Optimal Contract, Imperfect Output Observation, and Limited Liability

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Journal of Economic Theory � ET2171

journal of economic theory 71, 514�531 (1996)

Optimal Contract, Imperfect Output Observation,and Limited Liability*

Jacques P. Lawarre� e

Department of Economics, Box 353330, University of Washington, Seattle,Washington 98195-3330, and ECARE, 1050 Brussels, Belgium

and

Marc A. Van Audenrode

De� partement d 'e� conomique, Universite� Laval, Que� bec, Que� bec, G1K 7P4, Canada

Received February 10, 1994; revised October 24, 1995

We study the optimal contract when output is imperfectly observed. When allparties are risk neutral and the agent has unlimited liability, the optimal contractremains the same regardless of whether the output is perfectly or imperfectly observed.When the agent's liability becomes limited, the optimal contract changes profoundlyunder imperfect information. We show that the high-productivity agent neverproduces at first-best and may be required to produce an effort lower than the onerequired from the low-productivity agent, and both types of agents can earn rents.Journal of Economic Literature Classification Number: D82. � 1996 Academic Press, Inc.

1. INTRODUCTION

In the last few years, economists have paid considerable attention to thetheory of optimal contracts under asymmetric information. Most of thedevelopments in this field have primarily focused on situations where theprincipal perfectly observes some variable correlated with the agent's privateinformation. In the traditional agency model, it is generally assumed thatthe production technology is of the form X=X(e, :), where X is the output;e the agent's effort, and :, a stochastic variable of known density, an intrinsicproductivity characteristic of the agent. In models of both adverse selection1

article no. 0131

5140022-0531�96 �18.00Copyright � 1996 by Academic Press, Inc.All rights of reproduction in any form reserved.

* We thank Patrick Bolton, Mathias Dewatripont, Fahad Khalil, Pierre Lasserre, ShellyLundberg, Bentley McLeod, the participants at the Montre� al Micro workshop, and the 1995World Congress of the Econometric Society for comments. An anonymous referee providedespecially insightful comments. Van Audenrode thanks Canada's SSHRC and Que� bec's FCARfor generous financial support.

1 In adverse selection models, the agent knows the realization of : when choosing e.


and moral hazard,2 the output is often assumed to be observable while theeffort and the productivity characteristic are private information of the agent.

In reality, however, the output of the agent cannot always be observedperfectly. One can think of many situations where the output has certainimportant features difficult to observe. For instance, it may be difficult toascertain whether an auto mechanic did a good job fixing our car. TheDepartment of Defense may never find out whether a particular weapon iseffective in a nuclear war. Consumers often have difficulties in evaluatingthe quality of goods prior to buying them. Many employers have difficultiesin evaluating the output of individual workers.3 While the assumption ofimperfect observation of the output seems to be relevant in reality, thisquestion has only drawn limited attention from economists (Maskin andRiley [10] and Khalil and Lawarre� e [5] being the main examples).

A reason for this lack of interest might be that, in many principal-agentmodels, imperfect output observation has no effect on the optimal contract(see, e.g., Laffont and Tirole [7, 8]). This result is due to the assumptionsof risk neutrality and unlimited liability of the agent. The risk neutrality ofthe agent allows the principal to ``average out'' the observation errors inthe expected transfers specified in the contract. When the output observedis low (``in a bad year''), the agent will simply receive a low transfer. In fact,the optimal transfer is often so low that the agent's utility can be negative.This type of contract is only possible if the agent's liability is not limitedin adverse states of nature. When any of the assumptions of risk neutralityand unlimited liability of the agent is abandoned, the imperfect observationof the output is likely to have important consequences on the optimal con-tract. This is illustrated in Baron and Besanko's [3] model of noisy outputobservation when agents are risk averse in an adverse selection framework.

The goal of this paper is to derive the optimal incentive scheme in anadverse selection model when observation errors occur and when a riskneutral agent has a strictly limited liability. It is indeed a common featureof many contracts that the liability of the agent is limited. This fact isillustrated by the relatively low criminal penalties applied in the case ofbankruptcy. In the case of employment contracts, many legal restrictionsimplicitly or explicitly limit the worker's liability. Among these restrictions,one can point out minimum wage laws, or laws exonerating the workerfrom liability for damages caused during the execution of the contract.For instance, in the United States, the employer is responsible to thirdparties for a worker's torts committed within the scope of the worker'semployment, even though the worker's conduct was wanton and malicious(Prosser et al. [11]). Some courts also limit the ability of an employer

515OUTPUT MONITORING AND LIMITED LIABILITY

2 In moral hazard models, the agent chooses e before observing the realization of :.3 This assumption is at the root of the so-called efficiency wage literature.


from recovering against its own employee, when the employer is injuredbecause of employee's negligence (American Law Institute [1]).

We show that, when agents have limited liability and output is imperfectlyobserved, the agent, regardless of his intrinsic productivity characteristics,can receive a positive rent and never produces at a first-best level. Thetraditional result of adverse selection models known as ``efficiency at thetop'' no longer holds.

This model allows us to compare the effect of limited liability underimperfect input or output monitoring. In a related paper (Lawarre� e andVan Audenrode [9]), we have shown that when output is perfectly observed,while input is not, the only effect of introducing limited liability forthe agent is to prevent auditing from limiting the rent paid to the high-productivity agent. The optimal levels of effort and production are leftunchanged. Under imperfect output monitoring, imposing limits on theagent's liability not only serves to limit the rent that the principal canextract from the agent but also creates important distortions in the structureof the optimal contract and brings it farther away from the first best.

2. THE MODEL

We analyze a standard adverse selection model with two risk neutralplayers, a principal and an agent. The principal owns a productive unit(firm) and hires an agent to run it. The agent has private informationabout his productivity : and exerts an effort e which together with theproductivity parameter determines the output X=:e, with e�0 and:>0. Effort is costly for the agent. We assume that the cost of effort is.(e) with .$(e)>0, ."(e)>0, .(0)=0, .(e)>0, lime � � .$(e)=�, andlime � 0 .$(e)=0. The principal collects the output and compensates theagent with a transfer t. The agent's utility function is U=t&.(e), while theprincipal's utility function is 6=X&t.

We assume that the agent observes the realization of : before choosinghis effort. This assumption makes our model an adverse selection model.For simplicity, : can only take two values, :0 with probability q and :1

with probability 1&q. The output X, the effort e, and the productivityparameter : are private information of the agent. The principal canonly observe X imperfectly. We assume that he correctly observes X withprobability r and incorrectly with probability (1&r), with r>1�2. Since :can only take two values, X will also take at most two values in theoptimal contract,4 X0 and X1 . As usual, the high-productivity agent will

516 LAWARRE� E AND VAN AUDENRODE

4 Given the binary nature of our model we can impose restrictive beliefs on values of Xdifferent from X0 and X1 . The principal believes that X=X1 if he observes X>X1 , thatX=X0 if he observes any X # (X0 , X1), and that X=0 if he observes X<X0 .


have the opportunity to shirk by mimicking the low-productivity agent.However, since output is imperfectly observed, both agents now have anadditional opportunity to shirk by exerting no effort at all and producingzero. Indeed, an agent can always claim that a no-output observation isdue to the imperfect monitoring technology. We assume that, when anerror is committed, the principal is equally likely to observe the two alter-native outputs.5 Denoting X� as the observed output, we can summarize ourimperfect monitoring technology with the conditional probabilities

P(X� =Xi | X=Xi)=r

P(X� =Xj | X=Xi)=1&r

2(with i{j),

where X� is the observed output, X the true output, and both Xi andXj # [0, X0 , X1].

To conclude this section, we present the timing of the model.

1. Nature chooses the productivity parameters :. The agent learnsthe value of :.

2. The principal offers contracts specifying the transfers (t) for eachtype of agent as a function of the observed output.

3. The agent chooses effort (e). Production takes place.

4. The principal imperfectly observes the output.

5. Transfers take place.

3. THE RESULTS

3.1. First Best Contract

As a benchmark, we present the first-best solution. If the principal canperfectly observe : and the output,6 he will design two different contracts,one for each value of the parameter : by choosing ei and ti to

Max :iei&ti , (1)


5 Although this hypothesis is clearly restrictive, we will argue that it is not crucial to ourresults. In addition, such a monitoring technology is not uncommon. In the case of employeeperformance evaluation, for example, the employee's performance is often rated by his super-visor as average, above average, or below average. There are no a priori reasons to believethat a supervisor would err more often in one direction than another, evaluating theemployee's performance systematically too high or too low relative to his true one.

6 He can therefore deduce the agent's effort perfectly.


subject to the agent's individual rationality constraints: ti&.(ei)�0,(i=0, 1).

The solution to the problem implies that at the optimum .$(ei)=:i andti=.(ei).

3.2. Second-Best Contract

Assume now that the principal cannot observe :, but observes freely andperfectly the agent's output. For simplicity, and without loss of generality,we will assume from now on :1=1 and :0<1. The problem for theprincipal becomes

Max q } [:0e0&t0]+(1&q) } [e1&t1], (2)

subject to

t0&.(e0)�0 (IR0)

t1&.(e1)�0 (IR1)

t0&.(e0)�t1&. \e1

:0+ (IC0)

t1&.(e1)�t0&.(:0e0). (IC1)

The first two constraints (the individual rationality constraints) ensure thatthe value of the contract will be positive for each type of agent, while thetwo incentive compatibility constraints ensure that agents will choose thetype of contract that has been designed for their type. As is usual in thistype of problem (see, for instance, Laffont and Tirole [8]), only IR0 andIC1 are binding at the optimum. It can be verified ex post that the othertwo constraints are satisfied when they are ignored in the maximizationprogram.

The solution to this program gives

.$(e1)=1 and .$(e0)=1&1&q

q[.$(e0)&.$(:0e0)]. (3)

Equation (3) shows that the highly productive agent continues to produceat the first-best level. The low-productivity agent produces less here than inthe first-best solution. As is customary in these models, the principal has topay a rent to the high type to prevent him from shirking. The low-productivityagent receives no rent.

3.3. Imperfect Observation of Output

We now consider the case where the principal can only observe theoutput imperfectly (but still costlessly). With a probability r, the monitoring



process will provide the correct answer. With a probability (1&r), it willyield a wrong one. We define t~ as the transfer paid to the agent said tohave produced zero effort.

Given the monitoring technology described in Section 2, the problem forthe principal becomes

Max q } _:0e0&rt0&1&r

2t1&

1&r2

t~ &+(1&q) } _e1&rt1&

1&r2

t0&1&r

2t~ & , (4)

subject to

rt0+1&r

2t1+

1&r2

t~ &.(e0)�0 (IR0)

rt1+1&r

2t0+

1&r2

t~ &.(e1)�0 (IR1)

rt0+1&r

2t1+

1&r2

t~ &.(e0)�rt1+1&r

2t0+

1&r2

t~ &. \e1

:0+ (IC0)

rt1+1&r

2t0+

1&r2

t~ &.(e1)�rt0+1&r

2t1+

1&r2

t~ &.(:0e0) (IC1)

rt0+1&r

2t1+

1&r2

t~ &.(e0)�rt~ +1&r

2t0+

1&r2

t1 (IC00)

rt1+1&r

2t0+

1&r2

t~ &.(e1)�rt~ +1&r

2t0

1&r2

t1 . (IC01)

(IC1) and (IC0) are the traditional incentive compatibility constraintspreventing either type from shirking by imitating each other. (IC00) and(IC01) are the constraints preventing the low and high types, respectively,from producing zero effort. By inspection, it is clear that (IC01) is alwayssatisfied when (IC1) and (IC00) are imposed.

Solving the first-order conditions, we find

.$(e1)=1 and .$(e0)=1&1&q

q[.$(e0)&.$(:0e0)],

which implies that the introduction of imperfect observation of the outputleaves the required level of effort and output unchanged. On average, trans-fers to both types of agent remain unchanged. However, the agent who is



incorrectly audited can receive a negative utility. Imperfect observationdoes not add any additional costs for the principal. Its expected surplus isexactly the same with perfect or imperfect observation of output.

This result, well-known in the optimal contract literature (see, forinstance, Laffont and Tirole [7]), is hardly surprising given that all partiesare risk neutral. We have also established in a related work (Lawarre� e andVan Audenrode [9]) that the only effect of limited liability in a traditionaladverse selection model where output is perfectly observed was to removethe principal's ability to limit the rent he pays to the high-productivityagent. As a consequence of these two established results, we know that thechanges in the optimal contract which will be observed here under imperfectoutput monitoring and limited liability are caused by the simultaneouspresence of these assumptions and not by any one of them alone.

3.4. Limited Liability

The previous section assumed that the agent has risk-neutrality as wellas unlimited liability. If an agent is risk averse, he will obviously be reluctantto engage in a contract in which he may suffer a net loss. In addition, inthe case of employment contracts, for example, institutional limitations��like minimum wage or bankruptcy laws��will clearly limit the possibilityof placing the whole burden of imperfect monitoring upon the agent. In thenext model, such a possibility is explicitly ruled out. A limited liabilityclause is introduced.

Different forms have been used in the literature to introduce limitedliability rules. For instance, in an audit model, Baron and Besanko [2]interpret limited liability as an exogenous upper bound on the penalty thatthe principal can impose on an agent misreporting or cheating. Sappington[12] imposes a limited liability constraint to prevent situations where ``theagent can do no better under this contract than suffer a loss in utility belowthe level achieved in autarky i.e., if he had not contracted with the principalat all ).''

Practically, two techniques could be used to introduce limited liability inour model. One could impose a monetary constraint on the transfersbetween the principal and the agent (in our notation, ti�L, L being a non-negative constant) or one could impose a constraint on the utility of theagent (i.e., U=ti&.(ei)�L). Both techniques are used in the literature.For instance Ines [4] imposes a limited liability constraint on the transfersand Stole [13] on the utility.

In addition, Sappington [12] shows that both are equivalent in a modelwith perfect output observation. With imperfect output observation, thisequivalence between the two definitions of limited liability no longer holds.The reason is that the concept of agent's utility is now different whether



measured before the output is observed (expected utility) or after (effectiveutility). When the principal guarantees ti�L, he also ensures that the agentwill receive an expected utility greater than L. The agent can always chooseto exert no effort and therefore receive at least L. However, if the agentexerts an effort, he cannot be sure that he will not suffer ``a loss of utilitybelow the level achieved if he had not contracted with the principal.''Indeed, the contract could specify that an observed output correspondingto a zero level of effort should be paid no more than L. However, when azero-effort is observed, an error can have been made. An agent who haseffectively exerted a positive effort would receive a utility level below L.

Like Sappington, we are interested in studying zero-liability contracts.Therefore, we impose the strict limited liability constraint that the effective(ex post) utility be above L. Of course, such a constraint also implies thatthe transfers be above L.

Zero liability is a strong restriction on the ability of the parties to contract,and weaker forms of limited liability could generate results similar to ours.However, we can justify such a strict constraint either because the agent isinfinitely risk averse or because some law prohibits contracts which canpotentially generate loss of utility for the agent.

We interpret limited liability as an ex ante constraint on contracts. Thefact that the agent's level of utility and effort are unobserved does not limitthe applicability of this constraint: all it does is forbid the principal and theagent from entering into contracts which could potentially result in lossesfor the truthful agent.

Finally, note that our definition does not require per se that an agentshirking and getting caught receives a positive utility. When the principalconvicts an agent, he cannot tell with certainty whether he is a high-productivity agent shirking or a low-productivity agent mistakenly accused.We require that the latter receives at least zero utility. This, of course,results in the shirking agent earning at least zero utility as well.

In our framework, this will imply six new constraints to account for allpossible mistakes in monitoring. We label them LL, for limited liability:

t0&.(e0)�0 (LL0)

t0&.(e1)�0 (LL1)

t1&.(e0)�0 (LL2)

t1&.(e1)�0 (LL3)

t~ &.(e0)�0 (LL00)

t~ &.(e1)�0. (LL01)



The principal maximizes the same objective function subject to a new setof constraints. The problem [P] is

Max q } _:0e0&rt0&1&r

2t1&

1&r2

t~ &+(1&q) } _e1&rt1&

1&r2

t0&1&r

2t~ & , (5)

subject tot (IR0), (IR1), (IC0), (IC1), (IC00), (IC01), and (LL0) to(LL01). The lemmas in Appendix 1 indicate that the only binding constraintsin solving program (5) will be (IC1), (IC00), and, depending on the valueof efforts, (LL00) and (LL01). We ignore IC0 as it can be verified ex postthat it is satisfied.

We can identify four cases.

Case 1. Only (LL01) binds.In this case, e1>e0 . The solution is given by (see Appendix 2 for first-order

conditions)

.$(e0)=q:0

3r&1r+1

+:0

q+3qr+2rr+1

.$(:0e0)

and

.$(e1)=(1&q)3r&1

q&3qr+5r&1.

Case 2. Both (LL01) and (LL00) bind.In this case e1=e0 and solving the problem imposing the equality

constraint gives the following answer:

.$(e1)=.$(e0)=.$(e)=(1&q+:0 q)(3r&1)

6r+q&3qr+

q&3qr+2r6r+q&3qr

.$(:0e).

Case 3. Only (LL00) binds.In this case, e0>e1>:0e0 . The solution is given by (see Appendix 2 for

first-order conditions)

.$(e0)=:0q(3r&1)

4r+

q&3qr+2r4r

:0 .$(:0 e0)



and

.$(e1)=(1&q)3r&1

q&3qr+2r.

Case 4. (LL00) binds and e1=:0e0 .Finally, in this case, by inspection of the constraints, t1=t0 and (LL00)

is the binding limited liability constraint. The solution is given by (seeAppendix 2 for first-order conditions)

.$(e0)=3r&1

4r(:0q+(1&q) :0) and .$(e1)=

3r&14r

.

Note that since e1�:e0 and IC1 is binding, IC0 must be satisfied in thefour cases.

By introducing limited liability, we have implicitly made costly errors inthe observation of output. If the liability of the agent is unlimited, thetransfers offered when the output is correctly observed can always adjustto compensate the agent, in expected value, for the gain or losses he canpotentially suffer when observation errors occur. In expected value, asalready noted, the agent receives the same surplus as under perfect observationof output.

Under limited liability, this scheme is no longer feasible and the principalmust find an alternative way to limit the costs resulting from observationerrors. This is practically achieved through reducing the differences betweenthe high and low productivity transfers, particularly when the monitoringis unreliable. Indeed, by incentive compatibility, the principal must give arent to the high type agent. However, it is too costly to give a high transferto the high type when observation errors are frequent (since this transfermight have to be paid after X0 or even after nothing is produced). So theprincipal decreases t1 . In fact, when r is very low (case 4), t1 is lowered atthe level of t0 . The high type can only obtain a rent exerting an effort e1

lower than e0 . If r improves, the principal can raise t1 and, therefore, alsoe1 . However, e1 will never be raised above e0 (except in case 1��see theintuition below). Doing so would force the principal to increase both thehigh type transfer (due to incentive compatibility) and the low type transfer(due to limited liability). But then the principal should also increase e0 andkeep e0=e1 . This is our first proposition. To guarantee existence, we usea quadratic functional form for the effort function.

Proposition I. Suppose .(ei)=e2i �2. There exist conditions under which

the high-ability agent will never exert more effort than the low-ability agentat the solution to [P].


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Proof. If .(ei)=e2i �2, the solution when (LL01) is the only binding

limited liability constraint is given by e0=&q:0((3r&1)�(:20(q&3qr+2r)&

r&1)) and e1=(1&q)((3r&1)�(q&3qr+5r&1)). If &q:0((3r&1)�(:2

0(q&3qr+2r)&r&1))>(1&q)((3r&1)�(q&3qr+5r&1)), case 1 isimpossible (otherwise e1<e0) and we must be in one of the other threesituations with e0�e1 . Q.E.D

We can explain the intuition behind this proposition. Case 1 is only validfor relatively low values of q and :0 , i.e., when there are few low types (lowq), or these low types are very unproductive (low :0). In that situation, theprincipal will simply ignore them and require an effort from them which isclose to zero, in order to avoid increasing the rent he pays to the manyhigh types. Yet, as the principal will still have to pay a transfer to the lowtypes, the effort he will require from the high type will be below its first-best level. Case 1 applies.

When :0 and q increase, however, it becomes interesting for the principalto elicit effort from the low types. There are many of them and they are

Fig. 1. Optimal contract when .(ei)=e2i �2 and cases 2, 3, and 4 apply.



more productive. This situation, described in Proposition I, corresponds toa case where q and :0 are high for a given r, i.e., when case 1 is impossible.Our definition of limited liability implies that the worst possible outcomeof the contract for the agent is now always the relevant constraint for theprincipal. Figure 1 shows an example of the evolution of the optimal levelof effort as a function of r, the quality of the information obtained whenobserving the output, in the most interesting situation, i.e., when both qand :0 are high. As a general rule, effort increases with r. When r is low,observation errors occur so often that, to eliminate the potential cost ofthese mistakes, the principal has to give up trying to have the workers selfselect. All workers are required to produce the same output and receive thesame transfer. Case 4 applies. When r increases, i.e., when the probabilityof costly mistakes decreases, it becomes profitable for the principal tointroduce some state-contingent transfers. The principal can indeedincrease the high-productivity agent's production by raising his effort andhis transfer. Case 3 applies. When r is close to one, it becomes interestingfor the principal to increase the levels of effort, respecting the constraintthat the effort of the high-productivity agent cannot be larger than the oneof the low-productivity agent. Case 2 applies.

In addition, because of the limited liability constraint, the effort requiredfrom the high productivity agent does not tend to the first-best level asr tends to one. Remember that it is only because the principal cannotdistinguish with certainty between a high-productivity agent shirking and ahigh-productivity agent mistakenly accused of shirking that our limitedliability constraint becomes relevant. Therefore, as soon as r is strictlysmaller than one, the requirement to always ensure a positive utility to theagent sets a minimum level on t~ and t0 which justifies reducing e1 .

Another consequence of the principal's incentive to reduce the effort ofthe high type will be that the well-known result of ``efficiency at the top''does not hold in this model.

Proposition II. At the solution to [P], the high-productivity agentnever produces at first-best level.

Proof. It is easy to verify that in all four cases, e1 increases mono-tonically with r and that e1�1 when r=1. Therefore e1<1, \r<1, and thehigh productivity agent never produces at first best.

Proposition III. At the solution to [P], both low- and high-productivityagents earn rent.

Proof. In all four cases, neither (IR0) nor (IR1) are ever binding. Inother words, both agents receive a rent.



Again, the reason is the limited liability constraint. On the one hand, theprincipal must motivate the agent to exert a positive effort. On the otherhand, the agent said to have exerted a zero effort must also receive a non-negative utility. This double requirement will force the principal to leave arent to all types of agent. It is worth noting that this rent earned by thelow-type agent is generated by the limited liability constraint and not bythe principal's desire to deter him from producing zero effort. The low-typeagent did not earn any rent when imperfect output monitoring alone wasintroduced in the model.

Finally, it can be verified that our main qualitative conclusions continueto hold if the limited liability constraint is modeled as a minimum transferconstraint. Even though the optimal value of e0 changes (as expected), ourthree propositions still hold.

4. CONCLUSIONS

We have shown in a related paper (Lawarre� e and Van Audenrode [9])that the only effect of limited liability in a traditional adverse selectionmodel where output is perfectly observed was to remove the principal'sability to limit the rent he pays to the high-productivity agent. In thispaper, we have studied the joint effects of imperfect observation of outputand limits on agent's liability in an adverse selection model. Three majorresults have been derived under these assumptions:

1. The high-productivity agent may be required to produce a level ofeffort which is strictly lower than the one required from the low-produc-tivity agent.

2. The high-productivity agent never produces at first-best level.

3. Both high-and low-productivity agents can earn rents.

These results have been derived by adding two very realistic hypothesesto a simple model of adverse selection, the hypotheses of imperfect outputobservation and of limited liability. While neither of these two assumptionsalone fundamentally perturbed the results of the model, their joint additionhas profoundly changed its qualitative predictions. The importance of thechanges in the results generated by these assumptions clearly show theneed for further research in the area.

Our model could be extended to allow for a more general monitoringtechnology. The probability of mistakes in the evaluation of the output couldbe endogenized. For example, the probability of mistakes could decreasewith the gap between the actual and observed output. Such a formulationwould call for a continuum of values for the adverse selection parameter, :.



Our conjecture is that our basic conclusions will continue to hold whenmore general imperfect monitoring technologies are admitted. Indeed, aslong as any agent can be incorrectly monitored and is at risk of sufferinga net loss as a result of the mistake in monitoring, we will continue toobserve the key conflict between the incentive compatibility constraints andthe limited liability constraints. This conflict induces the slackness of allindividual rationality constraints and generates our conclusions.

Finally, the constraint of limited liability can entail large rents for anagent receiving t1 by mistake. Suppose that the observation of output isactually made by an auditor hired by the principal. The agent would haveincentive to bribe the auditor so that he reports only favorable observationsto the principal.7 An interesting extension of our model would study howthe principal can limit such collusion.

APPENDIX 1: BINDING CONSTRAINTS

The following lemmas establish under which conditions the constraintsbecome binding.

Lemma 1. (IR1) is never binding.

Proof. If (IR1) is binding, the RHS of IC1 is negative or zero. The RHSof IC1 is necessarily strictly bigger that the LHS of IR0, which is positiveor zero. So the LHS of IC1 must be strictly positive, which contradicts IR1binding. Q.E.D

Lemma 2A. IR0, IR1, LL2, LL3, and IC1 cannot all be slack at theoptimum.

Proof. If this was the case, the principal could decrease t1 and increaseprofit. Q.E.D

Lemma 2B. IR0, IR1, LL0, LL1, IC00, and IC0 cannot all be slack atthe optimum.

Proof. If this was the case, the principal could decrease t0 and increaseprofit. Q.E.D

Lemma 3A t1�t0 .

Proof. For both (IC1) and (IC0) to be met, we need e1�:0e0 . Indeed,(IC0) and (IC1) together imply .(e0)+.(e1)�.(e1 �:0)+.(:0e0). This


7 See Kofman and Lawarre� e [6].


condition cannot be met if e1<:0 e0 , by property of convex functions.This means that the high-type agent when exerting e1 always provides an

effort higher than he would have to provide when imitating the low-typeagent. Therefore, if t0 was larger than t1 , the high-productivity agent couldalways obtain a higher transfer at a lower level of effort by imitating thelow-productivity agent. Q.E.D

Lemma 3B. t0�t~ .

Proof. Immediate, by inspection of (IC00). Q.E.D

Lemma 4. (IR0) can be ignored in the maximization problem as it isalways satisfied.

Proof. Since t1�t0�t~ , rt0+((1&r)�2) t1+((1&r)�2) t~ �t~ . Now, assume(IR0) is not satisfied, then rt0+((1&r)�2) t1+((1&r)�2) t~ <.(e0). Together,these two conditions imply t~ <.(e0), which contradict (LL00). Q.E.D

Lemma 5. (LL0), (LL1), (LL2), and (LL3) can be ignored in the maxi-mization problem as they are always satisfied.

Proof. Immediate, since t1�t0�t~ and given (LL00) and (LL01).Q.E.D

APPENDIX 2: FIRST-ORDER CONDITIONS

First-Order Conditions of Case 1

In this case the Lagrangian can be written as

l=q } _:0e0&rt0&1&r

2t1&

1&r2

t~ &+(1&q) } _e1&rt1&1&r

2t0&

1&r2

t~ &+*1 _rt1+

1&r2

t0+1&r

2t~ &.(e1)&rt0&

1&r2

t1&1&r

2t~ +.(:0 e0)&

+*2 _rt0+1&r

2t1+

1&r2

t~ &.(e0)&rt~ &1&r

2t0&

1&r2

t1&+*3[t~ &.(e1)]. (A1)

And the first-order conditions are given by

:0q+*1 .$(:0e0) :0&*2 .$(e0)=0 (A2)

(1&q)&*1.$(e1)&*3.$(e1)=0 (A3)



&qr&(1&q)1&r

2+*1 _1&r

2&r&+*2 _r&

1&r2 &=0 (A4)

&q1&r

2&(1&q) r+*1 _r&

1&r2 &=0 (A5)

&1&r

2+*2 _1&r

2&r&+*3=0. (A6)

First-Order Conditions of Case 2


l=q } _:0e&rt0&1&r

2t1&

1&r2

t~ &+(1&q) } _e&rt1&1&r

2t0&

1&r2

t~ &+*1 _rt1+

1&r2

t0+1&r

2t~ &.(e)&rt0&

1&r2

t1&1&r

2t~ +.(:0e)&

+*2 _rt0+1&r

2t1+

1&r2

t~ &.(e)&rt~ &1&r

2t0&

1&r2

t1&+*3[t~ &.(e)]. (A7)


:0q+(1&q)+*1[:0.$(:0e)&.$(e)]&*2.$(e)&*3 .$(e)=0 (A8)

&qr&(1&q)1&r

2+*1 _1&r

2&r&+*2 _r&

1&r2 &=0 (A9)

&q1&r

2&(1&q) r+*1 _r&

1&r2 &=0 (A10)

&1&r

2+*2 _1&r

2&r&+*3=0. (A11)

First-Order Conditions of Case 3.


l=q } _:0e0&rt0&1&r

2t1&

1&r2

t~ &+(1&q) } _e1&rt1&1&r

2t0&

1&r2

t~ &+*1 _rt1+

1&r2

t0+1&r

2t~ &.(e1)&rt0&

1&r2

t1&1&r

2t~ +.(:0e0)&

+*2 _rt0+1&r

2t1+

1&r2

t~ &.(e0)&rt~ &1&r

2t0&

1&r2

t1&+*3[t~ &.(e1)]. (A12)




:0 q+*1 :0.$(:0e0)&*2.$(e0)&*3.$(e0)=0 (A13)

(1&q)&*1.$(e1)=0 (A14)

&qr&(1&q)1&r

2+*1 _1&r

2&r&+*2 _r&

1&r2 &=0 (A15)

&q1&r

2&(1&q) r+*1 _r&

1&r2 &=0 (A16)

&1&r

2+*2 _1&r

2&r&+*3=0. (A17)

First-Order conditions of Case 4.


l=q } _:0 e0&1+r

2t&

1&r2

t~ &+(1&q) } _:0e0&1+r

2t&

1&r2

t~ &+*1 _rt+

1&r2

t~ &.(e0)&rt~ +1&r

2t&

+*2[t~ &.(e0)]. (A18)


:0q+(1&q) :0&*1.$(e0)&*2.$(e0)=0 (A19)

&q1+r

2&(1&q)

1+r2

+*1 _r&1&r

2 &=0 (A20)

&1&r

2+*1 _1&r

2&r&+*2=0. (A21)

REFERENCES

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7. J.-J. Laffont and J. Tirole, Using cost observation to regulate firms, J. Polit. Econ. 94(1986), 614�640.

8. J.-J. Laffont and J. Tirole, ``A Theory of Incentive in Procurement and Regulation,'' M.I.T.Press, Cambridge, MA, 1993.

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Optimal Contract, Imperfect Output Observation, and Limited Liability

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