Pi (ICH)

(l-/3)ivL-P/>0 {IRL)

At the optimum, the two constraints bind5 and uniquely pin down thetwo choicevariables;that is,

n=fLf!L md\302\253=^+<t\302\243k

so that the seller'smaximum net expectedrevenue from an efficient

auction is

j3vH +0.-P)vL (7.7)

5.This means that (ICL)will besatisfied:

pkVH +(i_p)vH

-flf =(l-p)lVH -PI

implies

P^vL+(l-p)vL-Pfi<(l-p)\302\261vL-P\302\243

264 Multilateral Asymmetric Information: Bilateral Trading and Auctions

The sellerobtains lessrevenue than the expectedtotal surplus from

trade, which equals [1- (1- (3)2]vH + (1- j3)2vL, because,just as in

Chapter2, the sellerhas to concedean informational rent to each high-valuation buyer, equal to (1- j3)(vH - vL)/2, the probability of the low-valuation buyer obtaining the objecttimes the differencein valuations. In exante terms, this information rent coststhe seller2)3(1-(3)(vH-vL)/2, which

is in fact the differencebetweenthe expectedsurplus from trade and the

expectedrevenueof the sellerin expression(7.7).In the next subsectionweshall ask the questionwhether distortions from efficiencymight berevenueenhancing for the seller,that is,whether, just asin Chapter2,the sellermightwant the low-valuation buyers to have a lowerprobability of receivingthe

object,soas to reducethe informational rent of the high-valuation buyers.Before dealing with that question,let us consider the casewhere the

sellerfaces n > 2 buyers with independently and identically distributed

valuations:

(vH with probability /J

\\vL with probability 1\342\200\224 /3

As in the two-buyer case,the individual-rationality constraint (IRL)andthe incentive constraint (ICH)together imply that

and

Peu = n-l^n-V\\^-\\\\-pf\\ (1-jSr1i J n \342\200\224 i

yR (yH _yL)

The first condition saysthat the low-valuation buyers obtain zero rents,sincethey pay vL times the probability that they get the object(which is the

probability that all other buyers are also low-valuation types, times IIn,sincethen the objectis allocatedat random).And the secondcondition saysthat the high-valuation buyers pay vH times the probability that they get the

objectminus the information rent. The (somewhat involved) probabilitythat they get the objectdependson how many other high-valuation buyersthere are,sincethe object is allocatedat random among all of them. Thekey term, however,is the secondone, that is, the information rent, which

again is the probability that a low-valuation buyer getsthe object,times the

Static Multilateral Contracting

differencein valuations. But note that, asn \342\200\224>\302\260\302\260,

thanks to competition,theinformation rent tends to zero,so that the selleris able to extractalmostall the gains from tradeas the number ofbiddersbecomeslarge.6And sincethe objectis then allocatedalmost for sure to a high-valuation bidder,7 theseller'sexpectedrevenueconvergesto vH.

Optimal Auctions with IndependentValues

Let us return to the casewith two buyers, wherethe sellerhas to concedeapositive information rent to the high-valuation types.As saidbefore,thesellermay be able to increaseexpectedrevenue beyond the maximumrevenueobtainablewith an efficient auction, /JvH +(1-)3)vL,by refusing tosellto low-valuebuyers.Just as inefficiently high pricing may beoptimal forasellerfacing a singlebuyer, inefficient auctionsmay yield ahigher expectednet revenuethan efficient ones.Indeed,by reducing xLL below\\ the sellercanrelaxthe incentive constraint (ICH)and thus extracta higher paymentfrom high-value buyers.We now turn to the analysis ofoptimal auctions anddeterminewhen it is optimal for the sellerto setup an efficient auction.

Assuming again that only constraints (IRL) and (ICH)bind at the

optimum, the seller'soptimal choiceof xl} isgiven by the solution to the

following optimization problem:

max2[^+(l-i3)P/]XH'P\302\260

subjectto

[PxHH +(1-(3)xHL]vH -Pfi=[(3xLH+(1-(3)xLL]vH -P[ (ICH)

WxLH+(l-P)xLL]vL=Pl (IRL)

2xHH<l, xHL +xLH<l, 2xLL<\\ (feasibility)

ReplacingPeH andP\302\243

in the seller'sobjectivefunction using theconstraints (IRL) and (ICH),we obtain the reducedproblem:

6.Each of the n buyers receives, with probability fi, an information rent equal to

n

And n times this expression tends to zero when \302\253\342\200\224>\302\253=.

7. It goesto a high-valuation buyer whenever there is at least one, that is, with probability1- (1-j3)\"_1, which tends to 1when \302\253->\302\253=.

266 Multilateral Asymmetric Information: Bilateral Trading and Auctions

max2PQfixHH +(l-P)xHL]vH-[fixLH+(1-P)xLL](vH-vL)}

\"+20--P)Wxlh+0--P)xll]vl

subjectto

2xHH <1, xHL +xLH <1, 2xLL <1 (feasibility)

Theoptimal valuesfor xi} cannow bedeterminedby evaluating the signsof the coefficientsof x^ in the seller'sobjectivefunction:

\342\200\242 Considerfirst xHH. Its coefficientis 2j32vH, which is strictly positive.It isthereforeoptimal for the sellerto set xHH =

\\.\342\200\242 Next, note that the coefficientof xHL, 2)3(1- fi)vH, is always positive sothat it is optimal to set xHL = 1-xLH. Moreover,it is biggerthan thecoefficient ofxLH, 2[-fi2(vH-vL) + j3(l-j3)vL],so that it is optimal to set xHL =1and xLH =0.\342\200\242

Finally, the coefficientof xLL is

2[-j3(l-j3)(vH-vL)+(l-j3)2vL]so that it is optimal to set xLL = ~ if and only if (1- /J)vL > /J(vH

- vL);otherwise,it is optimal to set xLL =0.

The optimal auction thus again implies \"efficiency at the top,\" that is,maximum probability of trade (subject to feasibility) for high-valuation

buyers.When the sellerfacestwo low-valuation buyers, the goodis tradedif (1-P)vL > fi(vH

-vL), in which casethe seller'sexpectedpayoff is fivH +(1-P)vL. If instead(1-/J)vL< /J(vH

-vL), then the goodis not traded andthe seller'sexpectedpayoff is then [/J2 +2/J(l-(3)]vH.

Note that when the sellerfacesonly a single buyer, he decidesto excludethe low-valuation buyer under the samecondition [(1-fi)vL < /J(vH

-vL)],so that monopoly distortions are the samewhether there is a single buyeror multiple buyers competing for the object.The only differenceis that

with multiple buyers the likelihood of inefficient exclusion of low-valuation buyers is lower.Interestingly, if a sellerfacing two buyers hasthe option of producing a seconditem at zero cost, then he will alwayswant to producethe seconditem and sellboth items at a priceP=vLwhen

(1-P)vL > fi(vH- vL), and when the opposite inequality holds at price

P= vH.

Static Multilateral Contracting

Standard Auctions with IndependentValues

It is interesting to comparethe optimal auction (from the point of view ofthe seller)with standard auctionsfrequently usedin practice.Thereare atleastfour widely known and usedauctions\342\200\224the English auction, the Dutchauction, the Vickrey auction, and the first-price sealed-bidauction.Each ofthese auctions definesa game to be playedby buyers (or bidders),wherethe strategiesof the buyers areeitherbids(priceoffers)or continuation/exit

decisions.

English Auction TheEnglish auction is,perhaps,the best-known auction.Themovesofeachbuyer at any stageof the auction are to announcea priceoffer that is higher than any previously announcedbid, stay silent, or dropout. The auction endswhen no new priceoffersare forthcoming. The goodis soldto the highest bidderat the price offeredby that bidder,providedthat the highest bid exceedsthe introductory price set by the seller.Eventhough this is in many ways a rather simple auction, it is sometimesusefulto simplify its rules even further and to introduce an auctioneerwho

continuously raisesthe asking price.Thebuyers'movesthen are simply to\"continue\" or to \"exit.\"

Dutch Auction The auctioneerstarts the Dutch auction game by callinga very largeasking price and continuously lowering the priceuntil a buyerstopsthe processby acceptingthe last quoted asking price.Just as in the

simplified English auction, biddershave only two moves:\"stop\" or\"continue.\" The first buyer to stop the processgets the good at the last pricequotedby the auctioneer.

Vickrey Auction Herebuyers simultaneously offer a price in a sealedenvelope.The auctioneer collects all the bids and sells the good tothe highest bidderat the second-highestprice offer, provided that the

second-highestoffer is above the introductory price.If it is belowthe minimum price offer,but the highest bid is above it, then the highestbiddergets the good at the introductory price.If two or more biddersoffer the highest price, then the good is allocated randomly to one ofthem at the highest price.In this auction buyers'strategiesare simply priceoffers.

First-PriceSealed-BidAuction All the rules are the same as for the

Vickrey auction, exceptthat the goodissoldat the highest priceoffer.

268 Multilateral Asymmetric Information: Bilateral Trading and Auctions

Thesefour auctions seemrather different at first sight, and a sellerfacingthe choiceof any ofthesefour auctionsas the sellingprocedurefor his goodmay be at a lossdeterrnining which isbestfor him.

If the sellerand buyers are risk neutral and if the buyers'valuations areindependently drawn from the same distribution, it turns out that it doesnot matter which of these auctions the sellerchooses:as we will see,theyall yield the sameexpectedrevenue.What ismore,despitefirst appearances,the English auction and the Vickrey auction yield the samesymmetricequilibrium outcomein this case.TheDutch and first-pricesealed-bidauctionsare strategically equivalent.

Considerfirst the English and Vickrey auctions,whereeachbidderhas aunique (weakly) dominant strategy: In the English auction the dominant

strategy is to \"continue\" until the ascendingasking priceprocesshits thereservationvalue of the bidderand then to \"stop.\" This strategy clearlydominatesany strategy involving \"continuation\" until after the asking pricehasbeenraisedabovethe bidder'sreservationvalue.It alsodominatesany

strategy where the bidder\"stops\" before the asking pricehashit herreservation value.

In the Vickrey auction eachbidder'sdominant strategy is to offera priceequalto her reservationvalue.With such an offerthe bidderendsup gettingthe good at the second-highestbid\342\200\224a price independent of her own

announcedoffer\342\200\224whenever shehasmadethe highest offer.If a priceequalto her reservationvalue is not the highest offer,then the goodisnecessarily

soldat a pricegreaterthan or equalto her reservationvalue. Clearly,nobidderwould want to acquirethe goodat such a high price.Thus the

strategyof bidding her reservationvalue clearlydominatesany strategy where

the bidder offersa higher price.It also dominatesprice offersbelow thebidder'sreservationvalue:such bidseither do not affectthe final outcome(when they remain the highest bid, sothat the goodis still soldto the highestbidderat the second-highestbid),or they adverselyaffect the outcome(when they are no longerthe highest bid, so that the bidderlosesthe goodto another bidder).

In sum, the equilibrium pricesand the final allocationsin the uniquesymmetric dominant-strategy equilibrium are the sameunder the English andVickrey auctions, so that the sellerought to be indifferent betweenthem.What is the seller'sexpectedrevenue?With probability /J2 both buyers havehigh valuations and offer vH. In that caseeach buyer gets the good with

probability \\ at price vH. With probability 2/J(l-/J) one buyer has a high

269 Static Multilateral Contracting

valuation and the other a low valuation. In this event the high-valuation

buyer getsthe goodat the second-highestpricevL. Finally, with probability

(1-P)2both buyers have a low valuation. Theneitherbuyer getsthe goodwith probability ^ at pricevL.Thus the expectedrevenuefrom the Vickreyand English auctions is

pvH+(l-p2)vL

Also, the expectedpayoff of a vL buyer iszero,while a vH buyer getsan

expectedpayoff of (1-P)(vH -vL).8Considernow the Dutch and first-pricesealed-bidauctions.They are

equivalent in the sensethat the strategy setsand the allocationrules arethe samein both auctions.Indeed,in the Dutch auction, the game stopsassoonas someoneacceptsa price,so that the strategy set canbe thought ofas a priceone would be ready to pay given one'svaluation. And theallocation rule in both auctionsis that the goodgoesto the highest bidderatthe highest offer.

In order to determine the outcome of these auctions, note first that,unlike in the Vickrey and English auctions, there is no dominant strategyfor eitherplayerin the Dutch and first-pricesealed-bidauctions.It is then

lessclear what the outcomeof the game is likely to be for the latter two

auctions.EversinceVickrey's pioneeringarticleappeared,the (symmetric)Bayesianequilibrium has beensingledout as the most plausibleoutcomefor thesegames.

Moreprecisely,in each auction a player'sstrategy is given by b,(v), the

bidding function chosenby player i. Becausethe distribution of valuations

is discretein our simple two-type model,we must allow for mixed

strategies.The reason is the following: Supposethat bidder1choosesthe pure

strategy \302\243>i(v)

= v [or \302\243>i(v)

< v with bi(vH) sufficiently closeto vH]; then

bidder2'sbestresponseis to set b2(vL) =\302\243>i(vL)

= vL and b2(yH) = bi(yL) +e (with e> 0but arbitrarily small).If bidder2 hasa high valuation, she then

8. Note that there also exist asymmetric Bayesian equilibria in the Vickrey and Englishauctions. For example, bidder 1may play a \"preemptive\" bidding strategy and always bid vH. Abest response of the other bidders is then to always bid vL. But if all other bidders bid vL, thenone best response for bidder 1is to bid vH in the Vickrey auction and to \"continue\" until the

price has reached vH in the English auction.As we discuss later in the chapter, these two auctions do not yield equivalent outcomes in

general. For example, when bidders have imperfectly known (common) values, then the twoauctions produce different outcomes.

270 Multilateral Asymmetric Information: Bilateral Trading and Auctions

getsan expectedreturn of (1-(3)(vH-vL-e).But bidderl'sbestresponseto bidder2'spure strategy then is to set bi(vH) = vL + 2e,and soon.Thepoint is that, sincewinning biddersmust pay a priceequal to the winning

bid, they have an incentive to reduce their bid belowtheir valuation evenif they thereby run the risk of losing the good to the other bidder.Thus ahigh-valuation buyer may want to make a bid that isjust marginally higherthan vL. But if her behavioris anticipated,another high-valuation bidderwill outbid her.Thus high-valuation buyersmust randomizeto keepotherbiddersguessing.Sincethey must be indifferent between any bids they

choose,they will, with positive probability, chooseonly bidsin the interval

(vL, b), whereb solvesthe equation

(X-PXvH-vL)=(vH-b)As for low-valuation buyers, their equilibrium strategy issimply the pure

strategy b(vL) = vL. Toseethis point, observe,first, that a bidderwill neverwant to chooseb(yL) > vL.Second,the bestresponseto any pure strategy

b(vL) < vL is b(yL) +e.Finally, the bestresponseto any mixed strategy with

support [bL,vL], where bL < vL, is to choosesome (mixed) strategy with

support [bL, vL], where bL > bL.In sum, the game playedby low-valuation

biddersisessentiallyidenticalto a Bertrandprice-competitiongame,which

inducesthem to competeaway all their rents.Consequently, consider the symmetric (mixed strategy) Bayesian

equilibrium, where b(vL) = vL and f(b I vH) is the equilibriumdistribution ofbidsof high-valuation buyers, with b belonging to the open interval

(vL, b).By symmetry, the equilibrium probabilitiesof winning the auctionfor a low- and high-value buyer are,respectively,(1-(3)/2and /J/2 +(1-/J).These are the sameas in the English (and Vickrey) auction.In addition,low-valuation buyers get a net expectedreturn ofzero,just as in the English

(and Vickrey) auction.As for high-valuation buyers, their expectedreturn

from bidding b \342\200\224 vL is approximately (1- (3)(vH- vL). Sinceexpectedpayoffs from any bid b e (vL, b) must be the samein a mixed-strategyequilibrium, we canconcludethat a high-valuation buyer'sequilibrium expectedpayoff in the Dutch (and first-price sealed-bid)auction is (1-(3)(vH-vL),the sameas in the English (and Vickrey) auction.And given that, for eachtype, the bidders'equilibrium expectedpayoffs and probabilitiesofwinningthe auction are the same in all four auctions, we concludethat the seller'sexpectedrevenuemust alsobe the samein all four auctions!This expectedrevenueis

271 Static Multilateral Contracting

pvH+{l-p)vLNote that this expectedrevenueislowerthan the revenuegeneratedby the

optimal efficient auction, which is

j3vH+(l-j3)vL

The reasonfor this differencein revenueis that the high-valuation buyerobtains more rents in the standard auctionsthan in the optimal efficientauction; that is,her incentive constraint is not binding: she strictly prefersher own bidding strategy to the bidding strategy of the low-valuation buyer.Thisisnot the casewhen the sellerexcludesthe low-valuation type, that is,setsan introductory price of vH. In this case,the standard auctionsgenerate the samerevenueas the optimal auction whereone setsxLL =0.Therefore, exclusionof the low types is a moreattractive optionif one is restrictedto standard auctions insteadof being able to choosethe optimal efficient

auction.9In the two-type case,therefore,we have revenueequivalencefor the four

standard auctions but not revenueequivalencebetweenthesefour standardauctionsand the optimal auction.It turns out that revenueequivalenceisstrengthenedwhen we go from two types to a continuum of types, as thenext sectionshows.

7.3.4 Optimal Independent-ValueAuctions with a Continuum ofTypes:The RevenueEquivalenceTheorem

Let us comeback first to the bilateralcontracting problemwith oneuninformed sellerand one informed buyer consideredin Chapter2.Note that

it canbe extendedin a straightforward manner to a situation with severalbuyers if the sellerhasan unlimited number of items for sale.In the extremecase where there is a very large number of buyers with identically,

independently distributed types, all it takes then to adapt the one-buyerone-sellertheory is to reinterpretthe cumulative distribution function F(6)as a population distribution indicating the fraction of buyers with

preference types no greater than 6. If there is a smaller number of buyerswith independently distributed types, the seller'sproblemcan simply be

9. Remember that xLL =0is part of an optimal auction if (1- fi)vL <p(vH- vL). Exclusion willthus occur if one is restricted to choosing a standard auction even in somecaseswhere

(1- j3K>K?h-vL).

272 Multilateral Asymmetric Information: Bilateral Trading and Auctions

reformulated as the one-buyerproblemby defining F(0)=IT^iF^d,),wheree=(eu...,en).

However,if the sellerhas only a limited number of items for sale,theseller'sproblem is not just an optimal nonlinear pricing problem,but

becomesan optimal multiunit auction problem.In this subsectionweconsider the optimal multiunit auction problemwith risk-neutral, independent-valuation buyers of Maskin and Riley (1989)and show how it can beanalyzed more or lessas an optimal nonlinear pricing problem.This workextends the pioneeringcontributions of Myerson (1981)and Riley andSamuelson(1981)on revenueequivalenceand on optimal auctions.

Supposethat the sellercan sellonly q0 units of a given good.There aren buyers (n > 2) with preferencesdefinedas in Chapter2:ut(q, T)= 9{v(q)-T,where 0,-is distributed on [0,0] with cumulative distribution function

F,(0;).Applying the revelationprinciple,the seller'sproblem is then to

designa multiunit auction{^r\302\243(0\342\200\236 0_,),T^d-,, 0_,)},where(0;,0_;)=0,qt denotes

the quantity of items obtainedby buyer i, and T;is buyer i'spayment to theseller.Of course,the quantity of items soldcannotexceedthe total

quantity available,which we denoteby q0:

n

X^eu^foraiue^eui=l

Buyer z'sexpectedpayoff in a Bayesianequilibrium for this auction is then

n\\ei,e^Ee_\\ei^qi(oi,e_^-Tl(e^where7X0,)=E^T^di,0_;)].Indeed,in a Bayesianequilibrium, eachplayerexpectsthe otherplayersto play the equilibrium, that is,truth-telling,

strategies,so that expectationsof payoffs are computedusing the true type

distributions of the other players.Truth telling in the multiunit auction isincentive compatibleif and only if

rc;(0;,0;)=max7^(0,,0;)8,

Onecancheckthat, just as in Chapter2 and in section7.2.2,incentive

compatibility is equivalent to the monotonicity of E^q^d-,,0_,)} with respectto0;plus localincentive compatibility, which after integration implies

7ti(el,ei)=Kl(e,e)+Eg_{\\\\[ql(y,e_l)]dy\\ (7.8)

From the definition of njid^d)and from equation(7.8)we thus have

273 Static Multilateral Contracting

Tm= Ee_,{eriqtf;,0_;)1-7tX6,6) -J* v[qt(y, 6J]dy\\

so that the seller'sexpectedrevenuefrom buyer i is given by

Tf =\302\243

Eg_, [eiV[qt(0)]-jUi (0,0)-

\302\243

v[qt (y, 0_,-)]dy]f(0;)ddt

Integrating the secondterm by parts, we get

v[qmrT'=LE\302\260->

Ov[qi(e)]-n,<&&~hiOi) J

WddBi

where/z,(0;)=/,(0;)/[(l-F&9ij\\ ls tne hazard rate.The seller'sproblemcannow be expressedas

max\302\24377=E9

\302\243

subjectto

0^(0)]-7^(0,0)-viqMkm j

J,qm<q0

^(0,0)>O for all z

Ee.,iqi(Q)} nondecreasingin 0,-

(feasibility)

(interim IR)

(monotonicity)

Thisreformulation of the problemshows that the expectedrevenueof theselleris fully determinedby the rents granted to the lowesttypes ofbuyers,that is, Kt(6, 0),as well as the allocationrule for the good,that is g,{0),andthat all this is true independently of the specificpayments made by the

parties for each vectorof types 0.Thisoutcomeis the result of incentive

compatibility, which, in the caseof a continuum of types {with /,(#,\342\200\242)> 0 for

all 0/son [0,0]},pins down uniquely the increasein rents ofeachtype onceti;(05 0) and qi(6)have beenchosen.Thisresult is the revenue equivalencetheorem, due to Myerson(1981)and Riley and Samuelson(1981).It doesrely on the independenceofvaluations (which determinesthe way in which

we have taken expectations)and on buyer risk neutrality, two assumptionswe will relaxin the examplesthat will follow.Theprecedingresult, however,doesnot dependon any symmetry assumption in the distribution ofvaluations, but this assumption will beneededin order to comparethe optimal

274 Multilateral Asymmetric Information: Bilateral Trading and Auctions

auction with standard auctions (symmetry will berelaxedin an examplein

a subsequentsection).Tocomputethe optimal auction and compareit with standard auctions,

let us for simplicity restrict attention to the casewhere buyers have unit

demandsand their preferencesare given by

HWt forq>l

Moreover,we assume that F^di) = F(6i) and satisfiesthe regularitycondition

J(9i)=di is mcreasmgm 9tJ\\Pi)

so that we neednot considerthe possibility of bunching, and we can safelyignore the monotonicity condition, which will be automatically satisfiedatthe optimum. We refer the interestedreaderto Maskin and Riley (1989)for an expositionof the casewhere the regularity condition fails.It sufficesto say here that the nature of the solution in this caseis qualitatively similarto the solution in the one-buyerproblemconsideredin Chapter2.10

Given these assumptions we can rewrite the seller'soptimizationproblemas

'. r ., nrin-fo-,1}

m)maxEJy,

subjectto

10.It is worth highlighting that the optimal auction problem we started out with, whereeach bidder has only two possible valuations for the item, is in fact the limit of a problemwhere each bidder has a continuous distribution of valuations on some interval, but wherealmost all the probability mass is centered on only two values. For such extreme distributions

J(6t) is not increasing in6\342\200\236

so that the optimal solution involves bunching. Bunching canbeunderstood asa general form of conditional minimum bid restriction. It is generallypossible for the seller to raise expected revenue by imposing such restrictions. It is for thisreason that the optimal auction may yield strictly higher expected revenue than the fourstandard auctions in the two-types case(when the latter do not impose such rrimimum bidrestrictions).

275 Static Multilateral Contracting

Note that this reformulation takesinto accountthe fact that at the optimumfliC& 0)=0 for all i.

Substituting for J(9),we have

maxE9{X[/(e,-)2i(0)]}

subjectto

O<?i(0)<l and X?i(0)^^

Define \302\247

= inf{0, I /(0;)> 0}.Underour assumption that /(0;)isincreasing,

we have /(ft) > 0 if and only if ft > 6.Thus the optimal selling strategyis to sellup to q0 units to thosebuyers with the highest valuations in excessof 0.

How does this optimal selling procedure compare with standardauctions?In an English auction and a Vickrey auction, if the sellersetsan

introductory priceof ft this optimum will bereached,sinceit will meanthat

(1)buyers with valuations below6 will earn zero rents, and (2) the objectswill be allocatedto the highest-valuation buyers in excessof ft And the

samewill be true of the first-pricesealed-bidauction and the Dutch auctionwith an introductory priceof ft if we canshow that bidsare increasingwith

valuation (so that the objectswill be allocated to the highest-valuation

buyers in excessof 6) in the symmetric Bayesianequilibrium. Let us show

this result, for example,in the casewherebuyer valuations are uniformlydistributed on [0,1]and where the sellerfacestwo buyers and has a singleobject to offer.Let gt(b) be the probability that buyer i expectsto obtainthe objectif shebidsbr If her valuation is ft, she solves

maxiOi-bi)gi(bi)

Indeed,she trades off the probability of winning with the surplus uponwinning. Thefirst-ordercondition of this problemis

(ei-bi)gXbl)-gi(bi)=0

Let us now look for a symmetric equilibrium where bidsare increasingwith the valuation, that is, \302\243>,(ft)

such that dbjdd-, > 0.This means that

the equilibrium probability of winning the object, #(\302\243>,),for somebody

Multilateral Asymmetric Information: Bilateral Trading and Auctions

with valuation 9t is in fact g[bl{d)\\ = 6t (given that the 0;'sare uniformlydistributed on [0, 1]).Replacing0, by g(b^) in the precedingequationyields

[g(bl)-bl]g'(bl)-g(bi)=0

The solution of this differential equationisg(\302\243>;)

=2b\342\200\236

and since9t =g(b),this implies bt = 6J2:each buyer bidshalf her valuation, so that higher-valuation buyers get the object, thereby generating the same expectedrevenuefor the seller(given the sameintroductory price,here normalizedto zero) as the English or Vickrey auctions, and the same expectedrevenue as the optimal auction if the appropriate introductory price ischosen.And this result holds despitethe fact that, for specificvectorsof valuations, these auctions do not imply the samepayments. (Inparticular,

the English auction extractsmore revenuewhen the two highestrealized valuations happen to be \"closeby,\" and the Dutch auction extractsmore revenuewhen the two highest realizedvaluations happen to be \"far

apart.\All four standard auctions are thereforeoptimal if they set the

appropriate introductory price,in this symmetric setup with risk-neutral buyersand independently drawn valuations. Let us now look at exampleswherethis revenueequivalencemay fail.

Optimal Auctions with CorrelatedValues

Let us now drop the assumption of independenceof buyer valuations. Aswe shall see,relaxing this assumption has a crucialimpact on the optimalauction. This can beseenin a two-type case,but is much more general,asshown by Cremerand McLean(1988).Denote by fa the probability that

buyer 1has valuation v; and buyer 2 valuation v; (wherei =L,Hand;'=L,H),and supposethat the buyers'valuations are correlated,so that PhhPll-PhlPlh* 0 [instead,under independence,thereexistsj3such that j32= f3HH,

(1-P)2= pLL, and j3(l-P) = pLH= pHL\\.

What is the revenueof the four standard auctionsin this case?Underthe

English (and Vickrey) auction the seller'sexpectedrevenueis /WvH +(1-Phh)vl- It is straightforward to checkthat under the Dutch (and first-pricesealed-bid)auction the sellerobtains the sameexpectedrevenue.Indeed,in the latter two auctions a buyer with valuation vL obtains an expectedreturn ofzeroand a buyer with valuation vH obtains(1-&hh)(vh

-vL) justas in the English (or Vickrey) auction.Hence,the sellermust alsoreceivethe sameexpectedrevenue.11\"

277 Static Multilateral Contracting

While correlationdoesnot imply dramatic changesfor the four standardauctions, the sameis not true for the optimal one:the sellercannow obtainan expectedrevenueof f5LLvL + (1-Pll)vh, that is, extractall the surplusfrom the buyers!To seethis result, consider the general contract (orauction) specifying the following:

\342\200\242 A payment Ptj from buyer 1when her value is v; and buyer 2'svalue isv;-

(Pji then denotesthe payment from buyer 2).\342\200\242 An efficient allocationof the goodx^, that is,xHh =%ll =

\\, *hl = 1,and*lh =0.

Sucha contractextractsall the buyers' surplus if it is incentivecompatible and if the buyers'individual-rationality constraints are binding for all

types, or if

PLL^~PLLyPLHPLH^PLL(vL-PHL )+PlH(y \" PHH ) (KX)

Phh(~--

Phh]+Phl(vh -Phl) >-PhhPlh +Phl[~-Pll] (ICH)

and

Pll[^-P^)-PlhPlh=0 (IRL)

Phh[y-p\342\204\242)+P^(vh-phl)=o (irh)

Sincecorrelationmeansthat pLLpHH-PlhPhl *0,we can in fact find P;/s

such that these four conditionsare satisfied.Toseethis point, first rewritethe two participation constraints as follows:

PLH=~fa&-PLL) (IRL)

Phl=vh+^-(^-Phh) (IRH)Phl V / J

11.This revenue equivalence between the four auctions with correlated values is valid onlyin our special example with two possible valuations for each buyer. The appendix of this

chapter (section 7.7)shows that it breaks down with three different valuations.

Multilateral Asymmetric Information: Bilateral Trading and Auctions

Next, define\302\243

= (j3hhj3ll)/(j3hlj3lh)5 which differs from 1becauseofcorrelation.Using the two preceding equations and the fact that the

equilibrium surplus of eachbuyer is zero,we can rewrite the two incentiveconstraints as follows:

0>\302\243f-(vL-PvH)-pLL(vH-vL)+pLH\302\253;-l)PHH (ICL)

0>^(vH-PvJ+PhlG-DPu. (ICH)

As we can see,it is possibleto satisfy these incentive constraints andtherefore to extract the entire first-bestsurplus for the sellerwheneverthere isdependencebetweenvaluations, that is,whenever^1.In the caseof positive correlation(% > 1),we just needto set PHHand PLLlow enough.Intuitively, we deter deviations from truth telling by the high-valuation

buyer by requiring from her a low enough PLU and therefore,by the

participation constraint for the low-type (IRL),a high enough PLH. This helpsto satisfy the incentive constraint for the high-type (ICH)because,when agiven buyer i has a high valuation, not telling the truth becomescostlier:the probability that the other buyer, /, has a high valuation too, and thus

that the payment to be made is PLH instead of PLL,is higher than when

buyer i has a low valuation. The sameis true, mutatis mutandis, for a low-valuation buyer. And the sameprinciple appliesin the caseofnegativecorrelation (% < 1),but then it means setting PHHand PLLhigh enough:onceagain, the idea is to set the higher payments in the statesof nature that arelessprobableunder truth telling than under deviations from truth telling.

We have thus presenteda mechanism that allows the sellerto extractthefull efficient surplus. Note,however,that it works under the doubleassumption that (1)the buyers are risk neutral, and (2) they have no resourceconstraints. Indeed,note that the payments will have to vary a lot acrossstatesof nature when correlationbecomessmall, that is,when

\302\243

\342\200\224> 1.The Role of Risk Aversion

Let us now seewhat happenswhen we relax the assumption of risk

neutrality. We shall limit ourselveshere to consideringa two-type exampleand

comparing the performanceof standard auctions.This approach,however,will give us the intuition of the role of risk aversionmore generally.

279 Static Multilateral Contracting

Supposefor now that both buyers' preferencesare representedby the

strictly increasingand concavevon Neumann-Morgensternutility function

u(-).Howdoesrisk aversionaffectthe buyers'bidding strategy? Note,first

ofall, that the buyers'bidding behaviorin the English (and Vickrey) auctionis unaffectedby their attitudes toward risk. Whether a buyer is risk averseor not, it is always a bestresponseto \"continue\" until the asking pricehits

the reservationvalue (or to bid the reservation value in the Vickreyauction).Thus the seller'sexpectedrevenuein theseauctions is not affectedby buyers'risk aversion.

In the Dutch (and first-pricesealed-bid)auctions,however,risk aversionaffectsequilibrium bids.Toseehow, comparethe high-value buyer'sequilibrium (mixed) strategy under risk neutrality and risk aversion.12When the

buyer is risk neutral, herexpectedpayoff from abid b e (vL,b) isgiven by

[l-P+PF(b)](vH-b)whereF(b)denotesthe cumulative probability distribution function ofbidsof the high-value buyer

F{b)=\\b f{y\\vH)dy

When the buyer offersb \342\200\224 vL, her expectedpayoff is (1-P)(yH-vL),andsincein a mixed-strategy equilibrium all bidsthat are made with positiveprobability must yield the same expectedpayoff, the following equationholdsin equilibrium:

[l-P+j3F(b)](vH-b)=(l-l3)(yH-vL) (7.9)When buyers are risk averse,this equationbecomes

[l-P+PF(b)]u(vH -b)=(l-P)u(yH-vL) (7.10)whereF(b) now denotes the cumulative probability distribution function

of bidsof the high-value buyer when she is risk averse.Rearrangingequations (7.9)and (7.10)we obtain

j3F(b) ^vh-Vl(l-j3) vH-b

12.Recall that for low-value buyers (whether they are risk averse or risk neutral), theequilibrium strategy is simply the pure strategy b(vL) = vL.

280 Multilateral Asymmetric Information: Bilateral Trading and Auctions

and

PF(b) ^u{vH-vL)+(l-j3) u(yB-b)

Sinceu(-) is a strictly concavefunction, we know that

u(vh-Vl) Vh-Vl

u(vH -b)i vH-b

so that F(b)< F(b),with F(b)< F(b)for b belonging to the open interval

(vL, b), or 1-F(b)> 1-F(b).In other words, a high-valuation risk-aversebuyer is more likely to make high bidsthan a high-valuation risk-neutral

buyer. Becauseshe is risk averse,the high-valuation buyer wants to insureherselfagainst the bad outcomeof losing the item to the other buyer by

raising her bid.Therefore,the Dutch (and first-pricesealed-bid)auctionraisesa higher expectedrevenue than the English (and Vickrey) auctionwhen buyers are risk averse.

It canbeshown that a sellermay beableto raisean evenhigher expectedrevenuefrom risk-aversebuyers by charging a fee for participating in theauction (seeMatthews, 1983;Maskin and Riley, 1984b).Sucha fee amountsto a positive penalty on losers.This penalty has the effectof inducing risk-aversebuyers to bid evenhigher soas to insure themselvesagainst the badoutcomeof losing.13

7.3.7 The Role of Asymmetrically DistributedValuations

What happenswhen we drop the assumption of symmetry? Onceagain, letus start with the two-type case.Supposethat the two buyers have different

probabilitiesof having a high valuation for the good.Without lossofgenerality, assume j32 > Pi- Then the expectedrevenue of the sellerin the

English (and Vickrey) auction is given by ft/^vn +(1-P\\Pt)vl- Under theDutch (and first-pricesealed-bid)auction, however,expectedrevenueswill

be lower.Toseethis result, considerthe equilibrium bidding strategiesofthe two playerswhen they have low and high valuations for the good.Wecan say the following:

13.Note that (in contrast to the situation with a single buyer) this result holds irrespective ofwhether the buyer has preferences that exhibit decreasing or increasing absolute riskaversion. The point here is that while the penalty for losing lowers the equilibrium bid of a low-value buyer, this is only a second-order effect, while the resulting increase in the high-valuebuyer's bid is of first order.

281 Static Multilateral Contracting

\342\200\242 First, either buyer, when she has a low valuation, bidsvL, for the samereasonsas in section7.3.3.Therefore,by bidding just above vL, buyer 1obtains at least (1-{$i)(vh-vL) > 0 when she has a high valuation, while

buyer 2 obtains at least(1-j3i)(vH -vL) > 0 when shehasa high valuation.\342\200\242 Second,eachhigh-valuation buyer'sequilibrium bidding strategy is again,as in section 7.3.3,a mixed strategy. And both buyers must have thesamemaximum bid, bH (<vH), in equilibrium: otherwise,sinceequilibrium

strategiesare correctlypredicted,the buyer with the higher maximum bidcan, without lowering her probability of winning, lower her maximumbid toward the maximum bid of her opponent.Moreover,this maximumbid cannot be chosenby eitherbidderwith strictly positive probability: if

one'sopponentbidsbH with strictly positiveprobability, bidding bH isdominated by bidding just abovebH: then one obtains the goodfor sureinsteadofwith probability \\. Consequently,whenevera bidderbidsbH, sheobtainsas payoff vH -bH.\342\200\242 Third, obviously, eachbid that is made with positive probability by oneof the buyers in equilibrium must yield the sameexpectedpayoff to that

buyer. Equating the payoffs betweenbidding bH and bidding (just above)vL, we obtain

vH-bH=(l-Pi)(yH-vL) = (1-P2+p2a)(yH -vL)

wherea is the positiveprobability that buyer 2 bidsvL. Indeed,we needtohave buyer 2 bidding lessaggressively against the weakerbuyer 1in orderto have both payoffs equalto vH -bH.Thismeansthat buyer 2 will bid with

strictly positiveprobability like low-valuation buyers.Note that, under

symmetry, a =0.We are now ready to derivethe seller'sexpectedrevenue.It is equal to

the total surplus from trade minus the rents accruing to the buyers. Sincethe auction is efficient,14 the expectedtotal surplus equals

[1-(1-A)d-Pi)]vH +(1-A)d \" &)vl

Moreover, low-valuation buyers earn zero rents, while high-valuation

buyers earn (1-Pi)(vH -vL).The sum of expectedrents of the two buyersis therefore

14.\"Efficiency\" requires that if bids are equal, the higher-valuation buyer obtains the good:this in fact strengthens the result that the expected revenue here is lower than that of an

English auction.

282 Multilateral Asymmetric Information: Bilateral Trading and Auctions

(j31+j32)(l-j31)(vH-vL)

and the seller'sexpectedrevenueis

which is strictly lessthan the expectedrevenue under the English (andVickrey) auction.

Intuitively, the reason why revenue equivalencebreaksdown when

buyers'expectedvaluations differ is that in the first-pricesealed-bidauctionthe extent of competitionbetweenbiddersis detennined by the weakerbidder.This result, however,is not very robust.For example,the Vickreyauction may yield lowerexpectedrevenue than the first-pricesealed-bidauction when the two bidders'valuation supportsdiffer.Toseethis

possibility, supposethat the only differencebetween the two biddersis that

vi? > vh; otherwise,v\\ = v\\= vL and j3x= /32= j3.Theseller'sexpectedrevenue

in the Vickrey auction isthen j32v# +(1-j32)vL. But in the first-priceauction

expectedrevenue is strictly higher becausethe secondbuyer bidsmoreaggressivelywhen v# > v# than when v# =

vjj. In fact, under asymmetry, thereis no generalresult on the relative merits of first-priceversus Vickreyauctions (seeMaskin and Riley, 2000,for a more completeanalysis ofasymmetric auctions).

7.4 Auctions with Imperfectly Known CommonValues

In many auctions,buyers may not have perfectinformation about the valueof the item for sale.If all buyers are risk neutral and their valuation of theitem is independentof otherbuyers'valuations, then the theory developedin the previoussubsectionis still relevant.The only changethat is requiredis to replacethe bidders'true value by their expectedvalue.

However,if a buyer's final value dependson information other biddersmay have, then the auction-designproblemis fundamentally different. Inthis sectionwe considera simple exampleof an auction problemwherebuyers have imperfectly known common values.Many if not most auctionsin practiceare of this form. For example,art, treasury bill, or bookauctionsare effectively imperfectly known common-valueauctions:most biddersdonot know exactly how much the items to be auctionedare worth to others,and part or all of their own valuation for the item dependson what theybelieveto be the resalevalue of the item.

Static Multilateral Contracting

In our examplethe item for salehas a value v e {H,L},whereH>L>0.Nobodyknows the true value of the item.Thesellerand two buyers havea prior beliefi that the item has a high value, H.The buyers receivean

independentprivate estimateof the value or signal, st e {sH,sL},and thus

have private information about the value of the item. When the value ofthe item is H,the probability of receivingsignal sH isp > \\. Similarly, when

the value is L,the probability of receivingsignal sLisp > \\. Thus a buyer'sexpectedvalue upon receivingthe signal sH is given by vH = E[v\\sH] =pH+il-p)L.Similarly, vL = E[v I sL]=pL+(l-p)H.

TheWinner's Curse

Themain changeintroducedby buyers'imperfectknowledgeof their valueis that the bidding processand the outcomeof the auction may revealinformation that inducesbiddersto revisetheir valuation of the item. Onepieceof information that is revealedto the winner of the auction\342\200\224the highestbidder\342\200\224is that she receivedan estimateof the value that was morefavorable than the estimatesof all other bidders.That information leadsher tolowerher valuation of the item.Thisphenomenonis known as the winner'scurse:Winning is in a sensebad news,sinceit revealsthat the winner

overestimated the value of the item.In our examplethe winner's cursetakesthe following form: Supposethat

the sellerputs the item up for sale in a Vickrey auction, and supposethat

buyers behaveasbeforeand bid their expectedreservationvalue vH or vL.

Then the highest bidderwins and gets the item at the second-highestbid.When onebuyer getsthe signal sH and the otherthe signal sL,the winner getsthe item at the pricevL, but her value conditionalon

winning\342\200\224vw\342\200\224isnow

L+ Mvw =E[v\\(sH,sL)]=

\342\200\224-\342\200\224<pH+(l-p)L=vH

Even if winning brings about a downward revision of the value, thewinner of the auction still makesa positivenet gain, since

L+ Mvw =E[v\\(sH,sL)]=

\342\200\224-\342\200\224>pL+(l-p)H=vL

When both buyers get the signal sH, each one has a 50% chanceofwinning and paying the price vH. In this casewinning actually results in an

upward revision of the value, and the winner's net payoff is strictly positive.In other words, in this event there is a \"winner's blessing\":

284 Multilateral Asymmetric Information: Bilateral Trading and Auctions

vw=E[v\\(sH,SH)] =p2HH1P\\L>pH+a-p)L=vHp2+a-p)

Finally, when both buyers get a signal sL,the winner's net gain isnegative, since

vw=E[v\\(SL,sL)]=p2L+{1~P\\H<pL+0--p)H=vL

P2+(1-P)and in this casewe have the winner's curse.

Severallessonscan be drawn from this analysis. First, winning is not

entirely bad news in the Vickrey auction, even if the winner may be led toreviseher valuation downward. In fact,winning may evenbe a blessingif

the second-highestbid (which is the price the winner must pay) revealsahigh-value estimatefrom the other bidder.Second,winning is always badfor those bidding vL. Consequently,buyers with a value vL will bid belowtheir ex ante value, sothat bidding one'sown (expected)value is no longera dominant strategy. Third, the buyers with an ex ante value vH are betteroff bidding above their value vH. Indeed,this bid would not affect their

final payoff in the event that the other bidderhas a low-value estimate,but it would raise their probability of winning against a high-value buyerbidding vH.

It should be clear at this point what the Bayesianequilibrium in the

Vickrey auction is in our example:

1.Buyersreceivingthe signal sLoptimally bid

bL=E[v\\{sL,sL)]=p2L+(l-p)Hp2+0.-p)

With any higher bid they would make a negative expectednet gain, as weillustrated previously. Thebestresponseto any lowerbid, b < bL, is to raisethe bid by e> 0 soas to outcompetethe otherbidderin the classicBertrandfashion.

2.Buyersreceivingthe signal sH bid

u vx u m P2H+q-p)2Lbn =E[v | (sH,sH)]= \342\200\224\342\200\224

p2+(l-p)

Static Multilateral Contracting

Any higher bid would producea negative expectednet gain. And the bestresponseto any lowerbid, b < bH, is again to raise the bid by e > 0 in theclassicBertrandfashion.

Hence,the expectedrevenuefrom the Vickrey auction in our exampleis

L pl+(l-p)Standard Auctions with Imperfectly Known Common Values in the 2x2

We saw that when buyers have perfectlyknown valuesfor the item to beauctioned off, then the English and Vickrey auctions yield the sameexpectedrevenueand are in essenceequivalent. The same is true of theDutch and the first-pricesealed-bidauction.Moreover,when the buyers'valuesare independent,these four auctionsall yield the same expectedrevenue.

With imperfectly known buyer values, only the Dutch and first-pricesealed-bidauction are equivalent in general.The equivalencebetweenthe English and Vickrey auctions breaksdown with more than two buyersand more than two types, essentiallybecausethe English auction

aggregates the dispersedestimatesof the buyers more efficiently and, thus,reducesthe winner's curseproblem.Thus in the generalmodelwith n > 2biddersand N > 2 types the Vickrey auction generatesexpectedrevenueno greater than the English auction.Also, the Vickrey auction generatesahigher expectedrevenuethan the Dutchauction when thereare iV> 3 types.

Unfortunately the 2x2model (two bidderswith two types) consideredhere providesa somewhat misleading picture, since in this specialcaserevenueequivalenceobtains even though biddershave imperfectly known

common values.However,the virtue of this specialcaseis to highlightthe subtlety of the social-learningeffectspresent in imperfectly known

common-valueauctions.The readersinterested in seeinga generalcomparison of standard auctions when biddershave imperfectly known

common values are referred to the classicpaper by Milgrom and Weber

(1982).In this chapterwe consideronly the 2x2case.We beginby showing that revenueequivalenceobtains in the 2x2case.

We have alreadydeterminedthe seller'sexpectedrevenuein the Vickreyauction.Thus considerin turn the English and Dutch auctions (recallthat

286 Multilateral Asymmetric Information: Bilateral Trading and Auctions

the first-pricesealed-bidauction is strategically equivalent to the Dutchauction).\342\200\242

English Auction: It is a dominant strategy in the English auction for abuyer with ex ante value vL to stay in the auction until the price hits thelevel

ttf if m p2L+a-p)2HE[V | {SL,SL)\\ =

r\342\200\224=VLL

p2+a-p)and to drop out when the priceexceeds

E[v |On,sL)]= = vLH

and for a buyer with ex ante value vH to stay in the auction until the pricehits the levelvLH and to drop out when the pricereaches

E[v I Oh,sh)]= r\342\200\224 = vHHP2+(1-P)

Thus, given the dominant strategiesof vH buyers, it is a (weakly) dominant

strategy for a vL buyer to drop out when the price reachesvLL. Assumingthat this is the equilibrium bidding behaviorof vL buyers, one immediatelyinfers from the fact that a buyer decidesto stay in the auction when the

price exceedsvLL that this buyer has obtaineda signal sH}sHence,when

both buyers receivea signal sH, they are able to infer each other'ssignalfrom their respectivebidding behavior,and they stay in the auction until

the price hits vm. Therefore,the seller'sexpectedrevenuein the Englishauction with only two biddersis the sameas in the Vickrey auction.\342\200\242 DutchAuction: As in the casewherebuyers have perfectlyknown values,the equilibrium in the Dutchauction involves mixed strategies.It is easytoverify that in equilibrium vL buyers bid vLL and vH buyers randomizeoverbidsb e [vLL, b],where b < vHH. To verify that the equilibrium takes this

form, note first that any mixed strategy FH(b) must satisfy the equation

1 11-j{vhl-vLl)=-(vhl-b)+-FH{b){vHH -b) ,?^15.It is a straightforward exercise to show that a vL buyer will never bid above vLL in anyequilibrium.

287 Static Multilateral Contracting

Hence,b, which isgiven by the solution to

(vffL-

vLL) = (vHL -b)+(vHH -b)

is strictly lessthan vHH. Next, note that given the vH buyer's equilibrium

strategy, it is a bestresponsefor a vL buyer to bid vLL. Indeed,any b e [vLL,

b] would yield an expectedpayoff equal to

1 1-(vLL -b)+-FH(b)(vHL -b

From equation (7.11)we have

VHH-b

Substituting for FH(b), we can seethat any bid b e \\yLL, b] would yield anegative expectedrevenuefor a vL buyer:111-(vll-b)(vHH ~b)\342\200\224z(vLL -b)(vHL-b)=-(vHH -vHL)(vLL-b)<0

Finally, just as in the casewith independent(perfectlyknown) values, thereis no purebestresponsefor vH buyers and the unique mixed strategyequilibrium has support [vLL, b].Now, to seethat the seller'sexpectedrevenueis the sameas in the otherstandard auctionswehave discussed,note simplythat the payoffs of the two types of biddersare the sameas in the othertwo auctions [vL buyers get zero and vH buyers get (vHL

- vLL)/2]and the

equilibrium allocationsare the same (vL buyers get the item with

probability ^,and vH buyers get the item with probability |).Therefore,the seller'sexpectedrevenuemust be the same.

We have thus shown that revenueequivalenceis maintained in the 2x2model even though buyers have imperfectly known common values.It isworth pointing out that this equivalenceextendsto the nx2model.16Also,the equivalencebetweenthe Vickrey and English auctions extendsto the2xN model.Indeed,the two bidders'symmetric equilibrium strategiesin

the English auction then simply take the form of a reservationprice(conditional on the signal and history ofplay).Thebidderwith the highestreservation pricethen getsthe item at the second-highestreservationprice.With

16.We leave it as an exercise to show revenue equivalence in the n x 2model. This result canbeestablished by adapting the arguments used in the 2x2model.

288 Multilateral Asymmetric Information: Bilateral Trading and Auctions

two biddersthere is no additional information generated in the Englishauction overthe observationof the second-highestbid, so that the Englishauction is then informationally equivalent to the Vickrey auction.With

more than two bidders,however,some information is lost in the Vickreyauction by observing only the second-highestbid, so that the Englishauction should be expectedto generatemore revenue.This conclusionisindeedwhat Milgrom and Weber (1982)establish.

7.4.3 Optimal Auctions with Imperfectly Known CommonValues

Becausethe buyers'estimatedvaluesbasedon their private signalsarecorrelated when there is a common (unknown) componentto all buyers'valuations, it ispossiblefor the sellerto extractall the buyers'surplus in an optimalauction just asin section7.3.5.On the onehand, sincethis is a common-valueenvironment, maximizing the surplus simply meanstrading with probability

one,while it doesnot matter who is allocatedthe good.On the otherhand,

buyers with a higher signal will bereadyto pay morefor receivingthe good,sothat it will beoptimal to give it to them with ahigher probability.

Specifically,let [P(si,s2);x(si,s2)}denote an arbitrary (symmetric)contract offered by the sellerto the two buyers where P(si,s2) denotes apayment from the buyer and x($i, s2)an allocationof the item to the buyerwho has reported Si conditional on the reported vectorof signals (sl5 s2).Let us focuson x(Si,s2)such that x(sH, sH) =x(sL,sL)= \\, while x(sH,sL)=1and x(sL,sH) = 0.Call Pi; the payment to be made by a buyer who hasannouncedsignal i while the other buyer has announcedsignal;'.Then theselleris able to extractall the buyers' surplus if both buyers' individual-

rationality constraints are binding while both incentive-compatibilityconstraints are alsosatisfied:

-(l-p')PLH+pii^ll-PllU(1-W-^lh-Phh\\+p'(vll-Phl) qCL)

p\\^vHH -PotJ+d-.p'XvHL-PHl)^-p'Plh+(X-P,)(-^Vhl-PllJ(ICH)

-0--p')PLH+P\\2Vll~Pll]=0(IRL)

p\\-^vHH~PhhJ

+(1-pKvhl-Phl)=0 (IRH)

289 Static Multilateral Contracting

wherep'is the probability that a buyer who hasreceiveda low signalattributes to the other buyer having alsoreceiveda low signal (seebelow).

Theintuition behind theseconstraints is asfollows.Take,for example,thefirst one, (ICL).On the LHSis the payoff of the buyer when she hasreceiveda low signal and tells the truth about it, while on the RHSis herpayoff when she falsely claims to have receiveda high signal. In the first

case,she receivesthe good with probability \\ when the other bidderhasalsoreceiveda low signal, in which caseshepays PLLwhile the goodhasvalue vLL. What is the probability of this outcome?It is

Pt(Sl\\sl) =Ft(sl\\L)Ft(L\\sl) +Ft(Sl\\H)Ft(H\\sl)=p1+(X-pjl=p'

which is lowerthanp but higher than \\. Instead,when the otherbidderhasreceived a high signal [which happens with probability (1- p')\\, the

payment is PLH and the other bidder gets the good for sure.The RHSof(ICL)takesthe sameprobabilitiesp'and (1-p')and valuesfor the

object\342\200\224

that is, vLH and vLL\342\200\224but the probabilitiesof trade and the paymentscorrespondto the high announcement becausethe buyer liesabout her signal.

Just as in section7.3.5,we can first rewrite the two participationconstraints as follows:

^-\"^a^oPf-fl\") (IRH)

Next, defineE,= [p7(l-p')]2,which is greater than one.Using the two

precedingequationsand the fact that the equilibrium surplus ofeachbuyeris zero,we can rewrite the two incentive constraints as follows:

0 >^p-(vLH-&m)-p\\vHL

~Vll) +(1-P0(\302\243-DPhh (ICL)

0>^p-(vHL-&LL)+(X-p'X\302\243-l)PLL (ICH)

Just as in section7.3.5,it is possibleto satisfy theseincentive constraintsand thereforeto extractthe entire surplus for the seller.Sincethis is a caseof positive correlation,we needto set PHH and PLLlow enough. Intuitively,we deterdeviations from truth telling by the buyer who hasreceiveda good

290 Multilateral Asymmetric Information: Bilateral Trading and Auctions

signal by requiring a low enough PLLand, therefore,by the participationconstraint (IRL) for the low type, a high enough PLH. Doing sohelps to

satisfy the incentive constraint (ICH)for the high type because,when agiven buyer i hasreceiveda high signal, not telling the truth becomescostlier: the probability that the other buyer,;',has a high signal too,and thus

that the payment to be made is PLH instead of PLL,is higher than when

buyer i has receiveda low signal.The sameis true, mutatis mutandis, for alow-valuation buyer who has receiveda low signal.

We have thus presenteda mechanism that allows the sellerto extractthefull surplus. Again, it works under the doubleassumption that (1)the buyersare risk neutral, and (2) they have no resourceconstraints. Beyondtheseassumptions, the full rent extractionresult isvery general,ashasbeenshown

by Cremerand McLean(1988).All that is requiredis someform ofcorrelation betweenbuyer valuations, which obtains naturally in settings wherethere is a common componentto buyers'values.Thisresult isyet anotherillustration of the fragility of the optimality of the four standard auctionsconsideredin this chapter.On the whole, the conclusionthat emerges\342\200\224that

the widely observedstandard auctions are optimal only for a very narrow

set of parameters\342\200\224is disappointing for the theory of optimal contracts,forit suggeststhat some important considerationshave been ignored that

would explain why the English and sealed-bidauctionsare usedsooften.

7.5 Summary

In this chapterwe have consideredtwo canonicalstatic contractingproblems with multilateral private information. The first problemis one wherethe contracting parties are complementary to each other and where,as aconsequence,trade is fundamentally of a public-goodnature. The secondproblemis onewheresomeof the contracting parties (the buyers) are sub-stitutable (at least partially) and compete with one another. A classicexampleof a problemof the secondtype, which we have focusedon, is anauction organizedby a seller.We have highlighted the fact that, in contrastto bilateralcontracting problems(with one-sidedasymmetric information),the outcomeof optimal contracting is more difficult to predict,as the

contracting partiesare involved in a game situation that may have severalequilibria. There is by now a large literature on mechanism design that isconcernedwith the questionof how to structure contractsso that the gamethey induce results in a unique, hopefully dominant-strategy, equilibrium

291 Static Multilateral Contracting

outcome(seeFudenbergand Tirole,1991,chap.7; Palfrey, 1992,for two

extensivesurveys of this literature).Thisliterature isbeyondthe scopeofthis book,as it can be significantly more abstractand technicalthan thetheorieswe have coveredin this chapter.Rather than delveinto this

difficult topic,we have taken the shortcut followedby many and madethereasonable assumption that optimal contractual outcomesare given by the

Bayesianequilibria of the gameinducedby the optimal contract.Underthat assumption, we have seenthat key insights and methods

developedin Chapter2extendto moregeneralmultilateral contractsettings.For

example,the revelationprinciplecanbeinvoked to characterizesecond-bestcontractswithout too much lossof generality. Similarly, under mildadditional assumptions about preferences(the \"single-crossing\" condition) thesetof binding incentive constraints takesan appealingly simple form.

Somefundamental ideasand resultshave emergedfrom our analysis in

this chapter:\342\200\242 We have learnedthat asymmetric information persemay not bea sourceof allocativeinefficiency when hidden information is about private values.Rather, it is the combination of interim participation and incentiveconstraints that gives rise to allocativeinefficiencies.There is an importantlessonfor policy in this basic insight\342\200\224namely, that it may be desirabletomake public decisions,and lock individuals in, before they learn their

private information and can exploittheir vestedinformational rents.\342\200\242 Likewise,we have confirmed the basicgeneralintuition that competitionreducesinformational monopoly power,at least when hidden information

is about private values.Thus we have seenthat in an auction the seller'sexpectedrevenuerisesas more buyers competefor the item.\342\200\242 A remarkableresult of auction theory, which we have focusedon in this

chapter,is that severalstandard auction procedures,such as the English,

sealed-bid,and Dutch auctions, yield the same expectedrevenuefor thesellerwhen biddersare risk neutral, have independently and identicallydistributed private values, and play the unique symmetric equilibrium in

eachbidding game.Furthermore, theseauction procedurescombinedwith

minimum bid constraints may maximize the seller'sexpectedrevenue.As

significant as this result is,however, we have alsoshown that revenueequivalence breaksdown when any of theseassumptions donot hold.Thus,when

biddersare risk averse(or are budget constrained),the standard auctionsdo not generate the same expectedrevenue.What's more,none of the

292 Multilateral Asymmetric Information: Bilateral Trading and Auctions

standard auctions are then optimal. Similarly, when bidders'hiddeninformation is about common values, the standard auctions producedifferent

expectedrevenues for the seller.A basic reason for the breakdown ofrevenue equivalencein this caseis the winner's curse,which may induceexcessivelycautiousbidding in the sealed-bidor Dutch auction relative tothe English auction.

These,are basic insights about auctions, which highlight the potentialimportanceof the designof auction procedures.There is by now a largebody of literature on auction design,which has beenpartly stimulated by

government decisionsin a number of countriesto organizespectrumauctions, by the wave of privatization following the collapseof communism in

the former SovietUnion, and, more recently, by the advent of the Internetand online auctions like eBay.It is clearlybeyondthe scopeof this book tocoverthis literature in any systematic way. We refer the interestedreader,for example,to Klemperer(2003),Krishna (2002),and Milgrom (2004).

7.6 LiteratureNotes

The theoriescoveredin this chapter have a long ancestry.The foundingarticle on auction theory and more generally the theory of contractingunder multilateral asymmetric information is Vickrey (1961).Many of theideascoveredin this chapter are lucidly discussedin his article,which is

highly recommendedreading.One of the key insights in Vickrey's article\342\200\224

that it is a (weakly) dominant strategy for biddersto revealtheir true valuesin a second-priceauction\342\200\224has found a useful later applicationto the

problemof efficient provision of public goods,when the value of the goodto individual agents is private information. Thus Clarke (1971),Groves(1973),and Grovesand Ledyard(1977)have shown that in public-goodsproblemsit may alsobea dominant strategy for individuals to revealtheir

true valuesif their contribution to the costof the public goodis a form ofsecond-pricebid.Thisline of researchhas culminated in the major treatiseon incentives in public-goodssupply by Greenand Laffont (1979).Oneweaknessof what is now referred to as the Groves-Clarkemechanism isthat generally the sum of the contributions requiredby all individuals tofinance the public goodand truthfully elicittheir preferences(in dominant

strategies)exceedsthe cost of the public good.This finding promptedd'Aspremontand Gerard-Varet(1979)to show that it is possibleto imple-

293 Static Multilateral Contracting

ment the efficient supply ofpublic goods,obtain budget balance,and ensuretruthful revelationof preferences,provided that one concentrateson thelessdemanding notion of Bayesianequilibrium rather than insisting ondominant-strategy equilibrium. d'Aspremont and Gerard-Varetrequireonly exante participation constraints to hold.Moreover,they establishtheir

result under the assumption that individual valuations are independentlydistributed, but later research by themselves,Cremer and McLean,andothers has revealed that this result holds under much more generalconditions.The final stage in this line of research has been reachedwith the contributions by Laffont and Maskin (1979)and MyersonandSatterthwaite (1983),who have shown that public-goodprovision undermultilateral private information is generally inefficient when interim

(or expost) participation constraints must hold.The public-good-provisionproblem is technically very similar to the

bilateral-trading casewe focusedon in section7.2,although the economicinterpretation is different.Thissimilarity has allowedus to illustrate in

particular the contributions of d'Aspremont and Gerard-Varet(1979)and

Myersonand Satterthwaite (1983),as well as the optimal double-auctionmechanism of Chatterjeeand Samuelson(1983),in this context.

Subsequentto this researchon public goods,the 1980sand 1990shavewitnesseda major revival in researchon auction theory. Myerson(1981)and Riley and Samuelson(1981)provide the first general treatments ofoptimal auction designand rigorously establishthe revenue-equivalencetheorem for the first time. In later work, Matthews (1983)and Maskinand Riley (1984b)characterizeoptimal auctionswith risk-aversebidders.Wilson (1977),Milgrom (1979),and Milgrom and Weber (1982)breaknew

ground by analyzing equilibrium bidding behaviorand information

aggregation when biddershave private information on common values and their

bidding is affectedby the winner's curse.In related research,CremerandMcLean(1985,1988)show how the sellercan extractall the bidders'informational rent in an optimal auction when bidders'valuesare correlated.

More recently, the optimal auction designliterature has exploredasymmetric auctions (Maskin and Riley, 2000),auctionswith budget-constrainedbidders(Cheand Gale,1998),efficient auctions(Dasguptaand Maskin,2000),multiunit auctions (Armstrong, 2000),and auctions with externalities(Jehiel,Moldovanu, and Stachetti,1996),among severalother topics(for asurvey of the recentresearchon auctions seealsoKrishna, 2002,Klemperer,2003,and Milgrom, 2004).

294 Multilateral Asymmetric Information: Bilateral Trading and Auctions

7.7 Appendix: Breakdownof RevenueEquivalencein a 2 x3 Example

Although the English and Vickrey auctionsare equivalent in the 2 x Nmodel, the equivalencebetween the Dutch and Vickrey (or first- and

second-price)auctions breaksdown under correlation,as the following

example due to Maskin (1997)illustrates. In this example the two

biddershave threepossibletypes {vL = 0, vM =1,vH = 2).The buyers have

perfectlyknown correlatedvalues:These values are pairwisepositivelycorrelated (that is, using the terrninology of Milgrom and Weber (1982),they are affiliated). Specifically,the distribution of joint probabilitiesis

given by

Pr(vL ,vL)=Pr(vH, vH) =-cc

Pr(vL ,vM) =Pr(vw, vL)=Pr(vw, vH) =Pr(v#,vM) = \342\200\224

Pr(vw,vw) =--aPr(vL,Vtf) =Pr(vtf,vL) =0

wherea is positive but small.Theexpectedrevenuein the Vickrey (second-priceauction) is then given by

[Pr(vw, vM) +Pr(vtf, vM) +Pr(vw, vH)]+2Pr(vH, vH) =

Thus at a =0 the derivative of expectedrevenuewith respectto a is -\\.Considernow the expectedrevenue in the Dutch (first-price)auction.

Recallthat in this auction the (symmetric) equilibrium is in mixed

strategies,at least for types vM and vH. Thus let types vM and vH, respectively,

randomizeoverbidsin the supports[0,bM] and [bM, bH] with thecumulative distribution functions FM and FH.Then the seller'sexpectedrevenueis

given by

f f6MbdFu(b) +{\\-a\\Mbd[FM(b)f+2\302\260 ^ ;*

(7.12)^yF^jyiFMf

295 Static Multilateral Contracting

whereFM{b) is given by the indifferencecondition

^(l-b)+(\302\261-a\\FM(b)(l-b)

for all b e [0,bM]. Hence,

3ctbFM(b) = and bM =4-12a

(4-12a)(l-fc) \"\" 4-9aSimilarly, FH(b) is given by the indifferencecondition

^(2-b)+\302\261FH(b)(2-b) =^(2-bM)

for all b e[\302\243>w, fr^].Hence,

FH(b)= ,Ab-bM) andbH=\342\200\224\342\200\224-\342\200\224

4(2\342\200\224

Z?) 4+3a

(7.13)

(7.14)

Substituting equations (7.13)and (7.14)into expression(7.12),oneobtains the rather unwieldy expressionfor the seller'sexpectedrevenue

a 4-12g/ 3ab

\342\226\240\302\243

\342\200\224rr^ 4 I 4-4+3a ^ v ^a9 J4-12a

4-9a

(1-12a^'b+^~a

Jo\"9\"2Jo l(4-12a)(l-6)2/V3 JJo (,(4_i2a)2(l-&)3(9a2)(262) db+

9a J

(2-ft)dft + (7.15)

0 \342\200\236r 4-120^8+3a -\342\200\224-

-1 I 4-9aJ_ f 4+3aa J4-12o

9az6f 4-12aY4-6aY4-9aA4-9a(2-bY

db

Now the derivatives with respectto a of the first and third terms ofexpression(7.15)at a=0 are finite. However,the derivatives of the secondand fourth terms are both infinitely negative, as

lima->0

1 \\n Jj4-12a\\2V\\

(4-12a)2(l-''4-12oa4-9aJ.

d (4-12a\\da{4-9a/

296 Multilateral Asymmetric Information: Bilateral Trading and Auctions

and

f

(9)

lim-

8+3a 4-9a)4+3a <r

8+3a 4-12a\\A-9a) 4-Yla

4-9a4+3a4-6a4-9a

8+3a2 \342\200\224

4-9a)4+4a

Moreover,at a = 0 expectedrevenuesfrom the first-priceand second-priceauctions are both 1,ascanbeseenfrom expressions(7.12)and (7.15).Sincethe derivative of the seller'sexpectedrevenuewith respectto a is

infinitely negative for the first-priceauction but finite for the second-priceauction, the latter must generatestrictly more revenuefor a near zero.

This exampleis an illustration of the generalresult ofMilgrom and Weber

(1982)that the Dutch(and first-pricesealed-bid)auction generatesnomoreexpectedrevenue than the Vickrey auction when buyer's valuations areaffiliated. One acceptedintuition for this result is that the Vickrey auction

providesmore information to the winning bidderthan the Dutch auction

(the highest and second-highestbid as opposedto the highest bid only).As a consequence,biddersare encouragedto bid more aggressively in

the Vickrey auction.17

17.Note that Milgrom and Weber show only that expected revenue in the Dutch auctioncannot exceedexpected revenue in the Vickrey auction (Theorem 15).They are silent on whenor whether the Vickrey auction provides strictly higher revenues than the Dutch auction.Maskin's example thus provides a reassuring illustration of the breakdown of revenueequivalence in the casewhere bidders' values are affiliated.

Multiagent MoralHazardand Collusion

In this chapterwe considercontracting situations wherea principalinteracts with multiple agents, each taking hidden actions.Thisextensiontakesus from the generaltopicof optimal incentive provision, studied in Chapter4, to the field of organization designand the theory of the firm. The mainissueswe shall beconcernedwith here are the extent to which competitionor cooperationamong agents should be fostered,how agents should besupervisedor monitored,how to dealwith corruption or collusion,and how

hierarchicalan organization of multiple agentsshould be.Though the maineconomicquestionsaddressedin this chapter are different from thosestudied in Chapter4,the coreideasand generalmethodology are the same.

As betweenChapters2 and 7, a key differencebetweenthe single-agentcontracting situation covered in Chapter 4 and the multiagent situation

coveredhere is that in the former casethe principal'sproblemis one ofdesigning an optimal incentive contract, while in the latter his problemis to

designa whole organization of multiple agents that strategically interactwith eachother in a game.As in Chapter7, weshall limit our analysis hereto the most common equilibrium notions of thesegamesconsideredin theliterature: the Nash and Bayesianequilibria.We again refer the readerinterestedin other solution conceptsto the surveys by Moore (1992)and

Palfrey (1992).In an important articleAlchian and Demsetz(1972)proposedthat a firm

is in essencean organization that is setup to dealwith a moral-hazard-in-teams problem.That is,a firm is an organization (or a multilateral contract)set up to mitigate incentive problemsarising in situations involving

multiple agents.The specificsetting that Alchian and Demsetzconsideris oneinvolving a team ofworkerswhereeachworker wants to rely on the othersto do the work. As in situations involving public-goodsprovision under

private information mentionedin Chapter7, eachworker wants to free rideon the work of others.The employer'srole in Alchian and Demsetz'sviewis then one of supervising employeesand making sure that they all work.

Interestingly, to ensure that the employerhimself has the right incentivesto monitor his team, Alchian and Demsetz argue that he should be theresidualclaimant on the firm's revenuesand that workersshould be paidonly fixed wages.

Alchian and Demsetz'sperspectiveon the firm is the starting point ofthis chapter.In another important contribution, Holmstrom (1982b)provides a first formalization ofAlchian and Demsetz'smoral-hazard-in-teamsproblem and derives an optimal multilateral incentive contract for the

298 Multiagent Moral Hazard and Collusion

team when only the team's aggregateoutput is observable.We begin by

examining his model and his main conclusionthat incentive efficiencyrequiresa budget breaker\342\200\224that is, an individual who holdssomeclaims onthe output of the team but is otherwisenot involved in production.

We then proceedto analyze optimal incentive contractsand the role ofcompetition betweenagentswhen individual performanceis observable.Firms often provide incentives to their employeesby comparing their

individual performanceand promoting the better performers to higherranks or more desirablejobs.We detail in particular the contributions byLazearand Rosen (1981)and Green and Stokey(1983)and examinetherationaleand optimal designof such tournaments. In particular, weevaluate in detail both the benefits of such tournaments, in terms of reducingthe overallrisk agentsare exposedto, and the costs,in terms of reducingcooperationamong agents(asstressedby Holmstrom and Milgrom, 1990,among others).

A potential factorlimiting the effectivenessof tournaments is collusionamong agents.We examinecollusionboth betweencompeting agentsandbetweenagents and their supervisors.Following the important paper byTirole (1986),we derive in particular the optimal collusion-proofcontract and considerhow the organization can be structured to reduce the

potential for collusion.We closethe chapter by touching on the fundamental question of the

boundariesof the firm. As Alchian and Demsetz,and before them Kaldor(1934),have plausibly argued,a natural limit on the size of firms would

appear to be the employer'sdirrnmshing ability to monitor effectively alarger group of employees.However,as Calvo and Wellisz (1978)haveshown, a firm may still be able to grow infinitely large if the employersetsup an efficient internal hierarchy composedof supervisors,who monitorother supervisors,who monitor workers.Their theory providesa simplerationalefor the existenceof hierarchiesbut raisesthe difficult questionof what limits the sizeof the firm if it is not the CEO'slimited ability to

supervise.Following Qian (1994),we show how this challengecan beresolvedif, asWilliamson (1967)has argued,there is a greaterand greaterlossof control of the top the deeperis the hierarchy (that is, the morelayersit has).

299 Static Multilateral Contracting

8.1MoralHazard in Teamsand Tournaments

Considera principal engagedin a contractual relationwith n agents.Theprincipal is assumedto be risk neutral, and eachof the i =l,...,n agentshas the usual utility function separablein incomeand effort

whereu,(-) is strictly increasingand concave,!//;(\342\226\240)

is strictly increasingandconvex,and at e [0,\302\260\302\260).

Theagents'actionsproduceoutputs, which in the most generalform canbewritten as a vectorof random individual outputs:

q =(qi,---,qn)with joint conditional distribution

F(q\\a)

where

a ={au---,an)denotesthe vectorof actionstaken by the agents.

Following Holmstrom (1982b),we shall considerin turn two

diametrically oppositecases.In the first case,the output producedby the agentsisa single aggregateoutput Q with conditional distribution F(<2lfl).This is the

pure caseof moral hazard in a team. Contributions by multiple agentsarerequired,but a single aggregateoutput is obtained.In the secondcase,eachagent producesan individual random output qt, which may be imperfectlycorrelated with the other agents' outputs. In this secondcasewe shallassumethat the principal caresonly about the sum of the individual agents'outputs.

Tooffera concreteexample,in the first casethe aggregateoutput mightbe the revenuegeneratedby a multiagent firm, while in the secondcasetheindividual outputs might bethe cropsof individual fanners,or the salesgenerated by individual marketersor traders.

When only aggregateoutput is observed,the n agentsbasicallyface aproblemof private provision of a public good:each agent contributes acostly action to increase a common output. All agents benefit from anincreasein the effort of any one of them. As Holmstrom (1982b)points out,

300 Multiagent Moral Hazard and Collusion

optimal effort provision can then be obtainedonly when one breaksthe

budget constraint of the team, that is, the requirement that the sum ofindividual output-contingent compensationsmust be equal to the aggregateoutput. Or, to use the language of Alchian and Demsetz,optimal effort is

providedonly if there is also a residualclaimant on the output after all

agentshave beencompensated.This argument is developedin detail in

section8.1.1.Holmstrom'sargument is very general,but it relieson limited

information usedby the principal.Indeed,Legrosand Matsushima (1991)and

Legrosand Matthews (1993)have shown that it may bepossibleto achieveor approximateincentive efficiencywithout breaking the budget constraint

of the team by relying on the statistical information about the agents'choiceof individual actionscontainedin the realizedoutput. That is, as in the

single-agentcase,it is possibleto achievefirst-best efficiencyjust by relyingon the informativeness principle.We cover several examplesillustratingsuch incentive schemesin section8.1.2.

We then move to the oppositecasewhereindividual performancelevelsare observableand contractible.The novel issueto be consideredin this

caseis how the principal might benefit by basing individual agents'compensation on their relative performancerather than only on their absoluteperformance.Section8.1.3addressesthis questionby consideringsituations

where individual agents' outputs are imperfectly correlated in a simpleCARA utility function, normally distributed output, and linear-contractframework, where the optimal incentive schemecan easilybe computed.This simple formulation allows us to highlight two relatedideas:first,relative performanceevaluation serves the purposeof filtering out common

output shocks,and there is otherwiseno point in making agentscompetewith eachother;second,by reducing the overalluncertainty agentsface,theuse of relative performanceevaluation inducesmore effort provision in

equilibrium, sothat relative performanceevaluation is positively correlatedwith equilibrium effort levels.

Section8.1.4considersa simple and widely usedform of relativeperformance evaluation: tournaments. Thisclassicalform of relativeperformance compensationhas the particularity of using only an ordinal ranking ofperformance.As Lazearand Rosen (1981)first pointedout, when agentsare risk neutral, tournaments can be seenas an alternative to piece-ratecontractsin achieving first-best levelsof effort. Theyinvolve a lottery with

a probability of winning that each agent can influence through her effort

level.While the prize the winner obtains seemsunrelatedto her marginalproductivity expost and thus seemsarbitrary, from an ex ante perspectiveit can be reconciledwith optimal incentive provision.When agents arerisk averse, however, tournaments are generally suboptimal relative-performancecompensationcontracts,as they donot optimally tradeoff risk

sharing and incentives.However,as Greenand Stokey(1983)have shown,the efficiencyof a tournament increaseswith the number of participants,as the law of largenumbers helpsreduce the uncertainty in compensationgeneratedby the opponents'random performance.

UnobservableIndividual Outputs:The Needfor a BudgetBreaker

Considerthe situation where n > 2 agentsform a partnership to producesomedeterministic (scalar)aggregateoutput

Q=Q(a1,a2,---,an)

by supplying a vectorof individual hidden actions(or efforts) a = (au...,an).Supposethat the output productionfunction is a strictly increasingandconcavefunction of the vectorof actionsa = (ah...,an):

dat'

da}'

da^a^and the matrix of secondderivatives Q,y is negative definite.

Supposealsothat all agentsare risk neutral:

ut (w)=w

Formally, we definea partnership to be a vector of output-contingent

compensationsfor eachagent, w(0= [wi(0,w2(Q),...,w\342\200\236(0]such that

n

5><(2)=2 for each<2[=1

For the sakeof argument, we take each w,(<2) to be differentiable almost

everywhere.We focushere on the externality that arisesin a team of self-interested

individuals when effort levelsare unobservableand no measureofindividual performanceis available.In such a situation rewarding an agent for

raising aggregateoutput means rewarding her higher effort and also tl

higher effort of the other agents!Thus the difficulty forja princinal .feying

302 Multiagent Moral Hazard and Collusion

to motivate severalagentsis now compoundedby free riding among the

agents.As we know from Chapter4,a risk-neutral individual agent will supply

first-best effort, given that all other agentssupply first-besteffort, if she is

compensatedwith the full marginal return of her effort.We shall take it

that the first-best vectorof actionsa* = (af,a%,...,af) is uniquely definedby the first-orderconditions

\342\200\224-i\342\200\224'- =^-(flf), for eachi

Consideran arbitrary partnership contract w(<2).Underthis contracteachagent i independently choosesher actionat to maximize her own utility

given the other agents'actionsa_t = (ax,.. .,a^, al+u . . .,an). Thus agent i

will choosean effort at to satisfy the first-ordercondition

dwl[g(a\302\253,a-\302\253)]^G(a\302\253.a-\302\253) ,, x\342\200\224 =

VH\302\253i)dQ da,

If all agentsget the full marginal return from their effort, at least locally,it appearspossiblethat a Nash equilibrium where all agentssupply first-

besteffort levelsmight exist.Proceedingwith this lead,the first-best outputlevelmay then be achievedin a Nash equilibrium if each agent i gets acompensationcontractwi[Q(al,a_)~\\ providing her with the full marginalreturn from her effort, when all otheragentsalsosupply the first-best effort

level

rfH/,[6fa,a*)]dQ

But this condition amounts to setting w,(0= Q (up to a constant).Obviously this cannot bedonefor every agent in the partnership if wemustalsosatisfy the budget constraint

n

^wXQ)=Q for eachQ[=1

As Holmstrom (1982b)notes,however,it is possibleto specifysuch acompensationschemefor all agentsif oneintroducesa budget breakerinto

the organization.That is, if a third party agreesto sign a contractwith all n

303 Static Multilateral Contracting

agentsoffering to pay eachof them w,(0)=Q.It is straightforward to checkthat if a budget breakeragreesto sign such a contractwith the n agents,then there indeedexists a Nash equilibrium where all agents supplytheir first-best action and where the budget breakerpays out nQ(a*) in

equilibrium.This would bea profitable contractfor a budget breakerand eachof the

n agents if the n agentsalso agree to hand over the entire output to the

budget breakerand make an up-front payment z, to the budget breakersuch that

n

^zl+Q(ft*)>nQ(a*)[=1

and

z.<G(\302\253*)-V.(\302\253?)

Becauseat the first bestwe have

G(\302\253*)-i>.-K)>0

it is straightforward to checkthat there is a vectorof transfers z = (zi, \342\226\240..,

z\342\200\236)

that satisfiestheseconditions.Obviouslythe up-front payment z,will notaffect an individual agent'sincentivesto supply effort.

Holmstrom'sobservationthat a team facing a multiagent moral-hazardproblemrequiresa budget breakerhas beenvery influential. It has beenseenasa fundamental reasonwhy a firm needsa residualclaimant and why

a firm needsto seekoutsidefinancing to be able to breakits budgetconstraint. Note,however,that Holmstrom'sbudget breakeris very different

from Alchian and Demsetz'sresidualclaimant, who holdsequity in the firm

to inducehim to monitor the team's agentseffectively.At first glance,the

budget breakeralso looksvery different from any claimants on firms weseein reality. One feature of the budget breaker'scontract in particularseemsodd:he would losemoney if the firm (or team) performsbetter.Indeed,the budget breaker'spayoff wBB(Q) under the precedingcontractsatisfies

\342\200\224 =-{n-l)Qat\302\243(a*)

304 Multiagent Moral Hazard and Collusion

Interestingly, Holmstrom'slogicis entirely basedon the idea that a risk-neutral agent's incentives are optimized when she is a residualclaimant.But we know from Chapter4 that theremay beotherways to providefirst-

bestincentives by exploiting the information about actionchoicecontainedin realizedoutput. In the contracting situation we have considered,eachagent may, for example,be compensatedwith a Mirrleescontract, which

rewardseachagent with a bonus bx if output levelQ(a*)is realized,and bepunished with a penalty k to be paid to the budget breakerif any otheroutput is obtained.If the bonusesbt and k are such that

bi>\\}fi(a*)-k

for all i, then thereindeedexistsa Nash equilibrium whereall agentssupplytheir first-best actionunder this contractas well.In addition, this contractmay not requirea budget breakerto sustain it in equilibrium if

n

<2(a*)>5>[=1

It does,however, requirea budget breakeroff the equilibrium to collectallthe penaltiesnk.

It is not uncommon for firms to pay their employeesbonuseswhen

certainprofit targets are reached.Thesebonusschemesmay thus beseenas efficient incentive schemesto solvea moral-hazard-in-teamsproblem.

Another possibleinterpretation of the Mirrleescontractis debtfinancing by the firm. Under this interpretation the firm commits to repay total

debtsof

D=Q(a*)-Jjbl

and salariesbt to eachemployee.If the firm fails to meet its obligations, it

is forcedto default, the creditorscollectwhatever output the firm hasproduced, and the employeespay a bankruptcy costof k.

Although both the bonus and the debt financing schemesappear to bemore realistic than Holmstrom'sproposedincentive scheme, they arevulnerable to an important weakness.Theseschemesgenerally give risetomultiple equilibria in agents' actionchoice.For example,anotherequilibrium under theseschemesbesidesa* = (af,...,a%) may be for all agentsto do nothing. Indeed,if all otheragentsdo nothing [#_,-

=(0,...,0)],agent

305 Static Multilateral Contracting

Vs bestreply is to alsodo nothing, sinceon her own she cannot meet the

output target Q(a*)at a reasonableeffort cost.That is, if all other agentsdo nothing, we have

bi-\\ffi{fti)<-k

wherea, is such that

Q(0,---,ai,---,0)=Q(a*)

To rule out this inaction equilibrium the incentive schemewould have tosetpunitive penaltiesk [sothat bL

- iff^) > -k],but then it is easyto checkthat thesepenaltieswould support equilibria wheresomeagentsshirk.

In contrast, Holmstrom'sschemesupportsa unique efficient equilibrium.Toseethis point, observethat if all otheragentsdo nothing, agent i choosesat to solve

maxj2(O,-\",0\302\253,---,O)=Vi(ai)a,

so that her best responseis some at > 0.But since d2Q/daldaj>0,this

inducesall other agentsto bestrespondwith higher actions.Iterating bestresponsesin this way, it is easy to seethat when the first-best vector ofactionsa* =(of,a%,...,af) is uniquely definedby the first-orderconditions

as we have assumed,then a* = (af, a%,...,#*) is a unique Nashequilibrium under the Holmstrom scheme.

Note,however, that the fact that the Mirrleescontractwe have describedmay support multiple equilibria, someofwhich are highly inefficient, is nota fatal objectionto the applicationof theseschemesto solvemoral-hazard-in-teamsproblems.First, the efficient equilibrium under theseschemesmaybe a natural focal point. Second,as the next section illustrates, theseschemescan be easilyadapted to ensure that the efficient equilibrium is

unique. In addition, these schemescan alsobe adapted to satisfy budgetbalanceon and off the equilibrium path.

8.1.2UnobservableIndividual Outputs:Using Output ObservationstoImplement the First Best

In Chapter4wepointedout that when someoutput realizationsareimpossible if the agent choosesthe right action,then the first-best outcomecan

306 Multiagent Moral Hazard and Collusion

be implemented through a form of forcing contract, where the agent is

punished severelyfor output realizationsthat are not supposedto arise.Inmultiagent situations such simple contractsare lesseasilyappliedbecauseit is not always possibleto identify the culprit among the agentswho didnot do what she was supposedto do.Legrosand Matsushima (1991)and

Legrosand Matthews (1993),however,have shown that it is often possibleto implement small deviations from the first-bestactionprofile, which letthe principal identify moreeasilythe agentresponsiblefor a unilateral

deviation. What is more, they show that it is then possibleto implement orapproximatethe first-best actionprofile with incentive schemesthat always

satisfy budget balance.We now illustrate the basicideasbehind theseincentive schemesin three

simple examples.The first two examples(basedon Legrosand Matthews,

1993)assumethat aggregateoutput is deterministic, sothat shirking by anyof the agentsis perfectlydetected.These examplesrely on the idea that,

upon observing a deviation from the requiredaggregateoutput, the

principal may beable to identify at leastone agent who did not shirk. Theprincipal

is then able to preservebudget balanceby imposing penaltieson theother agents (who are presumedto have shirked) to be paid to the non-

shirking agent.The third example(basedon Legrosand Matsushima, 1991)considersa situation whereaggregateoutput is random, so that detectionof shirking is imperfect.This examplebuilds on ideassimilar to those ofCremerand McLean(1985),discussedin Chapter7, and extendsthe two

previousones by exploiting differencesin probability distributions overoutput realizationsthat arisewhen different agentsshirk.

Example1:Deterministic Output with Finite Action Space Considerthesituation where three agentswork together in a team.Each agent has abinary actionsetand caneithersupply effort (a,=1)or shirk (a, =0).Supplying effort is costly [y/;(l) > y;(0)],but it is first-bestefficient for all threeagentsto supply effort: af = a% =af =1.

The first-best output is denoted Q*, while the output levelwhen only

agent i shirks [whereat = 0 and a_t= (1,1)]is denoted Qt. LegrosandMatthews argue that in most casesone has <2i * Qi *\342\226\240 Qz (they arguethat this assumption is true generically).When this is the case,a singleshirker is identifiable and can be punished. It is easy to check that a

307 Static Multilateral Contracting

forcing contractcan then beusedto implement a* while preserving budgetbalance.

Legrosand Matthews point out that it is alsopossibleto implement thefirst-best outcomewhile maintaining budget balance,as long as a non-shirker canbe identified and rewarded.Thus in our exampleit is alsopossible to implement the first-bestoutcomewhen <2i = Qi *\342\226\240 Qi-The onlysituation where the first best is not implementableuniquely occurswhen

Q\\ -Qi= Qi (weleave it to the readerto verify theseobservations).

Example 2:Deterministic Output with Approximate Efficiency It is muchharder to identify a deviating agent when agents'actionsetsare largeandallow for many possibledeviations. The deviating agent can then moreeasilyselectan actionthat preventsthe principal from identifying the devi-ator.Legrosand Matthews show, however, that in such situations it may still

bepossibleto approximatethe first best.Hereis the argument: Considerateam of two agentswho can eachpick an actionat e [0;+00).Supposethat

the team'saggregateoutput is given by

Q=ax+a2

and let the effort costfor eachagent be

yfM =a?/2so that the first-best actionprofileis given by

af =af =1Theincentive schemeproposedby Legrosand Matthews is basedon the

idea that one agent\342\200\224for example,agent 1\342\200\224randomizes her actionchoiceand choosesax = 0 with a small positive probability e.Thisrandomizationlets the principal detecta deviation by agent 2 with probability e and thus

allowsthe principal to approximatethe first best.Toseethis result, considerthe incentive scheme

1.For Q > 1,w1(Q)=(Q-l)2/2and w1(Q)=Q-wl(Q)2.For Q < 1,wi(Q)=Q+k and w2(Q)=0-k

308 Multiagent Moral Hazard and Collusion

This incentive scheme supports a Nash equilibrium where agent 2choosesa2= 1with probability 1,and agent 1randomizesby choosingax =1with probability (1-e) and ax =0 with probability e.

Toseethat this is indeeda Nash equilibrium, note the following points:\342\200\242 Givenactionchoicea2= 1,agent l'sbestresponseis given by thesolution to

max =maxa\\

'tL.2

2 ~\\

2_=0

In otherwords,agent 1is indifferent betweenany actionau sothat a choiceof ai = 1with probability (1- e) and ax = 0 with probability e is a bestresponsefor agent 1.\342\200\242 As for agent 2,a choiceof a2> 1guaranteesan aggregateoutput that isno lessthan 1:Q>1.Sincewx(Q)is increasingin fl2wnen Q ^ 1>a choiceof fl2= 1is optimal in the actionsubset[1,+oo) and gives agent 2 a payoff

d-e)2 \342\200\224 +\302\243[1_0]-I =1_|A choiceofa2<1,however,implies that Q < 1with probability e.Thisgivesagent 2 a payoff of

d-e)1+^2- a\\-ek-\342\200\224 <1+a2-a2-ek

which is maximized at a2=\\.At a2=^,agent 2'spayoff is then

\342\200\224 -Ek

Therefore,a2=1is optimal if

, 1 12 4e

The incentive schemethus works as follows:If aggregateoutput is toolow, agent 2 is punished and agent 1is rewarded (to maintain budgetbalance).But, given that agent 1getsto collectthe fine k, shehas anincentive to try to loweraggregateoutput asmuch aspossible.It is for this reasonthat the penalty is imposedon agent 2 only when aggregateoutput is below

309 Static Multilateral Contracting

<2(0,1).Then the agent who is punished for low output is deterredfrom

underproviding effort, and the agent collectingthe fine ispreventedfrom

strategically lowering output. Thus a critical assumption in this exampleisthat at leastone agent has a finite lowestaction.

Under this incentive scheme,agent 1effectively actslike a monitor who

randomly inspectsagent 2.The monitoring cost is then the lossof outputthat agent 1couldhave contributed.But, oddly, here the monitor doesnotseeanything directly. Shecanonly infer agent 2'sactionindirectly from therealizedoutput.

This is undoubtedly an ingenious scheme,but it seemstoo fragile and

specialto be appliedin practice.For one thing, the principal must know

exactlywhat eachagent'seffort costis.In addition, it is not clearhow easilythis schemecan be generalizedand appliedto teamswith three or moreagents.Finally, this schemebreaksdown when aggregateoutput is random.

However,when aggregateoutput is random, as shown by LegrosandMatsushima (1991),other incentive schemesare availableakin to theCremerand McLeanlottery mechanisms discussedin the previouschapter.

Example 3:Random Aggregate Output Considera team of two agents.Asin the first example,eachagent has a binary actionset,ate {0,1},and first-

bestefforts aregiven by a* =(1,1).In contrastto the first example,however,there are now three possibleaggregateoutput realizations,QH> QM> Ql-

Aggregate output is increasingin the agents' effort choices.We shall

supposethat when both agentsshirk, aggregateoutput is QLwith

probability 1,and that the conditional probability distributions overoutput satisfyfirst-orderstochasticdominance.Specifically,we shall supposethat theconditional probability distributions are as follows:

Qh Qm Qla1=a2=l a J fa1=0,a2=lJ 1 ia1=l,a2=0\\\302\261 J f,

Note that eachoutput levelcanoccurwith positive probability under thethree actionprofiles,(1,1),(1,0),(0,1).But when one of the agentsshirks,the highest output is lesslikely. Moreover,the probability distribution isworsewhen agent 2 shirks. With the knowledgeof Cremerand McLeanschemesit is easyto seethat this differencein distributions, dependingonwho shirks, canbeexploitedto implement the first-best pair of effort levels

310 Multiagent Moral Hazard and Collusion

a*. What is more, the first best can be implemented with an incentiveschemethat satisfiesbudget balance.

Concretely,considerthe incentive schemewith transfers t(Q)definedasdeviations from equalsharesof output, tt(Q) = w\\(Q)

-QI2,and such that

h(Q)+h(Q)=0 for eachoutput level.Let y/; = y/;(l) > y/;(0)=0 denote theeffort costof action at =1.Then,preventing shirking by any agent requires

6

and

-[t2(QH)+t2(QM)]--t2(QL)>W26 3

It isstraightforward to checkthat we canfind transfers tl that satisfy theseincentive constraints while alsomaintaining budget balance:

tl(QH)+t2(QH)=tl(QM) +t2(QM)=tl(QL)+t2(QL)=0

For example,if

ti(QL)=t2(QL)=0

ti(QH)=6yr1=-t2(QH)and

t2 (Qm) =6(Vi +\302\2452)

=~k(Qm)

then all the constraints are satisfied.As in the Cremerand McLeanschemes,the idea is to reward eachagent

in the output realizationswhereher effort contribution makesthe greatestdifference.Hereagent 1is disproportionatelyrewardedwhen QHisrealized, while agent 2 is disproportionatelyrewardedwhen QM is realized.Interestingly, under this scheme,payments to each agent arenonmonotonic even though the probability distributions overoutput satisfy MLRP.

To summarize, these three examplesillustrate that it isoften possibletoprovide first-best incentives to a team without having to break budgetbalanceand without requiring that all agentsbe made residualclaimantson aggregateoutput. Thus Holmstrom'sinitial insight that participation ofthird parties(or outsidefinancing of the team) is essentialto provideadequate incentives to teamsneedsto bequalified.Oftenthe roleof the outside

311 Static Multilateral Contracting

party canbeplayedby one of the team members,whoseincentive problemis easyto isolate,and often the information containedin output realizationscan be usedto designeffectiveincentive schemes,as in the single-agentmoral-hazardproblems.However,the schemeswe have discussedhere,asingenious as they may be,may not be easilyimplemented in reality andindeedare not commonly observed.

8.1.3ObservableIndividual Outputs

When moving from a situation where only aggregateoutput of a team ofagentsis observableto onewhereindividual agent outputs are observable,onemovesfrom a problemof eliciting cooperation,or avoiding free riding

among agents, to a problemofharnessing competition betweenagents.Thisstatement isactually somewhat of an oversimplification, aswe shall see,but

the basicpoint remains that the coreissuebecomesoneof controlling

competition among agents.Relativeperformanceevaluation is a widely usedincentive schemein

multiagent moral-hazardsettings.Whether at school,at work, or in sports,individual agent performancesare regularly ranked.Theserankings areusedto selectand promoteagents, or generally to reward the bestperformers.A very common relative performanceevaluation schemeis the so-calledrank-ordertournament, which measuresonly ordinal rankings ofperformance. That is, it takes into accountonly the rankings of individual outputsfrom the highest to the lowest(or only the sign of the differencebetweenany two individual outputs) and otherwiseignores the information

contained in the value of the differencein outputs. We shall beinterestedin the

efficiencypropertiesof theseschemesand ask when there is no lossinefficiency in ignoring the levelsof individual outputs. Totake an examplefrom

sports,when is it efficient to take accountof only the rank of the racersatthe finish line,as in sailing regattas or Formula 1races,and when is it efficient

to alsotake accountof absolutetime (ordistance)performance,as in mosttrack-and-fieldevents,whererecordspeedsor distancesare alsorewarded?

Beforeanalyzing tournaments in detail,we setoffby consideringa simplesetting involving relative performanceevaluation of two agents.FollowingHolmstrom (1979,1982b),we can apply the informativeness principle ofChapter4 and begin by determming when an optimal (linear) incentiveschemefor agent i alsotakes accountof agent ;'sperformance(J *i).Inother words, we ask when agent z's compensationalso dependson herperformancerelative to agent ;'.We carry through this exercisein the

312 Multiagent Moral Hazard and Collusion

somewhat special setting that proved so tractable in the single-agentproblem,whereboth agentshave CARA risk preferences,wheretheir

individual outputs are normally distributed random variables,and where their

respectiveincentive contractsare linear in output.Considerthe multiagent situation with two identicalagents, each

producing an individual output qt by supplying effort ah where

qi =ax +Ei +ae2

q2=a2+e2+aex

The random variablesex and e2are independently and normally distributed

with mean zero and variancea2.Relativeto the single-agentproblemconsidered in Chapter4, the only innovation is the parametera.When a*0,the two outputs are correlated.As one would expect from the informa-tivenessprinciple, when the outputs are correlatedit should be optimal tobasean individual agent's compensationon both output realizations,asboth provideinformation about an individual agent'sactionchoice.

We shall take the principal to be risk neutral, but both agentsare riskaverse and have CARA risk preferencesrepresentedby the negative-exponentialutility function

u(w,a) =-e-T,[w-y,{a)]

wherea denotesthe agent'seffort, rj the coefficientof absoluterisk

aversion, and y/{-) the agent's effort cost,which we assumeto be a quadraticfunction:

1\\ff(a)

= \342\200\224 ca2

Finally, we restrict attention to linear incentive schemesof the form

w1=Zi+v1q1 +u1q2

w2=z2+v2q2 +u2qx

where the Zi are fixed compensationpayments, and the v( and ut arevariable performance-relatedcompensationcoefficients.

The absenceof relative performanceevaluation here is equivalent to

setting Ui =0.Given the symmetry of the principal'sproblem,we needonlyto solve for an individual optimal scheme (ah w,). A principal trying tomaximize his expectedpayoff in his relationwith agent 1will then solve

313 Static Multilateral Contracting

max E(qi-Wi)Ol,Zl,Vl^l

subjectto

jE[_e-i[\302\253^-^\302\253i)]]> m(vj7)

and

flieargmaxEt-e-\"^1-^1]a

where u(w) is the default utility level of the agent and w is her certainmonetary equivalent outsidewealth.

As we highlighted in Chapter4, maximization of the agent'sexpectedutility with respectto a is equivalent to maximizing the agent's certainty

equivalent wealth Wi(a) with respectto a, wherewx{a) is definedby

_g-nvi(a) _ jg/_g-r[ivi-v(a)]'l

Thus, as

Var[vi (ei+ae2)+ux(e2+asx)]=o2[(vi+ccui f +{ux +a\\>x) ]the agent'soptimization problemis to choosea to maximize her certainty-

equivalent wealth:

max\\zi+via +uia2-\342\200\224ca2\342\200\224

[(vi+awi) +(w1+av1)]lNote that, in the Nash equilibrium of the two agents' effort provision

game,eachagent correctlyanticipatesthe effort choicea,-ofthe otheragent.As in the one-agentproblem,the solution of the agent'sproblemhere is

easilyobtained:

Vifli= \342\200\224

c

Substituting for ^in eachagent'scertainty-equivalent wealth formula, weobtain eachagent'sequilibrium payoff.For agent 1this equilibrium payoffis given by

1Vj2 UiV2 Tj(72X, N2 , N21Zi+-\342\200\224+

\342\200\224

(vi +aa*i) +(wi+avi)2 c c 2 L J

314 Multiagent Moral Hazard and Collusion

Theprincipal'sproblemthen reducesto solving the following constrainedmaximization problemfor eachagent:

max{jt_(a+2t+\302\253!aj

subjectto

1V? UiV2 r\\G2 r 2 , N21 _Zi +-\342\200\224+ r-[(vi+aa*i) +(u1 +ccv1)

\\

=w

Or, substituting for Zi,

maxJ \342\200\224--\342\200\224

\342\200\224{(yx+ccuxf+iux +avif]nn yc 2 c 2 L J

The principal'sproblemcanbe solvedsequentially:

1.For any given v1; ux is determinedto rmnimize risk.

2.The variable vx is then set to optimally trade off risk sharing andincentives.

Proceedingin this way and minimizing the variance

rj<72

-[(Vi+OMi) +(Ui +OCVi) J

of agent l'spayoff with respectto u\\, keepingvx fixed,yieldsthe formula

Onelearnsfrom this formula that the optimal ux is negative when the two

agents' outputs are positively correlated,or a > 0 (assuming, as we shall

verify, that the optimal vx is positive).In other words, one learnsfrom this

expressionthat under the optimal incentive schemean individual agent is

penalizedfor a better performanceby the other agent.Why should an individual agent bepenalizedfor the other agent'sgood

performance?The reasonis that a better performanceby agent 2 is likelyto be due to a high realizationof e2,which alsopositively affectsagent l'soutput. In other words,both agents'high output is then partly due to goodluck. Therefore,by setting a negative uu the optimal incentive schemereducesagent l'sexposure to a common shockaffecting both agents'

315 Static Multilateral Contracting

output, and thus reducesthe varianceof agent l'scompensation.If both

agents'performanceoutcomeswerenegatively correlated,ux would insteadbe positive to reduceeachagent'sexposureto an exogenousshock.

From this formula we thus learn that the real reason why an agent'soptimal incentive scheme is basedin part on the other agent's outputrealization(in otherwords,the reasonfor introducing a form of relative

performance evaluation) isto reduceeachagent'srisk exposureby filtering out

the common shockcontribution to each agent'soutput. That is,as stressedby Holmstrom (1982b),the reasonbehind relative performanceevaluationis not to induceagents to competebut to use competition among agents(i.e.,relative performanceevaluation) to rewardeach agent'seffort while

exposingagentsto lessrisk. We alsoseefrom the precedingequation that

relative performanceevaluation is usedif and only if individual outputs arenot independent(a&0).

In our secondstep we substitute for the optimal ux in the principal'sobjectiveand solvefor the optimal v^

Jvi V\\ J](72 2(1-CC2)2}max! \342\200\224

v\342\200\224 \342\200\224>

vi [c 2c 2 1+a2 J

Differentiating with respectto Vi and solving the first-orderconditionthen yieldsthe formula

1+a2vi= y1+a2+ric<j2(l-a2)

Note that when a=0 this formula reducesto the familiar formula at theend of section4.2of Chapter4.The changeintroducedby the correlationin the two agents'outputs is to reducethe overallrisk exposureof any

individual agent and thus to enablethe principal to give stronger incentives toboth agents.Remarkably, when a approaches1(or-1),then each agent'soutput is almost entirely affectedby a single, common sourceof noise.Byfiltering out this common shock,the optimal incentive schemecan then

almost eliminate eachagent'sexposureto risk and thus approximate first-

bestincentives by letting Vi tend to 1.Finally, it is worth highlighting that although the true reasonfor relative

performanceevaluation is not to inducecompetition\342\200\224and through greatercompetitionto inducemore effort provision\342\200\224the end result is still highereffort provision.But the reason why equilibrium effort is higher is lowerrisk exposure.

316 Multiagent Moral Hazard and Collusion

8.1.4Tournaments

We now turn to the analysis ofperhapsthe most prevalentform of relative

performanceevaluation encounteredin reality, tournaments. As we have

alreadypointedout, tournaments do not make use of all the outputinformation available,sincethey basecompensationonly on an ordinal rankingof individual agents' outputs. One obvious advantage of tournament

schemesis that ordinal rankings of output are easyto measureand hard to

manipulate. Perhaps,most importantly, the principal has little incentive to

manipulate the outcomeof a tournament, as he has to rewarda winner nomatter who wins (that is,unlessthe principal cansecretlycolludewith oneagent, promising to favor her in decidingwho the winner is and extractingsomesurplus from her against this promise).However,when compensationis basedon output levels,the principal has a strong incentive to cheatandto report output levelsthat requirelowerpayments to the agent.Therefore,in situations where the principal is bestplacedto measureoutput and isable to massageoutput data, tournaments are well suited for reducing the

principal's incentives to manipulate reported output. As Fairburn andMalcomson(2001)have argued,one situation where this issueis likelyto be important is incentive compensationof employeesby midlevel

managers.They argue that a midlevel manager's ability to manipulateperformancemeasuresof his underlings is an important reason why atournament, which rewards the best employeeswith bonusesand jobpromotions, is an efficient incentive scheme.

In situations whereoutput is easilymeasuredand is difficult tomanipulate, however, it is less clear that tournaments are optimal incentiveschemes.As one might expectfrom the analysis in Chapter4,when agentsare risk neutral, tournaments (among other incentive schemes)may allowthe principal to achievefirst-best efficiency.Thisis indeed the case,as thearticleby Lazearand Rosen(1981)corifirms. We beginour analysis oftournaments with their articlenot somuch to highlight this point, which is afterall not too surprising, as to discusssomeinteresting economicimplicationsthey draw from their analysis.

8.1.4.1Tournaments under Risk Neutrality and No CommonShock

Lazear and Rosen (1981)considerthe situation where two risk-neutral

agentsproduce individual outputs that are independently distributed (nocommon shock).Basedon our precedinganalysis, there would appear tobeno reasonwhy tournaments would beefficient incentive schemesin this

317 Static Multilateral Contracting

setting. However,Lazearand Rosen show that the first-best outcomecanbe implemented using a tournament.

Specifically,they assumethat

q, =ai+Eiwherethe random variablese,-are identically but independently distributed

with cumulative probability distribution F(-),with mean 0 and varianceo2.Agents have effort cost y[ai),and the first-besteffort levelfor each agentis given by the first-ordercondition

1=yf'(a*)

Lazearand Rosencomparetwo methodsof payment: purely individual

performance-relatedpay (say, piecerates)and relative-performance-basedpay, tournaments. Thefirst method implements the first bestwhen pay is setas follows:

wt=z+qt

and

z+E[qt]-y/(a*) =z+a*-y/(a*) =u

whereu is the agents'reservationutility.

The tournament is structured asfollows:Theagent with the higher output

getsa fixed wage z plus a prize W, while the agent with the loweroutput

getsonly the wage payment z-Underthe tournament the expectedpayofffor an agent i exerting effort at, when agent;' exertseffort af, is then

z+pW-yf(at)wherep is the probability that agent i is the winner. That is,p is given by

p =Vr{qi >q}-)=Pr(a,-a, >\302\243/

-\302\243,-)

=H(at -a,)whereH{-)is the cumulative distribution of (e;--e,),which has zeromeanand variance2<72.Thebestresponsefor an agent under the tournament isthen given by

Wp-=yf'{a,)oat

or

Wh(at -aI)=y/'(al)

whereh(-) is the density of(\302\243y

-e,).

318 Multiagent Moral Hazard and Collusion

In a symmetric equilibrium, effort levelsby both agents are identical.Thus, in order to implement the first-bestactionchoicea* in a symmetricequilibrium of the tournament game, the tournament must specifya prize:

\302\2530)

The fixed wage z can then be set to satisfy each agent's individual

rationalityconstraint:

z+lm-\302\245ia)=u

We seehere that a tournament can do as well as a piece-ratesystem in

implementing the first bestunder risk neutrality. Themechanisms by which

the first bestis achieved,however, lookvery different. Under the piece-ratescheme,it lookslike the agent has direct controlover her compensation,up to a random shock:she isdirectly controlling the meanof hercompensation. Instead,in a symmetric tournament, both agentschoosethe sameeffort level,and it is solelychancethat determineswho the winner is!Incentives to work under the tournament, however,are driven by each agent'sability to control the probability of winning. The optimal tournament

choosesthe levelof the prize soas to provideexactly the samemarginalexpectedreturn from effort as in the piece-ratesystem.

Onecommon form of tournament ispromotionsin firms. One is at times

surprisedto seepromotedindividuals obtain very high raises:why shouldthat be, given that the individual remains the same from one day toanother?Hasher marginal productivity risensomuch from onejob to theother?The precedingsimple tournament exampleshows how

compensation canbedivorcedfrom marginal productivity becauseof lottery featuresthat look ex post like pure randomnessbut can be justified for ex anteincentive reasons.In particular, this exampleshows how a winner-take-allincentive-basedsystem may actually reflectex ante expectedproductivity-relatedpay.

The examplecan alsobe easilyextended to situations with more than

two agents to show why prizes must be increasingin each round ofelimination of a tournament (seeRosen,1986).To illustrate the basiclogicof the argument, consider a tournament among two pairs of agentswhere,just as in tennis, the two best agents are pickedin a first round

319 Static Multilateral Contracting

of elimination and move on to a secondround to competefor the number-one slot.Let Wi and W2, respectively,denote the prizes of the first- and

second-placedagentsfollowing the two rounds of competition.As before,agentsget paid a fixed wagez plus these prizes,and we solvefor asymmetric equilibrium in each of the three tournaments: the first two

tournaments to pick the besttwo performersand the subsequent\"final\" to selectthe overallwinner.

Proceedingby backwardinduction, we beginby solving for the optimallatter tournament. Basedon the precedinganalysis, it is easyto seethat wemust have

Moving back to the first round of elimination, we then seethat the

expectedpayoff for an agent i exerting effort a,-, given the other agent;'5*i effort ah is then

Z +pyVw-yifli)

where px is the probability that agent i wins in the first round, and

Vw =H(0)(Wx -W2) +W2- iff (a*)

is agent z'sexpectedcontinuation payoff of \"moving to the final,\" and thebestresponsefor an agent in the first round is then given by

dpi u \\

dcii

or

[HWWi ~W2) +W2- \302\245(a*)]^-=yf'ia,)

Thus, in order to implement the first-bestaction choicea* in asymmetric equilibrium of the first round, the tournament must specifya prizestructure such that

n(v) J

320 Multiagent Moral Hazard and Collusion

In this expression,h(0) is, as before,the partial derivative of the

probabilityof winning with respectto effort, given symmetric equilibrium effort

levels,while

MO)V

is the expectedequilibrium rewardfrom competing in the final. Sincethis

rewardis positive, it means that

MO)

That is, the differencein prizesbetweenthe first and secondagentsmustbe strictly higher than the differencein prizesbetween the secondagentand the last two. The reasonwhy the rewardsfor winning must beincreasing

in each round of elimination of the tournament is that after passingaround the remaining competitorsactually losethe option value ofsurviving. Thus,to preservetheir incentives to work, they needto becompensatedwith higher monetary rewards.

8.1.4.2Tournaments with Risk-AverseAgents and Common Shocks

We have just seenthat when agentsare risk neutral and when their outputsare independent,piecerates and tournaments are equally efficient

incentive schemes.In this subsectionwe discussthe analysis ofGreenand Stokey(1983),who show that this equivalencebreaksdown when agentsare riskaverse.Theyestablishthat tournaments are then dominated by piece-rateschemeswhen agents' outputs are independent,but tournaments maydominate piecerates when a common shockaffectsagents'performance.Tournaments are then also approximately second-bestoptimal incentiveschemeswhen the number ofagentsis large.Thereasonwhy a tournament's

efficiency improves as the number of agents in play increasesis that the

sophistication of the tournament increaseswith the number ofagents:it canspecifyfiner rankings and a richer reward structure {Wt} basedon thoserankings. With a largenumber of agents, the distribution of individual

performances, for a given realizationof the common shock,convergesto afixeddistribution. Eachagent then controlsherexpectedranking by choosinghereffort level.Thus, in the limit, the tournament system is like a piece-ratesystem that has filtered out the common shock.

321 Static Multilateral Contracting

We illustrate these ideasin our simple examplewith two output levelsq( e {0,1}.Each agent i can increasethe probability of success(qt = 1)by

exerting moreeffort.We modelthis ideaby letting the probability ofsuccessbe given by

where\302\243

e (0,1].The common shock affecting all agents' output is thus

introducedby letting all agents' output be zero {q{= 0) with probability

(1- \302\243).As we have donemany times, we shall take the cost-of-effort

function to be a simple quadratic function:

We begin by consideringthe situation with two identicalrisk-averseagentswith utility function

u(w) -iff (a)

whereu(-) is a strictly increasing,concavefunction.

We shall comparethe two incentive schemesby evaluating the cost tothe principal of implementing a given effort profilefor both agents(a1}a2).

We beginby consideringthe second-bestcontractin the artificial single-agent problemwhereone observes(1)whether output was affectedby acommon shock(in which casea wage wc is paid),(2) the individual agent'soutput realizationqL e {0,1},and (3) a referenceoutput realizationq, e{0,1}on which the single-agentcontractmay bebased.In this situation afeasiblecontractthus has five components:w = {w00,ww, wQ1, wn, wc} where,in the absenceof the common shock,agent i receiveswage w,y when herown output is i and the referenceoutput is;'.

We alsolet

Pr(g;=l)= \302\247fl2

As we have shown in Chapter4,when an individual agent'soutput cantake

only two values, then the agent'sincentive constraint

ai eargmaxj<j|[a(l-a2)u(wi0)+aa2u(wn)

+(l-a)(l-a2)u(wQo)+(l-a)a2u(wQ1)]+(l-^)u(wc)--ca2\\

322 Multiagent Moral Hazard and Collusion

canbereplacedwithout lossofgenerality by the first-ordercondition of the

agent'sproblem:

\302\243[(1

-a2)u(ww) +a2u(wn) -(1-a2)u(w00) -a2u(w0i)] =cax

Thus, assuming that the principal wants to implement actionax, his cost-rrrinimization problemreducesto

min_

K(w) ={Qal(X-a2)(ww)+ala2wn +{l-al)(l-a2)wm{h'OO.h'IO.h'OI.h'Ii.h'c }

+Q--a1)a2w01]+Q.-g)wc}subjectto

^[a1(l-a2)u(w10)+a1a2u(wn) +0--a1)0--a2)u(woQ)+0--a1)a2u(w01)]

+(l-t;)u(wc)>\302\261ca? (IR)

and

%[(l-a2)u(ww)+a2u(wn)-(l-a2)u(w00)-a2u(w01)]=cax (IC)Denotingby X and \\i the Lagrangemultipliers of the (IR) and (IC)

constraints, the first-orderconditionsof the principal'sconstrainedminimization problemthen are

*-0dw10

which implies

<Sjfli(l-a2)=<H(l-fl2)[Aai +/J]u'(ww)

and

-^-=0dwn

which imphes

Z,axa2 =\302\247fl2[M

+li\\u'(wn)

Therefore,the optimal contractmust set

Wio =Wu

and, similarly,

W00 =Vf0i >

323 Static Multilateral Contracting

We are thus able to draw the following observationsfrom this problem:

1.When either the common shockcanbe filtered out (wc &wi;) or there isno common shock (E, = 1), then any relative-performanceevaluation

scheme,which would result in either ww & wn or w0o & Woi or both, is

suboptimal. Only a piece-rateschemeis then optimal. As we have alreadyemphasizedin Chapter4,the reasonissimply that a relative-performance-evaluation scheme would then only increase the agent's risk exposurewithout improving her incentives.2. If a generalrelative-performance-evaluationschemeis suboptimal, then

a fortiori a tournament is suboptimal. Indeed,a tournament would specifya reward structure such that wn = wm =wc=T(where T stands for \"tie\,Wio = W (whereW stands for \"winner\,") and w0i = L (where L stands for

\"loser\.")UnlessL = T=W, this rewardstructure couldnot satisfy the

preceding optimality conditions.But when L =T=W, then the agent obviouslyhas no incentive to put in any effort.3.In the presenceof a common shock

(\302\243

< 1),a simple piecerate is

suboptimal. Indeed,the optimal contractmust satisfy^=0dwc

which implies

(!-\302\243,)=

X{l-\302\243,)uXwc)

and

-**-=0dwoo

which implies

Z(l-ai)0.-a2)=ZQ.-a2)[A(l-a1)+ii\\u'(w0o)

Theseconditionsthus require that w00 *\342\226\240 wc whenever /i **0,while a simplepiece-rateschemewould be such that w0o = wc.

4. In the presenceof a common shock,a tournament may dominate apiece-ratescheme.To seethis possibility, supposethat the first-bestoutcome for the principal is to implement the actionprofile ax =a2=1,sothat

success(ql =1)isguaranteedfor both agentsin the absenceof a common

(negative) shock(this is the first-best outcomewhenever the agents'effort

324 Multiagent Moral Hazard and Collusion

costparameterc is low enough).Considerthe principal'sproblemof providing agent 1with incentives to

chooseactionax under a tournament schemewhen agent 2'sactionchoicea2 = 1 is secured.Then the principal choosesthe tournament's rewardstructure {L,T,W} to solvethe problem

min fa.T+%0-ax )L+(1-\302\247T

T,WyL

subjectto

&ik(D+\302\243(1 -a{)u(L)+(1-\302\247u{T)

-\\-cal > 0

and

%[u(T)-u(L)]=cai

Thus, to implement actionax =1,the principal sets

u(T)=u(L)+^

As can be immediately seen,in that casethe agent gets T for sure and is

perfectlyinsured.In otherwords,the tournament then implements the first-

bestoutcome.That is, it implements a Nash equilibrium whereboth agentsset al =1and are both perfectlyinsured in equilibrium.

However,under the optimal piece-rate scheme that implements theactionprofile ax =a2=1,eachagent'scompensationis such that

Wn =Wio > Woi =W00 =Wc

In other words, each agent's compensationis risky, as wc is paid with

probability 1-^ > 0 whenever a negative commonshockoccurs.Thisextremecasethus illustrates how, by filtering out the common shock,

the tournament may exposeagentsto lessrisk than a piece-ratescheme.Itis for this reasonthat tournaments may dominate piece-rateschemeswhen

agents'outputs are affectedby a common shock.

Having shown that a tournament can dominate an optimal piece-rateschemein the presenceof common shocks,we now turn to the last majorresult establishedby Green and Stokey\342\200\224that a tournament mayapproximate a second-bestrelative-performance-evaluationschemewhen thenumber of agentsgrows large.

325 Static Multilateral Contracting

Supposethat n + 1identicalrisk-averseagentsparticipatein atournament, sothat eachagent competeswith n other agents.Greenand Stokey'sinsight is that, as n grows large,eachagent'srelative positionin the output

rankings will becomealmost entirely a function of her own effort supply.

Indeed,by the law of largenumbers, the distribution of the other agents'outputs conditional on the vectorof actions{ah \302\243_,)

and on the realizationof the common shockconvergesto a fixed limit distribution as n growslarge.A relative-performance-evaluationschemecan then filter out thecommon shockalmost perfectly.

To seethis result, considerthe following simple tournament: (1)if all

agentsproducethe sameoutput, everybody receivesT;(2)otherwise,thosewho are successfulreceiveW, and the othersreceiveL.Assuming that allother agentschooseeffort levela*, an agent choosingeffort a, obtains

Twith probability 0\342\200\224^)+tflla1 +

^0\342\200\224ai)0\342\200\224a*)n

W with probability ^(l-a?)and

L with probability \302\243(1

-at )[l-(1-a*f ]

From the precedingargument we know that a tournament is first-best

optimal when aL =a*=l.Thus,considerthe situation whereat =a*<l.Then,as n -->

\302\260\302\260,a* \342\200\224> 0, and (1-a*)n \342\200\224> 0, so that the tournament tends to the

second-bestcontract,in which agent i gets

wc =Twith probability (1-\302\243)

Wi =W with probability <^a,

and

w0 =L with probability \302\243(1

-aL)

Tosummarize, we have learnedin this sectionthat tournaments are sub-optimal when a small number of risk-averseagentscompete.Theyaredominated by piece-rateschemes,which donot basecompensationon any formof relative performanceevaluation, when agents'individual outputs arenotaffectedor are barelyaffectedby a common shock.When common shocksare a largecomponentof individual outputs, however, relative performanceschemesstrictly dominate any piece-rateschemes,but tournaments aregenerally dominated by more general relative performanceschemes,which

326 Multiagent Moral Hazard and Collusion

fully exploitall the information about actionchoicescontainedin outputrealizations.Only when there are a large number of agents involved dotournaments approximatethe second-bestoptimal contract.

8.2Cooperationor Competitionamong Agents

Thegoodsideofrelative-performance-evaluationschemesis that they helpreduce agents' risk exposureby filtering out common shocksthat affecttheir individual performance.But there may alsobe a dark sideto theseschemes,asLazear(1995)and othershave suggested.By fosteringcompetition among agents, tournaments and relative-performance-basedincentive schemesmay undermine cooperationamong agents and in extremecasesevenfosterdestructive behaviorsuch as \"sabotage\" of other agents'outputs. We take up this issuein this sectionand ask how a principal maybe able to induce agentsto cooperateor help eachother accomplishtheir

tasks when such cooperationis desirable.We shall cover two different

approachesto modeling cooperation that have been consideredin theliterature. In the first approach,cooperationtakes the form of agent i

helping agent j accomplisha task. In the secondapproach,cooperationisin the form of coordinationthrough contracting of agents' actionchoices.In this latter approach,when the principal triesto elicitcooperationamongagents, he lets the agentsform a partnership (or firm), and only contractswith the firm as a whole, as opposedto contracting with each individual

agent separately.

8.2.1Incentivesto Helpin Multiagent Situations

The first approachto fostering help among agentshasbeenexploredin Itoh

(1991).We shall illustrate his analysis and main findings in the by-now-familiar and highly tractable setting where agents' individual output cantake only two values:qt e (0,1}.We shall considera situation with a risk-neutral principal and only two risk-averseagents, i =1,2.The new featurein Itoh'ssetup is that eachagent now hasa two-dimensional actionset.Thatis,eachagent must now choosea pair

(*.,&.\342\226\240)\342\202\254 [0,+~)x[0,+oo)wherea, representsagent i'seffort on her own task and b, representsherhelpon the other agent'stask. Each agent'sutility function takes the usual

separableform

327 Static Multilateral Contracting

Ui(yv)-yfi(ai,bi)

where ut is strictly increasingand concave,and y/t is strictly increasingandconvex.In our illustration we shall also assumethe following functional

forms:

Ui (w)= \"*W

and

\\ffi (#,, \302\243>,)

=a?+b}+Ikciibi

where k e [0,1].As for the probability distribution over each agent's output, we shall

assumethat there are no common shocksaffecting individual outputs andthat outputs are independently distributed, with

Fr(qi=l)=ai0-+bj)

Theseassumptions imply that the two agents'efforts at and b, arecomplements and that, therefore,theremay bebenefitsin inducing cooperationamong the two agents.When k > 0,however,eachagent bearsan additionalcost in not specializingentirely in her own task and in helping the otheragent.

Thequestionwe shall beconcernedwith is determining when or whether

it is desirableto encouragemutual help despitethe fact that the sharing in

each other'soutput requiredmay alsoinduceagentsto free ride on eachother.

Theprincipal caninducecooperationby giving eachagent a stake in theother agent'soutput. That is, the principal can offereach agent a contract

Wi ={w'ooMlMoMl}wherewlmn denotesthe payment to agent i when q^m and q,-=n.The onlyrestriction on the contractsthe principal canofferto eachagent is,ofcourse,that all payments must benonnegative (wlmn

> 0).Giventhat the two agentsare identical, we can without lossof generality restrict attention tosymmetric contracts,w = {w0o,Woi>wio>wn}-

We set off by deriving equilibrium payoffs when the principal doesnot

attempt to induceany cooperationamong the two agents (\302\243>,

= 0).This isthe casewhenever the contractofferedto eachagent is such that w00= w01

and w10=wn.We then comparethesepayoffs to thosethat canbeobtainedunder cooperation(bt > 0).

328 Multiagent Moral Hazard and Collusion

8.2.1.1No HelpWhen the principal doesnot attempt to elicit any cooperation,he offerseach agent a simple piece-ratecontract rewarding output qt = 1with apayment wx and output qt =0with a payment w0. As we know from Chapter4, it is then optimal to set w0 =0 and to set wx to maximize

a,(l-wi)

subjectto

1

and

ai=rWi (ic)

a^-af=^->0 (IR)

Substituting for wu the principal'sproblem reducesto choosinga{ tomaximize the payoff

at -4a?

The optimum is then reachedfor

T12

and the principal'spayoff is then

2 IT3 U2

8.2.1.2Inducing HelpConsidernow contracts that induce agents to help each other (bi > 0).Optimalcontractsthat induce help are necessarilysuch that

Wn > wio and wQi > w0o =0

That is, agent i getsa higher reward when agent;'is successful.Only suchan increasein rewardscanencourageagent i to help agent;'.Note that by

conditioning agent i'spay on agent ;'soutput in this way, such contractsexposeagent i to greater risks and require that the principal pay a higherrisk premium. The questionfor the principal then is whether the higher

329 Static Multilateral Contracting

output obtained through cooperationpays for the higher risk premium.Tobe able to determinewhether that is the case,we needto solvefor the

symmetric Nash equilibrium in actionchoice(a, b) inducedby a contractof the form wn > w10 and woi > w0o =0.

In equilibrium, (a,b)must be a bestresponsefor each agent and solvethe individual agent problem

max a,(1+b)a(l+b,)Vv^7 +a,(1+b)[l-a(l+b.)]Vv%\"

+a(l+bj )[1-a,(1+b)Uw -a2-b2-Ikab

Differentiating with respectto b and a, we obtain the first-orderconditions at (aj,bj) = (a,b):a2(1+b)(4w -4w^)+a[l-a(l+b)\\4w^ =2(b+ak) (8.1)and

a(l+bf (Vm^~ -4Wi) +(1+b)[l-a(l+b)Uw^=2{a+bk) (8.2)Thesefirst-orderconditionsyield the answersto our main question:

1.When k > 0, there is a minimum threshold in help b* > 0 such that any

lesserhelp b (for which 0 < b<b*) yieldsa strictly lowerpayoff to the

principalthan no help at all (b =0).In otherwords,when k > 0,it pays to induce

eithera lot ofcooperationor noneat all.Thisobservationmay help explainwhy we often seein organizations a culture of eitherstrong cooperationornone at all.

Toseewhy the precedingfirst-orderconditionsyield this result, note first

that under no help the optimal contractexposeseach agent to minimumrisk by setting wn -ww = w, say, and w0i = w0o =0.

Now, as canbeseenfrom condition (8.1)[with a RHSequalto 2{b+ak)and a > 0],when k > 0, any b > 0 requiresa significant deviation from thecontractwn -ww = w and w01 = w00 = 0.Sucha discontinuous deviationfrom optimal risk sharing cannotbe optimal if only a very small changein

output is obtained(that is,when b is closeto zero).Only a significant changein output, inducedby a significant levelof b, can justify the increasedrisk

exposurefor the agents required under any contract seeking to inducepositivecooperation.2.When instead k = 0, contracts that induce help are always strictly

optimal. In fact, raising helping effort b abovezero is certainly optimal if

effort on the own task a were to remain fixed, since,with k = 0, the mar-

ginal costof raising b at 0 is zero.Consequently,to establishthe result, we

just have to make surea doesnot fall in the process.Thisproperty canbecheckedfrom the first-orderconditions(8.1)and (8.2),asfollows.Start from

the cornersolution where b =0,wn = w10= w, and w0i = Woo =0;then raiseVwn and Vw0i by du, while keepingVw10 constant; let da and db denotethe changesin a and b inducedby this changein compensation.From thefirst-orderconditions(8.1)and (8.2)we then observethat

{a+daf (1+db)du +(a+da)[l-(a+da)(l+db)]du=2db

which implies(a+dd)du =2db> 0 (i)

and

(a+da)(l+dbf^w+(1+db)[l-(a+da){\\ +db)]-fw =2(a+da)

which implies(l+db)-Jw =2(a+da) (ii)

Sinceda, db, and du are small, equation (i) implies that 2db~ adu > 0.Inaddition, at b =0 we have Vw/2 =a, so that equations(i) and (ii) togetherimply that

2da=db^w> 0

which establishesthe result.

Itoh'sanalysis thus highlights another reason why in multiagentsituations it may beoptimal to basean agent'scompensationalsoon the otheragents'performance.Thisapproachis meant to inducecooperationamongagents.By giving individual agentsa stake in other agents'output, it ispossible to elicitmorecooperation.But this may comeat the costofmorerisk

exposurefor individual agents.One difficulty with this basiclogicis that it

might predictmuch more cooperationthan we seein reality. It is at this

point that Itoh'semphasison the potential cost in terms of lackofspecialization (when k > 0) becomesrelevant.When there are gains tospecialization,

then it is worthwhile to encouragecooperationonly when it makesasignificant difference.

Note that when a common shockaffectsindividual agents'outputs, then

it is no longerobvious that inducing cooperationnecessarilyimplies that

agentshave a greaterrisk exposure.In the presenceof common shocksitmay, however, bemoreexpensiveto elicitcooperationif it meansabstaining

from arelative-performance-evaluationincentive schemethat niters out

the common shock.Thenext sectionconsiderscooperationin the presenceof common shocksbut in a somewhat different setting.

Static Multilateral Contracting

Cooperationand Collusionamong Agents

The secondapproachto cooperationamong agents (taken in Holmstromand Milgrom, 1990,Varian, 1990,Macho-Stadlerand Perez-Castrillo,1993,Ramakrishnan and Thakor,1991,and Itoh, 1993)doesnot allow agentsto

help each other directly to accomplishtheir tasks, but lets them signcontracts to coordinatetheir actions.The important and obvious questionin

this setup is,Why doesthe principal gain by letting agentsjointly determinetheir effort supply through contracts?Pursuing the logicof the revelationprinciple,why can the principal not do as well or better by contracting

directly with the agents?What doeshe gain by allowing for a stageofindirect contracting?

The answer to these questionsproposedin the literature rests on the

generalidea that when agentsmay be able to observeactionchoicesthat

the principal cannot see,then theremay begains to letting agentswrite sidecontractsthat coordinatetheir actionchoices.Interestingly, however, thesegains arenot availableif agents'individual outputs are affectedby commonshocks,for then it is bestto filter out the common shockthrough a relative

performanceincentive scheme,which inducesagentsto compete.Cooperation would undermine competitionunder sucha schemeand thus preventthe filtering out of the common shock.

We follow here the analysis of Holmstrom and Milgrom (1990),andconsiderthe situation involving two risk-averseagents, each producing arandom output

qt =al+siby supplying effort at. Supposealso that the random variables e, arenormally distributed with meanzero and variance-covariancematrix

<7i2 C\\ J

sothat the correlationcoefficientbetweenthe two variablesis p= Oi2/(oi<72)-

Finally, as is by now familiar, supposealso that the two agents haveCARA risk preferenceswith coefficientrjt representedby the negativeexponentialutility function

ui(wl,al) =-e-71'[w'-y,M]

where y/^ai) is the strictly increasing,convex,monetary-cost-of-effortfunction.

332 Multiagent Moral Hazard and Collusion

We shall considerin turn the contracting situations with no cooperation,where the principal contracts with each agent separately and agentsrespondby playing a noncooperativegamein actionchoices(no sidecontracting), and the situation with cooperation,wherethe principal contractswith the team of agentsthat contractually coordinatetheir actionchoices{full side contracting). In each situation we restrict attention to linearcontractsof the form:

wx = zx+vxqx+uxq2

w2 =z2+v2q2+u2qx

8.2.2.1No SideContracting

Underlinear incentive contractswe obtain the simple formula for the

agents'certainty equivalent wealths:

CEX (ax,a2)=zi+v1a1+uxa2 -yx(ax)--\302\261- (vfai2 +uxa\\ +2vxuxox2)

and

CE2(ax,a2)=z2+v2a2+u2ax-y/2 (a2)-~(v2o2 +u\\o\\ +2v2u2oX2)

A risk-neutral principal, contracting separatelywith each agent, then

chooses{z\342\200\236

V,-, u] (i =1,2)to maximize expectednet profits:

(1-Vj-u2)ax +(1-ux

-v2)a2-zx-z2subjectto the incentive constraints that the actionchoicesof the two agents(ax, a2)form a Nash equilibrium in effort levels:

at =argmax CEt {a{\342\226\240, a]) i =1,2,;'=2,1and the individual-rationality constraints

CEiia^a^O i =l,2,;=2,lAs is by now familiar, the equilibrium action choicesat are tied down

entirely by the agent'sshare in own output v; through the first-ordercondition of the agent'sproblem:

vt=yf,M i =l,2

333 Static Multilateral Contracting

Consequently, the sharesut in the other agent's output should be set tominimize risk exposurefor eachagent.That is

Oi ...ut =-vt \342\200\224 p for i &]

As we have already noted, when outputs are positively correlated(p > 0), the optimal linear contractpenalizeseach agent for a goodperformance by the other agent.

Substituting for ut in the agents'certainty-equivalent wealth formulas, wefind that the total risk exposureof the two agentsfor a given own

incentive (vi, v2) is

27fc[v?ff?(l-p2)] (8.3)

We shall comparethis expressionwith the total risk exposureunder full

sidecontracting at the same(au a2).

8.2.2.2Full SideContracting

The type of team the principal faceswhen he allows for sidecontractingbetweenthe two agentsbefore signing on the team of agentsdependsonwhat the agentsthemselvescan contracton.The literature considerstwo

situations: one,wherethe agentsdo not observeeachother'seffort choicesand can contractonly on output; and the other,where they can alsocontractually specifytheir actions.We now considereachone in turn and

highlight the following general observations,derived by Holmstrom and

Milgrom (1990):1.Ifthe agents can contract only on their respectiveoutput levels, then sidecontracting by the agents can only make the principalworseoff.1

Herethe agentsdonot bring any new contracting possibilitiesto the

principal. Theycan only use their sidecontracting at the expenseof the

principal. Indeed,note that all the agentscan do here under sidecontracting isto specifya transfer (0gi+%q2) from agent 1to agent 2.Theprincipal canalways undo any such transfer by resetting

1.Seealso Varian (1990).

334 Multiagent Moral Hazard and Collusion

Vi tO Vi -0

u2 to u2 +0

\302\253ito ux -X

and

v2 to v2 +x \342\226\240

and restricthimself to \"coalition-proofcontracts.\" But this approachforcesthe principal into a more restrictedproblem than the no-side-contractingproblem.Therefore,by letting agentswrite such sidecontractshe may bemadeworseoff.We will seethis samepoint again in the next sectionwhen

we discusssidecontracting betweenan agent and a supervisor.2.When the two agentscan contracton both their effort levelsand their

respectiveoutputs, then when they are facedwith a given incentive scheme(v;, u) for i =1,2, they respondby maximizing their joint surplus:

max(v! +u2)ax +(v2 +ux )a2-Yi (ai)-V2 (a2)to-Y[(vi

-0)2o\\ +(wi -xf<yl+2(vi -0Xki -Z)<7i2 ]

-~[(v2+xfol+(\"2 +<t>fol +2(v2+x)(u2 +<f)ol2]

As can beseenfrom this problem,a given incentive scheme(v,-,u) by the

principal then maps into a pair of effort levelsand a risk-sharing scheme:(0,Z> fli> ai)-Sincewe have transferableutility, we can think of the

principalas maximizing total surplus, optimizing over (v;, ut), i =1,2,and

anticipating the (0, %, au a2) that follows.This formulation of the principal'sproblemunder this form ofsidecontracting brings out the following useful

observation:

Theprincipal'soptimal contract is the samewhether the agents cancontract only on (ah a2) or on both (ah a^ qi, g2/).

This insight can be understoodintuitively as follows:For a fixed actionpair (au a2) the two agentsdo not add any new risk-sharing opportunitiesthrough their sidecontract.Therefore,once the agentshave contractedon

(ai,a2) they cannot improve the contracting outcomeby alsowriting a sidecontracton (qu q2).3.Thisobservationleadsto the next useful remark:

335 Static Multilateral Contracting

Theprincipal'soptimal contracting problemunder sidecontracting with

the two agents can be reducedto a single-agentcontracting problem,wherehe facesa single agent with coefficientof risk aversion

1/77 =1/77!+l/r72

supplying effort (ah a2) with cost-of-effortfunction1

\\ir{al,a1) =yl{al)+y2{a1)

To establishthis claim, we need to do a little algebra and derive the

optimal (0,x) for a given incentive scheme (v,-, ut), i = 1,2.We can then

computethe optimal total risk exposureof the two agents.Once the agents' supply of effort correspondingto the incentive (v,-,ut)

has beenfixed in the sidecontract,the sidetransfer (0gi+%q2) can be setto minimize the agents' total risk exposure.Then,

d(CE1+CE2)_Q

implies

*7i[(vi-

0)<7i2 +(ki -x)<?u]=ri2[(u2+(l>)<jf+(v2+^)<7i2] (8.4)and

d(CE1+CE2)_Qd%

implies

*fc[(\"i ~X)ol+(vi -0)<Ti2]=T72[(v2+%)ol+(u2+0)<7i2] (8.5)

Multiplying both sidesof equation(8.5)by <7i2/<72 and subtracting it from

equation(8.4)then yields

?7i(vi-0)=T72(w2+0)

Similarly, multiplying equation(8.5)by o\\loi2 and subtracting it fromequation (8.4)yields

li(u1-x)=ri2(v2+x)

Both conditionsthen imply

2.This result is due to Wilson (1968).

336 Multiagent Moral Hazard and Collusion

T]lVl-T]2U20=

771+772

^T]lUl-T]2V2

and therefore

Vi-0= (V1+M2)77i+r72

U\\-% =\342\200\224(\302\253i+v2)

771+772

v2+Z =\"\342\200\224(\302\253i+v2)

771+772

u2+(b =\342\200\224(vi+a2)

771+772

Thus the optimal total risk exposurefor the two agentsfor a givenincentive

(v\342\200\236 Ui) is

-\342\200\2247(772 (vi +u2fol+T]Kui +v2fol +2t]Kvi +u2){ul +v2)<7i2]2(7]!+772)

J

772

2(77!+172)2\\rfi{ux +v2)2<7| +r712(v1 +w2)2<7i2 +2r\\Kui +Vi){vi +\"2)0-12]

which isequal to

I[ML\302\261ML ]{(vi+M2)2a2+(Ml +V2)2022+2(\302\253! +V2XV! +^2)^2} (8.6)ZV (771+772) J

Finally, we can then define

tm!+772771 77^2 ( . 1 1 1>=T] since \342\200\224 = \342\200\224 +\342\200\224

V 77 Th. ti2J(771+772)2 ^+T72'

^ V Vi 772.

Therefore,the total risk exposurefor the two agentscan be reinterpretedas that of a single agent with absoluterisk aversionT], facing a contract(Vi + U2,Mi + V2).

4. With this simplification in hand, we are, at last, able to coiifirm our

337 Static Multilateral Contracting

central conjecture,derived by Holmstrom and Milgrom (1990)and alsoRamakrishnan and Thakor(1991):Full sidecontracting dominates no sidecontracting if and only if p < p.To establishthis result, observethat to implement the samepair of effortlevels(al5 a2)under both contracting regimes(with and without sidecontracting) we must have the samepair of incentive levels(v1;v2) under both

regimes.Undersidecontracting, in addition, we have ux = u2 =0 and (0,%)

set,as before,to miriimize risk exposure.Comparing the respectiveformulas for the risk exposuresunder no sidecontracting and sidecontracting onactions,equations(8.3)and (8.6),we then observethat (1)the riskexposure (8.3)starts above(8.6)for p = 0, and is equal to 0 for p =1;and (2)the risk exposure(8.6)is below(8.3)at p=0,but is positive and grows with

p.The result then follows from these two observations.The intuition for this result is as follows:At p = 0 cooperationbetween

agentscan only be good:it simply means that they work harder by

monitoring eachother.For p> 0,however, cooperationalsoundermines relative

performanceevaluation, which resultsin a higher costfor the principal the

higher the correlationp.We end this sectionon a note of caution made by Itoh (1993)about the

approachtaken here to the problemofcooperationamong agents.As iswellknown from mechanism design,when one allows for moregeneralmechanisms than we have here, the principal ought to be able to always do(weakly) better than letting agentscolludewith a sidecontract,by elicitingthe hidden information about agents'effort levelsdirectly through reportsfrom individual agents.The principal could,for example,elicitthe agents'information in an efficient,incentive-compatibleway by designing aMaskin-

type messagegame (we review Maskin schemesin Chapter 12).Oneexampleof a model taking this approachis that of Ma (1988).The theoryof collusiondiscussedpreviously thus seemsto be vulnerable, as thereappearto beoptimal contractual solutions that donot involve any collusion.

However,Maskin schemesare rarely observedin reality. An obvious

question then is, What makes them nonoperationalin reality? The onlylimited answer we shall offer for now is that the Maskin messagegameswould be powerlessif the agents could also collude in sending reportsand if utility were transferable.The reason is that, as the only goalof the

principal here is to extract informational rents from the agents througha messagegame, a coordinatedaction by all agentsought to be able to

338 Multiagent Moral Hazard and Collusion

defeat that purpose,as in the modelof collusionofTirole(1986)to which

we turn next.

8.3Supervision and Collusion

One important costof having agentsmonitor eachother,which we did notconsiderin the previoussection,is that monitoring distractsagentsfromtheir productionactivities.If there are returns to specialization,then it maybemore efficient to have someagentsspecializein monitoring orsupervision activities while others specializein production.Indeed,manyorganizations are structured this way in practice.However,as our discussiononcollusionamong agentsin the previoussectionindicates,the efficiencyofmonitoring and supervision by specializedmonitors may beconstrainedbythe possibility of collusionbetweenthe supervisorand the supervisee.Inthis sectionwe turn to the analysis of collusionin vertical organizational

structures, which in their simplest form involve one principal, onesupervisor,

and one agent.Suchvertical structures are ubiquitous, and collusionin these structures

isa fundamental concern.Whether in auditing, tax collection,law

enforcement, or regulation, a major concernis that monitoring activities areweakened or subvertedby collusionbetween the supervisorand the agent.Afundamental question concerningthese structures then is how theincentives of supervisorand agent ought to be structured to maximize theeffectiveness of monitoring activities and to niinimize the risk of collusion.Wenow turn to the analysis of this broadquestion.Tokeepthe analysis astransparent

as possible,we shall limit our discussionto situations where,as in

Chapter 5, the information about the agent producedby the monitor ishard, verifiable information. In such a setting, as in a typical auditing

problem,collusionbetweenthe supervisorand agent takes the particularly

simple form of suppressingdamaging information about the agent.

8.3.1Collusionwith Hard Information

The basicproblem we shall considerinvolves three risk-neutral parties:principal, supervisor,and agent.The supervisorobtains information aboutthe agent and providesit to the principal.The latter hires the agent and

supervisorunder a contractthat baseseach of their compensationon thesupervisor'sverifiable reports on the agent'sproductivity. An exampleof

339 Static Multilateral Contracting

such a situation is the shareholdersof a firm (collectivelyacting as the

principal) hiring an auditing firm to produce and certify the firm's annual

incomestatements, on which the firm manager'scompensationis based.Clearly, a concernshareholdersought to have in such a situation is that

the auditor and firm managermight colludeand \"manipulate\" the firm's

accounts,for exampleby underreporting lossesor instead by overstatingcosts.

Thissimple three-tiercontracting relationwas first analyzedin aninfluential paperby Tirole(1986).He allows for an extremeform of collusionbetweenthe auditor and the manager:collusionthat is enforceablethroughsidecontractsbetweenthe supervisorand agent, albeitat a higher costthan

normal contracts.In the presenceof such collusion,Tiroleshows that the

principal will designan optimal contractfor the two parties that iscollusion-proof.ThdX is,the principal canwithout lossofgenerality restrictattention to contracts that do not involve any collusionin equilibrium. Thisobservationis akin to the revelation principlediscussedin Chapter2,which

states that the principal can restrict attention to contracts that do notinvolve any lying by the agent under the contract.

One consequenceof restricting attention to collusion-proofcontractsisthat the principal will have to dull the agent'sincentives relative to asituation wherecollusionisnot possible,soas to reducethe agent'sincentivesto colludewith the auditor. As obvious as this observationis, it has

apparentlynot beentaken sufficient noteofby compensationcommitteesin large

U.S.firms that have granted such high-poweredincentives to their CEOsthat they have given them strong incentivesto manipulate reportedearnings. At the sametime, by letting accounting firms engagein both auditingand consulting activities, U.S.financial regulatorshave given these CEOsthe meansto \"bribe\" auditing firms with the prospectof lucrative

consultingcontracts.

We begin our analysis of this contracting problemby characterizing first

the outcomeunder no collusion.Thecontracting problembetweenthe

principaland agent takesthe following simple form: the principal buys a service

from the agent that he valuesat V > 1.The agent producesthis serviceat

(unobservable)costc e {0,1}.Theagent knows the value ofc,and the

principal's prior beliefsabout the agent'scostsare Pr(c= 0) =\\. The principal

canhire a monitor to get a better estimateof the agent'scost.Considertheextremecasewhere by paying a fee z to a supervisorthe principal canobtain exactproof of the agent's true costswith probability p when c = 0

340 Multiagent Moral Hazard and Collusion

and otherwiseno proofat all of the agent'scost[moreexplicitly, themonitor seesnothing with probability 1when c=1,and with probability (1-p)when c =0].When would the principal want to hire such a supervisor?

If the principal doesnot hire a supervisor,he facesa standard screeningproblemand offersthe agent a priceP=1for the serviceifV-l>V/2andP=0 otherwise.For the remainderof the analysis we shall take it that V >2,sothat the priceP=1is indeedoptimal. In that case,the principal'spayoffis simply (V -1).If the principal hires a monitor at cost z, and if there isno collusion,then, when c=0,the principal getsto seethe agent'scostwith

probability p and canofferthe agent a priceP=0.When he getsnoinformation from the supervisor,the principal now has even more reason tobelievethat the agent has high costs(c= 1)and then optimally offersthe

agent a priceP=1.This contractgives the principal a payoff of

\\pv*[i-\\P)y-T)-zWe shall assumethat z is small enough that

\302\261Pv+{i-\302\261Pyv-i)-z>v-i

In that casethe principal is better off hiring a supervisorto monitor the

agent.How is this contractaffectedby the possibility of collusionbetweenthe

monitor and agent?The supervisorand agent can gainfully colludeunderthis contractwhen the agent'scostis c=0 and the supervisorhas obtainedproofof the agent'slow costs.In that casethe agent will obtain only a priceP=0 if her costsare revealedby the supervisor,while if the supervisorpretends he has not obtained any proof the agent gets a priceP= 1for herservices.Supervisorand agent can then share a potential rent of 1by

suppressing the information.A self-interestedsupervisorcolludeswith the agent only if he benefits

from such behavior.Considerthe following collusiontechnology:if the

agent offers the supervisora transfer T for suppressinghis information,the supervisorobtains a benefit of kT, wherek < 1.The idea is that, sincethe agent's transfer to the supervisormust be hidden (through, say, somecomplexfinancial transactions in a tax haven on some Caribbeanisland),the supervisorendsup getting lessthan what the agent paid.

To avoid such collusion,the far-sighted principal has to promise thesupervisora reward w for proving that c = 0.This reward must be high

341 Static Multilateral Contracting

enough so that the following collusion-proofnessor coalition-incentive-compatibility constraint is satisfied:

w>k (CIC)When this constraint holds,the agent cannot gain by inducing the

supervisor to suppressinformation, sinceshe would have to pay the supervisormore than 1to get him to suppressthe information.

The principal would not incur higher expectedcostsfrom hiring the

supervisorif he can subtract from the supervisor'spay the amount \\pwwhen the supervisorreportsno information. Thenthe only consequenceforthe principal of the possibility of collusionbetween the supervisorand

agent is that he has to incentivize the supervisorby adding a \"bounty\" tohis pay for delivering valuable information. This is not a very surprising

finding. Nevertheless,it is worth pausing and pointing out that theremuneration of auditing firms in reality is far from resembling this structure.

Typically, auditing firms are not rewardedfor exposingaccountingirregularities. Instead,their incentives to performcomefrom the penaltiestheyface if they are found to colludewith the firm's manager.Unfortunately,these latter incentives are effectiveonly if the probability of detectionofcollusionis sufficiently high.

When, moreover,asTiroleassumes,the supervisorhas a limited wealth

constraint, sothat \\pw cannot beentirely subtractedfrom his pay when hedoesnot report any information, then collusionraisesthe principal'scostsofhiring a supervisorwhenever\\pk > z.In that case,the principal'spayoffwhen hiring a supervisorunder a collusion-proofcontractis

ip<y-k)+o.-ipW-i)Thissimple examplecapturesthe essenceofTirole'sanalysis.It highlights

in a stark way that the principal couldnot gain from allowing collusiontotake place in equilibrium. Collusioncouldoccurin equilibrium only if the

supervisor'slimited wealth constraint is binding, if k isa random variable,and if most of the time the realizationsof k are small. In other words,collusion couldoccurin equmbrium in his modelonly if it would not pay the

principal to attempt to deter the rare caseswherecollusionmight beprofitable for the supervisorand agent becausethey happento have accesstoa cheapenforcementtechnology. If, for example,the principal thinks that

k =\\ with probability 1-e and k = 1with probability e,where e is small,

Multiagent Moral Hazard and Collusion

then setting w =\\ is optimal, and equilibrium collusionoccurswith

probabilitye.

Our simple example also highlights how the possibility of collusionincreasesthe costsofmonitoring for the principal.Interestingly, in asomewhat richerexampleit canbe shown that when it is costly for the principalto pay a reward w to the supervisor,it may pay the principal to slightly dull

the agent'sincentives soas to reducethe collusionrent betweenthesupervisor and agent.We leave it to the readerto extend our simple setup toallow for this possibility.

Application: Auditing

Tirole'sbasic three-tier structure with collusionhas been applied to an

auditing problemand extendedfurther by Kofman and Lawarree(1993).Theirstarting point is the observationthat, in reality, firms and their

external auditors are themselvessubjectto random audits by supervisoryauthorities. The basictheory of collusionwe have just outlined suggeststhat the

supervisory authorities, or external auditors (as Kofman and Lawarreedenotethem), canserveasa controlmechanism not only on the firm's

managers but alsoon their auditors.Kofman and Lawarreeconsiderthe situation whereexternalauditors are

more expensivethan internal onesbut are immune to collusion.This is aplausibledescriptionof financial regulation in the United States,whereSECstaff are reputed to be incorruptible but resourcesare so stretchedthat SECstaff can investigate only a small fraction of listed firms at any

time (this fraction has at times beenso small that SECsupervision hassometimesbeenreferred to as the SECroulette).

Kofman and Lawarree'smodelis built on the following simple principal-agent structure:

A risk-neutral managerproducesoutput (or profits) with the following

technology:

q =9+a

where9e {&i,$2}is the firm's type, with 0i<&2and Pr(^)= \\. Themanagerhas a convexcost-of-effortfunction given by yf(a)

=a2/2.A risk-neutral

principal contractswith the agent.The principal'sfirst-best problem,when

he can observeboth 6 and a, is thus to choosetype-contingent transfers Ttand actionsa, to maximize his expectedprofit:

343 Static Multilateral Contracting

max-(0L+fl!-7i)+-(02+fl2 -Ti)a\342\200\236T, 2 2

subjectto

7]>-y fort =1,2 (MIRt)

where (MIRt) denotesthe individual rationality constraint of the managerwhen her firm is of type i.The solution to this problemis clearlyto set at =1and Ti= \\.

Theprincipal'ssecond-bestproblem(when 0,is private information), in

the absenceof any supervision, is to maximize the samepayoff but with anadditional incentive constraint:

T2_\302\243.iTijw{o,<h-m)2 (MIC2)

(As we know from Chapter2,only one of the two incentive incentiveconstraints will bind at the optimum.) We alsoknow from Chapter2 that thesecond-bestcontractspecifiesefficient effort provision for type fy, under-provision of effort for type &i, and positiveinformational rents for type Oi

only.For simplicity, supposethat Ad = fy

-&i = 1.Sinceeffort ax is no greater

than 1,this assumption means that type &z is able to produce qx at zeroeffort.Using this observation,the incentive constraint canbewritten in the

simpler form

atT2\342\200\224y->7i

(MIC2)

Combining constraints (MIR1)and (MIC2),substituting for T, in the

objective,and maximizing with respectto aL then yields the second-bestsolution ax =

\\ and a2=1.8.3.2.1Supervision

Howcan auditing improve on this outcome?Kofman and Lawarreemodelauditing by assuming that the auditor observesan imperfectsignal y, ofmanagerial productivity. As before,we assumethat this signal is hardinformation. This assumption simplifies the auditing problem,in particular by

ruling out the possibility that the auditor blackmails or extortsthe manager

344 Multiagent Moral Hazard and Collusion

by threatening to fabricatedamaging reports.As in Tirole'sproblem,the

only concernwith auditing is the possibility ofcollusionbetweenthe auditor

and the manager.Kofrnan and Lawarreeallow for both type-1and type-2errorsand assumethat the signals observedby the auditor are such that

Pr(yJ|0J) =C>i>l-C=Pr(y/l^)

Again, to avoid any complicating issuerelating to the threat ofextortion,assumethat both the manager and the auditor(s)observethe signal yt.

We now turn to the interaction of internal and externalauditors.Toseehow the threat of an externalaudit can be usedto reduce the risk ofcollusion betweenthe internal auditor and the manager, it is helpful to proceedin three steps.Thesestepsalsoallow us to coverother importantcontributions on optimal incentive contracting with auditing on which Kofrnan andLawarree'sanalysis builds. In a first step we analyze the optimalcontracting problembetween the manager, an honest-but-costlyauditor, and the

principal.In a secondstep we consider optimal contracting between acheap-but-corruptibleauditor, the manager, and the principal.Finally, in athird stepwe considerthe optimal contracting problemwhen both types ofauditors are combined.

Honest-but-CostlySupervisor

The contracting problem involving an honest-but-costlysupervisor issimilar to the auditing problemin Baron and Besanko(1984).It involvesthe following four stages:

1.Themanager learnsher firm's type 9t.

2.Theprincipal offerscontractsto the managerand auditor.

3.Themanager optimizesher payoff under thesecontractsby choosingherhidden action.4. The contract the principal has signed with the auditor specifiesan

output-contingent audit probability and triggers a stochasticaudit afterrealizedprofits qt are observed.

As in Tirole'ssetup,the objectivefor the principal is to reduce theinformational rent of the efficient manager.This can be achievedwith apunishment to be imposedon the efficient managerif she is found under-providing effort.That is, the optimal contract imposesa penalty on themanager when output is low, but the signal receivedby the auditor is theone that is positively correlatedwith the efficient type.

345 Static Multilateral Contracting

Let /bethe probability of an audit when output is low, and let K denotethe penalty imposedon the managerwhen signal y2 is observed.Let thecost of sending the supervisorbe z as before, and let K* denote themaximum penalty the principal can imposeon the manager.

Theprincipal'scontracting problemthen takes the form

max-{el+al-Tl+y[0.-QK-zb+-{e2+a2-T2)a,F\342\200\236YJC 2 2

subjectto

7i-r(l-0K>y (MIR1)

T2~>y (MIR2)

T2--|->71-r^ (MIC2)

In comparisonwith the second-bestproblemwithout supervision, threenew effectsarise:First, the principal facesnew auditing costs:sending theauditor with an ex ante probability y/2 costsyz/2.Second,auditing comeswith a type-1error:the inefficient manageris wrongly punished with

probability (1- \302\243).As the manager is risk neutral, this effectwashesout: it

involves a reward for the principal that is exactlycompensatedby a rise in

the transfer 7\\. Third, supervision reducesthe information rent of theefficient manager, who is punished more often than the inefficient one if shechoosesto producelow output. Toseethis result, note that when constraints(MIR1)and (MIC2)are combined,then

2 2

r2 -y *y -7(2\302\243

-W (MIC2')

Interestingly, when the penalty K is high enough, the secondindividual-

rationality constraint (MIR2)may alsobe binding. When this is the case,the efficient manager'sinformational rent will have beeneliminated.

Another straightforward, but neverthelessimportant, observationis that

the benefit of an audit riseswith K, so that maximum deterrenceis optimal.In otherwords, it is optimal to setK = K*.Thisobservationdatesback atleast to the early economicanalysesof crime by Becker(1968).In his

original contribution Beckershowedthat maximum deterrenceis efficient

346 Multiagent Moral Hazard and Collusion

when there are no type-1errors,as it minimizes the probability ofapprehension ofcriminals neededto deter crime,and thereforelaw enforcementcosts.As the analysis hereindicates,this logicextendsto the situation whereauditors may make type-1errors.3As is intuitive, the benefit of maximumdeterrencealsorisesin

\302\243,

the accuracyof the audit signal.The overalltrade-offfacing the principal involves rent extractionfrom

the efficient manager versus the audit costz.As canbe easilyverified, the

optimal contractfor a given costz has the following features:

1.When the signal has low accuracyand/or the maximum punishment ofthe manageris low, there is no audit at all and ax =

\\.

2.When the signal has somewhat higher accuracyand/or the maximum

punishment of the manager is higher, an audit is worthwhile. Giventhe

linearity of the contracting problem,the audit takesplacewith probability 1when a low output is observed.The audit reducesthe informational rent ofthe efficientmanager, but as long asshestill retainspositive rents, the effortchoiceof the inefficient managerremains at ~.3.When the accuracyof the signal \302\243

is very high or the punishment K* is

large,the efficient manager'sinformational rent vanishes [and (MIR2)is

binding]. At that point the inefficient managergradually raiseseffort fliupto the first-best level(as \302\243or

K* increases).Thereasonis that it takeslower-effort distortions to reducethe informational rent of the efficient manager.4. Once the first-best effort has beenreachedfor the inefficient manager,further increasesin signal accuracyor punishment make it possibleto start

reducing the probability yof an audit.

A number of important insights for the regulation of audits arecontainedin this simple analysis.In particular, this analysis highlights how the optimalfrequency of audits dependsin a nonmonotonic way on the accuracyofaudits and the levelof punishments for accounting fraud. We now turn tothe analysis of optimal audits in the presenceof collusion.

Cheap-but-CorruptibleAuditor

The contracting situation involving a cheap-but-corruptibleauditor is quitesimilar to Tirole'ssetup.It differs from the previousone in two respects:

3. Interestingly, maximum deterrence may even be efficient in the presence of type-1 errorswhen the agent is risk averse.The reason is that maximum deterrence may actually reduce the

agent's overall risk exposure if it results in a sufficiently low probability of audit (seeBolton,1987).

347 Static Multilateral Contracting

First, collusioncannow take placebetweenthe supervisorand the manageroncelow output has beenchosenand the \"high-productivity signal\" y2 hasbeenobserved.As in Tirole'smodel,we let the two partieswrite

enforceable sidecontractsthat maximize their joint payoff.Toavoid collusion,the

principal must again reward the auditor for revealing incriminatingevidence on the agent.Let w be the reward the auditor obtains when

revealingthe signal y2 to the principal.This reward must then besetto satisfy the

collusion-proofnessconstraint:

w>K (CIC)The seconddifferencefrom the previousproblem is that now audits

comefor free,so that the principal pays no costz.However,if we assumethat the auditor is resourceconstrainedand thus cannot be punished forfailing to revealthe signal y2, the principal still facesan expectedaudit costof yw(l -C)I2.That is, as before, the audit takes place only when qx isobserved(which happenswith probability \\ in equilibrium), and then theaudit is triggeredonly with probability y. In addition, the auditor then getsthe signal y2 only with probability (1-Q.

The contracting problemthe principal facesis thus similar to theprevious one,except that now w(l-Q replacesz in the principal'sobjectivefunction, and an additional constraint (CIC)is imposedon the principal.

Note that the principal's audit cost is now increasingin the agent'spunishment K.Thus the principal now faces the following trade-offbetween audit costsand informational rent extraction:in expectedterms, theauditor gets

which results in a reductionof the manager'srent, again in expectedterms,

by

y(2C-i)fas can be verified from constraint (MIC2').Therefore,the principal now

callsan audit if and only if2\302\243

-1> 1-\302\243,

or\302\243

> ^.When the accuracyof the signal is sufficiently high that this condition

holds, there will bean audit when qx is observedand, asbefore(and as canbe easilyverified), the optimal outcomewill dependon the value of themaximum punishment K* that canbe imposedon the agent.

348 Multiagent Moral Hazard and Collusion

If K* is small, effort provision by both types of agentswill be the sameaswithout monitoring, but the manager'sinformation rent will bereduced.For higher values of K*, the informational rent vanishes, and the low-productivity manager gradually raisesher effort level up to its first-bestlevel.

One fundamental differencefrom the previousproblem,however, is that

maximum deterrenceceasesto be strictly optimal here.When the efficientmanager'sinformational rent hasbeeneliminated and first-besteffort levelshave beenreached,a further rise in the punishment K brings no additionalbenefits, as the rise in the punishment is entirely offsetby a rise in theauditor'sreward,w. In the previousproblemthis was not the case.Indeed,the probability ofa costly audit, y couldbereducedwhile yK was keptconstant. Here,instead,the rent of the collusivesupervisordependson yK, andthis costdoesnot decreasewith K. It is for this reason that adding anadditional honest auditor can help, even if he is costly.The honest auditor canplay a useful role in monitoring collusionand thereby helping reduce thereward that must begiven to the corruptibleauditor.

Once again, this simple contracting problemhas yieldedan important

insight\342\200\224that Becker'sprincipleof maximum deterrence emphasizedsomuch in the law and economicsliterature breaksdown when the monitorand agent cancollude.

Cheap-but-Corruptibleand Honest-but-CostlySupervisors

Finally, considerthe contracting problemwhereboth types ofauditors maybe hired.Again, an audit takes placehere only when output qx has beenobserved.This problemis somewhat more complex,as the principal has achoicebetweena number of different audit patterns.We needto introducenew notation to describethis problem.

As before,y denotesthe probability of a (cheap-but-corruptible)internal audit. Let Kt denote the manager's punishment when the internal

auditor revealssignal y2, let (p denote the probability of an (honest-but-costly) external audit instead of an internal one, and let Kedenote themanager'spunishment when the externalauditor revealssignal y2. Finally,

let v denote the probability of an external audit when the internal audit

results in a reportedsignal yx. In this event, if the reported signal yx is dueto collusionbetweenthe internal auditor and the agent, then collusionisdetectedfor sure by the externalauditor and the respectivepunishments

Kie and St are imposedon the managerand the internal auditor.

Most of the difficulty in analyzing this contracting situation lies in thedefinition of the new contracting variables.Oncethesehave beendefined,the principal'sproblemcanbe straightforwardly setup in the usual form:

max ifo+ax-Tx+y[(l-Q(Kt-w)-^vz]+(l-y)(p[(l-OKe-z\\)

+^(e2+a2-T2)

subjectto

ri-a-0fo(l-7)tf.+l*itey (MIR1)

T2>y (MIR2)

T2-|->71-acpO--r)Ke+yKi] (MIC2)

and

w^Ki-v^Kie+Si) (CIC)Note that the probability of an audit when output qx is realizedis now

[y+ (p(l -y)] insteadof y As before,the efficient manager must becompensated for being unfairly punished in equilibrium with probability

(1- \302\243).This observationexplainsthe form of the constraint (MIR1).

Therearenow two potential benefitsofexternalaudits.Besidesthe usual

benefit of reducing managerial rents, there is the addedbenefit ofdeterringor reducing the benefit of collusionbetweenthe internal auditor and the

manager.Next, note that maximum deterrence continues to be optimal under

external audits, so that Ke = Kie = K* and St = ST,where S?denotes themaximum punishment that canbe imposedon the internal auditor.

What do we learn from this problem about the optimal frequency ofexternalaudits, that is, about v and (pi Considerfirst the optimal choiceofv. When the internal auditor starts with no resources,we can rewrite the

collusion-proofnessconstraint as

w =max{0,Kt -v(K*+$)} (CIC)Given that the principal already relieson an internal audit, the net

expectedbenefit of an externalaudit policy v is then

350 Multiagent Moral Hazard and Collusion

(l-0(X,-w)-C\302\253z=(1-0mm{Ki ,v(K*+ Sf)}-tpz=min{(l-0^-\302\243vz,v[(l-0(K*+ Sf)-&]} (8.7)Thus the choiceof v, which has no impact on the individual-rationality

or incentive constraints of the manager, is driven by the followingconsiderations: Either

(l-O(K*+S*)-&<0in which caseit is optimal to set v = 0, or

and equation(8.7)implies that the optimal v is given by

Ktv =

K*+S*

For this value of v, in turn, it is optimal to setKt =K*, which meansthat

w =0:wheneveran externalaudit takesplacewith positive probability, thecostof the internal audit is reducedto zero (and thereforey= 1is optimal).Since,moreover,externalaudits do not occurwith probability onewhen qxis realizedand an internal audit is triggered,this result implies that theoverallaudit costsfor the principal are reducedcomparedwith thesituation wherehe couldrely only on the externalauditor. Note that this

analysis alsoimplies that q>=0 is optimal; that is, it is inefficient for the principal

to rely only on a costly externalaudit.

The Kofrnan and Lawarreeanalysis thus rationalizesthe simultaneoususeof internal and externalauditing, whereinternal audits take placeon acontinuing basis,while externalaudits occuronly intermittently on a randombasis.This applicationhighlights the richnessof the principal-supervisor-agent framework when potential collusionis introduced.

Note,however, that this approachleavesopen the issueof theenforcement of sidecontracts.Severalattempts have beenmade to addressthis

issue.First, in Tirole (1992)and Martimort (1999)collusionis possiblethrough self-enforcingside contracts when auditor and agent interact

repeatedlyover time. An interesting implication of that analysis is that the

principal may gain by forcing frequent rotation ofauditors to preservetheir

independenceand undermine collusion.Regular rotation of auditors hasbeenproposedasa new regulatory requirement in the United Statesin the

351 Static Multilateral Contracting

wake of the accounting scandalsof 2001and is mandated in Italy. Second,in Leppamaki(1998)and Laffont and Meleu (1997)collusionis modeledas an exchangeof favors, with the supervisorsuppressingdamaginginformation about the agent in exchangefor the agent suppressingharmful

information about the supervisor.

8.4 Hierarchies

Much of the theory of organizations coveredsofar in this chapteris mostlyapplicableto small-scalefirms. We now addressoneof the most difficult andleast well understood questions in economics:What accountsfor largeorganizations, firms with more than 100,000employees,say?What is their

role?Why do they exist?And how should they be organizedinternally?Therepresentativefirm in micro-or macroeconomictextbooksis still the

small businessof the preindustrialization era.Yet, as many economichistorians have documented(most notably Chandler,1962,1977,1990),muchof the economicdevelopmentof the industrialization and postindustrial-ization erashas beendriven by large-scaleorganizations, such as the largerailroadsof the 19th century (somewith more than 100,000employees)orlarge automobile manufacturers like General Motors (with more than

700,000employees).Despitethe enormously important role these organizations play, formal

economictheoriesoflargefirms and their internal organization are still fewand far between.In this sectionwe covertheoriesofhierarchiesand large-scalefirms that build on the basicmultiagent moral-hazardparadigmdiscussed in this chapter.In our summary of the chapterwe shall alsopoint tothe other recent formal theoriesof hierarchies,which are not basedon abasicagency relation.

We shall beconcernedwith the following questions:Why do hierarchiesexist?How are efficient hierarchicalorganizations designed?What

determines the number of layers (or tiers) in a hierarchy, the pay structure, and

employees'incentives along the hierarchicalladder?Much of this sectionbuilds on the model by Qian (1994),which itself

integratesseveralearliermodelsby Williamson (1967),Beckmann (1977),Rosen (1982),and, most importantly, Calvoand Wellisz (1978,1979).TheCalvoand Wellisz approachto hierarchiesbuilds on the basicefficiency-wagemodel discussedin Chapter 4 and adds monitoring by hierarchicalsupervisors.The modelsby Williamson and others formalize the notion of

352 Multiagent Moral Hazard and Collusion

lossof controlin largemultilayer firms\342\200\224that is, the idea that monitoringand management of employeesbecomesmore difficult as the number oflayersthat separate the manager from his employeesbecomeshigher. Asweshall see,when oneaddsthe notion of lossofcontrolto a modelofhierarchies a la Calvoand Wellisz, as Qiandoes,one can formulate a theory ofthe optimal sizeand internal organization of firms.

Considera hierarchy composedof M layersor tiers.For notationalconvenience, our convention will be to denote the highest tier, occupiedby the

principal, as tier number 0,and the bottom layer ofworkersas tier numberM. For simplicity, we shall also ignore potential integer problems andassumethat the identicalnumber ofsubordinatesfor any given managerin

any given tier (obtainedby dividing the number of employeesin tier / +1by the number ofemployeesin tier /) is always an integernumber. We shallrefer to this number as the span of controlof the managersin a given tier.Thus, let xi be the number of agentsin tier / and m/+1 the spanof controlintier /; then the number ofagentsin tier / +1is given by xtmM. Also, in what

follows we assumethat there are N workersin the bottom layer:xM =N.With this notation in hand, let the total output of the hierarchy be

BNqM

where 6denotesthe profitability of the business,N the number ofproductive workers, or scaleof the business,and qM the effectiveoutput perproductive worker in an M-layeredhierarchy. Following Qian,we model lossof controlby assuming that

qi =atqt_i, whereat <1so that

qM =aMaM-l ''* #1^0

Thereis, thus, a lowerlossofcontrolthe higher are the a^s.In a first-bestworld the organization would have no lossof controlbecauseall the layerswould set at = 1.However,lossof controlarisesin a second-bestworld asthe variablesat are hidden actionschosenby each employeein layer /. Inthe second-bestproblemQian analyzesonly the principal facesnoincentive problemand setsa0=1.For all otherlayers,at<l,and an efficiency wagedeterminesthe levelof at as follows:Each (risk-neutral) employee'spayoffin layer / is given by a fixedwage,wt, which is paid if she is found not to beshirking, minus her costof effort y/(ai). Eachemployee'seffort provision is

353 Static Multilateral Contracting

monitored by her supervisorwith probability ph If the employeeis found

to be shirking, then she getsno compensation.Therefore,as we have seenin Chapter4,an employee'smaximum levelofeffort given the wagewt andthe probability of monitoring pt is determinedby the incentive constraint

wt -y/(ai)= (l-pi)wiwhere the RHSdenotes the employee'sexpectedpayoff when she shirksand setsa =0.

Theprobability ofmonitoring any given agent in layer /, in turn, isdetermined as follows-. When a supervisorhasmt agentsto monitor, shecanonly

superviseeachagent in her span of controlwith probability

1mt

In otherwords, the time a supervisorcanspendcheckingon a subordinateis inversely related to the number of subordinatesunder her control.Thisrelationis quite intuitive. Substituting for ph we obtain the followingfundamental equationlinking pay to span of controland performance:

wi =yf(fli)mi

From this equationwe can seethe fundamental trade-offthat shapestheform of the organization and the number of hierarchicallayers.Supposethere were only two layers, layer 0 occupiedby the principal and layer 1filled by N productive workers.Then,to achievea levelof effort aM for allthe workers, the principal might have to pay a largewage

wM =yr(aM)N

when N is large.The principal might then bebetter off setting up a three-layer hierarchy with an intermediatelayer of supervisors,eachwith a spanof control of mi workers.Setting up such an intermediate layer would

reduce the principal'sown span from N to N/mi and would increasethe

probability of inspectionof eachworkerfrom l/N to l/rai.The benefit ofsuch a deepeningof the organization is that now a levelofeffort aM for any

worker canbe elicitedwith a much lowerwageof

wM =yr(aM )ml

Thecostof this deepening,however,is that now the principal must alsopaya wagewx to N/mi supervisors.In addition, thesesupervisorsmust eachbe

354 Multiagent Moral Hazard and Collusion

incentivized to reducethe lossofcontrolresulting from the addition of theintermediate layer.Thus, if the principal wants to have no lossof controlwhatsoever,he needsto remuneratehis supervisorsat an efficiencywageof

mi

Thus the principal'sorganization-designproblemwhen moving from a two-to a three-layerhierarchy is to determinethe optimal spanofcontrolin themiddle layer and the optimal lossofcontrolat the two bottom layers.If hegets more supervisors,he can reduce his own span of control and thus

reduceboth the wagepersupervisorand the lossof controlin the middle

layer.However,he has to pay moresupervisors.4More generally, the principal's organization-designproblem for an

exogenousscaleN takes the following form:

Max\\ 6NqM -Y, \302\245(ai )mixi r

m{,ai,X{,M [ /=1 J

subjectto

X[ =Xi-\\Vfli

qt =qi-idi

x0 =q0=lxM =N

and

0<fl/<lfor all/

As is easyto see,this is a rather complexoptimization problem,involving

both integervariables(the number of tiers) and continuous variables.

4.Note that Qian specifies two separate activities in middle layers, supervision of employeeson the one hand and reduction of loss of control on the other. Moreover, only one of theseactivities involves an incentive problem. Supervision of employees in his model is a purelymechanical activity. Other, perhaps more natural, models specify a single supervisory activityin middle layers, which, however, involves an incentive problem. This is the approach taken inCalvo and Wellisz (1978),for example.The advantage of Qian's formulation, however, is thatit leads to a more tractable optimization problem.

355 Static Multilateral Contracting

Unfortunately, optimization problemsinvolving the designof hierarchiesare inherently difficult ones.This is, perhaps,an important reason why

hierarchieshave not beenstudied moreby economists.Progressincharacterizing optimal hierarchieshas been made in the literature either by

formulating a problemthat is sufficiently regular so that an explicitsolution to a differenceequationcanbefound (see,for example,Radner,1993)or, as Qian has chosen to do, by taking M, the number of layers in the

hierarchy, to bea continuous variable, so that the optimal hierarchy canbecharacterizedusing calculusof variations.5

Rather than attempt to providea more technically involved discussionof the solution to the generalcontinuous problemconsideredby Qian,weshall limit ourselveshere to a simple exampleof hierarchieswith only two

or three tiers (M =1or M =2).Considerfirst the caseof a two-tier hierarchy with M =1.Recallingthat

w = y/(a)N, which is paid to N agents, the principal'soptimization problemfor this hierarchy is fairly simple:

max{6Na-\\)/(a)N2}a

subjectto

0<a<lThefirst-ordercondition for this problemis given by

It is convenient to take the following functional form for the effort costfunction:

\\j/(a)=a3

For this functional form, the first-order condition yields the simpleexpression

5.As Van Zandt (1995)points out, however, formulating a continuous problem as Qian hasdone is not without conceptual problems. Heshows that, as much as continuousapproximations can bejustified with respect to xt, they are not necessarily valid with respect to M.

356 Multiagent Moral Hazard and Collusion

We shall take 6 to be low enough sothat the optimal a is always strictly lessthan one.6We can alsobackout the formula for the efficiencywage,

to observethat productivity perworker is decreasingwith the number ofworkers, while at the same time wagesare increasingin the number ofworkers.'

Consider now the three-tier hierarchy with M = 2.The principal'sproblemis now the much more complexproblem

max {6Na1a2-y(fli)>TCi*i-y/(a2)m2N}

ai,a2.,xi,mi,mz

subjectto

N =x1m2

jci =m1

fli<land

a2<lSubstituting for mx and m2, we obtain the unconstrained problem,

N2'max

\\6Na1a2 -v(\302\253i)^i

-y(a2)

which canbe solvedin the following three steps:First Step:Takeax and a2as Given, and Optimize with Respectto x\\. With

the functional form \\j/{a) = a] the first- and second-orderconditions with

respectto x\\ reduceto

a2\\\342\200\224

I =ai2x1

and

-all-lalN11'x\\<0

6. Notice that for simplicity we have suppressed the principal's own effort cost in settingoo=1. ^

357 Static Multilateral Contracting

From the first-ordercondition we thereforeobtain that

Xl=^LN^2-^ (8.8)

SecondStep:Substitute for the Value ofx\\ in Equation (8.8)into the

Objective Function and Maximize with Respectto ax. The reducedmaximandnow is

max {ONthOi-a^N^3(2~2/3+21/3)}

or

max {ONa^-a^KN413}01,02

where

?c=(2-2/3+21/3)= (6.75)1/3<2

From this maximand, it is easy to seethat the principal'sobjectiveislinear in alsso that ax = 1is optimal given that

eNa2-alKN*13

must bepositiveat the optimal a2.

Third Step:Substitute for ax =1in the Principal'sObjectiveand Solvefor a2.Theprincipal'sobjectivenow is

max {eNa2-alNAl3K}

subjectto

0<a2<land the first-ordercondition to this problemyieldsthe solution

Again, we shall take 6 to be small enough that a2<l.This procedureprovidesa completecharacterizationof the solution, and

we are now ready to comparethe two hierarchieswith M =1and M =2.A first observationis that optimal effort provision is lowerat lowertiers

in the hierarchy: a2< ax. That is, agentswork harder the higher up in the

358 Multiagent Moral Hazard and Collusion

hierarchy they are.This is a generalobservationthat is valid in the generalproblemconsideredby Qian.There is a fundamental economicreasonforthis result:the higher up in the hierarchy one is, the more worker outputsone'seffort entersinto. Sincesupervisors'effort entersmultiplicatively into

eachworker'soutput, the marginal return ofeachsupervisor'seffort ismagnified by the spanof controlof that supervisor,while the marginal costofthe supervisor'seffort is the same as for workers.It follows that

supervisors should work harder the higher up the ladder they are.A secondobservationis that the three-tierhierarchy (M = 2) is more

likely to dominate the two-tier hierarchy (M=1)when 6and/or N increases.Toseethis relation,note that, after substituting for the optimal efforts andcontrolspansin eachhierarchy, weobtain the following expressionsfor the

principal'sprofit under both hierarchies:

Na(6-a2N)=(2-3-z/2)6z/2N^2= 0.5e3l2N^2

for M =1,and

Na2(6-a2N1/3K) =(4k)\"162N2/3

= 0.1562N2/3

for M =2.Therefore,M =2 dominatesM =1when ^N116> 3.Again, this

observationis valid for the generalproblem.When the scaleof the firm

increases,it becomesprofitable to add more layers to the hierarchy. And

the depth of the hierarchy grows faster with scalewhen the overallproductivity of the firm is higher.

This latter observationis easy to understand when combinedwith thethird observation that in each hierarchy effort increaseswith 6 anddecreasesin N. This is intuitive and again a generalconclusion:When 6increases,the return on effort is higher, and since the cost-of-effortfunction is strictly convex,it pays more to introduce an intermediate layerat higher efforts to cut down on wage costsat lower layers.Similarly,an increasein N reducesthe levelof supervision in hierarchy M = 1and,as we have shown, results in a cut in effort. Consequently,a rise in Nincreasesthe gain of an intermediatelayer, to saveon wagesat the bottomlayer.

Our final observationabout wagesin hierarchy M =2 is that Wi increases

359 Static Multilateral Contracting

with both 6 and N, while vv2 increaseswith 6but decreaseswith N. Indeed,for the optimal three-tierhierarchy we have

Wi =v(fli)*i =*i =a2N2/32-V3 =-^-1/32-1/3

and

w2=^(fl2)\342\200\224=alN^2llz={-^\\N-^2^

The intuition for these results is that a rise in 6 makes higher effort

optimal at the bottom, which requiresa higher wagevv2 and thereforemoresupervision.But when adding more supervisorsthe principal'sspan ofcontrolrises,and to compensatefor the resulting supervision lossthe

principalmust pay higher wagesw1 to his supervisors.In contrast,while a higher

N resultsin a higher wagefor supervisors,to compensatefor thesupervision lossin the middle tier, it resultsin a lowerwageat the bottom layer assupervision at that level improves with the addition of more supervisors.Interestingly, therefore,when N increasesthere is an increase in wageinequality in the organization\342\200\224that is, a risein w1/w2.1

Tosum up, Qian'stheory as illustrated in this simple exampleyields many

empirically consistentpredictionsabout organizations:(1)The predictionthat effort decreaseswhen onegoesdown the hierarchicalladderis broadlyconsistentwith casualobservationon how hard employeesin large firms

work. (2)Thepredictionthat the optimal number of tiers of theorganization increaseswith profitability and scalealso appears to be in step with

reality. (3)The conclusionthat, for a given number of tiers, wagesincreaseas the profitability of the organization increasesis consistentwith reality,and more interestingly, the predictionthat wagesat the bottom layerdecreasewith scale,which at first sight seemscontradictory, alsoappears tobe borne out in reality. (4)The result that for the three-tierorganization(and more generally for organizations with more than three tiers) wageinequality within the organization increasesas its scaleof operationincreases is also in stepwith casualobservationon how pay scalesin firms

vary with size.

7. Note that when N > 4,we have w1 > wz as a2 <1.

360 Multiagent Moral Hazard and Collusion

Finally, an important result in Qian,which wehave not beenabletoillustrate in our simple example,is that, when the firm can optimize over thenumber ofworkersN, then in generalthere is a determinateoptimal finite

sizeof the firm, which is increasingin 6.Themain reasonwhy the optimalsizeof the firm is finite is that the principal has to incur a lossof controlashe addsmore layers.Thus Qian'smodel articulatesand extendsKaldor'stheory for why firms eventually facedecreasingreturns to scale\342\200\224because

they are unable to perfectlyduplicatethe scarcemanagerial input of the

top manager.As compelling as this explanationis,it unfortunately doesnot

fully resolvethe questionof the optimal sizeof firms. In Qian'smodellossof controlis assumedand not fully explained.In particular, Qian remainssilent on the questionofwhy firms would beunable to avoid lossof controlthrough selectiveintervention by top management. Tobeable to articulate

rigorousanswers to this fundamental question,weneedto moveto Part IV,

dealing with incompletecontractsand control.

8.5 Summary

In this chapterwe have introducedthe main approachesto the theory ofthe firm that are basedon a multiagent moral-hazardcontracting problem.We have seenhow the introduction of competitionbetween agents canimprove their incentive contractand how competitioncanbeundermined

by collusionamong agents,but alsohow competitionitself may undermine

cooperationamong agents.We have pursued the analysis of collusionfurther in the contextof a principal-supervisor-agentrelationand analyzedhow auditors'incentives must be structured to minimize collusionorcorruption. Finally, we have exploredhow a theory of firm size and internal

hierarchicalstructure can be built on the idea of monitoring and limitedattention.

The main insights of our analysis for the internal organization of firms

are the following:

\342\200\242 As Alchian and Demsetz(1972)first proposed,the main purposeofa firm

may be to mitigate incentive problemsarising in situations involving

multiple agents.A firm may be describedin simple terms as involving a bosswho supervisesworkers'effort provision, possiblyassistedby a group ofsupervisors.

361 Static Multilateral Contracting

\342\200\242 As Holmstrom (1982b)first pointed out, to maximize incentives for allteam membersit may be efficient for the firm to write a financial contractwith a third party, the budget breaker.Interestingly, this contractmayresemble a debt contract.Thus the theory ofmoral hazard in teamsmay provideyet another rationalefor debt financing.\342\200\242 When agents'individual performanceis observableit may be efficient tobasetheir compensationin part on their relativeperformance,as is donefor examplein tournaments, studied by Lazearand Rosen(1981)and Greenand Stokey(1983),in particular. As stressedby Holmstrom (1982b),themain advantage of relativeperformanceschemesis that they exposerisk-averseagentsto lessrisk when agents'individual performanceis subjecttocommon shocks.The strength of tournaments as a particular relative-performance-evaluationschemeis that they reducethe principal'sincentive to manipulate the outcomeex post,as he must pay out the prizes tosomewinners anyway

\342\200\242 The main drawback of relative-performance-evaluationschemesis that

they may fosterdestructive competitive behavior,such assabotageofotheragents'output, and generally undermine cooperationamong agents.Thus,as detailed,for example,by Holmstrom and Milgrom (1990),in situations

wherecooperationis likely to bring largebenefitsand whereagents'individual performanceis only subject to small common shocks,it may beoptimal to move away from relative-performanceschemesand insteadhave

agentsshare the returns of their actions.It may then evenbeworth lettingthe agentscollude.\342\200\242 In vertical principal-supervisor-agentstructures, however,it is generallynot optimal to let the agent colludewith the supervisor.As detailedbyTirole (1986),collusion-proofincentive schemesin such structures

generallyresult in dulled incentives for the agent, relative to situations where

collusionis not feasible,but enhanced incentives for the supervisor(orauditor).The analysis of optimal incentive contracting to prevent collusionin thesestructures yieldsa number of important insights on how incentivesof the accounting and auditing industry ought to be structured.

Theseare someof the main lessonsto be drawn from this chapter.Thereis a rapidly growing literature on organizations and the multiagent moral-hazard perspectiveon firms. We have coveredonly someof the foundational

studiesin this chapter.For a more wide-ranging treatment of the themesdiscussedhere we refer the reader to the book on the economicsof

362 Multiagent Moral Hazard and Collusion

organization by Milgrom and Roberts(1992)and the survey on the theoryof the firm by Holmstrom and Tirole(1989).

8.6LiteratureNotes

As we have alreadyemphasized,the founding articleson the multiagentmoral-hazardperspectiveon the firm are by Alchian and Demsetz(1972)and Holmstrom (1982b).We have already discussedmany of the

subsequent contributions they have spurredin the body of the chapter.An

important related general analysis that we have not covered in the chapteris Mookherjee(1984),which considersa more general framework than

Holmstrom (1982b).We have also not had the spaceto cover several different interesting

areasofapplication of contracting with multi-agent moral hazard.Thus,onesetof applicationsis sharecropping,partnerships, and franchisecontracts,which all have beenmodeled as contracting problemswith multi-agentmoral hazard (see,in particular, Eswaran and Kotwal, 1985,for an earlysuch modelof sharecropping;Mathewson and Winter, 1985,and Lai,1990,for models of franchising contracts;Demskiand Sappington, 1991,and

Bhattacharya and Lafontaine,1995,for models of partnership contractswith double-sidedmoral hazard).Another important area of application,which we have not beenable to cover,is joint ventures and in particularresearch joint ventures (see,in particular, Bhattacharya, Glazer, and

Sappington, 1992,for a modelof multi-agent moral hazard and disclosureof information in a researchjoint venture).A third large area ofapplication is the auditing and accounting literature (see,for example,Demskiand

Sappington, 1984,1987,and Melumad, Mookherjee,and Reichelstein,1995).Finally, an important omissionfrom the chapter,and which possiblymight bethe culmination of this approach,is the contribution ofHolmstromand Milgrom (1994)on the firm viewedas an incentive system.

Two areason which there is now a substantial body of economicliterature and on which we have barely touchedare,first, collusionin

organizations and, second,hierarchies.Following the original articleon collusioninprincipal-supervisor-agentstructures by Tirole (1986),the literature hasgrown in a number of interesting directions.Laffont and Martimort (1997,2000)have exploredcollusionunder asymmetric information and dealtwith

issuesrelating to the modeling of mechanism designof sidecontracts.Felli (1996),Strausz (1997),Baliga and Sjostrom (1998),Laffont and

363 Static Multilateral Contracting

Martimort (1998),and Macho-Stadlerand Perez-Castrillo(1998)explorethe idea that delegationto the supervisoror decentralizationmay bea way

of reducing the scopefor collusion.The difficult problemof collusionwith

soft information has also been exploredby Baliga (1999)and Faure-Grimaud, Laffont, and Martimort (2003).

Similarly, the economicliterature on hierarchiesincludesa number ofalternative approachesbesidesmoral hazard and lossof control,that Calvoand Wellisz (1978,1979),Williamson (1967),and Qian (1994)haveemphasized. Optimal hierarchiesto minimize the costsof information processingand communication have beenanalyzedin Kerenand Levhari(1979,1983),Radner (1992,1993),Bolton and Dewatripont(1994),Van Zandt (1998,1999),Prat (1997),and Vayanos (2003).Another approachto the designofhierarchiesby Cremer (1980),Aoki (1986),Geanakoplosand Milgrom(1991),and Marschakand Reichelstein(1995,1998)is basedon boundedrationality and the limited attention of managers.Another role ofhierarchies exploredin Garicano(2000)and Beggs(2001)is the facilitation of the

handling and allocationof moreor lessspecializedtasks to generalistsand

specialists.Finally, Sah and Stiglitz (1986)explore the question of the

optimal decisionstructure (hierarchicalor polyarchical)for the approval ofrisky projectsby firms, when managerscanmake mistakes in their projectevaluations. Hart and Moore(2000)alsodealwith the issueof the optimalallocationof decisionrights in an organization and explorethe ideaofhierarchies as precedencerules.

All these approachesattest to the richnessof the economicquestionofthe internal organization of firms. Theypoint to the complementary role ofhierarchicalinstitutions to markets, and also to the richnessof alternative

ways of allocating tasks, goods,and servicesbesidestrading in organizedmarkets.

REPEATEDBILATERALCONTRACTING

Thethird part of this bookdealswith ongoing,or repeated,contractualrelations betweentwo parties.New conceptualissuesarisehere that relate torenegotiation of long-term contractsand the inability of contracting partiesto always commit to or enforcelong-term contractual agreements.

Thefact that a contractual relationmay beenduring affectsincentiveprovision and information revelationin fundamental ways. For example,in the

early stagesof a contractual relationthe agent may hold backon revealinginformation that couldbe usedagainst her in later stagesof the relation.Alternatively, the agent may engage in reputation building activities toenhanceher future value to the principal.In short, repeatedinteraction

opensup a whole variety of new incentive issues,but it alsopermits morerefinedcontractual responses.

In principle, long-term contractscouldbe ever increasingin complexityas the contractual relationpersistsand optimal contracting problemscouldbecomeincreasingly intractable.This is all the more to be expectedif the

contracting parties may renegotiatethe contractalong the way. We shall

explain, however, that paradoxically,optimal long-term contractscouldactually take a simpler form in an ongoing relationthan in a one-shotrelation. They may, for example,specifya simple linear relationbetweenoutput

performanceand compensation,or they may just specifysimpleperformance targets.

At a more conceptuallevel,a major differencewith the static optimalcontracting problemsanalyzedin Part I is that in repeatedcontractingsituations the revelation principleno longergenerally applies.As a result, thecharacterizationof optimal long-term contractsunder asymmetricinformation is often significantly more complex.However,if the revelationprincipleno longerapplies,there is neverthelessa closelyrelated generalprinciplethat helps in determining the optimal form of the contract:Therenegotiation-proofnessprinciple.Accordingto this principleoptimal long-term contractstake the form of contractsthat will not be renegotiatedin

the future. Thedetermination ofoptimal renegotiation-proofcontractsthen

generally involves solving an optimization problemwherethe by now

familiar incentive constraints are replacedby tighter renegotiation-proofnessconstraints.

The first chapter in this third part (Chapter9) considersoptimal long-term contractsunder hidden information. Thesecondchapter(Chapter10)considersoptimal long-term contractsunder hidden actions.Much of thematerial in these two chapters draws on fairly recent research and is

366 RepeatedBilateral Contracting

synthesized here for the first time. Unlike the material in Part I,the

conceptsand methodsexploredin this part are not as well digestedand may

well evolvesignificantly in responseto future researchbreakthroughs.

Dynamic AdverseSelection

In this chapterwe considerrepeatedinteractionsbetweentwo contracting

partiesunder one-sidedasymmetric information. We analyze two oppositecases:onewherethe agent'stype is drawn onceand remains fixedovertime;the otherwherethere is a new independentdraw every period.In the first

situation, dynamic contracting issuesarisebecauseof the gradualelimination of the informational asymmetry over time. The secondsituation leadsto questionsof intertemporal distribution of allocativedistortions as a way

to reduce the agent'sinformational rent.We begin by consideringthe casewherethe agent'stype remains fixed over time.As in Chapter3,the game-theoreticconceptwe rely upon is perfectBayesianequilibrium.

9.1Dynamic AdverseSelectionwith FixedTypes

Thebasicissueat the heart of dynamic-adverse-selectionmodelswith fixed

types can be understoodwith the following example,which considersasellerfacing a buyer whosevaluation he doesnot know. As seenin Chapter2, in a static setting, the sellermay want to seta pricesohigh that he doesnot sellwith probability one,evenif his productioncostis belowthe lowestpossiblebuyer valuation. This is an exampleof the classictrade-offbetweenallocative efficiency and informational rent extraction under adverseselection.

What happens,however, once the buyer has madeher buying decision?If shedid not buy, the sellernow knows that her valuation is low.Theinformation revealedthrough the executionof the contractthereforeopensupanew trading opportunity, at a lowerprice.While this opportunity is Pareto-improving ex post, it endsup hurting the sellerfrom an ex ante point ofview, becausehigh-valuation buyers will anticipate that an initial

unwillingness to tradewill prompt the sellerto lowerhis price.Recontractingthus

limits the ability of the contractto rely on expostallocativeinefficiency forrent extractionpurposes;to put it differently, this is a casewheresequential optimization differs from overallex ante optimization.

This commitment problemunder adverseselectionis relevant in a varietyof settings that we shall consider.In fact, mutually efficient renegotiationariseswhenever the uninformed party has an interestin becoming\"softer\"

with the informed party. It happens,as we have stressed,with monopolistsfacing buyers with uncertain valuations (bargaining/durable-goodmonopoly problem).It also ariseswith creditorswho do not want to terminate

9

368 Dynamic Adverse Selection

entrepreneurswith badprojectsbecauseof previously sunk expenses(soft-budget-constraint problem).Moregenerally, the issueofrenegotiationariseswhenever the parties are protected by a long-term contract but cannotcommit against sequential,mutually profitable, alterationsof this contract.

A secondcommitment problem in dynamic-adverse-selectionsettingsconcernsunilateral contractual gains:When the partiescansign only short-term contracts,ex post opportunism can alsoarisebecauseof information

revealedby contractexecution.Thisis the caseof the ratchet effect,by which

the uninformed party offers the informed party \"tougher\" contractsif it

learns that its productivity or valuation is higher. The mechanicsof short-term contracting is somewhat different from that of long-term contractingwith renegotiation.Onceagain, however,what is an expost unilateral gainfor the uninformed party turns out to bedetrimental from an ex ante pointof view, becauseit is anticipated by the informed party.

How can the uninformed party mitigate these two commitmentproblems? Severalideashave beenexploredin the literature:

\342\200\242 First, onecandesigncontractswhoseexecutionlimits the amount ofinformation revealed about the informed party's type, that is, contractswith

(partial) pooling in the allocativechoicesof the informed party. Thisapproachlimits expost recontractingopportunities.\342\200\242 A more extremeversion of this idea is to make it impossiblefor theuninformed party to observe in timely fashion the contract choicesof theinformed party, by not investing in the appropriatemonitoring systems.Thisleads to \"simpler\" and more \"rigid\" contracts than the optimal staticcounterparts.\342\200\242 Moregenerally, the literature obviously stressesthe benefitsof signing

long-term contracts,even if the parties cannot commit not to engage in

future Pareto-improving renegotiation.While a long-term contractprotectseachparty against expostopportunistic behaviorby the otherparty, short-term contractscan insteadhelp alleviateexcessiveuninformed-partysoftness. For example,making it hard for an initial creditor to providerefinancing canlimit the soft-budget-constraintproblemmentioned earlier.

In section9.1.1we first detaila dynamic buyer-sellerproblemand discussthe consequencesof the two potential problems facing an umnformed

seller,that is,lackof commitment to engagein Pareto-improvingrenegotiation of long-term (orsale)contractsand lackof commitment not to engage

RepeatedBilateral Contracting

in opportunistic behaviorunder short-term (or rental) contracting.We then

turn to various other applications.Section9.1.2considersthe renegotiationof insurancecontractsand discussesthe desirability of restricting theinformation available to the uninformed party about the behavior of theinformed party. Section9.1.3turns to creditmarkets and the soft budgetconstraint, and the optimaliry of relying on a sequenceof uninformed

parties to alleviate commitment problems.Finally, section 9.1.4revisitsthese insights in a much-studied application,the regulation of amonopolist with private information on its intrinsic productivity.

CoasianDynamics

An important potential effectof repeatedinteractions is to reducemonopoly power by creating intertemporal competition between current andfuture incarnations of the monopolist.This effecthas beennoted first in

durable-goodsmarkets, whereit hasbeenobservedthat a monopoly sellermay face competition from old buyers who sellthe good in secondhandmarkets. A relatedpoint hasbeenarticulated by Coase(1972),who

conjectured that a monopolist selling a perfectlydurablegoodin a world with nodiscounting would be forced to sellessentiallyat marginal cost.The logicbehind his conjectureis that early buyers would not acceptmonopoly pricesif they anticipatethat future priceswill declinefollowing their purchaseofthe durablegood.Theexpectedpricedeclinewould indeedfollow if there isa fixed total demand for the durablegood.Coase'sbasiclogicis relevantmoregenerally to any dynamic contracting situation wherethe agent'stypeis fixedovertime, asemphasizedby Stokey(1981),Bulow(1982),FudenbergandTirole(1983),Gul,Sonnenschein,and Wilson (1986),and Hart andTirole(1988).In this section,wefollow the analysis ofHart and Tirole(1988),who

have detailedthe comparisonbetweenshort-term and long-term contracts.We considera problemof a buyer and seller(\"she\" and \"he,\"

respectively)with bilateral risk neutrality, two periods,two buyer types, and a

single unit ofgoodto exchange.We then briefly extendthe analysis to threeperiods.

Specifically,assumea buyer that has valuation vt per period ofconsumption of the good [with 0 < vL < vH and, initially, Pr(v#) = /3, which iscommon knowledge]while the sellerhas zero valuation for the good.Thesellermaximizes the net present value of his expectedrevenue,while the

buyer maximizes the net present value of her consumption minus herpayment to the seller.Both partieshave a common discount factor8<1.

370 Dynamic Adverse Selection

Finally, we callxu the probability that the type-z' buyer consumesthe goodin period t.

9.1.1.1Full Commitment Solution

Considerfirst what the sellercouldachieveby making a singletake-it-or-leave-it two-periodcontract offer. Let us define net present values asfollows:A= 8\302\260 + 81=1+ 8,Xi=Xii + 8xa,and Tt is the net presentvalue ofthe payment of the buyer of type i to the seller.

Under full commitment, becausebuyer types are fixed, the seller'sproblemreducesto the static problem.Indeed,the selleroffersthe buyera menu {(XL,TL),(XH, TH)}that solves

max(l-P)TL+PTH

subjectto

ViXi-Ti^Q i =L,HvJb-T^vJCj-Tji,j=L,Hand

0<Xi<A i =L,HAs explainedin Chapter2, at the optimum the participation constraint

for i =L and the incentive constraint for i =Hand j=L arebinding, sothat

the seller'sproblemreducesto

max(l -P)vLXL +P\\yHXH -(yH -vL)XL]

The solution is XH =A, and XL = A if

J3< \342\200\224 =J8'

in which caseTL= vL = TH = vL. Indeed,selling to both types meanscollecting the low valuation only. Instead,if /3 > /3',that is, if the probability offacing a low-valuation buyer is low enough, it is bestto excludeher and setXL =0=TL,which allowsthe sellerto setTH=vH. This is the trade-offhighlighted in Chapter2.Sincethe problemhere is linear,we thus have eitherex post efficiencyor a cornersolution with zero consumption.

Tomake things interesting, we now assume/3> j3'.Thisassumption makesa differenceafter the period-1choiceby the buyer:onceshehas declined

371 RepeatedBilateral Contracting

to consume,the sellerknows that her valuation is vL. It will then be in his

interestto lowerthe pricein period2.This response,however,will beanticipated by the buyer initially.

9.1.1.2Selling without Commitment: The Durable-GoodMonopolyProblem

If, at eachperiod t, the sellercan make only spot-contracting offersto thebuyer, that is, offers to buy the good in periodt at price Pt, what is the

perfectBayesianequilibrium of this game?If the buyer buys in period1,the gameis over.If the buyer doesnot buy

in period1after a price offerPi, call /3(Pi) the probability assessmentbythe sellerthat he is facing a type-Hbuyer.This assessmenthas to becompatible with Bayes'rule.

The continuation equilibrium in period2 has to be P2= vH if /3(Pi)> /3'and P2=vL otherwise,by sequentialrationality In any case,the type-Lbuyeris left with zerosurplus in period2.This buyer thus acceptsa period-1offerif and only if

Pi<vLA

What about the type-Hbuyer? If she expectsP2= vH, she is getting zerosurplus in period2 and acceptsa period-1offer if and only if

Pi<v#A

If insteadsheexpectsP2=vL, she is more reluctant in period1and acceptsan offer if and only if

Pi <vHA-8(vH-vL) =vH+8vL=P'

Theselleris thus facedwith three choicesin period1:\342\200\242 First, he canseta pricePx < vLA = vL(l + 8) and sellwith probability 1in

period 1.The maximum revenuehe can obtain in this range is by settingthe highest such price.Hisrevenueis then vL(l + 8).\342\200\242 Second,he cansellin period2 to the consumerof type L.Thenthe secondperiodpricewill be vL. Consequently,the highest possiblefirst-periodpriceis P',and the seller'srevenueis

(1-(S)8vL+PP'=(1-P)8vL +(S(vH+8vL )=($vH+8vL

372 Dynamic Adverse Selection

Sincewe have assumedthat /3 > vL/vH, this option is better for the sellerthan the first one.\342\200\242 Third, the sellercan set a first-periodprice higher than F,but lessthan

or equalto v#(l +5).In this case,the continuation equilibrium has to be in

mixed strategies:for P2= vL, the type-Hbuyer doesnot want to buy in

period1,soP2= vL cannot be a continuation equilibrium; and for P2= vH,

the type-Hbuyer wants to buy in period1,soP2=vH cannot be acontinuation equilibrium either.In order to make the sellerindifferent betweensetting his second-periodprice at vL or at vH, we must have vL = [5(Pi)vH.This means having the type-H buyer accept the first-periodoffer with

probability y such that

P'=

or

v \342\200\224J8-

jsa

\342\226\240J8'

jsa--y)-y)+a-\342\226\240J8)

' jsa-jsoIn turn, the buyer has to be indifferent betweenacceptingand rejectingPi,implying that the probability that P2-vH given P1}which we calla(Pi),hasto satisfy

vH(l +8)-P1=8[l-o(P1)}(vH-vL)which means

vHa+S)-PiC7(Pi)=l-

8(vH-vL)

A first-periodprice P1thus leadsto randomization by the type-Hbuyer,with (constant) probability y of acceptingthe selleroffer, and second-period randomization by the seller,with probability o(Pi)of offering thehigh second-periodprice and probability [1- a(Pi)]of offering the low

second-periodprice.Theexpectedrevenueof the selleris then

j8yPi +[j8(l-y)+(l-j8)]*Lwhich is maximized for the highest possibleprice in this range,that is, forP\\= vh0- + 8) and for a(Pi)=1.Theseller'srevenueis then

373 RepeatedBilateral Contracting

PyvHa+d)+[jsa-r)+a-P)]&l

which, using the value of 7we computedearlier,equals

This expectedrevenuehas to becomparedwith the seller'srevenueunderthe secondpricing strategy wherehe sellsfor sure to the type-Hbuyer onlyin period1,that is, fivH + 5vL. Theprecedingexpressionis increasingin /3:for /3 -> /3', selling for sure is better than randomizing, becauserandomizing

meansselling with a very small probability in period1.Instead,for /3 ->1,selling for sure in period 1is worse than randomizing, becauseby

randomizing the sellercan sellwith a high probability at the high price vH in

period1.Thereis thus a cutoff value of /3 abovewhich randomizing ispreferred and belowwhich selling for sure to type-His preferred.

Lack of commitment thus means that the sellercannot refrain from

cutting priceswhen hebecomesmorepessimisticabout the valuation of the

buyer. Selling to the high-valuation buyer for sure in period 1thus meanshaving to settle for a lower first-periodprice than under commitment,becausethe buyer understands the price will be low in period 2.And

keepingpriceshigh meansacceptingthat a salewill belesslikely in period1.This last strategy is profitablewhen the sellercannot hope to get muchrevenue from a low-valuation buyer in period2, becausethe probabilitythat he is facing such a buyer is low.

9.1.1.3Renting without Commitment: The RatchetEffect

Theprecedinganalysis assumeda sale,that is, consumption with

probability

1in periods1and 2 by a buyer who had made her decisionto buyin period 1.Another strategy is spot rental; that is, the buyer pays i?j in

period 1just to consumein period1,and the sellermakesa new offerR2in period2.

As is known from the durable-good-monopolyliterature, renting couldbe an attractive alternative to selling for the monopolist(see,for example,Tirole, 1988).This result, however,relieson the assumption of buyer

anonymity: the firm is supposedto facea continuum of anonymous buyers.Under a sale,the firm first serves the top part of the demand curve and

374 Dynamic Adverse Selection

cannotcommit not to subsequently lower the price,thereby making high-valuation consumersreluctant to buy early, just as before.Instead,a rentalsolution getsaround this problembecausethe sellerfacesthe samedemandcurve period after period and becausehe cannot keep track of who hasalready bought.

If we assumeinsteadthat the sellerfacesa nonanonymous buyer, things

changedramatically: In the caseof a sale,the only problemfor the sellerin the previoussituation was his inability to commit not to lower the pricein period 2 when he thinks he is facing a type-L buyer. Now, a secondproblemis the seller'sinability to commit not to raise the rental price in

period2 when he thinks he is facing a type-Hbuyer. This ratchet effectwill

alsobe taken into accountby the type-Hbuyer.What is the perfectBayesianequilibrium? In period2 it is the sameas

before for a given j3(i?i):R2=vH if /3(i?i)> /3' and R2=vL otherwise.What

about period1?As in the previoussection,we have three cases:\342\200\242 Note that the type-Lbuyer doesnot acceptany i?x abovevL. One optionfor the selleris thus to set R^=vL, rent to both types in period1,and setRi = vh (because/3 > vJvH = /3').Hisrevenueis then vL + 8fivH.

\342\200\242 In a fully separatingequilibrium, the highest first-periodrental price the

type-Hbuyer acceptsis such that

Vfi-R^divff-VL)

Indeed,separatingby renting in period 1implies obtaining no surplus in

period2 (i?2=Vh), while not renting in period 1meansobtaining the lowrental price in period2 (R2 = vL). In this case,the revenuefor the selleris

J3[v*-8(yH -vL)]+d[($vH +(1-j3)vL]

This equals

fivH+SvL

which is higher than the expectedrevenue under the first strategy for8 < 1,since/3 > vJvH = j3'.This outcomeis the sameas the one obtainedwith full separation when the buyer is offered a sale contract.Note,however,that here the constraint on i?x is

R^d-d^+dVLThis meansthat separationis possibleonly for 8<1.

375 RepeatedBilateral Contracting

\342\200\242

Finally, the sellercan inducesemiseparation.As in the previoussubsection,

this meansinducing the type-Hbuyer to rent with probability

J8-J8'jsa-jso

in order to make the sellerindifferent betweensetting R2 = vL and R2 = vH

after a rejectionofR^. In turn, the buyer has to beindifferent betweenaccepting

and rejectingi?i.Sinceacceptingmeansgetting no surplus in period2,the sellermust setR2 = vH with probability a(i?i) to satisfy the equation

vH-B1=8[l-(jm](yH-vL)to keepthe buyer indifferent betweenacceptingor rejectingi?1.Thismeansthat

cr(i?i)=l--8(vH-vL)

Once again, the bestsuch outcomefor the selleris the highest possiblerental price,that is, i?x = vH. In this case,the seller'srevenueis

fir vH (1+5)+[j8(l-7) +(1-P)]8vL

which, using the value of 7 we computedpreviously, equals

j3-j3' (, J8-J8'V-\342\200\2247rrvH+\\ 1--\342\200\224^- \\8vL1-J8'H I 1-P'JWith two periods,we thus obtain the same outcomeas with a sale.Thisresult ceasesto be true, however, with more than two periods,as we now

show.

9.1.1.4MoreThan Two Periods:On the Suboptimality of Rentingwithout Commitment

With more than two periods,the ratchet effectcan make the rentalsolution strictly worse than the sale solution, becausethe combination ofdynamic incentive problemsfaced by the seller\342\200\224his desireto lower therental pricefor the type-Lbuyer and to raiseit for the type-Hbuyer\342\200\224Limits

the separationof types that canoccur.Indeedonecanshow that, asa resultof the seller'sdynamic incentive problems,the type-H buyer cannot bemadeto acceptearly separating rental offers in equilibrium with probability 1.

Tobe specific,call j3( the seller'speriod-?probability assessmentthat the

buyer has a high valuation given that shehas rejectedhis offersin periodsl,2,...,t-l(we thus have j3i = /3).Thenwe have the following result:Ina T-period rental problem, if 8 + 82 > 1,we have /3, > /3' for allt < T-1.This result canbe provedby contradiction.Take the first periodt > 2 such that fit<fi'.Jit<T-land the buyer has rejectedthe previousoffers,the sellerhas an incentive to set the rental price equal to vL from t

on.If the buyer hasacceptedthe previousoffer,the sellerfrom then onsetsthe rental priceequalto vH. For a type-Hbuyer to acceptthe offerat t -1,we must thereforehave

vH-R^>{vH-vL){8+...+8T-^)>{vH-vL){8+82)

But this implies R^ < vL, which means that both types of buyers take thei?(_i offer, a contradictionwith /3(_i

> /3' > J3t.

In the rental case,there is thus a limited amount ofrevelationoftypes that

is possiblebefore the last two periods.This limitation implies in particularthat, when full separationis optimal in the two-periodproblem(which is thecasefor /3 closeenough to /3')>the saleoutcomeis strictly better than therental outcomein the three-periodproblem.Indeed,in the latter problem,the sellercouldsellin period1to the type-Hbuyer at a price

P,=vH+{8+82)vL

and in period2 to the type-Lbuyer at a price

p2=a+s)vLIn contrast, in the rental outcome,he only has the following two choices:\342\200\242 First, set i?i > vL, soas to have /^=j3'.Thecontinuation payoff asofperiod2,however,is only (1+ 8)vL (becausethe sellerrandomizesand thus gains

nothing when he setsa rental pricedifferent from vL). Moreover,in period1,we have i?i < vH and a probability of renting strictly lessthan the

probability of facing the type-Hbuyer. This outcomeis thus worsethan the saleoutcome.\342\200\242 Second,seti?x = vL, soasto have the two-periodfull-separationoutcomestarting in period2.But this is like the saleoutcome,with period2replacing period1.This is worseagain, however,because8 < 1and /3v# > vL.

The suboptimality of renting with nonanonymous buyers relative toselling illustrated in this three-periodexampleis a generalphenomenon,asshown by Hart and Tirole (1988).To summarize, in the durable good

377 RepeatedBilateral Contracting

monopoly problemrenting dominatessalesif buyers remain anonymous,but otherwisesalesdominate renting.

9.1.1.5Long-TermContractsand Renegotiation

Theprevioussubsectionscontrastedthe full-commitment solution and theno-commitment, spot-contractingsolution.In betweenthese two casesisthe casewhere long-term contracts are feasible but the sellercannotcommit not to offerthe buyer a new contractat the beginning of period2that would replacethe initial contract if the buyer finds it acceptable.Thiscasediffers from the full-commitment solution, wherethe selleris assumedto be able to refrain not only from engaging in opportunistic behavior(theratchet effect)but also from making Pareto-improving offers (cutting the

price to sellat mutually agreeableterms)/This intermediate caseis very realistic.Indeed,by contractual

enforcement, we typically mean protection against opportunistic contractviolations that hurt a contracting party. Contractualenforcementis Limited

to situations where one party complainsabout a violation and doesnot

prevent the partiesfrom simultaneously agreeingto tear up the agreementand sign a new one.1Pareto-improving renegotiationis a concernbecause,onceagain,what is Paretooptimal asofperiod2may not beParetooptimalas of period1.

What is the optimal long-term salecontractwithout commitment not to

renegotiate?SequentialPareto optimality means that the sellerhas to seta second-periodprice that is sequentially optimal, that is,P2= vH if /3(Pi)>

/3' and P2=vL otherwise.This is exactlythe same constraint as in

subsection 9.1.1.2.Indeed,there is no way the sellercan do better than the \"no-

commitment outcome\" under sales.Undera long-term contractwith renegotiation,one way of

implementing

the outcomeunder salesis as follows:

\342\200\242 In the \"full-separation case,\"the buyer canbeofferedin period1a long-term contractwith two options:she can consumeeither in both periodsata priceof vH + 5vL or only in period2 at a first-periodpriceof 5vL.

1.Onecould try to prevent voluntary renegotiation by including a third party in the contractthat would receive a large sum of money were the contract to berenegotiated. This plan willnot work, however, unless this third party has exogenous commitment powers. Indeed, thethird party will understand that, unless it accepts to renegotiate its payment downward, Pareto-improving renegotiation will not take place. If there exists a Pareto-improving renegotiation,it will then accept it in order to receive a share of the positive surplus it generates.

378 Dynamic Adverse Selection

\342\200\242 In the \"semiseparationcase,\"the buyer canbeofferedin period1a long-term contractwith three options in total: in period1,she has to choosebetweenconsuming and not consuming the good;if sheconsumesin period1,sheis to consumein period2 also,and to pay vH(l +5) overall(in period-1money); if shedoesnot consumein period1,shethen hasa secondchoicein period2:eithershe doesconsumeand pays 5vH overall,or she doesnotconsumeand pays nothing.

This is an illustration of the renegotiation-proofnessprinciple:In bothcases,we implement the solution through a renegotiation-proofcontract.This implementation canbeaccomplishedquite generally, becausethe result

offuture (anticipated) renegotiation canalways be included in the initial

contract. The precedingcontractsare indeed renegotiation-proof:Underfull

separation,renegotiationentails consumption for both types, which is expost efficient and thus not susceptibleto Pareto improvements; and in the

semiseparatingcase, strict Pareto improvements are also ruled out,

provided the high-valuation buyer choosesto consumein period 1with

probability

7 /3(i-/nIn this case,while a high price in period2 is of coursenot expostefficient,it is interim efficient: Giventhat only the buyer knows her type, there is noway the sellercanraisehis payoff by offering a new contract.Inducing moreconsumption would indeedmeanlowering the priceto vL, which would not

strictly improve the seller'ssecond-periodprofit relative to the initial

contract.While there is no differencebetweenspotcontracting and long-term

contracting with renegotiationin the caseof a sale,this is not true in the rentalcase.Indeed,long-term contracting gets rid of the ratchet effect,and the

precedinglong-term contractcaneasilybereinterpretedasa renegotiation-proof long-term rental contract:

\342\200\242 In the \"full-separation case,\"the buyer canbeofferedin period1a long-term contractwith two options:shecan consumeeither in both periodsata rental priceof vH in period1and vL in period2,or only in period2 at arental priceof vL.\342\200\242 In the \"semiseparationcase,\"the buyer canbeofferedin period1a long-term contractwith three options in total: in period 1,she has to choose

379 RepeatedBilateral Contracting

betweenconsuming and not consuming the good;if sheconsumesin period1,she is to consumein period2 also,and pays a rental price of vH eachperiod;if she doesnot consumein period 1,she then has a secondchoicein period2:either she doesconsume,and pays a rental priceof vH; or shedoesnot consume,and pays nothing.

This solution is exactly like the precedingone,oncewe properlydiscountwhat the buyer pays.We thus have two results here:2First, with long-termcontractsand renegotiation,the saleand rental outcomecoincideboth with

one another and with the spot-contractingsale outcome.Second,optimallong-term contractswith renegotiationcan be computedas renegotiation-proof contracts without lossof generality. This renegotiation-proofnessprincipleshould not be interpreted too literally: While optimal outcomescanbeachievedin this setting without equilibrium renegotiation,they canoften also be reached through contractsthat are renegotiatedalong the

equilibrium path. Therenegotiation-proofnessprincipleseemsnonethelessat oddswith reality, where contract renegotiationoften \"looksunavoidable.\" Accounting for this idea,however,requiresimposing limits on the setof initial contracts,a road that has been taken by \"incomplete contracttheory,\" and a topicwe shall discussin Part IV.

9.1.2Insuranceand Renegotiation

This sectionrevisits the insights of the previousone in the caseofinsurance contracts.Here,ex postmutually profitable renegotiationlimits the

ability of the contractto rely on allocativeinefficiency in order to improveinsurance.As in the case of Coasian dynamics, imperfect information

revelationemergesas an optimal contractual responseto this commitment

problem.This applicationalsoallows us to show how simple, lessinformative contractsreducethe roomfor renegotiationand may thereby improveupon contractsthat rely on the observationofmorenumerous choicevariables taken by the informed party.

Considera simple model,adaptedfrom Dewatripontand Maskin (1990;seealsoDewatripont, 1989),wherea risk-aversefirm seeksinsurance froma risk-neutral insurer. The sequenceof eventsis as follows:first, aninsurance contract is signed;second,the firm privately learns its profitability;

2.These results are general in the T-period version of this game, as emphasized by Hart andTirole (1988).

third, the firm makespublicly observableinput choices,and the contractisexecuted.The profit of the firm, k, equals

K =6Q(k,l)-rk-wlwherek is capital,with rental rate r; I is labor,with wagerate w; O0(k,I)is revenue,with

\302\243)(\342\200\242) increasingand concavein inputs; 6 e {6L,6H), where6L < 6H and both 0/sare equiprobableexante;and k and I are contractible,but 6 and revenueare not.

Expostefficiencycannow bedefinedasfollows:l*tand k*t are the expost

efficient input choicesif and only if 6iQk(k% /*) = r and 6iQi(k*, /*) = w

[whereQfcOan<3 QK')are the first derivatives of output with respectto kand I,respectively].Supposethat the payoff of the firm is v(k +I),where/ isthe insurancepayment to the firm and v is an increasingconcavefunction.

9.1.2.1Second-BestContracting (No Renegotiation)

The commitment optimum is the solution to

1H

max -V viOiQih,h)-rkt -wlt +/,]a\302\253,w. 2 i=L

subjectto the participation (or zero-profit)constraint of the insurer

1 H

z i=L

and the incentive constraints

OlQQci,ll)-rkl-wh+I^e.QikjJ^-rk,-wl,+1, i,j=L,HThe first bestinvolves ex post efficiencyand full insurancefor the firm,

that is, kl + IL= KH + Iff. This equation,however,violatesincentive

compatibility when 6 = 6H, so,as seenin Chapter2, the secondbestinvolves

setting

kff =kH, lH =lnand

kL<kt, lL<llIndeed,we have efficiencyat the top, in state 6H, but 0LQk(kL, lL) > r andOlQ^l,D> w, that is,underutilization ofboth inputs in state 6L, in orderto improve risk sharing.

\"

381 RepeatedBilateral Contracting

9.1.2.2SequentialInput Choicewith Renegotiation

What if input choicesare now sequential;say, capitalhas to beset first andlaborsecond,with the possibility ofa take-it-or-leave-itofferby the insurerto the firm after the first input has beenchosen?Onceagain, without lossof generality, we canconsidera renegotiation-proofcontract.

Focusing first on contractswith pure strategies, renegotiation-proofnessimplies one of two possibleconstraints:eitherpooling,that is, kL = kH, orseparationfollowedby ex post efficiency, that is, 6LQi(kL, lL) = w. Indeed,if kL = kH, there is no room for renegotiation,since the insurer has notlearnedanything from the choiceofcapital,sothe labor-input choiceis notconstrainedby renegotiation.But if kL *kH, there is symmetric information

after capitalhasbeenchosen,so the insurer canmake the firm an offerto

renegotiateaway any employment level lL such that 6LQi(kL, lL) *w.Thatis, the insurer can offerto move to ex post efficient employment (given 6Land kL) while pocketingthe efficiencygain.

Renegotiationhurts exante welfare,sinceit limits the expostinefficiencythat the contracting parties would like to commit to ex ante in order to

improve risk sharing.What about mixed-strategy contracts,which entail partial and gradual

information revelation?Intuitively, having someunderemployment oflaborin state 6L requiresat least somepoolingof capital levels.Considerthe

following contract:

(kHlHlH\\

kLrBikJcLl'LlLJ

wherethe first two rows of the contractmatrix are chosenwhen 6=6H andwith respectiveprobabilities1-7and 7while the third row is chosenwhen

6 = 6L.The first column of the contractmatrix concernsthe choicemadeby the firm in the first stage,that is, the capital-choicestage;the secondcolumn concernsthe (second-stage)laborchoice;and the third columnconcerns the net payment by the insurer associatedwith eachcapital-laborpair.In the first stage,state 6H is thus only partially revealedby the choiceof k,since,with probability 7 the firm in that statechoosesthe capitallevelassociated with 0L.The goalof this partial poolingis the desireto maintain atleast some level of underemployment of labor for state 6L in order to

improve risk sharing.

Indeed,renegotiation-proofnessrequires

382 Dynamic Adverse Selection

y{eH-eL)Qi{kL,iL)>BLQi(kaL)-w (RP)

This condition can be understoodas follows:Start from an inefficiently

low lL from an expostperspective.If 6=6L,raising employment by A/

generates an expost efficiencygain equal to

[eLQi(kL,lL)-w]M (9.1)But what if 0= 0H [note that, if kL is chosen,the relativeprobability of 0H

becomes(f/i) =7]?It is optimal not to inducethe firm to chooselL in this

caseinstead of l'H. Sincethe insurer has, by assumption, full bargaining

power in the renegotiation,it can retain the full benefit of the efficiencygain (9.1).Doingsostill leavesa net gain of

(0H-eL)Q^kLJLW

for the firm if 6= 6H, which has to beconcededthrough a higher insurancepayment in order to keepthe firm from choosinglL if 6 = 0H. This result

explainswhy, if (RP) is satisfied,the insurer doesnot find it profitable toraiseemployment abovelL through renegotiation.

The optimal renegotiation-proofcontractthus solves:3

1H

max -^yj^iQ^i^d-rki-wli+Ii]KUJ'hJ'h 2~[such that

|/l+|[7/W(1-7)/h]<0eHQ(kH,lH)-rkH-wlH+IH=eHQ(kL,l'H)-rkL-wl'H+I'H>

0HQ(.kL,lL)-rkL-wlL+ILand the renegotiation-proofnessconstraint (RP).

What canwe say about the optimal contract?It is easyto first show that

optimality implies efficiencyat the top, that is,kH = k% and lH =1%, but also

a level l'H such that 6HQt (kL, l'H) = w. Second,note that optimal insurancemeans trying to have a high IL,a low IH, and a low I'H. The cost ofpooling

at kL < k% when 6 = 6H is that it raisesl'H which, by the individual-

rationality constraint, limits the levelof lL.However,the benefit ofpooling

3.Note that, since the firm randomizes between kL and kH in the capital-choice stage when6= dH, it has the samepayoff under (kL,I'h, I'h) and (kH, lH, IH), and thus the same maximandaswithout renegotiation. -:

383 RepeatedBilateral Contracting

at kL when 6= 6Histhat it allows sustaining a lowerlL, as shown by (RP):if y =0,yI'H disappearsfrom the individual-rationality constraint, but expostefficiency is imposedon lL, as shown by (RP).Dewatripontand Maskin

(1990,1995a)show that, if | QuIQi| is not too large,the optimal contractinvolves 7 > 0, that is, imperfectinformation revelation.Intuitively, Q()should not be too concave,so that sustaining some underemployment oflabor doesnot requiretoo high a 7

Beyond imperfectinformation revelation,what are the other insightsfrom imposing renegotiation-proofhess?A first immediate consequenceconcerns the existenceof benefits from \"contract rigidity\": if adjusting

employment is hard after the beginning of the relationship, renegotiation-proofhesscouldbecometrivially satisfied.This insight is part of the generalidea that, with fixed types, there is a benefit from commitment.

A more interesting consequenceof renegotiation-proofhessis theexistence of benefits from \"limited observability\" and thus from \"simplecontracts.\" Here,observing k for the insurer has a benefit (to obtain IL, theinsureehas to setboth lL and kL), but alsoa cost,becauseofrenegotiation-proofhess.When is observing k worse than not observing it? Dewatripontand Maskin (1995a)providesufficient conditionsfor this insight. Hereis asimple example:assumek can take only two values:k e {k, k}, and assumethe secondbestinvolves k and /**< /* for 6L and k and 1% for 6H.Add the

following assumption:

eHQ(k,lF)-rk>eHQ(k,lt*)-rkthat is, if the firm can have only /** as employment level,k is the profit-

maximizing capitalchoiceeven if 6= 6H.Then,under this assumption, thesecondbestcanbe achievedif the choiceof capitallevelby the firm is notobservableby the insurer, but not if it is.The intuition is as follows:underthis assumption, controlling laboris sufficient to \"control\" capitaltoo,while

simultaneously getting rid of renegotiation.Instead,for /**</*,renegotiation is problematic,sincethe choiceof k leadsthe insurer to offer to

renegotiate away all ex post inefficiency in labor choice.Consequently, a\"simple\" contractwhere only the levelof labor enters the insurancecontract dominatesa morecomplexcontractwith moreobservability. Notethat

4.We also need the assumption that the insurer doesnot know exactly when capital is chosen;otherwise, it would beenough to know that capital has been chosen for sure to makerenegotiation profitable (seeDewatripont and Maskin, 1995a,and also Fudenberg and Tirole,1990,in a dynamic-moral-hazard context).

384 Dynamic Adverse Selection

this result begsthe questionof how the insurer can commit not to observecapitalchoices.4One idea is that observing such choicesrequiresex ante

information systems, which couldbehard to set up during the relationship.The insurer could thus decidenot to adopt such systems even if they arefree ex ante, since their informational value is negative at that point,becauseof the commitment probleminducedby renegotiation.

9.1.3SoftBudgetConstraints

\"Soft budget constraints\"\342\200\224the refinancing of loss-making state-ownedenterprises\342\200\224have

been portrayed as a major inefficiency in centrally

planned economies,most notably by JanosKornai (1979,1980).But thenotion of the potential lackof credibility of the termination threat of loss-making projectsis alsoseenas a pervasiveproblem in corporatefinance.As Dewatripontand Maskin (1995b)have highlighted,5 soft budgetconstraints can be seenas resulting from a dynamic incentive problem:they

representan inefficiency in the sensethat the funding sourcewould like tocommit ex ante not to bail out firms but knows that it will be tempted torefinancethe debtor ex post becausethe initial injection of funds is sunk.In such a world, hardening the budget constraint requires finding acommitment devicethat makesrefinancing expost unprofitable. The literatureon transition from plan to market has analyzed severaltransition strategiesas ways to harden budget constraints.6In this sectionwe focuson one ofthem, the decentralizationof credit,as an illustrative commitment deviceagainst soft budget constraints.

Soft budget constraints arise becausethe uninformed party cannotcommit to remain tough enough with the informed party (just like the sellerwho cannot commit not to lowerhis pricein the future). It is thus the mirror

image of the ratchet problemwhere the planner cannot commit not to beexcessivelytough with the firm it controls.Theoriginality of this sectionliesin the specificdeviceconsideredto get around this commitment problem,namely, the relianceon a sequenceof uninformed partiesas a way to limit

the amount of information generatedin the courseof the relationship.

9.1.3.1Soft BudgetConstraints as a Commitment Problem

Considerthe following adverse selectionproblem:A (private or state-

5.Seealso Schaffer (1989).6. See,for example, Dewatripont, Maskin, and Roland (1999)and especially the book byRoland (2000).

385 RepeatedBilateral Contracting

owned)creditorfacesa population of (private or state-owned)firms, eachneedingone unit of funds in initial period 1in order to start its project.A proportion /3 of these projects are of the \"good, quick\" type: afterone period, the project is successfullycompletedand generatesa gross(discounted)financial return kg>1.Moreover,the entrepreneurmanagingthe firm obtains a positivenet (discounted)private benefit EG.In contrast,there is a proportion(1-/3)ofbad and slowprojectsthat generatenofinancial return after oneperiod.If a projectis terminated at that stage,the

entrepreneur obtains a private benefit ET.Instead, if refinanced,each projectgeneratesafter two periodsa gross(discounted)financial return tCb and anet (discounted)private benefit EB. Initially, /3 is common knowledge, but

individual types are private information. A simple result easilyfollows:if 1<

7\302\243b

< 2 and EB > 0,refinancing bad projectsis sequentially optimal forthe creditor,and bad entrepreneurs,who expectto berefinanced,apply forinitial financing. Thecreditorwould, however,bebetter off if he wereableto commit not to refinancebadprojects,sincehe would thereby deterentrepreneurs with bad projects from applying for initial financing, providedET<0.

Tennination is here, by assumption, a disciplinedevicethat allows theuninformed creditor to turn away bad types and finance only goodones.7Theproblemis that termination is not sequentially rational if ^f is biggerthan one:once the first unit has beensunk into a bad project,its netcontinuation value is positive, so that, in the absenceof commitment, the soft-budget-constraint syndrome arises.In this setup,becauseirreversibility ofinvestment is such a generaleconomicfeature,the challengeis to explainwhy hard budget constraints prevail rather than why budget constraints aresoft in the first place.The next subsectionfocuseson the decentralizationof creditas a reasonbehind hard budget constraints.

9.1.3.2Decentralizationof Credit

The setup of the previoussubsectionis compatiblewith a ^f that reflectspure profit-maximization motives. Indeed,in the presenceof sunk costs,sequentialprofit maximization can be inferior to ex ante profitmaximization. Toavoid such an inferior outcome,the decentralizationof creditmaybehelpful, working through a reductionin ^f.

7. This differs from a static problem a la Stiglitz and Weiss (1981)where creditors can at bestfinance all types, and at worst finance only bad types (seeChapter 2).

388 Dynamic Adverse Selection

long-term projectsfrom being started.It is thereforePareto-dominatedbythe soft-budget-constraintequilibrium.

9.1.4Regulation

Just as the threat of termination of inefficient projectsby creditorsmay lackcredibility, the samemay be true of a regulator'sattempt to commit to aprice cap or a given requiredrate of return. This strategy can indeedbefrustrated eitherby lobbying from the regulatedfirm for priceincreasesorby public pressureto lowerprices.

We illustrate the dynamic incentive problemsfacedby a regulator of anatural monopoly in the framework popularizedin the bookby Laffont andTirole (1993),discussedin Chapter 2.There, we considereda natural

monopoly with an exogenouscostparameter6 e {6L,6H] and A6= 6H-6L> 0.The firm's costof producing the good is c = 6-e,wheree stands foreffort, with associatedcost y/(e) = [max {0,e}]2/2,increasingand convexin

e.The government wants the good to be producedfor the lowestpossiblepayment P = s + c (wheres is the subsidy paid to the firm in excessofaccounting costc).Thepayoff of the firm is P-c-y/(e) =s-y/(e), which

has to be nonnegative for the firm to be willing to participate.If thegovernment observesthe value of the costparameter 6,we have as first-bestoutcomee*=1and s* =0.5.

Chapter 2 consideredthe one-periodproblemwhere the governmentcannot observe the cost parameter of the firm, but has prior beliefPr (6= 6L)=pi.Calling (sL,cL)the contractchosenby type 6L (withassociated effort eL=6L-cL),and (sH,cH) the contractchosenby type 6H (withassociatedeffort eH=6H-cH), the government solves

min{/3i(sL-eL)+(1-frXsH -eH)}

such that

5L-[max{0,eL}f/2>0sH-[mzx{0,eH}f/2>0sL-[max{0,eL}fll>sH-[max{0,eH-A0}f\\l

sH -[max{0,eH}fll>sL-[max{0,eL+A0}f/lAssuming away corner solutions and noting that the two relevantconstraints are

389 RepeatedBilateral Contracting

sL-e2L/2=sH-(.eH-&e)2/2sH ~ejj/2=0

we obtainedin Chapter2 the following first-orderconditions:

eL=land

We thus have ex post allocativeefficiencyat the top, as well as underpro-vision of effort for the inefficient type, in order to reduce the rent of theefficient type [which equalse2Hll -(eH-A0)2/2].

9.1.4.1Two Periodsand Full Commitment

Assume now that the government facesa two-periodproblem,with aconstant costparameter6, an identicaltechnology ct=6-eh and an identicaleffort-costfunction y/(et) in eachperiodt=l,2.Normalize to 1the lengthof the first period,and call 8 the length of the secondperiod.Under full

commitment, the government offerstwo-periodcontracts(sm, cm, sm-, cm)and (sLi,cL1,sL2, cL2)simply to minimize the expectedsum of actual costsunder the participation and incentive constraints of the firm; these statethat, for each type of firm, (1)the sum of subsidiescoversat least the sumof effort costs,and (2) the two-periodcontractcorrespondingto its own

type is more attractive than the two-periodcontractcorrespondingto theother type.As observedin earliersections,in this stationary environment,the best the government can do is to replicate the one-periodcontractdescribedpreviously. Indeed,observefirst that the first bestis to replicatethe one-period first-best contract. In this solution, it is the incentiveconstraint of the efficient type that is violatedunder adverseselection,sowe have expost efficient effort for the efficient type (eL1= eL2= 1)and

underprovision of effort for the inefficient type. Could it be optimal tohave em *eml In fact, no:considerthe following contractchange:replace(em,em) by a unique effort levele such that

(l+5)e=eH1+5eH2

390 Dynamic Adverse Selection

while keepingthe inefficient type'spayoff unchanged:

Sm-e2m/2+8(sH2-e2H2/2)=(l+5){s-e2/2)wheres is the per-periodsubsidy paid to the inefficient type.By the

convexity of the effort-costfunction, this contractchangeallows thegovernment to reduceits total payment: while the expectedoveralllevelof effortremains constant by construction, the expectedtotal subsidy goesdown,since the stabilization of the effort level over time reducestotal effortcost.The only questionremaining concernsthe incentive the efficient typemight have to mimic the inefficient type. The efficient type'spayoff from

mimicking the inefficient type when em *emis

sm -{em-A0)2 k+S\\sH2-(eH2-A0)2 jl]

which equals

sHi-e2Hi/2+8(sH2-e2H212)+A6(eH1+8eH2)-(A6)2(1+8)/2Similarly, under the constant e, it is

(1+8){s-e2/2)+A0(1+8)e-(A0)2 (1+8)/2Since,by the construction of (s,e),the payoff of the inefficient type staysunchanged after this contractchange,then sodoesthe incentive of theefficient type to mimic the inefficient type. The result\342\200\224that stabilizing theeffort of the inefficient type while keepingits payoff constant is worthwhile

for the principal\342\200\224comes from the convexity of the effort-costfunction, that

is, from its positive secondderivative. As for the result that such a movepreservesthe incentive constraint of the efficient type, it canbe shown that

this relieson the nonnegativity of the third derivative of this effort-costfunction (seeLaffont andTirole,1993).Thiscondition is (just) satisfiedhere,sincea quadratic effort costhas a zero third derivative.

9.1.4.2Pareto-ImprovingRenegotiation

What if, after the firm has made its period-1choice,the governmentcameback and made a new contractoffer at the beginning of period 2?Assume that the initial contractremains the default contract;that is, thefirm can reject the new contract offer and stick to the original contractterms.As in earliersections,the firm's period-1contractchoicerevealsthefirm's type.Thisknowledgecanprovidethe government with an opportu-

391 RepeatedBilateral Contracting

nity to make a renegotiationoffer that is Pareto improving as of period2but that exacerbatesincentive-compatibility problems.In particular, the full

commitment optimum cannot be reachedanymore.As in earlier sections,there is no lossof generality in restricting

attention to renegotiation-proofcontracts,and three types of contractsarepossible: separating, pooling,and semiseparating contracts.

\342\200\242 Separating contracts:When (sm, cm) *(sL1,cL1),period-2outcomeshaveto be allocatively efficient for both types.The government thus solves

min{/3i(sLi-eL1+8[sL2-eL2])+(1-pi)(sm -em+5[sH2-eH2])}

subjectto

sm -e2m/2+8[sH2-e2H2/2] =0

sn ~eh/2+8[sL2-e\\2 /2]=sm-(em-A0)2 /l+d[sH2-{eH2-A0)2/l]

and

en =eH2=\\While eL2=1alsoobtains in the full commitment optimum, em=1doesnotand is thereforea binding renegotiation-proofnessconstraint. Optimizationwith respectto first-periodefforts yields the same outcomeas in a one-periodproblem,namely eL1= 1,and

em=l--^r-A61-Pi,

Type0Lobtains morerents than in the full-commitment outcomethanks toa higher level of em, which is anticipatedwhen the first-periodcontractchoiceis made.This rent concessionis costly for the government. However,the cost is proportional to 8, the length of the secondperiod, so that

this contractapproximatesthe full commitment optimum for 8 tending tozero.\342\200\242 Poolingcontracts:When (sm, cm) = (sL1,cL1)= (s-l, cx) (with associatedeffort levelsem= 6H-cx and eL1= 6L-c-j)at the beginning ofperiod2, thebeliefof the government is still Pr(0 = 6L)= j3i, and sequentialoptimalityis in fact not a constraint on the optimal contract.Specifically,thegovernment solves

min{/3i(si -eL1+8[sL2-eL2])+(1-#)fo-em +8[sHi-em])}

392 Dynamic Adverse Selection

subjectto

si -em/2+8[sH2-e2H2/2] =0

8[sL2-e2L2/2]= 8\\sm-{eH2 -AOffr]

This problemcanbesolvedperiodby period.In period2,we have

eH2=l--^-rA61-Pi,

as in the full commitment optimum, and this relation holds becausethe

government has an unchanged beliefabout the firm's distribution of types.Concerningperiod1,lowering the actual costCxfor both types meanslowering effort levelsone for one.This requiresraising s\\, to keepthe

participationconstraint of type 6L satisfied,by an amount em. Optimality thus

implies em=land eL1= 1-A0: here,first-periodunderprovision of effortoccursfor the efficient type.Theoptimal poolingcontractminimizes the rentthat is concededto type 6L.Thelossfor the government relativeto the full-

commitment optimum is due to the first-periodmisallocationof effort, andit consequentlytends to the full-commitment optimum when this periodbecomesrelatively short (that is,when 1/5tends to zero).\342\200\242 Semiseparatingcontracts:In order to reducethe period-1underprovisionof effort of type 6L that occursin the poolingcontract,one canreduce the

poolingprobability. Doingso,however,raisesthe rents that have to beconceded to type 6L.Specifically,assumethat type 6L mixes between(sn,cL1)with probability 1-7and (sm, cm) with probability 7,while type 6H chooses(sm, Cm) with probability 1.Call (s^,cL2) the continuation contractafter

(sli,cli)has beenchosen,and {(s^,cm), (sHL2,cHL2))the continuation menuafter (sm, cm) has beenchosen[where(sHL2, cHL2) is chosen in period 2by type 6L and (sm, cm) by type 6H].In this latter case,the regulator'sposterioris

The sequentially optimal period-2contractentails eHL2 = 1and

393 RepeatedBilateral Contracting

for the efficient and inefficient types, respectively.As before, only therequirement on em is a constraint, whenever& *A- The optimum is thusthe result of

min/3i{(l-y)(sLi-eL1+8[sL2-eL2])+Y(shi ~[eHi~A0]+8[shl2-eHL1\\)}+(1-/3i){sm-eH1+8{sH1-eH2)}

subjectto

Sm ~em/2+8[sH1-e2H1 /2]=0

Shli ~eitt,2/2 =sm -(eH2-AG)2/2

sLi ~eljl+8[sL2-e2L2/2]=sm-(em-A6)2 /l+8[sHL1-e2HL2/2]

and

eHi =1-7 A

i-AA6

Thefirst constraint is the participation constraint of type 6H; the secondoneis the incentive constraint of type 6L for period2, given that it has chosen(sm, cm) in period1;the third onestates that type 6L is indifferent between(sLi,cL1,sL2, cL2)and (sm, cm, Shli, cHL2), a necessarycondition for mixingto beoptimal; and the fourth one is the renegotiation-proofhessconstraint.As before,it is optimal to set the allocatively efficient effort levelfor type6L whenever it is not poolingwith the other type, which implies in

particular eL1= eHL1. After straightforward substitutions, one can rewrite the

optimization problemas

minjjSj-\342\200\224-eL1+8eh

\342\200\224 eLi

\342\200\224 --(eH2-A6)+8

+ a-A)

+\342\200\224\342\200\224-Om-A0)

(em -A6)2-(eH1-A0)-(~--eL1

\302\243h\\ , oC.R2 r.\342\200\224\342\200\224em+d\342\200\224\342\200\224deH2

subjectto

eH2=l-7i-AA6

The maximand is the sameas under full commitment, exceptfor the effi-

394 Dynamic Adverse Selection

ciencylossfrom poolingin period1for type 6L,which comesfrom settingeffort level (em -A6) insteadof eLh with ex ante probability j3iy. Solvingthis optimization problemyields,asbefore,

eu = eL2= eHLi =1while

i-A+Ar

Effort em thus riseswith 7sinceit is an inefficiently low effort for type 6L.Moreinteresting is the optimality condition on 7 and em.Taking thederivative of the maximand with respectto 7while having em move with 7 and

equating it to zero,yields

[p18A6+a-Pi)8(eH2-l)]-Q-Ae=l-pi

A* (em-A6)2-(eH1-A6)-[^-eL1TheRHSof this equationis positive:It representsthe allocativelossfrom

poolingin period1for type 6L.Therefore,the LHSof the equation has tobepositivetoo; that is,e^hasto behigher than 1-/3iA0/(l -Pi).But this

result canbesequentially optimal only if /32< Pi,sothat full poolingis never

optimal. Poolingbecomesasymptotically optimal only when 8 ->\302\260\302\260,

that is,when the costofpoolinggoesto zero in relativeterms. Otherwise,the ideais that, at effort level

em'1 l-A(which canbesustainedonly through full pooling),a risein this effort levelbrings only a second-orderloss,while any cut in 7 (from the value of 1)brings a first-ordergain.

9.1.4.3Short-TermContractsand the RatchetEffect

Assume now that the parties can sign only one-periodcontracts,that is,neither the regulator nor the firm can commit in advanceto an outcome(52,c2).What changesin comparisonwith the previoussubsection?Sequential optimality still inducesthe samesecond-periodeffort, that is, an effortof 1for the efficient type and an effort equal to

395 RepeatedBilateral Contracting

for the inefficient type under the belief that Pr(0L)=$i-As before,beliefsdependon the period-1outcome,and thus on whether the contractinvolvesfull separation, full pooling, or partial separation.The key differencebetweenlong-term renegotiation-proofcontractsand short-term contractsconcernspayments to the firm: the government cannot commit to giving thefirm morerents than the participation and incentive constraints require.Inparticular, under full separationthe government knows exactly which typeit faces at the beginning of period 2.Consequently, as in the previoussubsection,unit effort is offeredat that point. But, unlike in the previoussubsection,the government offersto give both types of firms exactly zerorents in period2.

The two incentive constraints for the firm are now

eh^ 1 x2 e-f-\\^-mand

sm ~ \342\200\224 ^ sa--{eL1+A0)

The first constraint is the usual incentive constraint for the efficient type,with two differencesfrom the full-separationcontractof the previoussubsection: (1)sincethe government gives the firm zerorents in period2 (that

is,choosessL1= elil'Z), rents can only be given in period1,through sL1;and

(2) if the inefficient firm receiveszero rents, the efficient firm obtains thesum over the two periodsof the effort-costdifferentials at (em, em);however,the government may now be forced to give the inefficient typepositiverents.Indeed,the risein sL1describedpreviously may inducetheinefficient firm to \"take the money and run,\" that is,pretend to be theefficient firm in period1[evenif that implies an effort levelof (eL1+A0)],while

not producing at all in period2.Laffont and Tiroleshow that indeed the

optimal contractmay, under someparametervalues,requireboth incentiveconstraints to be binding. In comparisonwith long-term renegotiation-proofcontracts,morepoolingmay be optimal, and full poolingcan in

particular bean optimum. The generalinsight from this model,as from models

396 Dynamic Adverse Selection

of long-term renegotiation-proofcontracts,is that information revelationthrough contractexecutionhas a costfrom an ex ante point of view.Institutional featuresthat limit the consequencesof information revelationcanthereforebe desirable.Thisis the casewith \"regulatory lags,\" whereby the

regulatory scheme is reconsideredonly infrequently by the regulatoryauthorities: While the government or the regulatedfirm might wish

otherwise ex post, ex ante social welfare can be enhanced by limiting the

frequency of regulatory reviews.Of course,a key assumption behind this

insight is the fact that we have concentratedon purely constant types:Inreality, the environment may changeover time, leading to the need forsomewhat flexibleregulatory schemes.For moreon this issue,seethe bookby Laffont and Tirole(1993).

9.2RepeatedAdverseSelection:Changing Types

One lessonemerging from our analysis of repeatedadverseselectionwith

fixed types is that there are no gains from an enduring relationship when

the informed party's type is fixed.Moreprecisely,the uninformed party'smaximum averageper-periodpayoff islowerwhen the two partiesinteractovertime than when they engagein one-timespotcontracting in ananonymous market. The best the uninformed party can hope to achievein arepeatedrelationis to repeat the optimal static contract.

In contrast,when the agent'stype changesover time, there aresubstantial gains to be obtainedfor the uninformed party from a long-termrelationship. First, when the agent's type changesover time, the agent is lessconcernedabout revealing her type in any given periodbecausedoing sodoesnot necessarilyundermine her bargaining positionin future periods.Moreover, the gains from trade to be shared from future contracting

providean additional instrument to screenagent types in early periods.The classicexampleof a dynamic contracting problem under adverse

selectionwith changing types is that of an individual household(or firm)facing privately observedincomeorpreferenceshocksovertime anddetermining how bestto consumedynamically thanks to financial contracting.We cast the analysis of our dynamic contracting problemin the contextofthis application.As will becomeclear,this somewhat abstractformulation

of intertemporal consumption (or investment) in the presenceof transitoryshockscanshedconsiderablelight on two fundamental economicissues:(1)

RepeatedBilateral Contracting

the effectof liquidity shockson consumption and investment; and (2) the

implications for long-run wealth distribution ofrepeatedpartially insurableincome(or preference)shocks.

We begin by formulating and analyzing the optimal long-termcontracting problem, then compare it to two benchmark situations: (1) nocontracting (or autarky) and (2) borrowing and lending in a competitivesecuritiesmarket. The optimal contractcan be thought of as an efficient

arrangement between a financial intermediary and householdssubjectto liquidity needs.By comparing this institutional arrangement with the

equilibrium outcomeobtained through trading in a competitive securitiesmarket, weare ablein particular to contrast,at an admittedly abstractlevel,the strengths and weaknessesof institution-based financial systems versusmarket-based financial systems (to borrow terminology introduced byAllen and Gale,2000).In one sentence,one important trade-offbetweenthese two systems that emergesfrom this analysis is that the former mayprovidebetter intertemporal risk sharing or consumption smoothing, but atthe risk of creatingfinancial instability through banking crises.

In this section,we treat three successiveproblems.The first has a two-

periodhorizon where the individual learns in period1whether shewould

rather consumeimmediately or is happy to wait until period2 for herconsumption. Thisproblem,first analyzedby Diamondand Dybvig (1983),thus

considersa singlepreferenceshock,an assumption that simplifies theanalysis and alreadydeliversquite a number of insights. Thesecondproblemweconsideralsohasa two-periodhorizon but with two independentlydistributed incomeshocks.Thisproblem,pioneeredby Townsend(1982),allowsus to considera more generalcomparisonbetweensimple debt contractsand insurance contracts.It also serves as starting point for the third

problem,which considersan infinite horizon.

Banking and Liquidity Transformation

It is often arguedthat one of the main economicrolesofbanks is totransform illiquid long-term investments into liquid savings. Thebasicideaisthe

following: Investments with the highest return are often long-term projectsthat generatepositivenet revenuesonly after a few years.An exampleofsuch a projectwould bea dam (togetherwith a hydroelectricplant), which

typically takesmore than a decadeto complete.Mostindividuals would shy

away from suchprojects,despitetheir high final return, becausethey cannotafford to tie up their savings for such a long period.In practice such

398 Dynamic Adverse Selection

long-term projectsare financed by banks (when they are not undertaken

by governments), who themselvesobtain the funds from their depositors.Thebanks act as intermediariesbetweenthe depositorsand the debtorby

providing a liquidity transformation service:they allowdepositorsto benefitfrom the future returns of the projectbefore these are realized,by

allowing depositorsto withdraw their savings at any time.Banksare ableto offerthis liquidity-transformation serviceby relying on the statistical regularitythat not all depositorswish to withdraw their funds at the sametime.

Oneof the first explicitanalysesofthe demand for liquidity and thetransformation function of banks is due to Diamondand Dybvig (1983).Theirpaper builds on the generalmethodology of optimal contracting under

asymmetric information outlined in Chapter2.We considera slightly moregeneralversion of their model,which consistsin an economywith acontinuum of ex ante identicalconsumerswho live for two periods.Theseconsumers have one dollarto invest at date zero in one of two projects:\342\200\242 A short-term project,which yields a grossreturn r > 1in period1.Thisinvestment canbe rolledover to yield r1 in period2.\342\200\242 A long-term projectwith the following cashflow overperiods1and 2:\302\243=0 t=l t=2-1 0 R>r2

The long-term project can also be liquidated at date 1and generate aliquidation value of L.

At date zero,consumersare uncertain about their future preferences.They canbe of two different types in the future: type-1consumers(0=0i)are \"impatient\" and prefer to consume in period 1;type-2 consumers(0i= 0z),are \"patient\" and are willing to postponetheir consumption if it

is worth doing so.These preferencesare representedby the following

type-contingent utility function:

TTf a\\ /\"(C1+77C2) when 0=0!]U(c1,c2;0)=<\\

(u(ji cx+c2) when 0=02J

where

\342\200\242

77 < 1and p.< 1areparametersrepresentingthe lossin utility from,respectively,

late consumption for type 1and early consumption for type 2.

399 RepeatedBilateral Contracting

\342\200\242 u(-) is twice continuously differentiable, strictly increasing,and concave.8

Theuncertainty about preferencesis resolvedin period1.Eachconsumerlearns exactly what her type is at that point. Naturally, this information

remains private.Consumers have independently, identically distributed preference

shocks;the ex ante probability of being a type-1 consumeris given by

Pr(0 = #i) = 7 Given that there is a continuum of ex ante identicalconsumers, there will bea proportion7of impatient and (1-7) ofpatientconsumers at date 1.A bank will be able to take advantage of this statistical

regularity to invest in the long-term projectdespitethe preferenceriskconsumers faceat date zero.

Thegeneral-resource-allocationproblemin this economyis to determinethe optimal mix of investments and the optimal revenue-sharingrule toinsure consumers against their preference uncertainty. This problemcan be representedas an optimal contracting problemunder asymmetricinformation. The problem is to maximize a representative consumer'sexpectedutility at date zero,subjectto a resourceconstraint and incentive-compatibility constraints.

As usual, it ishelpful to beginby solving the first-bestproblem,whereaconsumer'stype is assumedto be common knowledge.Thisproblemconsists in determining the amount jc \342\202\254 [0,1]to invest in the short-term projectand the consumption of each type.We denoteby cu the amount ofperiod t

consumption of a type-z consumer,t=l,2,i=l,2.Giveneach type'spreferences, it is obvious that a first-best optimal consumption allocationc% fort =1,2and i = l,2must specify

en = c*i = 0

The optimal mix of investment isalsoeasyto see:On the one hand, if r< L, then everything should be investedin the long-term project,and anamount

8.Diamond and Dybvig make the stronger assumption that

u\"(c) ,-c\342\200\224\342\200\224 >1

u\\c)

400 Dynamic Adverse Selection

should be liquidated in period l.9On the other hand, if r > L (and 12<R),then a fraction

= \302\243ii7

r

should be invested in the short-term projectand the remainderin the long-term project.We proceedwith the assumption that r>L.

Finally, optimal insurancerequiresthe equalizationof the marginal rateof substitution betweenperiod-1and period-2consumption with the

marginal rate of transformation, or

*/(<\302\243)

=.Ru'(c\302\243)

Thefirst-best contractis not attainable in generalin a second-bestworld.To be feasible,this contractmust satisfy the following incentive

compatibilityconstraints:

u\\cnj>u\\nc^j

and

u(c*2)^ u[rcnj

The first constraint ensuresthat an impatient consumerisnot better off

pretending to bepatient soas to obtain a higher return on her investment.

Similarly, the secondconstraint ensuresthat a patient consumercannot gain

by pretending to beimpatient. Note the important differencebetweenthesetwo constraints. A patient consumerneednot consumein period 1when

mimicking an impatient consumer;shecansimply withdraw her funds fromthe bank in period1and reinvest her savings in the short-term project.

The two incentive constraints are satisfied only if

1>77r

It is obvious that this condition doesnot always hold.But, evenif it doeshold, the first-best contractis not always attainable.Suppose,for example,that 77

=0,so that the first incentive constraint is always satisfied.Then thefirst-best contractsatisfiesthe secondincentive constraint only if c|>> rc\\vBut this assumption implies that

9. Diamond and Dybvig assume that r =L=1,and therefore their solution is such that all theconsumers' savings are invested in the long-term project.

ra'(cu)\302\243i?u'(rcu)

and this inequality cannot hold, for example,if the utility function hasdecreasingrelative risk aversion.10This observationestablishesthat thefirst-best contract is not always feasible in a second-bestworld.

Consequently, the second-bestcontractmay provideimperfectinsuranceagainst

preferenceshocks.It canbe shown in particular that it may result in an expost inefficient allocationof consumption such that c21> 0.

Four important economicinsights follow from the precedinganalysis:

1.As stressedfirst by Diamondand Dybvig (1983),the second-bestoptimalallocation,(cff,cff,cff, cff) (which may or may not coincidewith the first-

bestallocationc\302\243)

canbe implemented by a bank investing the consumers'savings in a suitable portfolioofprojectsand offering them demand-depositcontracts,such that the consumersarefree to determinein period1whether

to choosea consumption pattern (cff,cff)or (cff,cff).In otherwords,financial intermediation may emergeendogenouslyin this economyto providea (constrained)efficient liquidity transformation service.2.As stressedfirst by Jacklin (1987),the sameallocationcanbe achievedby setting up a publicly tradedfirm issuing equity to consumersat datezeroand paying dividends

ycff+(l-y)cffin period1.Impatient and patient consumersthen trade dividends for ex-dividend sharesin the secondarystock-marketin periodl.Thusaninstitutional setupwhereconsumersinvest in mutual funds, and wherethesefunds

hold shares in publicly traded firms, is in principleas efficient at

transforming liquidity as a bank.

10.Indeed, the inequality

ru\\c) <Ru\\rc)

implies that

u\"{c) \342\200\236 u\"(rc)-c\342\200\224\342\200\224 <-rc\342\200\224*\342\200\224-

u\\c) u'(rc)

With r > 1this inequality implies that the utility function u(-) has increasing relative riskaversion.

402 Dynamic Adverse Selection

3.The second-bestallocationis in generalstrictly better when consumerscannot reinvest the funds they withdraw in periodl.Toseethis point, it

suffices to comparethe two incentive constraints

u(iici?+c$g)> uiiicff +cf)and

u(iics2?+eg)> u{rcsnB +eg)The first constraint appliesif the reinvestment option is not available.Thesecondone allows for reinvestment and is thereforemore demanding. AsJacklin (1987),Diamond(1997),and Allen and Gale (2000),among others,have argued,a bank-basedfinancial system may be able to preventreinvestment of savings in period1and thus be able to providea better form

of insuranceagainst liquidity shocksthan mutual funds holding shares in

publicly traded firms and operating in the secondarymarket. In otherwords, it isoptimal to provideinsurancewith nontradedinstruments.

4. As stressedby Diamond and Dybvig (1983),however,the optimaldemand-depositcontractofferedby a bank may give riseto bank runs. Thebasicidea here is that if all type-2consumersdecideto mimic type-1consumers, then theremay not beenough money in the bank to meet allwithdrawal demands.When such an event happens,the bank fails, and the

anticipation ofsuch a failure is preciselywhat may trigger a run on the bank

by all type-2consumers.To the extent that the secondarymarket solutionis run-proof, it (weakly) dominatesthe solution with financial

intermediation,where depositorsare allowedto reinvest their savings in the short-

term investment in period1.9.2.2Optimal Contracting with Two IndependentShocks

We now extendthe previousanalysis by assuming that the consumerfacestwo independentincomeshocksrather than one preferenceshockat the

beginning ofperiod1.We closelyfollow the treatment in Townsend(1982).Considerthe dynamic contracting problembetween two parties:a risk-

averseconsumerand a risk-neutral bank.The consumer'stime-separableutihty function is u(c^)+u(c2),wherect denotesconsumption in period t =1,2.We assumefor now only that m(-) is strictly increasingand strictlyconcaveand that u'(0)= +\302\260\302\260. Laterwe considerspecificfunctional forms toobtain closed-formsolutions.The consumerfacesrandom incomeshocks.

403 RepeatedBilateral Contracting

Herincomeis independently and identically distributed in the two periodsand takes the value w = 1with probability p and w = 0 with probability

(1-p).The bank has sufficient wealth to be able to lend to the consumerwhatever is required.For simplicity we set the equilibrium interestrate tozero.In this setup the optimal contracttakes the form of an endowment-contingent transfer {b^w^;b2(w1;w2)}.Let w1 = (h^) and w2= (w1;w2); then

the first-best problemis the solution of

maxE{u[w1+b1(w1)]+u[w2 +b2(w2)]}

subjectto the individual-rationality constraint

EfoM+Mw2)]^Sinceinterestrates are equal to zero,the individual-rationality constraint

reducesto a condition that net expectedtransfers from the bank to theconsumer should be nonpositive.Assigning Lagrange multiplier A to the

individual-rationality constraint and differentiating with respectto ^(w'),we obtain the following first-orderconditions:

u'[wt +bt(wt)] =k for?=1,2so that, unsurprisingly, the optimal consumption stream of the risk-averseconsumermust beconstant:

wt+bt(wt)=p

which implies

b1(0)=b2(w1;0)=pb1a)=b2(w1;l)=p-lBesidesthe fact that the first-bestcontractinvolves perfectinsurancefor

the consumer,the other important property is that there appearsto benogain from writing a long-term contract.The repetition of the staticcontract\342\200\224that is,spotcontract[b(0)=pand b(l)=p- 1]\342\200\224achieves

the sameallocation.

Supposenow that the incomerealizationwt is private information to theconsumer.It is clear then that the first-best spot contractisno longerfeasible, sincea consumerwith incomerealizationwt =1would have anincentive to claim wt = 0 in order to hold on to her endowment and receiveapositivenet transfer bt(fit) =p.

404 Dynamic Adverse Selection

If the first bestis not implementable through a sequenceof spotcontracts, what canbe implemented through such a sequence?The answer tothis question leads to a striking insight: the only incentive-compatiblesequenceof spot contractsis no insurance.Toseethis, beginwith the last

period.Any differencein net transfers b2(0)*b2(l)would lead to aviolation of incentive compatibility, the consumerhaving a strict incentive to

always collectthe higher net transfer. Hence,there canbeno insuranceatdate t =2.Working backward to period1,it is straightforward to observethat there cannot be any differencein net transfers in period1either.Doesthis observationimply that there cannot be any scopefor insurancewhen

incomeshocksare privately observed?Fortunately no:Someinsuranceis

possiblewhen the consumerand the bank can write a long-term contract.To show this possibility, Townsendfirst considersa simple debt contract

with zero interest,that is, b2(w1;w2) =-\302\243i(wi). Facedwith such a contract,

the consumerchoosesb^w^) to solve

max u\\wx +bi {wx)]+E{u[w2 -bx {wx)]}

The optimal b-Syv-^) then satisfiesa familiar Euler equation:

uTwi + bi(w{j\\ = E{u'[w2-biiwi)]}

In general,the solution to this equationis such that\302\243i(l)

< 0 <\302\243i(0),

sothat, through this borrowing and lending behavior,the consumerends up

\"purchasing partial insurance\" in period1.This insurance, however,comesat the expenseof forgoing someintertemporal consumption smoothing, in

the following sense:The consumerstrictly prefersmore intraperiodinsurance in period1to an equalexpectedconsumption acrossperiods.

Townsend'snext important insight is that simple borrowing/lending

arrangements with b2(wi,w2) = -b^Wi) are not second-bestefficient in

general.To seethis, note that, by revealedpreferenceand the strict

concavity of m(-), the consumer'sincentive constraints

u[l + fetCl)] +E{u[w2 -^(1)]}> u[l+^(0)]+E{u[w2 -/^(O)]}

and

ufo(0)]+E{u[w2 -/^(O)]}> ufo(l)]+E{u[w2 -Ml)]}

are both slack.Therefore,it is possibleto find a better long-term contractthan a simple borrowing/lending contractb2(wi,w2) =-b^w-i).

405 RepeatedBilateral Contracting

To characterizethe second-bestlong-term contract,note first that any

incentive-compatiblelong-term contractrequiresa net transfer in period2that is independentof the consumer'stype:

b2(w1;0)= b2(yv1;l) = b2(w1) forWi =0,1As pointedout previously, the consumerotherwisehas an incentive to mis-

report period-2income.Therefore,the second-bestcontracting problemreducesto

max p{u[l+b,(1)]+[pu(l+b2 (1))+(1-p)u(b2 (1))]}+

(1-pMbL\302\256)] +[pu(l+b2(0))+(1-p)u{b2 (0))]}

subjectto

u[l+^(1)]+{pu[l+b2(1)]+(1-p)u[b2(1)]}>

u[l +^(0)]+{pill+b2(0)]+(1-p)u[b2 (0)]} (IC1)

uMO)]+{pill+b2(0)]+(1-p)u[b2 (0)]}>

uMl)]+{pill+b2(1)]+(1-p)u[b2 (1)]} (ICO)

and

pMl)+b2(1)]+(1-plhiO)+b2(0)]< 0 (IR)

An educatedguesssuggeststhat only (IC1)is likely to bebinding at the

optimum.We have alreadyestablishedthat an optimal second-bestcontractisnot

of the form of a simple borrowing/lending contract such that b2(w{) =-biiyv-i). This simple contractform is optimal only conditional on therealization of wx and doesnot provide sufficient coverageof incomerisk in

period1.Toseehow the optimal second-bestcontractcanimprove risk sharing by

trading off intraperiod risk at date 1against intertemporal consumptionsmoothing, it is helpful to considerthe following simple exampleproposedby Townsend:Let

u(w) =w -yw2

with y<\\-Let/?= \\, and considerthe classof contractswith balancedtransfers accrossincomestatessuch that

bt(X) +bt(0) =0 for?=1,2

406 Dynamic Adverse Selection

This classofcontracts,wherethe (IR) constraint is always satisfied,introduces some flexibility not availableunder a simple borrowing/lendingschemeby removing the requirement that transfers over time must net out

[bi(wi) + b2(w1)=0].It is,however, morerestrictivethan a simpleborrowing/lending contract in that it requires transfers to balance out acrossincomestates within a period.Note that while this contractclasscan beshown to improve upon the simple borrowing/lending schemeconsideredearlier,it isnot necessarilysecond-bestoptimal.

As it turns out, the completecharacterizationof the second-bestoptimalcontractis rather involved and not particularly insightful, even if theconsumer is assumedto have a simple quadratic utility function. We thereforelimit ourselveshere to showing that the contractwith balancednessacrossincomestatesdominatesthe contractwith intertemporal balancedness.

Letting b2 =-aib\\ with 0< a< 1and assuming balancednessacrossstates,the consumeroptimization problemreducesto

max^ 1-bi-yO\342\200\224 brf +\342\200\224[1+0*! -yil+abx)1+abx-y{ab1)1\\\\+

[bi-7(bi)2+|[l-ob,-ra-ab,)2-abx-yC-c^)2]}

subjectto

l-&i-y(l-6i)2+\342\200\224[l+afei -/(l+afci)2-1-0*!-y(aZ?i)2]=

1+fci -y(l+/3i)2+-[l-c(b1-Y(l-ab1)2-o(b1-yi-abrf]This assumesthat constraint (ICO)is not binding. Differentiating the

Lagrangeanwith respectto bx and a then yields, after straightforward

computations,

l-2y . , . 1a = \342\200\224-J- < 1 and b^ =1-7 2(l+a2)

Instead,under a simple borrowing/lending scheme,the consumercouldchooseshow much to borrowafter the first incomerealization.After beingunlucky in period1,the consumer'sproblemis

mSaib1-7(b1)2+^[l-Ih-ra-b1)2-b1-y(-b1)2]

407 RepeatedBilateral Contracting

which yields b1=\342\200\224. After being lucky in period1,the consumer'sproblemis

max l-h -yd-hf +i[i+h-rd+hf+h -yfo)2]

which onceagain yieldsb^= \\. With quadratic utility, the consumertherefore savesasmuch after beinglucky assheborrowsafter being unlucky. Herbehavior therefore satisfiesbalancednessacrossincome states,and is aspecialcaseof the previouscontract,with d=1by construction, and bi =

\\.It is now immediately clearwhy simple borrowing and lending is subopti-mal: It provideslessthan optimal insuranceagainst period1incomerisk:

1 14

<2(1+a1)

since

a = -<11-7when 7<7.

Intuitively, when balancednessacrosstime isnot required,the consumerendsup consuming more in period1after an unlucky incomerealization,sincepart of this consumption is \"compensated\" by lowerconsumption in

period1after the lucky incomerealizationand not solelyby lowerperiod-2 consumption (whether or not period-2 income is high). Insurance,however, is only partial in period1(and this remains true for second-bestinsurance):For period-1insuranceto be incentive compatible,it must still

be accompaniedby some \"compensating transfers\" in period 2, which

(becauseofincentive compatibility again) cannot dependon the realizationofperiod-2income.Sincethesetransfers destabilizeperiod-2consumption,they involve a costand thereforemake full period-1insuranceunattractive.

To summarize, the analysis of the two-periodproblemhighlights how

second-bestrisk sharing requirestrading off more intertemporalconsumption smoothing for intratemporal risk sharing than under a simpleborrowing/lending scheme.This analysis echoesand complementsDiamondand

Dybvig'sobservationsdiscussedearlier.It illustrates in a simple and

striking way that equilibrium risk sharing obtainedby trading debt claims in acompetitive securitiesmarket resultsin suboptimal risk sharing in general.That is,better risk sharing isobtainableby nontradableclaims issuedby a

Dynamic Adverse Selection

financial intermediary that is free to imposeany rate of intertemporaltransformation that is desirable.

Another conclusionemerging from this simple two-periodproblem isthat, given that second-bestinsuranceis only partial, endogenousincomeinequalitiesemergein this setup betweenthoseconsumersthat had a luckyrun of incomeshocksand those that were lesslucky. A natural questionraisedby this analysis ishow the incomedistribution evolvesunder second-bestrisk sharing as the number of periodsis increasedbeyond t = 2.Weturn to this questionin the next subsectionby extending Townsend'scontracting problemto an infinite-horizon setting.

Second-BestRisk Sharing betweenInfinitely LivedAgents

As the analysis of the two-periodproblemmakes clear, it is difficult toobtain sharp characterizationsof the optimal contractfor generalconcaveutility functions u(-).This difficulty is compoundedin the infinite-horizon

contracting problem.We therefore limit attention to the most tractablefunctional form for the agent'sutility function and assumethat the infinitelylived consumerhas CARA risk preferencesrepresentedby the utility

function

u{c)=-e~rc

We closelyfollow the treatment of this problem in Green(1987)andThomasand Worrall (1990)[seeAtkeson and Lucas,1992,for thetechnically

moreinvolved analysis of this problemwhen the consumerhasCRRArisk preferencesrepresentedby a utility function of the form u(c) =(<*\342\226\240+-1)1(1-r)].

The risk-neutral financial intermediary and the risk-averseconsumerhave the common discount factor5e (0,1).As in the two-periodproblemconsideredearlier, this consumeris faced with an i.i.d.binomial incomeprocess,such that in any given periodher incomeisw = 1with probability

p and w = 0 with probability (1-p).The consumercan sign a long-terminsurancecontractwith the risk-neutral bank. If incomeisobservable,then

the bank would breakevenby offering the consumera constantconsumption flow ofp and taking on all the residualincomerisk. Hence,the first-

bestflow utility levelattainable by the consumeris given by u(p) =-e'^,and the first-best discountedpresent value of this steady consumptionstream is given by

The worst outcomefor the consumeris autarky, where no insurance isobtained.In that casethe expectedflow utility of consumption is given by

-[pe-'+{l-p)]and the presentdiscountedvalue of the random consumption streamunder

autarky is

Note that an important implicit assumption here is that incometakes theform of a perishablegood.

Even if incomeshocksare private information, we know from theprevious sectionthat it is possiblefor the consumerto improve expectedflow

utility of consumption by writing a long-term incentive-compatibleinsurance (or loan) contract.Let 5*= {0,1}'denote the set of samplepaths ofincomefrom period t= 0 to t = t and let H = (w0; w-c...; wt) e 5*represent a history ofincomerealizationsup to time t; then a long-term contractis a sequenceof contingent net transfers b^h')for t =0,...,\302\260o. Although

solving for an optimal incentive-compatiblelong-term contractisapotentially daunting problem,it is possibleto obtain a reasonablysimplecharacterization of the optimal contract in the specialsetting consideredhereby exploiting the stationarity of the underlying problem.

In the previoussectionwe formulated the optimal contracting problemas a constrainedmaximization problemfor the consumer.In the infinite-

horizon problem it turns out that the dual problem of maximizing thebank'sexpectedreturn subjectto incentive-compatibility constraints andan individual-rationality constraint for the consumeris more tractable.Denotethe consumer'soutside option by v. The consumer'sindividual-

rationality constraint can then be written as

E X5'MM/*')+w,]} >v

As one might expect,this constraint is always binding at the optimum.Turning to the consumer'sincentive-compatibility constraints, denote by

&i(\" ) and\302\243>o(\" )tne transfer from the bank to the consumerin period t

when the consumerhas reported a history of incomerealizationsup to

410 Dynamic Adverse Selection

period t -1of A'-1 and reports incomerealization1and 0, respectively,at date t. Sincethe consumer'sincome is identically and independentlydistributed over time, the expectedfuture discountedvalue of the contractat time t can dependonly on past transfers basedon reported history andnot the actual history H. Indeed,since income is perishable,the only

intertemporal link here concernsthe transfers the consumercan expecttoreceive in the future, and these transfers dependonly on past reportedhistory of incomerealizations.We can, therefore,write the consumer'scontinuation value under the contractat date t as,respectively,v^A'-1) =v[\302\243i(A'-1)] and v0(AM) =

v[\302\2430(A'_1 )],so that the consumer'sincentive

compatibility constraints are

u[bi(h'-1)+1]+Sv^h'*1)> u[b0 (A'\"1) +l]+5v0(A'\"1)

and

liboih'-1)]+dvoih'-1)> u^A'\"1)]+dv^h1-1)

for all A'-1 e 5M and all t > 0.It is instructive to stressthe similarity betweentheseincentive constraints and (IC1)and (ICO)in the previoussubsection.

Having determinedthe bank'sconstraints, the next stepis to describethebank'sobjective.We can definethe bank'scontinuation value under thecontractas

-E I>W)In other words, the bank'spayoff is the expectedpresent discounted

value of future repayments.Taking advantage of the recursivestructure ofthe underlying contracting problem,we can definethe bank'svaluefunction to be the unique solution of the Bellmanequation

C(v)= min Mb1+8C(y1)]+(X-p)[b0+Se(y0)1b

subjectto

plufa +1)+8Vl] +(1-p)[u(b0) +5v0]= v (IR)and

uQh +1)+5vi > u(b0 +1)+dv0 (IC1)

411 RepeatedBilateral Contracting

u(b0 )+8v0>u(]bl) +8v1 (ICO)When the incomeprocessis observableto the bank, sothat incentive

constraints canbeignored,a flow utihry levelu (lessthan 0)canbeguaranteed atminimum costto the consumerby choosingnet transfers h\342\204\242 and b(Bsuch that

-r{b[B+l) -rb\342\204\242-e K1 '=-e \302\260 =u

or

i+b[B=--\\og{-u)=biBr

Therefore,the flow costto the bank of guaranteeing utility levelu equalto (1-5)v (lessthan 0) is given by

pb[B+d-p)b0FB =--log(-u)-Jp=--[log(l-5)+log(-v)]-pr r

Hence,the first-best discountedexpectedcost to the bank is

logCl-+logC-v)-^\"1[pb[BHl-p)b$B]=-1-8 {l-8)r =CtB(v)

When the incomeprocessis unobservable,the bank cannot offer an

incentive-compatiblecontractthat perfectlyinsuresthe consumer.Itscostis then higher becauseit needsto pay an extra insurancepremium tomaintain the same level of flow utility u. The incentive constraints (IC1)and

(ICO)canbe rewritten in slightly moreconvenientform as follows:

e~rU\\ +8v\\ > e~ri(o +dv0

and

Uq+5v0>ux -I- dvx

whereu0= -e~rbo and u1=-e~rbi.The two constraints canbe rewritten as

\302\253o

-\"i ^ 5(vi -v0)> e~r(ito -ih)

Note that, just asinTownsend'ssetting, weexpectto have at the optimum

U\\ < Mo and v0 < vx

that is, the consumerreceivesa net transfer when unlucky, but, toguarantee incentive compatibility, it comesat the cost of a lower continuation

utility. Both constraints are thereforeunlikely to bebinding simultaneously

412 Dynamic Adverse Selection

at the second-bestoptimum. Moreover,sinceb(B <b\342\204\242,

it appearsthat therelevant incentive constraint is (IC1),the one preventing the high-incometype from mimicking the low-incometype,

-e\"r(u1-ito)=5(v1-v0)

Consequently,the bank'soptimal-contract-designproblembecomes

C(v)= min\\ p \342\200\224logc-uo-i+secvi)r +a-p)\342\200\224log(-M0)+5C(v0)r

subjecttov =p(e~ru1+8v1)+(X-p)(u0+5v0)

and

-e~r(mi-u0) =5(vi -v0)

The form of the first-best costfunction CFB(v)suggeststhat a likely formof the second-bestcost function C(v)is

C(y)=k-log(-v)Q--8)r

wherek is a constant that remains to be determined.Indeed,substitutingfor this functional form in the precedingBellmanequation, we obtain

C(v)=mim p

+(X-p)

Ulog(_Ml)_1+A M^. r \\ (X-5)rJ_

-ilog(-H\342\200\236)-l+^-^).. r V (l-d)r).or, after somemanipulations,

C(v)=min<{ 8k-1\342\200\224 ^tM^Mt\342\231\246M^SftW;

log(-v)d-8)r

Letting

f(k,v,Vi,Ui) =8k-l\342\200\224\\p \342\226\240OteM?+(X-p)*TH\302\243H7

413 RepeatedBilateral Contracting

it is then apparent that the solution to the Bellman equation has the

requiredfunctional form

C(y)=mm<f(k,v,vi,ui)-L,. , , ., ., {1_d)rSinceC(v)is now expressedin terms of utilities relativeto v, that is, in

terms of ratiosujv and v,/v, it is convenient to further simplify the bank'sproblemby denoting11

v,- = fl;v and u{ = g,-v

Note that all utilities are negative, so that the a,'sand g,-'sare positive.The bank chooses(ai,gi, a0;g0) to solve

. 1Jmm \342\200\2241 p(ai;gi); r L(ao;go)

loggi+f1_5Jloga1+0--P)loggo+f Jlogflo

subjectto

d(a1-a0)=-e-r{g1-g0)and

pie-'gi+Sd!)+(X-p)(go+8ao)=l

(IC1)

(IR)

Note that the solution of this convexminimization program is

independent of k. Moreover,it canbecheckedthat the optimal solution entails

ax < a0 and g0 < gi

which, sincethe utilities are negative, implies ux < u0 and v0 < vx. Thissolution alsosatisfies(ICO),the other incentive constraint:12

11.Since the consumer has a CARA utility function, one can summarize the value of thecontract in terms of certainty equivalent. In other words,one can write v(b) = u(b, +n) (where ndenotes the risk premium). This means here that

v, =\342\200\224u,

8i

12.This constraint can bewritten as

g-0v +&z0v>g1v+&1v

Since v is negative, this is equivalent to

O\"(fl0-fli)<(gi-g0)

which is true, given the binding incentive constraint (IC1).

414 Dynamic Adverse Selection

As in the two-periodproblem,the consumercan thus obtain intraperiodinsurance against adverse income shocks(u0 > Ui) at the expenseofintertemporal consumption smoothing (v0 < v-j).A number of otherstriking implications emergefrom this analysis.First, sincevx > v0, the

inequalityin welfarebetweenlucky consumers,who have high incomeshocks,and

unlucky consumersgrows over time. Second,if we denote by vt theconsumer's utility at the beginning of period t, then vt convergesto -\302\260\302\260 almost

surely (seeThomasand Worrall, 1990).Theproofof this result relieson theobservationthat C'(vt) is a martingale or, in other words, that

\302\243[C'(v,+1)]=C'(v,)

Hence,by the martingale convergencetheorem, C\"(v()convergesalmost

surely to somelimit. Thomasand Worrall show that this limit must bezeroand thereforethat lim^vt \342\200\224\302\273 -\302\260o. Intuitively, the reasonwhy the limit mustbe -oois that any finite limit can occuronly with probability zerobecausefuture v,'s would always spreadout from that point. One way ofunderstanding the economiclogicbehind this striking result it to think of the

infinitely lived consumeras living permanently aboveher meanswhen shehasa negative incomeshockand continually postponing the pain of having to

repay her accumulateddebtsto the future. Such behaviormakes sensewhen the consumerdiscountsthe future. A compounding effectcomesfromthe convexity of C(v).Other things equal,the bank'scostis higher the morespreadout are the v,'s (to maintain incentive compatibility), but thecurvature of C(v)is flatter at low valuesof v. Therefore,the costof mamtaininga spreadbetweenvx and v0is lower,the loweris v\342\200\224hence the incentive forthe bank to let vt drift downward. There are two potential constraints onthe second-bestoptimal processvt. First, asdebtsaccumulate,the consumermay be tempted to default. Toavoid default the bank will limit the amountof debt the consumercanaccumulateand thus limit her ability to consumeaboveher means.As a result, there would belessintraperiod insuranceandmore intertemporal consumption smoothing. Second,if there wascompetition betweenmultiple banks, and consumerscould switch at any timebetweenbanks, then banks would also be limited in the extent to which

they can trade off intertemporal consumption smoothing to increaseintraperiod insurance.Indeed,the downward drift in vt would be reducedby banks'attempts to stealaway previously unlucky consumersby offeringthem better savings terms.

415 RepeatedBilateral Contracting

9.3Summary and LiteratureNotes

Thischapterhas first dealtwith dynamic adverseselectionmodelswith

constant types. Lackof commitment can arise in \"mild\" form when long-termcontracts protect each party against unilateral violations but the partiescannot commit not to engagein future Pareto-improving renegotiations;itarisesin moresevereform when only short-term contractsare available(forexample,becausethe principal is a government that cannot commit not to

unilaterally changethe law).In the first case,the principal cannot commitnot to be too soft with the agent in the future (this is the durable-good-monopoly problem or the soft-budget-constraint problem),while in thesecondhe canalsonot commit not to be too tough (this is the ratchet effect).In both cases,the principal is hurt becauseoutcomeshe finds sequentiallyoptimal are not ex ante optimal.

Commitment problemsarisebecauseinformation revealedthroughcontract executionleadsto recontractingopportunities that hurt the principalfrom an exante point ofview. A generallessonfrom this literature thereforeconcernsthe desirability of limiting the information revealedin the courseof contractexecution,that is,of signing contractswith (partial) pooling.Theform ofoptimal contractshasbeenanalyzed in detailby Freixas,Guesnerie,and Tirole(1985)and Laffont and Tirole(1988a)for the caseof short-termcontracting, by Dewatripont(1989)for the caseof long-term contractingwith renegotiation(and the renegotiation-proofnessprinciple),and by Hartand Tirole(1988)and Laffont and Tirole(1990)for both casessimultaneously.

For further theoreticaldevelopments,seein particular (1)Beaudryand Poitevin (1993)for an analysis of instantaneous renegotiationin

signaling games,(2) Rey and Salanie(1996)for a discussionof the possibility ofachieving long-term commitment through asequenceof(renegotiated)two-

periodcontracts,(3) Besterand Strausz (2001)for a generalizationof therevelationprincipleto the caseof imperfectcommitment (where \"truth

telling\" occurs only with probability less than one, due to equilibrium

pooling);and (4)Bajariand Tadelis(2001)for an analysis of imperfectcommitment and costly renegotiation.

The chapterhas alsolookedat specificapplications,the most influential

being the regulation setting due to Laffont and Tirole(seetheir 1993book).In terms of contracttheory, we have stressedseveraleconomicinsights:

\342\200\242 In the caseof a buyer-sellermodel(\"Coasiandynamics\,") the optimality

416 Dynamic Adverse Selection

of salecontractsoverrental contracts,to avoid the ratcheteffect(Hart andTirole, 1988).This is in contrast with the result obtained under buyer

anonymity, where renting is a way of avoiding all commitment problems,by making the problem of the sellerstationary (on this versionof the

durable-monopolyproblem,see,for example,Stokey,1981;Bulow, 1982,orGul,Sonnenschein,and Wilson, 1986).\342\200\242 The potential optimality of exogenouslyrestricting the observability ofcontract execution for the principal.While this is never optimal in

the absenceof commitment problems,becauseit worsens incentive-compatibility constraints, it can be desirableas a way to limit the harmcaused by limited commitment. This problem has been analyzed by

Dewatripontand Maskin (1995a)in the contextof insurancecontracts.Arelated insight is due to Cremer(1995),who stressesthe benefitsof \"arm's

length\" relationships.Heconsidersa dynamic modelwherea principal hiresan agent who has private information about her intrinsic productivity. Theagent'soutput dependson this productivity and alsoonher effort.Theprincipal

would like to incentivize the agent by threatening to fire her in caseoflow pasteffort.Thisthreat may not becredible,however,becauseCremerassumesthat, onceeffort hasbeenchosen,the principal'sbenefit from notfiring the agent solelydependsonher intrinsic productivity, not onher pasteffort. If the principal observeseffort or productivity and cannot commit in

advanceto a firing rule, the agent then knows shewill not bepenalizedforlow effort and therefore shirks. Keeping the agent \"at arm's-length\" isuseful in this case:Sincethe principal observesonly output and not effortor productivity, he may rationally attribute low output to low intrinsic

productivity, which makes the firing rule credible.This is thus another casewherereducing observability alleviatescommitment problems.\342\200\242 Reduced observability also helps alleviate the soft-budget-constraintproblem, that is, the refinancing of loss-making projects.As argued by

13.For a comprehensive treatment of the soft-budget-constraint literature, seethe survey byKornai, Maskin, and Roland (2003).This issue has received a fair amount of attention in the\"transition economics\" literature (seeRoland, 2000).Note that dynamic adverse selectionmodels have also been used to analyze aspects of the political economy oftransition, namely,the question of optimal industrial restructuring under political constraints (seeDewatripontand Roland, 1992,which builds on Lewis,Feenstra, and Ware, 1989).An interesting feature ofthis setting is that political constraints represent an intermediate form of long-termcontracting

between a government and its electorate: Prior agreements remain in force as long as amajority of the electorate refuses to change them. An agenda-setting reform-mindedgovernment, however, will try to undo them by endogenously targeting specific groups of thepopulation that can form a proreform majority.

417 RepeatedBilateral Contracting

Dewatripontand Maskin (1995b),this problem can naturally arise when

initial investments are sunk by the time the creditorrealizesthe projecthas a negative net present value.13In this setting, making it necessarytohave the refinancing performedby a new, lesswell informed creditormayharden the budget constraint and improve overallefficiency.As stressedby Dewatripontand Maskin (1995b)and vonThadden(1995),however,this

gain comesat the expenseof long-run risk taking. Theconnectionbetweenthe number ofcreditorsand the probability of refinancing with commitment

problemshas alsobeenanalyzed by Boltonand Scharfstein(1996).The main driving factorof dynamic-adverse-selectionmodelswith

constant types is the information revealed through contractexecution.When

types are independentacrossperiods,things change drastically, and themain focusis the trade-offbetweenintraperiod risk sharing and

intertemporal consumption smoothing. Threecaseswereconsidered:\342\200\242 First, a simple two-periodconsumption modelwhere the agent learnsatthe beginning ofperiod1that he needsto consumein period1or canwait

until period 2.This simple model of liquidity shocks,put forward byDiamondand Dybvig (1983),allowsone to highlight the benefitsofbankingand of deposit contracts\342\200\224but also,as stressedfirst by Jacklin (1987)concerning equity/bond contracts\342\200\224as a way to pool independently distributed

individual liquidity shocks.Theselatter instruments seemto bepreferable,sincethey are not subjectto bank runs, a potential equilibriumphenomenon in the Diamond-Dybvigsetting. However,one canarguethat a \"bank-

basedsetting\" may be able to allow for potentially more risk sharing, by

limiting the reinvestment opportunities of agentswho considerwithdrawing

their funds early while they need to consumeonly later on.This issuehas beenstressed,for example,by Diamond (1997)and Allen and Gale(2000).TheDiamond-Dybvigmodelhas in fact generateda largeliteratureon banking and liquidity provision; seeBhattacharya and Thakor (1993)and Freixasand Rochet (1997)for surveys.\342\200\242 The secondcasewe consideredis that of two independently distributed

incomeshocks.Pioneeredby Townsend(1982),this casehas allowedus tostressthe trade-off between insurance and intertemporal consumptionsmoothing. Specifically,in a static setting, there isno way to induceseparation of types, sincethe agent would always report the incomerealizationthat allowsher to receivethe highest net transfer. Instead,with two incomeshocks,one canmake the (constant)second-periodtransfer contingent on

418 Dynamic Adverse Selection

the incomereported in period1.In caseof a negative (positive)period-1incomerealization,the agent then optimally receivesa positive(negative)transfer, which then leadsto a negative (positive)transfer in period2.Thistransfer is beneficialbecause,in the caseof a negative (positive)incomeshockin period1,the agent'speriod-1marginal utility ishigher (lower)than

her expectedperiod-2marginal utility Note that some of this protectionagainst temporary incomeshockscanbeobtainedby simple borrowing and

lending. Townsendshows, however,that optimal long-term contractsgobeyondpure borrowing/lending to allow for more intraperiod insurance,atthe expenseof intertemporal consumption smoothing. As one might have

guessed,the distinction between situations with preference shocksandincome shocksis somewhat artificial. Indeed, the banking model ofHaubrich and King (1990)highlights how these two casesgive riseto

fundamentally similar economicproblems.\342\200\242 The third casewe consideredextendsthe Townsendmodelto an infinite

horizon.Intuitively, as in the two-periodmodel,agentswho experiencenegative income shocksreceive positive net transfers at the cost of lowerexpectedfuture consumption. Green(1987),Thomasand Worrall (1990),and Atkeson and Lucas(1992)have providedmfinite-horizon extensionsof the Townsendmodel.We have seenhow to computea stationarysolution of this problem.This solution leadsto increasingwealth dispersionovertime. Moreover,as shown by Thomasand Worrall, the probability ofreaching

zerowealth tends to 1in the long run! Lucas(1992)discussestherelevance of thesepredictions.Let us stresshere that it isnatural to think that

limits to contractenforcement(e.g.,the possibility of default) would

partially call into questionthese limit results (see,Kocherlakota,1996,and

Ligon,Thomas,and Worrall, 2002,for analysesof the implication of limitedcontractual enforcement).

Dynamic MoralHazard

In reality most principal-agentrelationshipsare repeated or long-term.Repetitionof the relationcan affect the underlying incentive problemin

three fundamental ways. First, repetitioncanmake the agent lessaversetorisk, sinceshecanengagein \"self-insurance\" and offseta bad output shockin one period by borrowing against future income. More generally,

repetitionintroducesthe possibility of intertemporal risk sharing. Second,repeatedoutput observationscan provide better information about the

agent's choice of action.Both of these effectsamelioratethe incentive

problemand increasethe flow surplus from the contract.A third

countervailing effect,however, is that repetitionincreasesthe agent'sset ofavailable actions.Shecan now choosewhen to work, and she can offseta badperformancein one periodby working harder the next period.

As one can easilyimagine, the interaction of these three effectsresultsin a considerablymore complexcontracting problem than the one-shotproblem.As we saw in Chapter4, the one-shotproblemis alreadytoosensitive to the underlying environment to be able to predictin a robust way

the simple spotcontracts(suchaspiece-rateor commission contracts)that

are observedin practice.However,although repetitionincreasesthe

complexity of the optimal contracting problem,it can paradoxicallyresult in

simpler optimal contractsin somespecialsettings, as we shall see.In general,optimal long-term contractscan unfortunately be

considerablymorecomplexthan simple one-shotcontracts.Indeed,the optimal

contract will dependin generalon the entirehistory of output realizationsandnot just on someaggregatemeasureofperformanceoversomefixedinterval of time. When an output outcomeaffectsthe agent'scurrent

compensation, it will alsoaffecther future rewardsunder an optimal contract.Inlight of this potential addedcomplexity, it is all the more pertinent to

inquire whether there are plausibleconditionsunder which optimal long-term contractsare simple.We shall devotesignificant spaceto this question.

Following the characterizationof the optimal long-term contract in ageneralsetting, where the one-shotcontracting problemis repeated twice,weshall determinethe conditionsunder which a seriesofshort-termincentive contracts,which regulatethe agent'ssavings, canreplicatethe optimallong-term contract.We then proceedto inquire under what conditionssimple incentive contractssuch as linearcontractsor efficiency-wage-typecontractsare optimal or nearly optimal. Next, weconsiderrenegotiationoflong-term contractswith moral hazard,when the agent'sactionshavepersistent effectson output. Finally, having consideredpartial deviations from

fuU-commitment contractswith renegotiation,weshall considerthe caseofno commitment whereimplicit as well as explicit incentivesare importantin affecting the agent'schoiceof actions.

Unfortunately, the literature on repeatedmoral hazarddoesnot considera common framework. In particular, different assumptions about when the

agent consumes,accessto credit,and controlof savings are made,and as aresult it is difficult to relate the different papersto one another.We basepart of this chapter on the unifying treatment proposedby Chiappori,Macho,Rey, and Salanie(1994).

10.1TheTwo-PeriodProblem

Supposethat the one-shotcontracting problemconsideredin Chapter4 is

repeatedtwice.That is,at the beginning ofperiodt=l,2the agent privatelychoosesan actiona e A. Thisactionthen mapsinto n possibleoutputrealizations in period t. The probability distribution of output q\\ in period t is

given by Pi(at), where at denotesthe agent'sactionchoicein period t andi =1,...,n.As in Chapter4,we assumethat pip)> 0 for all a e A Sincethe probability distribution over output outcomes in any given perioddependsonly on the actionchosenby the agent in that period,and sincethe actionset is the samein both periods,the two periodscanbe thoughtof as technologicallyindependent.The only possiblelink betweenthe two

periodsis through changesin preferencesofprincipal and agent in responseto output outcomesor actionchoicein the first period.Only in such a setupis it possiblethat a seriesof two spot contractsmay be optimal.

Tomaintain this separationbetweenperiods,we assumethat the

principal'sand agent'spreferencesare time separable.In eachperiodthe agent's

objectivefunction is,asbefore,

u(c)-iff (a)

where c denotes the monetary value of consumption. As in Chapter 4,we assumethat u{-) is strictly increasingand concaveand y/{-) is strictly

increasingand convex.We alsoassume thatlimM\302\243 u{c)= -\302\260\302\260 for somec> -\302\260\302\260.

Moreover,the agent has an identicalreservationutility in both periodsu.Theprincipal'sobjectivefunction is V(q -w), wherew is the wagepaid tothe agent.Again, V(-) is strictly increasingand weakly concave.That is, the

principal canbe risk averseor risk neutral.The timing of the contracting gamebetweenthe principal and agent is as

follows:

RepeatedBilateral Contracting

\342\200\242 At the beginning of period t =1the principal makesa take-it-or-leave-itofferof a long-term contractto the agent.Thiscontractspecifiesan actionfor the first periodax and an actionplan for the secondperioda2(q\\), aswellas a contingent wagescheduleWi(q\\) and w2(q], ql).\342\200\242 The agent can acceptor reject the offer.If the agent rejects,the gameends.If the agent accepts,she proceedsto choosean actionplan ax in thefirst period.\342\200\242

Following the realizationof first-periodoutput, the long-term contractcanbe renegotiatedand replacedby a new continuation contractspecifying

an actiond2 and compensationw2(q$).Again, the principal makesa take-it-or-leave-itofferand the new contractreplacesthe old one only if both

principal and agent are strictly better off under the new contract.

As we shall see,the optimal long-term contractofferedat the begiriningof period 1will be \"renegotiation-proof\" under fairly generalconditions.Also, under similar conditionsthe optimal long-term contractcanbeimplemented by a sequenceof short-term contracts(which regulatethe agent'ssavings),wherethe principal makesa take-it-or-leave-itofferin period1ofa short-term contract{au wi(q\\)}, followedby another offer in period2 ofa contract{a2, w2(qj)}.

some definitions To be more precise,we provide the followingdefinitions:

A long-term contractis renegotiation-proofif, at every contracting date,thecontinuation contractis an optimal solution to the continuation

contracting problemfor the remaining periods.A long-term contract is spot implementable if and only if there existsaperfectBayesianequilibrium of the spot-contractinggame, the outcomeofwhich replicatesthe long-term contract.A contractexhibits memory if second-periodwagesdependon first-periodoutput outcomes.

A new issuearising in repeated-moral-hazardproblemsis the agent'sborrowing/saving decisionsand consumption smoothing. If the agent isbetter off smoothing her consumption over time, then the optimal long-term contractmight providesome intertemporal risk sharing. Also, if the

agent'sattitude toward risk dependson wealth, then the principal must beableto track the evolution of the agent'swealth after consumption todetermine the agent'srisk preferences.What is more,the principal may want to

422 Dynamic Moral Hazard

control the agent'ssaving decisionssoas to providebetter incentives.Weshall considerin turn three different scenarios.

In the first, the agent hasno accessto creditmarkets and must rely on the

principal to transfer wealth over time. Although this scenariomay appearsomewhat unrealistic, it providesa useful benchmark.In the secondscenario, the agent has accessto credit,but we let the principal monitor the

agent'ssaving behavior.Finally, in the third scenariothe principal doesnotobservethe agent'ssaving decisions.

10.1.1No Accessto Credit

The results detailed in this sectionhave been first derivedby Rogerson(1985b).Assume here that the agent hasno accessto creditmarkets.In otherwords, the agent is forcedto consumewhatever she earns in any period.

The optimal long-term contract then specifiesn + n2 contingent

consumption levels,where

wt =wageassociatedwith q\\ in period1wtj =wageassociatedwith (q],qf) in period2

The agent respondsby choosingthe actionplan:1

fli = actionin period1at =actionin period2 contingent on q\\

Assuming away discounting, the optimal long-term contractthen solvesthe following program:

, ,P^,\302\243pMVtft-wd+^PjiadViqf-Wjf)(w,),(w,y),ai,(a,)~ |_

~subjectto

[flj, (a,)]eargmax ^ pSfa) u(wt) -if/fa) +^Pj(at)u(wv) -y/(at)5i,(5i) j=l L /=1

and

^ Piifh.) u(wt) -yriai) +\302\243 pjiaMwq)-yrfa) >2u

1.Note the slight abuse of notation here: The action in period 1ax is not the sameasthe actionin period 2 contingent on q\\, even though it is also called av

423 RepeatedBilateral Contracting

As is apparent from this program, the two-periodoptimal contracting

problemis significantly more complexthan the one-shotproblem.Nevertheless, by consideringmarginal intertemporal transfers for every givenfirst-periodoutput realizationq],a fundamental relationship betweencontingent wagesacrossperiodscanbeestablished(wedrop time superscriptsbecauseat any time the output distribution only dependson effort):

\342\200\22477-t-

=\302\261,pM)

V\\qj-Wil)

u'{Wij) _

(10.1)

Or,more strikingly, if the principal is risk neutral, this equationreducestoa form of Euler equation,

-77\342\200\224-:

=Ep/fa)-U'iWij).

The optimal contractsmoothesthe agent'sconsumption soas to equate(the inverseof) her marginal utility in the first periodwith the expected(inverse)marginal utility in the secondperiod.

To seewhy this relationship between contingent wagesacrossperiodsmust hold, supposeto the contrary that the optimal contract[(vv;),(wj,)] issuch that equation (10.1)doesnot hold, and considerthe modified contract[(vv<), (vv,,)] such that, for all (i,j),u(wi)=u(wi)-ei

and

u(wi])=u(wij) +ei

This new contractprovidesthe agent with the same level of expectedutility as the oldcontract(for any actionplan chosenby the agent) for everyfirst-periodoutput realizationq\\. Therefore,the agent acceptsthis contractand choosesthe sameactionplan as before.But the new contractchangesthe principal'spayoff by

^pM)V(qt -w^-Vfa-w<)+\302\243/?,(a,)[V(gy -w^-Viq,-Wq)]<=i L /=i

Now, for small e/s,we have, approximately,

Wi-Wi =\302\243i/u'(wi)

and

Wv-HV=-\302\243,/h'(w,;,)

so that the net benefit to the principal is approximately

2p/(\302\253i)

V\"(<7,-w,) V'(9i-wj)\302\243i (10.2)

The principal can thus increasehis payoff by choosingsmall \302\247'sof the

samesign as the term in brackets.In other words, unlessthe term in

brackets is equalto zero [sothat equation(10.1)holds]there are profitabledeviations from the contract

(vv\342\200\236 w^). Therefore,equation(10.1)is one of theconditionscharacterizingthe optimal long-term contract.

Another important observationemerging from expression(10.2)is that

in generalthe optimal second-bestcontracthas \"memory.\" It iseasyto seethis fact by applying an argument almost identicalto the precedingone.

Supposeby contradictionthat the optimal contract is memorylessandtakes the form (wt, w,), so that the second-periodwageis independentoffirst-periodoutput. Thiscontractis optimal only if

u'(Wi)=tp,\302\253.,)^%for an.\"

7=1 u'(Wj)

But note that under the memorylesscontract,at is independentof the

first-periodoutcomeqt, sothat the right-hand sideof the aboveequationisa constant independentof qt. But

U.'(Wi)=k constant

implies that the agent enjoysfirst-best insurancein period1.Thiscontractcannot be optimal in general,for then the agent has inadequateincentivesto supply effort in period1.

A somewhat disappointing implication of this observationis that in

general the optimal long-term contract will be very complex.Thiscomplexity

is difficult to reconcilewith the simplicity of long-term incentivecontractsobservedin reality. Long-termemployment (or incentive)contracts observedin practicedo not appear to fine-tune the agent'sintertemporal consumption to the extent predictedby the theory.

As we shall discussin later chapters,long-term contractsobservedin

reality alsoappear to be simple in part becausethey leavemany aspectsof

425 RepeatedBilateral Contracting

the long-term relation unspecifiedor implicitly specifiedin an informal

agreementamong the parties.In other words, the explicit part of observedlong-term contractsis often highly incomplete.A difficult questionfor the

theory is then to explain what is left implicit and what is written explicitlyinto the contract.

Beforetackling this difficult issue,however,it is important to understand

the generalstructure of optimal long-term contractsand to ask whether

thereareplausibleenvironments in which optimal long-term contractshavea simple structure.

Oneimportant insight about optimal long-term contractsemerging fromthe fundamental relationship betweencontingent wagesacrossperiodsinequation(10.1)is that it isalways in the principal'sinterestto try to \"front-

load\" the agent'sconsumption. In otherwords,it is optimal to forcethe agentto consumemorein earlierperiodsthan shewould like expostand to reduceher savings. Intuitively, by keepingher continuation wealth low, the principalcanensurethat the agent'smarginal utility of money remains high. Hecanthus reducehis costofproviding the agent with effectivemonetary incentives.

To show that the optimal long-term contracthas this property, we shall

hypothetically let the agent borrowor save(at zerointerest,as we assumeno discounting) after the realizationof the first-periodoutcome.We shalluse condition (10.1)to show that the agent neverwants to consumemorethan what sheearnsunder the contractin that periodand if anything wants

to savesomeof her first-periodincome.Toseethis result, note first that the agent'smarginal expectedutility with

respectto savings following the realizationof qt is

n

Substituting for u'{w^) using condition (10.1)(and assuming for

simplicitythat the principal isrisk-neutral), this becomes

\" 1HpifaVK)--\342\200\224\342\200\224

Now, since1/jcis a convexfunction of x, by Jensen'sinequality we have

1 \"

\342\200\224 <2P;ifk)u\\wV)

y PM) y=i

Dynamic Moral Hazard

so that

n

]Tp;faVOv,) -u'iyvt) > 07=1

In words, under the optimal long-term contract the agent's marginalexpectedutility with respectto savings is always nonnegative, so that the

agent would like to savemore at the margin than the contractallows her.One important simplification onemight hopefor is that the optimal long-

term contractbe implementable by a sequenceof spot contracts,for then

the one-shotmodelof moral hazard consideredin Chapter4 might easilybeadaptedto the repeatedcontracting problem.Unfortunately, spotimplementation requires that the optimal long-term contractbe memorylessifthe agent hasno accessto a savings technology.We shall see,however,that

if the agent can saveher incomeindependently and if the principal canmonitor her savings, then spot implementation is feasible under fairly

generalconditions.This approach is possibleunder monitored accesstosavings, sincethen the principal can separate the consumption-smoothingpart (by controlling the agent'ssavings decision)from the standard incen-tives-versus-risk-sharingpart of the contract.

MonitoredSavings

Underthe monitored-savings scenario,the agent has accessto credit,but

the principal can monitor the agent'ssavings. Tobe consistent,we alsoletthe agent monitor the principal'ssavings. A long-term contractcan now

specifynot only a profileof output-contingent wage payments but alsooutput-contingent savings for the principal and agent, respectivelytt and st.Therefore,the optimal contracting problemnow takes the form

max Ta(bi)v(<li -Wt-O+^p,(\302\253<)V{q}-w^ +tt)(>v,),(w,;),U),U) .\342\200\242_\342\200\242,

\"TT\302\253l,(a.)

'-1 \"- '-1subjectto

[fl!,(a,)]eargmax\302\243 Pi(ax)

a\\lat) i=i

and

i(.Wi-Si)-i//(A) +2Pi&i)u{Wij +st)-i//(A-)

y=i

2M\302\253i) u(wt -sl)-\\if{ax)+^pj{ai)u{wlj+Sj)-i//(a\302\253)

7=1

>2u

427 RepeatedBilateral Contracting

Observethat only total savings (st + tt) are determinedat the optimum,sincethe principal can always undo the agent'ssaving decisionthrough an

adequatechoiceof (wt), (wy), and (tt). Therefore,we can set the savings ofone of the parties to zero without lossof generality.

When we set tt equal to zero and comparethe precedingprogram with

the optirnization problem in the \"no access\"scenario,it appears that thesituation where the agent's savings decisionscan be monitored isequivalent to the no-accessscenario.The only differencenow is that the optimalcontract can exhibit \"memory of consumption\" without requiring any

\"memory of wages.\" Indeed,by setting s,so that

and

wi=c*-slthe long-term contractcan implement a consumption plan by controlling

savings directly rather than indirectly through history-dependentwageplans.In other words, when savings can be monitored,a long-termcontract with memory is now in principlespot-implementable.In fact, the

principal may now solvethe optimal long-term contractby separatingthe

consumption-smoothing part from the standard static incentive-contractingproblem. This central observation has been made independently byMalcomsonand Spinnewyn (1988),Fudenberg,Holmstrom, and Milgrom(1990),and Rey and Salanie(1990).

We shall illustrate this separability property more explicitly under the

simplifying assumption that the principal is risk neutral.

Supposethat the agent savess in the first period.Then the spotcontractin period2 is the solution to

n

maxTpj (a)(qj-w)

subjectton

a earg max ^Tpi (S)u(yVj +s)-yia)n

^Pj(a)u(Wj +s)-\\j/(a)> u(c +s)-\\j/(a)

where(c,a) are the outsideoption consumption leveland action,such that

u(c)-y/(a) = u.

428 Dynamic Moral Hazard

Now, given any realizationqt in period1,the optimal long-term contractmust solvethe following continuation problem:

n

md^y,p,{a){q]-wj)(*/),\302\253 /=1

subjecttoJ n

I a eargmax ^Tp}(a)u(Wj)-1/^(5)n n

^Pi(a)u(wj)-yia) >^Pj(af)u(<^)-y\"Of)Iy=i /=i

To replicatethe solution to this continuation problemwith an optimal

period-2spotcontract,it thus sufficesto set period-1savings st such that

n

uic+sd-y(a)=2Pi(af)u(tf) -y/-(af)

Note that a solution st to this equation always existsif the agent's utility

function u(-) is unbounded below.To summarize, when the principal can monitor the agent's savings, it is

possibleto implement the optimal long-term contractwith a sequenceofspotcontractseven if the optimal contracthas \"memory of consumption.\"Thereasonis simply that with monitored savings the principal canseparatethe consumption-smoothing problem by controlling the agent's savingsseparately.

Thisobservationis important for at least three reasons.First, it providesa rationalefor short-term incentive contracting.As long as the principal cancontrolthe agent'ssavings, there isno value in committing to a long-termcontract.Second,it pinpoints the main sourceof gain from a repeatedrelation: the opportunities createdfor intertemporal risk sharing. Third, it leadsto a simple recursivestructure for the long-term contracting problem.With

spot implementation, the principal's problem can be formulated as adynarmc-programming problem,wherethe principal maximizes the sum ofhis current-flow payoff and the expecteddiscountedvalue of future spotcontracting by offering spot contractsto the agent, which the latter canacceptor reject.

The result that optimal long-term contracts are spot implementableunder monitored savings extends to any finitely repeatedcontracting

RepeatedBilateral Contracting

problemif the agent'spreferencesare additively or multiplicativelyseparable. This conclusionhas been establishedin Fudenberg,Holmstrom,and Milgrom (1990).Besidesadditive (ormultiplicative) separability of theagent'spreferences,severalother assumptions that we have madeimplicitly

here are important to obtain this result.Theseassumptions guaranteethat there is no asymmetric information betweenthe principal and agent at

recontractingdates.Thus,to ensurethat there is common knowledgeofproduction possibilitiesand preferencesat each recontractingdate,we havenot only assumedthat the agent'ssavings canbemonitored by the

principalbut also that each productionperiod is time separable,so that the

agent'sactionchoicein any given period affectsonly that period'soutputand hasno impact on the agent'sfuture actionsetsor future costsof taking

particular actions.An important corollaryof spot implementation is that when it is spot-

implementable the optimal long-term contractis alsorenegotiation-proof.In other words, at every recontractingdate,the continuation contractis an

optimal solution for the remaining periods.This conclusionfollows

immediately from the observationthat once contingent savings s, have beendetermined,the second-periodoptimal spot contractis always an optimalcontinuation contract.We shall seein the next sectionhow spotimplementation or renegotiation-proofnessbreaksdown when there isasymmetric information betweenthe principal and agent at recontracting dates.

Free Savings and Asymmetric Information

The principal may not always be able to observeand monitor the agent'ssavings. In that casethe continuation contracting problem may involvesome asymmetric information. All the issuesconsideredunder dynamicadverseselectionthen becomerelevant.In particular, the optimal long-termcontractmay no longerbe renegotiation-proofor spot-implementable.

We shall illustrate this statement with an exampleinvolving a singleactionchoicein period 2 but (hidden)consumption by the agent in both

periods.

Example The agent only supplieseffort in period2.\342\200\242 a e {H,L) with y<#) =1and y<L) =0\342\200\242 q e {0,1}with px(H) =pH>Pi(L)=pL>0

430 Dynamic Moral Hazard

We restrictattention to parametervaluessuch that it isoptimal to inducethe agent to chooseactionH.A long-term contract then simply specifiessecond-periodoutput-contingent wages(vv0, Wi).

We shall denoteby c\" the agent'sfirst-periodconsumption if she plansto choosea =Hand by c\\ her first-periodconsumption when sheplans tochooseactiona =L.That is,for / =H,L,c{maximizes

u(c)+p,u(yv\\ -cj+il-p^uiwo-c)The agent's(binding) incentive constraint in period1is

u(c?)+pHu(wx -c(I)+(1-pH)u(w0 -c?)-1=u(ct)+pLu{wx -ct)+(1-pL)u(w0 -ct)

but by revealedpreferenceand strict concavity of w(-) we know that

u(ct)+Plu(w1-ct)+{l-pL)u(w0 -ct)> u(ci )+pLu(w1-c?)+(1-pL)u(w0 -ci )

so that in period2 we have

u{c\\ )+pHu(wx -c(i)+(1-pH)u(w0 -C?)-1> u{c\+pLu(w1-c?)")+(X-pL)u(w0 -c\In otherwords, in period2 the agent'sincentive constraint is slack.There

is thereforean opportunity for both parties to renegotiatethe initial

contract at that point. Note that if the principal couldmonitor the agent'sconsumption (or savings), this problemwould not arise,for then the principalcouldimposea particular consumption choiceon the agent in period1suchthat the incentive constraint in period2would remain binding. It is the

principal's inability to monitor the agent'sconsumption that createsarenegotiation problemin period2.

There is one specialcase,however,where the principal's inability tomonitor the agent's savings is not problematic,namely, when the agent'soptimal savings are independentof her choiceof action.Thisproperty ofthe agent'ssavings behaviorobtains when shehas CARA risk preferences.With such preferenceswealth effectsare absentand history canno longeraffect the agent's savings behavior.Fudenberg,Holmstrom, and Milgrom(1990)show that with CARA risk preferences the optimal long-termcontract (with no monitored savings) is then spot implementable and

renegotiation-proof.

431 RepeatedBilateral Contracting

10.2The T-periodProblem:SimpleContractsand the Gainsfrom EnduringRelations

The previoussectionhas highlighted the importanceof intertemporal risk

sharing as a benefit of long-term contracting. Two other sourcesof gainsfrom long-term contracting have beensuggestedin the literature:bettermonitoring of the agent'sactionsand more \"punishment\" optionsfor poorperformance.For instanceRadner (1986a)identifies the gains from

enduringrelationsunder moral hazardas follows:

Theserepetitions give the principal an opportunity to observethe results of the

agent's actions over a number of periods,and to use some statistical test to inferwhether or not the agent was choosing the appropriate action. The repetitions alsoprovide the principal with opportunities to \"punish\" the agent for apparentdepartures from the appropriate action, (p.27)

A possiblecountervailing effectof repeatedinteractionsmay alsobe toworsen the incentive problem the principal faces,for now the agent hasmore opportunities to shirk, try her luck, and make up for bad outcomesby working harder.In otherwords,in an enduring relationthe agent'sactionset is richer,and the principal'sincentive problemmay beworseas a result.This effect has been emphasizedby Holmstrom and Milgrom (1987)and has led to the idea that with repetition incentive contracts maybecomesimpler even though the underlying incentive problemgrows in

complexity.Much of the literature on repeatedmoral hazard,which allows for

infinitely many periods,hasbeenconcernedwith approximation resultsof thefirst-best outcomewith simple incentive contractsunder no (or almost no)discounting. In particular, Radner (1981,1985),Rubinstein (1979),Rubinstein and Yaari (1983),Fudenberg,Holmstrom, and Milgrom (1990),and Dutta and Radner (1994),among others,providesuch results.Thereare small differencesin the typesofsimple contractsconsideredand in their

interpretation. All contractsshare the feature that as long as the agent'saccumulatedwealth isabovea given threshold, the agent eithermaintainsa given mean levelofper-periodconsumption or continuesto beemployed.Should the agent'swealth fall belowthat threshold, then dependingon thecontractconsidered,the agent cuts down on her consumption in order tobuild up her stockof wealth, or the contractis terminated. Thesecontractsare interpretedto be a form of either employment contractor bankruptcy

432 Dynamic Moral Hazard

contractor,alternatively, as self-insuranceschemesfor the agent.Althoughthe basicunderlying argument and method of proof are similar, theintuitive explanationsgiven for the approximation resultsdiffer substantiallyfrom one paperto another.Onestrand of the literature (e.g.,Fudenberg,Holmstrom, and Milgrom, 1990)arguesthat the gains from the (infinitely)

repeatedrelationstem exclusivelyfrom self-insuranceby the agent, while

the other (e.g.,Radner,1986a)alsoemphasizesthe benefitsfrom enhancedstatistical inferenceand punishments.

In order to disentangle the relativeimportanceof thesefactors,we shall

attempt to isolate each one by repeating only some elements of the

principal-agentrelationship and not others.We thus distinguish five specialcasesin increasingorder of complexity:

Case1:RepeatedOutput In the specialcaseof repeatedoutput, the agentmakes a single action choice,but there are multiple output realizations.Moreover,the agent consumesonly at the end of the repeatedrelation.Herethe only benefit ofrepetitionisbetter statistical inference.Thisspecialcasecanbeseenas a simple representationof a problemwhere the agentmust be given adequateincentives to undertake a risky investment.

Case2:RepeatedActions In the caseof repeatedactions,only the agent'saction choice is repeated.There is a single output realization,and

consumption takesplaceat the end.Thiscasecanbeseenas an intertemporalanalogueof a multitask problem.The benefit (orcost)of repeatedactionchoiceis that the agent may be able to spreadher effort over time moreefficiently.

Case3:RepeatedConsumption Again, in the caseof repeatedconsumption,

the agent makesa single actionchoice,and there is a single outputrealization.But now the agent may consumein severalperiods.In this casethe only potential benefits from long-term contracting arise from

consumption-smoothing opportunities. We shall not discussthis casebelowbecausethe exampleconsideredin subsection10.1.3alreadyprovidestheintuition for this specialcase.As the examplehighlights, the gains from

long-term contracting dependto a large extent on whether the agent'sintertemporal consumption canbemonitored by the principal.If it cannot,then the opportunities for consumption smoothing open to the agent mayworsenthe underlying incentive problem\342\200\224so much so that the gains from

intertemporal insurancemay be outweighed by the greater costsof pro-

433 RepeatedBilateral Contracting

viding incentives to the agent.Thus unmonitored consumption smoothingmay comeat the expenseof the incentive efficiencyof the contract.

Case4:RepeatedActions and Output In this caseboth actionchoicesand

output are repeated,but consumption takesplace only at the end.This isthe setup consideredby Holmstrom and Milgrom (1987).The focusherewill benot somuch on the benefitsof an enduring relationas on the formof the optimal long-term contract.

Case5:Infinitely RepeatedActions, Output, and Consumption Thecaseofinfinitely repeatedactions,output, and consumption isthe most generalcaseconsideredin the literature. Here,as in the previouscase,we investigate towhat extent simple contractscanbe approximately optimal.

10.2.1Repeated Output

In the repeated-outputcasewelet the agent choosean actiononly once,att =0.We shall supposethat there are only two possibleoutput outcomesinany given period:q e {0,1}andp(a)=Fr(q = lla),with p(a)strictly

increasingin a.

We shall contrast two extremescenarios.One,where there is a singleoutput realizationat date t = 0,was consideredin Chapter4.In this

scenario wehave seenthat the second-bestcontract,in general,cannotapproximate the first-best outcome.

In contrast,in the otherextremescenario,wherethereare infinitely many

independentincrementsin output, a second-bestcontractcanapproximatethe first-best outcomearbitrarily closely.Toseethis point, considerthesituation where there are infinitely many independent,binomial incrementsin output over a finite interval of time T.That is, considerthe situation

where,over the interval of time T,cumulative output follows a Brownianmotion:

dq=a-dt+^o \342\226\240dZ

wherea e A denotesthe drift of output, oisa constant volatility

parameter,and dZ is a standard Wiener process.The drift is chosenby the agent

at time t = 0.Even if the principal and agent can observeonly final

accumulated output Q(T)and can write a contractcontingent only on Q(T),they can now approximatethe first-best outcome.Indeed,as Q(T) is

normally distributed with mean a \342\226\240T and variance o-T, a Mirrlees-type

434 Dynamic Moral Hazard

contract\342\200\224where the principal choosesQ such that for all Q(T)< Q the

agent gets punished and otherwiseshe gets a sure payment of w*\342\200\224can

approximatethe first-best outcome(seeChapter4).This examplecaptures in a simple and stark way the idea that more

precisestatistical inferenceismadepossibleby the observationof the sum

of repeatedindependentoutput observations.Herewe have made heavyuse of the fact that the normal distribution approximatesthe sum ofindependently,' identically, and binomially distributed random variables.Butthe generalpoint that repeatedobservationof independently distributed

output outcomesimproves statistical inferencedoesnot dependon this

construction. To the extent that an ongoing principal-agentrelation involvesmorefrequent output observationsthan actionchoices,there is a statisticalinferenceeffectthat amelioratesthe efficiencyof the contract.

10.2.2RepeatedActions

Take T = 2 and supposenow that there are n possibleoutput outcomesat the end of period2 and that the probability of outcomeqt is given by

pX^i +a2),whereat denotesthe agent'sactionchoicein periodt =1,2.Thatis,here the agent canchoosetwo effort levelsin succession,and the

probabilitydistribution over output is influenced by the sum of the agent's

efforts.In all other respectsthe staticprincipal-agentproblemconsideredin Chapter4 remains the same.Underthis scenariothe principal'soptimalcontracting problembecomes

n

max y,pl{a1+ajWiqt-w,)

subjectton

\342\200\242

(flj, a2)eargmax^PtQk+^MveJ-^fli)-^)and

n

2Piifh +\302\2532 )u(wt) -yifn) -

\\jf(a2)> u

- t=i

In the first-bestproblemthere is an obvious benefit from letting the agentspreadher effort over time. Since the effort-costfunction y/(-) is strictly

increasingand convex,it is optimal to supply an equal amount of effort in

each period.\"When the agent can spreadher effort over time, the overallcostof inducing a total effort supply of a is then reducedby

435 RepeatedBilateral Contracting

yf(a)-2y\\-\\

Doesthis benefit from lowercoststranslate into an improved second-bestcontract?Unfortunately, it is not clear that it will. The reason is that

when the agent can spreadher effort over time, the principal may faceworseincentive constraints. Cleancomparativestaticsresultson the effectsof a changein the agent'seffort coston the second-bestoutcomeare notobtainable.Grossmanand Hart (1983a)provideexampleswhereareduction in the agent's cost of effort makes the principal worseoff.They areable to show only a limit result that the first-best outcomeisapproachedin

the limit when the agent's (marginal) costof effort goesto zero (seetheir

proposition17).10.2.3RepeatedActions and Output

The previoustwo specialcaseshave highlighted, first, that if an ongoing

principal-agentrelation involves more frequent output observationsthan

actionchoices,thereisa statistical inferenceeffectthat amelioratesthe

efficiency of the contract;second,that if the frequency of output observationsislessthan the frequency ofactionchoices,then the optimal long-termcontract may producea worseoutcomethan if the frequenciesofactionchoicesand output observationswereconstrainedto be equal.

In this subsectionwe first explorethe issueof the relativebenefits andcostsof a higher frequency of output observationsand revision of actionsmore systematically by consideringan insightful exampleof a repeatedprincipal-agentrelation adapted from the repeatedpartnership game ofAbreu, Milgrom, and Pearce(1991).

In a secondstep we shall alsoexplorethe important theme that as the

agent's action set gets richer (with more frequent action choices),this

processleadsto a simpler optimal incentive contract,since,with moreincentive constraints potentially binding, the principal'sflexibility in

designingthe contract is reduced.This general idea was first explored by

Holmstrom and Milgrom (1987),who suggestedthat in somespecialsettings the agent'sopportunities for shirking over time may besovaried that

an optimal responseis to providethe agent with incentives that are linearin output, sothat the agent issubjectedto a constant incentive pressurenomatter how her performanceevolvesover time.

436 Dynamic Moral Hazard

10.2.3.1RepeatedActions and Output: RicherAction Setversus MoreFrequent Information

This example is best interpreted as a machine (or asset)maintenance

problem, where the agent must continuously exert effort to keep themachine in goodrunning order.The machine yieldsa flow payoff of n > 0to the principal as long as it is well maintained by the agent.However,ifthe agent shirks, the principal'sflow payoff is zero.The agent has only two

actionsavailable,a = 1(\"maintain\") and a =0 (\"shirk\") at any time t.Sheincurs a flow cost 0 < y/ < n when she choosesa = 1,but no cost when

a =0.Both principal and agent are assumedto be risk neutral and to discount

future payoffs at rate r > 0.For simplicity we assumethat principal and

agent cansign only efficiency-wagecontractssuch that the agent getsa flow

wagepayment w>y/as long as she remains employed.Otherwise,she getsa flow payoff of zero.

Theprincipal-agentrelationis modeledin continuous time, but principaland agent are assumedto be able to revisetheir actionsonly after a fixedinterval of time t has elapsed,so that effectively the problemis a discrete-time one.Thequestionwe shall investigate is how the efficiencyof thecontract is affectedwhen we vary t or r.

PerfectInformation

Supposeto begin with that the principal observesperfectlyhis accumulatedflow payoff at the end of each interval of time t.Then,as long as the agenthas chosento maintain the machine (a = 1),his mean flow payoff over theinterval of time t is given by

r\\te-rt7tdT=7r(l-e-rt)

Theoptimal efficiency-wagecontractthen takes the simple form that the

principal continues to employ the agent at wagew as long ashis meanflow

payoff is 7r(l -e~n) and otherwisefires the agent.Undersuch a contractthe agent'sincentive constraint takes the form

w(l-e~n)<w-\\i/TheRHSrepresentsthe agent'smean flow payoff when she never shirks.TheLHSis her payoff when she shirks and is fired at the expirationof the

437 RepeatedBilateral Contracting

time interval t. The optimal wage w is then set so that the incentiveconstraint is binding. It is instructive to rewrite the incentive constraint asfollows:

rt =log \342\200\224

wIt is clearfrom this expressionthat an increasein the frequency of revisionof the contract(a reductionin t) has the samebeneficialeffectas areduction in the discount rate r.We shall see,however,that when the principalcannot observeperfectlyhis accumulatedflow payoff at the end of eachinterval of time t, it ispossiblethat a greaterfrequency ofrevision canmake

things worse.The reason is that more frequent revisionsprovidenot onlythe principal with more scopeto fine-tune the incentive contractbut alsothe agent with greater opportunities for shirking. As we shall illustrate, if

greater frequency of revisionsentails a reductionin the quality ofinformation obtainedby the principal at the end of each interval of time t, then

the incentive problem may well be worse when information is availablemore often.Then,as in the exampleby Dewatripontand Maskin (1995a)describedin Chapter9,it may bepreferableto choosean information

technology whereperformancemeasuresare availablelessfrequently.

ImperfectInformation

Supposenow that the principal canno longerperfectlydeterminewhether

the agent maintained the machine over the interval of time t. Supposethat

the principal can observeonly machine breakdownsor failures over theinterval and that the Poissonarrival rate of a breakdown per unit time isX > 0 when the agent maintains the machine or fi > X when the agent shirks.The probability distribution of k breakdownsover the interval of time t isthen given by

Pk(f) = -

k\\

when the agent maintains the machine and

when she shirks.

438 Dynamic Moral Hazard

The efficiency-wagecontractmust now condition the principal'sfiring

decisionon the number of observedfailures.This is an imperfectsignalbecausefailures canoccurevenwhen the agent maintains the machine.Thehigher the number of observedfailures overthe interval of time t, the morelikely it is that the agent shirked, sincefi> X. This relationship canbeseenmore formally by observing that

Pk vA

is increasingin the number of failures k.It is therefore intuitively plausiblethat an optimal efficiency-wage

contract will choosesome cutoff number of failures k such that the agent isretainedfor all k < k but is fired wheneverthe number of failures isequalto or greater than k.We shall show that this is indeed the case.

The fundamental difficulty with shortening the time interval t is now

apparent.The shorter the interval, the lower the probability that asignificant number of failures occur,which would give the principal a sufficientlyaccuratesignal that the agent has shirked.In other words, the shorter thetime interval, the worsethe quality of the information availableto the

principalto basethe incentive contracton.Therefore,in this example,

increasingthe rate at which the principal and agent receiveinformation canmake

the incentive problemworse.Moreformally, let a=(a1,a2,...,ak,...,aK) denotethe principal'sfiring

probability as a function of observedfailures k. Undersuch a firing policyand a wagerate w, the agent'spayoff v if she always maintains the machineisgiven by

v ={l-e-rt){w-y)+e-nvXPfc(l-\302\253fc)

or

v(l-e-r')=(l-e-rt)(w-y)-e-rtv

since^k=1Pk=1.

\342\200\242K

2>afc (10.3)

439 RepeatedBilateral Contracting

If, however, the agent shirks over an interval t, her payoff is

w(l-e-rt)+e-rtv 2^fcd-\302\253fc)

so that the agent'sincentive constraint is now

w(l-e-rt)+e-rtv 2^fc(l-\302\253fc) <v (10.4)

Indeed,if this inequality is satisfied,a one-timedeviation doesnot pay.At the optimum, the wagerate w is setsothat constraint (10.4)binds.

Combining equations(10.3)and (10.4),the incentive constraint and the agent'spayoff under the contracttake the following simple form

\\ir{l-e-rt)=e-rtv(L-l)

and

K

ILfc=i^ttkPk (IC)

V =W \342\200\224

!//\342\226\240\342\226\240

where

\302\245

\302\243-1

\302\243= , \302\256kqk

UkPk

denotesthe likelihoodratio under the firing policya.Note that relative to the caseofperfectinformation the agent'spayoff is

reducedby y//(\302\243

-1).As canbeseenfrom the expressionfor v, the

principal's objectiveisnow to try to choosea firing policy a to maximize thelikelihood ratio \302\243 subjectto satisfying the agent'sincentive constraint.

Thisobjectiveis achievedby choosingthe lowestpossiblek compatiblewith the incentive constraint (IC),such that ak = 1for all k > k and ak =0for all k < k.Toseethis point, note simply that the solution to theunconstrained problem

440 Dynamic Moral Hazard

max\342\200\224|-

is given by ak =0 for all k < K-1and ak =1for k =K. If this solutionviolates the constraint

yer'(l-e-rt)= v(\302\243-l) ^CCkPk (ico

then obviously the principal must find the highest k such that constraint

(ICOholds.As we shall now illustrate, it is possiblethat when the time interval t is

short there may not existany k that satisfiescondition (ICOfor w < n. Butnote first that the LHSof condition (ICOis an increasingfunction of r andthat the RHSis independentof r.Therefore,a reductionin r allowsthe

principalto improve the contractby lowering v (or w) and increasingL.Thus,

as in the caseof perfectinformation, a lowerdiscount rate allows the

principalto improve the incentive contractby imposing scaled-uppunishments

on the agent if the number of observedfailures is greater than a giventhreshold.In the limit as r \342\200\224\302\273 0, the LHSis zero,so that the optimalcontract with imperfectmonitoring can let v \342\200\224\302\273 0.In other words, in the limit

when there is no discounting, the optimal contractwith imperfectmonitoring approximatesthe optimal contractwith perfectmonitoring. Note that

this statement is true for any time interval t.We shall now show that a reductionin the time interval t doesnot

necessarily improve the incentive problem in contrast to the caseof perfectinformation. Thecomparativestaticswith respectto t are significantly moredelicatethan with respectto r, aschangesin t affectboth the LHSand RHSof condition (ICO-To sidestepthe difficulties involved in this exercise,weshall insteadconsidera feasiblecontracting situation with t boundedawayfrom zero and show that for t closeto zero this situation is no longerfeasible.

Considera time interval t boundedaway from 0 and a discount rate rlow enough (but also bounded away from zero) so that an efficiencywage contractwith w < n is feasible.Note that such a contractexists if

441 RepeatedBilateral Contracting

k > yr, since the solution with perfect information can be approximatedwhen r \342\200\224\302\273 0.

Now, let t shrink. In the limit, as t ->0,the likelihoodratio \302\243

=fi/X. sincethe only eventsof order t are that no failure or one failure occurs.Moreover, when t \342\200\224\302\273 0, the agent's mean flow payoff v under the contract isapproximately

Jo Y r +OdX

Therefore,the incentive constraint reducesto

ry/= V- (Xi(ji-a.)

as t ->0.Now, if

Y>\342\200\224 7^-it is not possibleto satisfy the incentive constraint with a wagepaymentw < k when t ->0.Note that for fi closeto X this inequality can hold for

if/ < n, so that a contractcanbe feasiblefor t boundedaway from zero but

not for a t arbitrarily closeto zero.In other words, this exampleillustrates

that a high frequency of revision of actionsfor the agent can worsentheincentive problemto such an extent that a feasiblecontractno longerexists.The problemfor the principal is that when the time interval isvery short,he no longerhas a sufficiently accuratesignal to be able to deter one-timedeviations from the efficient actionby the agent.In other words, when the

agent canreviseher actionchoicemore and morefrequently, heropportunities for shirking increasefaster than the quality of the principal'ssignalabout her actionchoice.

10.2.3.2RepeatedActions and Output: RicherAction Setsand SimpleIncentiveContracts

We now explorethe idea that as the agent'sactionsetgetsricher(with morefrequent actionchoices),a simpler, possiblylinear optimal incentivecontract may be the result.Thisgeneralideahas beenexploredby Holmstrom

442 Dynamic Moral Hazard

and Milgrom (1987),who consideran examplewith T periods,where in

eachperiod t =1,...,T the agent choosesan actiona' e A.Thisactionaffectsonly the probability distribution p,((f) over the n

possible output outcomesat the end of that period.Tokeepthings simple,weshall allow for only two possibleoutput outcomesin eachperiod,n =2.Wedenote by p(cf) the probability of a high-output outcomeqH, and by [1-

/?(\302\253')]the probability of a low outcomeqL, (qH > qL).With this

simplification the agent'sactioncanbe taken to bedirectly the choiceof probability

p'=p(at),and her actionset the interval [0,1].Consumption by the

principalor agent takesplaceonly at the end of period T.

Sinceconsumption takesplaceat the end and the T productionperiodsare technologicallyindependent from one another, there appear to bebenefits neither from consumption-smoothing opportunities nor from

improved statistical inference.The only possibleintertemporal link between periodshere may be

through the wealth accumulation of the principal and agent, which mayalter their respectiverisk preferences.However,even this link is assumedaway in the Holmstrom and Milgrom setupwith the assumption that both

principal and agent have CARA risk preferences representedby the

respectivenegative exponentialutility functions

V{y) =-e-Ry

for the principal and

u(y) =-e-ry

for the agent.Therefore,there appear to beno benefitsfrom long-term contracting in

this example.Theremay, however,potentially bea cost,sincethe agent maybe able to modulate her actionsin any period t > 1on the observedrealized history of output up to t, Q'1=(q1,...,qtA).

Thepoint of this exampleisnot somuch to identify the benefitsof long-term contracting as to highlight how in this potentially complexrepeated-moral-hazardproblem the optimal long-term contract may have a verysimple structure, namely, that it is linear in final cumulative output.

It is important to emphasizeat the outset that a critical assumption forthis striking result is the assumption of CARA risk preferences.Without

this assumption Holmstrom and Milgrom's aggregation and linearity result

443 RepeatedBilateral Contracting

would not obtain.Undermore generalrisk preferences,the principal and

agent'sattitudes toward risk would evolvewith their accumulatedwealth,so that the optimal long-term contract (or sequenceof spot contracts)would be a nonlinear function of the history of output. With CARA risk

preferences,however,the optimal spot-incentivecontractis independentofthe agent'swealth, as we shall illustrate.

The principal'sone-shot optimal-contracting problem here takes theform

'max p \342\226\240V(qH -sH)+(1-p)V(qL-sL)

subjectto\342\226\240p earg max{pu[sH+w-yr(p)] +(1-p)u[sL+w-y(p)]}

P

and

.pu[sH+w~y/(p)]+(1-p)u[sL+w-y/(p)] > u(w)

where w denotes the agent's initial wealth and s, the agent's output-

contingent compensation,i =H,L.As we know from Chapter4,we can replacethe agent's incentive

constraint with the first-orderconditionsof the agent'soptimization problem

u[sH+w-y/(p)]-u[sL+w-y/(p)] -pu'[sH+w- \\if(p)]yf'(p)-(1-p)u'[sL+w-y/(p)]y/'(p) =0

or

_e-r[w+\302\273Mrt/0] +e-r[sL\342\204\242-Y(p)]+pXj/'(p)e-r[sH+W-\302\245{p)] +^ .^^'(pJg-K^+^^p)]=Q

Multiplying through by e\342\204\242,

we then note that the agent'sincentiveconstraint is independentof w:

_eMsH-v(P^+eMsL-v(P^+pY(p)e-r^-^p^+(i-p)\302\245Xp)e-r^-^p^ = 0 (10.5)

Substituting for the functional form ofu(-),the individual-rationalityconstraint canalsobewritten as

p{_e-rl\302\260HW-np)^ +(I_p)(_e-r[sL+W-\302\245{p)]}>_ e-rw

or,multiplying through by ew,

p(_g-r[W-*T(P)])+(1_p)(_e-r[sL-V) > 1 (10.6)

444 Dynamic Moral Hazard

Combining condition (10.5)with condition (10.6)now makesit apparentthat the optimal contract{sf,s\302\243}

isindependentof the agent'swealth w.Inother words, with CARA risk preferences,changesin w do not affectA =sh -s*,so that the optimal long-term contracting problem between the

principal and agent is effectively stationary. With this observationin mind,we can now turn to the T-periodversion of the problem.

Sincein any periodt < Tof this problem,the agent canobservethe entirehistory of output realizationsup to t, her strategy of play over the

contracting period is given by pl{QtA)- That is, she can condition her actionchoicein period t,pc,on the history of realizedoutputs up to t, Q~x.

Thus a long-term contract specifiesan incentive-compatiblehistory-

contingent action planp^Q'-1)and a history-dependentcompensations{QT).When the contractexpires,the agent'saccumulatedwealth is then given

by

s(QT)-tw)and the principal'saccumulatedwealth is

i<7'-*(<2r)(=1

TheT-periodproblemthus takes the form

max E\\V\\Y7\\q'-s(QT)\\]({\342\200\242(fl-W))

L lZ\"=1 /J

subjectto

pr(fiM)6argmax\302\243[M{W+j(Qr)-2''w[pr(j2M)]}l

p'(Q'-l)L '-* JJ

and

Givena contract{p'iQ1'1),siQ7)},we candefinethe value of the contractfor the agent to be

v0 = max\302\243[\302\253{j(Gr)-]\302\2431 tfp'(QM)]}]{p'iQ''1)}

445 RepeatedBilateral Contracting

and the continuation value at any date t to be

vt{Qt~1)= max ^[mRqO-I^Hp^Q7-1)]^-1}]Note that the continuation value excludesall the effort costs

accumulated by the agent up to time t, ^T =1v[pT(QT_1)]-The reason is, first, that

thesecostsare sunk by then and, second,that with CARA risk preferencesthe sunk effort costsenter multiplicatively into the agent'sobjectivefunction, so that they canbefactoredout.

Given the contract{p'(QM)>s(QT)},we can also define the agent'scertainty-equivalent wealth, w((Q(_1),in any periodt by solving the equation

u(wt) =vt(Q<-1)

Using the agent's certainty-equivalent wealth, we can rewrite the agent'soptimization problemgiven any history OtA in the following simple way:We can take pXQ1'1)to be the solution to the problem

max{pu[wt+1 (Q'-1;qH)-y/(p)] +(1-p)u[wt+1 (Q'\"1\\qL)-y/(p)]}p

Note that this problem has exactly the same form as the single-periodproblem consideredearlier.Therefore,the solution to the agent'soptimization problemdependsonly on the difference

and not on the agent'scertainty-equivalent wealth w((Q(_1).We cantherefore definea spotcontracts^Q)to inducethe actionchoicep'by the agent

Jr(GM;?r)=^+i(GM;?r)-^(GM)such that the first-ordercondition holds:

_e-^[*(G'-1;wW(pO]+e-^*(G'-1:\302\253W(pO]+py/(pt)e-'-[*(G'-1:wMp')]+NW^w=o (1\302\260'7)

Aggregating overall periods,we can then write the agent'sfinal wealth as

wT(QT)=S(QT)=f,St(Qt)+w0(=1

446 Dynamic Moral Hazard

where w0 denotes the agent's certainty-equivalent wealth at the time of

signing the contract.Or,denoting by A'L (and,respectively,A#) the number of realizationsqx

(respectively,q2) up to period t, we canrewrite s(QT)as follows:

^(<2r)=i^(Q(-1;^)Uk-^-1)+5(((2(-1;^)Ui-Ai-1)]+w0(=1

Thus the principal'sproblem reducesto choosing \\p\\ st] and w0 tomaximize

E\\V E?r-Jr(GM;?0-w0

subjectto equation(10.7)and

u(w0)>u(w)

Given that the long-term contracting problemis stationary, one might

expectthat the optimal strategy for the principal is alsostationary. That is,the principal chooses

P'(Q'-1)=P*and, therefore,

st{Q'-\\qL)=st and st(0'-u,qH)=s*

for (st,,s%) such that

_g-r[w-W)]+e-r[sL-tfp*)]+p!\302\245 \302\245>(p*)e-'[6-1'lP*)] +(I-p*)V'(p*)e-r[sl-'l'{P*)]=0

Thisisindeedthe caseand canbeseenfrom the following induction

argument: \"When T=1,the principal'soptimal strategy is obviously p* and (sf,s*).Supposenow that the principal'sstrategy is stationary for the repeatedproblemwith T = t, and let V% denote the value for the T-problemwhen

w0 = 0.Then,by the exponentialform of V, the principal'spayoff for theproblemwith T= t+1isgiven by

-eRwoE Vtf-sjEWIte'-sJQ1<_ eRm E]y(ql

_ Sl)y\342\226\240* j<_

\302\243Rmy*y*

447 RepeatedBilateral Contracting

The first inequality follows from the assumedoptimality of the

stationary strategy p* and (st,s%) for the repeatedproblemwith T= t.SinceV?entersasa constant in the remaining one-periodproblem,it doesnot affectthe solution to this problem,which is again p* and (st,s%).Thus thesolution to the principal'sT-periodproblemis stationary.

We can thereforewrite

In other words, the solution to the optimal long-term contracting problemis linear in aggregatefinal output.

The T-periodproblemconsideredby Holmstrom and Milgrom is specialin a number of respects.The most important restrictionis that both

principaland agent are assumedto have CARA risk preferences.We have also

assumedthat output follows a binomial process,but their aggregation and

linearity result extendsto the multinomial casewhereqt e {^i,q2,...,qn}-

The logicbehind their result is thus the following: When both principaland agent have CARA risk preferences(aswell astime-independentutility

functions and costof effort) the T-periodproblemis stationary, so that all

dynamics have beenremovedfrom the problem.With CARA riskpreferences it is possibleto break up the dynamic optimization problem into asuccessionofidenticalstatic problems.Next, if output isbinomial, the staticcontract is linear by construction.Finally, given the time-independentnature of the T-periodproblem,it is possibleto dispensewith disaggregateinformation on the output path and to write the contractas a linearfunction of aggregateoutput only.

While concedingthat the optimality of linear contractsis obtainedunder

strong assumptions, Holmstrom and Milgrom have argued that it providesa justification for concentrating on linear contractsin the \"normal-noise,

CARA-preference\"case.As seen in Chapters4, 6, and 8, this casehasprovedvery attractive in applications.

10.2.4Infinitely RepeatedActions, Output, and Consumption

Thepreviousexamplewith repeatedactionsand output providesoneillustration of how optimal incentive contractsin repeated-moral-hazardproblems can take a simple form: with (binomial) i.i.d.incrementsin output andCARA risk preferences,the optimal incentive contractwas shown to belinear (or affine) in final accumulatedoutput.

448 Dynamic Moral Hazard

In this subsection,we provideanother illustration of the optimality ofsimple contractsin a repeated-moral-hazardproblem.We illustrate how asimple no-insurancecontract,where the agent is madea residualclaimant,

may be approximately optimal in a general infinitely repeatedprincipal-agent relationship with little discounting.

Supposethat for any given actionchosenby the agent the distribution ofoutput qt has a compactsupport, and let q(a)denote the lowestpossibleoutput outcomeunder any given action choicea.Also denote by a* thefirst-best action,by q* = E[q\\a*] the mean output when the agent choosesthe first-best action,by q the lowerbound under the first-bestaction,and

by wt the agent'saccumulatedwealth up to periodt.

Supposethat the principal can perfectlymonitor the agent'sborrowing/saving decisionsand that, for simplicity, he doesnot allow the agent toborrow.However,the agent is allowedto saveas shepleasesotherwise.

Considerthe cutoff for the discount factor 8 < 1.Then,for any 8 > 8, afeasiblestrategy for the agent is to choosethe first-best actiona* in everyperiodand to consume

c=q*-68+0--8)wt Hwt>^-=-^8

where 6 e (0,q* -q) and

q*-qc=q +(l-8)wt ifw(<\342\200\224=-=- 8

Thus, as long as the agent'saccumulatedwealth remains abovea giventhreshold, wt > (q* -q)J8,she consumesapproximately her mean output

plus the interest earned on her accumulatedwealth (1- 8)wt, when 6 iscloseto 0.

Let H = (w0, wu ...,wt) denote the samplepath of the agent'saccumulated wealth under this action choice and consumption plan. Then the

agent's conditional expectedwealth in periodt + 1(under this first-bestactionchoiceand consumption plan) is given by

q*-qw, >\342\200\224=^=

8q*-q

wt <\342\200\224f^-

E[wt+1\\h<] = <

wt+6 if

q*-qwt+- if

449 RepeatedBilateral Contracting

Thus, since

f q*-q]E[wt+1 \\h']>wt+

minj6, -1

{wt} is a submartingale with drift boundedaway from zero.In other words,the agent'saccumulatedwealth grows on averageover time.

Moreover,the incrementsin wealth are uniformly bounded for 5 > 5(since the distribution for qt has compactsupport).Therefore,there is afinite time horizon T(e)for ee (0,1)such that

PrJ wt >^-j^for all t > T(e)[>1-e

In otherwords, the probability that the agent will beable to consumehermeanoutput plus the interestin every periodafter time T(e)isgreaterthan

or equal to 1-e.Therefore,the agent's flow payoff under this plan is atleast

(1-s)dT{e)[u(q*-6S)-yr(a*)]+[1-(1-e)dT{e)][u{q)-iff (a*)]

Hence,as e->0 and 5-\302\2731, the agent'sflow payoff tends to the first-best

payoff u(q*) -y/(a*) when 6 iscloseto zero.In other words, the contractwhere the agent is maderesidualclaimant

isapproximately first-best efficient when there is little discounting.Note that wehave not referred to the agent'sindividual-rationality

constraint in this argument. It should beclearthat the precedingargument canbeadaptedto allow for an adequateflow lump-sum transfer from the agentto the principal to satisfy the agent's (IR) constraint.

The simple contractwherethe agent ismaderesidualclaimant isnot the

only contract that is approximately efficient.Another approximatelyefficient contract is one resemblingan efficiency-wagecontract.Under this

contractthe agent is paid a fixed wages perperiodaboveher reservationflow utility as long as accumulatedperformanceQt= Q^ +qt-s remainsabovea given threshold Q.Should Qt fall belowthis threshold, the agentis fired and replacedby another agent.Using a limit argument similar tothe precedingone,it canbeshown that when there is little discounting, this

contractis approximately first-best efficient (seeRadner,1986a).

450 Dynamic Moral Hazard

10.3MoralHazard and Renegotiation

In the previoussections,dynamic considerationsarosebecauseofrepeatedeffort and output realizations.As this section indicates,dynamic issuescan ariseeven in the caseof a single effort choiceand output realization.Indeed,when there is a substantial lag betweenthe time when the agent'scostlyaction is sunk and the time when output is realized,there is scopefor renegotiationof the initial incentive contract.Oncethe actionis sunk,there is no needto exposethe agent to further risk. An opportunity then

arisesto improve risk sharing. In particular, if the principal is risk neutral

and the agent is risk averse,it is optimal to fully insure the agent onceheraction(or investment) is sunk. Of course,if the agent anticipatesrenegotiation after her actionhas beenchosen,her incentives may be altered.Twocasescanbe distinguished:

\342\200\242 If the effort choiceis not observedby the principal, the anticipation ofrenegotiationis problematic:the incentives of the agent may bereducedoreveneliminated.\342\200\242 If insteadthe effort choiceis observedby the principal (although still non-contractable),dynamic contracting can tackle sequentially the two

problems of inducing proper effort (through initial contracting) and providing

optimal insurance(at the renegotiationstage).We coverthese two casesin turn in the next two subsections.

10.3.1RenegotiationWhen Effort Is Not Observedby the Principal

When the effort choiceis not observedby the principal, the anticipationof renegotiationmay eliminate all incentives for the agent.To preventthis result, the outcome of full insuranceonce the action is sunk mustsomehowbe avoided.Avoiding this outcomeis possibleonly if the

principalremains unsure about which action the agent chose.Indeed,if the

principal knows for sure what the agent choseto do, he can determineexactly the expectedvalue of the underlying investment and insure the

agent at actuarially fair terms. However,if the agent is randomizing herchoiceover severalactions,the principal is put in a positionof an

asymmetrically informed insurer in the renegotiationphase.As we learnedfrom Chapter2, the optimal insurancecontractin this casedoesnot allowfor full coveragein general,so that the agent may remain exposedto somerisk.

451 RepeatedBilateral Contracting

In this subsectionwe shall consider this basic dynamic contracting

problemunder the assumption that the principal (who remains uninformed

about the agent's choiceof action)makes a take-it-or-leave-itoffer of amenu of contractsto the agent.This casehas beenanalyzed by Fudenbergand Tirole (1990)and Ma (1991).Under an alternative assumption,considered by Ma(1994)and Matthews (1995),it is the agent who makesa newcontractoffer,the form of which may conveyinformation about her actionchoice.We discussthis caseat the end of this subsection,and we stressthat

the problemsdiscussedwhen the uninformed party makesthe offer canbeeliminated under an informed-party offer.

We focushere on the analysis of Fudenbergand Tirole(1990).Thesimplest possiblesetup in which their problemcanbe formally analyzed is thetwo-outcomesmodel consideredin Chapter 4.In this problem,output is

high, qH, with probability p(a) and low, qL, with probability [1-p(a)].Suppose,in addition, that the agent'sactionset containsonly two elementsa e A ={0,1},and letpx=p(l)andpQ=p(0),with/?! >pQ.Also, let the costof effort y(l) = V> \302\260 and

\302\245fi)

=0.The dynamic contracting gamebetweenthe principal and agent unfolds

as follows:Theprincipal beginsby offering an initial (menu of) contract(s)C0to the agent.The latter acceptsor rejects(oneof) the contract(s)(in the

menu).If the agent decidesto reject the contract,the game endsand the

agent gets her reservationutility u = 0. If she accepts,she selectsactiona =1with probability jc and actiona = 0,with probability (1-jc).

Oncethe actionis sunk and beforethe outcomeisrealized,the principalis allowedto make a new offer of a menu of contracts{[wL(l),w#(l)];[wL(0),w#(0)]},which consistsof two pairs of output-contingent wages,being more attractive after a choiceof a = 1and a choiceof a = 0respectively.

Theagent canselecteitherof the two contractson the menu or rejectboth. If she rejectsthe new offer,the old contractis enforced,and if sheaccepts,the new contractis enforced.Finally, output is realizedand wagepayments are made.

Note that by the revelationprinciple,we can restrict attention to an

incentive-compatiblemenu of contracts{[wL(l),w#(l)];[wL(0),w#(0)]}inthe renegotiationphase.

As in the caseof dynamic contracting under adverseselection,we canrestrict attention at the initial contracting stage to \"renegotiation-proof\"contracts.Formally, an initial contractC0= {[wL(l),w#(l)];[wL(0),w#(0)]}is renegotiation-prooffor the mixed actionchoicejc if it solvesthe

principal's compensation-costminimization problem

452 Dynamic Moral Hazard

min x\\p1wH(l) +(l-p1)wL(l)]+0--x)\\p0wH(0)+(l-pQ)wL(0)]

subjectto the incentive constraints

pAwH(l)]+(1-pMwdD] Piu[wH(0)]+(l-PiMwdO)]

Pqu[wh(0)]+(1-p0)u[wL(0)]>pou[wH(X)]+0\342\200\224po)u[wL(X)]

and the participation constraints

PlM[wH(l)]+(l-Pi)M[Wf.(l)]^p1M[WH(l)]+(l-Pi)M[Wf.(l)]

p0u[wH(0)]+(1-pQ)u[wL(0)]p0u[wH(0)]+(l-p0)u[wL(0)\\

Two important observationsmust bemadeabout this problem:First, notethat the agent'scostof effort doesnot appear in the problem.It is becausethe agent's action is sunk at the renegotiationstage.Second,both types'individual-rationality constraints may be binding here.

To seewhy the principal can restrict attention to \"renegotiation-proof\" contracts,imagine that the principal offereda contractC0that wasnot renegotiation-proof.This contract would then be replacedby a newcontract C, which solves the precedingcompensation-costminimization

problem.Thenew contractCprovidesat leastthe samepayoff to the agent as the

old contract,C0,but it strictly improves the principal'spayoff. Moreover,sincethe agent anticipatesrenegotiation,her actionchoicejc remains thesamewhether sheis offeredC0,which getsrenegotiatedinto C,or whether

she is offeredC directly. Hencethere is no lossin restricting attention to

\"renegotiation-proof\" contracts.Just as in a standard adverseselectionproblemthe principal'schoiceof

an optimal menu of contractsto solvethe compensation-cost-minimizationproblem,given someanticipatedactionchoicejc by the agent, boilsdown

to a problemof extracting the informational rent of \"type a = 0\"\342\200\224who

always wants to pretend to be \"type a = 1,\" other things equal\342\200\224by

distortingthe efficient allocationof type a =1.Hencewe would expectthe optimal

menu of contractsto be such that type a =0 getsfull insurance(nodistortion at the top) but type a =1getslessthan full insurance.In other words,only contractsthat offer full insuranceto type a =0arerenegotiation-proof,so that we must have wL(0) = w#(0) = w*. As long as an action jc > 0 is

optimal, we must alsohave wL(l) <wH(l). Finally, and not surprisingly given

RepeatedBilateral Contracting

45\302\260 line

^h(1)

Figure 10.1Menu of Renegotiation-Proof Contracts

what we learnedin Chapter2,the interim incentive-compatibility constraint

for type a = 0 must be binding, so that

u(w*)=p0u[wH0)]+0\342\200\224po)u[wLQ.)] (IIC)It is instructive to represent the menu of renegotiation-proofcontracts

in a diagram, as seenin Figure 10.1.The indifferencecurve for type a = 1is steeperthan the curve for type a = 0 becausep1> pQ, so that type a = 0valuesinsurancemore than type a=l.

If the interim incentive constraint werenot binding, as at outcomec,the

principal couldoffer moreinsuranceto type a = 1without violatingincentive compatibility.

Themenu of contractssuch that type a = 0 is fully insured and such that

wL(l) \302\253\302\243 w#(l), with u(w*) =Pqu[wh(1)] + (1-Po)u[wL(l)],is not

necessarily renegotiation-proof.It is renegotiation-proofonly if the agent'sactionchoice,x, is not too high. To seethis point, consideragain the diagramdepicting the menu ofrenegotiation-proofcontractsand considerthedeviation shown in Figure 10.2.

454 Dynamic Moral Hazard

w,k45\302\260 line

wL{\\)

w* w* + 5

Figure 10.2Profitable Deviation for Small x

wH0)

Supposethat the principal offers to raise w* to w* + & This changeincreasesthe wagebill by (1-jc)5but allows the principal to offermoreinsuranceto type a = 1.

Let 5wH < 0 and 8wL > 0 denote the changein type a = l'scompensationschemethat keepsher indifferent:

p1dwHu(wH) +0--p0)dwLu'(wL)=0

Theexpectedgain in the wagebill of type a =1resulting from betterinsurance is then

xIpiSwh+O\342\200\224pJSwl]

Hencefor the contractto berenegotiation-proofwe must have

x\\plSwH+(l-pi)SwL]^0\342\200\224x)S

so that jc cannot be too high.

Having established that the principal can restrict attention torenegotiation-proofcontracts,we now turn to the derivation of the optimalex ante renegotiation-proofcontract.We shall take it that the principal

455 RepeatedBilateral Contracting

wants to inducea choice1> jc>0,inwhich casethe agent'sex anteincentive constraint is given by

Piu[wH(l)] +(1-PxMwlO)]-\302\245

=u(w*) (AIC)

Anticipating no renegotiation,the agent is indifferent betweenactionsa = 1and a = 0,and is then happy to choosethe prescribedjc.If the

principal expectsthe agent to choosethe prescribedjc, he will not renegotiatethe ex ante contract,so that the agent cannot gain by deviating from the

prescribedactionin order to inducerenegotiation.The two constraints (AIC)and (IIC)uniquely tie down wH(l) and wL(l)

as a function of u(w*). Similarly, for any w*, the principal wants to inducethe highest possiblechoiceof jc compatiblewith renegotiation,jc(w*), givenby

x(w*)[piWH (w*)+(1-px)wL(w*)] =1-x(w*)

Hencethe principal'sex ante problemreducesto

max x(w*){px[qH -wH (w*)]+(1-px)[qL-wL (w*)]}w*

+[1-x(w*)][pQqH+(1-po)qL~w*\\

Todeterminethe optimal choiceof w*, we thus needto investigate how

jc(w*) varieswith w*. Fudenbergand Tirole (1990)establishthat the signof dx/dw* may bepositiveif the agent'srisk preferencesexhibit decreasingabsoluterisk aversion.But if the agent has constant or increasingabsoluterisk aversion,then dx/dw* is negative.The intuition for this result is that,

by increasingthe agent's wealth w*, the principal may be able to reducethe agent'sdemand for insurancesufficiently so that he gains from it.

This analysis of moral hazard with renegotiation(through umnformed

party offers),while somewhat abstract,may provide some insights into

somecontractualclausesofexecutivecompensationcontracts.For example,an executivewho has made important long-run investments or strategicdecisionsshould beofferedthe option of scaling up or down her riskexposure. Similarly, an executive'scompensationafter retirement may besubstantially lesssensitive to the firm's performance(evenif this performanceconveysimportant information about the executive'sactions),if the CEOprefersthis option.In practice,CEOshave a lot of discretionin structuringtheir incentive packages.Also, they have the option of forfeiting their stock

Dynamic Moral Hazard

optionswhen they retire.Thisis consistentwith the theory ofmoral hazardwith renegotiation.

Let us end this subsectionwith a brief discussionof the casewhere theinformed party, that is, the agent, makesa take-it-or-leave-itrenegotiationoffer to the principal after effort has beenchosen.This case,analyzed byMa (1994)and Matthews (1995),differs from the onewe have just discussedin that the renegotiationoffercan act as a signal about the effort chosenby the agent.Instead,in the casewhere the principal makes an offer,hisbeliefabout effort is obviously independentof his offer.In the agent-offercase,low effort followedby full insurancefor the agent can be avoidedif

the principal'sbelief about effort becomesmore \"optimistic\" when the

agent makes a renegotiationoffer that is riskier for her.While this gamehas a multiplicity of perfectBayesianequilibria, as is typical in signaling

games,Ma and Matthews both show that the second-bestoutcome,that is,the outcomeof the moral-hazardmodelwhen renegotiationis impossible,can be sustainedas a unique equilibrium under appropriaterefinements.

Specifically,Ma assumesthat, if an effort is optimal for the agent both forthe initial contractproposedby the principal and for the renegotiatedcontract offeredby the agent, then this is the effort the principal believesthe

agent hastaken.And, in a similar vein, Matthews alsorulesout beliefsabouteffort where the agent would have played some types of dominated

strategies.2The lessonsone can draw from theseresultsare as follows:(1)the idea

analyzed by Fudenbergand Tirole (1990)and Ma (1991)\342\200\224thatthe

anticipation ofefficient risk sharing onceeffort hasbeenchosenreduceseffortincentives\342\200\224is quite intuitive; (2)its impact on equilibrium effort andinsurance depends,however, on the detailsof the renegotiationgame;and (3)under someassumptions, this impact canbecompletelyeliminated, but this

requiresstrong assumptions.

RenegotiationWhen Effort IsObservedby the Principal

When, after effort is chosenby the (risk-averse)agent, the effort is observedby the (risk-neutral) principalbeforethe uncertainty about output is realized,Hermalin and Katz (1991)show that the optimal contractcan achievethe

2.This approach amounts, in game-theoretic parlance, to a relatively weak form of forwardinduction, which allows selecting the best subgame equilibrium for the agent.

457 RepeatedBilateral Contracting

first best:the initial contractcanmake the agent residualclaimant, therebyinducing first-best effort, then renegotiationcan offerfull insuranceto theagent.Full insuranceheredoesnot destroyincentivesto exerteffort, in

contrast with either the modelwithout renegotiationor the modelwith

renegotiation but unobservedeffort by the principal.Indeed,here,the principaloffersthe agent a fixed wagethat dependson her levelof effort, sincehehasobservedit.The initial contractthus determinesa reservationutility forthe agent that dependspositively on her effort level.

Dewatripont,Legros,and Matthews (2003)add to this problem the

monotonicity and limited-liability conditionsof Innes,3detailedin Chapter4.Innesshowedthat theseconditionsled to the optimality ofdebtcontractsin a principal-agentmodelwith bilateralrisk neutrality, becausedebt wasthe effort-maximizing contract.Sucha contract,however,is not very goodin terms of insurancefor the agent in the casewhereshe is risk averse.Ina Hermalin and Katz setup, however,this problem disappearsthanks to

renegotiation,even though the optimal contractcannot be achievedwhen

the Innesconstraints are introduced.Toprovideintuition for this result, let us focuson an examplewith three

output or revenueoutcomes:

0= qL < qM < qH

Denotingby r, the repayments to the investor/principal given outcomeqhthe agent/entrepreneur'slimited liability constraint requires that rL = 0,while monotonicity of the investor'spayoff requires4

rL<rM<rH

In this case,a debt contractis definedby

f~M < qM =* f~H=

rM

or

either rH =rM or rM

=qM

3.Before them, Matthews (2001)performed the sameexercise for the casewhere effort is notobservable by the principal (seethe discussion below).4.Remember that, among the four constraints, it is the limited liability of the

entrepreneur/agent and the monotonicity of the payoff of the principal/investor that matter in the caseconsidered by Innes. The sameis true in this case.

458 Dynamic Moral Hazard

We now establish the optimality of a debt contract in the following

example,wherethe entrepreneur'sutility function is given by

2*Jwi -aThe agent's utility is thus the squareroot of the entrepreneur'smonetarypayoff, Wi = qt - rb minus the effort cost,which is assumedto be linear in

effort.Concerningthe productivity ofeffort, we makefor simplicity the

following assumptions on the probability pt{a) of qt (on top of MLRP;seeChapter4):

p'M(a)> 0 and p'u (a)< 0

and

p'H(a)>0 and Ph(o)<010.3.2.1First-BestContracts

Thefirst-bestcontractinvolves a constant monetary payoff w for the

entrepreneur and is the solution to

max2^vv-a

subjectto

^Ma)(qt-w)>I

where/ is the investment costthat the investor has to provide.Substitutingthe constraint (which is binding at the optimum) into the maximand yields

max2 j^pMq,-I-av ,

This is a concaveproblem(rememberthat qL = 0),whosefirst-ordercondition is

\302\243tPt'(g)g.

Since

^dpl(a)ql-I=w

459 RepeatedBilateral Contracting

this can alsobe rewritten as

^p'(a)qt- i

Calling a* the first-best effort level,we thus seethat the entrepreneur'spayoff is increasingin effort up until effort levela*.

10.3.2.2Second-BestContracts

Thesecond-bestcontractsetsthe r;'sexante.Giventhe effort choiceof the

agent, this determinesthe expectedpayoff of the investor.Assuming a take-it-or-leave-itoffer by the entrepreneur after effort choice,this investor'sexpectedpayoff remains the same,but the entrepreneurasksfor a constant

monetary payoff at that stage.This payoff w isdetenninedby

i i

while the second-bestcontractis the solution of

max2Vvv-a1.0

subjectto

2p,ifl)n > I, with rL =0<rM < rH

i

and thus

w =^Pi(a){qi-ri)i

What the entrepreneuroffersinitially is thus {rL,rM, rH], sothat her problemafter the investor has acceptedthe offeris

max 2JYJp,(a)(gi-rI)-a

This is once again a concaveproblem(rememberthat qL = rL = 0) whosefirst-ordercondition is

Y.Pi(a)(qi-rt)t^ =1 (10.8)

yZtiPiiaKqi-n)

460 Dynamic Moral Hazard

Since Jp,(a)(g[-r[)=w, this can alsobe rewritten as

^pKa&qi-r,)- i

Just like the first-best contract,the second-bestcontractallows the investorto recoupthe initial investment / but implies a different pair (w, a).Specifically,

sinceqL = rL = 0 and moreoverp'M(a) > 0,Pm(o) < 0,pii(a)> 0, and

Ph(o)< 0,the second-bestcontractmust imply a lowerconstant wage w anda lowereffort levelthan in the first-best contract.Indeed,in equilibrium,we have ^.pXcfri=Lwhich meansthat, for a =a*, the denominator of theLHSof equation (10.8)would be the same as in the first-best contract.Instead,its numerator is smaller [sincePm(o) > 0 and Ph(o) > 0,and sinceqM > rM and qH > rH]. A reductionof effort belowlevela* restoresequality,

since the denominator is increasingin effort while the numerator is

decreasingin effort [given Pm(o) < 0 and Ph(o) < 0].The second-bestcontract thus suffers from insufficient effort. The best

contractisconsequentlythe one that maximizeseffort.Thisisthe debtcontract, as can be shown by contradiction.Indeed,assumeit is not the case;that is,assumethe optimal contractimplies rH > rM and rM < qM. Callitsassociated effort levela.Thenconsiderthe following changein the contract:

drM= -\342\200\224drH>0

PmW

Sucha changewill leavethe payoff of the investor unchanged at unchangedeffort a.Moreover,given the initial contract, this change can be madewithout violating either the monotonicity or limited-liability constraint.

Finally, given MLRP, this changewill lead to a (Pareto-improving)increasein effort:at effort levela, the denominator of equation(10.8)is unchanged,while the changein the numerator is

Pm(o) H,^drH-p'H(a)drH=-drHPmW

Ph(o) p'M(a)

Ph(o) pM(a).Ph(o)>0

At effort levela, the incentive to invest is thus higher than 1,so that

equation (10.8)can berestoredonly by an increasein equilibrium effort.Thisprovesthat any nondebtcontractcanbe improved upon.

The intuition of the result is clear:just as in tones (1990),a debtcontractmaximizes incentives to exert effort subject to the limited-liability and

461 RepeatedBilateral Contracting

monotonicity constraints. And just as in Hermalin and Katz (1991),renegotiation after both partieshave observedthe effort choicebut before therealizationof output allowsone to combinemaximum effort incentives and

optimal insurance.The differencewith Hermalin and Katz is that the first

bestis here unattainable. Dewatripont,Legros,and Matthews (2003)showthat message-contingentcontractscannot improve upon simple debtcontracts given the constraint imposed by renegotiation-proofness,at leastwhen one restrictsattention to pure-strategyimplementation.

The precedingresult has an obvious parallel with the earlier result ofMatthews (2001),which shows that limited liability plus monotonicityimplies the optimality of an initial debt contract in the same model but

without observability of the effort choiceby the principal.Matthews (2001)in fact addsthe Innes(1990)constraints to the informed-party-offer modelof Matthews (1995).The result here is simpler, sincerenegotiationtakesplace under symmetric information, so that no refinement of perfectBayesianequilibrium is needed.

Furthermore, the resultsof Dewatripont,Legros,and Matthews (2003)and Matthews (2001)providea simple theory of the dynamics of the capitalstructure offirms: this theory fits, for example,very well the caseofan

entrepreneur who first obtains debt finance from a bank and then goespublic,that is,movesto equity finance.Sucha pattern is well documented(see,for

example,Diamond,1991b).Here,it resultsfrom the fact that the effort ofthe entrepreneur is especiallycrucialin the initial stage of the business.Note also that, here, the entrepreneurhas a constant wage, and thereforeno equity stake ex post,becauseshe is the only risk-averseparty. If both

parties were risk averse,we would have coinsuranceafter renegotiation,which couldbereinterpretedas giving the entrepreneursomeequity in thefirm.

10.4BilateralRelationalContracts

Sofar in this chapterwe have restrictedattention to court-enforceablecontracts. Theseare the only enforceablecontractswhen the relationbetweenthe contracting parties is finite. But when the principal and agent areengagedin a repeated,open-endedrelationship, they may beableto extendany formal court-enforcedcontractwith informal self-enforcedprovisions.Informal agreements are self-enforcingwhen some credible future

462 Dynamic Moral Hazard

punishment threat in the event of noncomplianceinduceseach party tostick to the agreed terms. A classicexampleof an informal self-enforcedprovision in employment contractsis a promiseof a bonus payment or apromotion as a reward for goodperformance.Mostemployment relationshave such informal promises.Even though employershave full discretionin whether to make the promisedbonus payment, they generally tend tohonor their promises.Onereasonwhy they donot renegeon their promiseis that they fear that disgruntled employeesmay leave or shirk. Another

exampleis dividend payments by corporations.Theseare entirely

discretionary payments, but corporationsare reluctant to renegeon their

promises out of fear that the stock price will drop substantially following theannouncement of the dividend cut. In Chapter3 we saw that initiation ofdividend payments may be a way of signaling expectedhigh future

earnings. We pointed out that one weaknessof the signaling theory is that it

assumesthat firms will not renege on their promise.Thus one way ofexplaining why they would stick to their promisedpayments is to seedividend payments as self-enforcingprovisions in a relationalcontract.

The early literature on relationalcontractsconfinesattention to

contracting situations with symmetric information (see,for example,Klein andLeffler,1981;Shapiro and Stiglitz, 1984;Bull, 1987).The most completeanalysis of relationalcontractsunder symmetric information is given byMacLeodand Malcomson(1989).

In this sectionwe provide an analysis of optimal relationalcontractsunder hidden actionsand information in a bilateralcontracting problem.For simplicity and for expositionalreasonswe shall restrictattention tosituations where both principal and agent are risk neutral. We shall alsoassumethat the performancemeasureon which the relationalcontractisconditionedis observableby both partieseven if it is not verifiable by acourt.This sectionis basedon the treatments ofMacLeodand Malcomson(1989)and Levin(2003).

We shall considera sequenceofspotcontractsbetweenan infinitely livedrisk-neutral principal and agent.Both have the samediscount factor8<1.In eachperiod t the principal'sand agent'sreservationutilities are,respectively,

V and u. If they engagein a spotcontract,the agent privately choosesan actiona e A. Thisactionresultsin n possibleoutput realizationsat theend ofperiod t, which we rank in order of increasingoutput: q1< q2 <...<qn. The probability distribution of output q\\ in period t is given by pi(a^),whereat denotesthe agent'sactionchoicein period t and i =1,...,n. As

463 RepeatedBilateral Contracting

usual, we assumethat pt(a) > 0 for all a e A and that the likelihoodratiopXa)lpi(a) is increasingin output q[ (that is, MLRPholds;seeChapter4).Following the output realization,the principal makesa payment Wt to theagent.Thispayment may comprisean explicit contractually agreedpaymentwt and a discretionarybonusbf Following the payment, the principal'send-of-periodflow payoff is

q't-Wt

and the agent'spayoff is

Wt-y/(at,6t)

where6t denotesa privately observedcostparameter.For simplicity again,weshall assumethat 6t takes two values, 6t e [6L,6H] with 6L < 6H. We shallalsoassumethat 6t is an i.i.d.variablewith /3 = Pr(0,= 6H). As before,weassumethat y/(-) is strictly increasingand convexin a.We alsoassumethat

y/(0, 6)= 0 and that y/e ^ 0 and y/a9 > 0.In a first-bestsituation the agent would choosea&Atomaximize

n

^Pi(a)qi-y/(a,e)i=i

Supposethat there is a uniquely definedfirst-best actionaFB(Q).To ensurethat we have an interesting problem,we shall, of course,assumethat

^Pl{aFB)qi-w{aFB,6)>s^V+ui=i

It is alsoconvenient to make the additional mild assumption that

n

^pA0)qt-iir(0,O)<isi=i

We are now in a positionto definethe relationalcontracting problem.Arelationalcontractis a perfectBayesianequilibrium of an infinite-horizon

game, where the players'movesin each stage game(or eachperiod)arethe following:

\342\200\242 A participation decisionand an actionchoice(subjectto participation)for the agent at

464 Dynamic Moral Hazard

\342\200\242 A participation decisionand an output-contingent bonus paymentdecision (subjectto participation and following the output realization)for the

principal b^qf)

Let oa and op denotea strategy for the agent and principal, respectively,over the irifinite-horizon game.A strategy specifiesthe agent'sand

principal'smove at any time t asa function ofobservedhistory ofplay and output

realizations.Moreprecisely,the principal and agent condition their

respective movesat date t on the history of output realizationsand past transfers.In other words, they condition their moves on the same observedpasthistory. The agent doesnot condition her move on her past actionswhen

her actionchoiceis not observableto the principal, as actioncostsincurredin the past are sunk and past actionsdo not affectthe principal'scontinuation play in any way.

The players' flow payoffs are determinedpartly by a court-enforcedpayment plan \302\243w\342\200\224which specifiesa payment from the principal to the agentat eachdate t as a function of verifiable history of play and outputrealizations\342\200\224and by a self-enforceddiscretionarypayment plan \302\2436\342\200\224which

specifies a bonus payment at eachdate t as a function of observablehistory ofplay and output realizations.For simplicity weshall assumethat outputrealizations are observablebut not verifiable, and that past payments areobservableand verifiable.

A perfectBayesianequilibrium of the irifinite-horizon gameis such that

the players'movesfollowing any history are bestresponses.The(normalized)payoff at any point in time t under a relationalcontract

(ct, 0=(c7fl, ap, Cw, C6)is then given by

Vt =(1-d)EtKa,0 25'-1[A(gT -WT) +a-A)V]

for the principal, and

ut =(1-5)EtKa,0 25'-1{A[WT-ys(aT, 6T)]+(1-A)u }

for the agent, where A is an indicator variable taking value one if bothpartiesdecideto participate,and zero if one of the partiesopts out.

465 RepeatedBilateral Contracting

Sinceopting out of the relationship is one of the movesavailableto eachparty at any time, any relationalcontract must give each party a payoff

greater than or equalto his or her outsideoption:

Vt > V and ut > u

Also, if a relationalcontractgeneratesa total surplus st > s,then it ispossible to enforceany division (ut, st -ut) of the surplus such that ut > u and

(st -u() > V as a relationalcontractby specifying a suitablefixed transferwt betweenprincipal and agent.Indeed,the fixed transfer affectsonly eachparty's participation decisionand not any other moves.As long as ut > u

and (st - ut) > V, participation is clearlypreferable,so that (ut, st - ut) is

clearlyalsoa relationalcontract.Hence,a relationalcontract(cr, \302\243)

at datet = 0 is optimal if it maximizes the joint surplus s0-Another implication ofthe result that it is possibleto enforceany division (ut, st - ut), such that

ut > u and (st-ut) > V, as a relationalcontractis that onecanrestrictattention to stationary contractswithout lossof generality to characterizethe

joint payoff under an optimal relationalcontract.

definition A relationalcontract (cr*, \302\243*)is stationary if in every period

on the equilibrium path at = a(6t), bt = b(qf), and wt =w.

We leave the proof of the above observationas an exercise.Theargument involves showing that onecanaverageout nonstationary transfers andactionsto \"obtain a stationary contractwith the samepayoff

Dynamic programming can be used to obtain a simple characterizationof an optimal stationary relationalcontract.Let u* and s*-u* denote the

agent'sand principal'spayoff under the stationary contract(cr*, \302\243*),let w*

and b*(q[) denote the court-enforcedtransfer and discretionary bonusunder the stationary contract (cr*, \302\243*),

and let a*(6) denote the agent'sactionchoice;then the joint value of the contracting relations* is given by

5*=max(l -8)Ee\302\253 [q-yf[a(0),0]I a(0)]+8Ee\302\253

[s*\\

a*(6)]a(9)

subjectto

a*(6)&a.rgm.ax\\Eq w*+b*(qi)+-\342\200\224~u*\\al \342\200\224 o -V(M)} (IC)

b*{qd+T^u*>-^-u (PCA)l\342\200\224o l\342\200\224o

466 Dynamic Moral Hazard

and

-bHqJ+-r^(s*-u*)>-^-V (PCP)1\342\200\224d 1\342\200\224d

The first constraint ensures that the agent has the correct incentive tostick to the prescribedactionplan a*(6).Thesecondconstraint ensurescontinued participation of the agent.Implicit in the constraint is theassumption that when the agent quits, she quits forever.There is no lossofgenerality in making this assumption. In fact the threat of a permanent quitis the strongestpossiblethreat for the agent and is the one that allows forthe largest possibleset of self-enforcingcontracts.The third constraint

ensurescontinued payment of the discretionarybonus by the principal.Ifthe principal were to withhold the bonus payment b*(qi) in any period,hewould receiveonly the flow payoff 8V /(1-5)in the future, as the agentwould quit forever following such a move.If the principal complies,the

agent doesnot quit and he gets the flow payoff 5(s*-u*)/(l-5)in thefuture. The third constraint thus ensures that complying with the bonus

payment results in a weakly higher payoff for the principal.Severalquestionsimmediately ariseconcerningthis contract.First, is it

clear that the agent's strategy of participating and supplying actiona*(6)in periodt as long as the history of output and payments has been{[q''1,w*+ \302\243*(g'-1)]}

in every periodand otherwisequitting forever,and the

principal's strategy of participating and paying w* + b*(qty as long as the historyof output and payments has been{[q''1,w* +

\302\243*(g'-1)]}in every periodand

otherwise quitting forever, are mutually best responsesfollowing any

history? If the principal decidesto quit foreverfollowing a deviation fromthe history of outputs and payments {[q1'1,w* + 6*(g'-1)]},then obviouslya bestresponsefor the agent is also not to participate,and vice versa.Therefore,if conditions(IC),(PCA),and (PCP)are satisfied,the contract(cr*, \302\243*)

doesindeedsupport a perfectBayesianequilibrium.Second,following a deviation by one party, is there not scopefor

renegotiation? It turns out that (stationary) relationalcontractssuch that u* >u and (s*-u*) > V can easily bemaderenegotiation-proof.Thereason isagain related to the possibility of enforcing any division (ut, s* -ut), suchthat ut > u and (s*-ut) > V, as a relationalcontract.If renegotiationprecludes enforcementof inefficient punishments such that sT< s* following adeviation in periodt < t, a relationalcontractcan still be enforcedwith

467 RepeatedBilateral Contracting

jointly efficient punishments by changing the division of the surplus s*following a deviation. Thus the following contractcan be specified:the

principal promisespayments w* and b*(qt) unlessone party deviates.If the

agent deviates,switch to a continuation contractwith payments vv + b(qt)such that u=u (and V =s* -u) following the deviation, and if the

principal deviates,switch to a continuation contractwith payments w and b(qt)such that V = V. If further deviations occur, treat them similarly. This

expandedrelationalcontract is obviously renegotiation-proofbecauseitlies on the constrainedPareto frontier. It should alsobe clear that, sincecontinuation payoffs are the samefor the deviating party as when

continuation involves nonparticipation, this expandedcontractcan enforcethesameoptimal joint payoff s*.

Third, is it clear that an optimal stationary contract (cr*, \302\243*) actuallyexists?When 6and q takeon a finite number ofvalues,aswe have assumedhere,existencecanbeestablishedusing similar arguments asusedfor static-adverse-selectionand moral-hazardproblems(seeLevin, 2003,for a proofof existence).

Interestingly, discretionary payments b*(qt) may bepositiveor negativedependingon how the net surplus from the relationalcontract(s*-s) isdivided betweenthe principal and agent.If the agent getsa small shareofthe net surplus, then discretionary payments b*(q^) must bepositive.Indeed,only largepositivediscretionarypayments, which may benecessaryto givethe agent adequateincentives to supply effort [orcostly actionsa(6)],cansatisfy the agent'sparticipation constraint (PCA)when (u* -u) tends tozero.In this casethe relationalcontractcanbe interpretedas a\"performance pay\" contract.At the other extreme,when the principal getsa smallshareof the net surplus, discretionarypayments b*(q)must benegative.Inthis casethe relationalcontractresemblesan \"efficiency-wage\" contract.

Themain differencebetweena relationalcontractand the fully

enforceable contractswe have seenin Chapters2,4, and 6 comesfrom theadditional self-enforcementconstraints (PCA) and (PCP).Letting b and b

denote the highest and lowestdiscretionary bonus payments, respectively,these two constraints can be combined into the following single self-enforcementconstraint.

(b-b)<-^-(s*-s) (SEC)l \342\200\224 o

468 Dynamic Moral Hazard

When this constraint is binding, relationalcontractsprovide lesshigh-

poweredincentives than fully enforceablecontracts.We now illustrate how

this constraint canmodify the form of the optimal contractin the two polaroppositecasesof pure moral hazard and pure adverseselection.

10.4.1MoralHazard

Thecaseof pure moral hazard ariseshere when 6L = 6H=0.As we saw in

Chapter4,an optimal incentive contractunder risk neutrality takesa verysimpleform: when there is no gain to sharing output risk, it is optimal tomake the agent a residualclaimant. More precisely,the optimal contractspecifiesan output-contingent payment to the agent of

W(q[) =w +qi

wherew is a fixed payment such that

w =u -max{\302\243? [q|a]-\\jf(a, &)}aeA

This contractis not self-enforcingif

(qn-qi)>^[Eq[qWFB]-\302\245(aFB,6)-s]

Whenever the self-enforcementconstraint (SEC)is violated under theresidualclaims contract,it is optimal to specifya relationalcontractsuchthat

b(q[)=b for qt>qk

b(qi)=b for q(<qk

for some qx< qk< qn. This result follows immediately from the MLRPassumption.We leaveit asan exercisefor the readerto prove.A useful hint

here is the analogy that canbedrawn betweenthis relationalcontractunder

pure moral hazard and risk neutrality and the fully enforceablemoral-hazard problemunder limited liability consideredby Innes (1990).Thusself-enforcingconstraints act like limited-liability constraints.

10.4.2AdverseSelection

Thecaseofpure adverseselectionariseswhen the agent'sactionchoiceatisobservableto the principal, but not the agent'scosttype 6.As in Chapter

469 RepeatedBilateral Contracting

2,we can apply the revelationprincipleto characterizethe optimalstationary relational contract. Observe also that the Spence-Mirrleescondition

de{ da J

is satisfiedunder our assumptions about the agent'scost-of-effortfunction

y/ag >0).Therefore,the optimal contractcan be characterizedas thesolution to the following program:

s*= max (X-5)Eea[q-}ir(a(0),0)\\a(0)]+8Eoq[s*\\a*(0)]

subjectto

b{eL)-xif[a{eL),eL]>b{eH)-xif[a{eH),eL]

b(6H)-y/[a(6H),6H]>uand

\\KeH)-b(eL)\\<-^-(s*-s)l \342\200\224 o

As weshowedin Chapter2,the optimal contractunder full courtenforcement would have \"no distortion at the top,\" which here means, on the onehand, that' the high-costtype providesan efficient levelof effort. That is,a(6H)satisfies

%p:[a{eH)]qi=xif\\a{eH),eH]1=1

On the other hand, the low-costtype underprovideseffort.This is adistortion imposedby the principal to reducethe low-costtype's informational

rent.Now, the basic\"no distortion at the top\" property doesnot hold

generallyunder an optimal relationalcontract,since the bonus scheduleb(6)

associatedwith the second-bestcontract under full enforceability mayviolate the self-enforcementconstraint (SEC).In that case,all cost typesunderprovideinputs. In other words, the self-enforcementconstraint putsan overall limit on the total incentives that can be given to the agent.Levin (2003)shows that when there are more than two cost types (say, a

470 Dynamic Moral Hazard

continuum), then not only doesthe optimal relationalcontract result in

underprovision of inputs for all cost types, but also \"bunching\" of costtypes (that is,equalinput provision for different cost types; seeChapter2)is more likely.

10.4.3Extensions

In this sectionwe have consideredonly pure relationalcontracts.But theframework canbe extendedto allow for both contractually explicitincentives and implicit incentives.Severalarticlestake a stab in this direction.Bernheim and Whinston (1998b)considera modelwhere the contracting

parties can write more or lesscompleteincentive contractssupplementedby implicit incentives.They show that it can be in the parties'interest torely lesson formal incentives than they could in order to exploitimplicitincentives as much as possible.Similarly, Baker,Gibbons,and Murphy

(1994)and Pearce and Stachetti(1998)explorea modelwith risk-averseagents combining both explicit and implicit incentives.Finally, MacLeod(2003)extends the setup consideredhere and analyzesa relationalcontracting problembetweena risk-averseagent and a risk-neutral principal,where the performancemeasuresof eachparty are subjectiveand imper-

\342\200\242

fectly correlated.Interestingly, he shows that in an optimal relationalcontract the agent'scompensationdoesnot dependon her own performanceevaluation, while her cooperationdoes.If accordingto her own evaluationthe agent feelsthat sheis unfairly treated, then shestopscooperating.

10.5Implicit Incentivesand CareerConcerns

In this sectionwe considerrelationalcontractsin multilateral settings andanalyze how the agent respondsto both explicitcontractual incentives andimplicit market-driven incentives.In most long-term contracting problemswith moral hazard in reality, agents'incentives are driven by a combinationof formal contractual rewardsand careerconcerns.This observationisespecially true of executivecompensation,as Gibbonsand Murphy (1992)have

emphasized,but it is alsotrue ofmost employment contracts.In contrasttocontractual incentives,market incentives are not easilycontrolledand canoften be excessivelystrong.

Holmstrom (1982a)providesthe classictreatment of incentives driven

by career concerns.The simplest versionof the modelhe considersis as

471 RepeatedBilateral Contracting

follows:Thereare two periods,t = l,2.The agent'soutput performancein

eachperiodis given by

qt =6+at +\302\243t

where6 denotesthe agent'sunknown ability, at is the agent'sunobservableeffort, and et is white noise(a normally distributed random variablewith

mean zero and varianceo2).A key departurefrom earliermodels(and, aswe shall see,a key simplification) is that the agent's ability is assumedtobe initially unknown to everybody,whether the agent or the principal.Onlythe prior distribution over 6 is commonly known and shared by allcontracting partiesex ante.Therefore,contracting at date 1takesplaceunder

symmetric information. Effort at is observedonly by the agent, who incurs

the usual hidden effort cost y/(at).Holmstrom assumesthat although output performanceqt is observable

by everyone,no explicit contractcontingent on the realizationof qt canbewritten. All the principal can do is pay the agent a fixed wageWx in period1and vv2 in period2.With fixed wagesit would seemthat the agent hasnoincentive to supply effort.However,although the agent has no explicitcontractual incentives,shehas market-driven incentivesthat operate through

changesin market wagesin period2 that follow from the observationoffirst-periodoutput.

Holmstrom takesthe second-periodmarket wagew2(q) to besetby

competition among principals for the agent's servicesand to be equal to themarket'sbeliefsabout the agent'sexpectedproductivity conditional on therealizationof first-periodoutput: E(6\\q1).Sincehigh first-periodoutput ismore likely for a high 6, the market updatesprior beliefsabout the agent'sability upward whenever a high output is observed.This result gives the

agent incentives to supply effort in the first period,soas to raise her first-

periodoutput and thus her second-periodwage.If 8 is the discount factor between the two periods,the (risk-neutral)

agent maximizes

wi-yr(fl)+&v2(g)

In a pure-strategyequilibrium, where the market anticipatesthat the

agent will put in effort a* in the first period,we thus have

Wi(q) =q-a*=6+a-a*

472 Dynamic Moral Hazard

and so

\\ff'{a*)=8

Notice,on the one hand, that when d>1the agent puts in moreeffort than

is first-bestoptimal in the first period.The first-best-optimaleffort levelindeed is given by \\if\\aFB)

=1.On the other hand, the agent puts in a2= 0in the secondperiod,asshedoesnot gain anything from exerting herselfatthat point. In other words, the agent may respondto market incentives by

exerting herselftoo much early on, in an attempt to build a reputation for

high ability, but then restson her laurels later on in her career.(Notice,however,that the market is not fooledby the agent in equilibrium and

correctly anticipatesand discountsher first-periodeffort choicea*.)Holmstrom

generalizesthis modelin a multiperiod setting and shows how, quitegenerally, the agent's equilibrium effort provision is excessiveearly on andinsufficient in later stagesof an agent'scareer.

Following these preliminary observations,we shall generalizethe one-shot career-concernsmodel following Dewatripont, Jewitt, and Tirole(1999a,1999b),and we shall comparethe career-concernsparadigm with

the explicit-incentivesone.We denote the agent'svectorof actionsor efforts by a =

(\302\253i,... , aN) eUN, which implies a cost y(a).The market observesa vector ofperformance variablesq =(qh ...,qM) e RM,which lead to a rewardw for the agent,whoseutility is then

w -y/(a)

The reward w reflectsthe market's expectationof an unknown (scalar)talent parameter6conditional on the observableq.Let/(0,q\\a) denotethe

joint density of talent and performancevariablesgiven effort vectora.Let

f{q\\a) =\\f{e,q\\a)dedenote the marginal density of the performancevariables.The agent'srewardfor performancevariablesq and equilibrium actionsa* (anticipatedby the market) is thus

w =E{6\\ q, a*)= 6\342\200\224 dOJ/folfl*)

The agent thereforechoosesa soas to maximize her expectedutility:

max \302\243[\302\243(01 q, a*)]-y/(a)

473 RepeatedBilateral Contracting

where the first expectationis with respectto performanceand the secondwith respectto talent.

It is helpful to derivea generalfirst-ordercondition for this problem.Assuming an interior solution, this first-ordercondition for an equilibriumis

_d_da

or

AJ f{q\\a

f(0,q\\a*) Jn ,j\342\200\224\342\200\224dO\\f(q\\a)dqf(q\\a*)

=Va(a*)

J J ff(g, gig*) faSq\\ a2dqd6=\302\245a(a*)

f(q\\a*)

where\\jfa

and fa denote the gradients with respectto effort of the agent'seffort-costfunction and of the marginal distribution. Using the fact that the

(multidimensional) likelihoodratio has zero mean [E(fa/f)=0],we canrewrite the equilibrium condition as

covk4Wfl(a*) (10.9)

where \"cov\" denotesthe covarianceof two random variables.5The vectorcov(#,fa If) describesthe agent'smarginal incentives. It expressesthe link

betweenperformanceand expectedtalent for a given equilibrium effort a*.We now considermorespecificapplicationsof this generalformulation.

10.5.1The Single-TaskCase

Consider the following generalizationof the additive-normal model ofHolmstrom: assumea single task and a singleperformancevariable(N =M = 1),and supposethat performanceis given by

q =6{lM+^)+ya +\302\243

5. Condition (10.9)for an implicit incentive scheme can be compared with the standardformula for explicit incentive schemes (Mirrlees,1999;Holmstrom, 1979;Jewitt, 1988).Supposethe agent receives an explicit transfer t(q) contingent on performance q and has utility u(t)from income t.Then the first-order condition for the agent is

cov\\u[t(q)],%-\\=Va(a*)

474 Dynamic Moral Hazard

where p., y and<j>

are positiveknown constants, while 6 and e areindependently distributed random variableswith Gaussiandistribution

0~N(0,<jI)and\302\243,\342\226\240~5V(0,c7\302\2432)

Holmstrom'sadditive-normal modelassumesthat \\i = 0 (and <j>= y=1).Apositive jU implies a complementarity between talent and effort (a puremultiplicative-normal modelimplies fi *0 and <j)=y= 0).In this case,talent

matters more,the higher the effort; that is, it makesa differenceespeciallywhen the agent \"tries to make things happen.\" This complementarity canthen lead to multiple equilibria: If the market puts a lot of weight on the

agent'sperformance,sheis inducedto work hard; in turn, her hard workleadsthe market to pay attention to her performance.Theconverseis truein the caseof low effort.

We can apply condition (10.9)to obtain, for a given expectedeffort a,

2\342\200\224 y v\"7

Oufl+r)+ A

(jM+<t>)(j\302\243

Themarginal costof effort (theRHS)is assumedto beincreasingin a.Thecovariancebetween6and the likelihoodratio (the LHS)dependson a onlywhen fi *0.Indeed,in the pure additive case(p.= 0), we obtain as RHS(assuming alsofor simplicity <j>= y=1):

that is, the signal-to-noiseratio, which is independent of effort. With amultiplicative effect (fi > 0) the derivative of the denominator of thecovariancewith respectto effort becomes

which is increasingin a.There existsthereforea levelof a, which we cancalla,such that this derivative is positivefor all a > a.Dependingonparameter values,a is strictly positiveor not. For ct|small enough, the covariancebetween6 and the likelihoodratio is always decreasingin effort. Take,forexample,the pure multiplicative modelwithout noise:q = 6a.For a givena*, the market estimates6asqla*,sothat a given increasein expectedeffort

RepeatedBilateral Contracting

Figure 10.3Additive Case(p.= 0)

by the agent leadsto a weakerupward revisionof the market'sestimateoftalent. For positiveof, however, there is another, oppositeeffect:higher

expectedeffort leadsthe market to put more weight on talent relativetonoisewhen observing goodperformance.In fact, for

<j>

=0,when the market

expectsa = 0, it attributes goodperformancesolelyto good luck, which isnot the casewhen it expectspositiveeffort. For low a's,this secondeffectcan dominate the first one for

<j>

small enough or o^loehigh enough. And

for a'shigh enough, the first effectalways dominates.Figure 10.3shows the unique equilibrium in the additive case(fi = 0).

Figure 10.4shows the unique equilibrium when \\i > 0 and<j) largeor d^a\\

low.Finally, the last caseis depictedin Figure 10.5,where two stableequilibria may coexist(the intermediateone beingunstable).

The Multitask Case

Let us now considera generalizationof the precedinganalysis to look atthe connectionbetweeneffort incentives and the set of (symmetric)activities pursuedby the agent.Assume that performanceon task i e{1,...,N}is given by

qt =6(fMi +<j>)

+yat +e;

476 Dynamic Moral Hazard

Va(a)

Figure 10.4Weak Multiplicative Factor (ji> 0;<t> large or alia}small)

Figure 10.5Strong Multiplicative Factor (p.>0;<p

small or g\\\\g\\ large)

477 RepeatedBilateral Contracting

where fi, y, and<p

are positiveknown constants, a-t> 0 is effort expended

on task i, and, as earlier, 0 and\302\243,-

are independently distributed randomvariables:

e~N(6,(j2e)and\302\243i

~3V(0,o\\)

Let us distinguish between the (exogenous)number of potential tasks

(N) and the number of tasks actually pursued (n) for which a{ > 0.Supposethat the principal caresonly about the aggregateoutput

N

so that only the total levelof effort matters, not its distribution acrossindividual tasks.Similarly, assume away any economiesor diseconomiesofscope,sothe costfunction canbewritten as a function, iff (a),of total efforta=^T cii [with Y > 0, Y' > 0= V'(0) ^ 0].We can now analyze the

relationship betweenthe number of tasks the agent pursuesand total effort.Considerfirst the casewhereonly the total performanceon a subset/ of

tasks

iel

is observedby the market. We are then back to the single-taskcase.Whatever the set of tasks the agent couldwork on, if the market observesonly

Qh the agent expendseffort solelyon the tasks includedin /.As for total

effort, d, if / includesn tasks,

iel

This expressionimplies that6

COVJ0,-7r\\ =J

(jM+n<j))+e

{fM +n<())(Je

6. Since all e,'sare uncorrelated, the distribution of Qiis normal with mean Q(pa +nb) + ydand variance ag(jjd +nb)2+na\\. Consequently,

k&) _ [(g-g)(^+w6)+X.6/e-][^+y1f{Qi) o2g{lia+nb)2 +ol

Dynamic Moral Hazard

Figure 10.6No High-Effort Equilibrium Beyond h

Positive equilibrium levelsof effort are thosewhich equate this covariancewith y/'(d).An increasein n lowersthe covariancebetweentalent and thelikelihoodratio, and thus equilibrium effort, in all three casesdepictedin

Figures 10.3-10.5.Themost interesting oneconcernsFigure 10.5,where,forn largeenough, the high-effort equilibrium disappears,as shown in Figure10.6.

In the (Holmstrom) additive-normal model, & decreasescontinuouslywith n, which may not bethe casewhen a multiplicative effectis introduced:As Figure 10.6shows, there is then a maximum n, call it n, that allows forthe high-effort equilibrium; beyondthat value, a can only bezero.

The intuition for why total effort decreaseswith n can be most easilyunderstoodby focusing on the pure additive case(fi = 0) and the puremultiplicative case

(<f>= y= 0).For simplicity, moreover,assume0= y= 1in

the pure additive caseand fi = 1in the pure multiplicative case.Considerfirst the \"noiselesscase\"o\\ = 0, where the pure additive and the puremultiplicative modelsbecome,respectively,

Qi=6n+aQi=6a

479 RepeatedBilateral Contracting

In the puremultiplicative case,n doesnot affecta.Instead,in the pureadditive case,incentives to expendeffort go down when n goesup, since themarket infers the following 6 from a performanceQjgiven effortexpectation a*:

Qi-a* n a-a*n n

There is,moreover,a secondeffectof an increasein n on effort, similar in

both cases:the varianceofX\"=i\302\243<> namely, no\\ increaseswith n. In total,

effort thus goesdown when n increases.Siuriniing up, when the market can observeonly the aggregate

performance on n tasks, the higher the number of tasks n, the lowerthe equilibriumtotal effort.Moreover,in the presenceof a multiplicative effect(p.> 0) andfor V'XO) > 0, there is a value of n beyondwhich the high-effort equilibrium

disappearsaltogether.7In the precedinganalysis we have restrictedpublic information to the

observationofaggregateperformanceQj.Supposenow that individual

performances are observed:

SI={qi\\ieI}Theresultsare then as follows:

1.In the pure additive case,Qris a sufficient statistic for 5/as far asupdating

0isconcerned,soobserving 5/or Qjmakesno difference.Consequently,the unique equilibrium effort a* is the samewhether aggregateperformance (Qi)or disaggregatedperformance(S7) is observed.2.In the pure multiplicative case,weinsteadobtain a multiplicity ofequilibria, sincethe agent will want to focuson the tasks the market expectsher

7. Concerning the robustness of these results, note that one could object, in the additive case,to the role of a rise in n in increasing the link between talent and performance Qf. if it meanssplitting the agent's total working time into more slices,one might want to have, in the\"noiseless\" case,

Qi=\342\200\224n +a=6+an

However, in this case,the results are again valid: for a]= 0,raising n leaves total effort

unaffected, and for a]> 0,it lowers total effort.

480 Dynamic Moral Hazard

to focuson!8Specifically,when at is expectedto be equal to 0,qt isconsidered as pure noise and disregardedby the market, making ax = 0 optimalfor the agent.For each/'c/, there is thus an equilibrium whereat = 0 fori <\302\243 I',and a{ is positiveand constant acrossz'se/',and where total effort isthe sameas that when only Qr =^ieI,qiis observed.

Following this analysis of multidimensional careerconcerns,we are now

in a position to ask:What results rely on career concerns,as opposedtoexplicit incentives?Theearlierresult ofa negative link betweenthe number

of tasks n and total effort when only aggregateperformanceis observablewould alsohold in a simple linear explicit-incentive-schemecontextwith

CARA preferences,sincea risein n meansmorenoise.It is thereforeour

\"technologicalassumptions\" that generatea benefit of focusunder

aggregate performanceobservability. Under explicit incentive schemes,the rolesof 6and e,however, aremoresymmetric than under careerconcerns,as thenext subsectionwill make clear.But here, the differencebetweencareerconcernsand explicit incentives lies in the casewhereSris observed,since(1)under explicit incentive schemes,the multiplicity of equilibria is not aconcern,and (2) in the additive case,which has a unique equilibrium,

expanding Srlowerstotal effort, while with pure explicit incentives,onecanalways disregardadditional information, which is at bestirrelevant.

A generallessonof the precedinganalysis is that there is a benefit of\"focus.\" Indeed,a principal interestedin maximizing Q=^ qi and thus

indifferent with respectto the distribution of individual performances(theg,'s)should try to inducefocus.As for the agent, in the caseof anexogenous date-1financial compensationshe prefers a broad mission, so as toreduceeffort; indeed,sherealizesshe will foolno one by working in

equilibrium and seesa broadmission as a commitment not to expendeffort.If,instead,the agent is able to obtain ex ante a wageequal to expectedproductivity, committing to a focusedmission through the observability of asingle qt maximizes her wage.

Theideaof giving agents\"focusedmissions\" is a celebratedtheme in the

analysis of government bureaucracies\342\200\224for example,in the bookby Wilson

(1989),which detailshow successfulgovernment agenciesare the ones that

8.In the absence of economies of scopein the effort-cost function, multitask equilibria are infact unstable: Were the market to put slightly more weight on any given task, the agent would\"react\" by concentrating her efforts solely on that task! However, the presence of economiesof scopecan ensure the local stability of these equilibria.

481 RepeatedBilateral Contracting

have managed to translate very broadobjectivesinto well-focusedmissions.Another important theme in that literature is the idea that missions haveto benot only focusedbut also\"clear.\" This canbe translated in this modelby looking at equilibria with random allocationofeffort acrosstasks by the

agent.In this case,Dewatripont,Jewitt, and Tirole(1999b)show that, in the

presenceof multiplicative effects,effort goesdown, ceterisparibus, when

one movesfrom a pure-strategyequilibrium to a mixed-strategyequilibrium. The intuition is that the market may now have to infer from the vectorof performancelevelsnot only talent but also which task(s)was (were)pursued.Consequently, a given variation in performancewill lead to lessupdating of the talent parameterin a mixed-strategy equilibrium than in apure-strategyequilibrium.

10.5.3TheTrade-offbetweenTalentRisk and Incentivesunder CareerConcerns

Let us finally consider task-specificproductivity and ask how effort isrelated to the specificbundle of tasks given to the agent.

Consider the additive-normal model, where performance for taski =1,2,is

qi =6i +al+\302\243l

where 0;and e[are normally distributed:

0,-0^(0,01)and\302\243,-

~3V(0,o2e)

with\302\243i

and &i independently distributed from oneanother and from the two

dimensions of talent; but the latter may be imperfectly correlated (until

now, we had assumedperfectcorrelation).Assume prospectiveemployerscare about 0X + &i. How is total effort ax + a2=a related to the correlationcoefficientp betweenQx and 0{lAggregate performance

q1+q2=(01+62)+a+\302\243i+\302\2432

has a normal distribution with mean 26+ a and variance2(1+p)aj+2o\\,so that

Mqi+qi) (e1+62-2e)+(\302\2431+\302\2432)

Hqi+qi) 2{i+P)ol+2ol

482 Dynamic Moral Hazard

and

fA 2(l+p)c792COV \302\2511+02, fj 2(l+p)(rl+2(j2e

Therefore,in equilibrium, total effort will be higher, the higher the p:sincethe mean of 0\\ + 02 is unchanged, a higher p implies a higher signal-to-noiseratio, becauseit meansa higher initial uncertainty about talent

relative to pure noisein performance.This observationimplies that, if one has to allocateN tasks in total to

agentswho caneachdo only n tasks, effort will bemaximized by groupingtasks that require \"similar\" talents. Consider,for example,four tasks that

must beperformed,with two agents,A and B, that can eachbe given two

tasks.Assume the output in task i to be

qi=OiK+ai+\302\243i

for k e{A, B}if agent k\\s allocatedto task i,has task-specifictalent 6iK, and

expendseffort at on task i.Assume that the 0iK'shave identicalmean 6 andvariancecri and all

\302\243,'shave zero mean and varianceo\\. Assume also that

tasks 1and 2, and 3 and 4,respectively,are thus related in that 6lKand 0zK

are positively correlatedand the same is true for &zK and 6^K,while otherdimensions of talent are otherwiseuncorrelated.Supposefurther that themarket caresonly about ^,.u)0iK, where I(k)is the set of the two tasksallocatedto agent k. In this case,incentives are maximized by allocatingtasks 1and 2 to one agent and tasks 3 and 4 to the other.9

This \"specializationresult\" is specificto the career-concernparadigm:In a modelwhere the agent caressolelyabout monetary incentives, as in

Holmstrom and Milgrom (1991),allocating tasks should reduce the total

variancedueboth to pure noiseand talent risk, and, by doing so,effort canbeincreased,sincethe trade-offbetweeneffort and risk hasbeenimproved.Thispositivecorrelationbetweeneffort and talent risk in career-concernmodels,a feature reversed in explicit-incentive-schememodels,is a keyinsight that helpsone to understand the importanceof \"focus\" in theliterature that tries to explaingovernment agencies'performance.

9. In a more general model, there may becosts to creating focus by clustering related tasksin this way, resulting, for example,from complementarities between unrelated tasks in the

principal's objective function, or from agent risk aversion (specialization will increase risk). Thesecosts have to beweighed against the beneficial incentive effect of specialization.

483 RepeatedBilateral Contracting

10.6Summary

In this chapter,we have first extendedthe classicalmoral hazard modeltothe caseof repeatedeffort, consumption, and output realizations.We haveisolated three potential gains from long-term contracting with moralhazard.First, the agent may be willing to take more risk, as she canpartially self-insureagainst a bad outcomeby smoothing consumption overtime. Second,the principal may gain more information about the agent'sactionchoicefrom repeatedobservationof the agent'sperformance.Third,an optimal long-term contractcan improve overrepeatedspot contracting

by monitoring the agent'ssavings and by forcing the agent to consumemorein earlierperiods.

Offsetting thesepotential gains, however, is the fact that the agent has aricher actionset in a repeatedrelation.The agent choosesa whole actionplan in a repeatedrelation,which canbe contingent on observedcumulative output performance.By adjusting her effort to past performance,the

agent can then \"slack off\" in moresophisticatedand lessvisible ways. This

greaterflexibility in effort supply overtime constrains the principal'sability

to extract the gains from self-insuranceor other sourcesof gains from an

enduring relationwith the agent.In extremesituations we have seenthat the agent'sactionsetmay beso

rich that the bestthe principal can do is to offer the agent a simpleincentive contractthat is linearin both output increments and cumulative output.In other settings, simple efficiency-wagecontractsor debt contractsmayalsobe approximately optimal. However,in general,the optimal contractpredicted by the theory can be extremely complex\342\200\224too complexto bedescriptiveor prescriptivefor incentive contracting in reality.

Oneof the reasonswhy most long-term contractsobservedin reality arerelatively simple is that enforcementcostsare likely to escalatesignificantlywith contractual complexity. Disagreementsand litigation are more likelyfor more complexcontracts,and judgesare more likely to make mistakesin interpreting and enforcing complexcontracts.In reality, formal long-term contractsare often written in vague terms, partly becauseit is oftenmore efficient to rely on self-enforcementthan on legalenforcement.Wehave seenhow the theory of optimal contracting canbe extendedto suchrelational contracts. In bilateral settings, the only differencewith legally

484 Dynamic Moral Hazard

enforcedoptimal long-term contractsis that someactionsand transfers areconstrainedby a self-enforcementconstraint. But otherwisethe structure

of the optimal contractis unchanged.However,in settings where there is

competitionfor the agent'sservices,implicit incentives or \"careerconcerns\"

may lead to substantial distortions, with the agent excessivelyrespondingto market incentives early on and later resting on her laurels.

Another limit to the gains achievablethrough optimal long-termcontracting is the inability of the contracting parties to commit not torenegotiate the contractin the future. We have, thus, seenhow expostrenegotiationcan undermine ex ante incentives.This issuehas gained particularprominence in debateson executivecompensation.Many criticsof executivecompensationhave singled out the common practiceof lowering the strike

price on out-of-the-moneyoptionsgranted to executives,following a dropin stock price.The resetting of the strike price obviously benefits the

company by restoring incentives expost,but the anticipation ofsuch movesundermines ex ante incentives.

By introducing relationalcontracts,implicit incentives,and renegotiationof long-term contracts,this chapter naturally paves the way toward thefourth major part of this book,on incompletecontracts.The trade-offbetweenex ante and expost incentives will take evensharperforms in thecontextof incompletecontracts,ascontracting partieshave evenlessroomto commit to expost inefficiencies.

10.7LiteratureNotes

Mostof the material in this chapteris basedon recentresearchthat has not

yet found its way into textbooks.While we have offeredour own

perspective on the literature, we have stressedthe following strands of work.First, on moral hazard with multiple effort and output realizations,we

have detailedthe pioneeringpapersby Rogerson(1985b)and by Radner(1981),and the subsequentwork by Malcomsonand Spinnewyn (1988),Fudenberg,Holmstrom, and Milgrom (1990),Rey and Salanie(1990),onthe one hand, and Rubinstein and Yaari (1983),Radner (1985),and Duttaand Radner (1994),on the other.The surveys by Chiappori,Macho,Rey,and Salanie(1994)and Radner (1986a)discussmuch of this literature. A

largerelatedliterature on repeatedpartnership gamesalsodealswith manyof the issuesin this chapter (see,most notably, Radner, 1986b;Abreu,

485 RepeatedBilateral Contracting

Pearce,and Stacchetti,1990;Fudenberg,Levine,and Maskin, 1994;Kandoriand Matsushima, 1998;and Compte,1998).Finally, an extensiveliteratureon long-term insurance contractscoverssimilar ground (see,e.g.,Dionne,2000).

Second,on the role of the frequency ofeffort and output realizations,wehave stressedthe contribution of Abreu, Milgrom, and Pearce (1991)and

especiallythat of Holmstrom and Milgrom (1987)on the optimality oflinearcontractsunder CARA preferences.As seenin Chapters4,6,and 8,such incentive schemeshave provedvery attractive in applications.

We have then turned to the issueof renegotiationof moral hazardcontracts. Pioneeringcontributions in this literature include FudenbergandTirole (1990)and Ma (1991)when the principal doesnot observe the

agent'seffort choice,and Hermalin and Katz (1991)when he does.Subsequent work includesMa (1994)and Matthews (1995,2001)in the first caseand Dewatripont,Legros,and Matthews (2003)in the second.Onecasewehave not coveredhere is the one where the agent cannot commit not tounilaterally breach the contract,e.g.,becauseof legal \"no-slavery\"provisions. In a simple insuranceproblem without hidden actionsor adverseselection,such lackof commitment leadsto interesting dynamics when thecontractplays an insurancerole,as shown by Holmstrom (1983)or Harrisand Holmstrom (1982).

The literature on bilateralrelationalcontracts,which we discussednext,is quite large.Early work includesKlein and Leffler(1981),Shapiroand

Stiglitz (1984),and Bull (1987).In this chapter, we have in particularstressedthe work of MacLeodand Malcomson(1989)and Levin (2003).Note that MacLeodand Malcomson(1988)extendthe setting to multilayer

organizations,while Bernheimand Whinston (1998b),Baker,Gibbons,and

Murphy (1994),and Pearce and Stacchetti(1998)combineexplicit and

implicit incentives. Finally MacLeod(2003)allows for subjective,imperfectly

correlatedperformancemeasuresobservedby the principal and the

agent.Finally, the literature on career concernswas pioneeredby Holmstrom

(1982a).Extensionsinclude Gibbonsand Murphy (1992),who have addedexplicitincentives to the picture;Meyerand Vickers (1997),who have alsoallowedfor relativeperformanceevaluation; and Dewatripont,Jewitt, andTirole(1999a,1999b),who have considereda multitask setting. Finally, Stein

(1989)and Scharisteinand Stein (1990)have appliedHolmstrom'sparadigm, respectively,to the questionsof short-termism and herd behavior.

INCOMPLETECONTRACTS

With this fourth part of the bookwe takeyet another important steptoward

reality. Recall that the first stageof contract theory\342\200\224as studied by Edge-worth\342\200\224was concernedwith the characterizationof efficient exchangecontracts in situations involving no uncertainty and no asymmetric information.It was only in the 1950sthat a secondstagewas reached,with the explicitconsiderationofuncertainty. At that time, problemsofoptimal coinsurance,optimal risk diversification, and portfoliochoicebecamethe main focusofattention.

About two decadeslater a third stage was reached\342\200\224considering

contractual situations with asymmetric information. This has led to theintroduction of incentive constraints alongsideparticipation constraints and tothe elaborationof a rich theory of incentive contracting. Parts I and IIattempted to providea comprehensivecoverageof this theory.

Following a decadeor so of intense research, the field reached yetanother stage in the 1980s\342\200\224concerned with issuesof repeatedor dynamiccontracting. Issuesof commitment and renegotiationwere systematicallyexploredin that decadeand later.This latest developmenthas led to theintroduction of \"renegotiation-proofness\"constraints alongsideincentiveand participation constraints. Part IIIcovered the main ideasemergingfrom this new wave of research.

The final stageof developmentis taken up in Part IV. As in Part III,this

stage is concerned with long-term contracts.However, the orientationis entirely different. Rather than optimizing over a generalcontractsetthat is subjectto incentive, renegotiation-proofness,and participationconstraints, long-term contractsare taken to be of a prespecified(incomplete)form, and the control variablesbecomeinstead ownership titles, controlrights, decision-makingrules, discretion,tasks, authority, and the like, to beallocatedamong contracting parties.Part IV discussesthis methodology,as well as, in Chapter 12,its relationship with the \"complete contract\"

methodology.

V

-i IncompleteContractsand Institution Design

11.1Introduction:IncompleteContractsand the Employment Relation

The introduction of incompletecontractsinvolves both a substantive and amethodologicalbreak.Thesubstantive changecanbe illustrated with aconcrete example,the conductofmonetary policy.A centralproblemin macro-economicresearchover the past two decades,as well as a major politicalissue,has beenthe questionof centralbank independence.Thisis an issueinvolving both incentive and commitment considerations.If one takes adynamic-contracting approachas outlined in Part III,one is inevitably ledto focuson the problemof financial compensationof the centralbanker.The main issuebecomeshow to structure the centralbanker'scompensation packageto inducehim to run monetary policy accordingto theobjectives stated in the law (e.g.,price stability). Someof the literature on the

subjecthasindeedtaken such a turn. For example,Walsh (1995)and othershave proposedcompensationcontractsfor centralbankerscontingent onthe rate of inflation. Somecountries\342\200\224most notably New Zealand\342\200\224have

evenconsideredimplementing such contracts.If, however,one takesan incomplete-contractingperspective,then other

aspectsbesidescentral bankers'compensationbecomerelevant, such asthe bank'sdecision-makingprocedures,discretionversusrules, and centralbank accountability. As this example illustrates, the substantive changebrought about by the introduction of contractualincompletenessis to shift

the focus away from issuesof compensationcontingent on outcomesto

proceduraland institutional-design issues.The analysis of institutions is clearlyof fundamental importance,and

an incomplete-contractingapproachoffersa vehicleto exploretheseissuessystematically. The fourth stage of research in contract theory is thus anatural development,which should eventually producean economictheoryof institutions asrich as the theory ofincentivesdevelopedin the past threedecades.

The introduction of contractual incompletenessalso involves amethodological break.Most of the literature assumesa particular form ofcontractual incompletenessand doesnot explain the form of the contractasthe outcomeofsomeoptimization problem.Optimization is confined to thechoiceof institution: the designof decision-makingrulesand the allocationofcontrolrights. Typically, contractual incompletenessisexplainedby alimitation in contractual language\342\200\224the inability to describeaccuratelycertain

Incomplete Contracts and Institution Design

eventsbefore the fact, evenwhen these eventsand their implications areeasily recognizedafter the fact.It hasbeena matter of debatehow

restrictive this constraint is in theory and in practice.We provide an extensivediscussionof this debate in Chapter12.For now, we shall simply assumethat the inability to describecertainfuture events is a binding constraint

and explorethe implications for institution designof this form ofcontractual incompleteness.

The Employment Relation

Perhapsthe first formal modelwith incompletecontractsis Simon's(1951)theory of the employment relation.In that model Simon comparestwo

long-term contracts,one where the serviceto be providedby a selleratsomepoint in the future is preciselyspecifiedin a contract,the otherwherethe serviceis left to the discretionof the buyer within somecontractually

specifiedlimits referred to as the \"acceptanceset.\" Simon identifies theformer contractasa \"salescontract\" and the latter asan \"employmentrelation.\" The choiceof contractis then determinedby the degreeofuncertainty at the time of contracting about preciselywhich servicethe buyerwould prefer in the future, aswell as the degreeofindifferenceof the sellerto providing a specificservicewithin the \"acceptanceset.\"Simon'stheoryraisesseveraldifficult questions.Perhaps the most important one is,Whymust the buyer and selleragree to a trade before the uncertainty isrealized? Why not wait for the uncertainty to be resolvedand then write aspot contract?A secondquestionrelates to the extremeform of the salescontract:Why can't the buyer and selleragree to some form of state-contingent delivery contract?Why is it cheaperto write a contractspecifying

an \"acceptanceset\" than a contract specifying a state-contingentdelivery plan?

It is possibleto addressmost of these questionsand to provideanupdated theory of the employment relationby introducing a \"holdup\"problem as in Klein,Crawford, and Alchian (1978)and more generally takingWilliamson's transactions-costperspective(Williamson, 1975,1979,1985).As a way of introducing the main ideasof the incomplete-contractsapproach to institution design,we shall beginby providing such an updatedtheory.

Supposethat the buyer can undertake an (unobservable)investment

today that raisesher payoff from the serviceto beprovidedby the sellerinthe future. This investment is sunk by the time buyer and sellernegotiate

Incomplete Contracts

over serviceprovision in a spot contract,so that the buyer may be \"held

up\" by the seller,to use a term favoredby Klein,Crawford, and Alchian.The needfor a long-term contract then arisesas a way of protectingthebuyer'sex ante investment against ex post \"opportunism\" by the seller(aterm favoredby Williamson). But prior to the investment stage the buyerdoesnot know which type of serviceshemay need,and it is not possibletodescribeunder what precisecircumstancesshe needsa particular service.The contractcan specify only the nature of a serviceto beprovidedunderall circumstances,or a menu of servicesthat the selleragreesto provideat

predeterminedterms (the latter arrangement can alsobe thought of as ageneraloption contract).

Williamson has argued that the incompletenessof contractsinevitablyinducescontracting partiesto attempt to interpret the contractto their own

advantage ex post and that this behaviorcan lead to both ex ante and expostinefficiencies.Thecontracting partiesnot only may make inefficient exante investments, if investment is not observableor verifiable and they

anticipate expostopportunism, but alsomay get involved in inefficient

contractual disputesexpost.Williamson seeskey institutional arrangements in

market economies,such as authority relations,to beresponsesdesignedtoovercomethe potential inefficienciesin long-term relationsgovernedby

incompletecontracts.An employment relation,which leavesmuch to thediscretionof the employer,is valuable if it canreducethe scopefor exposthaggling or allow the employerto appropriatethe rents createdby her exante investments.

In an attempt to simplify the formal analysis, most of the recentliterature abstractsfrom expostinefficienciesresulting from contractual disputesand focusesentirely on ex ante investment inefficiencies.We shall take thesameapproachin developingour updated theory of the employmentrelation. Thefirst formal modelof inefficient exante investment resulting fromex post opportunism is due to Grout (1984).Later the property-rights

theory of the firm formulated by Grossmanand Hart (1986)and Hart andMoore(1990)built on this generalideaalsodevelopedinformally by Klein,Crawford,and Alchian (1978)and Williamson (1975,1985).We shall discussthis theory in the next subsection.

A Theoryof the Employment RelationBasedon ExPost Opportunism

Considera risk-neutral buyer and a risk-neutral seller,neither ofwhom hasalternative contractual opportunities.There are n types of servicethat the

492 Incomplete Contracts and Institution Design

sellercanprovidedenotedby at(i =1,...,n).Thebuyer needsonly one oftheseservicesexpost.Provision of the serviceisobservableand verifiable,so that a contractcanbe written specifying a payment contingent onexecution of the service.Supposethat in the event of no trade both partiesgetzero,and if they trade servicea, at pricePt they get,respectively,

\302\243/(\302\253,-, 0,)-flfor the buyer, and

Pi-yfia^Bi)for the seller,where Oi denotesa state of nature. Supposethat there areN > n statesofnature (/ =1,...,N).Thus both buyer and sellerpayoffs canvary with the underlying state of nature.1

Thebuyer canincreaseU by increasingher ex ante investment /.That is,let U= U{an ft, I),with dllldl> 0 and d2UldI2 < 0.The costof investment istaken to be simply k(T) = I.To keep the analysis as simple as possible,assumethat

t/(fl,,0\342\200\236/)e{O,0(/),\302\256(/)}

and

^(fl,-,0,)6{O,C,C}with

0 < 0(1)-C<<p(D-c<0(1)-c< 0(1)for all/> 0

Furthermore, assumethat in any given state 6i

t/(a\342\200\2360,,/)=O=* y/(a[,el)=0

U(ai,ehI)=<p(I)^ ip(fl\302\2730i)=c

U(at,0hI)= \302\256(I)=* yr(ah0l) =C

In other words, in any given state 6h any valuable serviceiscostly,and thehigh-value serviceis alsothe high-costservice.

1.A slightly more general contract could specify apayment P0to the buyer in caseof no trade.This payment could be interpreted as a penalty clause. As will becomeclear, there is no lossof generality in setting P0=0here. However, in other settings the optimal penalty is nonzero,as we show in Chapter 12.

Supposealso that for any realizedstate of nature 9h there is always atleast one serviceex post at(6i) such that t/(a;, 6h 7) = 0(7) so that themaximum surplus <p(I)can be realized.Then the efficient levelof ex anteinvestment 7* isgiven by maximizing

max{</>(7)-7}/>o

and is characterizedby the first-orderconditions

f(7*)=lHowever,supposein addition that in any realizedstateofnature 0h there

are other servicesavailablea^Oi)and ak(6i) (j *i; k *i) such that\302\243/(\302\253,-,

6b

I) =<\302\243(/)

and U(ak, 6,,7) =0.We shall assumethat there is a basic technologicalconstraint that

prevents the buyer and sellerfrom writing an ex ante contractcontingent onthe realizedstate of nature. In other words, a precisedescriptionof eachstatewould besocostly that it would outweigh all the benefitsfrom writingsuch a fine-tuned contract.Giventhat contractscannot bemadecontingenton the state of nature, there are four basicoptionsavailableto the

contracting parties:\342\200\242 They canwait until the state is realizedand write a spot contract.\342\200\242 They canwrite a salescontract,which specifiesa single serviceto beprovided expost at prespecifiedterms.\342\200\242 They canwrite a buyer-employment contractspecifying a menu ofservices from which the buyer can chooseex post.This servicemust then bedeliveredat prespecifiedterms.\342\200\242 They can write a seller-employmentcontractspecifying a menu ofservices from which the sellercan chooseex post.This servicemust then bedeliveredat prespecifiedterms.2

Considereach type of contractin turn.

11.1.2.1SpotContract

If buyer and sellernegotiateover a serviceex post,we shall assumethat

they divide equally the gains from trade.These gains are maximized by

2.Note that these four options do not provide a fully exhaustive list of feasible contracts. In

particular, one can envision hybrid forms of employment contracts, where one of the partiesis randomly selected expost to pick a service within her prespecified menu of services.Asinteresting as these contractual forms are,we shall not pursue them here in order to save space.

494 Incomplete Contracts and Institution Design

choosinga servicea^Oi)such that U(at, 6h I) = 0(7) and!//(\302\253\342\200\236 60= c.Thus

buyer and sellereachget [0(7)-c]/2expost,sothat the buyer choosesherex ante investment P to maximize

2

and underinvests. The levelof investment is then given by

*'(/*)=2

11.1.2.2SalesContract

If buyer and sellerwrite a salescontractof the form{a\342\200\236 P],whereat denotes

the serviceto be deliveredand Pt the price,then expost the realizedstateis such that either

1.U(a\342\200\236 6h 7) =

</\302\273(/)and yfa, 60=c,

2.U(at, 0b 7) = yfo, $)= 0,or3.U(aH 6h 7) =$(7)and y/fo, 6) = C.

If the state of nature is such that U(at, 6h 7) = 0(7) and!//(\302\253;, 60=c, then

the contractis executed,and buyer and sellerget, respectively,0(7)-P,andPi -c.If, however, the other configurations of payoffs obtain, then the

parties are better off renegotiating the contractand switching to anotheractiona} such that U(ah 0b 7) = 0(7) and y/(a;, 60=c.

Renegotiationinvolves bargaining overthe surplus [0(7)-c-(<3>(7)-C)]

if U(at, 61,7)= <3>(7) and!//(\302\253\342\200\236 60 = C, and over the surplus (0(7)-c) if

U(at, 6b 7) = y/(at, 60=0.Assuming, as before,that this surplus is divided

equally among the parties,their respectivepayoffs in this event are

<&(/)-P,+|[>(/)-c-<&(/)+C] and p,-C+|[>(/)-c-<&(/)+C]

if

U(al,6,,I)=0(I) and y/(ai,el)=Cand

-Pi+\\W)~c] and Pi + \302\261[<p(I)-c]

if

U(Oi,et,n=0 and!//-(\302\253,, 6>,)=CK

495 Incomplete Contracts

Toproceedfurther we needto imposemore structure on this example.Supposenow that there are N = n equiprobablestatesof nature, and that

serviceat is the uniquely efficient actionin state Bt. In other words, U(ah 0h

I)= <p(I) and y(cli, 61)=c in state 0h and for i *I either

U(ai,6l,I)= \302\256(I) and y(a,,Gl)=C

or

U(ai,Ol,I)=0 and y/(al,Ol)=0

Moreover,all other servicesare equally likely to be a high-costserviceak or a worthless servicea^.

Given this probability distribution over action-staterealizations,the

buyer'sex ante payoff is given by

\342\226\240i[W-fi]+[^)[*0)-fl+|[W-c-\302\2560)+C]}

so that the buyer'soptimal levelof investment under this contractisgivenby the solution to the first-ordercondition

Note that the price P, is determinedex ante so as to split the overall

expectedsurplus from the transaction\342\200\224[<p(Is)-c].

11.1.2.3Buyer-Employment Contract

Now considerthe casewherebuyer and sellerwrite a buyer-employmentcontractofthe form {A, P(a)\\a&A}, whereA isasubsetofservices\342\200\224Simon's

acceptance set\342\200\224from which the buyer can choose, and P(a) are the

predeterminedterms for supplying servicea.Themain point ofwriting sucha contractis to give the buyer someflexibility to choosethe most desiredserviceexpost.Thecostof writing such a contract,however, is that it putsthe sellerin a positionwherehe agreesto providea servicethat may beverycostly ex post.Becausethere are both costsand benefitsin writing such acontract,it isunclearapriori whether this contractdominates asalescontract.

496 Incomplete Contracts and Institution Design

Sinceall servicesare symmetric ex ante, it is obviousthat the acceptanceset should contain all possibleservicesat, and that there isno lossofgenerality in setting P(a)=P.Under such an employment contract,the buyerwould always choosea servicethat maximizes

msa[U(a,ehI)-P]a

In other words, the buyer would always choosean inefficient actionyielding

a payoff of U{a,,0h I) =<3>(/) and imposing a cost Con the seller.Of

course,both partieswill renegotiateaway from this inefficient outcomeexpostand agreeon a new contractto supply serviceat in state 0hThe surplusfrom renegotiationis again divided equally, so that the buyer'sand seller'srespectivepayoffs become

fc-P+|[>(/)-c-<&(/)+C]

and

p-c+^W)-c-W)+QUnderthis contract the buyer choosesher ex ante investment IBE to

solve

maxj|foO)+*(/)]-/}Thus the levelof investment under the buyer-employment contractis givenby the solution to the first-ordercondition

<p'(IBE)+q>'(IBE)=2

Again, the terms of trade P are set so as to divide equally the ex ante

expectedsurplus from trade [<p(JBE)-

c\\.

11.1.2.4Seller-EmploymentContract

Finally, considerthe casewherebuyer and sellerwrite a seller-employmentcontract,wherethe sellernow choosesthe servicein the acceptancesetA.As before,the acceptanceset should contain all possibleservicesat, andsinceall servicesare symmetric ex ante, P(a)=P for all a e A. Underthis

employment contract,the seller'soptimization problemis

497 Incomplete Contracts

max{P-y(a,0i)}a

sothat the selleralways selectsthe leastcostly action a, such that y/(a;, 0,)=0.As under the buyer-employment contract,buyer and sellerwould then

renegotiateaway from this inefficient ex postoutcome.The surplus from

renegotiationwould again bedivided equally, sothat the buyer'sand seller'srespectivepayoffs become

-P+ \302\261[<p(I)-c]

and

The buyer then choosesher ex ante investment under the seller-employment contractISE to solve

max||[</>(/) -c]}so that the levelof investment under the seller-employmentcontractis thesameas under expost spot contracting.3

11.1.2.5Which ContractIsBetter?

Giventhat under eachcontractthe ex ante expectedsurplus from trade is

[<p(Ik)-c] (with k =s,S,BE,SE),the bestcontractis the one that induces

the most efficient investment levelby the buyer. It is easy to seethat, aslong as 0'(-O> c&'(-0-0> the sPot contractis (weakly) dominated by thesalesand buyer-employment contracts,sincethe latter two contractsinducehigher investment and sincethere is no overinvestment under any contract.4The buyer-employment contractdominatesthe salescontractwhen

-[f(/5)-<\302\243'(/5)]>0n

3.Note that this need not always bethe case.For example, if the least-cost servicehas astrictlypositive value, then the two investment levels would differ in general.4.When <&'(/) > <p'(I) the buyer may actually overinvest under the buyer-employment or thesales contract. In that case,spot contracting could actually dominate these two contracts.

498 Incomplete Contracts and Institution Design

Interestingly, the comparisonbetweensalesand buyer-employmentcontracts involves different considerationsfrom thoseSimon had in mind. Thedifferencestemsfrom the absenceof ex ante investment and expostrenegotiation in his theory. For Simon the comparisonbetween the two

contracts is only in terms of expost efficiency.Thebuyer-employment contracttends to inducetrade of excessivelycostly services,while the salescontractinducestradeofservicesthat donot maximize net surplus expost.Depending

on the extent of uncertainty and the variancein costsacrossservicesand statesof nature, one or the other contractmay dominate in his setup.

Here,instead,the contractsare evaluatedin terms of exante efficiency\342\200\224

that is to say, in terms of their effectson the buyer's incentivesto invest.If the marginal return on investment for the costly (inefficient) actionis higher than for the efficient action\342\200\224&(IS)

><t>'(Is)\342\200\224then the sales

contract may dominate the buyer-employment contract. Otherwise,the

employment contractis preferable.Note,however,that as n grows large,the differencebetween the two contractsbecomesnegligible.Thus, when

one allows for renegotiation,the comparisonbetweenthe two contractsismore subtle than the simple trade-offbetween\"flexibility\" and\"exploitation\" emphasizedby Simon.

While some important objectionsto Simon'smodelof the employmentrelationcan be addressedby introducing ex ante (nonobservable)investment by the buyer, other criticismsare harder to disposeof.One importantcriticism by Alchian and Demsetz(1972)of the notion of master-servantrelationimplicit in the employment contracthas beenthat there is nodifference betweenan employerorderingan employeearound and a customerorderinga grocerto delivera basketof goods.One differencethat we have

highlighted is that the employment contract is a commitment to serve in

the future as opposedto a spotcontractto sellgoodsto a customer.Whilethis is an important distinction, it is probably not the only defining one.

11.2Ownershipand the Property-Rights Theoryof the Firm

Just as an employment contractgives the employerthe right to determinewhat the employeeshould do (subjectto remaining within the law), anownership title on a property or assetgives the ownerthe right to disposeof the property as sheseesfit. Somelimits canbeput on the owner'srights

by law, and other limits may be agreedon contractually, but unlessthese

499 Incomplete Contracts

are explicitly specified,the ownercando what shewants with the asset.Inotherwords, the owner has \"residual rights of control\" overthe asset,to useterminology introduced by Grossmanand Hart (1986).In addition, anownership title gives the ownerthe right to all revenuesgeneratedby theasset that have not beenexplicitly pledgedto a third party.

It is instructive to contrast Grossmanand Hart'sdefinition ofownership\342\200\224based

on residualrights of control\342\200\224with Alchian and Demsetz'sandJensenand Meckling's\342\200\224which definesthe ownerof a firm only in termsof cash-flow rights (the ownerbeing the \"residual claimant\" on the cashflow)\342\200\224to appreciatethe importanceof the departure of the incomplete-contractsapproachfrom standard incentive theory.

Given the similarities between employment contracts and ownership

titles, it isnot too surprising that the next setof formal modelswith

incompletecontracts were concerned with ownership and residual rights of

control.The modelsof Grossmanand Hart (1986)and Hart and Moore(1990)explore the issueof the value of ownership and residualrights ofcontrolin situations wherepartieswrite incompletecontracts.

According to their theory, the ownerof a firm has the right in particularto excludeothersfrom using the firm's assets.Thisright servesas aprotection against ex post opportunism. In its simplest form the theory predictsthat ownership of productive assetsis allocatedto the party requiring themost protectionagainst expostopportunism. Thefollowing simple exampleillustrates the basictheory.

Considera situation with at most two separatefirms orproductive assets.Grossmanand Hart take the exampleof a publisher and a printer. Say firm

1is a printing pressproducing copiesofbooksor journals for firm 2,apublishing company.If the two firms areownedseparately,they may write long-term supply contracts,which are assumedto beincomplete.In otherwords,under thesecontractstheremay beeventsrequiring new decisionsnot pre-specifiedin the contract.In that caseit isassumedthat the printer andpublisher will negotiatea new expost efficient contract.

The expost negotiating positionwill differ from the ex ante situation, in

which the two firms have written the initial long-term contract,if in themeantime eitherfirm (or both) has taken actionsor madeinvestments that

are costly to reverse (or to switch to another supply relation).In otherwords, the ex post negotiating position of one of the parties may haveworsenedif that party is lockedinto the supply relationas a result of its

previousactions.For example,if the printing firm has spent vast sums

500 Incomplete Contracts and Institution Design

customizing its software to fit the specialneedsof the publisher, it will have

put itself in a positionwhere it has little choicebut to deal with thepublisher ex post.Recognizingthis fact, the publisher may be able to extractbetter terms ex post than ex ante when it was in competitionwith otherpublishers.

If the printing firm anticipatesthat it may end up in a weakernegotiating positionexpostby making publisher-specificinvestments, it may refrainfrom making these investments even if they are efficient.

In contrast,if firms 1and 2 wereintegrated,so that the printer was thesoleowner of both the printing pressand the publishing business,then hisexpost negotiating positionwith the publisher (who isnow his employee)would likely be lessaffectedby his ex ante specificinvestments. In fact, it

is conceivablethat under this ownership allocationthe printer might beableto appropriateall the gains from expostnegotiations.If that werethe case,the printer would ofcoursebeinclined to make any ex ante specificinvestments that are efficient.

But if the printer is the soleowner of both assets,then presumably the

publisher would have lessincentive to invest than under nonintegration orunder vertical integration with the publisher owning the integrated firm.

The property-rights theory of the firm predictsthat any ownershipallocation can arise in equilibrium dependingon the relative value of eachparty's ex ante specificinvestments. If investments in customizedsoftwareare most valuable, then it makessensefor the printer to own both printing

press and publishing business.If investments in authors, agents, ormarketersare most valuable, then it makessensefor the publisher to own

the publishing businessaswell as the printing press.Finally, if both types ofinvestment are important, it may bebestto have nonintegration.

Thisis in a nutshell the theory outlined for two firms in GrossmanandHart (1986).The important contribution of the property-rights theory ofthe firm is to both explain the value of ownership and delineatethe costsand benefits of integration. We now turn to a moreformal expositionof the

theory.

11.2.1A GeneralFramework with Complementary Investments

It is relatively straightforward to provide a formal treatment of theproperty-rights theory in a simple setting with only two agentsand two

assets.But how can the theory be applied to a large corporationwith

multiple divisions and thousands of employeesand suppliers?

Hart and Moore(1990)take a first stabat this questionasfollows.Just asin the two-firm caseconsideredby Grossmanand Hart, one canenvision afirst stagewhere/ agentsmake ex ante investment jc, at a cost i//(jc;) (theseinvestments may bemoreor lessspecificto somesubsetof assetsA in theset of all productive assetsavailablein the economy,A). Then,in a secondstage,the exante investments ofa subsetofagentsSc/ combinedwith thesubsetofassetsAc A cangeneratean expostsurplus from tradeV(S;A \\ x),wherex=(x1,x2,...,*/) denotesthe vectorofall agents'exante investments.

To simplify matters, Hart and Moore assumean extremeform ofcontractual incompleteness:no long-term contractsspecifying future trade canbe written ex ante.Only trade in ownership titles can take placeex ante.As in the casewith two firms, the division of the surplus from trade amongthe S agentsdependson who owns which subsetof assets.

The main modeling difficulty when consideringa generalsetup with /agentsis determining how the ex post surplus V(S; A

\\ x) getsdivided up

among the contracting partiesin multilateral negotiations\342\200\224in otherwords,how the terms oftrade get determined.Hart and Moore'sproposedsolution isto envision a centralizedmarketplacewhereall the negotiations get donesimultaneously expost.They assumethat the outcomeofmultilateral

negotiations is expostefficientunder any ownership allocationand that thesurplus is divided accordingto the Shapley value. Thus,as in the casewith two

firms, the only way ownership affectsthe final outcomeis through the division

ofexpostsurplus and its impact on exante investment incentives.Asemphasized earlier,the assumption that allexpostnegotiations are efficient\342\200\224in otherwords, that the Coasetheorem appliesex

post\342\200\224is mainly a simplifying

assumption. By consideringin turn the casewherethereisno renegotiation,which may involve extremeex postinefficiencies,and the caseof perfectrenegotiation,one is exploring two natural polarcases.Mostcontractualsituations in practiceare likely to bea combination ofthesetwo extremecases.

11.2.1.1The ShapleyValue

TheShapleyvalue assignsapayoff to an agent i possiblyinvolved in atransaction with S agentswho togetherown or controlQ)(S)assets.

definition: ownershipallocationLet S denote the set of all possiblesubsetsofagents/, and A the set of all possiblesubsetsof assetsin A. Thenthe mapping co(S)from S to A denotesthe subsetof assetsownedby thesubsetof agentsS. ______ \342\200\224

502 Incomplete Contracts and Institution Design

Hart and Mooreassumethat eachassetcanbecontrolledby at most oneof the groups of agentsS or its complementI\\S. In addition, they assumethat the assetscontrolledby somesubgroup S'cSmust alsobecontrolledby the whole group S.In effect,when a group of agentsSdecidesto forma firm, they agreeto poolall the assetsownedby any of the group members.Formally these assumptions translate into the following propertiesfor the

mapping co{S):

co(S)n(o(I\\S)=fl and ft)(50cft)(5) sothat ca(fi) =

fi

definition: the shapleyvalue Given an ownership allocationft)(5), avectorof ex ante investments x, and the associatedexpost surplus for any

given group of agents S, V[S, a>(S)\\x], the Shapleyvalue specifiesthe

following expectedexpost surplus for any agent i:B,((o\\x)=2p(S){VI5;fl)CSf)|j:]-VI5\\{i};fl)CSf\\{i})|j:]l- (11.1)

S\\ieS

where

(s-l)KI-s)\\P(S)= (11.2)

and s =\\S\\

is the number of agentsin S.In words,the Shapleyvalue is an expectedpayoff, whereexpectationsare

taken overall possiblesubgroups S that agent i might join expost.That is,each agent looksat ex post group formation like a random processwhereany order in which groups get formed is equally likely. It isfor this reasonthat the probability distribution p(S)is as specifiedin equation (11.2).Given any ex postrealizationof a group, S, the Shapleyvalue assignstoeachagent i in the group the differencein surplus obtainedwith the entiregroup S and with the group excluding agent i\\

V[S;(Q(S)\\x\\-V[S\\{i}-(Q(S\\{i\\)\\x\\

In other words, the Shapleyvalue assignsto each agent i the expectedcontribution of that agent to the overallexpost surplus obtainedthroughmultilateral trade betweenall agents.

11.2.1.2Example1:Printer-PublisherIntegration

Supposethat 1=2 and A = {a1;a2),as in Grossmanand Hart'sexampleofthe printer (agent1)and the publisher (agent 2).Each agent canmake exante investments xt in a first stage,and trade takesplacein a secondstage.

503 Incomplete Contracts

In this simple setup only the following three ownership allocationsarepossible:5

Nonintegration: <o(l) = {a^}, <o(2)= \\a2)

Publisher integration: <o(l) =0, <o(2)= {ah a2)

Printer integration: <o(l) = {a^a2), <o(2)=0

TheShapleyValue

Nonintegration Supposethat no expostsurplus canbegeneratedwithout

combining both assets.Thenunder nonintegration the expost surplus that

canbe generatedwith only one agent is

V({l};{ai}\\x)=V({2};{a2}\\x)= 0

wherex = (xi,x2).6If, however, both agentsform a group by trading accessto their

respective assets,they generatea strictly positivesurplus

V({l,2};{ai,a2}\\x)=V(x)>0Under nonintegration, the Shapleyvalue then assignsan expectedpayoffto eachagent of

Bi(NI\\x) =B2(NI\\x) =\302\261V(x)

(whereNIstandsfor nonintegration), sincethereareonly two equally likely

orderingsof group formation, {1,2}and{2,1}\342\200\224so that/?({l, 2})=p({2,1})

= i\342\200\224and

V({1,2};{fli, a2}\\x)-V({j};{aj}\\x) = V(x)

Printer Integration Under printer integration, it may be possibleforthe printer to generatean expostsurplus on his own, sincehe owns bothassets.The publisher cannot generate any surplus on his own, as under

5.Actually there is a fourth possible ownership allocation, where under nonintegration the

printer owns the publishing business and the publisher owns the printing press. Under this

ownership allocation, each agent holds the other agent's \"tools\" asa \"hostage.\" In the simpleexample considered here, this ownership allocation is equivalent in terms of equilibriumpayoffs and investments to nonintegration with the reverse asset ownership.6. This is a simplifying assumption. Grossman and Hart (1986)actually allow for a positivebut lower ex post surplus when both assets are not combined. As a result, nonintegration ismore likely to bean efficient ownership allocation in their setup than in this example.

504 Incomplete Contracts and Institution Design

nonintegration. Even if the printer can generatea positivesurplus on his

own, it seemsplausiblethat he might be able to do evenbetter by hiringthe publisher, so that the ex post surplus that can be generatedwith onlyone agent is likely to be as follows:

V({2};p\\x) =0, V({1};{a,,a2}\\x) =d^)with

<\302\243(*!)< V(x).

The Shapleyvalue under printer integration is then given by

Bi(PI\\x) =^[V(x)-$!(*!)]+<&!(*!) for agent 1

B2(PI\\x)=ItVM-fciOd)] for agent 2

wherePIstands for printer integration.

PublisherIntegration Similarly, one can take publisher integration to bethe mirror image of printer integration, so that the Shapleyvalue under

publisher integration becomes

Bl(pl\\x) = -[V(x)-\302\2562(x2)] for agent 1

B2(pi \\x) =-[V(x)-<\302\2432 (x2)]+ <\302\2432 (x2) for agent 2

wherepistands for publisher integration.

ExAnte Investments

Supposethat V(x) is strictly increasingand concavein x = (xi,jc2),that ^(x,)is increasingand concavein xt, and that the investment-costfunctions y/t(x^)

are strictly increasingand convexin xt. Undereach ownership allocation,agentschoosetheir exante investment noncooperativelyto maximize their

respectiveexpectedpayoff:

max^[fi)(S)|*i,x2 ]-y/i (xt)}

Investments are chosennoncooperatively, either becausethey are notobservable,as in the employment relation consideredearlier,or becausethey are difficult to verify by a third party. In eithercaseit would becostlyor impossibleto write an enforceableperformancecontractspecifying thelevelof investment eachparty must undertake.Note that if investments arenot observable,the expostpayoffs generatedby thoseinvestments are still

505 Incomplete Contracts

assumedto be observable.Theseare centralassumptions in the property-rights theory of the firm, which have generatedsomecontroversyTheconceptual issuesinvolved here are discussedextensively in the next chapter.

Suffice it to say here that if ex ante investments couldbe specifiedcontractually, then the allocationof ownership titles would not serveany rolein providing incentives to invest.TheCoasetheorem,stating that efficiencycanbeobtainedthrough private contracting nomatter how ownership titlesare allocated(providedthat ownership rights are completelydetermined),would then again obtain. In otherwords, the allocationof ownership rightswould be irrelevant for efficiency.

Proceedingunder the assumption that investments are chosen non-

cooperatively,the property-rights theory of the firm predicts that eachownership allocationresults in Nash-equilibrium investment levels.Theseequilibrium investment levelscan be obtained from the first-orderconditions of eachparty's optimization problem(assuming that it is concave):

dBt[co(S)1x^X2]3l, =1\"(xi)

Thus, under nonintegration, equilibrium investment levels(xf7, jcf7) aregiven by

1dVixi1 ,x^) _ m

(11.3)

\302\261dVifxfI]=v>(xn

Under printer integration, (jcf7, x\342\204\242)are given by

ldVtf',x?)lQ,{x[I)=,{xn2 dXi 2(11.4)

iayUff,jcff) ,\342\200\236.

2 dx~2-\302\245l{Xl ]

Finally, under publisher integration, (x{r, jcf7) are given by

(11.5)2 dx, ~\302\245l{Xl ]

\\^^+^{xt)=^{xf)

506 Incomplete Contracts and Institution Design

As can be seenby comparing conditions (11.3),(11.4),and (11.5),the

printer has greaterincentives to invest for any given levelof the publisher'sinvestment x2 under printer integration than he has under nonintegrationand publisher integration providedthat ^[(x^> 0.Similarly, the publisherhas greater incentivesto invest given xx under publisher integration than

he has under nonintegration and printer integration when <&2(x2)> 0.If,however, <&2(x2) 0, integration may have no effector a negative effectoninvestment.

Thus the property-rights theory of the firm proposesa simple analysis ofthe costsand benefitsof integration. Starting from a positionofnonintegration, the benefit of printer integration may be to inducemore specificinvestment by the printer, but the costof integration is then necessarilyalowerinvestment by the publisher.

Equilibrium OwnershipStructures

Theproperty-rights theory predictsan equilibrium ownership structure that

is ex ante efficient.Thereasoningis that, while contractson future trade ofinputs and servicesmay not be feasibleor may be too costly to write,contracts exchanging ownership titles are simple and easy to enforce.Two

strong underlying assumptions of the theory are,first, that eachcontracting

party has enough resourcesto buy any ownership title that it values themost (there are no wealth constraints) and, second,that once an efficient

ownership allocationhas been achieved,there are no further gains toretrading ownership titles. We shall relax each of these assumptions laterand considerthe implications for equilibrium ownership allocations.Fornow, however, we shall stick with these two underlying assumptions and

identify situations where,respectively,nonintegration, printer integration,and publisher integration are optimal.

Nonintegration is the equilibrium ownership structure if and only if

v(x^,xn-\302\245^n-\302\2452(xn>

m^Vix^1,xn-Yi(xn-W2(xnMxf^^-^(xn-^xf)^Otherwise,either printer or publisher integration obtains dependingon

which one yields the higher total net surplus. In determining which

ownershipallocationis optimal, it is helpful to beginby characterizingthe socially

efficient levelof investments (jcf,xf).A socialplanner maximizing the totalnet exante surplus would choose(x% ,xf)to satisfy the following first-orderconditions:

507 Incomplete Contracts

(11.6)

dx2

Assuming that the social planner's optimization problem is concave,thesearenecessaryand sufficient conditionscharacterizingthe sociallyefficient levelof investments. If the two parties' investments arecomplementary,

then d2V(xh x2)ldx2dx1 > 0.In that case,by comparing conditions (11.3),(11.4),and (11.5),we canobservethe following:

1.If <&'(*,)> 0,printer integration results in higher investment levelsforboth printer and publisher than under nonintegration.

Indeed,the printer invests more,since

2 oXi 2 2 aXi

and the publisher invests as much as or more than under nonintegration,sinced2V(xh x2)ldx2dx1 > 0.

Similarly, publisher integration results in higher investment levels forboth parties.Therefore,when c&[(jc,)> 0, nonintegration induces lowerinvestment levelsthan either printer or publisher integration.

2.If 3>'(x,)< 0,nonintegration resultsin (weakly) higher investment levelsthan under either mode of integration. In that casenonintegration is theefficient ownership allocation,sinceit inducesthe smallestunderinvestmentof all three allocations.

Note that situations where ^'(jc,)< 0 are not implausible. Anytime

specializedinvestments (such as customizedsoftware) adapted to the

specialskills of agent;are made,these investments could turn out to becounterproductivewhen agent / is not hired ex post.In other words,any customization toward agent / weakens the bargaining position ofagent i (even if he is the owner of all assets).In that caseintegration is

counterproductive,sinceit discouragesagent Vs investment (by weakeninghis ex post bargaining position) and thus in turn discouragesagent ;'sinvestment.

508 Incomplete Contracts and Institution Design

Whether one of the integration modesis efficient when <3>[(x,)> 0 cannotbe determinedfrom conditions (11.3),(11.4),and (11.5)alone.Indeed,if<3>'(x<) is large,either form of integration may result in overinvestment, in

which caseit is not obvious that integration dominatesnonintegration.7

Dependingon the specificfunctional form of V(x) and<&,(*,\342\226\240),

nonintegration or either integration modecanbe optimal.However,if 0 <

<&'(*,\342\226\240)

< [3V(jtj, x])]ldxl for all xh then integration alwaysdominatesnonintegration. In other words, if marginal returns oninvestment are always highest under expost trade betweenthe two agents, then

vertical integration dominates nonintegration.To summarize, this exampleillustrates that some form of integration is

optimal when marginal returns on investment are highest when all assetsand agentsare combinedin trade expost,and when the marginal return oninvestment remains positive if the owner ofall assetsdoesnot hire the otheragent.

11.2.1.3Equilibrium Investment Levelsand OwnershipAllocations in

the GeneralFramework

Extrapolating from the bilateralexample,there appearto beseveralsimplecharacterizationsof equilibrium investments and ownership allocationsina generalsetting.Theexamplesuggeststhat all ownership allocationsmightresult in underinvestment if marginal returns on investment are higherwhen more assetsand/or agentsare combinedin trade ex post.Similarly,the examplesuggeststhat if a subsetof assetsare worth more when

combined together,then they should be ownedby one party. Suchcharacterizations and others are indeedpossibleif a number of strong assumptionsare made about the underlying model.

Hart and Moore(1990)assumethe following propertiesfor V(S; A\\x),d[V(S;A | x)]ldxl = V'(S;A \\ x),and yifr):

Al. The function y/;-(Xi) is nonnegative, strictly increasing,and convex,limXi^oV'i(xd =0,and limJ.i^y^-(xi) = \302\260\302\260 for some0 < x < \302\260\302\260.

7. The idea that ownership may result in overinvestment incentives has been noted indifferent contexts by Bolton and Whinston (1993),Rajan and Zingales (1998),DeMeza andLockwood (1998),and Bolton and Xu (2001).The general idea that these papers have incommon is that overinvestment is the result of the owner's attempt to get a better deal forhimself in expost bargaining. Hewill beprepared to engage in socially wasteful investmentif this allows him to work out a better deal expost.

509 Incomplete Contracts

A2.The function V(S; A\\ x) is an increasingand concavefunction ofx, so

that V(S;A \\ x)> 0;moreover,V(S; A\\ x)> 0 and V(0; A\\x) =0.

A3. Vl(S; A | x)= 0 if i \302\243 S.A4. d[V'(S;A \\ x)]ldx,> 0 for all / *i.

Assumption A4 implies that all agents'investments are complements.

A5.For all subsetsS'cS,A'cA,

V(S; A |x) > V(S';A' \\x) +V(S \\S';A\\ A'\\ x)

Assumption A5 is a superadditivity assumption implying that a group ofagentscontrolling a collectionof assetscan always createas much surplusas or more surplus than the sum of the surplusesobtainableby subdividingthe group and assets.

A6.For all subsetsS'cS,A'cA,

Vi(S;A\\x)>Vl(S';A'\\x)

Assumption A6 is critical.As will becomeclear,it basicallyrulesout the

possibility of overinvestment under any given ownership allocation.When

appliedto the bilateralexampleconsideredpreviously, it implies in

particular that <3>[(j0< [dV(xh jc;)]/9jc[for allXj.

When thesesix assumptions hold, Hart and Moore(1990)show that the

following two-part propositioncanbe obtained:

1.For any ownership allocationa>(S)there is underinvestment. That is,the

unique Nash equilibrium investment levelsxe((o)satisfy x\\(co) <x* for eachi, wherejcf is the sociallyefficient investment level.2.If marginal returns on investment are higher for all agentsunder

ownership allocationcothan a, that is,

r) r)

\342\200\224\302\243,((\302\273|jt)>\342\200\224B,(fi)|jt) for all*oX[ a X{

then equilibrium investment levelsare higher under d> than (o,

xf(a))>xf((o)

and total ex post surplus increases,

W[xe(d>)]>W[xe((o)]

510 Incomplete Contracts and Institution Design

whereW(x) is definedby

W(x)^V(S,A\\x)-^i(x[)1=1

This propositionprovidesa simple method for evaluating the efficiencyof different ownership allocations.All that is requiredis to rank thedifferent allocationsby the levelof investments they induce.If allocationcb(S)induceshigher investments for all agents than allocationft)(5), then it is

clearlya more efficient allocation.It is instructive to seewhy this result obtains.As it is somewhat complex

it may help to skip the following sketchof proof on first reading.

Given assumptions Al and A2, Nash equilibrium investments xe(co)aregiven by the first-orderconditions

^-Bi((o\\x)\\x^{(a)=JJp(S)V'(S;(o(S)\\xe((o))=\302\245;(xe((o))

\302\256xi s\\ieS

for all i = 1,...,7. Or, using more compactnotation, Nash equilibriuminvestment levelsare characterizedby

V\302\253(*,fl>)!\302\253.(\342\200\236,

=0 (11-7)where

g(x,o))= (11.8)\302\243/?(S)nS;G)(S)|*]-2>U)- S i=i .

Exante efficient investment levels,however, are obtainedby maximizing

W(x)^V(S,A\\x)-Jj\302\245l(xl)f=i

and are characterizedby

y(5,A|jc)=^U)for all i =1,...,7.

Now considera changefrom allocationa> to d> such that

^-BAa>\\x)7>^-Bt(a>\\x)oXi dxl

511 Incomplete Contracts

for all x, and define

f(x,A) =Xg(x, ft)) +(1-X)g(x, ft))

for someAs [0,1].Letx(X)be the solution to

V/(*,A) =0 (11.9)Totally differentiating equation(11.9),we then obtain

H(x,X)dx =-[VgU,a))-Vg(x,co)]dX

whereH(x,A) is the Hessianoff(x, A) with respectto x (matrix of secondderivatives).

Now, by the concavity assumptions Al and A2, H(x,A) is negativedefinite. Moreover,by assumption A4, the off-diagonalelementsofH(x,A) arenonnegative.Togetherthesepropertiesimply that H(x,A)-1is a nonpositivematrix.8 Therefore,

dx(X) n

so that

xe(d))>xe(co)

Next, define

h(x, A) =XW(x) +(1-A)gU, ft))

8.Toseewhy H(x, X)1 is a nonpositive matrix, it is helpful to consider the following 2x2

example:

zJz\302\253Zn

wi \302\24322

where zu < 0,Zn < 0,z12 = z2\\ ^0,and ZUZ22-z2\\Zu >0,by assumptions Al, A2, and A4. By

Cramer's rule we then have

\342\200\236_i 1 i \302\24322

\342\200\224

\302\24312

detZ^-z2i Zn

anonpositive matrix. In the generalIxlcase,Cramer's rule can also beused to establish thisresult.

512 Incomplete Contracts and Institution Design

By assumption A6

VW(x)>Vg(x,a)

for all x,and therefore,replicating the argument madebeforewith/(jc, A),one can show that

x*>xe(co)

for all ownership allocations(o.

Finally, sinceW(x) is increasingand concaveand since

VW[xe(a)]>Vg[xe(G)),d)]and xe(a))>xe((o)it follows that W[xe(a>)]> W[xe(a>)].

A number of simple implications immediately follow from this

proposition

1.If only one agent invests ex ante, that agent should be the owner ofall assets.Thisobservationfollows from the fact that the marginal return

from investing is increasingin the number of assetsowned and by the

proposition.2.Any assetmust beownedby either group Sor its complementl\\S. Thisobservationfollows again from assumption A6 and the proposition.It hasthe important implication that ownership allocationswheremore than oneagent has vetopoweroveran assetare inefficient. In otherwords,joint

ownership of assetsis inefficient.

3.Strictly complementary assetsshould beownedtogether.Two assetsaresaid to be strictly complementary if accessto one assetgeneratesno valuewithout accessto another asset.Formally, assetsak and a( are strictly

complementary if

V'(S;A\\{ak}\\x) =Vl(S;A\\{al}\\x) =V'(S;A\\{ak,al}\\x)

Again this implication follows straightforwardly from assumption A6 andthe proposition.Another way of putting it is that separatingthe ownershipof two complementary assetsis like giving veto power to more than oneagent over the combinedassets.

11.2.1.4Example2:A Printer-Publisher-BooksellerExample

Example1and the accompanying generalframework providethe simplestpossibleillustration of the property-rights theory of the firm. They show

513 Incomplete Contracts

how the ownership allocationinducing the highest level of investments

emerges as an equilibrium ownership allocation.We now provide an

examplethat illustrates how the Shapleyvalue canbe derivedwhen threeagentsare bargaining. It also illustrates how the simple predictionof the

property rights theory\342\200\224that the equilibrium ownership structure is the oneinducing the highest level of investments\342\200\224can breakdown in a simplethree-agent,three-assetextension.

Supposenow that /=3 and that A ={a a2, a3),with agent 1as the printer,

agent 2 as the publisher, and agent 3 as the bookseller.Only the publisherand printer needto make exante investments xt in the first stage,but tradein the secondstagenow requiresall threeparties.9Any bookor journal canbesoldonly through a bookseller.The latter purchasesbooksexpost fromthe publisher, who may or may not be integrated with the printer.

Supposethat integration with the bookselleris either too costly or not

possiblefor regulatory reasons,so that the only integration decisionis, asbefore,betweenthe printer and publisher.10We shall only briefly illustrate

the Shapleyvalue in this three-agentexampleand discusshow the

integration decisionbetweenthe publisher and printer may be alteredby the

presenceof the bookseller.Supposeagain that no ex post surplus can be generatedwithout

combining all three assets.Then,under nonintegration,

V({1};Mix)=V({2};{a2}|x)=V({3};{a3}\\x) =0

and

VT{1,2};{a,,a2}\\x] =V[{1,3};{a,,a3}\\x] =V[{2,3};{a2,a3}\\x] = 0

If, however, all three agentsform a group by trading accessto their

respective assets,they generatea strictly positivesurplus from trade

V[{l,2,3};{a1,a2,a3}\\x]=V(x)>0Undernonintegration, the Shapleyvalue then assignsan expectedpayoffto each agent of

9. Ex ante investments are physical investments. There are no investments in human capitalhere for simplicity.

10.Suppose, in addition, that the courts would find it difficult to enforce a contract betweenthe bookseller and the other parties promising a payment to one or both other parties if theyintegrate.

514 Incomplete Contracts and Institution Design

B1(NI\\x) =B2(NI\\x) =B3(NI\\x) =^V(x)

since(1)there are six equally likely orderingsof group formation,

{1,2,3},{1,3,2},{2,1,3},{2,3,1},{3,2,1},and {3,1,2}(2)eachagent is last in two out of six orderings, and (3) only the last agentin the orderingmakesa positivemarginal contribution:

V[{1,2,3};{\302\253!, a2,a3}\\x]-V[{i,/};{a,,a,}|*]= V(x)

In other words, when two out of three agentshave alreadygiven accesstotheir assets,it is only the third agent that makes it possibleto generateasurplus of V(x) by alsogiving accessto his own asset.

Undereitherprinter or publisher integration the number of firms sminksto two. Now bargaining over accessto all three assetsis again bilateral(betweenthe integratedfirm {1,2},which can grant accessto assets

{\302\253!, a2),and the bookseller3).Therefore,the Shapleyvalue assignsan expectedpayoff to the integrated publisher {1,2} and the bookseller3 of,respectively,Bi2(I\\x) =B3(I\\x) =

\302\261V(x)

(where/ standsfor integration). Indeed,under integration there are againonly two equally likely orderingsofgroup formation, ({1,2};3)and (3;{1,2}).

Thus,under printer or publisher integration, the printer'sand publisher'scombinedshare of the ex post surplus is reducedfrom jV(x)toyVU).Clearly, it would not be in the publisher'sand printer's intereststo mergeif merging weakenedtheir ex post negotiating positionto such an extent,unless,ofcourse,they wereableto significantly increasethe expost surplus

by inducing significantly higher levelsof investment.Toillustrate this point, supposethat only the printer invests exante.Then

the printer would invest more and generatea higher surplus under printer

integration than under nonintegration, since\\V'{x1)>^V\\x1).However,printer integration would not occurin equilibrium if

1 2-V(xPI)-y/(xPI) <-V(xNI)- y(xm)

Thus the equilibrium ownership allocationin this exampleis not the sociallyefficient allocation(seeHeavner, 1999,for an extensiveanalysis of this

point).

Incomplete Contracts

11.2.1.5Summary

The property-rights theory of the firm thus providessimple elementsthat

explain how ownership affectseconomicdecisionsand what the costsandbenefitsof integration are. Ownershipis mainly viewed as a bargaining

chip in ex post multilateral negotiations.Synergiesin mergersare mainly

explainedas resulting from higher investments induced by the merger.Importantly, the property-rights theory of the firm doesnot view a mergerasresulting in a different form of transacting betweenthe mergedentities\342\200\224

as a shift from negotiation-basedtransactions to command-basedtransactions. All transactions take placein the samecentralizedmarketplaceunder

any ownership allocation.Only the terms of trade (the Shapleyvalue) varywith the ownership allocation.

Another limitation of the theory is that it is mainly a theory ofentrepreneurial firms run by owner-managers.As we highlighted earlier, the

theory predictsthat ownership ofassetsby shareholderswho are not in any

way related to the underlying businessof the firm is inefficient. Ownershipshould go only to agentswho make ex ante investments, and it should notbeshared.

We shall return to these important limitations in section11.3.Beforeturning to the modifications and extensionsto the property-rights theory

requiredto widen the scopeof applicationof the theory, we briefly turn toa discussionof ownership and integration with substitutable investments.

A Framework with Substitutable Investments

With substitutable investments, higher levelsof investment by agent / maydiscourageinvestment by agent i. We shall illustrate how this result canoccur in a simple setup with three agents first consideredby Bolton andWhinston (1993).In this setupthere is one upstream firm supplying scarceinputs to two downstream firms. Theex ante investments of the two

downstream firms are substitutable becausethe two firms competefor inputs.When inputs are in scarcesupply, only the highest bidders(with the highestexpostvalues)get to purchasethe inputs. Therefore,an increasein

investment by firm /,which tendsto raiseits expostvalue for the input, may havethe effectof lowering the expostprofit of firm i, and thus discouraging firm

i from investing.As we now illustrate, one then can no longer determineequilibrium

ownership allocationssimply by looking at the levelofexante investments.

!16 Incomplete Contracts and Institution Design

Moreover,equilibrium ownership allocationsmay be inefficient. Thisfinding

is not entirely surprising, sinceunder imperfectcompetition,concentrated ownership may serve the dual role of strengthening market powerand providing protectionagainst ex post opportunism.

Thesetupconsideredin Boltonand Whinston differs from the Hart andMooreframework in two main respects.

First, ex post surplus from trade is assumedto be random:downstreamfirm Dt, having madeexante investments

jc\342\200\236

cangeneratean expostsurplus

by teaming up with upstream firm U of V,(*i, 9),where 6 is a randomvariable with distribution fi(6) (i=1,2).Therandom shocksare such that undersomerealizations6,Vi(*i, 6)> V2(x2, 6),and under other realizations6,V2(x2, \302\247)

> Vfa,6) (seeFigure 11.1).Second,the expost multilateral bargaining solution is not the Shapley

value but something closerto a second-priceauction.That is, Bolton andWhinston show how noncooperativebargaining with outsideoptionsa laBinmore,Rubinstein, and Wolinsky (1986,seealso Sutton, 1986)resultsin equilibrium terms of trade for inputs identicalto those obtained in asecond-priceauction, when

|V,(*\342\200\236 6)-V,(x\342\200\236 0)|>^min{V,(;t\342\200\236

6),V,{x\342\200\236 6)} (11.10)

vm, 0)|l

/\\ .

\\(x2, 9)

v2(x2,e) ^^

V2(x2, 9)

Figure 11.1Ex-Post Surplus from Trade with Downstream Firms, Dt and D2

517 Incomplete Contracts

When condition (11.10)holds, the upstream firm sellsthe input(s) to thehighest bidderat a bid equalto the second-highestvalue

V,(x\342\200\236 6).AlthoughBoltonand Whinston alsoconsidercaseswherecondition (11.10)doesnothold, as well as caseswherethere is no scarcityof inputs expost,we shallfor the sakeof brevity restrict attention to situations where inputs arealways scarceand condition (11.10)is always satisfied.Theseare situations

ofextremecompetitionfor inputs by downstream firms, wherethe Bertrand(or Vickrey) equilibrium outcome,P = min{y,(jc,-, 6), V,{xh 6)}is a betterreflectionof the one-sidednature of negotiationsthan the Shapleyvalue,which gives the lower-valuebiddera strictly positiveex post surplus.

Supposethat only downstream firms make ex ante investments (such aspromotion or advertising of their respectivefinal products)and that thecost of investment, as before,is given by y/^x,). Suppose,in addition, that

subsection11.2.1.3'sconditionsin assumption Al hold for \\jft{x) and in

assumption A2 hold forV,(x\342\200\236 6) and V}(x},6).Let 0idenote the subsetof

realizationsof 6such that Vi(jtl5 6)> V2(x2, 0)and 02 the subsetofrealizations of 6 such that V2(x2, 6)> V^,6).11.2.2.1SociallyOptimal Investment Levels

The ex ante optimal investment levels(x%, x*) maximize the expectedexante net surplus given by

JL V1(x1,0)dn(0)-\\ir1(x1)+\\v2(x2,e)dii(e)-y/2(x2)

Givenour assumptions on y/,(xt) andV;(jt\342\200\236 6), this is a concaveobjective

function, sothat the exante efficient investment levelsare characterizedbythe first-orderconditions

\\6dv'^;e)dii(e)=

\302\245i(xl) (ii.il)11.2.2.2Equilibrium Investment Levelsunder Nonintegration

Remarkably, equilibrium investment levels under nonintegration satisfy

exactlythe sameconditionsand are thereforeefficient.To seethis result,we needto start from the expostbargaining outcomeand move backwardto the investment stage.The,expost bargaining outcomeunder

nonintegration is that the downstream firm with the highest value gets the inputsat a priceequalto the second-highestvalue.Thus expostpayoffs fordownstream firm Di are given by

518 Incomplete Contracts and Institution Design

and exante expectedpayoffs are

J8|IV, (*,,0)-Vj (x,,6)]dfi(e)-Yi U,)

Hence,Nash-equilibrium investment levelsxNI =(x\342\204\242, x2')under noninte-

gration are characterizedby the first-orderconditions

It is immediately apparent that equations(11.11)and (11.12)are thesame and therefore that equilibrium investment levelsunder nonintegra-tion areexante efficient.Thereis abasiceconomiclogicfor this result.Eachdownstream firm getsto appropriateexactly its entire expost rent when it

is positive(the amount by which its value exceedsthat of its rival). Soeachdownstream firm seeksto maximize the expectedexpost rent, which is thesameas the socialobjective.

Sincenonintegration inducesefficient equilibrium investments in this

example,it is tempting to concludethat nonintegration is necessarilythe

equilibrium ownership allocation.This is not the case,however.The basicreason,as mentioned earlier,is that the upstream firm U and one of thedownstream firms D, can strengthen their combinedmarket powerat the

expenseof the other downstream firm by vertically integrating. Or, to usethe terminology in Boltonand Whinston, vertical integration providesanticompetitive gains in the form of partial (or total) foreclosureof the unin-

tegrateddownstream firm.To seethis point, start again from the expost bargaining outcomeand

supposethat downstream firm Dxis vertically integrated with U.

11.2.2.3Equilibrium Investment Levelsunder Vertical Integration\\P*U\\

The integrated firm's ex postpayoff is

min^fo,0),V2(x2,6)}+max{0,V^x,, 6)-V2(x2,6)}where the first term is LPs payoff and the secondJDi'spayoff.Thisexpression is equal to

ViOci.0)

Incomplete Contracts

Thus the integrated firm's ex ante expectedpayoff is

and its equilibrium investment level is characterizedby the first-orderconditions

1 9*i4i(0)=vi'ur)

In other words, the integrated firm now overinvests, sinceit gets to

appropriatenot only its entire expectedexpost rent but alsopart of the other

downstream firm's expost rent, as Figure 11.2highlights. Sincethe

vertically integrated firm overinvests, the unintegrated firm D2now underin-

vests,sincethe set of states02 in which it getsa positiverent is now smaller

(seeFigures 11.2and 11.3for an illustration).As a result of the integrated firm's overinvesting, firms U and D1gain by

integrating, while firm D2loses(seeFigure 11.3for an illustration of theincreasedex ante expectedpayoff under vertical integration).

\"i(*i. 0)1

WSI9&

Iv2(x2,e)

Surplus of verticallyintegrated firm

Pi.U]

Surplus of

/ unintegratedfirm D2

Figure 11.2Expected Surplus from Trade for [Du U] and D2

520 Incomplete Contracts and Institution Design

Vfa, e)i nV2(x2

.Marginal social gainfrom an increasein investment dx1

Marginal \"expropriation\" ofD2's rent resulting from

an increase in investment dx1

Figure 11.3Marginal Expected Gain from Investment for [Dit U\\

Therefore,nonintegration cannot be an equilibrium outcomehere eventhough it supportsan efficient investment outcome.The three contracting

partiesmust be able to enforcean agreementex ante not to engagein any

retrading of assetsto ensure the stability of nonintegration. Agreementspreventing any future retrading of assets,however,are rarely seenand

appear to be impractical.In reality, an owner may needto trade his assetfor many different reasons,and a wholesaleban on future trades may behighly inefficient. The only possiblealternative then is to write a contractspecifying the contingencieswhen an assetcan be soldand thosewhen it

cannot.However,the samereasonsthat make it too costly to write

contingent long-term contractson future trade of inputs or other commoditieswould appear to make it prohibitively costly to write a long-termcontingent

contractspecifying future trades of assets.To be consistent,it isreasonable to assumethat it is simply not possibleto write contractsregulatingfuture trade of assets.

In that case,nonintegration cannotbe an equilibrium ownershipallocation, sincethe upstream firm and one of the downstream firms canachievea higher joint payoff by deviating and vertically integrating.

521 Incomplete Contracts

11.2.2.4Equilibrium OwnershipAllocations

Sofar we have not formally denned the notion of equilibrium forownership

allocations.A new difficulty arising from the possibility for future

retrades of assetsis that the desirability of the current trade dependsinpart on parties'expectationson future retrades.To deal with this issuein

a generalway, Bolton and Whinston introducea weak notion of stability

by imposing strict conditionson feasiblefuture retrades.Specifically,theydefinean equilibrium (or quasi-stable)ownership allocationto besuch that

there do not existany deviating trades that are guaranteedto strictly raisethe payoffs of all parties to that trade.A payoff increasefor an agent is

guaranteedif it is obtainedregardlessof whether any future trades occurthat do not involve that agent.

Accordingto that equilibrium notion, nonintegration is clearlynot stable,since,for example,U and Di can guaranteea higher joint payoff by

vertically integrating. Theirpayoff increaseis guaranteedbecauseno other tradeof assetswithout U and Di is feasible.

However,vertical integration by U and Di is an equilibrium allocation,sinceDi is not guaranteeda higher payoff by reverting back tononintegration.

If it doesso,Di individually exposesitself to a retrade resulting in

the integration of U and D2.Similarly, if U and Di deviatedby purchasingJD2'sasset,they would lowertheir joint payoff, sincethen D2would have noincentives to invest ex ante.

Tosummarize, the analysis in Boltonand Whinston points to an importantdifferencein the predictionsof the property-rights theory of the firm asthey apply to settings in which investments are complementary, or settingswherefirms are in competitionsothat their investments may besubstitutes.

Roughly speaking,in situations where there is little or no competitionexpost,ownership primarily servesthe roleofproviding a valuable bargaining

chip to protect ex ante investments. However,in situations where firms

competeexpost,concentratedownership servesthe dual role ofstrengthening

market powerand providing protectionagainst expostopportunism.In that case,equilibrium ownership allocationsmay well be inefficient.

11.3Financial Structure and Control

Theproperty-rights theory of the firm providesthe first formal elementsofan answer to the fundamental questionposedby Coase(1937)ofwhy there

522 Incomplete Contracts and Institution Design

are firms, what they do,and why there are organizations directingproduction and the allocationof goodsor servicesoutsidethe marketplace.

However,it doesnot accountfor the basicobservationassociatedwith

the namesofBerleand Means(1932)that in most corporationsownershipis separated(at leastpartially) from control.Professionalmanagers(withminimal ownership stakesin their firms) rather than shareholdersrun most

large corporations.As we saw in Part I,this separationof ownership andcontrolcreatesa fundamental agencyproblem.Though wehaveconsideredthe problemof how to optimally trade off risk sharing and incentives,wehave not addressedthe issueof why controlis separatedfrom ownershipand what the implications of this separationare for the theory of the firm.

Besidesrisk aversionand the desireto diversify risk, an obvious reasonwhy ownership is separatedfrom controlis that the agentswho benefit themost from the protectionof ownership do not always have the meanstoacquirethese valuable ownership rights. They are wealth constrained.Tobecomeownersthey must then raise the funds necessaryto purchasetheassetsfrom wealthy third parties.These third parties will have no specialconnectionwith the firm's business,but neverthelesswill insist on getting

protectionsof their own to ensurethat their investments are repaid.Theseprotectionsmay be in the form of voting or veto rights if they becomeequity owners,or collateraland other debt collectionrights if they becomecreditors.Thus managers'wealth constraints providea simple immediate

explanation for the separationof ownership and controlobservedin mostcorporations.

How doesthe answer to the questionof who should own a set ofproductive assetschangewhen the agent or agentswho needaccessto theseassetsare wealth constrained?We shall begin by consideringthis questionin the simplest possiblesetup, with only one productiveassetand two

agents, the manager and the financier. This setup was first consideredby

Aghion and Bolton (1992),who have pointed out that debt financing canbe seenas a way of allocating controlin a \"state-contingent\" fashion, with

equity holdersor managersretaining controlin nondefault statesandcreditors taking controlin default states.Aghion and Boltonassumethat

financial contractsare incompleteonly to the extent that they cannot be madefully contingent on all future states of nature. In particular, financial

contracts canbea function of realizedprofits, as in standard agencymodels.Astronger assumption is made in Bolton and Scharfstein(1990,1996)andHart and Moore (1989,1994,1998),where financial contractscannot be

Incomplete Contracts

basedon realizedprofits, as cash flows are either not observableto thefinancier or not verifiable by a court.We shall illustrate how the weakerassumption in Aghion and Bolton provides a richer theory of optimal(state-contingent)controlallocations.In contrast,the stronger assumptionof nonverifiability of cash flows in Bolton and Scharfstein,and Hart andMoore,gives riseto a dynamic theory of debt.

Wealth Constraints and Contingent Allocations of Control

Consider the start-up financing problem, where a risk-neutral wealth-constrainedentrepreneurE needsfunds K > 0 to start a venture. Thesefunds can be obtainedfrom a risk-neutral investor (say a venture

capitalist) /, who has unlimited resources.Once the venture is up and running, someevent 6 may occuraffecting

the future profitability of the project and requiring new actionsa to beundertaken. These actionsinvolve private costsor benefitsto the

entrepreneur, h(a, 6),which may vary with the realizedevent.Oncetheseactionshave beentaken, sometime elapses,returns r are realized,and the ventureis wound down.The utility function over incomeand actionsof the

entrepreneur is

UE(yE,a)=yE+h(a,e)The entrepreneurvaluesmonetary returns from the projectyE as well asother dimensionsof her job,such as personalsatisfaction from completinga project she created,her reputation, the benefitsof being her own boss,the ability to take time off when she wants, the benefitsof retaining thebusinessin the family, and so on.We label all these other dimensions as\"private benefits.\"

The investor, however,caresonly about money, sincehe doesnot takean activepart in the business.Therefore,his utility function is simply

UI(yI,a)=yI

Thisis the basicsetupconsideredby Aghion and Bolton(1992).uBecausethe entrepreneurvaluesother things besidesmoney, conflictsof

interestmay arisebetweenthe two contracting parties.Situations may arise

11.Note that there are no exante \"noncontractable\" investment decisions in this framework.

Herecontrol allocation is driven by considerations of expost inefficiency and not by exanteinvestment inefficiencies.

524 Incomplete Contracts and Institution Design

where the entrepreneurwants to take an action that doesnot maximizethe monetary return from the project.To some extent theseconflicts ofinterest can be resolved contractually ex ante or ex post.However,aswe shall see,contractual incompletenesstogether with the entrepreneur'slimited wealth constraint makes it impossibleto completelyresolve allconflicts contractually. As a result, the allocationof control\342\200\224who gets tomake the critical decisions\342\200\224is an important dimension of the financial

contract.As simple as this basicsetupis, there is neverthelessa fairly complexset

of financial contractsoverwhich the partiescan optimize.Hence,to keepthings as simpleas possible,Aghion and Bolton allow for only two

possible events, a \"good\" and a \"bad\" one, 6 e {6B,0G}.In addition, only two

coursesof action can be considered,say, \"liquidation\" or \"continuation,\"

following the realizationof the event 6, a e {aL,ac).Finally, realizedrevenues can take only two values, r e {0,1}.The ex ante probability of a\"good\" event is given by

Fr(6=eG)=pG[0,l]and the probability of a high return conditionalon event 6, (i =B,G) andactionaJ{j=L,C)is given by

y)='PT(r= l\\e =ei;a=a])

Also, it is convenient to write

/z(0,,a;)= /z;

To have an interesting problem, the required action must, of course,vary with the realized event. The convention is that continuation ismore efficient in the goodstate,while liquidation is better in the bad state,so that

yc+h8>yZ+h?and

yl+hl>y%+hE

Finally, the projectmust obviously yield a high enough expectedreturn sothat the investor iswilling to put up money, so that

pyS+(l-p)yl>K

525 Incomplete Contracts

11.3.1.1Financial Contracts,Bargaining, and Renegotiation

Financial contractsare assumedto be incompleteonly to the extent that

repayments to the investor and future actionchoicecannotbe madecontingent on the realizationof the event 6.The contractcan,however,prescribe an actionplan contingent on a signal \302\243

correlatedwith&\342\200\224a(Q\342\200\224as

well as repayments contingent on the signal \302\243,

the realizedreturn r, and the

prescribedactiona\342\200\2241(\302\243, r, a).

Consistentwith other simplifying assumptions, the signal can take onlytwo values, \302\243

e{\302\243G, \302\243b},

with

and

pB=-pi(\302\243=ZB\\e=eB)>$

Note that the distance

d = (\\l-pG\\ +\\l-pB\\)

can then be taken to be a measureof the degreeof incompletenessof thecontract.

Although Aghion and Boltonallow for generalcontractscontingent onactions,it is both simpler and morerealisticto restrictattention to contractsthat cannot prescribefuture actions or specifypayments contingent onactionchoice.Thiswould be the caseif the actionchosenwereunobserv-able to the other party, as in a standard setting with moral hazard.Hence,we shall consideronly contracts that allocatecontrol rights (but do not

prescribespecificactions),and specifyonly repayments f(\302\243, r)(independent ofactionchoice).Sincethere are only two possiblereturn realizations,r e {0,1},we can also restrict attention to affine transfers lower than orequal to r given the entrepreneur'swealth constraint

t{\302\243,r)= tF +kQ<r

Giventhat contractsare incomplete,it ispossiblethat the contractagreedon initially may result in an inefficient actionchoiceafter the realizationof6.In that casethe contracting partieswould want to renegotiatethe initial

contract.We shall supposethat the entrepreneur can make take-it-or-leave-it

offersboth at the initial contracting stage and at the renegotiationstage.Thisassumption reflectsa situation of extremeexpostopportunism by the

526 Incomplete Contracts and Institution Design

entrepreneur:oncethe investor hashandedoverthe money and theinvestment has beensunk, he is at the mercy of the entrepreneur.Hecan relyonly on legalenforcementof the initial financial contractto get his moneyback.

11.3.1.2EntrepreneurControl

Underentrepreneurcontrol,actionsare chosento maximize theentrepreneur's payoff

a'E =arg max{y) (l-ti)-ki+h)}

wherei e {G,B)refers to the stateofnature0\342\200\236

/ e {G,B)to the signal, and

je {C,L) to the chosenaction.Given that the entrepreneurgetsboth

monetary returns and private benefits, she may not always choosethe actionwith the highest expectedreturn y'r Sheis more likely to choosean actionwith a low y) to the extent that her share of monetary returns is lower.Ifshewerea 100%claimant on the project'sreturn, shewould always choosethe expostefficient action.However,in that casethe investor would getnocompensationfor his investment. Therefore,evenunder an optimal initial

contractit may not bepossibleto inducethe entrepreneurto choosethe expost efficient actionfor all realizationsof

\302\243

and 9.However,we shall shownow that even if the optimal ex ante contractinducesthe entrepreneurtochoosean inefficient action for somerealizationsof the signal \302\243

and thestate 9, expost renegotiationwill result in an efficient actionchoice.

Toseethis result, consider,for example,the situation where the optimalex ante contractinducesthe entrepreneurto continue (that is, choosea =ac)when C= Cgand 9= 9G,

yZO.-tG)+h8*y2<X-tG)+h2but not when when C= Csand 9= &g,

y?a-tB)+hG>yHl-tB)+hS

In that casethe entrepreneurand investor would renegotiatethe contractwhen C= Csand 9= 9G.That is, the entrepreneurwould make a take-it-or-leave-itoffer to the investor to chooseac insteadof aL in exchangefor anew transfer of

yc+hg-(y<\302\243+hG)-yG.tB

527 Incomplete Contracts

which leaves the investor with the same payoff as in the initial contractunder choiceof actionaL.Thisrenegotiationofferallows the entrepreneurto appropriatethe entire gain from renegotiation

yS+h8-(y2+h2)and to attain a strictly higher payoff:

yS-yftB +hg>y20.-tB)+h2since,by assumption, y2 +he >y2+hf..

It is easy to verify that under any initial contract,renegotiationwill

always lead to an efficient action choice.In other words, entrepreneurcontrol always achievesefficiency (possiblyfollowing renegotiation).Thegeneralprinciplehere is that, sincethe investor has unlimited wealth, hecan always bribe the entrepreneurto take the expost efficient action.

The problemunder entrepreneurcontrol thus is not inefficient

investment but inadequateinvestor protection.Thebasicreasonwhy the investor

may not be adequatelyprotectedunder entrepreneurcontrolis that evenif the entrepreneurdoesthe right thing, the investor doesnot get to sharethe returns from renegotiation.Venture capitalistsand other investors arewell awareof this problem.It isfor this reason that they generally insist onveto rights or majority controlas a condition for investment (seeKaplanand Stromberg,2003,for empiricalevidence).

Toseewhen the investor doesnot get sufficient protectionunder

entrepreneur control,supposethat

hS>h1but hl<hiIn that casethe entrepreneuris led to choosethe expost inefficient actionof continuing in the bad state unlessshe gets a sufficiently large financial

stake in the firm.Let

AB=(yZ+hBL)-(yE+h2)

denote the differencein total payoffs in the bad state (betweenliquidationand continuation), and let

ABy=yl-yc

denote the differencein monetary returns in the bad state.

528 Incomplete Contracts and Institution Design

Now the investor and entrepreneur have three possiblecontractualstrategiesto get the entrepreneurto chooseliquidation over continuation

in the bad state:1.Theycan rely entirely on expostrenegotiation; in that case,the contractproviding the highest return to the investor is the one giving all future

revenuestreamsto the investor, t%= 1and k%= 0 for\302\243= \302\243G> C^-Theinvestor's

ex ante expectedpayoff is then

UR=pyS+(X-p)yc

Note that the investor'spayoff isno more than TlR becausehe getsnoadditional gain from renegotiationin state 6B.

2.Theycanwrite a renegotiation-proofcontract giving the entrepreneurthe

right financial incentives to chooseliquidation overcontinuation in the badstate:the renegotiation-proofcontractproviding the highest return to theinvestor while maintaining the entrepreneur'sincentives is such that

l-k=^TT-iorC=CG,C*(and^=0)

The investor'sex ante expectedpayoff under this contractis

TVNR=\\pyc+Q.-p)yVfs

or

nNR=[py\302\247+(l-p)yl]-F

3.Theycanwrite a contractthat ispartially renegotiation-proofin. the badstate.Such a contractwould set

tB~~tfand tG=l (with k%

=0)

The investor'sex ante expectedpayoff under this contractis then

\342\200\224\342\200\236\342\200\236cnF* =pyc pG+(l-pGy +0--P)(i-PB)yE+PBy?BAf

Interestingly, renegotiation-proofcontractsdo not always give the bestprotection to the investor. The reason is that, to avoid renegotiation,the

529 Incomplete Contracts

investor must give up a large financial stake.Such a concessionisexcessivein the goodstate.Theinvestor canthereforedobetterby

attemptingto target a large financial return for the entrepreneuronly in the bad

state.In sum, entrepreneurcontrol always results in an efficient investment

policy,but, unfortunately, it doesnot providesufficient investor protectionwhen

max{UR,UPR, UNR} < K

11.3.1.3Investor Control

By giving control to the investor the contractcan ensure that the actionmaximizing monetary returns getschosenwithout having to bribethe

entrepreneur. Control thus gives addedprotection to the investor and ensuresthat he will invest in the venture. The investor'sexpostpayoff is

y)U+ki

so that he will want to choosethe action maximizing y\\ if and only if

ti>0.Although a financial contractwith U < 0 is feasible,the contracting parties

would never gain by having tt < 0 under investor control.Indeed,theinvestor would then at bestact like the entrepreneurby choosingactionsthat minimize y].Hewould then get a lowerpayoff than underentrepreneur control.Therefore,we can restrict attention to U > 0 under investorcontrol.The investor will then always choosethe actionwith the highest

expectedmonetary return y].By doing so,the investor doesnot necessarilychoosean expostefficient

action(he maximizes only monetary returns but not overallreturns). For

example,if

yt>ycthe investor may chooseto liquidate in the good state even though it isex post efficient to continue.This scenarioisnot implausible. Indeed,oneoften hearsentrepreneursand managerscomplain that investors areexcessively \"short-termist\"\342\200\224that they donot put enough value on preserving thefirm as a going concern.This exampleprovidesa simple illustration of this

conflict.

530 Incomplete Contracts and Institution Design

Now it would seemthat if the initial contract inducesthe investor tochoosean inefficient actionexpost,then\342\200\224as under entrepreneurcontrol\342\200\224

ex post renegotiationwould lead to an efficient choiceof action.This is,however,not the case.The reason is that the entrepreneuris wealth

constrained and doesnot necessarilyhave the financial resourcesto bribetheinvestor to continue rather than liquidate. The entrepreneur would beunable to effectively bribe the investor whenever

ySO-tJ-hKUiyZ-yg)TheLHSrepresentsthe entrepreneur'stotal pledgeablewealth in state 6Gunder the renegotiated action choice ac, and the RHS representstheminimum bribe required to get the investor to switch from liquidation tocontinuation.

Thus expostefficiencyunder investor controlis assuredonly if

ti*^r- (n.13)

Setting U = (yc-kijlyf., the investor'sex ante expectedpayoff is then givenby

p\\pGy2Hi-PG)yS]+'yc-kB(X-pXp

yGyl+kE +(i-PB)im^

Now, if the liquidation value is lower in the bad than in the good state,yi ^

y\302\243 (which seemsreasonable),then it is straightforward to seethat

the investor'sex ante expectedpayoff is strictly increasingin kG and kB.Hence,the optimal compensationscheme (ensuring ex post efficiency)under investor controlis to maximize kx subjectto the constraints that tx

>0 and ki < 0 (the latter constraint issimply the entrepreneur'sexpostlimitedwealth constraint when r =0).

Assuming that yl < yG (a sufficient condition),we concludethat theoptimal expostefficient compensationcontractis such that kt =0.In otherwords, under investor control the optimal ex post efficient compensationcontract is an equity contract with an equity stake for the investor oft = yclyl- The investor'smaximum ex ante expectedpayoff under an expost efficient contractis then

531 Incomplete Contracts

UI=[pyGL+a-p)yl]^yiNote that the result that equity is an optimal contract is not robust.

Indeed,if we were to relax the entrepreneur'sex postwealth constraint

from ki < 0 to kt < w (wherew > 0), then the optimal compensationcontract would bea combination of equity [t= (yG

-w)lyL.] and (collateralized)

debt (k =w).Thelargerthe entrepreneur'sinitial wealth w is,the more(collateralized) debt she would incur. This result is not entirely surprising. Bylowering the equity stake of the investor, the optimal contract is giving

lower-poweredincentives to the investor and thus rninimizing his ex postopportunism.

Unfortunately, even the bestex post efficient contractfor the investor

may not providea sufficient return to cover the investment outlay K. Inthat case,the contracting partiesmay be forced to write an expostinefficient contract\342\200\224one that sometimeslets the investor liquidate rather than

continue in the goodstate.If a contractwith tx< yGlyt for both realizations

of the signal I &{\302\243B, \302\243G}

is not feasible,then the next bestcontractis theone where tB = 1and tG = yGlyt- Thiscontractinducesan ex post efficientoutcomewhen the more likely signal \302\243G

is realizedin state 6G and aninefficient actionchoicewhen

\302\243s

is realized.Theinvestor'sexante payoff underthis contractis then

A/ =p\\p\302\260y\302\260c +(l-pG)yf]+(l-P)PByl+{l-PB)^ylyiNote that the maximum achievablepayoff for the investor here is

obtained by maximizing the ex post inefficiency in state 6G and settingboth tB and tG equal to 1.The investor'sex ante payoff under this contractis A/ =pyl + (1-p)yl\342\226\240

This highly inefficient contractis always feasible,since

pyG +(l-p)yB>pyS+(l-p)yl>K

11.3.1.4Contingent Control

Tosummarize our analysis sofar, wehave found that entrepreneurcontrolis always efficient but may not be feasible,while investor controlis alwaysfeasiblebut may not be efficient.We now show that when only inefficient

investor controlis feasible,a more desirableor efficient controlallocation

532 Incomplete Contracts and Institution Design

may beavailablethat allocatescontrolto the entrepreneuronly when\302\243G

isrealizedand to the investor when

\302\243s

occurs.Supposethat

yl > yG and hl<h$so that the entrepreneurand investor have conflicting objectivesin bothstates of nature. Underthis assumption about underlying cash flows and

private benefits, the entrepreneuris always in favor of continuation, while

the investor is always in favor of liquidation. It is only efficient to continuein the good state, so that an efficient state-contingentcontrol allocationwould be to give the entrepreneurcontrolin the goodstateand the investorcontrol in the bad state.Sincestate-contingentcontrolallocationsare notfeasible,the best approximation to this allocationis to have the signal-contingent allocationthat we describedpreviously.

Considerthis signal-contingent allocation,and supposethat tt = 1andki =0 for I=

\302\243s, \302\243g.Under this financial structure the investor'sand

entrepreneur's ex ante payoffs are,respectively,

Tisc=p[pGy2Hi-PG)yf]+(i-p)[pByBLHi-PB)yBc]and

nsc=p[pGh8+(i-pGm+(i-p)[pBhE+(i-PB)(yl-yE+hE)]

Indeed,

1.In the event (0G, \302\243G) (which occurswith probability ppG) the

entrepreneur has controland choosesthe expost efficient actionac.Thischoiceyieldsex post payoffs of yG for the investor and hG for the entrepreneur(given the cash-flow-sharingrule U = l and ki =0 for I =

\302\243B, \302\243G).

2.In the event (0G,\302\243B) [probability p(l-pG)]the investor hascontrolandchoosesthe inefficient actionaL (given the sharing rule tB = 1and kB = 0,this outcomecannot berenegotiated).Theexpostpayoffs are then,

respectively, yG for the investor and hG for the entrepreneur.3.In the event (6B, C,B) (probability (1-p)pB)the investor hascontrolandchoosesthe efficient actionaL.The ex post payoffs are then, respectively,yl for the investor and hf, for the entrepreneur.4. In the event (6B, \302\243G) [probability (1-p)(l-p3)]the entrepreneurhascontrol,threatens to choosethe inefficient actionac in the absenceofrene-

533 Incomplete Contracts

gotiation, and obtains a renegotiationrent of yl -yB. The ex-postpayoffsare thereforeyB for the investor and yl-yl+hi for the entrepreneur.

In contrast,the entrepreneur'spayoff under investor control(with tB =1and tG =yclyT) is

ni=p[pGh8Hl-pG)hZ]+{l-p){pBhE+{l~PB)yl{l-^]+h

Comparing nscwith %h it is straightforward to checkthat ksc> Ki whenever

y\302\261yji

yl yl

Similarly, a comparisonof the investor'sex ante payoffs under the two

controlallocations,n5Cand 1%,yields that I15C> 11/whenever

y\302\261y\302\261

yl yl (11-14)That is,contingent control(with U =1and kt =0) gives the investor a higher

payoff than investor control(with tB = 1and tG =yllyl) whenever thedifference in monetary returns betweenliquidation and continuation is higherin the good state than in the bad state.Vice versa, when the oppositeinequality holds, contingent control is preferred over investor controlbythe entrepreneur.

We can summarize the analysis of contingent control as follows.When

yllyl <yl/yland only investor and contingent controlarefeasible,then

contingent control is the preferredchoiceof the entrepreneur.When yllyl >

yllyl, then contingent control(with tB =1and tG =yllyl) is the equilibriumoutcomeonly if neither entrepreneurnor investor controlis feasible.Theremay indeedbe someinvestments with high setupcostsK that are feasibleunder the contingent-control allocationbut not under investor control(with tB = l and tG =

y\302\260clyG).

Comparing contingent controlwith entrepreneurcontrol,note that in thelimit aspG ->1and pB ->1,we have

n5C->pyc+(X-p)yl >max{n*,rwAlso, it is straightforward to checkthat n5C> n^.Therefore,aspG->1and

pB ->1,there are valuesof the setupcostK such that

534 Incomplete Contracts and Institution Design

Use>K>max{UR,UPR,UNR}

For thosevaluesof K, contingent controlis the contracting outcomeeitherwhen yllyl < yfyyl or when yHyl >

yGc\\yGL and K > ft/.12

11.3.1.5Comments

TheAghion and Boltonmodelprovidesone answer to the questionof who

should controla firm when the agentswho setup the firm are wealth

constrained. Their answer is consistentwith the result of Hart and Moore(1990)that the most efficient ownership or controlallocationis to give allcontrolrights to those agentsmaking ex ante investments or thoseagentshaving an essentialrole in production.In the Aghion and Boltonmodelthemost efficient allocationis entrepreneurcontrol.However,their analysis

points out that this allocationmay not befeasiblewhen thesecritical agentsare wealth constrained.Controlmust then be sharedwith investors.

Under some circumstances,the equilibrium-control allocation is acontingent-controlallocation.This situation canbe interpretedas a form ofdebt financing or, perhaps more plausibly, as a form of venture-capitalfinancing. Theventure-capitalinterpretation was first suggestedby Berglof(1994).More recentlyKaplan and Stromberg(2003)have argued that the

contingent-control allocationdescribesquite accuratelysomekey featuresof venture-capitalinvestment contracts.Their empiricalanalysis, which isbasedon 200venture-capitaldealsby 14venture-capitalpartnerships with

118firms, stressesthat cash-flowrights are generally allocatedseparatelyfrom voting or board rights and that future financing and controlrights areoften contingent on observablemeasuresof firm performance.Generally,the venture-capitallead partner obtains full control if the firm performspoorly, and if the firm performswell the entrepreneurcan increasehercontrolrights.

11.3.2Wealth Constraints and Optimal DebtContractswhen EntrepreneursCanDivert CashFlow

The Aghion and Bolton analysis can explain different forms of outsidefinancing when entrepreneurs are wealth constrained.In their model,

12.Somereaders may note that the analysis of contingent control here is somewhat differentfrom Aghion and Bolton (1992).The reason is that the comparison between the two controlstructures in that paper is incomplete (but seeVauhkonen, 2002, for a more completecomparison).

535 Incomplete Contracts

outside investors may take a controlling equity position,may holdnonvoting shares(which give them no controlrights whatsoever),or may write

a financial contractthat gives them contingent-controlrights. The last typeof financial contractcan take different forms. It couldbea contingent vetoright, an option contractlike a warrant, or a debt contract.

Debt is a specialform of financial contract inducing a specialform ofcontingent-controlallocation.First, debt is generally a fixed repaymentclaim on the firm, which is independentof realizedcashflow. Second,debtinvolves a transfer of controlfrom the entrepreneurto the creditor(s)onlyin statesof nature where the entrepreneuris unableor unwilling to honorher fixed repayment claim.In that casethe entrepreneuris said to be in

default of her debtobligation.Third, when the entrepreneurdefaults on herdebt repayment, the creditor(s)generally have the right to seizethe firm'scashflow up to the point when the repayment is fully met.

Tobeable to explain why debt financing is an efficient financial contract,one needsto introduceother featuresinto the Aghion and Boltonframework besidestheir form of contractual incompleteness.If one makes theadditional assumption that realizedcashflows are either privateinformation to the entrepreneuror nonverifiable to third parties,then debt canbean optimal financial contract.The reason is that with unobservable(orunverifiable) cash flows it is easy for the entrepreneur to engagein self-dealing by diverting cashflow from the firm. The investor'sonly responsesthen are to threaten to forecloseon the firm's assetsor to cut off future

lending if the entrepreneurdoesnot meet a fixed repayment.The first modelswith unobservableor unverifiable cashflow that derive

an optimal investment contract with the three main characteristicsofdebt\342\200\224a fixed repayment claim, a right to forecloseon the firm's assetsincaseof default, and priority repayment in caseof default\342\200\224are due toBolton and Scharfstein(1990)and Hart and Moore(1989,1998).We shallconsidera simple model of debt financing with nonverifiable cash flow,which is a specialcaseof both the Hart and Moore(1998)and BoltonandScharfstein(1990)models.

To set the stage,we beginby briefly discussing the earlierliterature onthe optimality of debtas a financial contract.As discussedin Chapter5,thefirst modelsof optimal debt are due to Townsend(1979),Diamond(1984),and Gale and Hellwig (1985).Cashflow is assumedto beprivateinformation to the entrepreneurin thesemodels,and investors are assumedto onlyobserverealizedcash flow at a cost.There is a positivemonitoring cost,which the risk-neutral contracting parties will attempt to minimize. It is

efficient to only monitor (or inspect the books)for a subsetof reportedearnings.Over the set of reported earnings for which there is noinspection, incentive-compatibletruthful reporting of earnings requires that

repayments to investors must be independentof reported earnings. This

requirementexplainswhy it isoptimal to havea fixed repayment claim, thefirst characteristicof debt.Then,in order to minimize monitoring costs,itis optimal to inspectonly low reportedearnings and to have creditorsclaimthe full realizedcash flows. This fact explainsa secondcharacteristicofdebt\342\200\224that creditorshave priority in default.However,the third

characteristic\342\200\224the right to forecloseon the firm's assetsin caseof default\342\200\224cannot

be adequatelyexplainedwith this approach.13Other modelsof debt as an

optimal financial contract,basedon moral hazard with or without

renegotiation (Innes,1990,Matthews, 2001,orDewatripont,Legros,and Matthews,

2003),do not accountfor creditorforeclosurerights either.

11.3.2.1A SimpleTwo-PeriodProblem

We now turn to the analysis ofa simple dynamic investment problemwherea critical feature of the optimal financial contractis the investor'sright toforecloseon the firm's assetsin caseof default. Considera risk-neutral

entrepreneurwith an investment project requiring an investment outlayK > 0 in period 0 and returning a random cash flow r e {0,rH) in period1.The probability that the high-cash-flow outcomeis realizedis given by

Pr(r = rH) =p e [0,1].Supposefor simplicity that there is no discountingand that the equilibrium interestrate is zero.We shall assumethat p andrH are large enough so that the investment has a positivenet expectedreturn and is worth undertaking:

prH >K

Supposethat the entrepreneur has no initial wealth and turns to awealthy, risk-neutral financier to fund the project.If cashflow r isobservable to both parties and verifiable by a judge, the wealthy financier will

agree to lend any amount L to the entrepreneursuch that

prH>L>Kin return for an rH-contingent repayment tH(L) > Lip.

13.As stressed in Chapter 5,the optimality of debt in these models hinges critically on theassumed risk neutrality of the parties.Also, to obtain the type of inspection policy that can beinterpreted asa risky debt contract with bankruptcy costs, random inspection andrenegotiation must be ruled out. Finally, it is tecrmically difficult to extend the result of optimality ofdebt to multiperiod settings (seeGale and Hellwig, 1989;Chang, 1990;Webb, 1992).

537 Incomplete Contracts

If, however, cashflow r is privately observedby the entrepreneur,or isnot verifiable in court, then this one-periodprojectcannot get any outsidefunding. Indeed,if r is privately observedby the entrepreneur,shewill neverdisclosethe realizationof rH, and the investor would never get his moneyback.Similarly, if rH is observableto both partiesbut not verifiable in court,the entrepreneurwould refuse to pay tH(L) when rH is realized,and theinvestor would not be able to enforcepayment in court.Either way, anextremeform ofcreditrationing obtainsbecauseof the inability to enforcerepayment in high-cash-flow states.

This extremeform of inefficiency, however,obtains only in the specialcaseofone-period-livedprojects.For multiperiod projects,wherecashflows

occurrepeatedlyovertime, it canbesubstantially reduced.Thebasicreasonis that when investment projectsgeneratecashflows repeatedlyover time,the investor can induceher to make repayments either by threatening toforecloseon the entrepreneur'sassetsin caseof nonpayment (as in Hartand Moore,1998)orby threatening to withhold future funding (asin Boltonand Scharfstein,1990).

Toseehow this threat works, considerthe sameinvestment problemasbeforebut now twice repeated.That is, the first investment projectwith an

outlay ofK> 0 in period0generatesa random cashflow r e {0,r11} in period1.Thesecondprojectalsorequiresan outlay ofK>0 in period1and generatesan identically and independently distributed cashflow r e {0,rH) in period2.14

In this longer-horizoninvestment problem,a long-term investmentcontract specifiesthe following:

1.The sizeof the initial loan,L0> K.2.Thesizeof the loancommitment in period1contingent on a highrepayment tH(L0)\342\200\224Li.

3.Thesizeof the loancommitment in period1contingent on a lowrepayment

tL(L0)\342\200\224L\\.

Sincethere are only two possiblecash-flowoutcomesin period1,thereis no needto specifya richerdeterministic loan commitment menu in that

14.Note that these two consecutive projects can also beseen as a single project with an outlayof K>0 in period 0,a random cash flow rx e [l, rH -K] in period 1,and second-period cashflow r2 e {0,rH}. The cash flow rx =I is given by / =-Kif the second project is undertaken evenafter a low-cash-flow outcome in period 1.It is given by / =0 if the second project is notundertaken. Hart and Moore (1998)consider a single project with two cash-flow rounds, whileBolton and Scharfstein (1990)consider two identical consecutive projects.

538 Incomplete Contracts and Institution Design

period.15Also, sincecashflows are eitherprivate information or nonverifi-

able,it will not bepossibleto extractany payment from the entrepreneurin period2,aswe have seen.Thus any long-term investment contractischaracterized by the five variables{L0,Lf, tH, L\\, tL).

Supposefor now that the parties can commit not to renegotiate this

contract.In its most basicform, we shall describea debt contractto be along-term investment contractsuch that

L?>K-(rH-tH)and

Lt<K+tLIn other words, a debt contractis such that the entrepreneuris granted

sufficient additional funding to be able to undertake the secondprojectwhen sherepaystH, but is deniedadequatefurther funding when shedoesnot meet her repayment obligation tH in period 1.The debt contractcanalsobe interpretedas giving the creditorthe right to foreclose,or \"pull the

plug,\" on the entrepreneur'sprojectwhen she defaults on the fixedrepayment obligation tH. Note that this contracthas the important property that

investment in period1is positively correlatedwith realizedcashflow.

We shall now show why long-term investment contractsmust be debtcontractsand why the risk-neutral investor may be willing to invest in thefirm in period 0 under such a contract.Applying the revelationprinciple,we observethat loan commitments in' any long-term contractmust satisfythe incentive-compatibility constraint requiring that the entrepreneurtruthfully report cashflow r11when it is realized:

rH -tH+Lf +yH(prH -K)>rH-tL+H+XH{prH -K) (11.15)where,yH and %H are indicator functions. The function yH takes the valueyH = 1when

rH -tH+L? >K

that is,when the entrepreneurhas enough accumulatedfunds to finance thenew project.Otherwise,yH =0.Similarly, %n =1when rH -tL +L\\ > K, andotherwisexH =0.

15.Bolton and Scharfstein (1990)allow for amenu of stochastic loan commitments such thatcontingent on a repayment t(LQ) the investor commits to finance the new project with prob-ablity ft, where i =L,H.

539 Incomplete Contracts

The other incentive-compatibility constraint\342\200\224that the entrepreneurtruthfully report cashflow rL =0 when it is realized\342\200\224always holdshere,asthe entrepreneurdoesnot have the funds availableto mimic a high-cash-flow' outcomewhen realizedcashflow is rL =0.The reader might wonderwhy the entrepreneurcannot borrowfunds from another lender to beableto mimic rH. The reason is simply that no lenderwould be foolishenoughto lend to the entrepreneurknowing that shewould never repay the newloan.

Besidesthe precedingincentive-compatibility condition, any long-terminvestment contractmust also satisfy the investor'sindividual-rationalityconstraint

-L0+p(tH-lJ?)+0-p)(tL-U{)>0 (11.16)as well as the wealth constraints

tH<rH+Lo-K (11.17)and

tL<L0-K (11.18)An optimal long-term investment contractthen maximizes the

entrepreneur's expectedpayoff

L0-K+p(rH -tH+If +YH[prH -K]) (n19)+0.-pX-tt<+Lt+XL[prH-K])

(where%L = 1when -tL + L\\>Kand otherwise%L= 0) subjectto

constraints (11.15),(11.16),(11.17),and (11.18).To find the solution to this constrainedoptimization problem, note

first that a long-term investment contract cannot satisfy the individual-

rationality constraint (11.16)if both yH and %n equal1.Toseethis point,note that the incentive constraint (11.15)and the limited-wealth constraint

(11.18)imply that

tH -Lf<tL-Ut<U-K-Lkwhen 7s=1and xH = 1-But then constraint (11.16)cannot hold.A similar

argument establishesthat long-term contractssuch that yH =0 and xH =0,or such that ^ =0 and %H=1,are alsonot feasible.Thus the only feasiblelong-term contractis such that yH =1and %n =

%L=0.In words,a long-term

contractis feasibleonly if it is a debt contract,wherethe investor commits

540 Incomplete Contracts and Institution Design

to refinancewhen the entrepreneurrepaystH, but not when the repaymentis only tL. In particular, note that financing the projectwith outsideequityis not feasiblehere,whether or not the investor gains controlof the project.Indeed,neither form of equity financing involves a commitment to investin the new project if and only if the first-periodcash-flowrealizationis rH.

Finally, note that financing the first project with a long-term debtcontract is feasiblewhenever

-K+p(rH-K)>0or

prH>K(l+p)When this condition is satisfied,the investor is willing to finance the first

projectwith a long-term debt contractsuch that L0= K,tH = rH, Lf = K,and tL = L\\ = 0.In addition, this contractis incentive compatiblefor the

entrepreneur.Thus creditrationing canbe substantially reducedfor multiperiod

projects wherecashflows occurrepeatedlyovertime. In the precedingexample,financing of the first projectthrough long-term debt is feasibleprovidedthe

project'spresentvalue is sufficiently larger than the setupcost.Moreover,renewedfinancing of the secondproject is guaranteedif the first projectpays off.Thereasonwhy financing is possiblewhen the projectisrepeatedtwice is that the lender can now use the threat of foreclosureortermination to inducerepayment of rH in period1.

This analysis raisesa number of questionsabout the robustnessof theresult of the optimality of debt as a financial contract.First, doesdebtremain an optimal financial contractif the partiesare allowedtorenegotiate the long-term investment contract?Second,is debt an optimal contractif there aremore than two cash-flowoutcomesin any given periodor when

cashflows are autocorrelated?Third, islong-term debt an optimal contractwhen there are more than two iterations in investment? Also, doesdebtremain optimal in a wider classof (message-game)contracts?We turn to adiscussionof eachof thesequestionsin the following subsections.

11.3.2.2Renegotiation

Notice first that a commitment to terminate the investment in period 1following a low-cash-flowrealization,r = 0, is credible.Indeed,theinvestor would have no reason to rescindthe contractin that casebecause

541 Incomplete Contracts

he can only losemoney by reinvesting at that point. Similarly, thecommitment to continue the investment in period1following a high-cash-flow

realization, r = rH, is also credible.Indeed,the continuation contractat that

point is Pareto efficient, so that no mutually beneficialrenegotiationisobtainable.

Doesthis conclusionmean that the precedinglong-term debt contractisrenegotiation-proof?Interestingly, as Hart and Moore (1989)first noted,the answer is no.The reason is that when rH is realizedthe entrepreneurmay be able to successfullynegotiatedown the potentially oneroustermsin the initial contractunder which she can get refinancing in period1.Toseehow this processworks, supposethat the initial long-term contractissuch that L0= K,tH = rH, Lf = K,tL = 0, and L\\ = 0.If the entrepreneurdecidedto repay tL when the cashflow was rH, shewould not beable to getnew financing for the secondprojectunder the initial contract.

Doesthis mean that she will not be able to undertake the new project?Not necessarily,sinceher retainedearnings at that point (rH -tL) = rH aresufficient to finance the entiresetupcostK on her own. It is,however,reasonable to supposethat investment in the new project in period 1by the

entrepreneurwould be sufficient evidencein court to prove that therealized cashflow in period 1was rH. As under existing law, it would then befeasiblefor the investor to block any new investment by the entrepreneurat that point. Note that if the investor couldnot stop the entrepreneurfrom

investing on her own following default on the long-term debtcontract,then

the contract would no longer be feasible,since the entrepreneurwould

always default following the realizationof rH.

Supposethat unless the contract is renegotiatedfollowing default the

entrepreneur would not be able to invest in the new project.Hart andMoore(1989)observethat following default (that is,a repayment of tL eventhough realizedcashflow is rH) the entrepreneurand investor can actuallyfind a mutually beneficialrenegotiation.Indeed,if they do not renegotiate,their respectivepayoffs are rH and 0.But if they do renegotiate,they

gain the additional surplus \\prH-

K\\ > 0, which they can divide betweenthemselves.

Hart and Moore(1989,1998)and most of the subsequentliterature on

optimal long-term debt contracting assumethat bargaining in renegotiationalways resultsin an efficient outcome.Thisassumption is not unreasonableif there is only onecreditornegotiating with the entrepreneur.We shall takethe bargaining solution in our simpleexampleto be any (a,1- a) split

542 Incomplete Contracts and Institution Design

of the renegotiationsurplus, where 1> a > 0 denotes the entrepreneur'sshareof the renegotiationsurplus.16 Thebargaining problemin our simpleexamplewith twice-repeatedinvestment is a standard problem,and the

bargaining solution doesnot require any specialjustification. As we shall see,however,in more generalmultiperiod problemsthe bargaining solution in

renegotiationis lessobvious and is sensitiveto specificassumptions aboutthe parties'rights while they are negotiating.

Underour assumedbargaining solution, a long-term debt contract is

renegotiation-proofonly if the following condition holds:

Lo-K+rH -tH+Lf +(prH-K)>rH+a[prH-K] (11.20)TheRHSof expression(11.20)denotesthe entrepreneur'spayoff when

(1)realizedcashflow is rH, (2) sherepaysonly tL, and (3) the originalcontract is renegotiated.If this payoff is lessthan what shewould get by

repaying tH, then the contract is renegotiation-proof.Notice that when a = 0,incentive compatibility of the long-term debtcontractguaranteesrenegoti-ation-proofness.But when a > 0 the renegotiation-proofnessconstraint

(11.20)may bind before the incentive compatibility constraint (11.15).Inthat case,the optimal renegotiation-proofcontractis obtainedby

maximizing expression(11.19)subjectto constraints (11.20),(11.16),(11.17),and

(11.18).As canbe easilychecked,the optimal renegotiation-proofcontractthen

is still a debtcontract.Theonly differencewith the caseof full commitmentconcernsthe feasibility of debt financing. A long-term debt contractisnow

feasibleif and only if

-K+p(l-a)(prH-K)>0Note that the repayment tH canalsobe interpretedas the purchaseprice

for the entrepreneurof the right to continue investing in the project,which

is soldto her by the investor. As long as this priceisset optimally by theinvestor, there will beno room for further renegotiation.

11.3.2.3GeneralCashFlows

In the precedingexamplewe have allowedfor only two cash-flowoutcomesin each period,with re {0, r11).The analysis is almost the same if one

16.Note that since rH > \\prH-

K\\, any division of the surplus from renegotiation is feasible

following the realization of r11.

543 Incomplete Contracts

assumesthat r e {rL, rH) with rL > 0, as in Boltonand Scharfstein(1990).They assumethat rL is pledgeablebut not (rH -rL),so that cashflows arein effectpartially observable(or verifiable).One differencewith the

preceding analysis is that under the new assumption, rL > 0,it may bepossibleto have projects that are worth funding, yet that do not allow the

entrepreneur to self-financein period 1following a high-cash-flow realization,becausethe inequalities

-K+prH+(l-p)rL>0and

rH -rL<Kare compatible.In that case,the investor is able to block any newinvestment by the entrepreneurin period1without having to enforceany legalrights to blocknew investments.

In fact, the analysis is, if anything, strengthenedif one allows forcorrelation in cashflows. Let E[r \\ r^ denote expectedperiod-2cashflows

conditional onperiod-1cash-flowrealizationrx.With positiveserialcorrelation,E[r | rH] > E[r \\ rL\\ Now the entrepreneur'sexpectedpayoff in period2 is

greaterwhen period-1cashflow is rH, so that shestandsto losemore if sheis not refinanced.This outcomereducesher incentive to underreportcashflow in period 1and strengthens the investor'sthreat of termination. Weleaveit as an exerciseto show that the optimal long-term contractremainsa debt contract.

The analysis alsoextendswith minor modifications to a situation wherecash flow is distributed continuously on the interval [rL,rH], with rL > 0,rH < +oo, and density functions/i(r)and f2(r) in periods1and 2,respectively.As shown in Faure-Grimaud(2000)and Poveland Raith (2004),when the

entrepreneurcanmake a take-it-or-leave-itofferto an investor, the optimallong-term investment contract under full commitment takes the formthat the entrepreneurcan refinancethe project if she repays D < E[r2]in period1 (where E[r2] denotesexpectedcash flow in period 2). Ifsherepaysonly t < D,then she is only allowedto refinancethe projectwith

probability

544 Incomplete Contracts and Institution Design

The probability of refinancing is such that the entrepreneur'sincentive isto repayt =r, whenever r <D.Thus the optimal contractis as if the investor

always had higher priority and the entrepreneurdid not get any return in

period1until the investor had beenrepaid in full. The optimal contractisdepictedin Figure 11.4.With a continuum ofcash-flowrealizations,it is evenmore striking how the optimal contractresemblesa standard debt contract.

Note that Hart and Moore (1998)considerthe related problemwherethere is a single up-front investment K > 0 in period 0 followedby two

t(r)k

Repayment

\302\243(0!

Probability of

continuation

Figure 11.4Repayment and Probability of Continuation under the Optimal Contract

545 Incomplete Contracts

observablebut not verifiable (random) cash-flowrealizations,rx e [rL,rH]and r2 e [rL,rH].Their framework is almost identicalto the one we haveconsidered,with one important difference:They assumethat all uncertaintyabout cashflows is realizedin period1.In this slightly different investment

problem,they explorethree additional extensions:First, they allow the entrepreneurto borrowmore than is neededto fund

the project.If the amount borrowedis B,then the surplus availableto the

entrepreneuris T=B-K. It isassumedthat when the entrepreneursavesT,it cannot beseizedby the investor. The entrepreneurcanuse T in period1either to meet repayments to the investor or to reinvest in the project.Ifhe reinvestsin the project,then shegetsa (random) unit return k, wherefor all k, 1< k < E[r2].In this problemthe entrepreneurmay want to carryforward T > 0 as a way of hedging against bad cash-flowoutcomesin

period1.Second,they allow for random verifiable cash flows (or liquidation

value) rL.

Third, they considermoregeneralcontractswherethe investor may havethe option to buy the new projectfrom the entrepreneurin period1.Whenthe investor becomesthe owner of the projectin this way, he becomesthe

recipientof the cashflow.

Restricting attention to standard debtcontracts(T,D,rL),whereD is the

required repayment in period 1and rL is the liquidation value in caseofdefault, Hart and Moorefirst ask which of two types of debt contractsis

optimal, afastestdebtcontractwith T=0 or a slowestdebt contractwith D> T + rH. Note that for any T there is a unique D such that the creditor'sindividual-rationality constraint binds.Also, the more the entrepreneurborrows(that is, the higher is T) the more sheneedsto repay (the higheris D) to keepthe creditoron his break-evenconstraint. Thus, the two debtcontracts\342\200\224fastest and slowest\342\200\224are the two extremefeasiblecontracts.

Under the slowestcontract (D > T + rH), the entrepreneurengagesin

maximal hedging but is always forcedto default in period1.Under this

contract she effectively \"rents\" the projectfrom the creditorfrom period0 to

period1.Under the fastestcontract(T=0) the entrepreneurborrowstheminimum amount and doesnot engagein any hedging.

Hart and Moorefind, on the onehand, that, when the unit return onreinvesting cashin period1is low (iz=1)and only rx is random, then the slowestdebtcontractis optimal. In otherwords,maximum hedging ofperiod1cash

546 Incomplete Contracts and Institution Design

flow is optimal. On the other hand, if only period-2cashflow, r2, is random,then the fastestdebt contractis optimal. When only r2 is random, hedgingis unnecessary.In addition, under this stochasticstructure the entrepreneurfacesa potentially high costof termination when r2 is high. Toeliminate this

cost,the optimal debt contractis such that the entrepreneuris always ableto repayD in period1.We leaveit asan exerciseto derivethesetwo results.

ThesecondquestionHart and Mooreaddressis,When are standard debtcontractsoptimal financial contracts?Theyshow that when the unit return

on reinvesting cashin period1is high (n=E[r2])and rL, rx and r2 arepositively seriallycorrelated,then the fastestdebtcontractis optimal in a largerclassof message-game(or option-to-buy)contracts.This debt contractisoptimal becauseit maximizes the likelihoodof continuation of the projectin the cash-flowstateswherecontinuation is most desirable.

11.3.2.4MoreThan Two Investment Rounds

The analysis and resultsof the dynamic problemwith two rounds ofinvestment also extend to situations with more than two rounds.In particular,debt contracts with foreclosure rights in the event of default remain

optimal. Also, the basiclife cycleof investment, with a developmentphasefollowedby a cash-flow-extractionphase\342\200\224which holdsby construction in

the two-periodproblem\342\200\224can now be derivedas an optimal policy in the

multiperiod problem.The main qualitative changein problemswith more than two rounds of

investment concernsrenegotiationand the credibility of the investor'sthreat of termination in the event of default. We shall illustrate this changeby consideringa problemwith three rounds of investment first analyzed in

Gromb(1994).Considerthe same investment problemas beforebut now

repeatedthree times.That is, eachof the three investment projectshas an

outlay of K > 0 in period t =0,1,2 and generatesan identically and

independently distributed cashflow r e {0,r11)in periodst+1.Tokeepthe analysis assimple aspossible,it is convenient to assumenow

that the investor has all the bargaining powerboth in the initial

contracting stageand in subsequentrenegotiations.Thus we shall assumethat theinvestor beginsby making a take-it-or-leave-itcontractoffer and that hecanat any time proposeto renegotiatethe contractwith the entrepreneur.

Considerfirst the optimal contractunder full commitment. This contractmust now specifycontingent loan commitments in periods1and 2.Proceeding by backward induction, it is easyto seethat there should benorefi-

547 Incomplete Contracts

nancing in period2 following a low-cash-flowoutcome.It is, however, lessclearwhether there should be no refinancing in period 1following a low-cash-flowoutcome.On the one hand, no refinancing in period1followingdefault is an evenmore effectivethreat than before,sincenow two roundsof investment arepassedup. On the otherhand, the costofpassingupvaluable investments is greater.Indeed,two morerounds of investment remainin period1.Our precedinganalysis shows that the optimal continuation

contractat that point is to renewinvestment for one round and to make aloan commitment on another round contingent on no default.

Gromb (1994)shows that it is (weakly) optimal to undertake the first

project.Indeed,the investor cannot do worsethan passingup the first

project becausehe can always replicatethe outcomein the two-investment-round problemby committing to neverfinance the third project.When p <

\\, it is optimal to cut off new lending for both remaining rounds ofinvestment in period1following a bad cash-flowrealizationrx = 0, but to alsocommit to finance both remaining projectsfollowing a high-cash-flowrealization ri = rH. The logicbehind this extremecontractis that the investorcanget the entrepreneurto makea maximum repayment following a high-cash-flowrealizationin period1equalto the presentexpectedvalue of thetwo remaining projects:tH =2prH

Sincethe entrepreneurstands to loseaccessto both continuation projectsif shereports a low-cash-flowoutcomein period1,sheis willing to pay asmuch as the whole continuation value of the two projectsfor the freeaccessto the two continuation projects.Note that when p <

\\, she is alsoable to

pay tH following a high-cash-flow realization rx = rH. Note also that theinvestor cannot extractmore than 2prH from the entrepreneurunder any

long-term contract,sinceshewill always run away with the money in thelast period.

Theother alternative for an optimal long-term contractwould be tocontinue to finance both remaining projects following a high-cash-flow

realization, but alsoto finance oneprojectfollowing abad cash-flowrealization.However,under this contractthe investor would have a higher investment

outlay and couldhope to get a total repayment of at most 2prH.As one might expect, the optimal full commitment contract is not

renegotiation-proof.But, more importantly, the threat to cut off any future

funding following default in period1isnow no longercredible.Indeed,as

548 Incomplete Contracts and Institution Design

we have pointedout, it isoptimal for the investor to makea renegotiationofferfollowing default in period1renewing investment for one round and

making a loan commitment for another round contingent on no default in

period 2.Anticipating such an offer, the entrepreneurwould nevermake

any repayment in period 1.Thus the investor's future monopoly rentsundermine his presentpower to extract rents by making his threat oftermination not credible.

There are two possiblerenegotiation-proofcontractsthat the investor

might then select.Under one contract,the investor would simply passupthe first round of investment and then offer the entrepreneurthe samecontinuation contractas in the problemwith two rounds of investment.17 Thiscontractwould yield a net expectedreturn of

p{prH-K)-K (11.21)Under the othercontract,the investor would fund the first project,committo fund both remaining projects if the entrepreneur repays tH = prH in

period1,commit to fund the secondproject if the entrepreneurdoesnot

repay anything in period1,and commit to fund the third project if the

entrepreneurrepays tH =prH in period2.Thiscontractwould yield a net

expectedreturn of

p{prH -2K)+0--p)p{prH-K)-Kor

[p+pO-p)](prH-K)-K0-+p) (11.22)Comparing expressions(11.21)and (11.22),we find that the latter contractisoptimal if and only if

K<(l-p)(prH-K)The analysis of this simple examplewith three investment rounds

highlights an important principle.If the future relationship is profitable, theinvestor has little incentive to terminate it following default. Thisfact, in

turn, undermines his ability to extractpayments from the entrepreneur.Inthe limit, when there is an infinite number of possibleinvestment rounds,

17.Note that since the investor has all the bargaining power, the two-period contractwhere he commits to renew funding only if the entrepreneur repays t\" = rH is renegotiation-proof

549 Incomplete Contracts

Gromb(1994)shows that evena monopoly lendercannot makea netpositive return on investment in a renegotiation-prooflong-term investmentcontract.Indeed,any renegotiation-proofcontractcannot rely on strictlyPareto-dominatedtermination threats. When lending is terminated, bothinvestor and entrepreneurget a continuation value ofzero.Therefore,if theinvestor were to obtain a strictly positivereturn under the long-termcontract, the termination threat would be Pareto-dominated.

Admittedly, this extremeresult may be sensitiveto underlyingassumptions on the bargaining processunder renegotiation.Hart and Moore(1994),for example,assume that when the entrepreneur defaults, theinvestor takes controlof the assetsand can collectthe project'scashflow

for that period.This ability, of course,significantly increasesthe investor'spowerto extractpayments from the entrepreneur.

Themultiperiod problemconsideredin Gromb(1994)hasbeenextendedin several different directionsby DeMarzo and Fishman (2002).Theyallow for investments of varying size,more general return distributions,

consumption and reinvestment of accumulatedcash flow, correlation in

returns, and different bargaining and liquidation scenarios.Their modelencompassesBulow and Rogoff (1989),Gromb (1994),and Hart andMoore (1994)as specialcases.Remarkably, they show that the optimalrenegotiation-prooflong-term investment contractisa debt financingcontract combining a bond with a long-term couponpayment and a line ofcreditthat the entrepreneurcandraw on in the event ofbad cash-flowoutcomes. Only when the line of credithas beenexhaustedwill theentrepreneur be forcedto default and transfer controlto the investor.

11.4Summary

In this chapterwe have introducedexogenouslimits on contracting, which

restrictthe contracting parties to write incompletecontracts.Theintroduction of theseexogenousconstraints is a major break from the earherchapters in the book.It isa methodologicalbreak,which allowedus to focusonand formulate otherkey notions oforganizations and firms, such asauthority

or employment relations,ownership, and controlrights. We beganwith

Simon'stheory of authority when long-term contracts are incomplete,which is essentiallya theory of flexibility: by delegating authority toan agent, the contracting parties let the agent make efficient choicesin

550 Incomplete Contracts and Institution Design

responseto changing circumstances.Thesechoiceswill be made with theinterestsof the controlling party in mind, but if the othercontracting partieshave aligned interests,they will on averageprefer the efficient choicesofthe controlling agent to a narrow and inflexible, contractually specifiedoutcome.We arguedthat much of Simon'stheory restson the assumptionthat long-term contractscannot berenegotiatedeasilyand that the theorymust be supplementedwith a holdup problemto explain why it ispreferable to write long-term contractsin the first place.

The introduction of a holdup problemled us next to the theory ofownership articulated by Grossman,Hart, and Moore,that ownership is aresidual right of control.This theory makes a fundamental breakwith earliernotions of ownership articulated by Alchian and Demsetz (1972)andJensenand Meckling (1976),among others,that ownership is only a rightto residualreturns. We showedhow Grossman,Hart, and Moorewereableto developa theory of the boundariesof firms and the costsand benefitsof integration basedon this notion of ownership rights. We alsohighlightedhow vertical integration decisionscouldbe sociallyinefficient and couldresult in monopolizationand reducedcompetition.

The notion of ownership as a residualright of controlcanbe refined by

introducing financial contracts.We saw how investments financed throughdebt basicallygive riseto a contingent ownership allocation.Controlof theinvestment project remains in the hands of the founder if all debtsarerepaid and is transferred to creditorswhen the entrepreneuris unable to

repay her debts.We determinedconditionsunder which a contingent-controlallocationinducedby debt financing dominatesany other form ofshared ownership or controlarrangement. Basically,when investors needprotectionsmostly in particular circumstances,and when the borrower'sinability to repay is a signal that is closelycorrelatedwith these states ofnature, then contingent controlis efficient.Thus venture capitalcontractsgenerally involve contingent-control allocations,as Kaplanand Stromberg(2003)have documented,becausecontrolin the hands of the venturecapitalist is more valuable in early stages,when successof the venture is still

uncertain. Oncesuccessis assured,the venture capitalist'sroleislessimportant, so that it is generally efficient to commit to relmquishing control in

thosecircumstances.Finally, we closedthe chapterwith a further refinement of contingent-

controlallocationsthrough debt financing in situations wherethe borrowermay be able to divert money from the firm, or may abscondwith the

551 Incomplete Contracts

loan and/or the proceedsof the investment. In the presenceof such awillingness-to-repay problem,lending for investment is feasibleonly if thelender can brandish the threat of cutting off the borrower from further

loansin the event ofa default.We argued that such a threat is effectiveonlyif the lenderhasan exclusiverelationwith the borroweror is ableto securecollateraland canforeclosethe firm's assetsin the event ofdefault. Inaddition, it must alsobe in the lender'sinterestto shut out the firm from further

lending or liquidate its assetsfor the threat to be credible.All in all,this incomplete-contractsperspectivecombinedwith the notion of limitedenforcementof repayment gives riseto a rich theory of debt financing andcreditorrights\342\200\224one that brings us closestto real debt contractsand to themain issuesthat debtor-creditorlaw and bankruptcy law are concernedwith.

11.5LiteratureNotes

Many of the ideasdevelopedin this chapter find their origin in Coase'sclassic(1937)articleon the theory of the firm. Thisarticlepoints out that

most of the economicactivity that takesplacein firms is not regulatedby

complete,explicit contracts.All this activity takes placeoutsidea marketenvironment by choice,and an explanation is required to show why sucheconomictransactions are superiorto market transactions.Marketscannotbeomniefficient, for otherwisetherewould beno firms. Similarly, economicactivity in firms cannot be omniefficient, for otherwisethere would be nomarkets. What determineswhether an activity should bea markettransaction or an authority transaction was in Coase'smind a fundamental micro-economicquestionthat remainedunanswered in his day.

The first attempt at developinga formal theory that couldaddressthis

basicquestionis due to Simon (1951).But, undoubtedly, the economistwho

has done most in taking up Coase'schallengeand in articulating a full-

fledgedeconomictheory of the firm and organizations is Williamson. Inseveralbooks(1975,1985)and classicarticles(1971,1979),he has

developedwhat is now commonly referred to as transactions costeconomics.

Interestingly, although Williamson's theory is basedon the notion ofincompletecontracts,which is the starting point for the property-rights

theory of Grossman,Hart, and Moorediscussedin this chapter,it providesa different perspectiveon firms. In his mind, firms are long-term governance

552 Incomplete Contracts and Institution Design

structures designedto dull the different contracting parties'drive to pursuetheir self-interest.Firms thus are able to achievemore efficient outcomesthrough better cooperation,wheremarket-basedtransactionsare likely tobreakdown due to excessivehaggling and excessivelyhigh bargaining costs.

Another classicarticle,which exploresthe consequencesfor the theoryof the firm ofholdup problems(a notion first articulated in Goldberg,1976)that arisewhen contractsare incomplete,is Klein,Crawford,and Alchian

(1978).Togetherwith Williamson's transactionscosteconomics,it providesthe main notions that have given riseto the more modern formal theoriesdiscussedin this chapter.

Much of the rapidly growing economicsliterature on the theory of thefirm subsequentto Grossmanand Hart (1986)and Hart and Moore(1990)is discussedin the survey article by Holmstrom and Tirole (1989),theClarendon lectures by Hart (1995),a book by Hansmann (1996),and

perspectivearticlesby Holmstrom and Roberts (1998)and Bolton andScharfstein(1998).

As discussedin this chapter,financial contracting hasbeena particularlyfruitful application,following the contributions of Aghion and Bolton(1992),Bolton and Scharfstein(1990),and Hart and Moore (1989,1994,1998).Beyondthe work discussedin this chapter,let us alsomention thearticlesby Berglofand von Thadden (1994)and Dewatripontand Tirole(1994a)that provide finer predictionson the optimal capitalstructure ofthe firm (seeSection12.4.1.2in Chapter12on this).

In the last few years there has also beenan explosionof researchonthe implications of incompletecontracting for the designof economicinstitutions such as ownership, governance,and the internal organizationof the firm. The most recent literature has exploredthe link betweenself-enforcing (relational)contracts and assetownership (see,in particular,Baker,Gibbons,and Murphy, 2002;Halonen,2002;and Stoleand Zwiebel,1996a,1996b).Another major line ofresearchhasexploredthe questionofthe governanceof larger firms or organizations with potentially a largenumber of owners (or voters) and also an administrative structure with

multiple managers(see,in particular, Gertner,Scharfstein,and Stein,1994,and Inderst and Muller, 2003,on internal capitalmarkets in

conglomerate firms; Hart and Moore,2000,Hart and Holmstrom, 2002,Rajan and

Zingales, 2001,and Wernerfelt, 1997,on internal hierarchiesof large and

growing firms; and Bolton and Rosenthal,2003,and Aghion and Bolton,2003,on constitution design,majority voting, and politicalintervention).

Foundationsof Contractingwith UnverifiaMeInformation

12.1Introduction

The literature on incompletecontractspresentedin the previouschapterhasallowedus to analyze a number of important economicissuesthat hadbeenleft untouched in the earlierparts of this book,which were devotedto \"classical\" or \"complete\" contract theory. This latter brand of contracttheory, however,hasalsoanalyzed the paradigm ofmutually observablebut

unverifiable information through the lense of mechanism design.Thisapproachhas led to a literature on Nash implementation, similar to the

Bayesianimplementation paradigm consideredin Chapter7.In the Nash implementation paradigm, Maskin (1977)hasmadethe point

that it is in principleeasyfor a mutually observedpieceof information, say,a state of nature 6, to be made verifiable and thus contractableto a third

party: just ask the parties that are known to have observed6 to announceit, and punish them heavily if they disagree in their announcements.This type of mechanism will generate truth telling about 6 as a Nash

equilibrium.This reasoning,however, is subject to several criticisms: (1)While

truth telling is a Nash equilibrium, so is coordinationof announcementson a single state of nature that is not the true one.(2)The punishment ofall agentsis clearlynot in their collectiveinterest:What if the agentsdecided to tear up the contractual mechanism after a unilateral deviationfrom truth telling, that is,what if they decidedto renegotiate?(3)Morefundamentally perhaps,this type of mechanism is not one that is observedin

practice.The implementation literature has mainly focusedon issue1,that is, the

uniquenessproblem.Thisabundant literature (surveyed in Moore,1992)issuccinctly coveredin the next section,with specialemphasison Maskin'stheoremand on its most successfulextension,subgame-perfectimplementation,

due to Moore and Repullo (1988).While this literature has beenquite successfulat characterizinggeneralimplementation results, it is fair

to say, however,that it has not really addressedcriticism 3.Issue2,that is,renegotiation,hasbeenaddressedin a general

implementation contextby Maskin and Moore(1999).It has alsobeenintroducedin

a number of applicationsof the observable-but-unverifiable-informationparadigm that have focusedon the \"holdup\" problemin bilateraltransac-

554 Foundations of Contracting with Unverifiable Information

tions with relation-specificinvestment (following Goldberg,1976;Klein,Crawford,and Alchian, 1978;and Williamson, 1975,1985).As wehighlighted

in the previouschapter,this problemhas beenat the heart of the

preoccupationsof incomplete-contracttheorists.Taking a mechanism-

design perspectivewith ex post renegotiationhas in fact generated anumber of interesting results:As shown in section12.3,it allows one,first,

to rely on implementation techniquesto definewhich outcomescan beimplemented and, second,to take advantage of equilibrium renegotiationto show that the optimum canbeachievedthrough a simple initial contractthat is later renegotiated.

A centralissueraised by incomplete-contracttheory is the importancenot only of observable-but-unverifiableinformation but also of ex antenoncontractableactions.This issuehas beenaddressedin the mechanism-

designperspectiveby Maskin and Tirole(1999a,1999b),who have arguedthat ex ante noncontractability of actionsdoesnot restrict implementabil-ity under certainconditions.Thisquestion, togetherwith the responsebyHart and Moore(1999),basedon Segal's(1999a)work, isalsosummarizedin section12.3.

We then goon to argue, in section12.4,that another route to addresstheconcernsof incompletecontracttheory is to assumenot only ex ante but

alsoexpostnoncontractability of actions.While at first glancethis analysisseemsto govery much \"the incompletecontractroute,\" it isquite standard:the effort variable in a classicalmoral-hazardmodel is exactly an actionthat is noncontractableboth ex ante and ex post!One implication of expost noncontractability of actions,moreover,is that this assumption leadsto a theory of authority, providedone isable to contracton \"who can takethe action.\" Section12.4detailsthe type of resultsone can obtain undertheseassumptions, in the contextof financial contracting and of delegationof authority.

Finally, we concludethe chapterby looking at a repeated-gamesettingwith ex post unverifiable payoffs that providesan alternative foundation

for the notion of authority: While spot-markettransactionscannot takeadvantage of punishment strategiesinvolved in repeatedinteraction, this

statement is not true for an authority relation\342\200\224where one party \"follows

orders\" and is compensated\"fairly\" for executing orders.It is shown in

section12.5that such a relationmay dominate spottransactionsin relationsthat are durableenough.

555 Incomplete Contracts

12.2Nash and Subgame-PerfectImplementation

12.2.1Nash Implementation: Maskin'sTheorem

The key theorem that started the literature on implementation with un-

verifiable information is due to Maskin (1977).Sincein his frameworkindividuals are all assumedto know the stateof nature and there isthus noprivate information betweenthem, implementation is in Nash equilibrium.Maskin starts with a setofstatesofnature 0and a function/correspondencef(6)to beimplemented for eachstateof nature 6e G.Theimplementationproblemis then the designof a gameor mechanism M [with strategies,ormessagesm, for eachplayer i and outcomesgim^ ...,m\342\200\236 ...),belongingto a set of outcomesY] that is to beplayedby the agentsand for which allNash equilibria of the gamebelongto f(0).This isparticularly challengingif f(6)isa function, thereby allowing only one acceptableoutcomefor eachstate of nature 6.

Maskin'stheorem focuses on two propertiesoffunctions/correspondences f(6) to beimplemented.The first one is monotonicity: f(6) is saidtobemonotonicif, for eachpair 0 and 6 belonging to 0and for eachy e Y

such that ye f(6),we have alsoy e f(6) whenever,for eachplayer i andeachoutcomez e Y, if y is (weakly) preferred to z by playeri in state 6,y

is also (weakly) preferred to z by player i in state 6.At first glance,monotonicity seemslike a very natural property for

functions/correspondencesto be implemented:It says that if an outcomey belongs to the acceptableset f(6) for state 6, it also belongsto f(6),the acceptableset for any state 6 for which, relativeto state 6, outcomey

never \"goesdown\" in the ranking of any individual relative to any otheroutcomez- But while this is indeeda natural efficiencycondition, the nextsubsectionwill indicatethat it is in fact quite restrictive in terms ofdistributional objectives.

The secondproperty of functions/correspondencesf{6)to beimplemented is referredto asweak no-vetopower(WNVP):/(0)satisfiesWNVP

if, for eachstate 6, outcomey must be in the acceptablesetf(6)wheneverat most oneagent doesnot have outcomey ashis or her preferredoutcomein Y.

This property is a weak form of nondictatorship, applying to the casewhereall agentsbut one agreeon the most preferredoutcome.It ismuchlessrestrictive than monotonicity, and, in particular, in the presenceof aprivate good, such as money, it is trivially satisfied.Indeed,there is no

allocationof money that is \"the bestone\" for more than one agent at a timeif they all value money.

We are now readyto stateMaskin'stheorem,which consistsin two parts:

(i) Necessary condition: If f{6) is Nash implementable, then it ismonotonic.

(ii) Sufficient condition:With at leastthree agents, if f(6)is monotonicandsatisfiesWNVP, then it is Nash implementable.

Let us first establishthe necessarypart (i) by contradiction.If f{6)is notmonotonic, then there existsa pair of statesofnature 6, 6 and an outcomey such that y e f(6)and y \302\243 f(6) even though, for each agent i and eachoutcomez & Y, if y is (weakly) preferredto z by agent i in state 6,y is also(weakly) preferredto z by agent i in state 6.

However,saying that f(0) is implementable and that y belongsto f(6)means that there existsa mechanism M whoseNash equilibrium set forstate 6includesoutcomey. But this statement meansthat outcomey is alsoa Nash equilibrium in state 6:Saying that y isa Nash equilibrium for state6, with equilibrium vectorof messagesm*, means that outcomey ispreferred to any other outcomeg(mi, m%) for each agent i and messagem,;however,given our assumption on y, 6,and 6,outcomey must alsobepreferred to any other outcomeg(mh m%) for eachagent i and messagem, in

state 6.Consequently,outcomey is alsoa Nash equilibrium in state 6;that

is,y e /(#),a contradictionwhich provesclaim (i).Let us now provethe sufficient condition (ii).Thiscanbeprovedby

construction: Take a mechanism M where the set of messagesfor agent i is{(6,y, n) I 6 e Q,y & Y, n e N), with the following properties:(1)If allannouncements concerning6and y are identicalacrossagentsand y e f(6),then the outcomeg(-)=y; (2) if all agentsbut i announcethe same 6 andthe samey e f(6),while agent i announcesa state 6 and an outcomez,then the outcomeof the mechanism is z if agent i (weakly) prefersy to zin state 6, and the outcomeof the mechanism is y otherwise;and (3) in allother cases,the outcomeof the mechanism is y where outcome y is theone announcedby the agent who has announced the highest positiveinteger n.

Clearly,for each state 6 and outcomey e f(6),y is a Nash equilibriumoutcome.The key questionis, Can an outcome y <\302\243 f(6)also be a Nash

equilibrium? It cannot, as may beseenas follows:

\342\200\242 First, assume that everybody announcesa state 6 and an outcomey, such that y e f(6) but y <\302\243 f(6).Then,by monotonicity, there exist an

agent i and an outcomez such that i (weakly) prefers y to z in state 6 but

strictly prefersz to y in state 6.Consequently,by property (2),i candeviateand imposez, so that y is not a Nash equilibrium.\342\200\242 Second,if y is the result of nonunanimous announcements about state6 and outcome y, then, by property (3), there is at most one agent who

cannot, by deviating alone, impose his or her preferred outcome.(Thiswould be the agent facing other agentswho all play the same strategy.)All other agents can do it, by announcing their preferred outcomeandannouncing an integer higher than the equilibrium announcements of theother agents.Consequently, for y to be a Nash equilibrium, by WNVP, it

must belongto f(6).\342\200\242 Third, by property (3),the argument just madealsoappliesin the caseofunanimous announcements about a stateof nature 6 and an outcomey \302\243

f(0).Claim (ii) has thus beenestablished.

The intuition for Maskin'sresult is as follows.Considerfirst the

necessarycondition (i).It simply saysthat, if an outcomey e f(6)is a Nash

equilibrium of a mechanism in state 6,sowill it be in any state 6 for which this

outcomeremains as attractive relativeto other outcomesas in state 6.Thesufficiency part (ii) is much lessimmediate. It shows how to construct amechanism that uniquely implements any monotonicfunction/correspondence that satisfiesWNVP.

Theidea of the construction isto get rid of undesiredNash equilibria by

appropriately enriching the strategy spacethe agentshave at their disposal.First, it gets rid of the possibility that all agentsagreeon a state 6 and anoutcomey e f(9)in the casethat the true state of nature is not 6.Thisistaken careof by allowing any single agent to imposeanother outcomethat

he or she is known not to prefer to outcomey if the true state is really 6.Bymonotonicity oif(O),this exactly getsrid ofall undesiredoutcomes.Second,the mechanism preventsother equilibria whereall agentseither agreeona 6 and a y \302\243 f(9)or fail to agreeunanimously. Here,the trick used to getrid of this equilibrium is to allow agents to single-handedly imposetheir

favorite outcomeby \"naming the largestintegerof all the integerschosenby the agents.\" This \"integer game\" works becauseany candidateequilibrium involves prespecifiedstrategies,and thus prespecifiedintegers, sothat

a deviation can always involve a largerinteger than thesecandidatestrategies.1 Equilibria thus fail to exist becausethe agentshave beenendowedwith unbounded strategy sets.

1.And it works in all possible casesbecause of WNVP.

558 Foundations of Contracting with Unverifiable Information

12.2.2Subgame-PerfectImplementation

Maskin'stheorem has generateda whole literature about uniqueimplementation under unverifiable information. For a very good survey, seeMoore (1992).Somestudiesmake the implementation problemmoredifficult by excluding unrealistic mechanisms,for example,those that include

integergamesas a way to eliminate undesirableequilibria. Other studiesinstead make the implementation problem less difficult by consideringapproximate(or \"virtual\") implementation or by consideringa subsetofNash equilibria.Among these,subgame-perfectimplementation, pioneeredby Mooreand Repullo(1988),isparticularly noteworthy, becauseit showsthat most desirableoutcomes are in fact uniquely implementable assubgame-perfectequilibria.

As an illustration, we focushere on a public-goodexamplewith two

agents, i = 1,2,and a decisiony e {0,1};y = 1means that the public goodis produced,while y = 0 meansit is not. The decisionrule can alsospecifyassociatednet transfers (tlt t2) e R2.Agent z'spreferencedependson aparameter 6h with the stateofnature 6=(61}&i) e 0.Specifically,agent l'spayoff is Oiy + h, and agent 2'spayoff is &$ + t2. Parameters#i and <% areinterpretedhere as the utilities the agentsderivefrom the public goodnet

of its productioncost.A decisionrule can be dennedas a triple [y(6),h(6),t2(6)].The mono-

tonicity assumption isvery restrictive here,in preventing distributional

concerns from being taken care of.Indeed,-notethat the allocatively efficientdecisioninvolves y(6) = 1if and only if 6i + fy > 0.If one takes a state ofnature (#i, ^) where the decisionis to produce the good and where thetransfers are [h(6),t2(6)],we cannot rule out this outcomefrom thedecision correspondencefor any other state where,for example,the onlydifference with (#i, &i) is that parameter fy has increased.Indeed,in this newstate of nature, the outcome[y(0),h(6),t2(6)]has not gone down in any

individual ranking of outcomesrelativeto state of nature 6.However,distributional concernswould imply that one would naturally want agent 1toreceivea lowernet transfer, sincehis or her utility from the decisionto

producethe goodhas increased.The strength of subgame-perfectimplementation is that the monotonic-

ity assumption is not required anymore: In fact any outcomerule [y(6),ti(6), t2(6)]can be easily implemented as a unique subgame-perfectequilibrium using the following two-stagemechanism:

559 Incomplete Contracts

\342\200\242 Stage1:(a) agent 1announcesa parameter Oi, (b) agent 2 \"agrees,\" in

which caseone goesto stage2,or \"challenges,\" that is,announcesaparameter 0x ^ 6i, (c)when challenged,agent 1then has to choosebetweentwo

options(and that choiceis then implemented):eitheran outcome(jc,tx-At,

-tx-At) or an outcome(z,tz-At, -tz+At), such that At is very largeand

61x+tx>61z+tz

and

G-i_x +tx <6-i,z+tz

\342\200\242 Stage2 (if agent 1hasnot beenchallenged):sameas stage1,but with therolesreversed:agent 2 first announces62,then if agent 1agrees,outcome[y((h, Oz),z\\(#i, fy), t2(Q].,&i)] is implemented; otherwise,move to a step (c)similar to the one in stage1.

The idea of the mechanism isas follows:Each agent isaskedin turn toreveal his or her preferenceparameter,and each is deterred from lying

because,by being challenged,he or she then has to pay the principal anamount At that is chosento be very large.However,the challengedagenthas the possibility to \"get back\" at the challenger,by \"sticking to the initial

choice,\"that is, choosingdecisionjc insteadof decisionz.In this case,the

challengeralso has to pay the principal At, while if the challengedagent\"concedes,\"by choosingdecisionz, the challengerpocketsAt. Themechanism inducestruth telling because,for eachpair ofparameters0,*6n thereexist x, z, tx, tz satisfying the conditionsof step (c)of stage1(this can bedoneby choosingjc = 1and z =0 if 0;> 6>;,and jc =0 and z = 1otherwise).And if the set of statesofnature 0isfinite, there existsan amount At largeenough to simultaneously prevent deviations from truth telling andludicrous challenges.

The strength of subgameperfection is that, in comparisonwith Nash

implementation, we add an off-equilibrium possibility of checkingindividual preferences:We needonly to find any outcomepair for which there isa preferencereversalbetweenthe parameter 0;announcedby agent i andthe parameter0;announcedby the challenger.Thisrequirement istypically

easyto satisfy in economicapplications.The mechanism has severaladditional attractive properties:(1)It is balancedin equilibrium if for eachstate6wehave ^(6)+12(6)=0;(2) if wehad at least three agents,wecouldhavebalancednessevenout of equilibrium, becausein the casewhereboth the

560 Foundations of Contracting with Unverifiable Information

challengedagent and the challengerhave to pay At, they couldpay it to athird agent; finally (3),for the mechanism to work, for eachagent i,weneedto identify only one other agent beyondi that has observedpara-meter0,.

Note, however,that the precedingmechanism reliesto a great extenton the uncompromising faith in rationality of all players that underliessubgame-perfectequilibrium: If in step(a) of stage1,agent 1deviatesfromtruth telling, agent 2 decidesto challengein step (b)becausehe or shehasfaith in the fact that agent 1will make a payoff-maximizing decisionin step(c).Thisreasoningis the key behind equilibrium truth telling in step (a).However,under this reasoning,a deviation from truth telling issure to costagent 1a very large amount At. It then takes quite some confidencefor

agent 2 to think that agent 1will \"comeback to his or her senses\"in step(c)and optimize overa stake that is typically much smaller.Thisis in facta general problem with subgame-perfectequilibrium in games whereplayers take actionsrepeatedly:Deviationsare always consideredto be\"one-shotdeviations from rationality\" that do not shatter the faith playershave in the subsequentrationality of their opponents.However,here, in

contrast to standard game theory, we have specificallydesigneda gameinsteadof analyzing one whoserules are derivedfrom economicstylizedfacts.And wehave specificallychosenthe costof the initial deviation to bevery large in casea challengeoccurs,and as a result, it is really the movein step (a) that is the crucialone for agent 1.Therefore,the precedingcriticism of subgame perfectionisparticularly relevant here.

A secondcriticism of the precedingmechanism isthat, oncean agent hasbeenchallenged,the continuation equilibrium may involve ex postinefficient outcomes,sothat it is subjectto the renegotiationcritique. In the nextsectionwepresentmodelsthat addressthis critique.We do it in the contextof a specificbilateral investment setting.

12.3The HoldupProblem

The notion of \"hold-up problem\" has first beendennedand addressedinthe seminal articleby Goldberg(1976).It has beendevelopedfurther by

Klein, Crawford, and Alchian (1978)and Williamson (1975,1985).Hereweconsiderthe formulation of the contracting problemgiving riseto aholdup problem due to Hart and Moore(1988):Two contracting parties,aprospectivebuyer and a prospectiveseller,canentera relationship in which

they can end up trading a quantity q e [0,1]at a priceP.The utility they

561 Incomplete Contracts

obtain from trading dependson the buyer'svaluation v and the seller'sproduction costc.Theseutilities are uncertain at the time of contracting andcanbe influenced by specificinvestments madeby eachparty at an earlierdate.Specifically,we make the following assumptions:

ve{vL,vH}, withvL<vH and Pr(vH) =;whereinvestment; coststhe buyer y/(J), and

cg{cl,ch},with cl<c#and Pr(cL)= i

where investment i coststhe seller<j)(i). Assume that the two investment-cost functions are increasingand convex,and that they are sunk whateverthe expost levelof trade.The expost payoff levelsare thus

vq-P-y/(J)for the buyer and

P-cq-<j)(i)for the seller.The timing is as follows:First, the parties contract;second,they simultaneously choose their investment levels i and /;third, they both learn the state of nature 6= (v, c);fourth, they executethecontract.

What is the first-best outcome?Assume for simplicity that

CH>VH>CL>VL

Underthis assumption, the expostefficient levelof trade is q =1if 6= (yH,

cL)and 0 otherwise.As for ex ante efficiency, sincethe partiesare assumedto be risk neutral, it is equivalent to investment efficiency;that is, i and jmust result from

max {(/(vH -cL)-v(/)-00}

Assuming an interior solution, the first-orderconditionsgive us the

optimal investment levelsi* and ;'*:i*(yH-cL)=y'(j*)and

/*(v*-cL)= </>'(/*)

Foundations of Contracting with Unverifiable Information

The contracting problem the literature has analyzed is one where thestate of nature 6 = (v, c) and the investment levels i and ;'are not con-tractable,although 6 is observableto both contracting parties ex post.Ifthere is spot contracting expost,after 6 is realizedand investments i and jare sunk, and if the gains from trade at that point are evenly dividedbetweenbuyer and seller,there will be underinvestment in equilibrium aswehave alreadynotedand asFigure 12.1belowillustrates.Thesolidcurvesrepresent the optimal investment functions i* and ;'* while the dashedcurvesrepresentthe bestresponsefunctions under spot contracting:

and

-7(v*-cJ=f(i6r)

The difficulty facedby the contracting partiesex ante is how toformulate an optimal long-term contractthat is independentof 0,which mitigates

'SB

Seller'soptimalinvestment function First best

Buyer's optimalinvestment functionno

Secondbest

Buyer's best responseunder spot contracting

Jsb I*

Figure 12.1Underinvestment with a Holdup Problem

Incomplete Contracts

this underinvestment problem.As explainedin the next subsections,thecontributions in the literature differ in the following respects:\342\200\242 First, they make different assumptions on the extent to which the leveloftrade is contractable:Chung (1991),Aghion, Dewatripont,and Rey (1994,hereafter ADR), and Noldekeand Schmidt (1995)allow for \"specific-performancecontracts,\" where the contract can specifya given level oftrade that partiescanrequestexpost,whether it is efficient or not. Instead,Hart and Moore(1988)focuson \"contracting at will,\" wherecourtsonlyenforcepriceschedulescontingent on the levelof trade,without being ableto identify who was responsiblefor the possiblefailure to trade.This issuewill prove crucial for the ability of the contract to achieve first-bestoutcomesor not.\342\200\242 Second,while all contributions assumethat the partiescannot commit notto undertake ex post Pareto-improving renegotiations,they differ in their

assumptions about the ability to contractually influence the renegotiationprocess.ADR go furthest in contractual \"renegotiation design,\" by

assumingthat relativebargaining powers in renegotiationcan be contractually

chosen.While they focuson specificexogenousbargaining games,Chung,

Noldeke-Schmidt,and Hart-Mooreend up with the same(one-sided)distribution of bargaining powersas ADR. Instead,Edlin and Reichelstein(1996)assumesimpleNash bargaining powersin the renegotiationprocess.It turns out, however, that assumptions about bargaining powersare not asimportant as the distinction betweenspecificperformanceand contractingat will.

\342\200\242

Finally, whereasall the precedingcontributions rule out directexternalities,

that is, any directeffectof the buyer's investment on the seller'scostor of the seller'sinvestment on the buyer's valuation, Che and Hausch(1999)allow for such externalities.Theyshow that not only is the first bestnot generally reachableanymore, but the null contract may be the optimalcontract.And Segal(1999a)obtains a similar result without such directexternalitiesbut for environments that are very \"complex.\"

SpecificPerformanceContractsand RenegotiationDesignLet us first assume that the contract can specify\"default options\" that

partiescanrequestwhenever trade is possible.In this case,one can definethe levelof trade q such that

564 Foundations of Contracting with Unverifiable Information

q(cH-cL)=$'(}*)and one can considerthe following contractual mechanism:Once6 hasbeenrealized,the partiesplay the following game:in stage1,the buyer canmake an offer(P,q) to the seller;in stage2,the selleracceptsthe offer(andtrade takesplaceat these terms), or rejectsit, in which caseq is traded,ata prespecifiedprice P designedto share the ex ante surplus accordingtoinitial bargaining strengths.

This mechanism implements the first best.Indeed,note first that the

buyer has full bargaining powerin the two-stagegame.Shewill thus offerto trade the ex post efficient quantity while leaving the sellerindifferent

between this trade and his default-optionpayoff. \"While ex post efficiencyis guaranteed,what about investment efficiency?The sellerwill anticipateobtaining his default option payoff whatever the ex post levelof trade,sothat he will solve

max {P-icLq-(1-i)cHq-<p{ij\\

By the construction of q, investment leveli* is the seller'soptimal choice,whatever the buyer's investment may be.Finally, since the buyer has full

bargaining power,she is residualclaimant on her investment and solves

max{i* j(yH -cL)-[P-i*cLq-(\\-i*)cHq]-yr(j)}

Shethus maximizes total surplus minus the payoff of the seller(whichdoesnot dependonher investment) and minus her costof investment.

Consequently, she chooses;=;*if the sellerchoosesi = i*.The precedingmechanism thus induces efficient bilateral investment

and circumvents the \"moral-hazard-in-teams\" problem a la Holmstrom

(1982b),that wediscussedin Chapter8.For the buyer, efficient investmentis achievedsimply by making her a residualclaimant.Moreintriguing is thecaseof the seller,who has the appropriateincentive to invest despitehavingno bargaining powerat all.Hisincentive to invest comesfrom his being ableto request the default option, whoseattractivenessriseswhen hisproduction costgoesdown. This option makesthe seller'spayoff sensitiveto hisinvestment, and, through this secondinstrument, both parties can have

properincentives to invest.Theprecedingmechanism is in the spirit of subgame-perfect

implementation,as it is a multistage mechanism with a unique subgame-perfect

equilibrium (where stage 1could be reinterpreted as having the buyer\"announce 0\.")Moreover,in this gameeachparty actsonly once,so that,

565 Incomplete Contracts

in comparisonwith the Moore-Repulloexample,the relianceon backwardinduction is lessobjectionable.Still, the mechanism relieson the ability ofthe partiesto commit to expost inefficient outcomes:If in stage1the buyerhas madea \"crazy offer,\" the sellermay face two expost inefficient

possibilities and no way out of this suboptimal choice.ADR, however, provide a reinterpretation of this mechanism that

involves a much-weakenedability of the parties to commit not to engagein Pareto-improving renegotiations:Assume that, in the absenceof acontract, the parties bargain\342\200\224starting at a date t after 6 has beenobserved\342\200\224

about the terms of trade in an alternating-offer bargaining game a laRubinstein (1982).As is well known, in state 6=(vH, cL)there is a unique

stationary subgame-perfectequilibrium of this game for any pair ofdiscount factors8B< 1for the buyer and 8S< 1for the seller.Indeed,when it

is her turn to make an offerPB, the buyer solves

min PB such that PB-cL>8S(Ps-cL)

Similarly, when it is his turn to make an offerP5,the sellersolves

max Pssuch that vH-Ps>8B{vH -PB)Sinceat the optimum thesetwo inequalities arebinding, uniquenessfollows

(with trade taking placeimmediately at time t). If the sellermakesthe first

offer, the price is

ps_ 1-\302\243b o 1-8S1-SBSS 1-SB8S

If insteadthe buyer canmake the first offer,the price is

pB=S ^~^B 1-fts1-SBSS 1-SB8S

Sincediscount factorsare lessthan 1and vH > cL,the priceofferedby the

buyer is lowerthan the priceofferedby the seller.If, however,both discountfactorsare equaland tend to 1,then both pricestend to (v# +cL)/2,so that

gains from trade are sharedequally. But if one party becomesvery patientrelative to the other (8k ->1while 8t remains boundedaway from 1),he orsheobtains the entiresurplus from trade.

How can the contract influence the bargaining process?First assumethat the contractcan specifya default option (P,q) that each party can

Foundations of Contracting with Unverifiable Information

enforcewhen it is his or her turn to react to the offerby the other party:

Beyondacceptingor waiting one periodto be able to make an offer,it canrequest the default option.This action turns the bargaining game into an

alternating-offer gamewith an \"outside\" option,which hasbeenstudied by

Binmore,Rubinstein, and Wolinsky (1986),who have definedthe so-calledoutside-optionprinciple:Call (UB, Us)the parties'payoffs associatedwith

the outsideoption.Call(C/|,Us)the parties'(expost efficient) payoffs in

the bargaining gamewithout the outsideoption.Then,if (U%, Uf) Pareto-dominates(UB, Us),the equilibrium payoffs of the gamewith the outsideoption are (U$, Uf).If (UB, Us)doesnot Pareto-dominate(UB, Us),theoutcomeof the gamewith outsideoption is expost efficient and givespartyk for whom Uk> Ut his or her outsideoption payoff.

In order to limit the sellerto his default-optionpayoff, one has to makesure this payoff is better for him than the outcomeof bargaining without

the default option.This outcomecanbe ensuredby introducing a penalty

for delayedtrade that the sellerwould have to pay the buyer. If this penaltyis big enough, the sellerwill immediately accept any offer (P, q) that isbetter for him than (P, g),in order to avoid delay.

The first-best outcomecan thus be implemented.Moreover,it can beachievedin a \"light\" way, that is, through a relatively simple contract,consisting in a default option and a penalty for delayedtrade.Simplicity is in

fact achievedbecausethe contractallowsfor equilibrium renegotiation and

only supplements the underlying bargaining structure the agentshave attheir disposalin the absenceof contracting.

One important assumption, however, is that renegotiationstopswhenever the default option has beenchosenby one party. This is reasonableif

thereis,for example,afixedcostto beincurred whenever the sellerproduces,sothat multiple tradesovertime would beexcessivelycostly.In such a case,\"requesting the default option\" simply means that the sellerunilaterallydecidesto produce.In some environments, however,the technology mayallow the parties,after the default levelof trade has occurred,to keepbargaining if the expostefficient tradeis higher than the default trade.This

possibility would undermine the precedingresults,assection12.3.2.2will show.

OptionContractsand Contracting at Will

The original holdup model of Hart and Moore (1988)did not introducerenegotiationdesign,but insteadconsideredthe following bargaining game(which we present in simplified form, following Noldekeand Schmidt,

567 Incomplete Contracts

1995):After 6 has beenrealized,the partiescan simultaneously sendoneanother new written trading offers.Assume now that trade can take onlythe values 0 and 1,so that an offerconsistsin at most two prices,sinceit

can concernonly \"trade\" and \"no trade.\" Assume the initial contractwas apair (P0,Pi) (for no trade and trade,respectively)and call the new offersby the buyer and seller(Po,Pf) and (Po,Pf). Once the stage where the

parties exchangenew offers is over, trade doesor doesnot take place(in a way that we will specify)and a payment is made,possiblyafter theintervention of courts.Theseare assumedto be able to observewhether

trade has taken placeor not and to enforcethe correspondingpayment.This payment is the original one, that is, Pk if a quantity k was exchanged(k =0,1),unless a party finds it in his or her interest to show the court anew written offermade by the other party. Theseassumptions guaranteethat the initial contractprotects each party against unilateral violationswhile allowing for Pareto-improving renegotiations.

In equilibrium, the only offersevershown to the court are thosesent bythe buyer to the sellersaying sheacceptsa higher priceand thosesent bythe sellerto the buyer saying he acceptsa lower price.Why would suchofferseverbe written? Becausethey may be the only way to ensure that

the other party acceptsthe expost efficient trade.The outcomeof this game dependson the ability to enforce default

options,as we now show.

12.3.2.1OptionContracts

Noldekeand Schmidt (1995)allow for specificperformancecontracts,as in

ADR.They consideroptioncontracts,where the sellerreceivesa priceP0if the good is not delivered,and has the option to deliverthe good andreceivean additional payment K (sothat Px =P0+K).

How doesrenegotiationproceedin this setting? Threecaseshave to bedistinguished:

\342\200\242 First, considerthe caseK < cL.Barring renegotiation,the sellerneverhasan incentive to deliverthe good in this case.This is expost efficient

whenever the buyer's valuation is low or the seller'scost is high (or both), in

which caseno trade takes placeand the equilibrium payment made bythe selleris P0.But what happensin state 6= (vH, cL),wheretrade isefficient? Achieving trade requiresraising the premium the sellerreceivesfor

delivery at least up to cL.In fact, the buyer can make sure not to have toraise it further: By sending a letter offering Pf =P0+cL,the buyer induces

568 Foundations of Contracting with Unverifiable Information

the sellerto deliverthe goodand knows the sellerhas the incentive to showthis letter to the court.Indeed,otherwisethe enforcedpricewould be thelowerPi.Moreover,any letter sent by the sellerrequiring a higher pricewould simply not beshown to the court by the buyer. Consequently,just asin ADR, one party has full bargaining power:In this game, the party who

behaves in equilibrium in a way that is suboptimal at the initial contract

priceshas no bargaining powerat all.And sincehere it is only the sellerthat makes the trade decision(unlike in the next subsection),it is alwaysthe buyer who has full bargaining power.\342\200\242 Second,considerthe casecL < K < cH.Barring renegotiation,ex postinefficiency occurswhen both the seller'scostand the buyer'svaluation arelow, since the sellerwould find it profitable to deliverthe good although

vL < cL.Onceagain, the buyer can extractthe full surplus from

renegotiation, by sending a letter agreeingto a higher price for no trade, that is,P^ =P,-cL.\342\200\242

Finally, when cH < K, the selleralways wants to deliverthe good,and this

caseis inefficient when the buyer's valuation is low or the seller'scost is

high (or both).As before,the buyer can extractthe full surplus from

renegotiation by sending a letter agreeingto a higher pricefor no trade,that is,Pq=Pi~c,wherec is the realizedcostof the seller.

This mechanism is similar to the one presentedin the previoussubsection: Thebuyer has full bargaining power,while the sellerreceiveshis initial

contractpayoff, which dependson the value of his costevenin caseswheretrade may not take placeexpost.Let us focuson the casewherecL<K <cH, wherethe sellerobtains P0when his costis high, and P1-cL=P0+K-cLwhen his costis low.Hethus chooseshis investment to solve

max {i(K-cL)-<j)(i)}

Toensurean appropriateinvestment choice,K has to be chosenso that

K-cL= f(z*)

This implies K < cH, sincefrom the definition of the first-best outcome,wehave

<P'(i*)= ]*(yH -cL)<vH-cL<cH-cLGiven the choiceof K, it is optimal for the sellerto make the first-best

investment choice,whatever the investment decisionof the buyer. As for

569 Incomplete Contracts

the buyer, as in the previoussubsection,sheis residualclaimant with respectto her investment

choice\342\200\224having full bargaining power in therenegotiation\342\200\224so shechooses;'= ;'* when the sellerchoosesi = i*. The first-bestoutcome is implemented again, this time with a simpleoption contract(P0,Pi=P0+iQ.12.3.2.2Contracting at Will

Assume now, as in the original Hart-Moore(1988)model,that specificperformance contractscannot be enforcedby courts.Think, for example,that,were the sellerto deliverthe good,the buyer couldalways claim it is notof \"appropriate\" quality. If quality is unverifiable, the court canonly observewhether trade took placeand enforcequantity-contingent priceschedules,but cannot distinguish who is responsiblefor the lackof trade.In this setup,trade takes placeonly if both parties want it to happen.Hart and Mooreusethe following model:After 6has beenrealized,the partiescanexchangewritten messageswith new price offers.Then both parties simultaneouslydecidewhether they want to go aheadwith trade.Only if both agreedoestrade take place,followedby the sameassociatedpayments as before.

Considera simplecontract(P0,Pi).Underthese terms, trade takesplaceif and only if v > Px -P0> c.If this condition is not met, at leastone party

prefersnot to trade,and no trade is the outcome.This is ex post efficient

unless 6= (v#, cL),in which casewe have the following two possibilities:\342\200\242 IfP1-P0>vH>cL,although trade is efficient,the buyer finds it too

expensive. In this case,it is the sellerwho has full bargaining power,sincehe canensure trade by sending the buyer a letter agreeingto P(= P0+ vH. With

this offer,the buyer is ready to trade and is unable to inducethe sellertoagree to any lowerprice.\342\200\242 If vH > cL> Pi -P0,it is now the sellerwho finds trade too costly for the

price difference.The buyer now has full bargaining power,being able toinduce trade by agreeingto Pf = P0 + cL.As before,the full bargaining

powergoesto the party who would benefit from trading at the initial

contracting prices.The precedingreasoningindicatesthat in this examplethere is no loss

of generality in choosingP0 and Px such that vH > Px -P0> cL,sincepricedifferencesthat do not satisfy these inequalities are renegotiatedsoas to

(just) satisfy them. Howmuch investment doessuch a renegotiation-proof

Foundations of Contracting with Unverifiable Information

contractgenerate?The key idea with at-will contracting is that, in caseofdisagreement,the starting point of renegotiationis always no trade,so that

the partiesbenefit from their own investment only when it is efficient totrade.The payoffs are therefore

i]'[vH-(Pi-Po)]-Po-wU)for the buyer, and

P0 +ij[(P^P0)-cL]-(t>(i)for the seller.Consequently,the classicalmoral-hazard-in-teamsproblemarises,with underinvestment as the outcome.Hart and Moorethus providefoundations to the arguments of Goldberg,Klein, Crawford, and Alchian,and Williamson that underinvestment is likely when long-term contractsareincomplete.

Note finally that the precedingresult assumesthat the parties sign asimplecontract(P0,Px).Couldthey improve upon it using messagegames?Hart and Mooreshow that they cannot, sothat, just as in subsections12.3.1and 12.3.2.1,the optimum is achievedthrough a simplecontractthat relieson equilibrium renegotiation.

DirectExternalities

Let us now introducedirect investment externalities,as done by Che andHausch (1999).For simplicity, assume that only the sellercan profitablyinvest in the relationship.Assume, moreover,that the seller'sinvestmenthas an impact not only on his productioncostbut alsoon the quality of the

product.Specifically,the seller'sinvestment influences not only his cost in

that Pr(cL)= /3i but also the buyer'svaluation in that Pr(v#) = yi. The first

bestis then the result of

max{/3yi2 (vH -cL)-(f>(i)}

Assuming an interior solution, the first-ordercondition implies

2j8y/*(vH-ci.)=0,(i*)

As shown by Cheand Hausch,directexternalitiescandramatically affectthe efficiencypropertiesof contracts.To illustrate, assumethe partiescaninitially sign a simple specific-performancecontract (P,q). FollowingEdlin and Reichelstein(1996),assume also that the renegotiation that

follows the investment choiceand the realizationof 6 can be representedby generalizedNash bargaining, leadingto ex post efficiencywith a share

571 Incomplete Contracts

a of the surplus from renegotiationgoing to the sellerand a share (1-a)going to the buyer. The seller'sexpectedpayoff from the contract(P,q)is his default payoff plus a share a of the surplus from renegotiation,that is,

E{v,c)u{P-cq+a[(v-c)q*-(y-c)q]}-<p(i)

whereq* is the first-best levelof trade [that is, 0 unless6 = (vH, cL)].This

expressionis thereforeequal to

P+apyi2(v/f -cL)-[(1-a)E{Vfi)V (cq)+aEiViC)U (vq)]-<j)(i)

or, equivalently,

P+apyi2(vH -cL)-q{(1-a)[cH-pi(cH-cL)]+a[vL+yi{vH-vL)]}-<p(i)

This latter expressionidentifies three effectsfrom an increasein

investment i on the seller'spayoff:\342\200\242 The first effect [a/3yi2(v#-cL)]refers to the fact that the sellercapturesa sharea from the surplus generatedby investing. For a < 1,this in itself isinsufficient to avoid underinvestment.\342\200\242 The secondeffect[q(l- a)fii(cH -cL)]arisesbecause,by investing, thesellerimproves his own default payoff. Just as in the earliercases,this

provides additional incentives to invest and the moresothe higher the default

option q.' The third, countervailing, effect[-qayi(vH-vL)], is due to the fact that,

by investing, the sellerimproves the default option of the buyer. This

externality results in a disincentive to invest. Onceagain, the effectis strongerthe higher the default option q.

The last two effectsare linear in q, which doesnot appear in the first

effect.Raising q thus raisesincentives to invest if and only if

ccy(vH -vL)<(l-a)/3(cH-cL)If this condition is not satisfied\342\200\224which happens if a or 7 is large or /3

is small, for example\342\200\224then setting q = 0 is optimal: The null contractisthe optimal initial contract.One casethat is very intuitive is j3 = 0:Whenthe seller'sinvestment only improves the valuation of the buyer and notthe seller'scost,positivedefault options only improve the buyer'sbargaining positionwhen the sellerinvests more.Consequently,they act as a

572 Foundations of Contracting with Unverifiable Information

disincentive to invest and are counterproductivein comparisonwith thenull contract.2

Che and Hausch,moreover,have shown that no message-contingentcontractcan improve on simple contractsof the form (P,q).This is thus

a general lessonfrom the entire holdup literature we have consideredhere,whether the optimal contractachievesthe first-best outcome(as in

ADR or Noldeke-Schmidt)or not (as in Hart-Mooreor Che-Hausch):Byrelying on equilibrium renegotiation,it is possibleto derivesimpleoptimalcontracts.

12.3.4Complexity

The contractswe have discussedso far in this chapter assumeunverifi-

ability of the state of nature but not of trades,a type of unverifiabilitystressedin the Grossman-Hart-Mooreincomplete-contractparadigm. Onereasontradesmay not be contractable,however,is the excessivecomplexity

involved in specifying exante the nature of transactions:Theexactspecifications of ex post transactions may not be known

yet\342\200\224for example,if we are talking of new productsand if investment concernsR&D.Themechanism-designquestion is then, What can contracts achieve when

actionsare contractableex post but not ex ante, especiallyin \"complex\"environments?

Maskin and Tirole(1999a,1999b)have identified conditionsunder which

ex ante noncontractability of actions is irrelevant, thereby questioningthe incomplete-contractmethodology a la Grossman-Hart-Moore.Theirkey observationis that, while Grossman,Hart, and Mooreassumenoncontractability of actionsexante, they assumethat the payoff consequencesof the various actions that couldbe taken ex post can be foreseen.This

foresight is indeednecessaryin order to be able to make rationalinvestment choicesbeforeuncertainty is realized.But, remarkably, it alsoallowsMaskin and Tiroleto construct a mechanism (which builds upon subgame-perfect-implementationresults) that doesnot requirespecifying actionsexante.Instead,the mechanism specifiestransfers and the right to make offersex post contingent on announcements about the state of nature (and its

payoff consequences).Maskin and Tirolemake an important methodologicalcontribution. Their

approach,however,is rather abstract,and we shall only discussit in this

2.Bernheim and Whinston (1998b)expand on this point in a more general setting.

573 Incomplete Contracts

chapter in the context of a specificholdup model developedby Segal(1999a).Segal'smain goalis in fact to definea notion of complexity of thetrading environment and to relate it to the effectivenessof contracting. Inhis setting, when complexity grows without bounds, contracting losesits

power and we are left with the null contract,just as in Che and Hausch(1999).Segalthus providesa foundation for contractual \"incompleteness\"connectedto the difficulty of specifying in advancethe realizationofuncertainty. In this section,we developSegal'sinsight while relying on theformulation, results, and proofs contained in Hart and Moore (1999),who

simplify the analysis and illustrate and qualify the resultsof Maskin andTirole(1999a).

Assume a contracting problembetweentwo risk-neutral agents, a buyerand a seller.Only the sellercan invest to raise the surplus from trade.Theproblemhas three stages:in stage1contracting takesplace,in stage2 thesellerinvests, and in stage3 the state of nature is observedby both partiesand trade takesplace.Assume it is always efficient expost to tradeone unit

of a good,or \"widget,\" but there is uncertainty exante about which \"type\"

of widget should betraded.Thereare initially N types ofwidgets.Typesarecontractibleex post.One type only should be traded.Callit the \"special\"

widget, generating constant valuation v for the buyer and random costc forthe seller.Exante, c e [cL,cH},with cL<cH<vand Pr(cj,)= i, where <p(i) is,as before,the seller'sinvestment cost.There are thus no directexternalities. The other widgets are called \"generic,\" with productioncost for thesellerequal to3

nc&=cL+-\342\200\224(cH-cL), forn =l,...,iV-l

N

The problemex post is that of recognizingwhich is the specialwidget

among the N possiblewidgets.TheparameterN is thus a measureofcomplexity. As will becomeclear,the key fact will be that, asN becomeslarge,cl\"fills\" the interval [cL,cH].

Assume completesymmetry ex ante acrosswidgets, both in terms of the

probability of being the specialone and in terms of production costwhen

generic.A state of nature is a cost realizationof the specialwidget and apermutation of the N -1genericwidgets plus the specialone.Each state

3.The seller's investment has no impact on the cost of generic widgets. This assumption ismade only for simplicity.

574 Foundations of Contracting with Unverifiable Information

ofnature is thus equally likely. Finally, assumethat c and the cf'sareobservable but unverifiable at stage3.

The first bestinvolves trading the specialwidget in all states of natureand setting investment soas to solve

max {i(y -cL)+(1-i)(v-cH)-<p(i)}i

What happensunder noncontractibility of i and unverifiability of c andthe cf's?Intuitively, the answer to this questionmay dependon whether the

parties can or cannot commit not to renegotiate,and on whether widget

types can or cannot bedescribedex ante.

12.3.4.1No Renegotiation

Even if the widgets cannot bedescribedat stage1(but rememberthey canbeat stage3),the first bestcanbeachievedprovidedthe partiescancommitnot to renegotiate:Just set up a mechanism in which the sellercan makethe buyer a take-it-or-leave-itofferat stage3.Sincehe is endowedwith full

bargaining power,the sellerwill choosethe first-best investment level.This result is consistentwith Maskin and Tirole'sclaim that ex ante non-

describability may not matter: Clearly,sincethe first bestcan be achievedwithout specifying exante any explicitmessage-contingentoutcome,4beingable to perform such ex ante descriptionaddsnothing.

While the no-renegotiationbenchmark is exceedinglysimple, thingsbecomemuch moreinvolved when theparties cannot commit not to engagein subsequentPareto-improving renegotiations.

12.3.4.2Renegotiation

Assume for simplicity that the buyer has full bargaining powerin

renegotiation. CallPi the expectedpriceobtainedby the sellerif his costis c,.Thesellerchooseshis investment i to solve

max {i(PL-cL)+(1-i)(PH-cH)-<p(i)}i

First-bestinvestment obtains if PL= PH, but investment decreaseswhen

PH -Plincreases.In particular, in the absenceofa prior contract,the buyer

4.With one exception: \"No trade\" has to be contractable ex ante, since the buyer must beallowed to decline the seller offer. Maskin and Tirole derive their results under theassumption that there is at least one level of trade (e.g.,no trade) that is contractable ex ante.

575 Incomplete Contracts

offersthe sellerPt = c,expost,which leadsthe sellerto choosei = 0.The striking result from this model is that, even if the widgets can be

describedat stage1,becauseofthe parties'inability to commit not to engagein Pareto-improving renegotiations,the optimal contractimplies i \342\200\224\302\273 0when

N -> \302\260\302\260. The gains from contracting thus vanish when com-plexitygrowswithout boundsand the partiesmight as well not bother to sign any initial

contract.Note that this result generalizesto any distribution of bargaining

powersin the renegotiationprocess,aswell asto bilateralinvestments.Let us now establishthis result. The underlying logicis to observehow

incentive-compatibility requirements translate into post-renegotiationpricesthat dependalmost one for one on the costrealizationof the seller,just as under the null contract.

Specifically,take any mechanism M.Definea stateofnature (L,t) asonewhere the specialwidget costscLand where the N -1genericwidgets arearrangedaccordingto a permutation t. Without lossof generality, take the

specialwidget to be widget 1,and have widgets 2,3,...,N cost

^4(0,-0,ei4(C,-Clx....c1+&,-ei)N N N

respectively.Denotethe equilibrium strategiesof the two parties when

playing mechanism M in stateofnature (L,t) asmB(L, t) and ms(L,t) (that

is, their announcements, truthful or not, of the state of nature that is

mutuallyrevealedto them). Definethe priceP(L,t) as the equilibrium priceat

which the specialwidget istradedin state(L,t), possiblyafter renegotiation.Thisyieldssurplusesv -P(L,t) for the buyer and P(L,t) -cLfor the seller.

Considernow state of nature (H,/),with costcHfor the specialwidgetand a new permutation of the widgets:Thespecialwidget becomeswidget

N, and widgets 1,2,3,...,N -1now cost

l(c\342\200\236_Cl), Cl+1(Ch-ClX...,cl^(ch-Cl)N N N

Denote the equilibrium strategies of the two parties when playingmechanism M in state of nature (H,x')as mB(H, x')and ms(H, t'),anddefinethe priceP(H,t')as the equilibrium priceat which the specialwidgetis traded in state (H,t'),possiblyafter renegotiation.

What doesincentive compatibility of truthful reporting of the state ofnature imply for pricesP(L,t) and P(H,t')? Let us focuson the followingtwo incentive constraints:

576 Foundations of Contracting with Unverifiable Information

1.In state of nature (H,t'),the sellershould not play as if the state ofnature were (L,t).2.In stateofnature (L,t), the buyer should not play as if the stateofnaturewere (H,t').

Both unilateral deviations amount to the players choosinga pair ofstrategies [mB(H, t'),rns(L, t)].Without lossof generality, assume that

mechanism M prescribes,upon such a pair of announcements, a starting

point of renegotiationwhere the buyer has to pay the selleran amount Pand wherewidget n is traded with probability jc\342\200\236,

while no trade happenswith probability

N

i-2>\342\200\236>o

Sincewe assumedfull bargaining powerfor the buyer in renegotiation,the incentive constraint in state of nature (H, t')to avoid deviation 1bythe selleris

P-^xn\\cL+\342\200\224(cH-cL) <P(H,t')-chI /V

\302\253=1

Similarly, in order to avoid deviation 2 by the buyer in stateofnature (L,t), and keepingin mind that the ex post efficient trade will result anyway,

what weneedisfor the sellernot to losefrom this deviation, or

- ^ T n-\\fP~2^xn\\cL+\342\200\224\342\200\224{cH-cL)

>P(L,T)-CL

Thesetwo inequalitiesimply

iV-1P(H,tO-P(L,x)>cH-cL-%^(cH-cL)>^\342\200\224i(cH-cL)

Thiscondition has to besatisfied for any pair (t, t'),and therefore,sinceall permutations are equally likely, the expectedprice the sellerreceiveswhen his costis cH, minus the expectedpricehe receiveswhen his cost iscL,is at least

N-l.\342\200\224icH-cL)

577 Incomplete Contracts

which tends to cH-cLwhen N \342\200\224\302\273 \302\260\302\260. Consequently, the sellerobtains noneof the gains from investing, just aswithout any prior contract.We have thus

establishedthe result.The lessonsof this result are twofold. First, as in Maskin and Tirole

(1999),we have a casehere where ex ante describability of widgets onceagain doesnot matter: dropping it cannot hurt, sincein the limit there isalreadyno value of contracting even with ex ante describability. Second,what matters here is not whether widgets are describableex ante but

whether the partiescancommit not to engagein expostPareto-improvingrenegotiations.

Let us end with an important point concerningthe interpretation of this

result. Theproofhasfocusedon incentive compatibility for statesofnature

(L, t) and (H,if). Thesestatesdiffer in the costlevelof the specialwidget:widget 1,the specialone in state (L,t), costscL,while widget N, the specialone in state (H,if), costscH.More importantly, the two statesofnature are\"extreme\" in the cost differencesof the these two widgets when they arenot special:widget N is the most expensivegenericwidget in state (L, t),while widget 1is the cheapestgenericwidget in state (H,-?).As a result, it

is particularly difficult to prevent the sellerfrom claiming that the state ofnature is (L, t) when it is in fact (H, if) (so that widget 1rather than Nshould be produced)and converselyfor the buyer. In this modelthe costdifferencebetween the most expensiveand cheapest generic widgetsincreaseswith N, the \"complexity\" of the environment. Reiche (2003a)makes the point, however,that the value of contracting in this kind ofsetting can alsogo to zero if the seller'sinvestment is \"ambiguous,\" that is,hasavalue that canbenegative if the wrong expostactionis taken (becauseof technologicalcomplementarity betweenthe investment and the specificwidget to beproduced).Ambiguity works, just like complexity, becauseitgeneratesthe samenegative correlationbetweenthe costof the expostefficient widget and the costof the \"associated\"ex post inefficient widget asin states(H, t) and (H,x').12.3.4.3Describability

Although describability did not matter in the precedinganalysis, Hart andMoorealsoprovidean examplewhereit does.This is a variation on theprevious analysis that has no uncertainty. Assume then, without lossofgenerality,

that widget 1is always the specialwidget, while widgets 2 to N are the

genericones,costing, respectively,

578 Foundations of Contracting with Unverifiable Information

cL+ \342\200\224 (ch-cl\\ cl+ \342\200\224 {ch-cl\\ ...,cL+\342\200\224\342\200\224-{cH

-cL)N N N

Here,with describability, it is easyto achievethe first-best outcome:write

a specific-performancecontractwherethe partiesagree to trade at stage3one unit of widget 1at a fixed price,for example,v.

Instead,without describability at stage1,we are in the samesetup as in

the previoussubsection,with no benefit of contracting when complexitygrows without bound, that is,when N becomeslarge.Indeed,assumethereare N \"names\" at stage1,each of which is equally likely to describethe

specialwidget at stage3, and these are the only ways to describewidgetsat stage1.So,as Hart and Moore (1999,p.125)write, \"Even though the

buyer and the sellerknow at stage1which widget is the specialone,theyhave no words to describeit, other than the N names, any one ofwhich mayturn out to beappropriateat stage3.\"This problemis thus the sameas theone consideredin the previoussubsectionwhere,without commitment

against Pareto-improving renegotiation,there is asymptotically no value ofcontracting when complexity grows.

In contrast to the Maskin-Tiroleresult, describability matters herebecauseof the combinedeffectof renegotiation and risk neutrality: When

renegotiationcan be assumed away, first-best implementation can beobtaineddespiteundescribability, as in section12.3.4.1.When the partiesare risk averse,renegotiation-proofnesscan to some extent becircumvented by \"creating risk,\" and thus expost inefficiency, following somemessages sent by the parties.In such a case,Maskin and Tirole manage toconstruct mechanisms that still achievefirst-best implementation despiteundescribability. As the precedingexampleshows, this result is notpossible anymore under risk neutrality.

12.4ExPost Unverifiable Actions

The previoussectionfocusedfirst on ex ante and ex post contractableactions,and then, in the last subsection,on ex ante noncontractablebut expost contractableactions.We now focuson ex ante and ex postnoncontractable actions.This type of actionshas in fact alreadybeenconsideredearlier:Think of the effort choiceof moral-hazardmodels, a choicethat is

Incomplete Contracts

indeed noncontractible,so that the agent has to be given an incentiveschemein order to exerteffort. While in classicalmoral-hazardmodelsthe

identity of the agent is exogenouslygiven, one can endogenizewho has tomake the effort decision,that is,who has the authority overa given action.Severalmodelsof the literature canbereinterpretedas generalizationsofthe moral-hazardparadigm in order to analyze issuesof allocationofauthority. Thenext subsectionsdiscussin turn optimal financial contractingand the distinction between\"real\" and \"formal\" authority.

Financial Contracting

12.4.1.1Financial Constraints and Contingent Control

To illustrate this point wenow providea somewhat different illustration ofthe Aghion and Bolton (1992)model, which was already discussedinChapter11.Namely, we reinterpretit as a \"complete-contract\"modelwith

expostunverifiable actions.In this versionof the model,at date 0 an

entrepreneur/manager needsan amount K from an investor to start a project,which yieldsa random verifiable profit that canbeeither0 (\"failure\") or 1(\"success\")at date2.The distribution of profit dependson the realizationof the state ofnature 6and on an actionto be taken at date 1,Assume that

at date 1the state of nature 6 getsrevealedand is verifiable.What is notverifiable, however, is the action that is to be taken at date 1.Think, for

example,ofactionslike businessstrategieswhere\"the devil is in the details\"

and wherethesedetailscannot be includedin a contract,so that the only

thing that can be done is to \"put somebodyin chargeof the job.\" Date-0contractscan thus specifydivisions of the final profit aswell ascontrolallocations (that is,who can take the actionat date 1),contingent on the stateof nature 6.

To be specific,assumethat two actionscan be taken: \"Reorganization\"

(i?),which for simplicity yieldsa probability ofsuccesspR whatever the stateof nature, and \"continuation\" (C),which yieldsa probability of successptin state Q,where/ =1,2,3,wherepx <pi<pzand all three 0/sareequiprob-able at date 0.Statesof nature are thus monotonic in the relativeattractiveness of the continuation action.Assume also that the entrepreneurobtains an extra private benefit h under continuation. For example,under

reorganization,shelosesexisting quasi-rents.Let us concentrateon the casewhere

p1+h<pR< p2+h

580 Foundations of Contracting with Unverifiable Information

but

Pi<Pr<Pz

Recall that the investor caresonly about financial returns, while the

entrepreneur\342\200\224who has no financial resourcesto start with\342\200\224has a payoff

equal to her private benefit plus her financial return. Underthe precedinginequalities, the ex post efficient action is C except in state #i, while theactionthat maximizesmonetary financial returns is actionR exceptin state&i- Finally, assumethat the projecthas positivefinancial NPV if the expostefficient actionis taken in all statesof nature, that is,

(pR+p2+p3)/3>KThe first-best contract is the one where the ex post efficient action is

taken at date 1while satisfying the individual-rationality constraint of theinvestor (that of the entrepreneurwill be automatically satisfied given the

assumption that shehas no funds of her own). Howcan one contractuallyachievethe first best?

One can first note that the first best can be achieved through acontingent-control contract, wherethe investor is given controlwhen 6= #iwhile the entrepreneuris given controlin state &i (who has controlin state&i is irrelevant), and where the entire financial return is given to theinvestor. Such a contract,which we alreadydiscussedin Chapter11,leadshere to efficient decisionmaking in all three statesof nature. Moreover,itsatisfiesthe investor'sindividual-rationality constraint becausethe projecthas beenassumedto have a positive financial NPV under ex post efficientdecisions.

Considernow contractsthat have noncontingent control.Entrepreneurcontrolrequiresgiving the entrepreneursubstantial financial returns if expost efficient actionshave to bechosenwithout renegotiation.Specifically,inducing the entrepreneurto chooseactionR in state #i requires

wpR > wpi +h

wherew is the entrepreneur'swagein state of nature fy when the projectsucceeds(it is optimal to give the entrepreneura zerowagewhen it fails).Intuitively, the entrepreneurhas to be given a big enough share of thereturn when the project succeedsto compensatefor the lossof privatebenefit when reorganizationis chosen.But this requirement conflicts with

investor individual rationality if

581 Incomplete Contracts

hPr |+Pi +PzPR-Pi

<3<K

in which casethe projectcannot start.However,insisting on a renegotiation-proofcontractis excessive,because

under entrepreneurcontrol,ex post efficient decisionsare alwaysguaranteed. Indeed,as discussedin Chapter 11,the investor has the financial

resourcesto buy backcontrolto prevent an inefficient expostactionchoiceby the entrepreneur.Note, however,that this type of contract producesequilibrium-contingent control.Moreover,it requiresgiving theentrepreneur in equilibrium at leastan expectedmonetary reward equalto h in stateft. This is the minimum neededto induceher to choosethe expost efficient

actionR and thereby give up private benefit h associatedwith actionC.Thismonetary concessionconflicts with investor individual rationality if

[(pR-h)+p2+p3]/3<KWhile allowing for renegotiationmakes it more likely to meet the

individual-rationality constraint of the investor, there are thus caseswhereit will not work.

What about investor control!Theinvestor'sindividual rationality canbesatisfied in this case\342\200\224for example,by giving him the entirefinancial return

of the project.However,as stressedin Chapter11,the problemis now toavoid inefficient reorganization.To avoid this outcomewithout

renegotiation would requirepenalizing the investor in state 6^ if the projectsucceeds!This approachis indeed the only way to inducethe investor to choosethecontinuation action in this state. Consequently,if we assume (following

Innes,1990)that individual financial payoffs have to be monotonicin thetotal financial return of the project,the contracthas to involve equilibrium

renegotiationin orderto induceexpostefficiencyIn turn, this requiresgivingthe entrepreneursomemoney in state &z to allow her to buy backcontroland choosethe efficient continuation action.Indeed,asstressedby Aghionand Bolton(1992),despitethe fact that there is no asymmetry ofinformation betweenthe parties,renegotiationcan fail to reachexpostefficiencyin

the presenceof wealth constraints. The only way to achieveefficientdecisions in all statesofnature is to give the entrepreneura sumpfl -p2in state&2, in order to allow her to \"buy back\" controlfrom the investor in this state.Through equilibrium renegotiation,this contractthereforeexactlyreplicatesthe outcomeof the contingent-controlcontractdescribedearlier.

582 Foundations of Contracting with Unverifiable Information

12.4.1.2Rationalizing OutsideEquity and Debt

The precedingmodelhas focusedon ex post actions that were not con-tractable.Pursuing the parallelwith moral hazard models,we now add anearlier stagewith an effort choicefor the entrepreneur.This allows us torationalizethe existenceof two outsideinvestors, rather than one as in the

Aghion-Bolton model,as shown by Dewatripontand Tirole(1994a).Consider the following timing:

\342\200\242 At date 0,the entrepreneurobtains funding K, and the financial contractis signed.\342\200\242 At date 0',the entrepreneurchooseseffort a e {aL, aH).\342\200\242 At date 1,first-periodprofit kx e {0,1}is realized,with Pr(^i= 1I aH) =Ph>Pl=PrOi=1I aL).

\342\200\242 At date 1',action A e {C,R) has to be chosen.As in the previoussubsection,action C stands for \"continuation,\" and action R stands for

\"reorganization.\"\342\200\242 At date 2,second-periodprofit n2 is realized.

In terms of payoffs, assumefor simplicity that the entrepreneurcaresonlyabout private benefits and costs,and that (1)effort aH involves an extracostiff

> 0 relativeto effort aL,and (2) as in the previoussubsection,actionCgives the manager an extraprivate benefit h relativeto actionR.As before,however, investors are risk neutral and care only about financial profits.Investorswill thus share among themselvesthe financial profits izx + ih-

As in the previoussubsection,assumethat financial profits are verifiablebut that the actionA and the effort of the entrepreneurare not. Contractscan thus specifydivisions of total profits it\\ + ih as well as allocationsofcontroloveractionA, both potentially contingent on first-periodprofit it\\.

We have here a doublemoral-hazardproblem:The entrepreneurhas tobe given incentives to exert effort, and the investor in controlof actionA

may have to be given incentives to take the properaction.Assume that it

is efficient to implement the high effort aH. Thisactionchoicewill be in theinterestof the entrepreneurif and only if the probability of continuation

dependssufficiently strongly on the effort level,or if and only if

MPr(C|flH)-Pr(C|aL)]>^ (IC)What about the choiceof actionAl Assume that, at date 1,an

unverifiable signal su is realized, simultaneously with first-periodprofit. This

583 Incomplete Contracts

unverifiable signal determinesthe density of second-periodprofit iti. Callthis density Ja(Pi I su) when actionA is chosen,and assumethe following:

\342\200\242 E{piI C,su) -E{%i I R, su) increaseswith su and takesvalue 0 for su= su.\342\200\242 Action Cis riskier than actionR for eachsu in the sensethat, controllingfor the differencein expectations,actionCinvolves an increasein risk a laRothschild and Stiglitz (1970)in comparisonwith actionR.

The first assumption is a normalization: It simply meansthat theunverifiable signals, the su% are ranked accordingto their differencesin expectedsecond-periodprofits acrossactions.The secondassumption is the crucialone:It saysthat continuation leadsto a higher variability of second-periodprofits than reorganization.It is a natural assumption wheneverreorganization involves selling divisions, reducing the size of operations,and thelike\342\200\224that is,actionsthat reduce the varianceof future returns.

Let us now turn to optimal contracting. Take the point of view ofinvestors, who want to induce high effort while maximizing expectedsecond-periodprofits. These profits are maximized by choosingaction Cwhenever su > su. However,if, for the sakeof exposition,the unverifiable

signal su isassumedto beuncorrectedwith entrepreneurialeffort, second-period-profitmaximization then fails to inducehigh effort from the

entrepreneur: Shefaces the same probability of reorganizationwhatever hereffort choice.Consequently, a trade-offarisesbetweeninducing effort and

inducing expostprofit maximization. Sincefirst-periodprofits are assumedto be correlatedwith effort, high effort can be inducedonly if the

probabilityof reorganizationis positively correlatedwith first-period profit. As

in classicalmoral-hazardproblems,the optimal contractminimizes expostinefficiency subjectto satisfying the incentive constraint (IC).Without

computingthe result explicitly, we can say that this contractwould lead to a

second-bestcontractdefinedby a pair (s*0,s$),wheres*0> su > s&and wherefirst-periodprofit ii\\

= 0 leadsto actionCbeingchosenwhenever su > s%0

while first-periodprofit K\\= 1leadsto action C being chosenwhenever

su > s*i.How can different controlstructures achievethis second-bestoutcome

when the actionsand the signal su are unverifiable? First, note that

entrepreneur controlsimply cannot, and is in fact very bad.Indeed,the

entrepreneur, who only caresabout h, then choosesto continue whatever the

signal su, a choicethat is not expost efficient and moreoverfails to induce

584 Foundations of Contracting with Unverifiable Information

high effort. Second,while single-investor control leadsto second-period-profit maximization, it fails to inducehigh effort from the entrepreneur,asexplainedearlier (this conclusionremains true wheneversu is correlatedwith entrepreneurialeffort, but insufficiently so).

A structure that can achievethe secondbestis dual, contingent investor

control, that is,giving controlto an investor with a \"bias\" in favor of actionC after high first-periodprofit realizationand to an investor with a \"bias\"

in favor of actionR after low first-periodprofit realization.5If thesebiasesare strong enough, the entrepreneurchooseshigh effort, in order tomaximize the probability of having high first-periodprofit.

Howdoesone generatesuch biaseswith risk-neutral investors?Since,by

assumption, actionCis riskier than actionR, a bias in favor of actionCisinducedby giving the investor in control a claim that is convexin total

profit, that is, an equity-like claim; similarly, a bias in favor of actionR isinducedby giving the investor in controla claim that is concavein total

profit, that is, a debtlike claim. And sincethe investor in controlhas to begiven incentives to choosethe right action,a secondinvestor is neededtoact as \"budget breaker,\" that is, to receivethe residualprofit. The modelthereforepredictseither noncontingent controlwith the investor in controlholding a contingent claim (similar to equity after nx =1and similar to debtafter nx = 0) or contingent control (equity control after K\\

= 1and debtcontrolafter

n\302\261

= 0) with noncontingent debt and equity claims.To give an example,considerthe following case,where actionR yields

second-periodprofit iiq, = r with probability y and 7% = 0 with probability

(1-7),while actionCyieldssecond-periodprofit 7% =1with probability suand 7V2

=0with probability (1-su).Assume 0 < r < 1,su e [0,su],and su > 7This assumption implies su = jr for some su< su, and if there is a singleinvestor, this is the cutoff usedto decidewhether to continue ornot. Instead,with debt and equity, if there is an amount of debt D, an equity holder in

controlcontinuesif and only if5\342\200\236(1

-D)> y(r -D),or if and only if

r-D F,_

Let us thus considerhaving two investors, an equity holder and a long-term-debtholderwith an amount of debt D*.The variableD* can be set

5.For simplicity, we assume away renegotiation here. SeeDewatripont andHrole (1994a)foran extension with perfect renegotiation.

Incomplete Contracts

to ensuresecond-bestimplementation after high first-periodprofit (^i=1).Specifically,giving controlto equity in this casemeans that reorganizationtakes place if and only if su > sti providedthat D* satisfiessfa = sft (D*).Howdo we then make sure that, after low first-periodprofit fa = 0),reorganization takesplaceif and only if su > s%07 If -s*o< 7, for any debt levelD< r, the debt holder is excessivelytough with the entrepreneur,sincehechoosesreorganizationif and only if su <

y. The equity holder,however,would beready to buy backcontrolfrom the debt holder if su > s%(D),andthe more so the higher the unverifiable signal su. Specifically,the equityholder is ready to makean early repayment of an amount PE toward long-term debt D whenever

su[l-(D-PE)]-PE>7(r-D)Intuitively, paying early means paying with a higher probability, but it

allows the equity holderto take the riskier actionC.Consequently,second-bestimplementation can be achievedafter low first-periodprofit fa = 0)asfollows:Thedebtholderis given controlinitially, unlessthe equity holderpays the debt holder an amount PE toward long-term debt, which is suchthat

sto[l-(D*-Pi)]-Pi=Y(r-D*)Underthis condition, the equity holderbuys backcontrolfrom the debt

holder after low first-periodprofit if and only if sa is such that y > su > s%0,

which yields second-bestimplementation.Though this is only a simple examplewhereexactimplementation of the

second-bestrule happensin a realisticfashion, the morerobust lessonfromthis subsectionis the following: In a world ofnoncontractableactions(concerning both the entrepreneur/managerand the investors)and contractableprofits, one can naturally find a role for the optimal coexistenceof two

financial securitiesthat combinecontingent-controlrights and nonlinearincomerights in a way that resemblesstandard debt and equity.

Formal and Real Authority

Aghion and Tirole(1997)alsofocuson noncontractableactions,which in

their setup amounts to choosinga project that the agent has to work on.Thereare initially N > 3 potential projects.Projectk e {1,2,...,N} givesthe principal a private benefit Hk and the agent a private benefit hk.

586 Foundations of Contracting with Unverifiable Information

Initially, thesevarious projectsare indistinguishable from one another.Onthe one hand, at least one of them is sufficiently bad that choosingoneproject at random is worse for both parties than undertaking no project(with associatedpayoff normalized to zeroin this case).On the otherhand,the partiesknow in advancethat the bestprojectfor the principal giveshim

H> 0 while it gives the agent fih > 0,and that the bestprojectfor the agentgives her h > 0 while it gives the principal aH> 0.Calla and /3 the

\"congruence parameters,\" and assumethat they are positivebut smaller than

one.The higher theseparameters,the more congruent the preferencesofthe two parties.Note that partial congruenceis built into the setup in any

case,sinceone has assumedthat (1)the two partiesagree that choosingaprojectat random is worsethan undertaking no project;and (2) they eachprefer to allow the other party to choosehis or her favorite projectratherthan undertaking no project(sincea and /3 are assumedto bepositive).

Theseassumptions about congruenceare crucialgiven the information-

acquisition technology:While both parties are initially uninformed aboutthe private benefits associatedwith individual projects,they caneachexerteffort to improve their information. Specifically,the principal can at costy/p(E) (increasingand convexin E) becomefully informed with

probabilityE about projectbenefits, while he remains fully uninformed with

probability1-E.Similarly, the agent can at cost y/A(e) (increasingand convex

in e) becomefully informed with probability e about projectbenefits,while

she remains fully uninformed with probability 1-e.The timing of the gameis as follows; Jn stage1,the parties contract;in

stage2, they exert effort to acquireinformation about individual projectpayoffs;finally, in stage3,a decisioncanbetaken on which projecttoundertake, if any. Assuming that efforts areprivately chosenand that partiesonlycareabout their private benefits,6 the contractonly consistsin an allocationof authority for stage3.As Aghion and Tirolestress,what can becontractually

allocatedis solely\"formal\" authority, that is,who has the right to takethe decision.This differs from \"real\" authority, that is, who actually takesthe decision.Indeed,given the partial congruencebuilt into the model,aparty endowedwith formal authority choosesto undertakea projectonlyif he or she is informed about projectbenefits; otherwise,he or she

transfers authority to the otherparty (or equivalently, asksthe other party for a

6. This assumption is made for the sakeof presentation and can begeneralized (seeAghionand Tirole, 1997).

587 Incomplete Contracts

recommendationand follows it). In turn, this otherparty choosesa projector makes a recommendationonly if he or sheis informed about projectbenefits.Otherwise,no projectis undertaken.

Given this continuation equilibrium in stage3, if the contractallocatesformal authority to the principal, the payoffs for the principal and the agent(Up and Ua, respectively)upon choosingtheir effort levelsare

UP = EH+(1-E)eaH-yrP(E)

uA = Efih +(1-E)eh-y A (e)

Theseconditionsreflect the fact that the principal chooseshis favoriteactionwhenever he is informed about individual projectpayoffs, while theagent'sinformation matters only when the principal is uninformed. In stage2, simultaneous effort choiceleadsto the following first-orderconditions:

(l-ae)H=y/p(E)

(l-E)h=xir'A(e)

As indicatedby the secondfirst-ordercondition, higher effort E by the

principal crowdsout effort e by the agent, who understands that her effortmatters with a lowerprobability. Theremay thereforebea gain for the

principalto commit to exerting lowereffort, for example,by choosingan agent

who is more congruent with him (that is, an agent \"with a higher or\.7") Inthis case,indeed,the principal exerts lesseffort, as indicatedby the first

first-ordercondition:Heis lessworried about being uninformed becausethe projectchosenby the agent when she is the only one informed leadstoa lowerrelativelossfor the principal.

Another way for the principal to inducethe agent to work harder is to

delegateher formal authority on project choice(this is equivalent, in theGrossman-Hart-Mooreapproach,to selling the underlying assetnecessaryto get the projectgoing). In this case,given the continuation equilibrium in

stage 3, the payoffs for the principal and the agent upon choosingtheir

effort levelsare

Up =eoH+(1-e)EH-y/P (E)

uA=eh+(l-e)Efih -yA (e)

7. Aghion and Tirole also look at increases in the \"span of control\" of the principal as away for him to commit to spend lesseffort per agent, since he has more projects to acquireinformation about.

588 Foundations of Contracting with Unverifiable Information

The agent now choosesher preferredactionwhenever sheis informed,while the principal'sinformation matters only when the agent isuninformed. Simultaneous effort choicenow leadsto the following first-orderconditions:

0.-e)H=yr'P(E)

(l-PE)h=yfA(e)

A comparisonof these first-order conditions with the ones wherethe principal has formal authority indicatesthat the agent exerts moreeffort and the principal exerts lesseffort under delegation.Indeed,thetwo effort levels are strategic substitutes, reinforcing the fact that theindividual endowedwith formal authority has more incentives to exerteffort, ceterisparibus, sincehe or she canhave the first go at choosingtheaction.

TheAghion-Tirolepaper deliversa rich setofpredictionsabout

authorityin organizations, all this in a very simple model.8From the point of view

of contract theory, it is also a goodexampleof the tractability of modelswith noncontractableactionsbut contractablecontrolallocations.

12.5ExPostUnverifiable Payoffs

In this final sectionwe explorea modelby Boltonand Rajan (2001)where,as before,authority restson an informational advantage, but unlike beforeis sustainedby an ongoing relationbuilt on trust. Authority is modelednotjust as a decisionright or action but as the act of giving ordersthat areexpectedto be executed.The model comparestwo modesof transacting,the negotiation/contracting modeand the authority mode.In the

contracting mode,the servicesor goodsto beprovidedby a \"seller,\" as well as theterms of trade,are spelledout in detail in a spot contract.In the authoritymode,the buyer writes a long-term employment contractwith the seller,specifying only the terms of employment, leaving the details of which

serviceto providein any given periodunspecified.In this mode,the buyerdirectsthe sellerto perform a specificservicein eachperiod.Theselleronlyhas the choiceof executing the order or quitting. There are no ongoingnegotiations about which serviceto provideor at what terms.

8.Seealso Burkart, Gromb, and Panunzi (1997)for a corporate finance application, whereshareholder dispersion acts asa commitment device to \"empower\" management.

589 Incomplete Contracts

What are the costsof contracting that make it possiblefor authority to

supersedethe contracting mode?In the Boltonand Rajan model,the mainsourceof contracting costs is private information and unobservability ofpayoffs. It is assumedthat the buyer (the \"boss\" in the authority mode)knows more about the costsand benefits of a particular servicethan theseller.The latter learnshis true costsfrom carrying out a serviceonly afterthe fact.Thespotcontractcannot specifyterms contingent on realizedcosts,as costs(in utility terms) are essentiallyunobservable.Themost the sellercanhope for then is to obtain compensationfor expectedcostsat the timeofsigning the contract.But a spotcontractwhereterms of tradereflectonly

expectedcostsand not actual costsof a particular servicemay inducethe

buyer to sometimesdemand excessivelycostly services.This is the sourceof inefficiency that an authority modemay beable to overcome.

How can the authority mode overcome this contractual inefficiency?Becausethe authority mode is basedon a long-term contract and an

ongoing relationship, the timing of the seller'spayments canbemademoreflexible.The buyer can now compensatethe sellerwith a bonus after thelatter has carriedout a particularly costly service.Thebuyer'sincentive to

pay such bonusesis supportedby the seller'sthreat to dissolvethe

relationship should the buyer not compensatehim adequately.Given that the

buyer is inducedto always fully coverthe seller'scostsex post, the buyeris also inducedto choosethe action that maximizes net surplus. In otherwords, the buyer choosesthe first-best action in the authority mode,andthus generatesan efficiency gain, which would be lost should the seller(employee)decideto quit. It is the prospectof losing this rent that

preserves the buyer'sincentives to fully coverthe seller'scostsand inducehimto stay.

Tobespecific,considera contractualsituation betweentwo risk-neutral

parties who may interact repeatedlyover time. At any given date t, the

buyer canwrite a contractwith the sellerspecifying the nature of the serviceto beprovided,as well as financial terms. For simplicity there are only two

types of services(or actions)that the sellercanprovide;denotethem by a\\

and a2.At any given time the buyer demandsat most oneof thesetwo

services. Provision of the serviceis observableand verifiable, so that a contractcanbewritten specifying a payment contingent on executionof the action.

The two parties'payoffs are normalized to zero in the event of no trade.If they agreeon the provision of a serviceat at somepricePi (i =1,2),their

payoffs are

590 Foundations of Contracting with Unverifiable Information

for the buyer and

Pi-cCa.,0)for the seller.Thesepayoffs dependon a state of nature 0 (which belongsto a finite set0).Specifically,assumethat

v(A,0)e{vL,vH} and c(al,0)e{cL=0,cH}with

0< cH < vL < vH

In addition, assumethat at eachdate t there is an independentdraw ofa new state 0,with eachstate being equiprobable.As in Segal(1999a),this

assumption capturesthe idea that the precisenature of a servicerequiredby the buyer at any given moment changesover time and in anunpredictable manner.

The setof statescomprisesall possibleconfigurations of payoffs for thetwo actions,{c(\302\253i, 0),c(a2,0), v(\302\253i, 0),v(a2, 0)},so that there are 16distinct

statesof nature. Under theseassumptions, provision ofsomeserviceisefficient in all states of nature. The (first-best)efficient servicein any givenstate is the one maximizing net gains from trade

v(a,-,0)-c(a,-,0)We shall assume,moreover,that

vH-cH<vl

That is, it is more efficient to produce the low-valuation, low-costservicethan the high-valuation, high-costservice.

As explainedpreviously, a central assumption underlying the entireanalysis is that the buyer privately observesthe realizedstate of nature at

any given date beforeany contractis drawn. Thesellerlearnshis true costof providing the serviceonly while providing it. In other words, the buyerhas superiorinformation about the seller'scostofproviding the service.Thecritical informational assumption to obtain an efficient modeof transactionbasedon command and authority is that one of the contracting parties(the one ultimately exercisingauthority) has superiorinformation. If, as isthe casein Chapter7, the buyer had private information about v(ah 0) and

Incomplete Contracts

the selleraboutc(a\342\200\236 6), then authority would not obviously emergeas an

efficient modeof transacting.The two partiesare drawn from a poolof anonymous buyers and sellers.

Once they are matched,they canchooseto engagein a spot transaction orto interact repeatedlyover time in an employment relation.Thereare,thus,two alternative modesof contracting: a market mode and an employmentmode.Under the market mode,the buyer and sellerare drawn from the

poolof buyers and sellersevery period.They meet in the marketplaceandwrite a spot contractfor the provision of a serviceat that period.Following provision of the service,the selleris compensatedand the contractual

relationship dissolves.Under the employment mode,the buyer and sellerwrite a long-term

contract. This is defactoan employment contractwherethe sellerbecomesthe

employeeand the buyer the employer.Thecontractspecifiesthe following:

1.A wagepayment in every period.2.A discretionarybonus to bepaid at the end of eachperiod,which varieswith the seller'scostof performing the prescribedtask.

3.A signing-up fee as well as a severancepayment.

The employment contract doesnot specifywhat tasks the employee/selleris required to perform.Instead,it is a mutual understanding that hewill executethe actionprescribedby the employer/buyer at any time in the

(foreseeable)future, as long asshesticks to the implicit agreementofcompensating him with discretionarybonus payments when costsincurred in

executing prescribedactionshappen to be high. This contractcan beterminated at will by either party, in which casethe contractual relationshipendsforever.

Whichever mode the parties choose,it is assumedthat their presenceinthe market may endforeverat the end of any period t with an exogenouslygiven probability y> 0.This probability of termination servesthe dual

function of discounting the future and measuring the expectedfrequency ofinteraction of contracting parties in this market.

The Spot-ContractingMode

At the beginning of any period,the buyer and sellerwrite a spot contractspecifying the servicethat the selleris to provide,as well as the terms oftrade.The buyer knows her own payoff and the seller'scost of providing

the service.Shemakesa take-it-or-leave-itcontractoffer,which the sellercan accept or reject.If the sellerrejects,he gets a reservationpayoffof u. Thus a spot contract is a pair {a, P(a)},where P(a)denotes the

payment to the sellerfor providing servicea.The spot-contractinggamecan be analyzed as a simple informed-principal problem as in Myerson(1983)or Maskin and Tirole(1990).It couldpotentially have manydifferent solutions.Boltonand Rajan (2001)arguethat a natural solution to this

contracting game canbecharacterizedas the solution to a slightly modifiedex ante problem,which admits a unique solution.

The modified problem is a contracting problem where the buyer andselleragree on an incentive-compatiblestate-contingentspot contractbefore the state of nature is realizedand observedby the buyer. Applyingthe revelationprinciple,an ex ante efficient spot contractis a pair of state-contingent action and compensationschedules{a(6),P[a(6)]}that

maximizes the buyer's expectedpayoff subject to meeting the following

incentive-compatibility and individual-rationality constraints:

max ^Xe{v[a(0),0]-P[a(0),0]}

subjectto

-^2e{F[a(0\\0]-c[a(6),0]}> u (IR)

and,for all(0,0)e0x0v[a(0),0]-P[a(6),0]> v[a(0),0]-P[a{6),0] (IC)

Note first that, in order to be incentive compatible,a contracthas to

specifya constant payment for the same action chosen in two different

statesof nature, that is,

P[a(0),0]=P[fl(0)]

Incentive-compatiblecontractsthus take the form \\a{6), P[a(6)]}.Second,an optimal contracthas to be such that P[a(6)]=P.Toseethis

point, note that any contractwith

AP =\\P(ai)-P(a2)\\>vH-vL

inducesthe samechoiceof serviceby the buyer in all statesof nature: Shesimply picksthe cheaperserviceavailable,say serviceax [if P(fli) < P(a2)-(vH -vL)].The net expectedtotal surplus from such a contractis

593 Incomplete Contracts

-rVL+-vH+-(vH-cH)+-(vL-cH)

or

1 1 1rL+-vH--cH

Thiscontractis dominated by a contractwith AP < vH -vL. Indeed,under

the latter contract,the buyer optimizes in a lexicographicorder, first, by

selectingthe servicein eachstate that maximizes v(a, 6)and, second,when

both serviceshave the samegrossreturn [v(ai, 6)= v(a2, 6)],by picking the

cheaperservice(sincesheis then indifferent). It is easilyshown that thetotal net surplus is then13 1

Note finally that picking the cheaper servicewith the lowerprice (asopposedto the one with the lowercost) is not costminimizing. The total

surplus from trade couldbe increasedin the latter classof contractsby

setting AP =0,so that the buyer, when indifferent betweenthe two actions,couldchoosethe lesscostly service.Straightforward computations indicatethat the total net surplus is then13 34Vi

+4V\"-8C\"

Consequently,the ex ante second-bestcontractis such that

P(a1)=P(a2)=P=-cH+u

Importantly, note that the contractwith P[a(6)]=Pdoesnot achievethesame surplus as a first-best contract,which implements an actionplan tomaximize v(a, 6)-c(a,6).As is easilyshown, the first-best outcomeyieldsan expectedsurplus of

3 5 1rL+-v\342\200\236--cH

In other words, under the ex ante second-bestcontract,there is ashortfall in profits of

3 5 1\342\200\224 Vr + \342\200\224

VH -\342\200\224 CH-8 8 413 3 =^[vl-(vh-ch)]>0

To sum up: the ex ante optimal spot contracttakes a very simpleform:It is a salescontractspecifying a state-independentidenticalpricefor eachservice.Thepriceis setat a levelsuch that the selleris compensatedfor thecostshe is expectedto incur. Under such a contract,an inefficiency arisesbecausethe buyer'sobjectiveis to maximize (v -P) insteadof (v -c).Wenow turn to the employment contractand show that one equilibrium underthis contractmay yield the first-best payoff.

12.5.2The Employment Relationand Efficient Authority

Considernow the situation wherea buyer and sellerhaveelectedto removethemselvesfrom the large pool of anonymous agents that constitute themarket and to enter into a long-term employment contract.Sucha contractentails an obligation on the part of the seller/employeeto carry out the

buyer/employer'sdirectionsor commands.In exchangethe buyer/employercommits to a flow wagepayment w, and alsopromisesdiscretionarybonus

payments b to bepaid in the event that the employeehas to carry out aparticularly oneroustask.Thecontractis openendedand at will. In otherwords,eitherparty is free to quit at any time. In particular, the buyer/employer isfree to fire the seller/employeeshould the latter fail to executehercommands. Thecontracthas an indefinite duration and endseitherwhen oneofthe partiesdecidesto quit or when an exogenousevent occursthat inducesseparation.Recall that the probability of such an event occurring at any

given time is denotedby y> 0.In sum, the employment contractdescribedhere is a simpleform of relational contract as analyzed in Chapter10.

The relation specifiesthe following sequenceof events in every time

period t:

1.The buyer learnsthe state of nature that prevailsin period t and directsthe sellerto providea given serviceat e [a\\, a2}.2.The sellerrespondsby either executing the buyer's demand,therebylearning the costc(at, 6) of doing so,or by quitting, having decidednot toexecuteher order.3.If the sellerexecutesthe buyer'sorder,the buyer must choosewhether

to pay a compensatorybonus to the seller.

Let ht = {(c0,a0),(ci,\302\253i),.. .,(c,_i,a^)}denote the history of theemployment relationup to period t shared.by both parties.It is the sequenceofexecutedactionsand realizedcostsfor the first t periodsof time, beginning

595 Incomplete Contracts

with time periodt = 0 (realizedbenefits couldalsobe part of the history,but these turn out to be redundant information here).Given (ht, 6)where6t is the prevailing state at time period t, the buyer's strategy is a pair{at(9\342\200\236 ht), bt(6t, ht)} and the seller'sstrategy is an execute/quitrule xt(at, ht)e {0,1},wherex =1when the sellerdecidesto executethe prescribedactionand x =0 when he decidesto quit.

As we know from Chapter10,the optimal employment relationalcontract maximizes the expectedsurplus from the employment relation,subjectto satisfying incentive and participation constraints. Obviously, when the

employment contract induces first-best equilibrium play, it is efficient.Boltonand Rajan characterizethe employment contractfor which first-best

equilibrium play can be supportedfor the largestinterval of termination

probabilitiesy. This contractis given by the following:

1.A bonus payment just large enough to inducethe employerto alwayschoosea cost-minimizing action.This requiresthat

b>vH-vL

When this inequality holds, it is not in the employer'sinterest to order ahigh-costactionin order to gain (vH -vL-b).2.A wagepayment just sufficient to ensureparticipation by the employee.Assuming u = 0, this means

w + \342\200\224 (b-cH)>04

3.First-bestaction choice\342\200\224at(6t, ht) =aFB(6t)\342\200\224and promisedbonus

payments\342\200\224bt(6t, ht) =min{c(fl(_1, 0M), b}\342\200\224as long as the employeehas executedall previousorders;otherwise,at(6t, ht) =maxfl v(a, 6t) and bt(6t, ht) = 0.4. Executionof employerorders\342\200\224xt(at, ht) =

1\342\200\224as long as the sellerhasmade the promisedbonus payments\342\200\224fe(_i(0(_i, ht_^)

= min{c(a(_2, 6t_2), b}\342\200\224

otherwiserefusalto carry out the order\342\200\224xt(at, h) =0.The employer'spresent expectedpayoff under the optimal relational

contractis then

lf3 5 1 \\

This expressioncompareswith the presentexpectedpayoff by engagingin a sequenceof spot contractsgiven by

596 Foundations of Contracting with Unverifiable Information

1(1 3 3

7UVL+4V*-8C\"

The equilibrium under the employment relation obtains only if it is in

the employer'sinterestto pay the bonus b =vH -vL whenever the employeewas orderedto choosean actionwith cost cH.If the employerdecidestorenegeon the promisedbonus payment, the employment relationdissolves,so that the most shecanhope to obtain from such a deviation is

b + \\-vL+-vH--cHy V4 4 8

If, however,shedoesnot renege,sheobtains at least

l-y(3 5 1

Consequently,the employermakesthe promisedbonus payment only if

1-7(3 5 1 A L 1-7(1 3 3 A /101N\342\226\240vL+-vH--cH\\>b + ~\\-rvL+-vH--cH\\ (I2.l)

7 V8 8 4 J 7 U 4

If this condition holds, it is a bestresponsefor the employeeto executethe prescribedactionas long ashe continues receivingcompensatorybonus

payments, and otherwiseto quit. Similarly, if this condition holds,it is a bestresponsefor the employerto chooseat(0t, ht) = aFB(6t) and bt(6h ht) =min{c(fl(_1, 9t-\\), b) as long as the sellerhasexecutedall previousorders[andotherwiseto set at(6t, ht) =maxfl v(a, 6t) and bt(0t, ht) =0].

If, however,condition (12.1)doesnot hold, the employment relation isnot sustainable.There is thus a cutoff y definedas

0<r= ^-(vH-cH)][8(yH-vL)+vL-(yH-cH)] K }

such that, if the probability of termination y is smaller than or equal to7, the internal-transaction mode organizedaround an authority relationdominatesthe spot-contractingmode.

Thisresult summarizes in simple terms the main factorsthat underliethechoicebetweenspot contracting and authority:

597 Incomplete Contracts

\342\200\242 The lessdurablethe relation,that is, the higher the probability oftermination 7, the lesslikely it is for a buyer and sellerto engagein anemployment relation.\342\200\242 The larger the scopeof \"exploitation\" of the seller/employeeby the

buyer/employer (as measuredby the sizeof vH -vL), the lesslikely is the

emergenceof an employment relation.

Remarkably, these factors are closelyrelated to those emphasizedbySimon (1951)in his theory of the employment relation (seeChapter11).Finally, note that, by allowing for generalprobability distributions overstates of nature, this theory would alsobe able to stressthat theemployment relation is lesslikely to emergewhen there is lessuncertainty as tothe nature of the servicedesiredby the buyer.

12.6Summary and LiteratureNotes

This chapter has focusedon the observable-but-unverifiable-information

paradigm, and beganwith the pioneeringcontribution by Maskin (1977).Hewas the first to highlight the powerof revelationmechanisms when it iscommon knowledgethat agentsshareinformation that is unavailable to the

principal, becauseone cancomparetheir announcements and reward them

on the basisof this comparison.Maskin, moreover,constructedmechanisms/contracts that achieve unique Nash implementation after havingidentified necessaryconditionsfor implementation. This work has inspireda large literature summarized in Moore(1992).Within this literature, thework by Moore and Repullo (1988)shows that the setof implementablefunctions is significantly largerwhen one movesfrom unique Nashimplementation to unique subgame-perfectimplementation.

This implementation literature has, however,been criticizedon two

grounds: first and foremostbecausethe mechanisms it has identified do not

seemto be usedin real-worldsettings, and secondbecausethey typically

commit individuals to play inefficient outcomeson someequilibrium paths,and are therefore not renegotiation-proof.Subsequent contributions,

however, have shown that the introduction of renegotiation\"may kill two

birds with one stone\":9 Relying on equilibrium future renegotiationpermitsimplementation of optimal outcomesusing simpleand realisticinitial

contracts.10 This possibility hasbeenshown in this chapterin the specificcontextof the holdup problem.In this setting, the specificsof the environment

598 Foundations of Contracting with Unverifiable Information

(potential availability of default options,potential ability to influence

bargaining powersin renegotiation,presenceor absenceof direct investment

externalities)determinewhether first-best outcomescan be achievedornot. As we have shown, one often cannot improve upon simplecontractslike an option to sell(asin Noldekeand Schmidt, 1995),a default optiontied to a penalty for delayedtrade (asin Aghion, Dewatripont,and Rey,1994),or a simple pair of pricesfor trade/notrade (as in Hart and Moore,1988).

In some extremecases,the null contractis even the optimal contract.For example,as shown by Cheand Hausch (1999),it may be inefficient

to sign an initial contractwith a positivedefault levelof trade in a settingwhere the seller'sinvestment directly increasesthe buyer's valuation

for the good,sincethe buyer benefits from the seller'sinvestment throughher ability to requestthe default trade.11An alternative setting in which thenull contractmay beapproximately optimal is the one consideredby Segal(1999a),where the type of good to be traded ex post is unverifiable: Inthis case,when the environment becomesvery complex,asmeasuredby the

unboundedly large number of possibletypes of goodsthat can be traded,the value of contracting goesto zerowhen commitment not to renegotiateis assumedaway.12

The preceding results have been derived under the assumption that

actions(e.g.,the levelor nature of trade)are contractableboth ex ante andexpost.Even under this assumption, the results of Che-Hauschand Segalcan be seen as providing foundations for the Grossman-Hart-Moore

9. Ofcourse, assuming away the parties' ability to commit not to take advantage of future

Pareto-improving opportunities clearly reduces the set of implementable outcomes. For ageneral analysis of implementation with renegotiation, seeMaskin and Moore (1999)and

Segal and Whinston (2002).10.Aghion, Dewatripont, and Rey (2002)make the point more generally that contracts canoften be seenas partially influencing an underlying game that the contracting parties arebound to play in the future. This \"partial contracting\" interpretation should becontrasted with\"classical\" mechanism design a la Maskin (1999)or Moore-Repullo (1988)where the gamethe agents play is completely endogenous.11.Bernheim and Whinston (1998b)consider amore general setting where signing a contractwith fewer clauses (i.e.,with more \"ambiguity\") improves the parties' incentives to behaveefficiently in the relationship.12.Reiche (2003a)explores a setting where investment is \"ambiguous,\" that is, has avalue that can be negative if the wrong expost action is taken. In this case,the value ofcontracting also goes to zero in the absence of increasing complexity. And Reiche (2003b)is the first attempt to analyze the issues raised by Segal in an asymmetric informationcontext.

599 Incomplete Contracts

incomplete-contractparadigm, which strictly limits the power of ex ante

contracting.The argument invoked to justify such limitations is that actionsare contractableex post but not ex ante.As shown by Hart and Moore(1999),what often matters, however, for the powerof ex ante contractingis not whether actionsare contractableex ante but whether parties cancommit not to engagein future Pareto-improving renegotiationof thecontract. This finding is consistentwith the generalresultsofMaskin and Tirole(1999a),who show that first-bestoutcomescanbeachievedunder full

commitment despiteex ante noncontractability of actions.13As shown by Hartand Moore(1999),the irrelevanceof ex ante noncontractability of actionsis not valid, however, if the contracting parties are both risk neutral andunable to commit not to renegotiatethe initial contract.

Theprecedingdebateon the \"foundations ofincompletecontracting\" has

proceededunder the assumption that actionsare noncontractableex antebut contractableexpost.An alternative route is to focuson actionsthat arecontractableneither ex ante nor ex post.Suchactionsare similar to theeffort variable in moral-hazardmodels,which the principal cannevercontract upon:The agent is by assumption \"in charge\" of choosingthe effortlevel.Assuming that actionsare not evencontractableexpostdirectly leadsto a theory of authority, when one is able to choosewho is in chargeof agiven action.We have shown in this chapterhow the pioneeringcontribution of Aghion and Bolton (1992)on debt as a contingent-controldevicecanbereinterpretedquite simply in terms of a \"completecontract\" settingwith exante and ex postnoncontractableactions.Other contributions that

have built on exante and ex post noncontractableactionsto investigateapplicationsin corporate finance or the theory of organizations include

Dewatripontand Tirole (1994a),Legrosand Newman (2000),Hart andMoore(2000),and Hart and Holmstrom (2002).Aghion and Tirole(1997)have enrichedthe paradigm by distinguishing between\"formal authority,\"

the right to take decisions,and \"real authority,\" the ability to take decisions,which often requiresappropriateknowledge.Burkart, Gromb,and Panunzi

(1997)and Dessein(2002)have used this setup in a corporate financecontext,while Dessein(2004)and Aghion, Dewatripont,and Rey (2004)have further exploredthe connectionbetween information transmissionand authority in organizations.

13.Seealso Maskin and Tirole (1999b)and Maskin (2002)for specific illustrations and Tirole(1999)for a general discussion.

600 Foundations of Contracting with Unverifiable Information

We have concludedthis chapterby discussinghow a repeatedgame with

expost unverifiable payoffs canprovidean alternative theory of authority.We have basedour discussionon a modeldue to Boltonand Rajan (2001).This approachis closelyrelated to morereduced-formmodelsof authoritywith implicit cooperationin relationalcontractsby Bull (1987),MacLeodand Malcomson(1989),Wernerfelt (1997),Kreps (1997),Baker,Gibbons,and Murphy (2002),Halonen(2002),and Levin(2003),among others.

Though wehave coveredquite an extensivesetofmodelsin this chapter,let us end by mentioning two additional approachesthat try to endogenizecontractual incompleteness.First, Anderlini and Felli (1994)have exploredthe role of boundson the complexity of contracts,a topicworth pursuingin the future together with other approachesto \"bounded rationality.\"

Second,various papershave considereda signaling explanationof contractincompleteness.For example,a footballplayer may forgo an injury

insurance clausein his contractin order not to signal that he thinks he is

accident-prone (Spier,1992);a supplier may forgo a penalty for breach ofcontractby his buyer in order not to signal that he is afraid of potentialcompetitors(Aghion and Bolton,1987);and, similarly, not signing a long-term laborcontractor debt contractmay signal that one is not afraid to goback on the labor or capital market (Hermalin, 1988;Diamond,1993).Finally, Aghion and Hermalin (1990)have arguedthat legalrestrictionsoncontractsmay limit this kind of (wasteful) signaling.

Marketsand Contracts

Thischapterconsidersa limitation on contracting that is very different fromthe one discussedin the previoustwo chapters:insteadof consideringthe

impossibility of contracting on somestatesofnature or actions,this chapterconsiderslimits on the number of partiesthat canbepart of the samecontract. Specifically,we shall consider competition between multiple

principalsthat eachsimultaneously offeragentsbilateralcontracts.In a way, we

alreadyconsidered,in Part III,a form of intertemporal competition in thecaseof a singleprincipal;for example,in Chapter9,a durable-goodmonopoly's desireto maximize profits in future conflictswith its overallprofit-maximization motive as of now. We were,however, able to reinterpretsuch

intertemporal competition in terms of additional constraints on the initial

\"grand contract.\" Hereinstead,with competitionbetweenprincipals,wearecloserto an industrial organization setup,wherebilateralcontractsare thestrategicvariablesof the game.

This chapter providesonly selectedcoverageof the large number ofmodelsof market competitionwith bilateral contracts.It starts by coveringby now classicalideason market breakdown and existenceof equilibrium

problemsunder adverseselection,work due in particular to Akerlof and toRothschild and Stiglitz. This material considersexclusivebilateralcontracting. The chapter continues with an explorationof competition with

exclusivecontractswhen contractoffersare made sequentially and earlycontractual agreementscanbesetup to createa barrier to entry for future

competitors.The chapter then proceedswith an analysis of nonexclusivecontracting, which has been the subject of more recent work\342\200\224with the

analysis of \"common-agency\" games in particular. Finally, we closethe

chapterwith an analysis of the link betweenproduct market competitionand incentives, treating in turn competition betweenprincipal-agentpairsand the disciplinary role of productmarket competitionon managers.

13.1(Static)AdverseSelection:Market Breakdownand ExistenceProblems

In this sectionwe coverthe classiccontributions ofAkerlof (1970)and ofRothschild and Stiglitz (1976).We offer a unified treatment of their ideasby focusing on a single application:an insurance problem.

Thus, considerthe insuranceproblemwith individuals dennedby their

probability pt of having an accident(with, say, px < p2<...< Pn) and type

pt being representedin proportion c^ in the population of individuals.

Specifically,an individual has wealth w and utility u(w) without an accident,

Markets and Contracts

whereashe suffers an incomelossL and has utility u(w -L) following anaccident.Utility is increasingand concave,so that individuals are ready to

pay an insurancepremium for being fully insured at actuarially fair rates.However,if there is perfectcompetitionbetweenrisk-neutral insurers andfull information, the insurancepremium It paid by individuals of type i issuch that

Ii=PiLIn this case,we know that risk-averseindividuals strictly benefit from

insurance,becausethey obtain a payoff of

u(w -It) > pi u(w -L)+(1-pi)u(w)

The Caseof a SingleContract

Assume now that only individuals know their own types, that is, their own

accidentprobability /?,.Moreover,assume, as in Akerlof (1970),that onlyone type of contract, herefor simplicity afull-insurance contract, is available.In this case,only one premium level/ will emergein equilibrium. For anindividual to accept the equilibrium contract,we must have u(w - I) >

p,u(w -L) + (1-pt)u(w). This means that only individuals with a high

enough risk of accident will seek insurance.Note also that, just asin the analysis of credit rationing under adverseselectionin Chapter 2,raising the price (here, the premium) of insurance worsens the pool ofinformed parties that acceptthe contract.With a finite number of types, atworst there canbe an equilibrium with insurancesolelyfor the highest-risktype pN. For another equilibrium to existwith, say, types p}to pN acceptingthe contract,we need

u(w-I)>pvu(w-L)+(1-pi)u(w) for i =j,j+1,...,Nu(w-I)<piU(w-L)+(l-pi)u(w)for i =l,2,...,;'-land

N N

^aj=y\302\243jaipiL

Intuitively, we needto find a break-eveninsurancepremium / that makesit profitable for all types;to N to acceptto pay /.The challengeis that thelowerthe risk of an accidentfor an individual, the lessvaluable is a given

603 Incomplete Contracts

insurancecontract.Any contract that poolsseveral types has lower-riskindividuals subsidizing higher-risk individuals, for the insurer to breakevenon average.On the one hand, if high-risk individuals are too costly (that is,have very high accidentprobabilities)or too numerous, then lower-riskindividuals will prefernot to be insured at all.On the other hand, if utility

is concaveenough and the p,'scloseenough, all types will pool;in this case,everybody will get insurancejust as with full information.

With adverseselection,things canthus differ from full information in that

(1)low-risk types may prefer not to be insured; (2) in the extremecase,only the highest-risk individuals will be covered(we have a \"degenerate\"

market); and (3) there canbemultiple equilibria; in this case,equilibria arePareto-ranked:any individual who is insured prefersto be pooledwith asmany lower-risk individuals as possible,to enjoy more favorablecross-subsidization;moreover,sinceinsuranceis voluntary, individuals are betteroff in equilibria where they chooseto be insured.

Note, however, that the equilibrium with maximum coverageisconstrained Pareto-efficient:given the constraint of a unique contract,thereis no way to satisfy the zero-profitcondition for insurers and to make it

attractive for lower-risk individuals to join in the pool, becausecross-subsidization would be too unfavorable for them. A government without

better information on individual types, however, could improve total

surplus by subsidizing all insurancecontracts,while having equallump-sumtaxation on all individuals: with perfect competitionamong insurers, the

subsidy would be translated into a lowerpremium, thereforeoffsetting thetaxation for thosewho chooseto be insured; moreover,the subsidy would

increasethe attractivenessof insurance,making more types ready to jointhe poolof insurees.1

The preceding analysis exactly parallels that in the original Akerlofmodel,whereuninformed buyers are facing sellersofusedcarsofunknown

qualities, but where it is common knowledgethat sellersknow the qualityof their own car and wherereservationpricesare increasingin quality, sincethe alternative to selling is to keepusing one'scar. Consequently, just ashere whereraising the insurancepremium worsensthe poolof insurees,inthe Akerlof model lowering the price of usedcars worsensthe pool ofsellers.Therefore,it may becomeimpossibleto trade any other cars than

1.Seethe recent paper by Bisin, Geanakoplos, Gottardi, Minelli, and Polemarchakis (2004)for a general statement of this result.

Markets and Contracts

the worst ones in a world where,under symmetric information, all carswould besoldin equilibrium.

One final remark:In our insurancesetup,we do not have \"full market

breakdown,\" that is, the absenceof trade in the worst possibleequilibrium.This outcomeis possiblein Akerlofsmodelbut doesnot ariseherebecausewe have assumedthat everybody trades under full information.

Consequently, it is always profitable to offer an insurancecontractacceptableonlyto the \"worst\" type. If insteadthe worst type werenot able to trade underfull information, the presenceof adverseselectioncan then give rise to

completemarket breakdown as the only equilibrium.

The Caseof Multiple Contracts

Following Rothschild and Stiglitz (1976),let us now allow for multiplecontractsand screening.In particular, one can introducethe possibility ofdeductibles.Two types (px < p2) suffice to illustrate the main insights fromtheir analysis.What Rothschild and Stiglitz have shown is that (1)if

insurers are allowedto engagein \"cream skimming\" by offering better terms tolow-risk individuals, poolingequilibria will fail to exist;(2)separatingequilibria will alsofail to exist if the proportionof high-risk individuals is low

enough in the population; and (3) if an equilibrium doesexist,it isconstrained Pareto-efficient.

The maintained assumption in Rothschild and Stiglitz's analysis is that

insurancecontractsare exclusive:each individual can take on only a singleinsurancecontract.Section13.3will look at nonexclusivecontracts,while

section13.2will endogenizethe degreeof exclusivity, by allowing forcontractual penaltiesfor switching contractual partners.

The game consideredby Rothschild and Stiglitz is as follows:In stageone,insurers canoffer individuals contractsthat are pairs (/\342\200\236 D),where,asbefore,7, is the insurancepremium while Dt is the deductible.If anindividual of type i acceptsthis contract,sheobtains an expectedpayoff

ptu(w -Dt-I,)+0--Pi )u(w -7,)

In stage two, individuals choosethe bestpossibleinsurancecontract ordecideto remain uninsured, in which casethey obtain an expectedpayoff

ptu(w -L) + (1-pt)u(w).

605 Incomplete Contracts

Rothschild and Stiglitz assumethat insurers are committed to honoringthesecontractsonce they have beenoffered,even if insurers end up losingmoney on them. In equilibrium, however,they have to earn zero profits,becauseinsurers competea la Bertrand:if an insurer is earning positiveprofits by offering somestage-oneinsurancecontracts[say,pairs(/,-,Dt), fori =1,2],an inactive insurer couldtake away theseprofits by offering exactlythe samemenu of contracts,but with a slightly lowerinsurancepremiumfor everybody (that is,ask for 7;-e,with e> 0but arbitrarily small andconstant acrossz's).Thisinsurer would now earn strictly positive profits.

What are the candidate zero-profitequilibria?First, these cannot bepooling equilibria becauseof \"cream-skimming\" behavior.To be precise,assumean activeinsurer is earning zero profits by poolingrisks, that is,by

attracting low-risk and high-risk individuals in proportions/3 and (1-/3)with the contract(Jh Dj),such that:

Ij=[p1(S+p20-[5)lL-D])An inactive insurer couldthen earn strictly positive profits by entering themarket and offering a lower-premium, higher-deductiblecontract (Ik, Dk)that is attractive only to low-risk individuals. For this to bethe case,the newcontracthas to simultaneously satisfy the constraints:

fruiw -Dk-Ik) +(1-p^iiw-Ik) > fruiw -Dj-Ij)+(1-pi)u(w -Ij)and

p2u(w -Dk-Ik)+(1-p2)u(w -Ik)<p2u(w -Dj-Ij)+0\342\200\224 p2)u(w -Ij)Theseinequalities imply that:

(pi~Pi)[u(w -Dj-Ij)-u(w -Dk-Ikj\\ > (p2-pi)[u(w -Ij)-u(w -Ik)]

Sincep2>p\\, it is possibleto find Dk > Djand Ik < I}such that theseinequalities are satisfied.An entrant canthereforemanage to attract only low-riskindividuals. Can it also make positive profits? Yes,and this result can beshown as follows:Considera contract (Ik, Dk) that would leave low-risk

types indifferent in comparisonwith the earliercontract(IpD).In this case,if the new contractis very closeto the old one,that is, if the premium andthe deductibleare not changedmuch, the insurer will make almost the same(strictly positive)profit on low-risk types asunder the poolingcontract,only

606 Markets and Contracts

slightly lessbecauseof slightly worseinsurance.Moreover,it will not losemoney anymore on high-risk types,sincethis contractdoesnot attract any

of those risks.This contractcan thus be profitably offeredby an entrant, aresult which provesthat there existno poolingequilibria.

Can there be separatingequilibria!As we learned in Chapter2, theseclearlyhave to involve full insurance for high-risk types:there is no reasonto distort their choiceofinsurance,becauselow-risk individuals donot have

any incentive to \"pretend\" to be high risk. Due to perfect competition,insurers will earn zeroprofits on high-risk individuals; simultaneously, low-risk individuals will be offeredthe bestpossiblepartial insurancecontractconditional onbeingunattractive to high-risk individuals. Specifically,equilibrium contracts(7lsA) and (72,D2)will have to be such that

D2=0and

I2=piLwhile (7i, A) solves

maxpiu(w -7\\-70+(1-p\\)u(w

-7i)

such that

and

u(w -I2)>p2u(w-A -/i)+(1-P2)u(w -

7X)

This pair of contractsis constrainedPareto-efficientamong the set ofseparating contracts:indeed,they support a \"least-costseparatingequilibrium,\"sinceinsurance is maximized for high risks, and maximized subjectto theincentive constraint for low risks.

The problemis that this pair of contractsmay alsofail to be anequilibrium, in which casewe have nonexistenceof a competitive equilibrium: by

looking at the precedingmaximization program, one cannote first that thecontractofferedto low risks doesnot dependon the proportions\302\253i

and Oiof low risks and high risks in the economy;and secondthat the contractwill

involve a larger deductiblethe larger the differencep2-p\\ betweenlowrisks and high risks.As a result, in order to induceseparationand to avoidthe cross-subsidizationbetween types associatedwith pooling,low risks

607 Incomplete Contracts

may end up facing a lot ofrisk. If the proportionofhigh risks is low enoughin the economy,low risks would in fact bebetter off beingpooledwith high

risks, in order to enjoy full insurance.Moreover,sincehigh risks can onlybenefit from pooling,an inactive insurer could earn positive profits by

entering and offering a single full-insurance contract that would Pareto-dominate the precedingseparatingcontracts.This move would, however,eliminate the candidateseparatingequilibrium. And this outcomewould

imply nonexistenceof equilibrium, becausea poolingcontractcannot bean

equilibrium either,as we have demonstrated.Recallfrom Chapter3 that a similar issuearisesin signaling models:the

Choand Kreps(1987)equilibrium is alsothe least-costseparatingequilibrium, and it canalsobePareto-dominated.It survives asa perfectBayesianequilibrium, however, thanks to the Cho-Krepsout-of-equilibrium beliefs.Hereinstead,in a screeningcontext,there are no off-equilibrium beliefstoconsider:individuals just have to compare two insurancecontractswith

known payoffs.However,our ruling out poolingequilibria, togetherwith the consequent

nonexistenceof equilibrium, doesdependon the fact that we allow firms

to steal customersfrom one another in a \"targeted\" way. This possibilityexistsin the Rothschild-Stiglitz setupbecausehigh risks stick with the initial

poolingcontract,which couldbankrupt the insurer that offersit, oncelowrisks have gone.Severalauthors have proposedsolutions to this

nonexistence problemby modifying this latter assumption:\342\200\242 Wilson (1977)considersthe notion of anticipatory equilibrium; under this

concept,deviations that would becomeunprofitable if the initial contractswerewithdrawn arenot allowed(the initial contractscouldbecomeunprofitable and be withdrawn when they are competing against the deviating

offer).With this restrictionon allowabledeviations, poolingequilibria maysurvive against deviations with separatingcontractsif the deviation causesthe initial poolingcontractto beunprofitable. Indeed,the withdrawal of theinitial poolingcontract then, in turn, causesthe deviating contract to beunprofitable (since following the withdrawal of the initial poolingcontract, the deviating contract attracts all types, and is unprofitable as aconsequence).\342\200\242 Riley (1979)insteaddefinesthe notion of reactive equilibrium; here,onedoesnot allow deviations that would becomeunprofitable if they ledcompetitors to react by adding new contracts.This in fact allows the least-cost

608 Markets and Contracts

separatingcontractto survive as a reactiveequilibrium, becausewe know

that it can be broken only by poolingcontracts,which themselvescan bebrokenby a separatingcontract.

Severalauthors have tried to provide game-theoreticfoundations fortheseanalyses.For example,Hellwig (1987)looksat a game where,in stage1,firms make offersto individuals, in stage2 offerscanbe withdrawn, andin stage3 individuals choosebetweenthe remaining offers.Heshows that

the Wilson equilibrium is a perfectBayesianequilibrium of this game, andeventhe unique \"stable\" one (in the tenninology ofKohlbergand Mertens,1986,who have built upon the \"intuitive criterion\" of Cho and Kreps).Incontrast,Engersand Fernandez(1987)lookat the samegame asHellwig'sexceptthat stage2 is replacedby successiverounds in which firms can addcontractsto the set of contractsalreadyon offer.They show that the Rileyequilibrium is a perfectBayesianequilibrium of this game but that theremay beother equilibria too,which canbesupportedby \"trigger strategies\"similar to thoseusedto support equilibria in repeatedgames.It remains an

open questionwhether either the Wilson or Riley equilibrium would

continue to existin the larger,morenatural, game wherein stage2 offerscanbe addedor withdrawn.

13.2Contractsas a Barrier to Entry

In many market settings buyers do not get all competing contractoffersatthe same time. Often buyers may, for example,already have insurancecoveragefrom an existing insurer when a new entrant appearswith moreattractive terms.Moregenerally, when sellersmake contractofferssequentially,

then early sellersmay attempt to lockcustomersinto long-termcontracts by imposing penaltieson the buyer in the event ofearly tennination.

It isnot obvious a priori that contractsthat lockin customersin this waywould everbe agreeableto buyers. After all, buyers only stand to loseby

limiting competitionamong sellers.As Aghion and Bolton (1987)haveshown, however, theremay still begains from tradefor the buyer and initial

sellerat the expenseof future moreefficient sellersin signing such\"exclusive\" contracts.In short, contractsthat specifypenaltiesfor earlytennination canbeusedto extractefficiencyrents from future entrants.

To seehow this works, considerthe following simple exampleadaptedfrom Ziss (1996).Supposethat there are two transaction periods,t =0 andt =1.In period0 an incumbent firm ean offer a serviceat costCj> 0.A (rep-

609 Incomplete Contracts

resentative)buyer is willing to pay at most v = 1for the service.Supposethat C/ <

\\ for simplicity. This assumption implies in particular that thereare gains from tradebetweenthe buyer and seller.In period1a new entrant

may be able to providethe serviceat cost cE.As there is only one firm in period0, the priceof the servicewill bep0=

1.In period1,however,entry will occurwhenever cE ^ ch ensuring that

competition will take place.Supposethat the incumbent firm has only signedaspot insurancecontractwith the buyer in period 0; then in period 1the

buyer is \"up for grabs\" when entry occurs,and Bertrandpricecompetitionbetweenthe incumbent and the entrant ensuresthat the equilibrium pricewill be/?!=C/.When entry doesnot occur,then the incumbent continues to

chargethe \"monopoly price\" pi =1.Assuming for simplicity that the prior probability distribution of the

entrant's future cost cE is uniformly distributed on the interval [0,1],the

buyer'sex ante expectedpayoff under spot contracting is given by

(l-c/taand the incumbent's payoff is

1-Cj+(1-Cj)2Supposenow that the incumbent and buyer sign a long-term contractin

period0 specifying a pricefor the servicein both periodsp0 and pi aswellas a penalty for early termination d > 0.Under such a contractthe buyerwould agree to switch to the entrant in period1only if the entrant's offerpE is solow that

l-pE>l-p1+d

The probability of entry under such a contractis then

Pr(entry) =Pr(c\302\243 <p1-d)=p1-d

Notice that in a Bertrand equilibrium in period 1the entrant would onlyneedto offer a pricepE =px -d to attract the buyer, so that the buyer'sexpectedpayoff under such a long-term contractis simply (1-p0)+(1-pi).

The incumbent's ex ante expectedpayoff, however, is

p0-cj+(pi -cj)(l-pi+d)+d(p!-d)Given that the buyer always has the option of only acceptinga spot

contractfrom the incumbent, the long-term contract (p0, pl5 d) isacceptable to the buyer only if

610 Markets and Contracts

(l-po)+(l-p1)>(l-cI)cITherefore,the incumbent seller'sproblemis to choose(p0, plt d) to solvethe following maximization problem:

max {p0-ci+(pi-ci)(1-p1+d)+d(px-d)}

(P0,Pl,d)

subjectto

(l-p0)+0-p1)>(l-cI)cINoticethat the incumbent sellercansetp0=1without lossof generality.

Therefore,the incumbent's problemreducesto choosingd to maximize

(1-Cl)+0-dfld-Cl)Cl+d]+d[l-(l-Cl)Cl-d]

Solving for the optimal d, one observesthat

i+q-c,xi-2^2

and

Pr(entry) =p1-rf*=Y

Two important observationsemergefrom this simple analysis.First, it is

always preferablefor the incumbent to sign a long-term contractwith the

buyer, which partially locksthe buyer in. Second,this contract tends toreducecompetitionoverallbecausethe probability of entry under the long-term contractis Cj/2,as opposedto C/under spotcontracting.

Thelong-term contractservesas a deviceto extractpart of the efficiencyrent of the new entrant. As in the classiccontracting problemsdiscussedinChapter2,the optimal contracttradesoff rent extractionand allocativeefficiency. Thedifferencehere is that it is a buyer-sellercoalitionthat doestherent extractionat the expenseof a moreefficient future seller.The generalimplication of this analysis is that \"contract markets\" may not give risetothe sameefficient competitive outcomeas competitive markets for goods.By writing exclusionarycontracts,buyer and sellercoalitionscan distort

competitionand produceinefficient equilibrium outcomes.Theseimplications gobeyondthe precedingsimple illustrative example.

Theyare alsorobust to a number of generalizations.As canbe readilyverified, the long-term contractcontinuesto give riseto the same inefficient

611 Incomplete Contracts

exclusionevenif it canberenegotiatedexpost,as long as the entrant's costcE remains private information. When the entrant's costis publicinformation,

then inefficient exclusioncan still arisewhen there are many buyersand when the entrant gains from increasingreturns to scale.As can bereadily checked,the incumbent sellercan then achieve an equilibriumoutcomewhereevery buyer acceptsa long-term exclusionarycontractin asubgame-perfectequilibrium.

13.3Competitionwith BilateralNonexclusiveContractsin the PresenceofExternalities

We now relaxthe assumption that contractsare exclusiveand considercontractual situations where the principal may freely sign multiple bilateralcontractswith different agents.Thelargerthe number of agentsthe

principalcancontractwith, the more complexthe multilateral contracting game

the parties are involved in. The fine detailsof such a contracting game in

reality\342\200\224who gets to make the first contractualoffer, what happenswhen

multiple offersare made and someare turned down, what is the order ofcounteroffers,and so on\342\200\224may be so complexthat they may. be virtually

impossibleto keeptrack of in a formal analysis of optimal contracting.Accordingly, one strand of the literature has taken a cooperative-game

approachto the analysis of these complexcontracting games,which

suppresses most institutional details of the contracting processand focusesinsteadon \"reasonable\" propertiesany equilibrium outcomemust satisfy

(seeRay and Vohra, 1997,for a notablecontribution in this vein).Another

approachhas beento generalizethe classicalWalrasian competitiveequilibrium framework to allow for asymmetric information and contractingwith nonexclusivecontracts.Underthis approach,the objectiveis tocharacterize equilibrium contractual outcomesand to determinethe existenceofcompetitive equilibria and their efficiency.Someof the most notablecontributions in this line are by Prescottand Townsend(1984),Gale (1992),Dubey,Geanakoplos,and Shubik (1996),Bisin and Gottardi (1999),andBisin and Guaitoli (1998).

In keepingwith most of the earlierparts of the book,we shall coveronlythe theoretical analyses that take a noncooperative game-theoreticapproachto modeling competitionin nonexclusivecontracts.We refer thereaderinterestedin the other two approachesto the works we have cited.

612 Markets and Contracts

The main advantagesof the noncooperativeapproachhave beennicelyformulated by Galeas follows:

Issuesof tractability aside,the (noncooperative) approach remains an idealtowardswhich economictheory ought to strive. Thevirtue of the noncooperative approachis that every detail of the environment ismade explicit. Theextensive form describesall the institutional details of the market, the information that is available to the

players and the actions they can take. It determines a unique feasibleoutcome for

every possibleprofile ofplayer strategies, not just the equilibrium strategies. Finally,it avoids the embarrassment of the invisible hand: everything that happens in

equilibrium canbe attributed to the actions of the players. (1992,p.229)

A potential drawback of this approachto keepin mind, however, is that

it unfortunately predictsoutcomesthat tend to be sensitiveto fine

institutional details,such as who makes the contracting offers,whether publiccommitments are available,and whether the partiesare engagedin a one-shot game or interact repeatedly.It is then always a concernthat an

inadequate institutional representationofa particular contracting situation mayhave led to the wrong predictionson the efficiency of particularcontractual arrangments.

The problemof nonexclusivecontracting arises in many contexts.Forexample,it is common in insurance markets, as has beenstressedin theclassiccontribution by Pauly (1974).Hisarticleis one of the first to arguethat an insuree'sinability to commit not to contractwith multiple insurerscan result in \"overinsurance\" and ultimately undermine the insuree'swelfare.It is alsoa well-recognizedproblemin creditmarkets, where therisk of \"debt dilution\" ariseswhen a borrowercan borrow from multipledifferent sources(see,e.g.,Fama and Miller, 1972;White, 1980).Othersituations include upstream firms supplying inputs to multiple competitors,afirm selling a productwith network externalities,and auctions and biddingwith externalities(e.g.,the spectrum auctions, takeoverbids, etc.).

Rather than analyze one of theseparticular settings, like insurancecontracting or borrowing and lending, we shall attempt to highlight the

principlesand factors that are common to all these situations by analyzing a

generalmodel of contracting betweenone principal and multiple agentsdue to Segal(1999b).

In this contracting problem, one principal can engage in a bilateralcontractual relationwith any of N agentsfacing him

(\302\243

=1,2,...,TV).2Let

2. In the literature on common agency problems the terminology is often the reverse, with asingle agent facing N principals (seeBernheim and Whinston, 1985,1986a,1986b).

613 Incomplete Contracts

xt e Xt cR+ denotethe trade with agent i and let tt the payment from agenti to the principal.Then,x = (xu ...,xN) denotesthe trade profile with all N

agents.Theprincipal'spayoff under this trade profile is given by

5>+/(*)1=1

while eachagent i'spayoff isK,(x)-f,

Importantly, we shall allow for an agent ;'strade x} to imposean

externality on another agent i ^j, in two ways. First, agent/strademay increaseor decreasethe principal'spayoff of trading with agent i. Second,agent;'strade may increase or decreaseagent i's payoff of trading with the

principal.We shall considertwo alternative one-shotcontracting games:

\342\200\242 Theoffergame,wherethe principal makesall the contractoffersand eachagent can only acceptor rejecther own contractoffer.Thisgame,considered in Segal(1999b),is descriptiveof the following contracting situations

in the literature:(1)a largeupstream firm selling inputs to multiple

competitors (as in McAfeeand Schwartz, 1994,or Hart and Tirole,1990);(2) afirm selling a productwith network externalities(as in Katz and Shapiro,1986);(3) an auction of an objectwherepossessionof the objectby oneagent imposesexternalitieson others(as in Jehieland Moldovanu, 1996,orJehiel,Moldovanu, and Stacchetti,1996);(4) a raider attempting a hostiletakeoverof a target firm by making a tender offer to the shareholdersofthe target company (asin Grossmanand Hart, 1980,1988,and HarrisandRaviv, 1988b,1989);(5) a big risk-averseindividual buying insurancefromseveralinsurancecompanies(as in Pauly, 1974,or Kahn and Mookherjee,1998);and finally (6) a borrower taking a loan from severalbanks (as in

White, 1980,and Bizerand DeMarzo,1992).\342\200\242 The bidding game, where the individual agentsmake bilateralofferstothe principal.This is the common agency game analyzedin BernheimandWhinston (1986b).This game-form has beenusedto describecontractingsituations involving (1)multiple manufacturers distributing their productthrough a common retailer (Bernheimand Whinston, 1985),(2)

multipleshareholders writing bilateral incentive contracts with a manager

Markets and Contracts

(Bernheimand Whinston, 1985,1986a),and (3) lobby groups influencing agovernment agency (Bernheim and Whinston, 1986a;Grossman and

Helpman,1994,2002;Dixit, Grossman,and Helpman,1997).We begin our analysis of bilateral,nonexclusivecontracting, with

externalities between a principal and multiple agentsby consideringfirst theoffergame.

The Simultaneous OfferGame

For expositionalpurposesit is helpful to restrict attention to situations

where

1.individual tradesare boundedone-dimensionaltrades:xl e Xt = [0,x].2.all N agentsare identical.

3.the welfareof the contracting parties dependsonly on the aggregatelevelof trade:

/(x)+2u,(x)sw(i>l-) (13.1)

with W(0) > 0 and W\"Q < 0.We considerin turn two contracting scenarios,onewhereall the contract

offersarepublicly observedby all agents,and the otherwherebilateralcontracts betweenthe principal and any agent;'are not observedby the otheragents i *j.As we shall highlight, the observability or nonobservabiliry ofbilateralcontractscan make a critical differencefor the efficiency of theoverallcontracting outcome.

13.3.1.1Publicly ObservableContracts

Theprincipal beginsthe contracting game by making a set ofbilateralcontract offers{(xt, tt)}, i=1,...,N. Eachbilateralcontractthus involves a levelof tradex[ with agent i and a transfer to the principal tr Importantly, noneof the bilateralcontractsis contingent on the levelof trade agreed to byother agents;& i. If we were to considercontractswith someagent i that

are contingent on the tradesagreedto by the other agents,we would

effectively allow for a form of multilateral contract.Thepurposeof this section,however, is to explorewhat would happenwhen the partiesdo not engagein completemultilateral contracting. For expositionalreasonsit is helpful

615 Incomplete Contracts

to focuson the extremecasewhere no form of interdependenceamongagentsis controlleddirectly through somemultilateral contractual clauses.We shall, however,briefly discussan intermediatecasewhen we considermenu auctions.

In practice,most multilateral contracting situations involve a nexus ofbilateral contracts, which attempt to deal with the interdependenceofagents' contracts only in a very limited form. There are many reasonsfor this contractual incompleteness,which we do not attempt to captureexplicitly here.Rather, as in Chapter11,we limit ourselveshere toexploring

the consequencesof this contractual incompletenessfor equilibriumoutcomes.

Having receivedthe principal'scontractoffers,the N agentsfollow by

simultaneously making an accept/rejectdecision.If all agentsaccept their

respectivecontract offers,they get payoffs u,(x) - tt for i = 1,2,...N.Instead,an agent;'who would have unilaterally rejectedher offer would

get a payoff u;(0,x_;),wherex_, denotesthe vectorof tradeswith all agentsother than agent;.

Noticethat given a setof bilateralcontractoffers{(jc\342\200\236 z1,)} there may be

multiple equilibria in the continuation accept/rejectcoordinationgame.This is the case,for example,if the offers

{(jc\342\200\236 U)} are such that, for all agents

Ui(x)-ti > ut (0,x_,-)

but also

ut(xt,0)-U<ut(0,0)

where0 denotesthe vectorx_, =(0,...,0).Whenever the contractoffersinducesuch multiple equilibria in the

continuation game, we shall assumethat all agentsalways coordinateon themost efficient equilibrium. Under that assumption there is no lossofgenerality in restricting attention to contract offers

{(jc\342\200\236 \302\243,)}sucn tnat every

agent i ends up acceptingher contract.For all such contract offers,the

optimal set of offersfor the principal keepseachagent on her reservationutility and specifiestransfers U such that, for all i,

ut (x)-ti=ut (0,x_,-)

Substituting for all the tt in the principal'sobjective,the optimal set oftrades {jc,} is then detenninedby maximixing the objective

516 Markets and Contracts

/(x)+5>1(x)-j;iil(0,x_l) (13.2)i=l 1=1

Comparing objectives(13.1)and (13.2),one then immediately observesthat the optimal setof contractsfor the principal implement a sociallyefficient outcomeif and only if ut(0, x_;) = u,(0, 0) for all i and x_;. That is, theCoasetheoremappliesin this multilateral contracting situation with

externalities if and only if the nontraders remain unaffectedby the tradesof theother agentswith the principal.In other words, whenever bilateraltradesimposeexternalities on nontraders\342\200\224be they positive[dut(0, x_t)/dXj > 0] ornegative [du-^O, x^/dx,< 0]\342\200\224the equilibrium of the bilateral contracting

game may depart from socialefficiency.An inefficiency arisesin this case,as the principal is attempting to extractthe externality rents on nontraders

u,(0, x_j). However,without such externalities,total welfare of the

contracting parties is maximized, as there is no asymmetric information andthereforeno information rents to be extracted.

Underour assumption that only the aggregatelevelof trade matters fortotal welfare,the principal'sproblemcanbereducedto solving

max W(X)-R(X)

where

-N

Xm,(0,x_,)-i=l

N

x*\302\253

i=l

-=x.

When externalitiesare positive (negative),R(X) is weakly increasing(decreasing)in X.Toseethis point, considerthe caseof positiveexternalities and take any X < X.Let x be the vectorassociatedwith X in R(X)and x'the vectorassociatedwith X.We then have ^jct=X and ^c- =X.Definex'{=xt-(X-X')/N,and x\" for the correspondingvector.With

positive externalities,it must then be the casethat

R(X')<2Ul(0,xZi) <2ut (0,x_,)=R(X)i=l i=l

which establishesthat R(X) is weakly increasingin X. The reasoningissimilar for negative externalities.

Therefore,if we denoteby X* the aggregatelevelof trade that maximizestotal welfareand by X the aggregatelevelof trade that maximizes the prin-

617 Incomplete Contracts

cipal'spayoff, it is easy to seethat under positiveexternalitieswe haveX < X*, while under negative externalitieswe have X > X*.3 In the caseofpositive (negative) externalitieson nontraders, the principal prefers tounderprovide(overprovide)the goodorservicein order to reducethe rentsassociatedwith nonparticipation, and thus get agents to acceptcontractswith higher transfers.

This general result helps unify a number of disparate insights in theliterature on multiagent contracting:

1.In the absenceofexternalitieson nontraders\342\200\224for example,in the caseofinsurance provision by multiple insurers or in the caseof a new technologywith network externalitiesand no prior substitutes\342\200\224there is no reason toexpectthe contracting outcomein a publicly observablecontracting processto be inefficient, sincethe principalcan extract all the rents associatedwith

equilibrium tradeswithout inducing any distortions.

2.In the caseofpositiveexternalitieson nontraders,asin a hostiletakeoverwhereefficiency is increasingin the proportionof sharestendered to theraider,4the principal is likely to commit to tradesthat are lower than would

besociallyoptimal, in order to reduceequilibrium externality rents to non-traders.3.In the caseofnegative externalitieson nontraders, as in the caseof input

provision by an upstream firm to competing downstream firms (anapplication that we will detaillater),the principal is likely to contracton tradesthat are higher than is sociallyoptimal, in order to reduceall agents'equilibrium externality rents.

13.3.1.2Privately ObservableContracts

In some of the applicationsthat we have mentioned it is unlikely that

the principal is actually engaging in publicly observable contracting

negotiations or is able to commit to observablelevelsof trade with eachofthe agents.Accordingly, the contract-theoryliterature has exploredthe

implications for equilibrium outcomesof secretbilateral contracting in

3. Things are in fact somewhat more complicated than it appears at first: as pointed outby Segal (1999b),in many applications X* and X will not be single-valued. The result hederives, however, points to equilibrium trades that are lower (higher) than the level oftrade that maximizes the total welfare of the contracting parties in caseof positive (negative)externalities.

4.SeeBurkart, Gromb, and Panunzi (1998)for a model of takeovers with this property.

618 Markets and Contracts

multilateral-contract settings involving externalities.In particular, Hartand Tirole(1990)and McAfeeand Schwartz (1994)investigate the caseofan input provider contracting with several downstream competitors,and Katz and Shapiro(1986)look at the caseof an industry with network

externalities.This section covers some of the main lessonsfrom this

literature.We now assume that each agent i observes only her own contract

offer (xjt).When agent i receivesan offer from the principal, she now

has to guesswhat other offersthe principal has made to the other agentsto decidewhether or not to accept her own offer

(*,-,\302\243,). Furthermore,the principal'soffer to agent i may be a signal of the form of the offersto agents;^ i, so that the contracting game with bilateralsecretoffersis, in effect,a complexsignaling game.We know from Chapter3 that

signaling gamesmay have a plethora of perfect Bayesianequilibria when

out-of-equilibrium beliefsare not restricted.To be able to make sharperpredictionsabout equilibrium outcomes,several restrictionson out-of-equilibrium beliefshave beenconsideredfor contracting gameswith secretbilateralcontracts(seeMcAfeeand Schwartz, 1994,for an extensivediscussion of reasonable out-of-equilibrium beliefs for such contracting

games).Themost common restriction(or \"refinement\") that hasbeenconsidered is the notion of passivebeliefs, first introducedby McAfeeandSchwartz (1994),which amounts to saying that, when observing an out-of-equilibrium contractoffer (x-,t[),agent i believesthat all other offersremainthe equilibrium offers.

We follow the literature here and characterizeequilibrium outcomesofthe contracting game when out-of-equilibrium beliefsare restrictedto bepassive.Underpassivebeliefs,when (i,t) denotesthe equilibrium contractprofile, agent i acceptsan offer

(jc\342\200\236 tt) if and only if

u,(j:I,i_I)-fI>MI(0,i_/)

Restricting attention, asbefore,to equilibrium outcomeswhereall agentsaccept their contractoffers,the principal choosesequilibrium tradesxl tomaximize the payoff

2>,+/(x)

subjectto eachagent'sparticipation constraint.

619 Incomplete Contracts

At the optimum again, all the participation constraints are binding.

Substituting for all the tt in the principal'sobjective,we find that optimal tradesin a passivebeliefequilibrium are given by

N N

Xi eargmaxgU,i_,)=/(x)+ \302\243Ui(*,,i_,)-

\302\243ut(0,i_,)

*\342\200\242 1=1 i=i

As always, when one imposesa sharp restrictionon out-of-equilibriumbeliefs,as we have donehere,the questionariseswhether a refined perfectBayesianequilibrium existsat all.As onemight expect,it ispossibleto showthat when g(x, x) is a continuous function and is quasi-concavein x, then apassivebeliefequilibrium doesindeedexist.

Note that here the reservationutilities u,(0, x_,) enter as constantsanddo not affect the equilibrium solution {je,}.This fact implies, in particular,that for every agent i the equilibrium trade % satisfies

xt earg max /(jc,, x_,)+ut (xt, i_,-)

Thiscondition for optimaliry is referredto aspairwiseproofnessby McAfeeand Schwartz (1994).The optimal trade for an individual agent cannot bedirectly altered by changing the trades offeredto the other agents.Thisisa critical differencewith the previouscontracting problem,wherecontractofferswerepublicly observable.Now eachbilateralcontractmaximizes thebilateral surplus between agent i and the principal.This result doesnotmean, however,that the overallmultilateral surplus is maximized. Indeed,total surplus is not maximized in the presenceof externalities at equilibriumtrade levels,for then it is not possibleto adjust an individual trade levelspecifiedin a bilateralcontractto changesin other agents' trade levels.

There is one specialcasewhere the welfareoptimum is attained.If atthe welfare-maximizing trade vectorx* all agents' payoffs are such that

ufe* x_() is independentofx_\342\200\236

then this trade vectorcanbe supportedasapassive-beliefequilibrium. Toseethis point, note that, by the definition ofa passive-beliefequilibrium x, we have

/(x)+|;u[.(x)>/(x*)+|;it[.u*,x_[.)i=i i=i

The condition that at x* all agents'payoffs are such that ufe% x_,) is

independent of x_, then implies that

620 Markets and Contracts

/(x*)+f>,(**,i_) =/(x*)+J>,(x*)1=1 1=1

Sincex* maximizes

N

/(x)+2\"<(x)1=1

x* is thereforealsoa passive-beliefequilibrium.In general,the distortion relativeto the welfareoptimum dependson the

sign of the externalitiesat the efficient tradelevel.Assuming again that onlythe aggregatelevelof trade matters for the total welfareof the contracting

parties,Segal(1999b)shows that when externalitiesarepositive(negative),aggregateequilibrium output X is below(above)the sociallyefficient levelX*. We leave it to the readerto verify this simple claim.

Finally, note that when one compares the outcomesunder publiclyobservableand privately observablecontracts,the relevant externalitiesthat the principal is attempting to internalize and appropriateare different.In the former contracting problem,the relevant externality to determinethe directionof the distortion ofequilibrium outcomesrelativeto efficiencyis the one on nontraders, while in the latter contracting problemtherelevant externality is the one affecting all tradersat the sociallyefficient tradelevel.Giventhat different externalitiesare involved, it is not too surprisingthat in generalthe ranking of the two contracting outcomesin terms ofefficiency is ambiguous. That is, in generalit is not possibleto say whether the

nonobservability of bilateralcontractsresults in greateror lessefficiency.An examplewill make this comparisonmore concrete.

13.3.1.3Application: Input Provision to CompetingDownstreamFirms

This exampleis adapted from Hart and Tirole (1990)and McAfeeandSchwartz (1994).An upstream firm, with a monopoly on an efficient input

productiontechnology, producesinputs at unit productioncosts,which wenormalize to zero,and may selltheseinputs to potentially two downstreamfirms (1and 2),who competea la Couraot in the final goodsmarket. Theinverse demand function in the final goodsmarket is the simple lineardownward-sloping function

p =l-x-i,-x2

621 Incomplete Contracts

wherexl is the output of downstream firm i = 1,2.One unit of output foreach downstream firm requiresone unit of input. While each downstreamfirm may purchasethe required inputs from this efficient upstream firm, it

can alsopurchaseits inputs from an inefficient competitive fringe at unit

price p > 0.Clearlyhere it is sociallyefficient for both downstream firms to procure

their inputs from the efficient upstream firm. Moreover,the joint profits ofall three firms,

n=(1-x1-x1){x1+x2)

are maximized by setting

(x1+x2)=-In our previousnotation this exampleis such that/(x) =0,sincecostsare

zero for the upstream firm, and

ul(xl,Xj) =(X-xl-Xj)xiWhat about externalitieson nontraders?They may arisehere as a result

of the downstream firms' option to buy inputs from the inefficient

competitive fringe. Specifically,if a downstream firm i rejectsthe contractfromthe upstream firm and expectsits competitorto producexp it purchasesinputs from the competitive fringe to maximize its profits

(l-xl-x]-p)xlso that its bestresponseas a \"nontrader\" with the efficient upstream firm

is given by

xl =max{0,(1-Xj -p)/2}and its payoff as a nontrader is

Ui (0,Xj) =[max{0,(1-xj -p)/2}]If the competitive-fringe productioncostsare sohigh that it is not

profitable for the downstream firms to procureany inputs from the fringe, then

the nontraders' payoff is always zero and there are no externalitiesoftrading by one downstream firm on the nontrading firm. In that case,contracting with publicly observablebilateralcontractsresults in an outcomethat maximizes the joint profit of all three firms.

622 Markets and Contracts

Otherwise,the equilibrium under publicly observablecontractsisinefficient and is given by the solution to

max 2 0--xi -Xj)xt -(1-x} -pf/4*\"*' i,;=l,2,i*;

As can easilybe seen,this solution involves setting (jcx +x2) above theefficient levelof ~.Specifically,equilibrium tradesare then given by

x -x -v'-I_\302\243X\\

\342\200\224 X2 \342\200\224 X \342\200\224

Note that this solution requiresp <\\, for when p =

\\ then x'= -j, so that

productionusing outsideinputs ceasesto be advantageousfor a nontrad-

ing downstream firm. Theupstream firm can thus sellhalf of the monopolyquantity at a priceof tt =~,which leaveseachdownstream firm with a zerorent if both acceptthis offer.For lowerlevelsof p,however,the upstreamfirm is better off expanding the volume ofsalesbeyondthe levelthat

maximizes joint profits, in order to reduce the rents eachdownstream firm canobtain by exploiting the outsideoptionof procuring inputs from the

competitive fringe.As Hart and Tirole(1990)have highlighted, under privately observable

contracts the upstream firm's monopoly position is undennined by its

inability to commit not to supply more than the monopoly inputs tothe downstream firms. To seethis point, observe that under privatelyobservablecontracts the upstream firm choosesthe level of trade with

downstream firm i to maximize total bilateralprofits from the trade with

firm i, taking as given the expectedlevel of trade with downstream firm

;'(which we can denoteby x,).Thus the upstream firm is setting (jc\342\200\236 tt) tosolve

max?,

subjectto

u,(*I,5:y)-fi>MI(0,5:y)

Downstreamfirm i'sparticipation constraint canbe rewritten as

(1-Xi -Xj )xt -*,>(!-Xj -pf/4

Incomplete Contracts

Sincethe expectedlevelof trade x} is like an exogenousparameter (in apassive-beliefequilibrium), the solution to the bilateralprofit maximization

problemis

1-X;x,=\342\200\224

By symmetry, we alsohave

1-Xix>=~Solving these two equationsso that % =

x\\ for i = 1,2, we obtain the

equilibrium trades x under privately observablecontracting. Thisequilibrium outcomeis none other than the Cournotcompetitionoutcomewith

Xi = Xz=^.

Comparing the outcomesunder publicly and privately observablecontracts, we thus observethat the joint profits of firms are higher under

publiclyobservablecontracts,as long as p > 0.Thisis not to say, however, that

welfareis maximized under publicly observablecontracts.Moreover,while

firms are better off under publicly observablecontracting, consumersareclearlybetter off under privately observablecontracting.

The SequentialOfferGame

In many multilateral contracting situations it is unreasonableto think that

the principal will engage in bilateral contracts simultaneously with all

agents.It is more likely that the principal first signs up one agent andthen turns to a secondagent and so on. For example,an insuree maycontractfirst with a primary insurer and in a secondstep take onsecondary

insurance.While severalof the insights obtainedin the previoussectionalsoapply

to situations where the principal contractssequentially with agents, thereare also some important differences,which we highlight in this section.In particular, a major differenceunder sequentialcontracting, which has

important consequencesfor equilibrium outcomes,is that the first agent(s)to sign a contractwill take into accountthe potential impact of a changein

their contracton future contracts.This assumption is in contrast with the

assumption ofpassivebeliefs,whereeach bilateralcontractis determinedas if it had no impact on other trades.

624 Markets and Contracts

Much of the discussionin this sectionis basedon the simple and eleganttreatment of an insurance-contractingproblemwith moral hazard undernonexclusivecontractsby Kahn and Mookherjee(1998).Considera risk-averseprincipal facing N perfectlycompetitive risk-neutral insurancecompanies (agents).Without insurance, the principal has wealth w in the eventofno accidentand (w -L) in the event of an accident.Hisutility of wealth

is a strictly increasingand concavefunction u(-).Theprincipal canexercisecaution to reduce the risk of an accident.If he is cautious, the probabilityof an accidentis pL, while if he is not, it is pH > pL.The principal incurs autility cost y/ > 0 for being cautious.We assumethat being cautious isprofitable for the principal in the absenceof insurance.That is,

pLu(w -L)+(1-pL)u(w) -y>pHu(w -L)+(1-pH)u{w)

We shall also assumethat when he is perfectlyinsured at actuarially fair

terms his payoff is higher when he can commit to be cautious:

u[pL(w -L)+0--pL)w]-\\)/>u[pH(w -L)+(1-pn)w]

Thus, under first-best contracting, the principal is perfectlyinsured andexercisescaution.Insurancecompaniescompeteto supply insurance andmake zeroprofits in equilibrium. Theprincipal then obtains a payoff of

u[pL{w -L)+(1-pL)w]-yUnder second-bestcontracting, when the principal'scautiousnessis not

observableand he cannot commit to becautious,we know from Chapter4that perfect insurance is generally suboptimal. Consider,first, thebenchmark situation where the principal can commit to an exclusiveinsurancecontractwith a single insurer. Without lossof generality, the insurancecontract canbe representedas a pair (/,D),where/ is the premium and D >0 is the deductible(which means that in caseof an accidentthe insurance

company pays the principal only an amount L -D).We shall assumeherethat being cautious is always efficient evenunder moral hazard.Therefore,the second-bestcontracting problemfor the principal canbe written as

max pLu(w -I-D)+(1-Pl)u(w -I)-y/

subjectto

I>pL(L-D)

625 Incomplete Contracts

and

pLu(w -I-D)+(l-pL)u(w -I)-\\j/>pHu(w -I-D)+(l-pH)u(w -1)The first constraint is a participation constraint for the insurance

company. It has to at least break even by providing insurance, under the

assumption that the principal takes propercare to avoid accidents.Thesecondconstraint is the incentive constraint for the principal, that is,here,by convention, the insuree.

It is convenient to work directly in terms of the principal'swealth in thetwo states of nature. For this purpose,define the no-accidentwealth aswN = w -1and the accidentwealth as wA = w -1-D.

The second-bestproblem can then be rewritten after some simplemanipulations as

max pLu(wA) +(1-pL)u(wN) -y/

subjectto

pLwA +(1-pL)wN < w -pLL

and

(Ph ~Pl\\u(wn)-u(wa )]>y/The first constraint is binding at the optimum, sinceotherwisewN canbe

raised,thereby raising the maximand and relaxing the secondconstraint.The incentive constraint will alsobe binding at the optimum, sinceotherwise we can improve insurancefor the principal at an unchanged expectedprofit for the insurer. Thus the second-bestcontract requires a strictly

positivedeductibleD (that is, wA < wN) in order to induce the principalto be cautious.This applicationis just another illustration of the classicaltrade-offbetweenrisk sharing and incentives,which we have discussedat

length in Chapter4.Considernow what happenswhen the principal cannot commit to writing

a singleexclusiveinsurancecontract.Assume that, nomatter what contractsthe principal has alreadysignedwith insurers, he can always go and offerany insurer a new, supplementary insurancecontractbefore choosinghislevelof care.All bilateralcontractsare assumedto bepublicly observable.As a result of the principal'sinability to commit to a singleprimary insurer,the second-bestoutcome we have just describedcannot be attained

anymore.

626 Markets and Contracts

More specifically,once signed, any contractwhere the incentiveconstraint binds can subsequently be improved upon by the principal.Thisresult canbe seenby the following reasoning:(1)in the second-bestcontract, the principal'spayoff is the samewhether he is cautious or not, sincethe incentive constraint binds; (2)supposethat, contrary to the second-bestoutcome,the principal decidesnot to be cautious; (3)he can then increasehis payoff by obtaining full insuranceby taking on supplementaryinsurance; (4) the secondaryinsurer realizesthat the principal will facea higherrisk of an accidentand will take that into accountin setting the premium,but the new contractis still profitable for the principal and the secondaryinsurer.Of course,by taking on secondaryinsurance,the principal and the

secondaryinsurer imposea negative externality on the primary insurer,who would strictly prefer the principal to be cautious.

If the second-bestcontract,and more generally any contractwhere theincentive constraint is binding, cannot be an equilibrium outcomeundernonexclusivecontracting, what contractscanbe equilibrium outcomes?Aswith renegotiationin Chapter9,wecan,without lossof generality, think ofthe equilibrium outcomeas implemented with a single primary contract,forwhich the principal hasno incentive to take out supplementary insurance.

There are only two possibilitieshere for a third-best contract(withnonexclusive contracting).Either this contract induces the principal to becautious, or it doesnot. It is obvious that if the principal is not exercisingcaution, then the optimal contractis a full-insurance contractwith a risk ofan accidentofpH- Giventhe zero-profitcondition for insurers, we have in

that case

WA =WN =W =W \342\200\224 PffL

which gives the principal a payoff of u(w -pHL).Twocasesare then possible:either u(w -PhL) is lower than the no-

contractpayoff

uN = pLu(w -L)+(1-pL)u(w) -yor it is higher.

Case1:u(w) < uN In this case,the third-best equilibrium contractinducesthe principal to exertcaution, and the optimal amount ofinsuranceis givenby the solution to

627 Incomplete Contracts

max pLu(wA )+(X-Pl)u(wn )-y/

subjectto

pLwA+(l-pL)wN<w-pLLand

pLu{wA)+(1 -pL)u(wN )-y>u[pHwA +(l-pH)wN ]

Note that the secondconstraint is now strongerthan our familiarincentive constraint. Thisconstraint implies that a contractinducing a high levelof caution can be an equilibrium contract only if it cannot be improvedupon by a supplementary insurancecontractwith no caution (and the bestsuch contractinvolves full insurance).When this third-best incentiveconstraint binds, our familiar incentive constraint is redundant: indeed,the

supplementary contractstrictly improves upon the contract(wA, wN) if the agentchoosesnot to exercisecaution.

In equilibrium, this third-best incentive constraint will be binding.

Otherwise, it would bepossibleto improve on the insurancearrangement while

mamtaining nonnegative profits for the insurer.

Case2:u(w) > uN In this case,it is possiblethat there isno way to reacha higher utility levelthan u(w). This is the case,for example,when the initial

wealth pair (w -L,w)involves an expectedutility for the principal that isnot much lower than the second-bestlevel.However,if under the initial

wealth pair (w - L, w) the principal has a much lower utility than thesecond-bestlevel,it is possiblethat the equilibrium outcomeisthe sameasin Case1.

Tosummarize:Theprincipal'sinability to commit not to enter into othercontractslowershis equilibrium payoff. However,the extremeoutcomeoffull insuranceand no caution is not necessarilythe only third-bestequilibrium outcome.Instead,anotherpossibleoutcomeis for the principal to limit

the amount of insurancehe getsfrom a primary insurer evenmore than hewould under a second-bestcontract.Hehas to do this to be able to resistthe subsequenttemptation to obtain supplementary insurance.

An analysis very similar to the precedingcan be performed for thesituation where the principal is a borrowerwho sequentially borrowstoinvest in a project that alsorequireseffort to raise its continuation value.Bizer and DeMarzo (1992)considersuch a contracting problem,where

Markets and Contracts

externalitiesare limited becausedebt is prioritized but are nonethelesspresent becausehigher total debt means lower effort by the principal(Myers',1977,familiar debt-overhang problemmentionedin Chapter4).Nonexclusivity then results in higher interest rates and indebtedness,andin lowereffort by the principal.

The Bidding Game:CommonAgency and Menu Auctions

In some multilateral contracting situations it is the agents who initiate

contract offers,or bids.3This is the case,for example,in procurementcontracting, where government agenciestypically invite bidsrather than

make offersto individual contractors.When agentsmake bids, thedimension of the coordinationproblembetweenagentsis significantly increased.Where in the offer game the coordinationproblem among agentsarisesonly in their accept/rejectdecisions,in the bidding game thecoordination problem arisesin the contracts they offer and the accept/rejectdecisionsof the principal they induce.Not surprisingly, given the greaterpotential for coordination failures in the bidding game, much of the

early literature on common agency and menu auctions has focusedon this

issue.Here,therefore,we concentrateon illustrating how the efficientcontractual outcomemay not arise in equilibrium becauseof a coordinationfailure among agentsin a simple exampledue to Bernheimand Whinston

(1986a).Theyconsidera multiobject auction without externalities,wherethe

principal initially owns M objectsand his valuation for each of the objectsisnormalizedto zero.HefacesN agentswho can,in a first stage,submit non-negative bidsfor any of the possiblecombinations of the objects.In asecondstage,the principal choosesthe allocationof these objectssoas tomaximize his total monetary payoff. This game has two noteworthyfeatures: First, wehave an auction setting whereeachobjectcanbe allocatedonly to a single individual. Second,given that all transfers are nonnegative,the principal is always weakly better off acceptingall offers,sothat we candispensewith all parties'participation constraints.

Tobemorespecific,let us focushere on a principal putting up two itemsfor auction, xa and xb.Two agentswith preferencesfor the two items arerepresentedin the following payoff matrix:

5.Note that the literature is known as the common-agency literature; that is, the namesprincipal and agent are reversed in comparison with the terminology used in this chapter.

629 Incomplete Contracts

Allocations Agent 1Payoff ux Agent 2 Payoff u2

Nothing 0 0xa 6 5xb 5 6xa and xb 8 7

Given this payoff structure, the efficient allocationisfor agent 1to get xaand agent 2 to get xb. Under this outcomethe total surplus from trade is12.There exists, however, an inefficient subgame-perfect-equilibriumoutcomeof this game wherebidder1getsboth items and a surplus oftradeof only 8 is obtained (while the principal getsa payoff of 7).Thisequilibrium outcomeis obtainedif the two agentseachsubmit the followingequilibrium bids[denotedby \302\243,(\342\226\240)]:

ti(xa+xb)=7 and ti{xa) =tl{xb) =Q

Toseethat theseare indeed equilibrium bids, note that if bidder1onlybidsfor both items, bidder2 can never gain by bidding for a single item

only, and viceversa.Thereare other equilibria, however.In particular,define a globally truthful schedulewith utilities u* for i =1,2,...,N as apayment schedulewherefor each individual i and eachallocationx, we have

ut[x, tt(x, u*)]=uf

Thisis a payment schedulewhereeachagent i endsup truthfully

revealingher marginal willingness to pay for all positivelevelsof trade,and

therefore obtains a constant payoff whatever the (positive)levelof trade.In the

precedingexample,we can rewrite globally truthful schedulesas

t1(xa+xb)=8-u*,t1(xa) =6-u*,and t1(xb)=5-u*for agent 1and

h(xa+xb)=l-u$,t20ca)=5-u$,and t2(xb) =6-u$for agent 2.

For globally truthful schedulesto be equilibria, they have to be mutual

bestresponses,given the optimal choiceof the principal.Note that, sincethe principal cannot make any offer, eachagent will give him a zeromarginal surplus from trading with her. If the principal trades only with agenti, who moreoveroffersa globally truthful schedule,he choosesto sellboth

630 Markets and Contracts

objects,and he thus obtains aspayment tt{xa +xb).Not giving any additional

surplus from trading with an additional agent means that the paymentstfea + xb) for i = 1,2 are equalizedacrossagents and are equal to therevenue the principal obtains from trading with both agents.But tradingwith both agentsunder globally truthful schedulesmakessensefor the

principal only if it involves sellingxa to agent 1and xb to agent 2.Thistransaction takesplaceat a total priceof

ti(xa +xb)-2+t2(xa +xb)-1=2tt {xa +xb)-3

By the precedingreasoning,theseconditionsimply

ti(xa+xb)=3

so that in equilibrium xa is soldto agent 1at a price of 1and xb is soldto

agent 2 at a priceof2.Thisis an equilibrium for the following reasons:

1.Theprincipal ismaximizing his surplus by selling the two objectsin this

way, given the two schedulesput forward by the agents.2.Agent 1has a utility levelof 5 and couldnot get moreby deviating and

trying to get both objectsfrom the principal.3.Similarly, agent 2 has a utility level of 4 and could not get more by

deviating and trying to get both objectsfrom the principal.

Onecanshow that an equilibrium with globally truthful schedulesalwaysexists,and that it is efficient in the absenceof externalities,that is, when

agents' payoffs dependonly on their own trades with the principal.Thisconclusionis intuitive, becauseunder globally truthful schedules,the

principalfacesconstant utility levelsfor all the agentshe is trading with.

Theprecedingexamplethereforeshows that going from offergamesto

bidding gameshas two implications: (1)coordinationfailures are likely tobe endemic,but (2) in the absenceof externalitiesthere existsan efficient

equilibrium whereagentssubmit globally truthful schedules.

13.4Principal-Agent Pairs

Introducing contracting questionsin industrial organization is very natural:

most situations of oligopolisticcompetition involve \"managerial firms\"

whereagencyproblemstypically arise.What is the impact of theseagency

631 Incomplete Contracts

problemson the equilibrium outcomeof oligopolymodels?And is there astrategic value for principal-agentcontracts?

Thesequestionshave beenstudied by a first generationof papersthat

have assumedcontractsto bepublicly observableand parties to be able tocommit not to renegotiate them. For example,Fershtman and Judd (1987)focuson a duopoly wherefirm i is representedby two risk-neutral,

symmetricallyinformed individuals: a shareholder/principaland a manager/agent.

The gameconsideredhas two stages.In stage 1eachprincipal-agentpairhas to agreeon a linearmanagerial incentive scheme

wi=aiqi+pi7Ci

where the managerial compensationw, dependslinearly on the output qtsoldby the firm and its profit level Kr Once these contractshave been(simultaneously) signed,in stage2 of the game the a,'sand /3,'sbecomepublic information and competition takes place, in (static) Cournot orBertrand6 fashion.

As is familiar from industrial organization, strategicconsiderationsdiffer

betweenCournotand Bertrandcompetition, becausethe first correspondsto the caseof strategicsubstitutability (downward-sloping reactionfunctions) and the secondto strategiccomplementarity (upward-slopingreaction functions):7 in the Cournotcase,it pays to commit to being tougherbecausedoing so inducesthe competition to becomesofter,while in theBertrand ease,getting the competition to becomesofter is achievedby

being softertoo.Consequently, the outcomeof the two-stagegame impliespositivesalesincentives (a,> 0) when stage2 is modeledas Cournotcompetition, while negative salesincentives (at < 0) result under Bertrandcompetition.By comparison,in the benchmark casewithout any strategicconsiderations,only profits would be remunerated(a,=0).

Theprecedingresult is not without problems.First, one can regret that,as is usual with thesestrategicmodels,the predictiondependscrucially onthe nature of competitionbetweenthe firms, which is itself often hard todetermine.Second,and this is a criticism more specificto this model,how

6.Assume that the two firms' products are differentiated, in order to keep reaction functionscontinuous.

7. See,for example, Bulow, Geanakoplos, and Klemperer (1985)and Fudenberg and Tirole(1984).

632 Markets and Contracts

canwe justify the assumption that the partiescan commit to an incentiveschemethat is not renegotiation-proof?Indeed,while it isin the interestofeachprincipal-agentpair to have the otherpair believethat they will deviatefrom pureprofit maximization, oncestage2 is reachedeachpair would liketo secretlyreturn to profit maximization for each given levelof the stage-2 choicevariable of the other pair. In fact, if secretrenegotiationwereallowedafter stage1,under symmetric information within eachprincipal-agent pair they would eachrevertback to ok = 0,and this movewould beanticipatedby the other pair, so that contractswould loseall strategicvalue.8

Introducing asymmetric information within each pair is a natural

solution to prevent perfect renegotiation.This idea is by now familiar from

Chapter9:ex post Pareto-improving renegotiationismore difficult in the

presenceof asymmetric information becauseit typically implies that theuninformed party has to concederents to (some types of) the informed

party. The strategicvalue of renegotiation-proofcontractsin the presenceof asymmetric information has first been investigated by Dewatripont(1988)in a setting where contracting takes placebetweenan incumbentfirm and a third party (e.g.,a laborunion) before the potential entry of acompetitor.Contracting is designedto commit the firm to high post-entryoutput. It is assumedto take placebefore the firm learnsits privateinformation (about its own cost,say).However,if the firm has learned its costby the time the entry decisionoccurs,renegotiating away the excessivelyhigh post-entryoutput is hard becausethe union will beafraid (and rightly

so)of losing rents in the form of lowerwages.Consequently,as long as theexpost allocativeinefficiency isnot too high, it will berenegotiation-proof.Contracts thus keepsome strategicvalue with asymmetric information,evenwhen expost Pareto-improving renegotiationis unavoidable.

One can thus rely on asymmetric information to restorethe strategicvalue of contracts in the presenceof renegotiation.Note two caveats,however:first, Dewatriponthas not addressedthe problemof ambiguityof the impact of contractsdependingon whether competitionis of theCournot or Bertrand type; second,the reasoninghas relied on a model

8.A third problem concerns the fact that Fershtman and Judd consider only a specific subsetof feasible contracts, namely, contracts linear in sales and profits. Katz (1991)shows that, byallowing more general contractual forms, one can generate many potential equilibria in thisgame.

633 Incomplete Contracts

whererenegotiationhappensat the interim stage(oncethe informed party-has learnedits type) while initial contracting tookplaceat the initial stage(beforethe informed party had learnedits type).What about secretrenegotiation just after the initial contracthasbeensignedand beforeany newinformation has arrived?This is what would make renegotiationeasiest.Couldit eliminate any strategicvalue of contracting?

Thesequestionsare analyzed in depth by Caillaud,Jullien, and Picard(1995).Let us illustrate their conclusionsusing one of their examples.Itassumesa Cournotduopoly with market demand dennedby p =0.5(5-a-b),wherea and b are the output levelsof the two firms A and B.Let usfocuson asymmetric information at firm A. Assume that the principal hasno private information and haspayoff idjx, b, t) =0.5(5-a-b)a-t, while

the agent has private information on her unit cost 6t and has payoffu(a, t, 6i) = t - Ofi. The principal thus receivesthe revenue of the firm

and pays a transfer t to the agent to coverher own cost.As for the privateinformation, assumethat it is common knowledgethat the agent canhavetwo equiprobablecostrealizations01=1-8 and 02 = 1+ 8,with 8not toolarge.9

The gameconsistsof three stages:\342\200\242 The first stage is the initial public contracting stage.In this setup, it will

be enlightening to focuson very simple contractsof the form \"The

principal acceptspaying the agent an unconditional amount R.\"

\342\200\242 Thesecondstageis the secretrenegotiationstage:the principal offersthe

agent a new contractof the form t(a, b).If the agent refuses,then the

principal pays the agent the initially agreed amount R (and, given the payofffunction of the agent, weshall have zerooutput from this firm). If the agentaccepts,we move to the third stageand to Cournotcompetition betweenthe agentsof the two firms. Note that, while the other firm hasobservedtheinitial contract,and thus R, it doesnot observethe new contractt(a, b) orwhether it is acceptedor not.\342\200\242 In the third stage,the agent learnsher type Qt. Shecanstay in the market,in which casethere is a simultaneous choiceof outputs (callit a, for the

type-f agent of firm A and b for firm B).The agent is alsoassumedto beable to quit at any moment and obtain a zeropayoff (while producing zerooutput).

9. Onecan verify that 8< 0.3is sufficient for the following discussion.

634 Markets and Contracts

Both the initial public contracting and the secretrenegotiationare thus

assumedto happenbefore the agent has learnedher type.This assumptionmakesrenegotiationas easyas possible;therefore,obtaining any strategicvalue for contracting is hardest.

Let us assumethat an initial contracthas beensigned,with associatedpayment R from the principal to the agent in stage3 if stage-2renegotiation has failed.Assume moreoverthat the principal canmakea take-it-or-leave-itofferto the agent in the secretrenegotiationstage.What will behisbestoffer?It will beconvenient to think not in terms of an outcome(th at),but in terms of

(u\342\200\236 a),noting that tt = ut + Ofli. Givenan expectedoutput bfrom the other firm, the bestofferfrom the point of view of the principalwill solve

maxY 0.5(0.5(5-ax-b-2Q^ax-ut}

subjectto

28a1>u1-u2>28a2 (IC)

Ul>0 fori =1,2 (EPIR)

Q.5ui+Q.5u2>R (EAIR)

The principal thus maximizes his expectedrevenue,net of the transfer tothe agent, subjectto the two incentive constraints [the agent should havean incentive to choose(tb at) when her type is 0;]and to the expost and exante participation constraints [constraints (EPIR)and (EAIR),respectively]. While expost the agent cannot have a negativepayoff, from an exantepoint ofview, the constraint concernsthe initial contractthat has givenher the possibility of obtaining a transfer R at no cost.

The full-information contract involves dropping the incentiveconstraints, which implies maximizing 0.5(5-at-b -lO^for each 0;.Thisoperationyields the standard Cournotbestresponse,which we candenoteby A* (b)=0.5(5-b) -6t. What if we start insteadwith asymmetricinformation and without an initial contract,that is, with R = 0? This approachmeans solving the preceding program without the ex ante constraint

(EAIR).From Chapter2 weknow that one type will have the sameoutputas without the incentive constraint but will earn rents, while the other will

have no rents but an output distorteddownward. Specifically,onecanverifythat the output levelswill be ax =A\\{b), while a2=A2(b, 0) = 0.5(3-b) -

635 Incomplete Contracts

35< A*(b). The rents for the agent of type #i are equalto what shewould

get by mimicking type <%, that is,28a2(sinceu2 =0 and u1 =u2 +28a2),which

we can denoteby 2R\302\260(b). From an ex ante perspective,the expectedrentsof the agent are thus

R\302\260(b).

As long as the initial contractgives the agent a transfer R <R\302\260(b),

it will

be irrelevant, and everybodywill expect secretrenegotiationto deliveran output response{A*(b), A2{b, 0)}.But what if the initial contractspecifiesa transfer for the agent in excessof

R\302\260(b)7Then the trade-off

betweenrent extractionand allocativeefficiencyis modified, becausetheinitial contractrequiresthat more rents beconceded,in expectedterms, tothe agent.Theefficientway to do soisfor the principal to give rents to type

Oi, which at least allows nim to reduce the underprovision of output by

type #2. If the public initial contractimplies a transfer R somewhat in

excess ofR\302\260(b),

secretrenegotiationwill imply u2 =0, =2R, ax =A*(b), anda2=R/5 (this last expressionis obtainedfrom the relation ux = u2 +28a2).For higher R's,output a2 can riseup to its first-best level,in which casehigher i?'sthen imply higher utility levels for both types at unchangedoutput levels.

We thus seethat, by modifying the trade-offbetweenrent extraction andallocativeefficiency, an initial contractoffering rents to the agent has an

impact on the output levels that are secretly(re)negotiatedbetween the

principal and the agent.Specifically,a more generousinitial contractleadsto lessallocativeinefficiency for type <%, and thus a higher expectedoutputlevelby firm A for any given levelof output b by firm B.UnderCournotcompetition, a higher output is attractive as it implies a reduction ofequilibrium output b.

Specifically,assuming that firm B has a marginal cost of 2, its reactionfunction is b = 1.5- (ax + a2)/4.In the absenceof an initial contract,onecancheckthat the equilibrium involves b = b0 =1+25/3, ax =A*(b0), anda2=A2(b0, 0).This contractgives the agent a levelof expectedrents

R\302\260(b0).

It is then strictly in the interestof the principal to give the agent an initial

contractwith a slightly higher payment thanR\302\260(b0):

it implies only asecond-order lossfor a given output b (given the upward adjustment of a2),while

delivering a first-ordergain in terms of reductionof the output b of theotherfirm (from the reactionfunction, b goesdown by 25%of the increaseina2)-

In the Cournotsetting, not signing an initial contractis thus not anequilibrium, despitethe fact that secretrenegotiationisexpectedright after the

636 Markets and Contracts

initial contracthas beensignedand thus in the absenceof any newinformation. The equilibrium initial contractoffersrents to the agent that limit

the needto reduceoutput belowits full-information Cournotlevel,therebyshifting upward the reactionfunction of the firm and lowering theequilibrium output of the other firm.

What about a (differentiated)Bertrandsetting now?If wekeepthe samegameasbeforeexceptfor having pricecompetition in the last stageinsteadof quantity competition,we have the samerole for initial contracting:by

offering rents to the agent initially, one limits the downward distortion ofoutput relative to its full-information level.Here,however,the strategiceffectof the initial contractisdetrimental: sinceit makesthe firm \"tougher,\"its competitoralsobecomestougher. Thereis no usefor an initial contractthat grants rents to the agent in order to limit the underproduction that

adverseselectioncreates.Caillaud,Jullien, and Picardshow that no stage-zerocontractwill be signedunder Bertrandcompetition.

Their approachcan thus be seenas having \"killed two birdswith onestone.\" By allowing for secretrenegotiation,it has enhancedrealismbut

alsoreducedthe ambiguity of the role of strategiccontracting: sinceit canonly make the firm tougher by reducing the underproduction due toadverseselection,it isonly usedunder quantity competitiona la Couruot.Their analysis also leadsto an unambiguous policy conclusion:when it iseffective,strategiccontracting raisesoutput and welfare.

13.5Competitionas an Incentive Scheme

The previoussectionshave considereda variety of environments wherecontracting has strategiceffects.We end here by asking how in turn

contracting and the size of agencycostsare influenced by the competitiveenvironment. This question is an old one, encapsulatedin the famous

saying by Hicks(1935):\"The bestof all monopoly profits is a quiet life.\"

The presumption that competitive pressureslimit agency problems andresult in higher efficiency is indeed widespread.10Theoreticalanalyses,however, offera cautionary perspectiveon this question:there is often an

ambiguous relationbetweencompetitionand agency costs,becausecompetition simultaneously has two effects:(1)a changein the agent'seffort

10.See,for example, the influential management work of Porter (1990).

637 Incomplete Contracts

for a given incentive scheme;and (2) a change in the incentive schemechosenby the principal.

This questionwill be investigated using a modeldue to Schmidt (1997)where the agent's effort is driven both by financial incentives and by athreat of liquidation (and the associatedlossof private benefits).A changein competitionthen affectsboth the threat of liquidation and theequilibrium incentive schemethe agent faces.Before analyzing this model,letus stressthat it doesnot exhaust the possibleeffectsof a changein

competition. For example,severalstudiesconsiderthe information effectof anincreasein competition:if a firm facescompetitors,relativeperformanceevaluation (asin Chapter8)becomespossible.Though exploiting this

information can,ceterisparibus, increasethe payoff of the owner of the firm, theeffecton the manager'seffort is ambiguous (seeNalebuff and Stiglitz, 1983;Hart, 1983b;Scharfstein, 1988;and Hermalin, 1992).Other effectscan alsobeambiguous: asshown by Hermalin (1992),this observationis true for theeffectof a changein competitionon a manager'sdisutility ofeffort andattitude toward risk.

Schmidt'smodelconsidersa principal-agentproblemwhereeffort by the

agent servesto decreasethe marginal costofproduction.Specifically,while

initially the costof the firm is cH, the agent candecreasethis costto cLwith

probability p e [0,1]at an effort cost y/(p). To make the problemwellbehaved,assumey/', y/\" > 0, y/\"(0)

= V'(0)= 0> and lirnp_*i yr(p) =\302\260\302\260.

Consider a three-stageproblem:first a contracting stage,then a stagewherethe

agent chooseseffort, and finally a stage where the principal decidestoliquidate the firm or to continue its operations.Under liquidation, wenormalize the financial value of the firm, aswell as the agent'sprivate benefit,to zero.In the absenceof liquidation, the agent obtains a positiveprivatebenefit h, and we define the financial value of the firm, in reducedform, as

k(c,<j), e)

where,ceterisparibus, tu(c, <p, e) decreasesin its costc e {cH,cL}and alsoin

the degreeofcompetition <p.The value of the firm is alsoassumedto dependon a noiseterm e,which has cumulative distribution F(e).To simplify the

analysis, assumethat liquidation is never attractive for the principal if thecostof the firm is low; that is, tu(cl, <p, e) > 0 for all

<pand e.

If the agent receivesa wagew, the payoffs of the principal and agent are,respectively,

638 Markets and Contracts

Up =max{0,7r(c,<j>, e)}-w

and

uA =w -\\j/(p) +Xh

whereA =1if the principal doesnot liquidate the firm and A =0 otherwise.Note that theseexpressionsimplicitly assumethat the liquidation decisionis noncontractable;therefore,the principal will always make the decisionthat maximizes the ex post financial value of the firm, and moreoverthe

wagethe agent will receivecannot dependon this decision.It is,however,assumedthat the costrealizationis verifiable,sothat the principal canofferthe agent a wagew, for costrealization

c\342\200\236

with i e {H,L}.Both partieshavebeen assumed to be risk neutral. In order to make the moral-hazardproblemrelevant, assumethat the agent is wealth constrained,in the sensethat w, > 0.

Definel(<f>) as the probability of liquidation under cost cH, and define

II;0/0=f max{0,K(ct, <j), \302\243)}dF(e)

The variable11,(0)is thus the expectedvalue of the firm under costc,and

competitionparameter<p. For a given degreeofcompetition0,onecan then

definethe principal'sfirst-best problemas

max p[ULdp) -wL]+(X-p)WLh (</\302\273)

-\342\204\242h ]

subjectto

p(wL +h) +(l-p){wH +[1-K<p)]h}-

\\jf{p)> u

where u is the agent'sexogenousex ante outsidepayoff. It implies aprobability of low costPfb{<P) such that

V Ipfb 0/0]=nLo/o-nH (</>)+i{<p)h

In words, the first bestrequiresthat the marginal cost of effort is equatedto its marginal benefit, in terms of (1)higher financial value of the firm and

(2) lowerprobability of liquidation and associatedlossof private benefitfor the agent.

Turning now to the secondbest,we have to add the following incentiveand lirnited-Uability constraints:

p eargmax p(wL +h) +(1-p){wH +[1-Z(</0M-yrip)

639 Incomplete Contracts

and

w,->0 for z' =#,LFor simplicity, assumethat this problemis globally concave(and thereforeadmits a unique optimum), which is ensuredby the following assumption:

2y/\"(p)+py/f'Xp)>0for allp (13.3)Sincewe considera problemwith only two costrealizations,the incentiveconstraint canbereplacedby the agent'sfirst-ordercondition if weassumean interior solution for p:wL-wH+l{$)h-Xjf'ip)

= 0

The optimum can then bederivedasfollows:First, if the presenceofmoralhazard preventsthe first bestfrom being reached,wehave an optimal high-costwagewH(<p)

=0.Second,the first-orderincentive constraint implies an

optimal low-costwage

wd(t>) = YXp)-K(t>)h

Third, assuming that u is low enough that the agent'sparticipationconstraint doesnot bind given the limited-liability constraints, the wholeproblem canbe rewritten simply as

max p[UL(<p)-yr'(p) +l(<p)h]+(1-p)UH(<p)

which implies that the second-bestlow-costprobability Psb(<P) is dennedby

nL (</\302\273)

-Y Ipsb m+idp)h-psbWy\" Ipsb (</>)]-nH (</>)

=o

From this optimality condition, one can now computethe effectof anincreasein competition, that is, of an increasein

(j>,on the agent's effort

choice:

dpSB(<p)= [dTlL(.<p)/d(t>-dTlH(.<p)/d<t)]+hdl(<p)/d<p

d<j) 2xif\"[pSBm+PsB^)w'\"\\PsB<4)\\

By the concavity assumption (13.3),the denominator of this expressionisstrictly positive.The numerator is the sum of two parts:\342\200\242 The secondpart hdl(<p)/d<p is positive.It reflectsa \"threat-of-liquidationeffect\":moreseverecompetitionmeans that the agent works harder for a

640 Markets and Contracts

given wagedifferencewL - wH, in order to reducethe risk of losing herprivate benefit.\342\200\242 Thefirst part of the numerator, dTlL(<j))/d<j)

-dn.H(<t>)ld<j), which has a

potentially ambiguous sign, reflectsa \"value-of-cost-reductioneffect.\" A changein competitionaffectsthe attractiveness for the principal of inducing the

agent to reducecosts.Thiswill affect the wagecontract (wH, wL) that the

principal will offerthe agent.

Two comments are in orderhere.First, the positivesign of the threat-of-liquidation effectrelieson the assumption that more competitionraisesthe impact of costdifferenceson the probability of liquidation (sincethis

probability has beenassumedto stay equalto zerofor a low-costfirm,whatever 0may be).Thegeneralpoint is that morecompetitionwill partly altereffort incentives under unchanged wagelevels,becauseof its impact on thethreat of liquidation. Second,the value-of-cost-reductioneffectcan easilybe negative and outweigh the other effect.Indeed,the value for the

principalofcostreductionby the agent cangodown when competitionbecomes

fiercer.This effectcan be illustrated in the caseof symmetric Bertrand

competition with N > 2 firms. In this case,profits are always zero,exceptwhen

there is a singlefirm that has a low cost cL,in which casethis firm earnsstrictly positive profits, which we denote by II.As for private benefits,assumethat a firm with costc is liquidated if and only if there existsat leastone firm with strictly lower cost than c in the market (otherwise,it will

producea positivequantity, whether it earnspositiveprofits or not).Undertheseassumptions, the probability of liquidation of firm i when it has a lowcostis zero,while when it has a high cost,it is /(</>)

=1-(1-p,)1*'1,assuming

that all other firms have an individual probability of low costprSimilarly,

when it has a low cost,firm i has a probability (1-p^'1of earningII.Consequently,in a symmetric equilibrium, the second-bestindividual

low-costprobability pN is dennedby

yXpn)+pnv'Xpn)=a-Pvf'1n+[l-a-pNf^]hIndeed,the LHSof the equationis the marginal costofan increasein effort,including the increasein rent that goesto the manager.TheRHSis the

marginal benefit of an increasein effort, which has two components:(1)the

profit IIwhen one is alone in having a low cost,times the probability that

no othercompetitorhas a low cost,and (2)aprivate benefit that oneobtains

641 Incomplete Contracts

with probability one rather than (1-Pn)1*'1(that is, only when no

competitor has a low cost).By the concavity assumption (13.3),the LHSis

increasingin pN. The RHSis insteaddecreasingin pN providedn > h. Inthis case,a risein competition,that is, in N, implies a decreasein the

probability of low costpN. This result followsbecause,when the number ofcompetitors increases,the value of exerting effort in terms of increasedprofits

goesdown: this value is zero when at least one competitoris successfulat

lowering costs,and this event occursmore frequently\342\200\224for a givenindividual effort level\342\200\224the higher the number of competitors.

Note that this argument assumesthat there are at least two firms in themarket. Schmidt shows that the comparisonbetween monopoly and

duopoly can go either way in this setup, dependingin particular on thelevelsof monopoly profits under cLand cH.When a duopoly implies moreeffort than a monopoly, we have a nonmonotonic relationbetweenproductivity and the number of firms in the market, with the highest

productivityfor two firms.

13.6Summary and LiteratureNotes

This chapterhas focusedon situations whereseveralindividuals sign

competing contracts,whereasin previouschapterswe were always looking at

\"grand contracts.\" We first stressedcompetitionin exclusivecontracts,with

the classicalanalysesofAkerlof (1970),when offering a menu of contractsis impossible,and Rothschild and Stiglitz (1976),when it is allowed.In thefirst case,multiple Pareto-rankedoutcomescancoexist.Theworst one caninvolve completemarket breakdown,while the best one is constrainedPareto-efficient.In the secondcase,the main issueis failure of existenceofequilibrium. This stands in sharp contrast with signaling models,whereextrememultiplicity is the norm. We have highlighted, however, that

nonexistence canbe avoidedif one constrains the ability of the partiesto stealonly the desirablecustomersfrom their rivals, asstressedby Wilson (1977)or Riley (1979).

Another effectof adverseselectionon competition, which we have not

exploredin this chapter,is to dampencompetitive forces(seeGreenwald,1986,for a particularly elegantanalysis ofreducedcompetitionin the labormarket when current employershave moreinformation about the

productivityof their employeesthan prospectiveemployers).

642 Markets and Contracts

Somemodels endogenize the degreeof contractual exclusivity. In

particular, Aghion and Bolton(1987)show that an incumbent firm and its

customersmay find it advantageousto sign a contractwhere customershave to pay a penalty for buying from an entrant rather than theincumbent, becausethis pushesthe entrant to offer morefavorableterms to thesecustomers.

A more recent literature has consideredcompetitionwith nonexclusivecontracts.A unifying perspectiveis put forward by Segal(1999b).Pauly

(1974)was the first to point out that an insuree'sinability to commit not tocontractwith multiple insurers canlead to overinsuranceand lowerinsureewelfare.More generally, we made the distinction between the offergame(the principal makesall the offers)and the bidding game (the competingagents make the offers).These two gameslead to the following insights:

\342\200\242 In the offer game, when contractsare publicly observable,inefficiencyarisesin the presenceof externalities on nontraders, which prompt the

principalto distort trade soas to relax individual participation constraints.

\342\200\242 When contractsare private in the offer game, what matters instead isthe presenceof externalities at equilibrium trade levels, which are notinternalized by the principal becausethey are unobservableto individual

agents.Note that private contracting in the offer gamerequiresspecifyingindividual beliefsfor each individual offer receivedfrom the principal.Wefocusedon passivebeliefs(see,for example,McAfeeand Schwartz, 1994),whereby each individual agent, when observing an out-of-equilibriumcontract offered by the principal, believes that the other offers remain

unchanged.\342\200\242 The precedinginsights were derived under simultaneous offers.With

sequentialoffers instead, an individual correctlypredicts the impact ofcurrent contracting on future contracting. In the insurance case,while

overinsurancecan result from nonexclusivecontracting, underinsuranceisalso possible,as a \"commitment device\"against eventual overinsurance(see,for example,Kahn and Mookherjee,1998).

\342\200\242 As for the bidding game, we have focusedon the key new issuerelativeto the offergame,namely the coordinationproblemthat canarisefrom thecoexistenceof offersmadeby different partiesand its associatedpotentialinefficiency. This inefficiency disappears,however, in the absenceofexternalities between agents, when one focuseson globally truthful equilibria

(see,for example,Bernheim and Whinston, 1986a).

643 Incomplete Contracts

Section13.3discussedthe many applicationsof this setting. Let us stressthat the bidding game,whoseanalysis hasbeenpioneeredby BernheimandWhinston (1985,1986a,1986b),has led to many important applications,for example,to politicaleconomyand trade (seein particular the bookby Grossmanand Helpman,2002).On the theoreticalfront, we shouldmention the work on common agencywith adverseselectionby Martimortand Stole(2002),aswell as the recentwork by Segaland Whinston (2003)on \"robust common agency.\"

Section13.4focusedon the strategicuse of contractsin an industrial

organization context, that is, on competitionbetween \"principal-agent

pairs.\" Early work (e.g.,Fershtman and Judd, 1987)11pointed out the

desirability of using contractsbetween shareholdersand managersas away to commit to non-profit-maximizing behaviorand thereby influencerivals. Katz (1991),however, has shown how this idea could lead toextreme multiplicity of equilibria. Moreover,while it is profitableto have an

opponent believe one is not maximizing profit, deviations from profitmaximization are not immune to renegotiation(seeDewatripont, 1988).In this section we developed the contribution of Caillaud,Jullien, andPicard (1995),who establishthe strategicvalue of public contractsevenwhen they canbe immediately and secretlyrenegotiated(seealsoBoltonandScharfstein,1990).Their usefulnesslies in altering the participationconstraints of the agent in various statesofnature in a way that endsup

reducingthe well-known allocativeinefficiency arising with contracting under

adverse selection.When reducing this allocativeinefficiency leadsto a\"favorable\" changein the behaviorof competitors,the contracthas

strategicvalue. In the setting of Caillaud,Jullien, and Picard,for example,an

initial contractthat reducesunderproduction is useful in a Cournotsettingbut not in a Bertrandsetting.

Finally, whereassection13.4focusedon the role of contractson productmarket competition,section 13.5instead lookedat the impact ofcompetition on contractsand agency costs.The literature, pioneeredby Hart

(1983b),has highlighted the ambiguous effectof competitiononmanagerial effort\342\200\224for example,with respectto the ability to rely on relativeperformance evaluation (seeScharfstein,1988,and alsoHermalin, 1992,for amoregeneralanalysis).Here,wehave not discussedthis effectand wehaveconcentratedinsteadon the relationbetweenthe intensity of competition\342\200\224

11.Seealso Bonnano and Vickers (1988)and Brander and Lewis (1986).

644 Markets and Contracts

simply parametrizedby firm profitability and the threat of bankruptcy\342\200\224

and managerial effort with endogenousmanagerial incentive schemes.As shown by Schmidt (1997),12more competitionhas two implications:First, the relationbetweeneffort and the probability of bankruptcy isaffected by a changein competition,which affectsmanagerial effort for an

unchanged incentive scheme;second,the value of effort for the

owner/principalis affectedby a changein competition,which leadsto a changein the

equilibrium incentive scheme.In general,both effectscan be ambiguous.In Schmidt'ssetting, reasonableexamplessuggestan inverted U-shapedeffort level, that is, one with highest effort for intermediatedegreeson

competition.

12.Seealso Aghion, Dewatripont, and Rey (1999)and Raith (2003).

APPENDIX

Exercises

In this chapterwe collecta setof exercises(classifiedby chapter) that aremeant not only to review basicconceptsand methods introducedin theprevious chapters,but alsoto exploreideasand applicationsthat wehavenothad the spaceto coverin the main body of the book.Someof thesequestions arebasedon articlesthat wehavenot fully coveredin the text. Othersare basedon teaching material accumulatedover the years by ourselvesand other instructors of contract theory at the UniversiteLibre deBrux-elles,the MassachusettsInstitute ofTechnology,and PrincetonUniversity.

Chapter2

Question1Considerthe following monopoly screeningproblem:A government agencywrites a procurementcontractwith a firm to deliverq units of a good.Thefirm has constant marginal costc,sothat its profit is P-cq,wherePdenotesthe payment for the transaction.The firm's costis private information and

may be either high (cH) or low (cL,with 0 < cL< cH).The agency'spriorbeliefabout the firm's costis Pr(c=cL)= /3, and it makesa take-it-or-leave-it offer to the firm (whosedefault profit is zero).1.If B(q)denotesthe (concave)benefit to the agency of obtaining q units,what is the optimal contractfor the agency?2.Compare this second-bestsolution with the first-best one,obtained if

costsare known by the agency.Discussthe results.3.What would the first-best and second-bestsolutions be if c, insteadoftaking two values,wereuniformly distributed on [c,c],with 0 < c < c?

Question2

Considerthe monopoly problemanalyzed in section2.1,but assumethat

the monopolist has one unit of the good for sale,at zero cost,while the

buyer canhave the following utility:

eL-Tor

log(0H-D

648 Exercises

Thebuyer'srisk aversionthus riseswith her valuation. Show that the sellercan implement the first-best outcome(that is,sellthe goodfor sure,leaveno rents to either type of buyer, and avoid any costof risk in equilibrium)by using a random scheme.

Question3

A monopolistcan producea good in different qualities. The cost ofproducing a unit of quality s is s2.Consumersbuy at most one unit and haveutility function

(6s if they consumeone unit of quality su(s\\0) =

\\

H J10 if they donot consume

The monopolistdecideson the quality (or qualities) it is going to produceand price.Consumersobservequalities and pricesand decidewhich qualityto buy if at all.

1.Characterizethe first-best solution.2.Supposethat the sellercannot observe6, and supposethat

(6H with probability 1-/3e=-WL with probability /3

with 6H > 6L>0.Characterizethe second-bestsolution and consumers'informational rents.3.Supposenow that 6 is iimformly distributed on the interval [0,1].Characterize the second-bestoptimal quality-pricing schedule.

Question4

Consideran economywith a continuum of agents who produceoutput qby supplying input a (for effort) with the individual productionfunction

q = 6a,where 6 is an idiosyncratic productivity parameter.Theproductivity density in the population is given by f(6) with support [6,6][wheref{6)> e> 0].All agents have the sameutility function u(c)-a, with u' > 0 andu\" < 0.1.What is the distribution of output and consumption in the economywhen all agents live and work in autarcy?

2.Supposethat all agents in this economycanwrite an insurancecontractbefore they know their productivity type. All agents are identicalex ante,

649 Appendix

and their future productivity is i.i.d.with density f{6).\"What is the optimalinsurancecontract when 6 and a are observableex post? What is the

optimal insurancecontractwhen only 6 is observable?What is the optimalcontractwhen neither 6 nor a is observableex post and /(0)/[l-F(6)]ismonotonically increasing?3.Interpret the last solution. Show that the marginal premium is given by

P'(0)=(0m'[c(0)]-1)c'(0)and

P'(6)=P'(J9)=0

Discussthis solution.

Question5

Consideran economyin which firms want to go public.A typical privatefirm is ownedby a risk-averseentrepreneurwith personalwealth Wo and

increasing,strictly concavevon Neumann-Morgensternutility function uE.Firms are worth 6+ e,where 6 and eare independentreal-valuedrandomvariables.The quantity e is realized in the future, and its realizationisunknown to everybody at the time of the interaction.It has mean 0.Therealizationof 6,however, is known to the entrepreneurbut to no one else.Assume 6can take two values, 0L < 6H,where 0L occurswith probability /3,and 6H with probability 1-j3.Thisis common knowledge,aswell as the factthat [$6L+ (1-P)6H> 1.There is a monopolisticinvestment bank that

proposes the terms of the initial public offering placedwith the market. Theoffering is a pair (x, T) e [0,1]x R+, where1-x is the fraction of the firm

soldto the market in exchangefor a payment of T to the entrepreneur.Theinvestment bank acts as if it is risk neutral and maximizes total expectedmarket profits from the sale:

UI(x,t,6)=a-x)6-TThe safe interest rate is normalized to 0. The strategic interactionconsidered is the following: first, nature chooses6, then the bank proposesthe terms of the offering, and finally the entrepreneurdecidesabout the

acceptanceor rejectionof the offer.

1.Write down the entrepreneur'sexpectedutility as a function of x, t,and 6.

650 Exercises

2.Determinethe first-best terms of the offering. \"Why will the first bestnotberealizedin the presentsetting?3.Write down the problemas a screeningproblem.Explain briefly.

4. Which constraints will bind at the optimum?

5. What allocationwill the type-0j, entrepreneurobtain at the optimum?

6.Solvethe screeningproblemfully.

7. What elementsof your solution are typical of screeningproblems?Isanything surprising?

8. Make a conjecturefor the outcomeunder competition betweeninvestment banks.

Chapter3

Question6

Considera firm that can invest an amount / in a project generating highobservablecashflow C> 0 with probability 6 and 0 otherwise:6e {6L, 6H}with 6H-6L=A > 0 and Pr[0= 6L]= /3.Thefirm needsto raise/ fromexternal investors who do not observethe value of 6.Assume that 0LC-1> 0.Everybody is risk neutral, and there is no discounting.

1.Supposethat the firms can only promiseto repay an amount R chosenby the firm (with 0 < R < C) when cashflow is C and 0 otherwise.Can agoodfirm signal its type?2.Supposenow that the firm alsohas the possibility ofpledging someassetsas collateralfor the loan:Should a \"default\" occur(the firm being unableto repay R), an assetof value K to the firm is transferredto the creditorwhosevaluation is xKwith 0 < x < 1.Thesizeof the collateralK is a choicevariable.Give a necessaryand sufficient condition for the \"best\" perfectBayesianequilibrium to be separating.How doesit dependon /3 and xlExplain.

Question7

Considerthe following modification of the Myers-Majluf (1984)model.Supposethat the asset in placecan take three ex post values, A = 0,1,2,and let y= Prob(A =1)and fi =Prob(A =2).Thenew project,however, hasa safevalue VN >I,where/ is the start-up costof the new project.Suppose

651 Appendix

that the firm canbeof two different types, G and B,where(y, ^) e {(ys,}iB),(Yg, jug)}, with jo > Yb, but

E[Ag]=Yg+ 2/ig =Yb+2/ib =E[AB]

Solvefor the setof perfectBayesianequilibria under, respectively,equity

financing and debt financing. Discuss.

Question8

A firm has a projectrequiring an investment of20at t =0 for a sure return

of 30at t= 1.There is no discounting. The investment costhas to beraisedfrom the financial market. Assume that a new equity issueis proposed.Potential new investors are uncertain about the value of the firm's assetsinplace:A e {50,100}with Pr[A = 100]=0.1.1.Supposethat investors believethat both types of firms invest.What

fraction of the firm's equity has to be issuedto new investors?What are the

payoffs to existing shareholdersif they undertake the project?Are thesebeliefsreasonable?2.Supposethat investors believethat only bad firms issuenew equity. Samequestions.3.Supposenow that shareholderscommit at t= 0 to a wasteful advertising

campaign at t =1after the projectreturn is realized.The advertisingexpenditure is ah irreversibleactionon the part of the firm that results in a dropin profits of K.The sizeof the expenditureis a choicevariable.Show that

a goodfirm can signal its type through such expenditures.Discuss.

Question9

Considera two-periodmodelwith no discounting. An entrepreneurhas aprojectgenerating a random cashflow Ce {C,C}at t =1whereC-C=A

> 0 and Pr[C= C]= 6.The (project)type 6 e {6L, 6H), with 6H > 0L, is the

entrepreneur'sprivate information, and the (common knowledge)market

prior is Pr[0= 6L]= /3.Theprojectrequiresan investment / at t =0 that the

entrepreneurneedsto raise from the (competitive)market. Assume that

C+ 6LA > I > C.The entrepreneur is assumedto be restrictedto issuingdebt only or equity only.

1.Show that there exist perfect Bayesian equilibria in which both

types issuedebt.Which one among these equilibria is preferred by 6H

652 Exercises

entrepreneurs?Show that there may existperfectBayesianequilibria in

which both types issueequity. Which one among these equilibria ispreferred by 6H entrepreneurs?And do 6H entrepreneursprefer this

equilibrium to their most preferreddebt equilibrium? Explain.2.In the remainderof the problem,assumethat the entrepreneurincurs anonpecuniary costoffinancial distressK> 0whenever sheis unable to meeta repayment at t =1.Giveconditions on K for a pooling equilibrium with

debt to exist.Giveconditionson K for a separatingequilibrium to exist.Can 6H entrepreneursbebetter off with K > 0 than with K=0 (in their

preferred equilibrium)?

Chapter4

Question10Takea standard moral-hazardproblemwherethe principal considersoffering

the agent a lottery rather than a fixed payment w[ if output qt isobserved.Specifically,the agent would receivein this casewtj with

probability pij > 0,with ;=1,2,...,m and

m

Assume the agent'sutility function is u(w) - y/(a). Show that such arandomizing incentive schemecannot beoptimal if the principal is risk neutral

and the agent is strictly risk averse.

Question11Considera risk-averseindividual [with utility function of money u(-)]with

initial wealth W0 who faces the risk of having an accidentand losing anamount x of her wealth. Shehas accessto a perfectlycompetitive marketof risk-neutral insurers who canoffer schedulesR(x) of repayments net ofthe insurancepremium. Assume that the distribution of x, which dependson accident-preventioneffort a, has an atom at x =0:/(0,a) =1-p(a)and

f(x, a) = p(a)g(x)for x > 0.Assume p\"(a) > 0 > p'(a).The individual's

(increasingand convex)costof effort, separablefrom her utility of money,is y(a). Determine the first-best and second-bestinsurance contracts.Discuss.

653 Appendix

Question12Considera principal-agentproblemwith three exogenousstatesof nature,

0i, 02, and 03; two effort levels,aLand aH\\ and two output levels,distributed

as follows as a function of the state of nature and the effort level:

Stateof nature 0i 02 03

Probability 0.25 0.5 0.25Output under aH 18 18 1Output under aL 18 1 1

The principal is risk neutral, while the agent has utility function ^w when

receivingmonetary compensationw, minus the costofeffort, which isnormalized to 0for aLand to 0.1for aH- Theagent'sreservationexpectedutility

is 0.1.1.Derive the first-best contract.2.Derivethe second-bestcontractwhen only output levelsare observable.3.Assume the principal can buy for a price of 0.1an information systemthat allows the partiesto verify whether stateofnature 03happenedor not.Will the principal buy this information system? Discuss.

Question13Considerthe modified linear managerial-incentive-schemeproblem,wherethe manager'seffort, a, affectscurrent profits, qx =a +e?l,and future profits,

q2 =a+\302\243q2,

wheree?rarei.i.d.with normal distribution N(0, cr^).Themanagerretires at the end of the first period,and the manager'scompensationcannot bebasedon q2.However,her compensationcandependon the stockprice P= 2a +

\302\243P,where

\302\243P

~ N(0,op).Derive the optimal compensationcontractt = w +fq1+sP.Discusshow it dependson dp and on its relationwith o].Compareyour solution with that in the chapter.

Question14

Considerthe following principal-agentproblem.There is a projectwhoseprobability ofsuccessis a {ais alsothe effort madeby the risk-neutral agent,at costa2).In caseofsuccessthe return is R, and in caseof failure the return

is 0.TheparameterR can take two values,X with probability X and 1with

654 Exercises

probability 1-A. To undertake the project,the agent needsto borrowanamount / from the principal.The sequenceof eventsis as follows:

\342\200\242 First, the principal offers the agent a debt contract,with face value DQ.The agent acceptsor rejectsthis contract.\342\200\242 Second,nature determinesthe value R that would occurin caseofsuccess.Thisvalue is observedby both principal and agent.Theprincipal can then

chooseto lowerthe debt from DQto Dh\342\200\242 The agent choosesa levelof effort a.This level is not observedby the

principal.\342\200\242 The projectsucceedsor not. If the projectsucceeds,the agent pays theminimum ofR and the facevalue of debt Dx.

Answer the following questions:

1.Computethe subgame-perfectequilibrium of this gameas a function of/, A, and X.2.\"When do we have Dr < DQ1Discuss.

Question15An entrepreneurhas two projectsavailable,each requiring an investment

outlay of 6 at t =0.The first projectgeneratescashflow Cx e {5,45}at t =1.The secondprojectgeneratescashflow C2e {0,48}.The probability ofgetting a high cashflow is in eachcaseequalto a,wherea alsodenotesthelevelof effort of the entrepreneur.The entrepreneurhas cost of effort of40a2and can choosebetweenthree effort levels:a e {0,j,\\}. The firm hasno assetsin place.Everybody is risk neutral, and there is no discounting.The two projectsare mutually exclusive.

1.If the entrepreneurcanself-finance,what levelof effort will she chooseunder eachproject?Which projectis worth investing in?

2.Supposenow that the entrepreneur is cash constrainedand that the

projectis entirely financed with debt.What facevalue D of debt should the

entrepreneurchooseunder eachproject?Which project doesshe end up

choosingif she can get an unconditional loan,that is, a loan that doesnot

dependon which of the two projectsshedecidesto invest in? Discuss.3.Can the entrepreneurdo better by issuing a fraction s of equity insteadof financing the projectwith debt?

655 Appendix

Question16Considera target firm with widely dispersedownership (atomistic risk-neutral shareholders)facing a takeoverbid from a risk-neutral \"raider.\"

Themonetary value pershareunder incumbent management is normalizedto zero.Once the raider has gainedcontrol,she obtains a private benefitZ > 0 and can alsoexert effort a at private cost ka2/2, which generatesavalue pershare of aVR whereVR > k.

Gaining controlrequiresacquiring 50% of the sharesof the target (thesecurity voting structure is \"one share,one vote\.")Assume the raider doesnot own any sharesprior to making the offer.There are no coststo thetakeoverother than the pricethe raiderhas to pay.Incumbent managementis assumedto remain passive.

Theraidermakesa public tenderofferofa pricepershareofb.Thisofferis \"unrestricted\" and \"conditional\"; that is,the raiderbuys all the sharestendered providedat least 50% of the sharesof the firm are tendered.Facedwith this offer, target shareholdersnoncooperativelychoosewhether totenderornot (assumethat they think their tendering decisionwill not affectthe outcomeof the takeover).

1.Derivethe posttakeovervalue pershare,given that the raiderhas gainedcontroland holdsa fraction of sharess > 50%.2.At what bid priceb will the raiderbeable to acquirea fraction ofshares5>50%?3.Derive the raider'soptimal offer b* and optimal posttakeoverownership

stakes*,respectively,when Z =0 and Z > 0.4. Is the raider'soptimal bid efficient; that is, doesit maximize the sum ofshareholderand raider returns?

Chapter5

Question17Considera sellerof a single item facing two potential buyers. The itemis either worth vH or vL < vH (the buyers agreeon thesevaluations; that is,it is a \"common-value environment\", as defined in Chapter 7) and thecommon prior probability is Pr(v = vH) =0.5.

656 Exercises

Supposethat the sellerprivately receivesan estimateof the value of theitem from an expert(in the form ofa signedwritten letter)prior to the sale.Thereare n possibleestimates

e\342\200\236with i =1,...,n. Assume -E[vleJ = v, such

that

Vi < V2 < \342\200\242\342\200\242\342\200\242<V\342\200\236

The buyers have no information about what the estimateis likely to be.Theirbeliefsare

Pr(e,)=-n

1.Show that the unique equilibrium of the game wherethe sellersellsthe

object through an ascending-priceauction (or \"English auction\"; seeChapter7) is for the sellerto first disclosethe content of the expert'sestimate to the bidders.2.What is the equilibrium in the English auction when the sellermust paya costK > 0 to obtain an expert'sassessment?

Question18Consider the following stylized exampleof trading by two risk-neutralmarket makers, each of them small enough so that the directinfluence oftheir trade on the stockpricesis negligible.It is common knowledgethat

one of the traders will becomeinformed about the true underlying valueof a stockwith probability /3, while the other always remains uninformed.Assume each market maker can trade up to one unit twice in succession.Thequestionis,should the market makers berequiredto disclosetheir first

tradebefore they initiate their secondtrade?Thequestionis consideredatan ex ante stage where both market markers are identicaland neitherknows who will becomeinformed (if any). So,at stage0,the market makersmust decidewhether to introducemandatory disclosureof trades at stage1.

Prior to stage1,the stockis worth either110or 90,with equalprobabilities,

so that the stockpriceis 100.At the begirrning of stage1,one marketmaker becomesinformed about the value of the stockwith probability j3,while the other remains uninformed.

For the first trade,assumethe uninformed market maker buys with

probability 0.5and sellswith probability 0.5.Following the first trade,theremay

657 Appendix

bemandatory or voluntary disclosure.Thena secondtrade canbe initiated

by both informed and uninformed market makers.

1.What are the ex postpayoffs of the informed and uninformed tradersunder mandatory disclosureand no mandatory disclosure,respectively?2.When is mandatory disclosurebetter from an ex ante perspectivethan

no mandatory disclosure?3.When there is no mandatory disclosure,when is there voluntarydisclosure? Discuss.

Question19Considera two-periodmodelwith no discounting. A risk-neutral

entrepreneur has a projectgenerating at t =1a random cashflow n e \\nL, nH} with

kh > 7rL and fi =Pr[7r=7rJ.Theprojectrequiresan investment / at t =0 that

is raised from a risk-neutral creditor in a competitive capital market.

Supposethat at t =1the entrepreneurobservesn, but the creditorcanverify

k only by incurring a monitoring costK (assumethat verification costsaresmall enough for financing to beviable).Let

(57tL+(l-[5)7rH>I+K

and I>kl>K,and assumethat repayments must benonnegative.

1.Assume that only deterrriinistic monitoring contractsarefeasible.Derivean optimal financial contract.Explain why there is no unique optimalcontract within this setting.2.Show that random verification isoptimal. Explain.3.Focusagain on deterrninistic contracts,and supposenow that the

entrepreneur is risk aversewhile the creditorremains risk neutral. Identify the

unique optimal contract.Explain.

Question20

Considera financial contracting problembetween a wealth-constrained,risk-neutral entrepreneurand a wealthy risk-neutral investor. The cost ofinvestment at date t = 0 is /.The project generatesa random return oninvestment at date t =1of tu{6,1)=2min {6,1},where6isthe stateofnature,

uniformly distributed on [0,1].

658 Exercises

1.Characterizethe first-best levelof investment, IFB.

2.Supposethat the realizedreturn at t =1is freely observableonly to the

entrepreneur.A costK > 0 must bepaidfor the investor to observek{6,I).Derive the second-bestcontractunder the assmptions of (a) detemiinisticverification and (b) zero expectedprofit for the investor, taking into

accountthat repayments cannot exceedrealizedreturns (net of inspectioncosts).3.Show that the second-bestoptimal investment levelis lower than IFB.

Chapter6

Question21A risk-neutral profit-maximizing monopolistproducing two goods,good1and good2,facesa consumerwith utility function vxqx + v2q2-T,whereTis the payment from the consumerto the monopolist,qt is the quantity ofgoodi(i =1,2) bought by the consumer,and v; is the consumer'svaluation

for good i.Themonopolist'scostis c(qi)+c(q2).Each consumervaluation

v, can take two values, vf and vf. DefineAv,- = vf -vf > 0.Assume that Vi

and v2 are independently distributed and Pr(v; = vf) =j3,-. The monopolist

doesnot observethe realizationof the v;'s,but their distribution is common

knowledge.

1.Supposethat the monopolist decidesto treat the contract-designproblemas two separateproblems,one for eachgood.\"What is the second-bestoutcome?2.Assume that the monopolistnow decidesto considera generalcontract instead of two separate ones.How many (IC)constraints doesthe monopolistface? How many (IR) constraints?(Donot write them

down.)3.Considerthe relaxedproblemwith only the following constraints:(a) the

(IR) constraint for type (vf-, v2); and (b)the following four (IC)constraints:for type (vf, v2) versustype (vi\", v2), (IC1);for type (vf, vf) versustype (vf,vf), (IC2);for type (vf, vf) versustype (vf, vf), (IC3);and for type (vf, vf)versustype (vf, vf), (IC4).Write down this relaxedproblem.

659 Appendix

4. Show that for this relaxedproblem the (IR) constraint for (vf-, vk) isbinding. Show that constraints (IC1)and (IC2)are alsobinding. Usetheseresults to eliminate from the monopolist'sproblem the payments it getsfrom types (vf\", v%), (vf, v%), and (vf\", vf).That is,write the relaxedproblemas a maximization problemwith respectto quantities and the payment the

monopolistgetsfrom type (vf, vf), subjectto the two constraints (IC3)and

(IC4).5.Write the Lagrangian of this last problem. Denote by Xx and ^the Lagrange multipliers of (IC3) and (IC4),respectively.Show that

Aj + Az = 1/[(1-A)(l -fii)].Show that the quantities of good i(i= 1,2)sold to the types who have a high valuation for that goodare as in part 1of this question.Discuss.6.Write down the first-orderconditionsfor the quantities of goodi(i=1,2) soldto the types who have a low valuation for that good.Show that Xx

and 7^i are strictly positive.Comparethem to the answer of part 1.Discuss.7. Considerthe symmetric casev\\ = v\\ = vL, vf =vf = vH, /3i = /32= j3. Showthat the solution to the relaxedproblemis also the solution to the initial

problem.

Question22

Consideran agent who works for a risk-neutral principal.The agent canallocate time to n + 1tasks.Call a, the amount of time spent on task

i(i= 0, 1,2,...,n).The principal caresonly about task 0, getting output

q = aQ +\302\243,

where e is normally distributed with mean 0 and variance<j2.

The agent, however, derivesa benefit v^a) from spending time on tasksi =1,2,...,n. Shehas CARA risk preferences,and her utility function is

_\302\243-T][\302\273'-y/{aQ+ai+-+an)+vi{ai)+V2{a2)+- +v\342\200\236(a\342\200\236)]

where v^a0 + ax +...+an) is the cost of time, with iff' and iff\"> 0 and

y/(0) =0.Assume alsothat v- > 0,vf < 0,and v,(0)=0,and that optimizationwith respectto the a,'sleadsto interior solutions. Callvv0 the agent'sreservation wage.

1.Derive the first-best outcome.Discuss.2.Assume that the principal doesnot observethe a;'schosenby the agent,but only q and whether the agent can engageat all in any of the tasks 1,2,...,n.Assume the principal canoffer the agent a linear incentive scheme

660 Exercises

t +sq and he can alsochoosethe subsetSof tasks the agent is allowedto

engagein (that is,for any other task ;*,the principal canforcea;=0).Determine the optimal subsetS as a function of s.What happenswhen 5 = 1?Comparewith the answer in part 1of this question.What happenswhen sdropsbelow1?Discuss.

Question23

Considera risk-neutral principal who has to take a decisiond e {A, 0,B}.Theoptimal decisiondependson a random parameter6=6A+ 6B.Exante,we have Pr(6U =-1)= j3 = Pr(0fl = 1)and Pr(6U = 0) = 1-j8

= Pr(0fl = 0).Moreover, 6A and 6B are independently distributed. Assume that the

optimal decisionis A if 0=-1,0 if 0=0,and B if 6=1.Let K be the costto the principal of not taking the optimal decisionfor any 0.

The principal has accessto a population of risk-neutral agentswho, by

spending oneunit ofeffort looking at 0;(i =A, B),canobtain hard evidencethat I0J =1.Specifically,if 0,=0,no evidenceis found, while if 10^1 =1,hard

evidence(that is,evidencethat cannot beforged)is found with probability

p.Agents' utility functions are w -ny/, wherew is their wage,n the number

of units of effort expendedby the agent (n =2 means that the agent looksat both 6A and 0B),and y/\"the unit costof effort.

1.What is the first-best effort levelin this case?When is it optimal to havetwo units of effort expended?2.Assume that agentsprivately chooseeffort but canbepaidperpieceofevidenceprovided.Can the first-best effort levelbeachievedwhile leavingno rents to the agent(s)?Doesit matter whether oneor two agents(expending, respectively,two or oneunits ofeffort) arehired by the principal?Doesthe outcomedependon agents' limited-liability constraint w > 0?3.Assume now that the hard evidenceis automatically transmitted to the

principal but that contractscanbecontingent only on the decisiontaken bythe principal, and not on the amount of information generated by the

agent(s).What is the optimal contractfor the principal [i.e.,the contractthat minimizes the expectedcostof wrong decisionsplus the wagebill tobeconcededto the agent(s)]when a single agent is hired by the principal?And when two agentsare hired by the principal?Again, doesthe outcomedependon agents' limited-liability constraints?

661 Appendix

4. Assume finally that, whenever the hard evidenceis generated,it is first

obtainedby the agent, who can then decidewhether or not to discloseit tothe principal.Does this new sequenceof moves change the conclusionreachedin part 3 of this question?

Question24

A risk-averseentrepreneur [with strictly increasingutility function u(-)]producesrandom output q, uniformly distributed on [0,q],with q > 0.Theentrepreneurwants to diversify risk by writing a risk-sharing contractwith

a risk-neutral financier (with initial wealth W > q),specifying a transfer tothe entrepreneurcontingent on realizedoutput.

Output isobservableto the entrepreneur.It is alsoobservableto thefinancier unlessthe entrepreneurfalsifies her accounts.After observing therealizationof q, the entrepreneurcanproducea falsifedoutput reportR atcost

\\{r(q,R) =\\(q-R)+\\c(q-R)1wherec>0.Supposethat the entrepreneuris protectedby limited liability

and that her reservationutility u is higher than ql2.

1.Characterizethe first-best contract.2.A contract with no output falsification is such that R(q) = q for all

q e [0, q]. Show that the first-best contract would lead to falsification.Derivethe entrepreneur'sequilibrium falsification in responseto the first-

bestcontract.3.\"What is the optimal no-falsificationcontract?Show that the optimalcontract with no falsification is linear in output q (hint: all incentive constraintsare satisfied as long as the contractis locally incentive-compatibleat any

q e [0,q\\).

4. Explain why a contract with falsification may dominate the optimalcontractwith no falsification when c \342\200\224\302\273 +\302\260o.

Chapter7

Question25

Consider a public-goodproblem representedby a decisiond e [0,1].Decisiond has an impact on N individuals: Individual i = 1,2,...,N has

(differentiable)utility function vt(d, 6t) - T\342\200\236where 6t isa privately known

parameterand T[is the payment made by the individual to the \"planner.\"

The sociallyefficient decisionis

d*{6)=argmax^,v, (d, 6t)

where 6= (6U...,0N).

1.Derivethe \"Grovesmechanism,\" that is,the directrevelationmechanismthat implements d*(6)as a dominant strategy equilibrium. Show that it is

unique. Todo so,show that socialefficiencyimplies

0, =argmaxftXv;[d*(0_/,0,),0;-]i

where0, is any announcement that individual i couldmake.Usethis resultand the ICconstraint for individual i to derivethe Grovesmechanism asthe solution of a differential equation,which is unique up to a constant.2.Show that budget balance[i.e.,^T^O)=0for all 0]in the Grovesmechanism canbe satisfied if and only if d*(6)is \"(n -Inseparable.\"Note that

a function F(m), with m = (mi, m2,...,m\342\200\236),is (n -Inseparableif and only

if it canbe written as

i

3.Rather than insisting on dominant-strategy implementation, focus onBayesianimplementation. Assume that the 0,'sare distributed

independently.Show that there always existsa budget-balancingmechanism that

leadsto truth telling asa Bayesianequilibrium. Todoso,start with a Grovesmechanism and redistribute the surplus/deficit among the agentsso as toleave the ICconstraints unaffected.

Question26

Considera continuum ofsellersand a continuum ofbuyers, eachofmeasure1.Each sellerowns initially one unit of the good.Sellervaluations for that

good(denotedby vs) are i.i.d.on [vy, vs] with density fs(ys).Each buyer is

potentially interested in buying one unit of the good.Buyer valuations

(denoted by vB) are i.i.d.on [vB, vB] with density /b(vb). Assume that

VB< Vs.

Callxs(vs) the seller'sprobability of selling given an announcedvs, and

xB(yB)the buyer'sprobability ofbuying given an announcedvB.Call T5(v5)

663 Appendix

the seller'spayment (to a \"planner\") given an announcedvs, and TB(yB)thebuyer's payment (to a \"planner\") given an announcedvB. Considerthe\"Walrasian mechanism,\" dennedby

xs(vs)=l and Ts(vs)=P ifv5<P

xs(ys)=0 and Ts(vs)=0 iivs>PxB(yB)=l and TB(yB)=-P iivB>PxB(vB)=0 and TB(vB)=0 ifvB<P

1.Show that the ICconstraints and IR constraints are satisfiedfor any

priceP.2.Show that there exists a price P such that trade is efficient andbalancednessis satisfied.3.Comparewith the Myerson-Satterthwaite (1983)result.

Question27

A decisionwhere to locate a hazardous-wastedump is taken through anauction betweenn towns in a given country. Calldt town z's disutility from

taking on the dump. Assume the d,-'sare uniformly and independentlydistributed on [0,1].Call Tt the transfer the town requestsfor taking thehazardous-wastedump. The lowestbiddergets the dump and receivesits

requestedtransfer, which is paid equally from the other towns. Computethe symmetric equilibrium of this auction.Is this an efficient allocationmechanism?Discuss.

Question28

Considera two-person,independentprivate-value auction with valuations

uniformly distributed on [0,1].Considerthe following assumptions onutilities: (a) bidder i(i= 1,2) has utility v, -P when she wins the objectandhas to pay P, while her outsideoption is normalized to zero; (b) bidderi(i= 1,2) has utility Vv,- -P when she wins the objectand has to pay P,while her outsideoption is normalized to zero.

1.Compare the seller'sexpectedrevenue in cases(a) and (b) for the

Vickrey auction.2.Compare the seller'sexpectedrevenue in cases(a) and (b) for the

(linear)symmetric bidding equilibrium of the first-price,sealed-bidauction.

3.Discuss.

664 Exercises

Question29

Consideran auction setting wherea risk-neutral sellerof a housefacestwo

potential risk-neutral buyers. Buyer 1is a real-estateagent who knows themarket value of the houseperfectly.Buyer 2 doesnot know the marketvalue of the house,and the samegoesfor the seller.Theselleris determinedto sellthe house,while eachbuyer is uninterested in buying at a pricehigherthan the expectedvalue of the house.Thevalue of the housev is either vH

or vL with vH > vL, and the seller'sand buyer 2'sex ante beliefsare givenby /3 = Pr(v = vL). Exceptfor the realizationof v, all the precedinginformation is common knowledge.

1.Show that buyer 2 obtains an expectedpayoff of zero in the first-pricesealed-bidauction.2.Which of the following standard auctions maximizes the seller'sexpectedrevenue:the English, Vickrey, or Dutch auction?3.What is the optimal auction?4. Assuming now that v is uniformly distributed on the interval [0,1],characterize the Bayesianequilibrium in the first-pricesealed-bidauction.What

is the equilibrium payoff of the uninformed buyer?

Chapter8

Question30

Considera risk-neutral principal who employs two agents, i =1,2,who canproducean amount qt =

/(\302\253,-, \302\243,),wherea, is agent z'seffort leveland e,is a

random shockwith (atomless)density g(ei).Thee,'sare identically and

independently distributed. Eachagent has an outsideopportunity levelof utilitynormalized to zero.Agents are not wealth constrained.

1.Assume first that each agent i is risk neutral, with utility equal to

monetary compensation,wt, minus a convexcostof effort!//(\302\253;). Assuming that

a positivelevel of productionis desirable,derive the first-bestoutcomewhereeachagent has a zeroexpectedpayoff. Show how it canbeachievedthrough an incentive schemewhereeach agent is rewardedaccordingtoher output only. Show how it can alsobe achieved(assuming that asymmetric pure-strategyNash equilibrium exists)through a simple tournamentwhere the agent who is \"behind\" in terms of individual output is paid anamount Wi and the one who is \"ahead\" is paid an amount Ww. Discuss.,

665 Appendix

2.Assume now that each agent i is risk averse,with utility equal to

u(wt)-

!//(\302\253,),where u(-) is strictly increasingand concave.Extend the

classof possiblerelative-performanceincentive schemesto W\\(qi, q2) andW2(qi, q2).Considerany pair (al9 a2) that canbe sustainedas a Nashequilibrium asa result oftheseincentive schemesand that alsogiveseachagenta nonnegative expectedutility level.Show that this pair of effort levelscanalso be sustainedin this way by a pair of incentive schemesW\\(qx) and

W2(q2) that lacksrelativeperformanceevaluation and that doesnot have ahigher expectedcostfor the principal.Discuss.

Question31Two agents can work for a principal.The output of agent i, i = 1,2, isqt =

fl; +e\342\200\236

whereat is agent z'seffort leveland e,-is a random shock.Thee,'sare independentof each other and normally distributed with mean 0and varianceo1.

In addition to choosinga2, agent 2 canengagein a secondactivity /32.This

activity doesnot affectoutput directly, but rather reducesthe effort costofagent 1.The interpretation is that agent 2 can help agent 1(but not theother way around).The cost functions of the agentsare

1 2Yi(a1,b2)=-(a1-b2)

and

Yi(a2,b2)=-a2l+b21

Agent 1choosesher effort levelax only after shehas observedthe levelofhelp b2.Agent i'sutility function is exponentialand equal to

wherew, is the agent'sincome.The agent'sreservationutility is -1,which

correspondsto a reservationwageof0.Theprincipal is risk neutral and isrestrictedto linear incentive schemes.The incentive schemefor agent i is

Wi=Zi+viqi+uiqj

1.Assume that ax, a2, and b2 are observable.Solvethe principal'sproblemby maximizing the total expectedsurplus with respectto ax, a2, and b2.Explain why ax > a2.

666 Exercises

2.Assume from now on that au a2, and b2 are not observable.Solveagainthe principal'sproblem.Explain why ux =0.3.Assume that the principal cannot distinguish whether a unit of outputwas producedby agent 1or agent 2.The agentscan thus engagein

arbitrage, claiming that all output was producedby one of them. Assume that

they will do sowhenever it increasesthe sum of their wages.Explain why

the incentive schemein part 2 leadsto arbitrage.What additional constraint

doesarbitrage imposeon the principal'sproblem?Solvethis problem,andexplainwhy ux > 0.

Question32

Two agents work for a principal.The output of agent i, i = 1,2, is q{ =al +

e\342\200\236

where a{ is agent z's (privately observed)effort level and e< is arandom shock.The e;'sare normally distributed with mean0 and variance-covariancematrix

Agent z'sutility function is

_e-T,[w:-ca?/2]

wherevv; is the agent'sincome.Eachagent'sreservationwageis O.Theprincipal

is risk neutral and is restricted to (symmetric) linear incentiveschemes.Specifically,the incentive schemefor agent i is

wt =z+vqi+uqj

Agents can colludeby writing a sidecontractbefore efforts are chosen.Assume agent 1canmake the following take-it-or-leave-itcontractoffertoagent 2:s=<j>(ql-q2)+(p

1.Derive the optimal side contract for an arbitrary incentive scheme(v, u), assuming that effort choicescannot bepart of the sidecontract.2.Show that the principal can without lossof generality restrictattention

to collusion-proofcontracts.

667 Appendix

3.Derive the optimal collusion-proofcontract.Discuss.4. Would the principal benefit if collusionwere made impossible?Howwould the answer changeif agentscouldwrite a sidecontractconditionalon effort choices?

Question33

A (risk-neutral) municipal government considersfunding an investment

project put forward by an association(also risk-neutral). The cost of the

project is known, but the government is unsure about its socialvalue, andits assessmentis at oddswith that of the association.Specifically,if the

projectis of \"good quality,\" its socialvalue (net of the costof the project)as assessedby the government is 6G > 0,while the associationwould derivea private benefit vG > 0 from seeingit go through. If insteadthe projectisof \"bad quality,\" its net socialvalue is 6B < 0, but the associationwould

derivea private benefit vB, higher than vG, if it went through. Theassociation knows the quality of the project,while the government's (common-knowledge)beliefis Pr(vB) = /3.

In the absenceof information, the government is ready to fund the

project,sincewe assume/30B+(1-j5)6G > 0.However,sincetaxation is dis-tortionary, the government has net value X > 0 for each unit of revenueraised from the association.However,the government would be unwilling

to allow a bad-quality project to go through even if it wereable to chargethe associationfor its full private benefit; that is,we assume6B + XvB < 0.

Assume that the government has accessto a (risk-neutral) \"expert\" who,when the projectis ofbad quality, managesto obtain an (unfalsifiable) proofof this fact with probability p, but observes\"nothing\" with probability

(1-p);\"nothing\" is alsoobservedwith probability 1when the projectis ofgood quality. The expert starts with no financial resourcesand cantherefore only be rewarded.The associationis assumedto observewhen the

expert obtains a proof of bad quality, while the government has to be\"alerted\" by the expert.1.Derivefirst the optimal schemefor the government when it cannot relyat all on the expert.2.What is the optimal schemewhen the government canrely on the expertand when collusionbetween the expert and the associationis impossiblebecausethe expert is \"honest\"?

668 Exercises

3. What is the optimal schemewhen the government canrely on the expertbut the expert is \"self-interested\" and the associationcan promise the

expert a sidepayment for not alerting the government when he obtains aproof of bad quality? Assume the collusiontechnology is such that, for

every unit ofmoney the associationpays,the expertonly collectsanequivalent of k < 1units of money.4. What is the optimal schemewhen the government is unsure about the

prospectfor collusion,becauseit believesthat with probability y the expertis \"honest\" and with probability 1-yhe is \"self-interested\"?

Chapter9

Question34

Consider a two-period durable-goods monopoly problem where asellerfaces a single buyer with reservationutility for the durable goodv e {vL,Vff} with vH > vL > 0.Theseller'sprior beliefabout the buyer'sreservation utility is Pr(v = vH) = 0.5.The seller'scost of producing the goodcan also take two equally likely values:c e {cL,cH) with cH>cL>0.Sellercostsand buyer reservationvalues are independently distributed and areprivate information. Assume

vH-cLvL-cL>\342\200\224-\342\200\224

and

vL-cH<- 2

The common discount factoris given by 8> 0.1.Underwhat conditionsdoesa pooling equilibrium exist in the first

periodwhere

(a) both types of sellerseta first-periodprice

Pi=VH--(yH-vL)

669 Appendix

(b) the rype-vH buyer acceptsthis pricewith probability

vH +cH-2vL7=vh-Vl

and the type-vL buyer rejects it with probability 1;and (c)following aperiod-1rejection,the type-cLsellersetsp2 = vL in the secondperiod andthe type-cnsellersetsp2 =vH'm the secondperiod?2.Explain why the sellergains from having private information about costswhen his cost is cL,but not when it is cH.

Question35

Considera two-periodregulation modelwhere the \"type\" of the firm is

endogenous.Initially, the regulator offersa revenue function R\\{q)

specifyingthe payment it offersfor output levelq.Then,the firm sinks a

(privately observed)amount / that determinesits per-periodproduction costc(q,I)[with c(0,1)=c?(0,1)=0,cqq(q, I)> \302\243> 0,and c7(g,I)< 0],and choosesto producequantity qx, which leadsto first-periodpayoff

RM)-c(qi,D-IIf instead the firm quits, it earns-Iand the game is over.If the firm haschosenqu the regulator'sfirst-periodpayoff is q\\

-R\\(q\\).

In the secondperiod,after having observedqlt the regulator offersRi(q),upon which the firm either quits (leadingto a zero payoff for both partiesin period 2) or choosesq2, with associated second-periodpayoffsi?2(^2)-c(q2,1)and q2 -^2(^2)for the two parties.

1.What is the full-commitment strategy of the regulator?2.In the absenceof commitment, show that this game has no pure-strategy perfectBayesianequilibrium with / > 0.

Question36

Considerthe soft-budget-constraintsetting discussedin Chapter9 (section9.1.3),with the following variation: Assume that each individual investor isinfinitesimal, but can, in a first stage of the game, join a \"small creditor,\"with one unit of funds in total, or a \"large creditor,\" with two units of funds.

Assume away agencyproblemsbetweenthe manager running this

undertaking and the small investors.Beyondthis initial stage,the setting is as in

section9.1.3.

670 Exercises

1.Assuming first a pure adverseselectionsetting for entrepreneurs(thatis,the goodentrepreneurcanchooseonly a \"quick\" project),determinethe

equilibrium (assuming that creditorsplaguedby the soft budget constraint

canneverhope to get a higher probability of goodentrepreneursthan the

population average/3\342\200\224to get rid of \"Rothschild-Stiglitz nonexistenceproblems\" discussedin Chapter13).2.Add moral hazard for good entrepreneurs;that is, they can choosebetweenthe \"quick\" projectand a \"goodbut slow\" project,whosepayoffsare as in section9.1.3.3.When is there is a single equilibrium of the game?When are there two (Pareto-ranked)equilibria?

Question37

Consider the following investment/insurance problem under privateinformation:

A risk-averseagent invests an amount p/2 in a project with randomincomeshocksin two periodst=l,2,with

fl with probability p10 with probability 1-p

and vv2 e {0,1},with

fPr(w2=l|w1=l)=7<Jp\\pr(w2=l\\w1=0) =ii>pwherey > 0.5and p < 1.Theagent'sutility function is time separable:U(ci,c2)= \"(ex)+u(c2)with u(c) taking the following piecewiselinear form:

u(c)= 2+2T 2)

\\[c for c < \342\200\224

Theagent canobtain insuranceagainst the incomeshocksat actuarially fair

rates at the beginning of every period.1.Characterizethe first-best optimal consumption allocationunder the

assumption that the agent cannot do any private saving.2.Assuming that incomeshocksare private information, show that when

only spot contractsare feasible,the agent cannot get any insurance.

671 Appendix

3.Supposethat the agent canborrowfrom and lend to a bank at zerointerest rate. Characterizethe agent's optimal payoff under borrowing and

lending.4. When is insurancein the form of borrowing and lending an optimalcontract?

Chapter10Question38

Considera two-periodprincipal-agentproblem,where, in period 1,the

agent chooseseffort levela,which producesindependently and identicallydistributed profit outcomesin eachperiod,qx e {qL, qH) and q2 e {qL, qii)-The profit outcomeqH occurswith probability p(a)\342\200\224a strictly increasingfunction of a\342\200\224and the outcomeqL with probability [1-p(a)],with 1>p(a)> 0 for all a e [0,a],where a < \302\253>. The agent'sutility function is u(w) -a,with u' > 0; u\" < 0.The agent can neither borrow nor save,so that she isforced to consumewhat she earns in each period.The principal is riskneutral and canborrowor lend at zerointerestrate.

1.Let {wL, wH, wLL, wLH, wHl, Whh) denote the agent's profit-contingent

compensationin periods1 and 2.Show that the optimal contingent-compensationcontractmust satisfy the equation

1 pifl) , l-p(g) , . ,\342\200\236

-7\342\200\224-=

\342\200\224^

-+\342\200\224;\342\200\224-, for* =L,#

u (wt) u {wiH) u (yvlL)

2.Using Jensen'sinequality show that

wt <p(a)wiH +[1-p(a)]wlL, for i =L,Hunder the optimal contract,when 1/m' is concave.3.Supposenow that under the precedingoptimal contract the agent isallowedto saveand borrowat date1following the realizationofqx. Explainwhy shewould want to save.

Question39

A firm has assetsin placeand a new investment opportunity, and lives forthree periods.In period t =0, the firm's debt structure is chosen(that is, its

levelof short-term debtDxmaturing at t =1,and long-term debtD2due at

672 Exercises

t =2).At t = 1,the assetsin placeyield a return C\\, and a new investment

opportunity appearsthat requiresan investment outlay /.At t =2,the assetsin placeyield a further return C2,and the new investment projectgeneratesa cashflow CN, if it hasbeenundertaken at t =1.At t =2, the firm isliquidated and proceedsare distributed to investors.That is, outstanding debtclaims are repaid,when feasible,and the residualproceedsare distributed

to shareholders.Thecashflows Cxand C2areknown at t =0,but CN remainsuncertain until t =2.Exante it is common knowledgethat CN e {CL,CH)with CH> / > CL> 0 and Pi[CN= CH]= y.

The firm is run by a manager who decideswhether or not to undertakethe projectat t =1.Themanager is an empirebuilder.Shealways choosesto undertake the project if she can.So,if there is sufficient financial slackat t =1,the manager will invest. If this is not the case,the manager will turn

to a new lender (bank) for a loan to fund the investment outlays. (Fundsborrowedat t =1canalsobeusedto repayZ>i.)Debtraisedat t =1is juniorto all existing debt,but senior to equity. The debts Dx and D2cannot berenegotiated.If the firm defaults at t = 1,there is a bankruptcy costk > 0and the firm is liquidated.(Therearenobankruptcy costsif the firm defaultsat t =2.)Finally, all agentsare risk neutral, and the risklessrate of return

is zero.

1.Supposethat d =D^Doesthis assumption prevent the manager from

undertaking projectswith a negative net presentvalue (NPV) at t =1?Thatis, how much additional funding can the manager raise,and what projectswill be undertaken?2.Relax the assumption that Cx=Dh State the condition (asa function ofDi and D2)for the manager to be able to undertake the new project.3. \"What are the optimal values of D^ and D{1\"What values of Dx andD2ensure that the investment will be undertaken if and only if its NPV is

positive? Explain the role of short-term debt in affecting managerialinvestment behavior.4. Now assume that Cx is a random variable independent of CN andwith the known probability distribution Cx e [Ct,C?)with C?> C{and

Pr[Ci = Cf] = 77. Assume also that all projects have a negative NPV

and that I>C?-C\\. What are the optimal values of D1and\302\243)2?

5.Drop the assumption in part 4 that all projectshave a negative NPV.

Supposeinsteadthat the uncertainty about CN isresolvedat t =1.That is,

Appendix

at the time the investment is made,the return is known. Assume further

that CH=2I= ACL. (Cicontinues to be a random variable).Show that forI > C?-Ct, it is optimal to setD^= C\\ and D2> C2.Explain why risky

long-term debt dominates risky short-term debt in this case.6.Assume now that the oppositecondition holds,that is,/< Cf-C[.Showthat risky short-term debt is necessaryto avoid overinvestment. What arethe costsassociatedwith avoiding overinvestment?

on 40

Considera variant ofHolmstrom's(1982a)career-concernmodel:Therearetwo identicalperiods.After the secondperiod the manager retires.Theoutput of the manager in period t is

qt =6+at\302\243t

+(1-at)et

where 6 is the manager'sunknown ability, at e [0,1]is the manager'sunobserved action, \302\243(

is an unobservedstochasticreturn term, and et is anobservedstochasticreturn term. Onemay think of the manager'sactionasa decisionto allocatea dollarbetweena firm-specificproject,which returns

\302\243(,and a market project,which returns et.The allocationis known only to

the manager.The market pays the manager her expectedvalue in each period

wt = E[qt\\It], which dependson the market's information It and its

expectation of the manager'saction.Assume that the market'sand the manager'sfirst period beliefsabout ability are such that 6 is normally distributed

with mean mg and precision(the inverseof the variance)he.Beliefsin thesecondperiodare updatedbasedon inferencesabout alsthe observedoutcome qx, and the observedmarket return e\\.

Assume that the returns e(and et are independentacrosstime as well asfrom one another.Each is normally distributed with zero mean and with

precisionshe and he, respectively.The manager is strictly risk averse.Herpreferencescanbedescribedby

2

m(wi , w2) =2OEM-T]var[w,])

where wt is her incomein period t, E is the expectationoperator,and varis the varianceoperator.The coefficientof risk aversion77 is greater than

674 Exercises

0.Note that there is no costassociatedwith choosingat. However,at isconstrained to lie in the interval [0,1].1.Write down the equationsthat characterizea rational-expectations(self-fulfilling) equilibrium for this model.2.Show that in the rational-expectationsequilibrium the manager will

necessarily choosethe first-periodallocationa.\\ = 0; that is, she will invest allthe money in the market project.3.Would this conclusionbe altered if we instead assumedthat the firm-

specificprojecthad an expectedreturn E(et) = 1?Discuss.

Chapter11Question41

An upstream supplierinvests x dollarsin periodt = 0 in acquiringtechnical skills to producecustomizedsoftwarein period t =1for a downstreamproducer.The type ofsoftwarerequiredand the terms of trade canonly bespecifiedin period t = 1.The total surplus from trading the softwarein

period t = 1is given by v(x), where v(-) is a strictly increasing,strictlyconcavefunction with v'(0)> 1,which is boundedabove [v(jc) \342\200\224\302\273 M < +\302\260o

as x \342\200\224\302\273 +oo].In period t =1the upstream and downstream producersarelockedin a bilateralbargaining situation, resulting in an efficient trade andin a 50/50split of the surplus from trade.

1.Explain why expostspotcontracting resultsin exante underinvestment.

2.Supposenow that a new computer to run the softwareis alsoavailablein periodt=l.With this new computer,total expost surplus generatedbythe softwareincreasesfrom v(jc) to V(x) =/[v(jc)]> v(jc), where/'> 0 and/\" < 0.What is the first-bestlevelof investment in skills in period t =0 giventhat the contracting partieshave accessto this new computerin periodt =1?Is this first-best investment levelalways higher than the first-best levelof investment obtainedwhen accessto the new computeris denied?3.Supposenow that the new computercan be ownedeither by (a) theupstream producer,(b) the downstream buyer, or (c)a third party owner.Under up- or downstream ownership there will be bilateral ex postbargaining resulting in a 50/50split of the total surplus.Underthird-party

ownership of the new computer,there is trilateral expost bargaining, and the

675 Appendix

owner of the computergetsthe full marginal contribution of the computer,V(x) -v(x), while the other two partiessplit the surplus v(x) in half. Showthat up- or downstream ownership dominatesthird-party ownership if /'>1,but third-party ownership dominateswhen \\ <f < 1.4. Explain why upstream, downstream, or third-party ownership maydominate when /'< -j.5. How would your answersto parts 3 and 4 changeif the trilateral

bargaining solution were given by the Shapleyvalue?

Question42

Considerthe following vertical integration problem:There are two risk-neutral managers, each running an asseta, where i = 1,2.Both managersmake ex ante investments. Only expost spotcontractsregulating trade arefeasible.Expost trade at pricePresultsin the following payoffs: JR(jci)

-Pfor manager 1and P-C(x2)for manager 2,where the jc,'sdenoteexanteinvestment levels.

If the two managersdonot tradewith eachother,their respectivepayoffsare

rUi,A)-Pm and Pm -c(x2,A2)

where At denotes the collectionof assetsowned by manager i. In this

problem,A[ = $ under /-integration, Al = {a^, a2} under z'-integration, andAl = [a]under nonintegration.

As in the Grossman-Hart-Mooresetting, it is assumedthat

R(x1)-C(x2)>r(*i, A1)-c(x2,A2)

for all (jci,x2) e [0,3c]2and allA\342\200\236

R'(x1)>r'(x1,{a1,a2})>rXx1,{al})>r'(x1,</\302\273>0

and

-C\\x2)>-c'(x2,{ai,a2})>-c\\x2,{a,})>-c'(x2,$)>0

1.Characterizethe first-bestallocationof assetsand investment levels.2.Assuming that the managerssplit the ex post gains from trade in half,

identify conditionson /(xt,A) andc'(jc\342\200\236 A) such that nonintegration is

optimal.

676 Exercises

3. Under the same assumption on ex post bargaining, identify conditionson /(xhA) and

c'(*\342\200\236 A) such that integration under the ownership ofmanager 1is optimal.4. Supposenow that

r(x1,A1)-Pm<

for all x\\ e [0,x] and all Ah Supposealso that

Pm -c(x2,A2)>

for all x2 e [0,x] and all A2. Under theseassumptions, bargaining under anoutsideoption would give the following outcome:

Equilibrium payoff of manager 1:R(x{)-C(xx) -[Pm-c(x2,A2)]

Equilibrium payoff of manager 2:[Pm-c (x2,A2)]

In other words, manager 2 getsher outsideoption.In what way would the

analysis and resultsabout optimal ownership allocationsand equilibriuminvestment levelschangeunder this new bargaining solution?

Question43

Considera firm seekingoutsidefinance.At date 0, the firm needsto raise/ and has no liquid funds. At date 1,it will have (verifiable) assetsin placeworth A and generateliquid returns Ci that are assumedto be nonverifi-

ableby outsiders(for example,becausethey are private information to the

firm). We have Ci= Cf with probability /3 and Cx= 0 with probability 1-/3, where /3Cf> /.The firm doesnot know the realizationof Ci at date 0.At date 2 the firm alsohas long-term returns (future earnings prospects)C2,which are alsounverifiable. The realizationof C2will be known to thefirm alreadyat date 1.We have C2= Cf with probability 7and C2= C2with

probability 1-7, and C2 > C2> A.A contractbetweenthe firm and an outsideinvestor is a pair of functions

(P,L),whereP=P (6) and L = L(0)dependon the information 6= (CltC2)availableat date 1.P< Cx is the firm's payment made out of Ch and L< A is the firm's payment made by liquidating assets.If an amount L ofassetsis liquidated, long-term earnings to the firm will be (1-L/A)C2.

Appendix

The firm and outside investors are risk neutral. At date 0, outsideinvestors have costsof funds equal to one and offer competing investmentcontracts.The firm then choosesthe most preferred one (randomizing if

offersare identical),and this contractis executed.Formally, therefore,atdate 1the firm announcesits information, and the payments specifiedin

the contractare carriedout.

1.Write down the parties'utility functions and the contracting problematdate0.2.What is the first-best allocation?3.Show that in the (second-best)optimal contracting problem thefunctions P and L must bemonotonein C2.Interpret.4. Solvethe contracting problemand discuss.5.Now supposethat the partieswill renegotiatedate-1inefficiencieswhenever possible,that is,wheneverthe firm has the funds to compensatetheinvestor for not liquidating. Supposealso,to simplify matters, that

information is symmetric ex post, that is, that the investor observes6 (but that

0isstill nonverifiable).Supposefinally that in theserenegotiationsthe firm

has all the bargaining power(that is,makesa take-it-or-leave-itofferto the

investor).Determinethe optimal renegotiation-proofcontract.6.Compareyour results in part 5 to those in part 4.

on 44

An entrepreneurwith no initial wealth has a projectthat requiresan initial

investment K and whoseoutput can take two values:q e {0,1}.Themarketinterestrate is normalized to zero.Theentrepreneuroffersa financial

contract to an investor. After the initial investment, both parties observetherealizationof the state of the world, 6 e {B,G), which, however,is notobservableby a court and thus is not contractable.Instead,a contractcanbe contingent on the realizationof a binary signal, s,which is verifiable in

a court.The signal is distributed as follows:If 0= G, then 5 = 1with

probability one.If 6=B,then 5=1with probability 7and 5=0 with probability1-7.Assume that 7is sufficiently small but strictly positive.

In eachstate of the world, an actiona has to be taken:a e {S,C),whereS is interpreted as \"stop\" (downsize,liquidate) and C is interpreted as\"continue.\" The probability of high output dependson the realizedstate ofthe world and on the actionchosen:Pr[g= l\\6,a]= a9.(Note that ag also

678 Exercises

expressesthe expectedmonetary return of the projectgiven actiona andstate6.)While the investor caresonly about monetary returns, theentrepreneur alsohas a private nonmonetary benefit h from choosingCrather than

S.Monetary and nonmonetary returns satisfy the following inequalities:

CG< SG< CG+h

CB+h<SB

SG\342\200\224 Cq<Sg\342\200\224 Cfl

Actions cannot bedescribedin an exante contract.Instead,a contractspecifies control rights, that is, it specifieswhich party has the right to choosethe action.Besides,the contractspecifiesthe entrepreneur'scompensationasa function of the realizeds and q.If the party in controlchoosesan actionthat is not Pareto optimal, the partiescan try to renegotiateto an optimaloutcome.Assume that the entrepreneurhas all the bargaining power in

renegotiation.

1.For each of the following control structures, find the values of K forwhich a contractimplementing the first-best actionchoiceis feasible:(a)entrepreneurialcontrol;(b) investor control;(c)contingent control(E hascontrolwhen 5= 1,7has controlwhen 5 =0).2.Under what conditionsdoeseachcontrolstructure dominate?3. Compareyour resultswith Proposition5 in Aghion and Bolton(1992).

Chapter12Question45

Considera public goodproblem.ThereareN > 3 agents.Theindicatorvariable y is equal to 1if the good is suppliedand 0 if not, and tt e R is thetransfer to agent i.The preferencesof agent i are quasi-linear:Qy + tt with

6i e RL We want to study socialchoicecorrespondences,/(\342\200\242),that we can

implement in Nash equilibrium.

1.Show that monotoniciry implies that/(\342\200\242)

satisfiesthe following two

conditions:

condition 1 Consider0=(01,...,6N) and (y, tlt...,tN) e / (6) such that

y = 1.Consideralso(j>

= (0i,...,<j)N)such that 0 > 6 (this means that Vi,

fr> 6t).We then have (y,tu...,tN) e /(0).

679 Appendix

condition 2 Consider0 and (y, tx,...,tN) e /(0)such that jc =0.Consideralso

<j)such that 0 >

<j> (this means that Vi, 0,> <ft). We then have (y, th ...,to) e M-2.Consider now

/(\342\200\242)

that satisfiesconditions1 and 2.Show that it ismonotonia3.Show that

/(\342\200\242)

satisfiesno veto power. Conclude that/(\342\200\242)

is imple-mentablein Nash equilibrium.

4. Show that we can implement an / (\342\226\240)

that satisfiesefficient supply [that

is, V0 and V(y, h,...,tN) e /(0),y = 1if and only if 2^0, 0] that doesnot involve transfers when y =0 and that is balanced[i.e.,V0 and V(y, tx,.\342\200\242\342\226\240,tN)ef(6), lf=1^=0].5.Explain why these results are satisfactory if one caresonly aboutefficiency (efficient supply and not throwing money away) but much lesssatisfactory if one caresabout a \"fair\" sharing of the costof the public good.

Question46

Consideran environment with three individuals (1,2,and 3) and five

outcomes, A,B,C,D,and E.Individual 1canbeof two types, which we call 0X

and 0!.Thesameis true for individual 2,who canbeoftype 02or (fa. Instead,individual 3 has constant preferences.Individual preferencesare asfollows:

\342\200\242 Individual l'spreferencesare summarized by A> B>C>D >E when

she is of type 0ls and by B >A>E>D >Cwhen she is of type fa.\342\200\242 Individual 2'spreferencesare summarized by A>B>C>D >E when

she is of type 02, and by B >A>E>D > Cwhen she is of type fa.\342\200\242

Finally, individual 3'spreferencesare summarized byE>D>OB>A.

Thesocial-choicefunction/(\342\200\242)

is definedas follows:

/(0l502)=A, /(0i,02)= \302\243>=/GM2), and/Xfa, <j)2)

=B

1.Show that/(-) is monotonicand satisfiesNoVeto Powerand is thereforeimplementable in Nash equilibrium.

2.Show that it cannot be implemented in dominant strategies,however.

680 Exercises

Question47

Consider an implementation setup without investment but with risk-

sharing. Assume that the parties,a buyer and a seller,have the following

utihty functions:

ub[v(q,0)-P]

us[P-c(q,6)]whereub, v, and us are increasingand concavefunctions, c is an increasingand convexfunction, and v(q, 0) =0 and c(q,0) =0 for all ffs. Contractingtakesplacebefore 6 is known, and trade (P,q) after 6 has beenobservedby both parties.

1.Describethe first bestin this setup.2.Assume that 6 is observablebut not verifiable, and that the partiescannot commit not to renegotiatebut can contractually agreeon message-contingent default optionsand allocationsof the entire bargaining powerto one party. Construct a revelationgame that implements the first bestwithout equilibrium renegotiation.3.Can a contract without messagesbut with equilibrium renegotiationimplement the first best?Interpret.

Question48

Considerthe contracting problemseenin sections12.3.1and 12.3.2,but,rather than assuming

CH > VH > CL > VL

assume

Vff > CH > VL > CL

1.Computethe first-best outcome.2.\"What is the optimal option contractunder the assumptions made byNoldekeand Schmidt (1995)?3.What happensif option contractsare not feasiblebecausethe buyer canalways claim that the sellerfailedto deliver the good,so that we are in theat-will-contracting world of Hart and Moore(1988)?

681 Appendix

4. \"What would happenif, starting from a Hart-Mooreworld, \"trade\" ratherthan \"no trade\" were to be the ex post disagreementoutcome?

Question49

Considera buyer-sellermodelwhere the sellercan make a costlyinvestment i in quality enhancement.Theirrespectivepayoff functions are

P \342\200\224 cq \342\200\224 i

for the seller,and

v(q,i)-Pfor the buyer, where vqi > 0; that is, the effect of a rise in quality-enhancementon the buyer valuation v(-) is increasingin the quantity tradedq.In addition, P and c stand for price and marginal production cost,as in

Chapter12.Prices and quantities are assumedto be contractable,while quality is

observablebut not contractable.The timing of the game is as follows:Instage0, the partieswrite an initial contract.In stage1,the sellerchoosesi(which then becomesa sunk cost).In stage2,with probability 0.5,the buyercanmake an offer (P,q),while with the remaining probability 0.5 it is thesellerwho can make an offer (P,q).Finally, in stage3, the party who didnot make the offer in the previousstage can either accept the offer orchooseto stick to the initial contract.

1.\"What are the first-bestlevelsof q and i?2.\"What are the equilibrium levelsof q and i in the absenceof an initial

contract[or,equivalently, with an initial contract(P0,q0) = (0,0)]?3.Supposethat the initial contractcanonly consistin a single pair (P0,qo)-\"What will the equilibrium contract be,as well as the associatedqualitylevel?4. Add now to the previouscasethe following at-will-contracting provisionfor the buyer. First, when the sellermakes the offer in stage2, the buyercanin stage3 acceptit, acceptthe initial contract(P0,qo),or chooseto walk

away, that is,chooseoutcome(0,0).Second,when the buyer makesthe offerin stage2,the buyer canchoosebetween(P0,qo) and (0,0)if the sellerdoesnot accept the buyer's stage-2offer.\"What is the impact of this at-will-

contracting provision on the equilibrium q and il

682 Exercises

5.If we assumeaway directexternalitiesand have payoff functions v(q) -P for the buyer and P-c(i)q- i for the seller[with c'(i)< 0],show that

choosinga single pair (P0,qo) as initial contractis now optimal.

Question50

Considera simplified versionof the problemin Segal(1999a)and Hart andMoore(1999),with only two widgets, widget 1and widget 2.Widget 1coststhe seller0 to produce,while widget 2 coststhe seller1to produce.Valuations are uncertain, however:in the \"good state of nature,\" widget 1givesthe buyer utility v > 1while widget 2 gives her zero utility; the \"bad stateof nature\" is the oppositecase,in that it is widget 2 that gives the buyer

utility v > 1while widget 1gives her zero utility (note that surplus is thus

higher in the goodstate).At investment costi2, the sellercaninduceaprobability i that the state of nature is good(and thus 1-i that it is bad).

1.Derivethe first bestin this environment.

2.Assume that the buyer has full bargaining powerin renegotiation.What

will be the investment level without the contract if payoffs are mutuallyobservableex post?3.Explain how a contractcan inducethe first bestif widgets can becontractually identified by their number ex post and if the partiescan commitnot to renegotiatethe contract.4. Assume widgets canbe contractually identified by their number both expostand ex ante, but the partiescannot commit not to renegotiatethecontract (and the buyer has full bargaining powerin the renegotiation).Towhat

extent doescontracting reduce the underinvestment of the sellerrelativeto the first best? (Following the lines of the proof detailed in section12.3.4.2,look at the two following incentive constraints:the one wherethe

buyer considersclaiming that the state is bad while it is in fact good,andthe one where the sellerconsidersclaiming the state is goodwhile it is in

fact bad).5.Discussthis result and compareit with Segal'sresult.

Question51Considerthe Aghion-Tirole(1997)modeldescribedin Chapter12.Assume

quadratic effort costs:

683 Appendix

1.Computethe principal'spayoff with and without delegation.Derivethe

comparativestaticsof the maximum of these two payoffs with respectto aand /3. Discuss.2.\"What is the effectof relaxing the assumption that a > 0?3.What is the effect of relaxing the assumption that, if uninformed, the

agent would not \"choosea projectat random\"? Specifically,call yP < 0 the

principal'spayoff from the agent'srandom choice,and yA > 0 the associatedagent'spayoff.What is the outcomeof the gameunder theseassumptions?

Chapter13Question52

Consideran agent with CARA utility function

-n(w-ca2/2)

where w is monetary compensationand a is effort.Hercertainty-equivalent reservationwageis vv0. Output q = a + e,whereehas a normaldistribution N(0, o2).Two risk-neutral principals (who do not observea) areinterestedin q:principal 1getsa benefit Bxq and principal 2 getsa benefit

B2q.The agent couldbe a government agency,and the principals couldbepoliticalinterestgroups.

1.Assuming that the principals can join forcesand offer the agent acontract w = t + sq to maximize (5X + B2)q,what is the optimal contractforthem (a)when they canobserveand contracton a and (b)when they cannotobservea?2.Assume that eachprincipal i independently offersa contractw, = tt +stq,that contractoffersare simultaneous, and that the agent can only eitheracceptboth contracts[in which casesheobtains h + t2+ (si +s2)q]or refuseboth. Computethe equilibrium levelofSi+s2,and compareit with the levelofs obtainedin part 1.Discuss.

Question53

Consideran agent who works for two risk-neutral principals (i =1,2) and

producesqt -at + e, for principal i where at is effort and the \302\243,'s are inde-

684 Exercises

pendently and normally distributed with mean 0 and variance of. Theagent'sutility function is

_e-T,(W-a2j2-al/2)

where w is monetary compensation.Principalsare restricted to makingsimultaneous noncooperativecontractoffersthat are linear in output levels.Theagent canaccept0,1,or both contracts.Hercertainty-equivalentreservation wageis 0.1.Assume eachprincipal canobserveonly his own output qt and thus offersa contractw, = tt +s^.Computethe Nash equilibrium contractsand effortlevels.Howdo they compareto the optimal contractwhere the principalscanjoin forcesand offercontractsthat maximize their joint payoff? Discuss.2.Assume eachprincipal canobserveboth output levelsand thus offersacontractwt = tt + stqt +fa.Computethe equilibrium contractsand effortlevels.Comparewith the solution in part 1.

Question54

Considera public project that a firm can build at cost 6 e [6,6].The firm

knows the realizationof 6,but regionalgovernments believeit to bedistributed accordingto a density f(6) and cumulative distribution function

F(6)[with F{6)lf{6)nondecreasingin 6].There are two regionalgovernments (i = 1,2),with socialbenefit 5, in

regioni (positive or negative, dependingon employment or environmental

considerations,for example).Eachgovernment i canseta (negative orpositive) transfer T, that the firm receivesif and only if it builds the project.

The firm's payoff is

u =d(T1+T2-6)

whered = 1if the project is built and 0 otherwise.Government z'spayoffis

Vl=d[Sl-(X+X)Ti] +u/2

whereA is the shadow costof public funds.

1.What is the first-best value 6*such that the projectshould bebuilt if and

only if 0<0*?

685 Appendix

2.Assuming from now on that 6 is private information to the firm,consider first the casewhere the regionalgovernments join forcesand chooseT= T\\ + T2soas to maximize Vi + V2. Derive the optimal transfer T andthe value 0\302\260 such that the firm builds the project if and only if 6 < 6C.

3. Now assumethat the regionalgovernments set the 7Vs noncooperatively.Derive the equilibrium 7Vs and the value 6nc such that the firm builds the

projectif and only if 6 < 6nc.

4. Discussand comparethe answersobtainedin parts 1,2,and 3.

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Author Index

Abreu, D,435,484-485Adams, W. X, 200-201,210,233Admati, A. R.,189-190Aghion, P.,42,127,522-525,534,552,563,

579,581,582,585-588,598-600,608,642,644

Akerlof, G.,16,57,601,602,604,641Alchian, A. A., 27-29,129,297-298,300,

303,360,362,490-491,498,499,550,552,554,560,570

Allen, F.,127,397,402,417Anderlini, L.,600Anderson, E.,219Aoki, M.,363Armstrong, M.,77,204,212-214,233,293Arrow, K.X, 1,8,25,129,169Asquith, P.,121Atkeson,A.,408,418Azariadis, C,67,70-71,96,97

Bagwell, K, 127Baily,M.N.,67Bajari, P.,415Baker, G.P.,170,470,485,552,600Baliga, S.,362,363Banerjee, A., 42, 62Baron, D.P.,74,77,96,97,344Beaudry, P.,415Bebchuk, L.,29,35,158Becker,G.S.,345Beckmann, M.X, 351Beggs,A.W.,363Berglof, E.,534,552Berle,A.A.,522Bemanke, B.,42Bernheim, B.D.,41,124,127,234,470,485,

572,598,612,613,628,642,643Besanko, D.,344Bester, H.,57,96,415Bhattacharya, S.,121,126,362,417Binmore, K.,240,566Bisin, A., 603,611Bizer, D.,613,627-628Bolton, P.,42,127,162,229,235,346,363,

417,508,515-517,521-525,534,535,537,538,542-543,552,579,581,582,588,589,592,595,599,600,608,642

Bonnano, G,643Boot, A. W.A., 234Borch, K.H.,131,169Brander, X, 643Brennan, M.X, 126Bull, C.A., 240,462,485,600Bulow, X, 369,416,549,631Burkart, M.,588,599,617

Caillaud, B.,97,234,633,636,643Calvo,G.,298,351,352,354,363Chandler, A., 351Chang, C,197,536Chari,V.,71,96Chatterjee, K.,255-257,293Che,Y. K,293,563,570,572,598Chiappori, P.,420,484Cho, I.K.,100,107-110,116,125,607Chone, P.,212,233Chung, T.Y.,563Clarke, E.H.,292Coase,R.H.,369,521-522,551Compte, O.,484-485Constantinides, G,126-127Cowan, C,77Crawford, R.G,490-491,552,554,560,570Cremer, X, 235,276,290,293,306,309-310,

363,416

Dasgupta, P. S.,96,242,293d'Aspremont, C,245,247,292,293Debreu, G,1,8DeMarzo, P.,549,613,627-628DeMeza,D.,57,508Demsetz, H.,27-29,129,297-298,300,303,

360,362,498,499,550Demski, X S.,129,362Dessein,W.,599Dewatripont, M.,168,223,227,234,363,379,

383,384,387,416,417,437,457,461,472,481,485,552,563,582,584,598-600,632,643,644

Diamond, D.W.,121,397-402,407,417,461,535, 600

Dionne, G.,485Dixit, A., 234,614Dubey, P.,611Dutta, P. K, 431,484Dybvig, P.,121,397-402,407,417

Edgeworth,F.Y,487Edlin,A.S.,563,570Engers, M.,608Eswaran, M.,362

Fairburn, XA., 316Fama, E.F, 612Farrell, X, 243-244Faulhaber, G,127Faure-Grimaud, A., 363,543Feenstra, R., 416Felli, L.,362,600Fernandez, I.,608Fershtman, C,631,632,643

710 Author Index

Fishman, M.J.,171,186,189-190,198,549

Freixas, X., 415,417Fried, X, 29,35,158Fudenberg, D.,104,107,110,238,240,290,

369,383,429^32,451,455,456,484-485,631

Gale, D.,171,172,190,196-198,397,402,417,535,536,611-612

Gale, X, 293Garicano, L.,363Garvey, G.,29Geanakoplos, X D.,363,603,611,631Gerard-Varet, L.,245,247,292,293Gertler, M.,42Gertner, R.,552Gibbard,A.,25,96,237Gibbons, R.,470,485,552,600Glazer, X, 362Goldberg, V. P.,552,554,560,570Gordon, D.F, 67Goswami, G,126-127Gottardi, P.,603,611Grafen,A.,126Green, E.X, 408,418Green, X R.,1,45,71,96,179,238,292,298,

301,320,324-325,361Greenbaum, S.,234Greenwald, B.C,641Gromb, D.,546-547,549,588,599,617Grossman, GM.,614,643Grossman, S.X, 38-39,71,73,97,129,152,

155-157,170,171,197,435,491,499,500,503,550-552,572,587,598,613

Grout, P. A., 491Groves,T.,292Grundy, B.D.,126-127Guaitoli, D.,611Guesnerie, R.,42,96,97,234,415Gul, F, 369,416

Hagerty, K.M.,171,186,189-190,198Halonen, M.,552,600Hammond, P.X, 96,242Harris, M.,96,137,169,485,613Harsanyi, X C,239Hart, O.D.,38-39,71,73,97,129,146-147,

152,155-157,171,197,234,363,369,376,379,415^16,435,491,499-503,508-509,516,522-523,534,535,537,541,544-546,549-552,554,560,563,566,569,570,572,573,577,578,587,598,599,613,618,620,622,637,643

Haubrich, X G,418

Hausch, D.B.,563,570,572,573,598Heavner, D.L.,514Hellwig, M.,96,171,172,190,196,197,198,

535,536,608Helpman, E.,614,643Hermalin, B.E.,127,456-457,461,485,600,

637,643Hicks,X R.,636HolmstrOm, B.,30,34-35,129,137,141,

146-147,159,162,169,218-220,222-223,233,234,297-300,302-305,310-311,315,331,333,336,361-362,427-433,435,441-442,447,470-471,473,478,482,484,485,552,564,599

Inderst, R.,552limes, R.D.,119,163-168,198,457,460-461,

468,536,581Itoh,H., 326-331,337

Jacklin, C.X, 401,402,417Jaffee, D.M.,57Jehiel, P.,293,613Jensen, M.C,27-28,129,133,163,499,

550Jewitt, I.,152,170,472,473,481,485Jin, L.,29John, K., 127Judd, K.L.,631,632,643Jullien, B.,96,633,636,643

Kahn, C.M.,71,96,170,613,624,642Kaldor,N.,298,360Karriien, M.I.,90Kanbur, R.,8Kandori, M.,484-485Kaplan, S.,527,534,550Katz, M.L.,456-457,461,485,613,618,632,

643Keren, M.,363Khalil, F, 235KiMstrom,R.,8Kim, S.K., 170King, R.G,418Kiyotaki, N., 42Klein, B,462,485,490-491,552,554,560,

570Klemperer, P.D.,26,292,293,631Knight, F,8,13Kocherlakota, N., 418Kofman,F, 342-344,350Kohlberg, E.,608Kornai, X, 384,416Kotwal,A.,362Kraus, A., 126

711 Author Index

Kreps, D.M.,100,107-110,116,125,129,600,607

Krishna, V, 292,293

Laffont, J.J.,2,8,45,74,75,96,97,228,234,292,293,351,362-363,388,390,395-396,415

Lafontaine, F.,362Lai, R.,362Lawarree, J.,342-344,350Lazear, E.P.,298,300,316-317,326,361Ledyard, J.,292Leffler, K.B.,462,485Legros, P.,168,300,306-307,457,461,485,

599Leland, H.E.,99,126,127Leppamaki, M.,351Levhari, D.,363Levin, X, 462,467,469^70,485,600Levine, D.K.,484-485Lewis, T.R.,229,234,235,293,416,643Ligon, E.,418Lockwood, B.,508Lucas,R.E.,408,418

Ma, C.T.A., 126,337,451,456,485Macho,I.,420,484Macho-Stadler, L.,331,363MacLeod, W. B.,462,471,485,600Majluf, N. S.,113,118-119,121,126,127,163,

172,174Malcomson,J.M.,316,427,462,484,485, 600Marschak, T.,363Martimort, D.,2,45,234,350,362-363,643Mas-Colell,A.,l,45,238Maskin, E.S.,39,47,66,67,73,77,82,96,

100,112,125,242,272,274,280,282,293,294,337,379,383,384,387,416,417,437,484-485,553-557,572-573,574,577,578,592,597-599

Mathewson, G.,362Matsushima, H.,300,306,484-485Matthews, S.A., 168,280,300,306-307,451,

456,457,461,485,536McAfee, R.P.,201,229,233,234,613,

618-620,642McLean, R.P.,276,290,293,306,309-310McMillan, J.,201,229,233,234Means, G,522Meckling, W. H.,27-28,129,133,163,499,

550Meleu, M.,351Melumad, N., 362Mertens, J.F.,608Meyer, M.,485

Milbourn, T.,29Milgrom, P. R.,34-35,127,142,147,171,197,

218-220,222-223,233,234,285,288,292-296,298,331,333,336,361-363,427-433,435,441-442,447,482,484,485

Miller, M.,112,120-121,612Minelli, E.,603Mirrlees, J.A., 47, 62,85,96,129,139-142,

147-149,151,155,169-170,473Modigliani, F, 57,112,120-121Moldovanu, B.,293,613Mookherjee, D.,196,362,613,624,642Moore, 1,39,42,297,363,491,499,501,502,

508-509,516,522-523,534,535,537,541,544_546,549-554,558,560,563,566,569,570,572,573,577,578,587,597-599

Morgenstern, O.,1,10-12,237Moulin, H.,237Muller, H.,552Mullins,D.W.,Jr.,121Murphy, K. J.,470,485,552,600Mussa, M.,47, 96Myers,S.C,113,118-119,121,126,127,

133-134,163,167,172,174,628Myerson, R.B.,74,77,96,97,240,250,273,

293,592

Nalebuff,B.J.,637Newbery, D.M.G,129Newman, A. F, 42,599Noe, T.,126-127Noldeke, G,126,563,566-567,572,598

Okuno-Fujiwara, M.,171,176-179

Palfrey, T.R., 238,240,242,290Panunzi, F, 587,598,617Pauly,M.V.,612,613,642Pearce,D.,435,470,484-485Perez-Castrillo, J.D.,331,363Pfleiderer, P.,189-190Picard, P.,633,636,643Pivetta, F, 229,235Png, I.,196Poitevin, M.,415Polemarchakis, H.,603Porter, M.,636Postlewaite, A., 171,176-179Povel, P.,543Prat, A., 363Prescott, E.C,42,611Pyle, D.H.,99,126,127

Qian, Y, 298,351,352,354,355,358-360,363

712 Author Index

Radner, R.,355,363,431,432,449,484Raith, M.,543,644Rajan, A., 588,589,592,595,600Rajan, R.G,508,552Ramakrishnan, R.T.S.,331,336Raviv,A.,96,137,169,613Ray, D.,611Rebello, M.,126-127Reiche, S.,577,598Reichelstein, S.,362,363,563,570Repullo, R.,553,558,565,597,598Rey, P.,415,420,427,484,563,598,599,644Rhodes-Kropf, M.,229,235Riley, J.G,47, 66,67,77,82,96,229,234,

272,274,280,282,293,607-608,641Riordan, M.H.,127,229,234Roberts, J.,127,234,361-362,550Roberts, K, 96Rochet, J.G,96, 204,212,233,417Rogerson, W.P.,151,170,420-426,484Rogoff, K., 549Roland, G,229,235,384,416Rosen, S.,47,96,298,300,316-318,351,361Rosenthal, H.,552Ross, S.,169Rothschild, M.,41,583,601,604,605,607,

641Rubinstein, A., 431,484,565,566Russell,T.,57

Saez,E.,96San, R.K.,363Salanie, B.,2,45,415,420,427,484Samuelson,W. F.,255-257,293Sappington, D.E.M.,170,229,234,235,293,

362Satterthwaite, M.A., 25,237,250,293Schaffer, M.E.,384Scharfstein, D.S.,417,485,522-523,535,

537,538,542-543,552,637,643Scheinkman, J.,162,170Schmidt, K.M.,563,566-567,572,598,637,

641,644Schmittlein, D.G,219Schotter, A., 240Schwartz, M.,613,618-620,642Schwartz, N. L.,90Scotchmer,S.,179Segal,I.,40,554,563,572-573,590,598,612,

613,617,620,642,643Shapiro, G,129,134,462,485,613,618Shavell, S.,137,169,171,181,189-190,198,

473Shin, H.,234Shleifer, 234

Shubik, M.,611Simon, H.A., 37-38,490,495^96,498,551,

597Sjostrom,T.,362Sonnenschein, H.,369,416Spence,A. M.,16,99,101-102,107,125,126,

129,169Spier, K, 127,600Spinnewyn, F.,427,484Stacchetti, E.,293,470,484-485,613Stein, J.,485,552Stiglitz, J.E.,16,41,57,70-71,96,97,129,

133,134,363,385,462,485,583,601,604,605,607,637,641

Stokey,N. L.,298,301,320,324-325,361,369,416

Stole, L.,96,233,552,643Strausz, R.,362,415Stromberg, P.,527,534,550Sutton, X, 516Suzumura, K.,171,176-179

Tadelis,S.,415Thakor, A. V.,234,331,336,417Thomas, J.,408,414,418Tirole, X, 39,74,75,97,100,104,107,110,

112,125,127,159,223,227,228,234,238,240,290,338,339,341-344,346-347,350,361-362,369,373,376,379,383,388,390,395-396,415^16,451,455,456,472,481,485,552,554,572-573,574,577,582,584-588,592,599,613,618,620,622,631

Townsend,R.M.,33,42,171,172,190,198,397,400-406,417-418,535,611

Tuomala, M.,96

van Damme, E.,125,126VanZandt,T.,355,363Varian, H.R.,331,333Vauhkonen, X, 534Vayanos, D.,363Vickers,X, 77,485,643Vickrey,W.,292Vishny,W.,234Viswanathan, S.,229,235Vohra, R.,611von Neumann, X, 1,10-12,237von Thadden, E.L.,386,387,417,552

Walker, D.,29,35,158Walsh, G,169,489Wantz, A., 124Ware, R.,416Webb,D.G,57,197,536Weber, R.X, 285,288,293-296

713 Author Index

Weigelt, K., 240Weiss,A., 16,57,96,126,385Weitzman, M.,32Welch, I.,127Wellisz, S.,298,351,352,354,363Wemerfelt,B.,600Whinston, M.D.,1,41,45,201,233,234,238,

470,485,508,515-517,521,572,598,612,613,628,642,643

White, M.,613Williams, X, 127Williamson, O.E.,38,222,298,351,363,490,

491,551,552,554,560,570Wilson, C,607,608,641Wilson, J.Q.,480-481Wilson, R.B.,96,100,169,293,335,369,416Winter, R., 362Winton,A.,197Wolinsky, A., 566Worrall,T.,408,414,418

Xiong,W,162Xu, C,508

Yaari, M.E.,431,484Yellen, J.L.,200-201,210,233

Zahavi, A., 126Zeckhauser, R.J.,129,169Zingales, L.,508,552Ziss,S.,608-609Zwiebel, J.,552

SubjectIndex

Additive-normal model (Holmstrdm),473^75,478-480,481-482

Adverse selection, 14-20.Seealso Dynamicadverse selection; Screening; Signaling

in bilateral contracting, 15-20,47-127in bilateral trading, 243-261bunching in, 88-92bundling in, 199-208combining with moral hazard, 228-232with continuum of types, 82-92,250-261,271-276

in corporate financing and investmentdecisions,112-120,126-127

in credit rationing, 57-62debt financing under, 116-119in dividend policy, 120-125,127in dynamic contracting, 31-34,36equity financing under, 114-116,119-120with finite number of types, 77-81implementation problem in, 83-85in implicit labor contracts, 67-74ironing in, 88-92linear pricing and, 49-50literature notes, 96-97,125-127,233-235,292-293

market competition and, 601-608multidimensional, 77-92,199-216,233-235in multilateral contracting, 25,238,239-296in optimal income taxation, 62-67in random contracts, 81-82in regulation, 74-77single two-part tariffs and, 49-52,214-216with two-type case,241-250

Anticipatory equilibrium, 607Asymmetric information. SeeInformational

asymmetryAt-will contracting, 562,569-570Auction theory, 2,26-27,238,255-259,

261-290asymmetrically distributed valuations in,280-282

auctions with imperfectly known values,282-290

auctions with perfectly known values,261-265

Bayesian equilibrium in, 257-259,270-271,284

optimal auctions with correlated values,276-278

optimal auctions with independent values,265-266

optimal efficient auctions with independentvalues, 262-266

revenue equivalence theorem and,271-276,285-290,291,293-296

risk aversion and, 278-280simple double auctions, 255-256,257-259winner's curse and, 26,283-285,291-292,293

Auditing, 342-351cheap-but-corruptible auditors and,346-351

honest-but-costly auditors and, 344-346,348-351

Autarky, 409Authority relation, 37-38,579,585-597,

600employment relation and, 594-597formal and real authority, 585-588informational advantage in, 588-597

Bankingliquidity transformation in, 397-402,405^08

Bargaining power, 26-27,102,568bargaining/durable-good monopolyproblem and, 365-366,371-373,415

Barrier to entry, contracts as,608-611Bayesian equilibrium, 238-242,290-291,

456,464in auction theory, 257-259,270-271,284

defined,240-241Bayes'rule, 107,125-126Bertrand equilibrium, 517,609,631-633,

636,640-641,643Bidding gamecommon agency and, 613-614,628-630,642-643

coordination problem in, 628,642globally truthful equilibria and, 629-630,642

menu auctions in, 628-630Bilateral trading, 243-261

continuum of types, 250-261two-type case,241-250

Bonuses, 304-305,462,464Borch ruleadverse selection and, 69defined, 13moral hazard and, 130-131,135-136,143-145

Breach of contract, 3Budgets

budget balance, 292budget breaking and, 30,297-298,299-300301-305,361

soft-budget-constraint problem, 367-368,384-396,415,416-417

Bunching, 88-92

716 Subject Index

Bundling, 199-208mixed, 200-201,202,233optimal contract in asymmetric model,211-212

optimal contract in symmetric model,208-211

pure, 200-201,210,214seller's relaxed problem and, 206-207,208-211

Buyer-employment contracts, 493,495^496,497-498

Capital structure, 112-120.Seealso Debtfinancing; Equity financing

control and, 521-549CARA model, 142,159,218,300,312,331,

408,430,442-447,480,485Careerconcerns,471-482,484focused missions and, 480-481literature notes, 485^486in multitask case,475-481in single-task case,473-475trade-off between talent risk andincentives under, 481-482

CDFC.SeeConvexity of the distributionfunction condition (CDFC)

Centrally planned economies,20,21,32CEOcompensation, 29,35,36,158,217-218,

339,455^56Certainty-equivalent compensation,

138-139,445Chebyshev's inequality, 215-216Choice under uncertainty (von Neumann

and Morgenstern), 1,10-14,239-240,279Cho-Kreps intuitive criterion, 107-110,116,

607Coalition-incentive-compatibility constraint,

340-341Coasetheorem, 7,250,501,505Coasian dynamics, 369-379,415^16

full commitment solution, 370-371renting without commitment, 373-376selling without commitment, 371-373

Cobb-Douglas function, 70, 73Coinsurance, 9-10,31,130-131Collusion, 30

among agents,331-338auditing and, 342-351with hard information, 338-343maximum deterrence of,345-346,348\342\226\240

Collusion-proof contracts, 298,339Commitment

full commitment solution, 370-371renting without, 373-376selling without, 371-373

soft budget constraints and, 384-385Common agencyin bidding games,613-614,628-630,642-643

Common shocksfiltering out, 28-29,315individual agents and, 330teams and, 314-315tournaments and, 320-326

Compensadon-cost-rninimization problem,452-453

Complex contracting environments, 40,572-578

describability and, 577-578for long-term contracts, 424-426renegotiation and, 574-577

Constrained Pare to-efficient contracts,606-607

Contingent control, 531-534,550-551,579-581,584-585

Contingent ownership allocation, 550Contracting at will, 563,569-570Contract renegotiation. SeeRenegotiationControl rights, 2,521-549.Seealso

Supervisioncontingent allocation of,531-534,550-551,579-581,584-585

debt financing and, 522,534-549,582-585entrepreneur control, 526-529,533-534,579-581,583-584

equity financing and, 522,582-585investor control, 529-531,581,584wealth constraints and, 523-534

Convexity of the distribution functioncondition (CDFC),151-152

Cooperation, incentives for, 298,326-338Coordination problem, 387-388,628,642Corporate financing decisions,112-120,

126-127.Seealso Debt financing; Equityfinancing

Costly state verification (CSV)approach,190-197,198

Cost-plus scheme, 75Cournot equilibrium, 631-633,635-636,643Creamskimming, 41,605Credit decentralization, 385-387Creditor rights, 551Credit rationing, 57-62

asymmetric information in, 60-62decentralization of credit and, 385-387differences in risk characteristics, 57-58for multiperiod projects, 540

Debt dilution, 612Debt financing

under adverse selection, 116-119

717 Subject Index

cash flow asprivate information and,534-549

control and, 522,534-549,582-585costly disclosure of information and,190-197,198

under moral hazard, 162-168rationalizing outside, 582-585renegotiation of,540-549standard debt contract, 196,198

Debt-overhang problem, 628Decentralization of credit, 385-387Describability, in complex contracting

environments, 577-578Dilution cost, 115,116-118Direct investment externalities, 570-572Disclosure. Seealso Mandatory-disclosure

laws; Private information

full-disclosure theorem, 175-178Dividend policy, 120-125,127,462Dutch auctions

asymmetrically distributed valuations in,280-281

Bayesian equilibrium and, 270-271compared with first-price sealed-bid

auctions, 268,269described, 267optimal auctions with correlated values, 276optimal selling procedure in, 275revenue equivalence theorem and, 293-296risk aversion and, 279-280seller's expected revenue in, 285-287

Dynamic adverse selection, 31-34,36,365^18

in banking and liquidity transformation,397-402

bargaining/durable-good monopolyproblem and, 365-366,371-373,415

with changing types, 396-414Coasian dynamics in, 369-379,415-416with fixed types, 365-396with infinite horizon, 408-414,418literature notes, 415-418long-term contracts and, 368,377-384ratchet effect and, 368,373-375,394-396,415

regulation and, 388-396renegotiation and, 365-366,370-371,377-384,390-394,415

short-term contracts and, 394-396,415soft-budget-constraint problem and,367-368,384-388,415,416-417

in two-period horizon, 397-402,417in two-period horizon with two

independently distributed income shocks,397,402-408,417-418

Dynamic contracting. SeeDynamic adverseselection; Dynamic moral hazard

Dynamic moral hazard, 34-35,419-486in bilateral relational contracts, 461-471,483^84

career concerns and, 471-482under free savings and asymmetric

information, 429-430implicit incentives and, 471-482with infinitely repeated actions, output, and

consumption, 433,447-449literature notes, 484-486long-term contracts and, 450-461under monitored savings, 426-429with no accessto credit, 422-426renegotiation and, 450-461with repeated actions,432,434-435with repeated actions and output, 433,435^47

with repeated consumption, 432-433with repeated output, 432,433^34summary, 483-484in T-period problem, 431-449in two-period problem, 419-430

Edgeworth box, 1,6Education, signaling and, 16,99,100-106,

126Efficiency-wage contracts, 74,134,436-447,

467under imperfect information, 437-441under perfect information, 436-437

Effort substitution, 218-228,234conflicting tasks and advocacy and,223-228

salesman job descriptions and, 219-223Employment contracts, 4-14.Seealso

Executive compensation packagesemployment relation in, 489-498explicit long-term, 35-36under hidden information, 18-19implicit labor contracts, 67-74incomplete contracts and employmentrelation, 489-498

ownership versus,499sales contracts versus,38salesman job descriptions and, 219-223under uncertainty, 7-14without uncertainty, hidden information, or

hidden actions,4-7Employment relation, 489-498

authority relation and, 594-597nature of,490-491theory based on expost opportunities,491-498

718 Subject Index

English auctions

asymmetrically distributed valuations in,280-281

Bayesian equilibrium and, 270-271compared with Vickrey auctions, 268-269described, 267optimal auctions with correlated values,276

optimal selling procedure in, 275-276revenue equivalence theorem and,293-296

risk aversion and, 279-280seller's expected revenue in, 285,286,288

Entrepreneurscontrol rights and, 523-534,579-581,583-584

debt financing and, 522,534-549,580-581property-rights theory of the firm and,498-521

risk profile of,8,13wealth constraints, 534-549

Equilibrium-contingent control, 581Equilibrium renegotiation, 566Equilibrium unemployment, 16Equity financing

under adverse selection, 114-116,119-120control and, 522,582-585disclosure of verifiable information,172-190

rationalizing outside, 582-585Ex ante investments, 38,504-506Ex ante noncontractable actions, 554Exclusive contracts, 40-41Executive compensation packages,21CEOcompensation, 29,35,36,158,217-218,339,455^56

managerial incentive schemes,157-162,218-223

moral hazard in, 157-162,217-218,455^156

option contracts in, 35,484,566,567-569Ex post noncontractable actions,554Ex post unverifiable actions, 578-588

authority relation and, 37-38,579,585-588,594-597,600

financial contracting, 579-585Ex post unverifiable payoffs, 588-597

authority relations in, 588-597,600spot contracting and, 589,591-594

Filtering out the common shock,28-29,315Financial contracting. Seealso Debt

financing; Equity financing

contingent control in, 579-581with expost unverifiable actions,579-585

financial constraints in, 579-581rationalizing outside equity and debt in,582-585

Financial structure, 521-549.Seealso Debtfinancing; Equity financing

Firm-specific risk, 8First-price sealed-bid auctions

asymmetrically distributed valuations in,280-281

Bayesian equilibrium and, 270-271compared with Dutch auctions, 268,269described, 267optimal auctions with correlated values,276

optimal selling procedure in, 275revenue equivalence theorem and,293-296

risk aversion and, 279-280seller's expected revenue in, 285-287

Forcing contracts, 305-306Foreclosure, 518,546Formal authority, 585-588Full-disclosure theorem, 175-178

Globally truthful equilibria, 629-630,642Golden handcuffs, 158Grossman-Hart-Moore (GHM)incomplete-

contract paradigm, 572,587-588,599Groves-Clarke mechanism, 292

Hazard rate, 87-89Hidden-action problem, 1,2,14-15,22,45,

238,365-366.Seealso Dynamic moralhazard; Moral hazard

Hidden-information problem, 14-15,23-24,45,238,365-366.Seealso Adverseselection; Dynamic adverse selection;Screening; Signaling

Hierarchies, 30,298,351-360Holdup problem, 38-39,490-491

complexity and, 572-578for contracting at will, 563,569-570for direct investment externalities,570-572

incomplete contracts and, 552,553-554,561-578

for option contracts, 566,567-569renegotiation design and, 563-566for specific-performance contracts,563-566,567-569

Implementationadverse selection and, 82-85Nash implementation, 39-40,553,555-557,559,597

719 Subject Index

Implicit incentives, 35,471-482additive-normal model (HolmstrOm),473-475,478-480,481-482

in multitask case,475-481in single-task case,473-475

Implicit labor contracts, 67-74contracting with adverse selection, 71-74contracting without adverse selection,67-71

Incentive-compatibility constraints, 1,16-17,21-23,53-56,132,205,452-453,539

in auctions, 277-278in bilateral trade, 249,251-255trade-off between insurance andincentives, 23

Incentive contracting, 30-36.SeealsoExecutive compensation packages; Teams;Tournaments

adverse selection in, 31-34market competition asincentive scheme,636-641

moral hazard in, 34-36Income taxation, 62-67Incomplete contracts, 2,36-42,487-644

bilateral nonexclusive contracts in

presence of externalities, 611-630competition as incentive scheme, 636-641contracts asbarrier to entry, 608-611employment relation and, 489-498expost unverifiable actions and, 578-588expost unverifiable payoffs and, 588-597financial structure and control, 521-549Grossman-Hart-Moore (GHM)incomplete-contract paradigm, 572,587-588,599

holdup problem in, 552,553-554,561-578implementation theory and, 39-40literature notes, 551-552,597-600,641-644market competition and, 601-644ownership rights and, 37-39,498-521property-rights theory of firm and, 498-521static adverse selection and, 601-608subgame-perfect implementation of,558-560,564-566

summary, 549-551,597-600,641-644unverifiable information and, 553-600

Individual-rationality constraints, 17-18,48-49,53-56,59-60,75,205,539-540

in bilateral trade, 245-255efficient trade under, 245-248inefficient trade under, 248-255

Information acquisitionincentives for, 180-181

Informational asymmetry. Seealso Moralhazard; Private information

bilateral, 20-24contracting under, 24in credit rationing, 60-62moral hazard and, 20-34,429-430multilateral, 25-30

Informational monopoly power, 19Informational rent, 32,56,205,259Informativeness principle, 169,300Informed-principal problem, 45, 99-100,110-112.Seealso Moral hazard;

SignalingInitial public offerings (IPOs),signaling

and,126-127Institutional design.SeeOrganizational

designInsurance, 2coinsurance,9-10,31,130-131under moral hazard, 21,129with multiple contracts, 604-608renegotiation and, 379-384self-insurance, 34,419with single contract, 602-604static adverse selection and market

competition, 601-608supplementary, 625trade-off between incentives and, 23unemployment, 7-11,67-74within-period, 33-34

Interim efficiency, 378Intertemporal consumption smoothing, 404,

414Intertemporal risk sharing, 419,428Introductory pricing, 127Investment under risk, 2Investor control, 529-531,581,584Involuntary unemployment, 70-71Ironing, 88-92

Laffer curve, 133-134Laffont-Tirole model, 74-75LDICs.SeeLocal downward incentive

constraints (LDICs)Least-cost separating equilibrium, 109-112,

125-126Limited-liability constraints, 638-639Limit (predatory) pricing, 127Linear contracts, 137-142cost-of-effort function in, 137-138maximizing expected utility, 138-139suboptimality of, 139-142

Linear pricing, 49-50Liquidation, 196,545,637,638-640Liquidity preferences

banking and, 397-402,405^08uncertain, 121-122

720 Subject Index

Local downward incentive constraints

(LDICs),78-81,83-85Local upward incentive constraints

(LUICs),79-81Long-term contracts, 2,23.Seealso

Incomplete contracts; Insurance;Renegotiation

complexity of,424-426and dynamic adverse selection, 368,377-384

and dynamic moral hazard, 450-461gains from enduring relations, 431-449

Lossof control, 30,351-360LUICs.SeeLocal upward incentive

constraints (LUICs)

Managerial compensation. SeeExecutive

compensation packagesMandatory-disclosure laws, 24,176,178,

180-186,197-198connection between market manipulationand, 189-190

incentives for information acquisition,180-181

potential costs of, 196Marginal tax rates, 65-67Market-clearing wages,16Market competition, 601-644

with bilateral nonexclusive contracts in

presence of externalities, 611-630contracts as barrier to entry, 608-611as incentive scheme, 636-641literature notes, 641-644principal-agent pairs and, 630-636static adverse selection and, 601-608summary, 641-644

Maskin schemes,40,337-338,553,555-557

Maskin-Tirole informed-principal problem,110-112Maximum deterrence, 345-346,348Mechanism design,290-291,554,572Memoryof consumption, 427-428in contracts, 421,424of wages,427

Menu auctions, in bidding games,628-630Mirrlees contracts, 304-305,433-434Mixed bundling, 200-201,202,233Mixed-strategy contracts, 381MLRP. SeeMonotone likelihood ratio

property (MLRP)Modigliani-Miller theorem, 112-113,

120-121Monitored-savings scenario, 426-429

Monopoliesbargaining/durable-good monopolyproblem and, 365-366,371-373,415

Coasetheorem and, 7,250,369-379,501,505

competition asincentive scheme and,636-641

informational monopoly power, 19multiproduct monopoly problem, 199-216regulation of natural, 74-77

Monotone likelihood ratio property(MLRP), 147-148,151-152,156,164-166,169,310,462-463

Monotonicityin bilateral trading, 244implementation problem and, 83-85optimization problem and, 87-88unverifiable information and, 555,558

Moral hazard, 2.Seealso Dynamic moralhazard

in bilateral contracting, 20-24,129-170,228-232

Borch rule and, 130-131,135-136,143-145combining with adverse selection, 228-232debt financing under, 162-168described, 14-15in dynamic contracting, 34-35efficiency wage models, 134in executive compensation packages,157-162,217-218,455^56

first-best outcome, 130-132,140,148-152,305-311

Grossman-Hart approach to, 152-157insurance and, 21,129Laffer curve effect and, 133-134linear contracts and, 137-142literature notes, 169-170,233-235,362-363in managerial incentive schemes,157-162,218-223

multidimensional, 216-232,233-235in multilateral contracting, 25,238,297-363optimality of debt financing, 162-168in organizations, 29-30renegotiation and, 450-461,484second-best outcome, 130-133,142-148shirking and, 134-135summary, 168-169,233-235,360-362in teams,27-30,238,297-298,299-315,563in tournaments, 28-29,298,300-301,316-326

value of information and, 136-137Multiagent situations, 326-338.Seealso

Supervision;Teams;Tournamentscollusion among agents in, 331-338incentives for cooperation in, 298,326-330

721 Subject Index

Multilateral contracting. SeeStaticmultilateral contracting

Mutually efficient renegotiation, 365-366,370-371

Nash implementation, 39-40,553,555-557,559,597

Nexus of contracts, 40Noncontingent control, 584-585Nonexclusive contracts, 40-41No-slavery constraint, 18Null contracts, 563

Offer game, 613,614-620,642sequential, 623-628simultaneous, 614-623

One-shot contracting. SeeSpot contractingOptimal control allocation, 523Optimal income taxation, 62-67Optimal nonlinear pricing, 52-56Option contracts, 35, 484,566,567-569Organizational design,489-552hierarchies and, 351-360supervision and, 298,351-360

Outside-option principle, 566Overemployment, 73Overinsurance, 612Ownership

defining, 499and property-rights theory of the firm,498-521

Ownership allocation, defining, 501-502Ownership rights, 2

equilibrium ownership structures and,506-512

incomplete contracts and, 37-39,498-521

Pairwise proofness, 619Pareto-improving renegotiation, 368,

377-379,390-394,415,563,565,575,577,578,632

Partnershipsarbitrary partnership contracts, 302defined, 301-302repeated partnership games,484-485

Passive beliefs, 618-620,623,642PBE.SeePerfect Bayesian equilibrium

(PBE)Pecking-order theory of finance, 118,

126-127Penalty for delayed trade, 566Perfect Bayesian equilibrium (PBE),

102-106,618Maskin-Tirole informed-principal problemand, 110-112

mixed-strategy, 183-184pooling, 104,105-106,108pure-strategy, 181-183semiseparating, 104,106separating, 104-105,109-110

Perfect price discrimination, 48-49Piece-rate pay system,317-318,324,328,

419Pooling contracts, 391-392,394Pooling equilibrium, 104,105-106,108,116,

605,607Price-cap scheme, 75Pricing

advertising and, 127introductory, 127limit (predatory), 127optimal nonlinear, 52-56perfect price discrimination, 48-49

Principal-agent pairs, 630-636,643Bertrand equilibrium and, 631-633,636Cournot equilibrium and, 631-633,635-636,643

Principal-agent problem. SeeAdverseselection; Moral hazard; Regulation;Supervision

Privacy laws, 176Private information, 1,2,171-198,244.See

also Adverse selectionbilateral, 20-24debt financing and, 190-197,198about employee characteristics, 16-18equity financing and, 172-190literature notes, 197-198mandatory-disclosure laws, 24,176,178,180-186,189-190,197-198

summary, 197-198unverifiable information, 172-173,553-599verifiable information, 173-178voluntary disclosure of verifiable, 172-178,197

voluntary nondisclosure, 178-180,186-190Privately observable contracts, simultaneous

offer game and, 617-620Promotion schemes,28-29Property-rights theory of the firm, 2,

498-521,551-552equilibrium investment levels under

nonintegration, 517-518equilibrium investment levels under

vertical integration, 518-520equilibrium ownership allocations, 521equilibrium ownership structures and,506-512

framework for complementaryinvestments, 500-515

722 Subject Index

Property-rights theory of the firm {coni)printer-publisher-bookseller integration,512-514

publisher-printer integration, 499-500,502-512

Shapley value and, 501-502,503-504,513-517

socially optimal investment levels, 517substitutable investments and, 515-521summary, 515

Publicly observable contracts, simultaneousoffer game and, 614-617

Pure bundling, 200-201,210,214

Random contracts, 81-82Rank-order tournaments, 28,300-301,311,

316Ratchet effect, 32,368,373-375,394-396,

415Reactive equilibrium, 607-608Real authority, 585-588Recessions

implicit labor contracts and, 69unemployment and, 8-9,69,73

Redistributive taxation, 62-67Regulation, 388-396.Seealso Supervisionadverse selection in, 74-77auditing in, 342-351Pareto-improving renegotiation and,390-394

Relational contracts,2,35-36,461-471,483^84

defined, 465dynamic moral hazard and, 461-471,483^84

dynamic programming and, 465-467employment relation and, 594-597fully enforceable contracts versus,467-468pure adverse selection and, 468-471pure moral hazard and, 468

Relative performance, 28-29,315.SeealsoTeams;Tournaments

collusion and, 338-343and cooperation or competition amongagents,298,326-338,361

supervision and, 343-360Renegotiation, 2,31,33,36,365,540-549

in complex contracting environments,574-577

of debt financing, 540-549in dynamic adverse selection, 365-366,370-371,377-384,390-394,415

in dynamic moral hazard, 450-461insurance and, 379-384moral hazard and, 450-461,484

of sales contracts, 494-495secret, 633-634sequential input choice with, 381-383unverifiable information and, 553-554,597-598

when effort not observed by principal,450-456

when effort observed by principal, 456-461Renegotiation design, 563-566Renegotiation-proofness principle, 31,33,

365,377-384,415,421,429,451-454,467,528-529,548-549,581

Rentingwithout commitment, 373-376informational rent, 32,56,205,259sharing of rents, 253-254

Repeated bilateral contracting. SeeDynamicadverse selection; Dynamic moral hazard

Reservation prices,249-250Residual rights of control, 38,499,550Revelation games,39-40,597Revelation principle, 16-18

in bilateral trading, 244,250-251collusion-proof contracts and, 339described, 16-17,241-242in dynamic moral hazard, 451equity financing and, 191in optimal nonlinear pricing, 53-54in repeated bilateral contracting, 365

Revenue equivalence theorem, 26,271-276,291

in 2x 2model, 285-290in 2x 3model, 294-296

Sales contracts, 38,490,494-495,497-498Savingsfree, 429-430monitored, 426-429

Screening,2,15,45,47-97.Seealso Adverseselection

in credit rationing, 57-62in implicit labor contracts, 67-74in optimal income taxation, 62-67in regulation, 74-77

Secret renegotiation, 633-634Self-enforcement constraint, 3,468-470Self-insurance, 34,419Seller-employment contracts, 493,496-497Selling, without commitment, 371-373Semiseparating contracts, 392-393Semiseparating perfect Bayesian

equilibrium (PBE),104,106Separating contracts, 391Separating equilibrium, 104-105,109-112,

115-116,125-126,606

723 Subject Index

Sequential offer game, 623-628Shapley value, 501-502,503-504,513-517Sharing of rents, 253-254Shirking, 134-135,306-311,439,462Shocks.SeeCommon shocksShort-term contracts, 394-396,415Side contracting, 331-338

full, 333-338lack of,332-333

Signaling, 2,15,45,462.Seealso Adverseselection

Cho-Kreps intuitive criterion, 107-110,116,607

in corporate financing and investmentdecisions,112-120,126-127

in dividend policy, 120-125education and worker productivity, 16,99,100-106,126

in industrial organization, 127Maskin-Tirole informed-principal problem,110-112

Signal-to-noise ratio, 482Simultaneous offer game, 614-623

and input provision to competingdownstream firms, 620-623

for privately observable contracts,617-620

for publicly observable contracts, 614-617Single-investor control, 583-584Soft-budget-constraint problem, 367-368,

384-388,416-417commitment and, 384-385decentralization of credit and, 385-387regulation and, 388-396,415risk taking and, 387-388

Spanning condition, 156Span of control, 30,351-360Specific-performance contracts, 563-566,

567-569Spence-Mirrlees single-crossingcondition,

54,78,94Spot contracting, 32,33-34,371-373,419,

420-430,493^94,497,589bidding game and, 613-614,628-630expost unverifiable payoffs and, 591-594offer game and, 613,614-628

Standard debt contract, 196,198Static bilateral contracting, 2-24,45-235adverse selection and, 15-20,47-127in bilateral trading, 243-261exclusive, 40-41incentive contracting, 41^42moral hazard and, 20-24,129-170,228-232

with multilateral exchange, 40-42

nonexclusive, 40-41under uncertainty, 7-14without uncertainty, hidden information, or

hidden actions,4-7Static multilateral contracting, 2,25-30,

237-363adverse selection and, 25,238,239-296moral hazard and, 25,238,297-363

Stock options. SeeOption contractsSubgame-perfect implementation, 558-560,

564-566Substitutable investments, 515-521Supervision, 343-360.Seealso Regulation

auditing as,342-351and cheap-but-corruptible supervisors,346-351

hierarchies and, 351-360and honest-but-costly supervisors,344-346,348-351

loss of control in, 30,351-360organizational design and, 298,351-360,489-552

span of control in, 30,351-360Supplementary insurance, 625

Task arbitrage, 233Taxes,62-67Teams.Seealso Tournamentsbonuses and, 304-305budget breaking and, 30,297-298,299-300,301-305,361

collusion and, 331-338first-best outcome with, 305-311free riders and, 297moral hazard in, 27-30,238,297-298,299-315,564

observable individual outputs and,311-315

unobservable individual outputs and,301-311

Tender offers,613Termination threat, 384-388Theory of the firm. SeeProperty-rights

theory of the firm

Third-best contracts, in sequential offer

game, 626-627Tournaments, 316-326,361.Seealso Teams

moral hazard in, 28-29,298,300-301,316-326

rank-order, 28,300-301,311,316with risk-averse agents and commonshocks,320-326

under risk neutrality and no commonshock, 316-320

Transactions cost economics,551-552

724 Subject Index

Unbounded strategy sets,557Uncertain liquidity preferences, 121-122Unemploymentbooms and, 8-9equilibrium, 16involuntary, 70-71recessions and, 8-9,69,73

Unemployment insurance

implicit labor contracts and, 67-74nature of,7-11

Uniform tax rebates, 65-67Unit deadweight cost of borrowing,

122-123Unraveling, 175-178Unverifiable information, 172-173,553-600expost unverifiable actions,578-588expost unverifiable payoffs, 588-597holdup problem, 560-578literature notes, 597-600Maskin's theorem, 555-557Nash implementation, 39^0,553,555-557,559,597

renegotiation and, 553-554,597-598subgame-perfect implementation, 558-560,564-566

summary, 597-600

Venture capital, 527-531,534Verifiable information, 40,172-178,197Vickrey auctions

asymmetrically distributed valuations in,280-281

Bayesian equilibrium and, 270-271,283-285

compared with English auctions, 268-269described, 267optimal auctions with correlated values,276

optimal selling procedure in, 275-276revenue equivalence theorem and,293-296

risk aversion and, 279-280seller's expected revenue in, 285,286,288winner's curse and, 283-285

Von Neumann-Morgenstern utility function,1,10-14,239-240,279

Wage rigidity, 69-70Weak no-veto power (WNVP), 555-557Wealth constraints, 523-549

and contingent allocation of control,523-534

and optimal debt contracts when

entrepreneur can divert cash flow,534-549

Willingness-to-repay problem, 550-551Winner's curse,26,283-285,291-292,293Winner-take-all system,318WNVP. SeeWeak no-veto power

(WNVP)Worker productivity, and education as

signal, 16,99,100-106,126