Electronic copy available at: http://ssrn.com/abstract=1932882

Group Decisions, Contracts and

Informational Cascades1

Praveen Kumar

C.T. Bauer College of Business

University of Houston

Houston, TX 77204

pkumar@uh.edu

Nisan Langberg

C.T. Bauer College of Business

University of Houston

Houston, TX 77204

nlangberg@uh.edu

September 22, 2011

1We thank Daron Acemoglu, Mark Armstrong, Ken Arrow, Salvadore Barbera, Ken Binmore,Philip Bond, Andres Caravajal, Michael Fishman, Michael Gallmeyer, Tom George, Itay Goldstein,Roger Guesnerie, Peter Hammond, Milton Harris, Ron Kaniel, Marcus Miller, Dilip Mookherjee,Adriano Rampini, K. Sivaramakrishnan, S. Viswanathan, Myrna Wooders, and seminar partici-pants at University of Houston, Texas A&M University, the Duke-UNC Corporate Finance Con-ference, the Utah Winter Finance Meetings, the Western Finance Association Meetings, and theMarie Curie Conference on Economic Theory (in the honor of Peter Hammond) at the Universityof Warwick for useful comments or discussions on the issues addressed in this paper.

Electronic copy available at: http://ssrn.com/abstract=1932882

Abstract

We analyze the (Perfect Bayesian) equilibrium in an observational learning model with com-

munication and bilateral incentive contracts when pairs of agents sequentially make decisions

based on internal communications and the observed history of decisions by other groups. One

of the agents in each pair receives a private noisy signal on an unknown state and we char-

acterize asymptotic learning through the design of e¢ cient bilateral contracts. Long run

information aggregation depends not only on the properties of the signal structure but also

on the intensity of the con�icts of interest between the informed and uninformed agents.

In contrast to the results from the herding literature, we �nd that information can be as-

ymptotically complete with bounded (private) beliefs while there may be herding even with

unbounded beliefs; in the latter case herding occurs when the con�icts are su¢ ciently high

that it is incentive-e¢ cient to divorce decisions from private signals. We derive su¢ cient

conditions on agents�preferences and the signal structure for learning to be asymptotically

complete as the number of agent-pairs gets large and illustrate their applications by adapting

models from the price-quality discrimination and investment contracting literatures.

Keywords: Group Decisions; Bayesian learning; Optimal contracting; Interest

con�icts; Pooling

JEL classi�cation codes: G32, D23

1 Introduction

The aggregation of dispersed information, with its attendant implications for e¢ cient re-

source allocation, has been a central concern in economics. Recently much attention has

been directed towards the possibility that agents completely ignore their useful private in-

formation and herd on wrong decisions if they make decisions sequentially after observing

decisions by other agents (Bannerjee, 1992; Bikhchandani, Hirshleifer and Welch, 1992).

However, in many situations decisions are made by groups based on intragroup communi-

cations and the observed decisions of other groups. For example, investors (e.g., venture

capitalists) allocate funds to start-up �rms in an emerging industry based on their commu-

nications with managers/founders and the investments made by other �rms in the industry

(Gompers and Lerner, 1999). Similarly, governments approve the marketing of new drugs

based on whether they have been licensed to be produced in other countries and based on

their own communications with local agencies that conduct safety trials.

Unlike the single-agent decisions typically considered in the herding literature, there is

often heterogeneous information within and con�icts of interest among group members �

impeding information aggregation both within and across groups because the informed group

members will have incentives to shade their information. However, in such environments, the

informed can be provided incentives for communication through contracting or mechanism

design. In the examples given above, uninformed investors and regulators can design incen-

tive contracts or mechanisms to induce information from managers and domestic producers,

respectively, and the contract design itself will depend on the observed decisions. Therefore,

the results from the herding literature represent only one aspect of the more general enquiry

into the type of economic environments that may (or may not) lead to an aggregation of

dispersed information toward e¢ cient decisions.

Motivated by these examples, we examine a model with a large number of pairs (the

simplest possible group) of agents that together sequentially decide on an action, when

their payo¤s (from various actions) depend heterogeneously on an underlying state that is

unknown. One of the agents in each pair receives a private noisy signal on the true state.

But because agents� interests are not perfectly aligned, the informed agent will generally

1

not communicate this information truthfully in the absence of speci�c incentives to do so.

We examine the sequence of interim e¢ cient decision rules (Holmstrom and Myerson, 1983)

implemented through the design of e¢ cient bilateral communication contracts, when agents

can observe the previous decisions made by other pairs. Our focus is to clarify the conditions

under which information is aggregated or is complete asymptotically (Smith and Sorensen,

2001; Acemoglu, et al., 2010); i.e., as the number of pairs becomes in�nite the Bayesian

estimate of the state converges in probability to the true state.

Our model therefore combines elements of both the herding and the mechanism design

literatures. As in the herding literature, we consider long term Bayesian learning when

agents sequentially decide on an action based on private noisy signals on an unknown state

(that a¤ects payo¤s) and on the observed history of decisions by other agents. However, here

agents�choices are governed by the optimal communication mechanisms designed by unin-

formed agents. Moreover, in our model the optimal contract design at any stage depends on

the observational learning from decisions (or contracting outcomes) made by other groups.

The dependence of the optimal contract design on the observation of prior contracting out-

comes is an important aspect of our model.

Prima facie, the ability to design incentives for truthful communication of private infor-

mation should reduce the likelihood of herding on the wrong decision. Indeed, in a group

decision with communication the informed agents might not suppress their information once

confronted with information-inducing incentive contracts designed by the uninformed agents.

Somewhat surprisingly, however, our analysis indicates that the con�icts of interest among

group members in the multi-agent decision making environment can impede the aggregation

of information. This is because, unlike the single-agent decision-making situation, the cru-

cial determinant of long run information aggregation when decisions are made by groups is

not the information content of the signals per se but whether the informed agents have the

incentives to reveal their signals to the uninformed agents.

To explicate, for complete learning it is su¢ cient that the optimal contracts induce truth-

telling from the informed agent for any given history of decisions by other groups. But if

there exist equilibrium paths where optimal contracts cease to induce information from the

informed agents, then partial learning (or herding) occurs. Indeed, we �nd that if groups

2

pool asymmetric information to make decisions, then herding on the wrong decision is still

possible even if there are unbounded private beliefs, in contrast to results in the literature

that information will be asymptotically complete with unbounded private beliefs (Smith and

Sorensen, 2001).1

We derive su¢ cient conditions on preferences and signal structures that ensure complete

learning in the long run. These conditions are related to (but are not equivalent to) conditions

used in the optimal contracting literatures for adverse selection and moral hazard (even

though there are no unobservable actions in our model). To guarantee complete learning

asymptotically, one requires not only the Spence-Riley single crossing property but also

the monotone hazard rate property of the posterior beliefs with the monotone and concave

likelihood ratio properties of signals (conditional on the state). The former condition on

posterior beliefs is more restrictive than the monotone hazard rate property of the prior

distribution of types (Myerson, 1981; Maskin and Riley, 1984).2 Meanwhile, the monotone

and concave likelihood ratio properties of noisy signals are used, for example, by Jewitt

(1988).3 Essentially, these conditions are su¢ cient to ensure that the history-dependent

interim-e¢ cient decision rules do not pool over the domain of the informed agents�private

signals.

To help understand the role of the aforementioned restrictions, we note two important

di¤erences between our set up and the standard nonlinear pricing problem with hidden

types.4 First, in our set up � as in the canonical herding model � the private information

is a noisy signal on the underlying state; the signal by itself does not a¤ect agents�payo¤s

(as in adverse selection models). Second, in our dynamic model the optimal contract design

depends on the endogenous posterior beliefs on the unknown state and not the exogenously

1Private beliefs are bounded if the likelihood ratio implied by individual signals is �nite and strictlybounded away from zero. That is, with bounded beliefs the information content of signals (at the margin)is bounded above. In contrast, with unbounded private beliefs agents can receive signals with unboundedinformation content.

2The monotone hazard rate property of the prior distribution of types plays an important role in guar-anteeing the monotonicity of the optimal decision rule in models with adverse selection.

3In Jewitt (1988) these conditions are used in validating the �rst order approach for the derivation ofoptimal contracts in moral hazard settings.

4Varian (1989), Fudenberg and Tirole (1991) and Wilson (1993) provide excellent expositions of thisliterature.

3

given distribution of types. In particular, the probability distribution of the private informa-

tion (i.e., the signals), conditional on the state, is invariant even as both agents sequentially

become more certain of the state.

But how economically appealing (or restrictive) are these su¢ cient conditions? In partic-

ular, what happens to asymptotic learning when they do not apply? To address these issues,

we present two applications of our analysis on asymmetric information problems that have

been analyzed widely in the literature. We �rst examine learning regarding the unknown

value to consumers of a product innovation by adapting the standard optimal price-quality

discrimination model (Mussa and Rosen, 1978; Varian, 1989) to our setting. Here, sellers

induce self-selection of the informed consumers through the design of price-quality discrim-

ination menus. In equilibrium this leads to asymptotically complete learning, i.e., the true

value of the product innovation eventually becomes known as the number of buyer-seller

interactions gets very large.

We also examine learning on unknown capital productivity in a new industry, a prob-

lem of long-standing interest in the literature (Zeira, 1987, 1994; Rob, 1991). In our set-up,

noisy signals on the unknown state are received by �rms that sequentially enter the industry.

However, because of separation of ownership and control, the signals are observed privately

by managers; there is an agency con�ict between owners and managers because the latter

privately value control over larger assets or investment (Stulz, 1990; Hart, 1995). We adapt

the standard incentive contracting formulation of this agency con�ict (Harris and Raviv,

1996) to our sequential observational learning setting. In this case, the single crossing prop-

erty does not apply for the managers�preferences. Unlike the previous example, here we �nd

that there is herding when the agency con�ict is su¢ ciently high, i.e., the managers�private

valuation is su¢ ciently large.

In sum, departures from the su¢ cient conditions for e¢ cient information aggregation

can occur in many important economic environments and there can be herding on the wrong

decision even when communication and incentive contracting are feasible; action spaces are

unrestricted; and private signals have in�nite information content. Thus, rich action or

signal spaces can not by themselves reduce drastically the probability of herding on the

wrong action, as is the case with single-agent decisions (Lee, 1993; Smith and Sorensen,

4

2001; Eyester and Rabin, 2009). In particular, if there is a substantial con�ict of interest

among the agents of a group, then the interim-e¢ cient decision rule may completely pool

over the hidden signals, allowing herding on the same ine¢ cient decision even when private

beliefs are unbounded.5

The remainder of the paper is organized as follows. In Section 2, we describe the ba-

sic economic environment and de�ne aspects of optimal contracts and learning that help

frame the subsequent analysis. Section 3 provides the main results, while Section 4 presents

applications of the analysis of Section 3. Section 5 summarizes the results and concludes.

2 The Basic Model

2.1 Environment

There are a countably in�nite number of stages indexed by n 2 N . To each stage is associated

a pair of individuals (xn; yn) who together decide on an action an chosen from a pre-speci�ed

action space A � R. For convenience, we label individuals fxng1n=1 and fyng1n=1 as type-X

and type-Y agents, respectively. Each individual�s payo¤s depend on its type (X or Y ),

the action an, an underlying state � 2 � = f0; 1g, and any monetary transfers between

them wn 2 W � R: We assume that the payo¤s for type-X and type-Y agents are given,

respectively, by:

UX(an; wn; �) = uX(an; �) + wn; U

Y (an; wn; �) = uY (an; �)� wn (1)

Here, ui : A�� ! R; i = X,Y, are von Neuman-Morgenstern expected utility functions

that are thrice continuously di¤erentiable on (the interior of) A and concave. We allow for

the possibility that the agents cannot agree or decide on an action, in which case they each

receive zero utility.

The true state � is unknown and all individuals have the common prior belief �1 = Pr(

5Indeed, it is possible that the private signals of agents are not used in decisions even in the early rounds,irrespective of whether there are bounded or unbounded beliefs. This is in contrast to the herding modelswhere agents use their information to determine their actions in the early rounds even with bounded beliefs.

5

� = 1): At each stage n; type-X agents privately receive signals receive signals sn 2 S; which

is a metric space. The signals fsng1n=1are generated independently conditional on � according

to the non-degenerate probability measures G�; � 2 �; on the common support S, and such

that G0; G1 are are non-identical. These assumptions imply that at least some signals are

informative, but no signal is perfectly revealing of �. We will denote the probability density

(when sn is absolutely continuous) and the probability mass functions (when S is �nite) with

g0(s); g1(s), respectively.

2.2 Bilateral Contracting

We focus on bilaterally interim e¢ cient decision rules at each stage n 2 N :6 The information

set at n is �n = (sn; hn) 2 �n ; where hn is the observed pro�le of decisions di = (ai; wi) for

i = 1; :::; n � 1; i.e., hn = (d1; :::; dn�1) = ((a1; w1); :::; (an�1; wn�1)). A decision rule is the

mapping �n : �n ! �[A�Wn] (the space of probability measures on A�Wn). We will let

Q(�n(�n)) denote the projection of �n(�n) on �[A�Wn] (i.e., the conditional distribution

of dn as governed by �n(�n)).

Now let �n = Pr(� = 1 hn): Then, by Bayes�rule, the private beliefs of type-X agents

given the signal sn are:7

pn(sn;�n) � Pr(� = 1 sn; hn) =g1(sn)�n

g1(sn)�n + g0(sn)(1� �n)(2)

The support of the private beliefs is [p¯; �p] � [0; 1]. Private beliefs are bounded if p

¯> 0 and

�p < 1; but they are unbounded if [p¯; �p] = [0; 1]:

The expected utility of xn when sn = s, but the decision rule �n(sn = s0; hn) is used, are:

V Xn (s; s0;hn; �n) �

ZA�W

�pn(s;�n)U

X(a; w; 1) + (1� pn(s;�n))UX(a; w; 0))�dQ(�n(s

0; hn))

(3)

6The concept of interim e¢ ciency is introduced and discussed generally in Hlomstrom and Myerson (1983).7For the discrete signal space, this specializes to pjn =

Gj1�n

Gj1�n+G

j0(1��n)

; j 2 fH;Lg:

6

A decision rule �n is (Bayesian) incentive compatible if for every s; s0 2 S :

V Xn (s; s;hn; �n) � V Xn (s; s0;hn; �n); s 6= s0 (4)

(For notational ease, let V Xn (s;hn; �n) � V Xn (s; s;hn; �n)): In addition, �n is acceptable for

xn if

V Xn (sn;hn; �n) � 0;8s 2 S (5)

�n is admissible if it is incentive compatible and acceptable.

Meanwhile, the expected utility of yn conditional on hn; namely, V Yn (hn; �n), is computed

as follows. Let n(s;hn) denote the probability density function of s 2 S conditional on hn;

i.e., n(s;hn) = g1(s)�n + g0(s)(1� �n). Also, denote the cumulative distribution of signals

at stage n by �n(sn = s;�n) = G1(s)�n + G0(s)(1 � �n). Then the expected utility of ynunder the decision rule �n is:

V Yn (hn; �n) =

ZS

ZA�W

�pn(s;�n)U

Y (a; w; 1) + (1� pn(s;�n))UY (a; w; 0)�dQ(�n(s; hn)) n(s)ds

(6)

��n is bilaterally interim e¢ cient if it is admissible and maximizes V Yn (hn; �n):8

By the revelation principle (e.g., Myerson, 1979) any admissible decision rule can be

implemented through a direct mechanism or bilateral contract with the message spaceMn =

S (with truth telling on signals) for xn and the (possibly randomized) message-contingent

decision rule �n(mn; hn) 2 �[A�Wn]; mn 2 Mn: With a straightforward adaptation of

notation, the incentive compatibility conditions (3)-(4) can be expressed as V Xn (s;hn; �n) �

V Xn (s; s0;hn; �n); s 6= s0 2 S; i.e., �n is incentive compatible if it induces truth-telling with

mn = sn. Similarly, acceptability (for xn) requires V Xn (s;hn; �n) � 0, for each s 2 S.

The optimal bilateral contract C�n(hn) = ��n(mn; hn) is then an admissible contract that

maximizes V Yn (hn; �n):

By construction, the various generations of agents (xn; yn) observe only the prior pro�le

8In general, interim e¢ cient rules maximize a welfare function that is a weighted average of the expectedutilities of the two agents (xn; yn), where the weights may depend on �n (Holmstrom and Myerson, 1983).Our formulation conforms to the standard principal-agent contracting framework.

7

of contract outcomes or decisions, i.e., f(at; wt)gt�n�1 : Hence, form the viewpoint of obser-

vational learning, it is the information content of prior decisions (rather than intra-group

messages) that is relevant. But the information gleaned from the optimal decisions can range

from complete revelation � where the optimal decision rule ��n is invertible in the signal sn

� to no revelation � when the decision rule is identical for all signal-types.

For our purposes, the following categorization of the informational content of contracting

outcomes is useful. Let ��n(mn = s; hn) denote the support of the probability distribution of

decisions Q(��n(mn = s; hn)) and recall that �n(s;hn) is the cumulative probability of sn = s

conditional on the observable history of contracting decisions hn. Then the optimal contract

is:

De�nition 1. A separating contract (denoted by CS�n (hn)) if

Pr(sn = s0jdn 2 ��

n(mn = s0; hn); hn) = 1; 8s0 2 S;

2. A bunching contract (denoted by CB�n (hn)) if there exist s; s0 2 S such that

Pr(sn = s0jdn 2 ��n(mn = s

0; hn); hn) < 1 &

Pr(sn � sjdn 2 ��n(mn = s

0; hn); hn) 6= �n(s;hn);

3. A pooling contract (denoted by CP�n (hn)) if for all s; s

0 2 S,

Pr(sn � sjdn 2 ��n(mn = s

0; hn); hn) = �n(s;hn):

While the strict monotonicity (or separation) and bunching in the optimal allocation

with asymmetric information has been widely studied, we reiterate the important di¤erences

between our set up, which builds on the canonical herding model, and the standard nonlinear

pricing problem with hidden types. First, the private information, i.e., the signal s, itself

does not a¤ect the payo¤s of either the informed or the uninformed agent; rather, signals

in�uence expected payo¤s only through their e¤ect on the Bayes-estimate (or posteriors)

on the unknown state parameter �. Second, there is a distinction between the posterior

8

beliefs on �, which are endogenous because they depend on the outcome of the contracting

game, and the likelihood that the (type-X ) agent will receive a given signal, which is �xed

exogenously by the signal structure fG1(s); G0(s)g:

2.3 Equilibrium and Asymptotic Learning

A sequence of bilateral contracts f�ng1n=1 generates a stochastic process of decisions f(an; wn)g1n=1 ;

where the randomization occurs through the realization of signals fsng1n=1 and the (possibly)

randomized decision rule. A Perfect Bayesian equilibrium (or just �equilibrium�) in the game

set up above is then speci�ed by a sequence of optimal bilateral contracts �� = f��ng1n=1 and

the sequence of communication strategies (for xn)m� = fm�ng1n=1 where m

�n = sn; n 2 N . It

is straightforward to show that a ��n = (��;m�) is an equilibrium because, by construction

��n(mn; hn) and m�n = sn are sequentially rational strategies.

9

An equilibrium is then the pro�le �� = f��ng1n=1. (We label the continuation equilibrium

at the root hn by �� hn :) We let P�� denote the probability measure generated by ��; while

the Bayes-consistent beliefs Pr(� = 1 hn) recursively generated by this measure are denoted

by ���n : Then, learning is asymptotically complete along �

� if

limn!1

Pr(���

n = � �) = 1 (7)

That is, with probability 1 the posterior measure is degenerate and the expectations of

� are consistent (with the true � 2 f0; 1g). And learning is asymptotically incomplete if

limn!1 Pr(���n = � �) < 1:

Whether learning is asymptotically complete (or incomplete) depends, of course, on the

information content of the contracting decisions. In particular, note that if at any information

set hn, the optimal contract is pooling (as de�ned above), then ���n+1 = �

��n and hence every

CP�n (hn+i); i = 1; 2::; along �

�hn will also be pooling:

9To complete the speci�cation of the PBE, we assume that upon observing any market outcome cn =2C�n(�n), for any n, �n+1 = �n:

9

Proposition 1 Along any ��, for any hn, n 2 N ; if C�n(hn) = CP�n (hn), then ���n+i = �

��n ,

i 2 N , with probability 1.

Thus, complete pooling of the optimal contracts will rule out complete learning. Con-

versely, if with positive probability the optimal contracts reveal some new information, which

will be the case if they are separating or partially bunching, then learning will be complete.

Corollary 1 Along any ��, for any hn, n 2 N ; if C�n(hn) = CP�n (hn) and 0 < �

��n < 1; then

learning is asymptotically incomplete along �� hn : However, if C�n(hn) = C

S�n (hn)�CB�

n (hn)

for every hn, n 2 N , then learning is asymptotically complete along ��:

3 Conditions for Complete Learning

In this section, we identify conditions that are su¢ cient to rule out complete pooling in the

optimal contract; i.e., these conditions ensure asymptotically complete learning (cf. Corollary

1). We will adopt the convention that ui(�; �); i =X,Y are strictly increasing functions, i.e.,

ui1(a; �) > 0 and ui(a; 1) > ui(a; 0), for every a 2 A.10 In addition, we will use the following

restrictions on preferences and the signal structure:

De�nition The preferences of the type-i agent, i = X,Y, satisfy the single crossing property

(SCP) if ui1(a; 1) � ui1(a; 0); for every a 2 A:11

The SCP is the well-known Spence-Mirrleees condition (with binary types) and plays

a basic role in theory of sorting and models of incomplete information (maskin and Riley,

1984; Milgrom and Shannon, 1994). Next, �x any information set hn, n 2 N ; such that the

posterior beliefs are �n. (Since the posterior beliefs �n are informationally su¢ cient for the

history data hn, we suppress h for notational ease.) Recall that, the cumulative distribution

of signals at stage n is �n(sn = s;�n) = G1(s)�n+G0(s)(1��n) with the associated hazard

rate:1� �n(s;�n) n(s;�n)

=1� (G1(s)�n +G0(s)(1� �n))

g1(s)�n + g0(s)(1� �n)(8)

10This is essentially a labeling convention since it interprets the high state to be �good.�11The SCP will apply strictly if the inequality is strict for each a 2 A:

10

We will say that:

De�nition The signal structure fG1(s); G0(s)g satis�es the posterior monotone hazard rate

property (PMHRP) if, for any given �n 2 [0; 1],@�1��n(s;�n) n(s;�n)

�@s

� 0, for each s 2 S.

In the literature, the monotone hazard property (MHRP) on the distribution of types is

widely used (Myerson, 1981). However, the PMHRP is di¤erent than requiring the MHRP on

G�(s) for each �. The following is an example where the PMHRP is more restrictive than the

MHRP for G1(s) and G0(s): Let, for � > 2 and s � 0; g1(s) = (��1) exp(�(��1)s); g0(s) =

� exp(��s), and g1(s) = g0(s) = 0 when s < 0: In this case, the hazard rates are constant,

i.e., 1�G1(s)g1(s)

= (��1)�1 and 1�Go(s)g0(s)

= �; thus, the monotone hazard rate property is trivially

satis�ed for G1(s) and G0(s): However, the ratio

1� �n(s;�n) n(s;�n)

= n(s;�n)

� n(s;�n)� exp(��s)�n(9)

is not a constant.12

Finally, we also require the monotonicity and concavity restrictions on the likelihood

ratio of signals (conditional on the unknown parameter �).

De�nition The signal structure fG1(s); G0(s)g satis�es the

1. Monotone likelihood ratio property (MLRP) if g1(s)g0(s)

is increasing in s.13

2. Concave likelihood ratio property (CLRP) if g1(s)g0(s)

is concave in s.

The MLRP is used in the literature on hidden actions or moral hazard (along with other

conditions) to justify the �rst order approach to the analysis of the principal-agent problem

(Rogerson, 1985). Jewitt (1988) uses both the MLRP and the CLRP in developing another

set of conditions to justify the �rst order approach. This connection with moral hazard mod-

els is noteworthy because, while there are no hidden actions in our set-up, agents do receive

noisy signals on unobserved parameters (i.e., �). However, in moral hazard models the noisy

12Nevertheless, straightforward computations show that the PMHRP is satis�ed strictly in this case.13Note that when the signal space is binary, i.e., sn 2 fH;Lg, then the strict MLRP is satis�ed if

g1(s = H) > g0(s = H):

11

signal (output) is publicly observable and endogenously generated by the agent�s unobserv-

able action; here, the noisy signal (s) is private information and generated exogenously by

the unobservable state.

We examine the application of these conditions for two polar assumptions on the signal

space S: when signals are continuous, i.e., S is an interval on the real line, and when S is

binary.

3.1 Continuous Signals

Suppose that S = [sL; sH ] � R: For expositional convenience, we �rst restrict attention to

non-randomized contracts; we then show that under the identi�ed su¢ cient conditions for

non-pooling decision rules, this restriction is not binding. Thus, at each n 2 N the decision

rule is �n(s) = (an(s); wn(s)), and given any �n, the expected utility of xn when it receives

the signal s and reports s0 2 S is:

V Xn (s; s0;�n; �n) � pn(s;�n)uX(an(s0); 1) + (1� pn(s;�n))uX(an(s0); 0) + w(s0) (10)

Under our assumptions on ui, the optimal decision rule will be piece-wise continuously di¤er-

entiable (e.g., Hadley and Kemp, 1971). Maximizing (10) over s0 at a point of di¤erentiability

yields the necessary and su¢ cient conditions for local incentive compatibility:

@V Xn (s; s;�n; �n)

@s0= 0,

@2V Xn (s; s;�n; �n)

(@s0)2� 0 (11)

As usual (Mirrlees (1971) and onwards), we use the indirect utility function (with truth-

telling) to eliminate transfers in the optimal decision rule. Put,

JXn (s;�n; �n) � maxs02S

V Xn (s; s0;�n; �n) = V

Xn (s; s;�n; �n) (12)

If the SCP applies, then a decision rule �n(s) is incentive compatible if and only if an(�)

is a monotone increasing function, i.e., _an(s) � 0 (a.e.). Using (11)-(12) and applying the

12

envelope theorem yields,

dJXn (s;�n; �n)

ds� _JXn (a(s); s;�n) = _pn(s;�n)[u

X(an(s); 1)� uX(an(s); 0)] (13)

But the MLRP implies that _pn(s;�n; �n) � 0. This, along with the fact that uX is increasing

in �, implies that _JXn (a(s); s;�n) � 0. It follows from (13) then that the participation

constraint for the type-X agent binds only for sn = sL and, from the fundamental theorem

of calculus,

JXn (s;�n; �n) =

Z s

sL

_JXn (a(r); r;�n)dr (14)

Thus, while the role of the MLRP in the moral hazard literature is to ensure that the

optimal sharing rule for risk-averse agents is non-increasing in output, here its role is to

ensure that the private posterior beliefs of the informed agent for the high state (i.e., � = 1)

are non-decreasing in the signal (s), which in turn yields a convenient monotonicity (in s)

of the informed agents�indirect expected utility.

Meanwhile, the expected utility of yn with the decision rule �n(s) is:

V Yn (s;�n; �n) =�pn(s;�n)U

Y (an(s); wn(s); 1) + (1� pn(s;�n))UY (an(s); wn(s); 0)�

(15)

To derive the su¢ cient conditions for an(�) to be monotone, we examine the relaxed problem:

max�n(�)2A�W

ZS

V Yn (s;�n; �n) n(s;�n)ds, s.t., (14) (16)

Now, put,

Kin(an(s); s;�n) � pn(s;�n)(ui(an(s); 1) + (1� pn(s;�n))(ui(an(s); 0); i = X; Y (17)

This formulation allows us to eliminate transfers (i.e., wn(s)) and write (from (12), (15) and

(17)):

V Yn (s;�n; �n) = KYn (an(s); s;�n) +K

Xn (an(s); s;�n)� JXn (s;�n; �n) (18)

13

Next, by substituting the incentive compatibility constraint (14) into into (18), we can re-

write the relaxed problem as:

maxan(�)2A

ZS

�KYn (an(s); s;�n) +K

Xn (an(s); s;�n)�

Z s

sL

_JXn (a(r); r;�n)dr

� n(s;�n)ds

= maxan(�)2A

ZS

24 KYn (an(s); s;�n) +K

Xn (an(s); s;�n)�

1��n(s;�n) n(s;�n)

_JXn (an(s); s;�n)

35 n(s;�n)ds (19)

(applying integration by parts). The optimality condition for the optimal action is therefore:

YXi=X

@Kin(an(s); s;�n)

@an=1� �n(s;�n) n(s;�n)

@ _JXn (an(s); s;�n)

@an(20)

An application of the implicit function theorem on (20) clari�es the su¢ cient conditions for

the optimal action rule to be monotone.

Proposition 2 At each n 2 N and given any posterior expectations �n; the optimal action

rule a�n(s) is monotone increasing if SCP, MLRP, CLRP, and the PMHRP hold.

We note that in justifying the �rst-order approach to the optimal contract in the moral

hazard problem, Jewitt (1988) uses the MLRP and CLRP to ensure that the inverse of the

agent�s marginal utility with the optimal sharing rule is nondecreasing and concave. Here,

the MLRP, CLRP, PMHRP, and the SCP are jointly used to ensure the desired compar-

ative static property of the optimal decision rule, i.e., _a�n(s) > 0:14 Finally, the conditions

annunciated in Proposition 2 are su¢ cient to rule out randomization in the optimal decision

rule.15

Using Proposition 2 along with Corollary 1, we conclude that:

Theorem 1 Suppose that the SCP, MLRP, CLRP, and the PMHRP hold; and at least one

of these conditions applies strictly. Then, in any ��; C�n(�n) = CS�n (�n) for every �n: Hence,

learning is asymptotically complete along ��:

14Of course, the SCP per se plays no role when only hidden actions are present.15Strausz (2006) shows that randomization is super�uous if the optimal decision rule is monotone.

14

3.2 Discrete Signals

Suppose now that S = fH;Lg: For notational ease, let gj� = g�(s = j); jn = n(sn =

j;�n); j 2 fH;Lg; � 2 f0; 1g: (The notation (ajn; wjn); j 2 fH;Lg; is interpreted similarly.)

The optimal contracting problem is:

C�n(�n) 2 arg maxhajn;wjniH

j=L2A�Wn

fHXj=L

jn�pjnU

Y (ajn; wjn; 1) + (1� pjn)UY (ajn; wjn; 0)

�(21)

subject to,

pjnUX(ajn; w

jn; 1) + (1� pjn)UX(ajn; wjn; 0) �

pjnUX(akn; w

kn; 1) + (1� pjn)UX(akn; wkn; 0); j; k 2 fH;Lg (22)

pjnUX(ajn; w

jn; 1) + (1� pjn)UX(ajn; wjn; 0) � 0; j 2 fH;Lg (23)

In the standard adverse selection model with �nite types, the SCP ensures that the incen-

tive compatibility constraints are binding only in one direction and the individual rationality

(or participation) constraint is also only binding for the boundary type. However, because

of the noisy signal structure in our setting, one also has to assume the MLRP in order for

these properties of the optimal contract to apply.

Proposition 3 If the SCP and the MLRP hold, then in any optimal separating contract

CS�n (�n) the incentive and individual rationality constraints (22)-(23) are only binding for

the lower-type, i.e., sn = L:

Using Proposition 3, we can substitute the binding IR and IC constraints for the low-type

in the objective function (21) to eliminate wLn and wHn (see the Appendix): The �rst order

necessary and su¢ cient condition for optimal actions are then:

pHn uY1 (a

Hn ; 1) + (1� pHn )uY1 (aHn ; 0) + pLnuX1 (aLn ; 1) + (1� pLn)uX1 (aLn ; 0) = 0 (24)

YXi=X

pLnui1(a

Ln ; 1) + (1� pLn)ui1(aLn ; 0) = 0 (25)

15

It follows therefore that aH�n > aL�n if the SCP and the MLRP hold. To see this, suppose

not and assume that aH�n = aL�n = a: Then subtracting (25) (evaluated at aLn = a) from (24)

(also evaluated at aHn = a), we obtain:

pHn uY1 (a; 1) + (1� pHn )uY1 (a; 0)� (pLnuY (a; 1) + (1� pLn)uY1 (a; 0))

= (pHn � pLn)(uY1 (a; 1)� uY1 (a; 0)] > 0 (26)

using the SCP and the MLRP.

Theorem 2 Suppose that S is �nite and the SCP and MLRP hold. Then, in any ��;

C�n(�n) = C

S�n (�n) for every �n: Hence, learning is asymptotically complete along �

�:

3.3 Unbounded Beliefs

Notice that the su¢ cient conditions for asymptotically complete learning apply independent

of the range of the posterior beliefs; i.e., they hold for both bounded and unbounded beliefs.

Here, we depict the situation with bounded beliefs and unbounded beliefs respectively as:

infs2S

g1(s)

g0(s)= �� & sup

s2S

g1(s)

g0(s)= �+ (27)

infs2S

g1(s)

g0(s)= 0 & sup

s2S

g1(s)

g0(s)=1 (28)

Of course, we know from the herding literature that there can be asymptotically complete

learning when beliefs are unbounded, even if there is herding when beliefs are bounded (Smith

and Sorensen, 2001; Acemogulu et al., 2010). However, in the situation at hand, the issue

is not whether the type-X agent has arbitrarily high or low private (posterior) beliefs on

�; rather, the question is whether this agent has the incentives to reveal such information

to the type-Y agent? Put slightly di¤erently: If the optimal contract is completely pooling

with bounded beliefs, will pooling unravel if there are unbounded beliefs?

Consider �rst the case where the signal space is discrete and the MLRP applies. Here,

unbounded beliefs correspond to the situation wheregH1

gH0= 1. But suppose that the SCP

does not apply and, at the optimum, uY1 (a; 1) � uY1 (a; 0): It is apparent from (26) then

16

that unbounded beliefs will not necessarily reverse the ordering of the optimal decision rule

because with unbounded beliefs (pHn � pLn) is simply maximized at 1. Similarly, when the

signal space is continuous, unbounded beliefs by themselves are not su¢ cient to guarantee

that the optimal decision rule will be separating, i.e., a�n(s) is strictly monotone on a set of

positive measure. This can be checked directly from evaluating (20) for the case (28). We

will illustrate these points in our examples below.

4 Examples

We apply the analysis of the previous section to two concrete settings, namely, price-quality

discrimination and managerial investment contracts. Our analysis is of independent eco-

nomic interest because we extend the literatures that have examined these topics in di¤erent

settings.

4.1 A Sequence of Buyer-Seller Contracts

We consider �rst a sequential optimal price-quality discrimination model adapted to signal

structure at hand.16 Suppose that building on some putative technological innovation a new

product is introduced, but a critical parameter of the product�s performance and utility to

consumers � for example, its long run reliability � is unknown. An in�nite sequence of

�generations�n 2 N produce and consume various possible versions or �qualities� of the

product. However, the utility derived from any quality depends on the underlying (and

unknown) reliability of the product.17

In this setting, xn is the buyer and yn is the producer or seller, where xn purchases only

one unit of the product. The seller chooses the product quality an 2 R+; produced with the

unit cost �(an)2=2, and o¤ers it at the price wn: (Hence, A�Wn = R2+). If xn agrees to

16Varian (1989) provides an excellent survey of the price discrimination literature. The preference andcost speci�cation used here is adapted from Kumar (2002, 2006).17The notion of quality here represents the bundle of desirable attributes that are distinct from reliability.

To �x ideas, consider the introduction of a new architecture of smart phones that are sold in a variety ofperformance bundles, but the reliability of the underlying architecture is unknown.

17

purchase, then the payo¤s to the agents are:

UX(an; wn; �) = �an � wn

UY (an; wn; �) = wn ��(an)

2

2(29)

Here � 2 f0; 1g is the unknown reliability parameter. In every generation n; xn receives a

private signal sn 2 fH;Lg regarding � with the probability � � gH1 = gL0 > 12: Hence, both

the SCP and the MLRP apply in this example.

Next, we compute,

Hn � Pr(sn = H hn) = ��n + (1� �)(1� �n) (30)

Moreover, given the signal sn and beliefs �n the type distribution is:

pHn = Pr(� = 1 sn = H;�n) =Pr(� = 1; sn = H �n)

Pr(� = 1; sn = H �n) + Pr(� = 0; sn = H �n)(31)

=�n�

�n�+ (1� �n)(1� �)

pLn = Pr(� = 1 sn = L; �n) =�n(1� �)

�n(1� �) + (1� �n)�: (32)

Notice that,

E(� sn = k; �n) = Pr(� = 1 sn = k; �n), for k 2 fH;Lg.

The optimal contracting problem is therefore:

C�n(�n) 2 arg max

hajn;wjniHj=L

2A�Wn

HXj=L

jn[wjn �

�(ajn)2

2] (33)

subject to the buyer�s incentive compatibility and individual rationality constraints:

pjnajn � wjn � pjna

kn � wkn; j; k 2 fH;Lg (34)

pjnajn � wjn � 0; j 2 fH;Lg (35)

18

As a benchmark, if sellers could observe the signals, i.e., fsigni=1 2 �n, then the optimal

(or expected pro�t maximizing) o¤er is: ~ajn = pjn=� and ~w

jn = p

jn~ajn when sn = j 2 fH;Lg.

It also follows from Corollary 1 that there is asymptotically complete learning.

But with incomplete information, the optimal contract may be either separating or pool-

ing. Under the preference assumptions of this example, in the optimal separating contract

the incentive constraints are only binding downwards (i.e., bind only for the high-type con-

sumer with sn = H), while the individual rationality constraint is binding for the low-type

consumer. Thus, we can simplify the IC and the IR constraints as pHn aHn �wHn = pHn aLn �wLn

and pLnaLn � wLn = 0:

It is well known that inducing separation thus requires distorting the price-quality o¤er

for the low-type consumer relative to the complete information optimal o¤er; and, in the

extreme, it may be optimal to �price out�such consumer-types by settingaL�n ; w

L�n

�= h0; 0i

(Mussa and Rosen, 1978). The following Proposition characterizes the optimal contract

C�n(�n):

Proposition 4 For any 0 < � < 1 and for any �n; n 2 N :18

1. If �n <�0; 1

2�

�; then C�n(�n) is separating with�

aHn =pHn�; wHn = p

Hn a

Hn � aLn

�pHn � pLn

��(36)�

aLn =pLn�� Hn p

Hn

� (1� Hn ); wLn = p

Lna

Ln

�(37)

2. If �n 2�12�; 1�; thenC�n(�n) is separating with the pricing out of the low-type consumer:�

aHn =pHn�; wHn = p

Hn a

Hn

�;qL�n = 0; pL�n = 0

�(38)

Proposition 4 indicates that separating is optimal for all posterior beliefs on the under-

lying state. Thus, it follows from Corollary 1 that:

Corollary 2 Learning is asymptotically complete along the �� speci�ed in Proposition 4.

18We remark that �� � [(�(1 + �)� 1]=[(2�� 1)�], so that �� = 0 if � 2 (0:5;p5�12 ):

19

4.2 A Sequence of Investment Contracts

We consider next a sequential model of optimal investment contracting under asymmetric

information. Suppose that a new technology is introduced but its productivity is unknown.

Interpreting �generations,�n 2 N ; as a sequence of �rms that enter the industry, let xn man-

age or control a production technology owned by yn; this technology stochastically relates

capital investment an to output zn = [�(�h � �`) + �`] f(an) where f is twice continuously

di¤erentiable, strictly increasing and strictly concave, while � 2 f0; 1g is the unknown pro-

ductivity parameter, and �h > �` � 0.

There is a con�ict of interest between the two parties because xn derives private bene�ts

of control. Speci�cally, agents�preferences are:

UX(a; w; �) = w + 'a

UY (a; w; �) = [�(�h � �`) + �`] f(a)�Ra� w (39)

Here, the parameter 0 < ' < 1 models the the private bene�ts of control or utility from

�empire building�(Stulz, 1990; Hart, 1995) and R > 1 is the cost of capital.19 The preference

speci�cation in (39) is standard in this literature, and is adapted from Harris and Raviv

(1996). Notice that the preferences of type-X agents does not depend on the unknown state

�; i.e., the SCP does not apply strictly here. We will consider both the cases of discrete and

continuous signals.

4.2.1 Discrete Signals

Prior to the investment, xn receives a private signal sn 2 S = fH;Lg with � � gH1 = gL0 >12: Hence, conditional on �n,

Hn is speci�ed in (30), while the posterior type distribution

fpHn ; pLng conditional on sn is given in (31)-(32). Furthermore, we put

mn(s) � E(�jesn = s; �n)(�h � �`) + �` = (�h � �`) pn(sn;�n) + �`, s 2 S (40)

19Note that the complete-information optimal level of investment afb(�) satis�es[�(�h � �`) + �`] f 0(afb(�)) = R.

20

The bilateral contract between xn and yn speci�es a menu of investment levels and wage

payments hajn; wjniHj=L 2 R2

+: Here, in the optimal separating contract only the incentive

constraint for the low-type agent is binding. Hence, the optimal contract solves:

C�n(�n) 2 arg max

hajn;wjniHj=L

2A�Wn

f Hn�mn(H)f(a

Hn )�RaHn

�+(1� Hn )

�mn(L)f(a

Ln)�RaLn � wLn

�subject to,

wLn = '�aHn � aLn

�The optimal separating contract CS�

n (�n) is given by:

f 0(aHn ) =1

mn(H)

�1� Hn

�'

Hn+R

!; f 0(aLn) =

R� 'mn(L)

, wLn = '�aHn � aLn

�; wHn = 0

(41)

Now, the proposed contract in (41) is feasible i¤ aHn > aLn , i.e.,

1

mn(H)

�1� Hn

�'

Hn+R

!<R� 'mn(L)

(42)

With some manipulation (see the Appendix), (42) can be re-stated as (where � � �`�h��` ):

Z(�n) ��n + �h

(1��)�n(1��)�n+�(1��n)

i+ �

>R

R� ' (43)

Otherwise, the optimal pooling contract CP�n (�n) is given zero wages and action apooln such

that,

f 0(apooln ) [E(�j�n)(�h � �`) + �`]�R = 0 (44)

Proposition 5 For any 0 < �n < 1 and for any �n; n 2 N :

1. If �n 2 (��; �+) ; then the equilibrium is a separating equilibrium, i.e., C�n(�n) =

CS�n (�n) as given by (41).

2. If �n =2 (��; �+) ; then the equilibrium is a pooling equilibrium, i.e., C�n(�n) = C

P�n (�n)

as given by (44) where wLn = wHn = 0.

21

Notice that pooling occurs at the extreme levels of �n because

lim�!0

Z(�) = lim�!1

Z(�) = 1 <R

R� ' . (45)

At �rst, this might seem similar to the result in the herding literature that information

aggregation ceases at the extremes of prior beliefs. However, note that if � = 0, which is

true for the case �` = 0; then there is still pooling for su¢ ciently high priors on �, but there

is separating for low priors on � provided that � is su¢ ciently high.

Corollary 3 1. The boundary is �� = 0 if (1) �` = 0 and (2) � > R2R�' , but �

� > 0

otherwise.

2. The boundary �+ (��) is increasing (decreasing) in R and decreasing (increasing) in

'.

Hence, it follows from Proposition 5 and Corollary 1 that learning can not be asymptot-

ically complete in this model because information aggregation ceases once posterior beliefs

enter the pooling region.

Corollary 4 Learning is asymptotically incomplete along the �� speci�ed in Proposition 5.

Finally, we examine the case where gH1gH0= �

1�� becomes unbounded, i.e., � " 1. We have

(for � > 0)

lim�"1

�lim�!0

Z(�)

�= lim

�"1

�lim�!1

Z(�)

�= 1 (46)

That is, pooling in the extreme region of posterior beliefs is robust to unbounded private

beliefs when a high or low signal becomes almost perfectly revealing of the true state �.

Moreover, even at the point � = 1 (i.e., perfectly informative signals) the optimal contract is

one of pooling when ' is su¢ ciently high (yet still below R).20 This illustrates the situation

where unbounded private beliefs do not lead to asymptotically complete learning because of

the con�ict of interest between the two parties.

20Namely when '2 + '(2R+ �n)� �nR > 0.

22

4.2.2 Continuous Signals

We consider next the case of a continuous distribution of signals fsng1n=1, with S =[sL; sH ].

Here, the private beliefs that � = 1 of xn given �n and the signal sn are given by (2) and

n(sn = s;hn) = g1(s)�n+g0(s)(1��n): The bilateral contract between xn and yn speci�es a

menu of investment levels and wage payments han(�); wn(�)i : S ! R2+: Here, in the optimal

separating contract the upward incentive constraint is binding. Hence, the optimal contract

solves:

C�n(�n) 2 arg max

an(s);wn(s)

Z n(s;hn) [mn(s)f(an(s))�Ran(s)� wn(s)] ds

subject to,

wn(s) + 'an(s) = wn(s0) + 'an(s

0) for all s; s0 2 S (47)

(where, mn(s) is de�ned in (40)). Since wages in (47) are determined by highest action an,

one can show that under the optimal contract there exists a maximum level of investment

�an such that:

han(s); wn(s)i =

8<: h�an; 0i for s 2 [�s(�an); sH ]

ha�n(s); ' (�an � a�n(s))i for s 2 [sL; �s(�an)](48)

where, a�n(s) is given by:

f 0(a�n(s)) =R� 'mn(s)

; s 2 [sL; �s(�an)] (49)

and �s(�an) satis�es:

mn(�s(�an)) =R� 'f 0(�an)

. (50)

One can now analyze the optimal �a�n, where �s(�an) and a�n(s) are given by (49) and (50) (see

Appendix). Pooling occurs when the solution is �an = aPn de�ned by:

f 0(aPn ) [E(�j�n)(�h � �`) + �`]�R = 0 (51)

23

We conclude:

Proposition 6 There exist 0 � ��� < ��+ � 1 such that for any �n; n 2 N :

1. If �n 2����; ��+

�, then C�

n(�n) = CS�n (�n) and is characterized by (48)-(50) for �a

�n >

aPn .

2. If �n =2����; ��+

�, then C�n(�n) = C

P�n (�n) and is characterized by (51), where wn(s) =

0 for s 2 S.

Hence, it follows from Proposition 6 that learning can not be asymptotically complete in

this model because information aggregation ceases once posterior beliefs enter the pooling

region.

Corollary 5 Learning is asymptotically incomplete along the �� speci�ed in Proposition 6

5 Summary and Conclusions

The aggregation of dispersed information has been a central concern in economics. The

literature on herding shows that if agents make decisions individually and sequentially af-

ter observing previous decisions, then there can be herding on the wrong decisions where

agents eventually completely ignore their private information even if it is inconsistent with

the decision. However, there are many situations where groups of informed and uninformed

agents sequentially make decisions based both on intragroup communications and the ob-

served decisions of previous groups. In such environments, agents with private information

on the underlying state can be provided incentives for communication by other uninformed

agents through bilateral contracting or mechanism design.

It would appear that the ability to design incentives for truthful communication of private

information reduces the likelihood of herding on the wrong decision. This is because, unlike

the single-agent decision making situation, in a multi-agent situation the informed agents

can not autonomously decide to suppress their information; the uninformed agents in each

group (who design the incentive contracts) must also implicitly agree to suppress by not

24

designing information-inducing incentive contracts. However, we �nd that contracting with

communication both facilitates and impedes (asymptotic) information aggregating relative

to the single agent decision making situation. While the received literature indicates that

there will be herding with bounded beliefs (where there is a maximum information content of

signals) but complete learning asymptotically with unbounded beliefs, we �nd, in contrast,

that information can be asymptotically complete with bounded beliefs while there may be

herding with unbounded beliefs. In particular, there is herding on the wrong decision because

there exist equilibrium paths where optimal contracts do not induce any information from

the informed agents, even though these agents can receive private signals of unbounded

precision. Indeed, it is possible that the private signals of agents are not used in decisions

even in the early rounds, irrespective of whether there are bounded or unbounded beliefs;

this is again in contrast to the results in the literature.

Long run information aggregation when groups with dispersed information take actions

based on intra-group communications observed decisions of other groups depends on whether

it is cost-e¢ cient to induce information from the informed agents (within the group). This

depends not only on the precision (or information content) of private signals but also on

the con�icts of interest between the informed and the uninformed agents. Of course, the

incentive e¢ ciency of information elicitation is endogenous and, in an observational learning

setting, depends on the observed history of contract-based decisions by other groups. We

highlight the dependence of the optimal contracts on the endogenous posterior beliefs on

the unknown state and derive su¢ cient conditions on preferences and signal structures that

ensure complete learning in the long run. Some of these conditions are novel (for example,

a monotone hazard rate restriction on posterior beliefs) while some of the other conditions

are used in the optimal contracting literatures for moral hazard even though there are no

unobservable actions in our model. Through examples we show, however, that these condi-

tions may not apply in many interesting economic settings and herding can will occur if the

con�icts are su¢ ciently high.

25

Appendix

Proof of Proposition 1: By de�nition each �� speci�es for each n 2 N the possibly ran-

domized decision rule ��n(mn; hn) and the corresponding support ��n(s; hn) of the conditional

distribution Q(��n(mn; hn)) of decision dn. It is convenient to assume initially that ��n(mn; hn)

is deterministic, for the argument is then extended straightforwardly to behavioral strategies

or mixtures over the deterministic decision rules. Now, for any decision at stage n 2 N ,

dn � (an; wn) 2 A�Wn, de�ne �n(dn;hn) = fs 2 S j dn 2 ��n(s; hn)g. Then by Bayes�

rule,

�n+1 = Pr(� = 1 dn; hn) =

�n

0B@ Z�n(dn;hn)

g1(s)ds

1CA�n

0B@ Z�n(dn;hn)

g1(s)ds

1CA+ (1� �n)0B@ Z�n(dn;hn)

g0(s)ds

1CA(52)

But if C�n(hn) = CP�n (hn), then by de�nition

�n(dn;hn) = S: Hence, from (52), it follows

that

�n+1 = Pr(� = 1 dn; hn) =

�n

0@ZS

g1(s)ds

1A�n

0@ZS

g1(s)ds

1A+ (1� �n)0@ZS

g0(s)ds

1A (53)

=�n

�n + (1� �n)= �n

Because this applies for any deterministic ��n(mn; hn); it follows that the statement is true

for any randomizes decision rule as well.

Proof of Corollary 1: Note that if C�n = CS�n (hn) � CB�

n (hn), then by de�nition,

�n(dn;hn) � S for every dn 2 A�Wn. And since the probability measures G1(s) and

G0(s) are non-identical (conditional on �) by assumption, it follows from (52)-(53) that for

26

any hn; n 2 N ;

Pr(���

n+1 = ���

n ) =

8<: 1 if C�n(hn) = C

P�n (hn)

< 1 if C�n(hn) = CS�n (hn)�CB�n (hn)

(54)

Thus, if hn; n 2 N ; is such that ���n < 1, for C�n(hn) = CP�n (hn), then ���n+i = ��

�n ;

i = 1; 2; :::;with probability 1. Hence, the �rst part of the Corollary then follows from

the de�nition of asymptotic complete learning (cf. (7)), since

limn!1

Pr(���

n = � �) < 1 (55)

Next, suppose that C�n(hn) = C

S�n (hn) �CB�

n (hn) for each n 2 N . Then, conditional on �,

the sequence of posterior beliefs f���n g is independent under the probability measures G1(s)

and G0(s): Furthermore, by the de�nition of conditional expectations, f���n g are Martingales.

It follows from the Martingale convergence theorem that f���n g must converge to some inte-

grable ���; and since G1(s) and G0(s) are mutually singular (see, e.g., Billingsley (1979, pp.

409-410), it must be the case that ���= 1 if � = 1 (or ��

�= 0 if � = 0).

Proof of Proposition 2: A straightforward application of the implicit function theorem

on (20) yields:

_a�n(s) /YXi=X

@2Kin(an(s); s;�n)

@an@s�@ 1��n(s;�n)

n(s;�n)

@s

@ _JXn (an(s); s;�n)

@an�

1� �n(s;�n) n(s;�n)

@2 _JXn (an(s); s;�n)

@an@s(56)

But if the MLRP applies then:

YXi=X

@2Kin(an(s); s;�n)

@an@s=

YXi=X

_pn(s;�n)[ui(an(s); 1)� ui(an(s); 0)] � 0 (57)

27

Next, if the MLRP and SCP apply, then

@ _JXn (an(s); s;�n)

@an= _pn(s;�n)[u

X1 (an(s); 1)� uX1 (an(s); 0)] � 0 (58)

Hence, in addition to MLRP and SCP if the PMHRP also applies then,

@ 1��n(s;�n) n(s;�n)

@s

@ _JXn (an(s); s;�n)

@an� 0

Finally, if the CLRP holds, then �pn(s;�n) � 0; and hence,

@2 _JXn (an(s); s;�n)

@an@s= �pn(s;�n)[u

X(an(s); 1)� uX(an(s); 0)] � 0 (59)

The Proposition then follows from (56).

Proof of Theorem 1 : It follows from (56) that _a�n(s) > 0 for each s in the interior of S

if at least of the assumed conditions holds strictly. Hence, ��n(s; hn) is separating in s and

since this holds for any �n; C�n(hn) = C

S�n (hn): The conclusion then follows from Corollary

1.

Proof of Proposition 3: We �rst show that if the incentive compatible constraint for the

low-type is binding, then the corresponding incentive constraint is non-binding. Now de�ne

for j; k 2 fH;Lg

�j;kn � pjnu

X(ajn; 1) + (1� pjn)uX(ajn; 0) + wjn �

pjnuX(akn; 1) + (1� pjn)uX(akn; 0) + wkn

Note that if the IC constraint is binding for the low-type then �L;Hn = 0: Hence, �H;L

n > 0

if �H;Ln +�L;H

n > 0: But

�H;Ln +�L;H

n = (pHn � pLn)�(uX(aHn ; 1)� uX(aHn ; 0))�

uX(aLn ; 1)� uX(aLn ; 0))�

(60)

28

But (pHn � pLn) > 0 from the MLRP and the expression inside the square parentheses in (60)

can be re-written as

(uX(aHn ; 1)� uX(aLn ; 1))� (uX(aHn ; 0)� uX(aLn ; 0)) > 0

using the SCP. It is then straightforward to show that the IR constraint is binding for the

low-type in any optimal separating contract.

Proof of Theorem 2: Using Proposition 3, we can eliminate the transfers wjn; j = H;L;

and write the optimal contracting problem as:

maxhajniH

j=L2Af Hn [pHn uY (aHn ; 1) + (1� pHn )uY (aHn ; 0) + pLnuX(aHn ; 1) + (1� pLn)uX(aLn ; 0)] +

(1� Hn )[YXi=X

pLnui(aLn ; 1) + (1� pLn)ui(aLn ; 0)]g (61)

The remaining argumentation is given in (24)-(26).

Proof of Proposition 4: Since the individual rationality constraint is binding for the low-

type consumer, we have, pLnaLn = wLn ; and since the incentive compatibility constraints are

only binding downwards, aHn (pHn � pLn) + wLn = wHn : Substituting these conditions in (33)

and ignoring, for the moment, the non-negativity restrictions on an; the optimal contract

maximizes:

OBJ = Hn

�pHn a

Hn � pHn aLn + wLn �

�(aHn )2

2

�+�1� Hn

��pLna

Ln �

�(aLn)2

2

�(62)

The �rst order conditions are

@OBJ

@aHn= Hn

�pHn � �aHn

�@OBJ

@aLn= � Hn pHn +

�1� Hn

� �pLn � �aLn

�Solving these and acknowledging the non-negativity constraints, we obtain the following

characterization of the optimal contract:

29

aHn =pHn�

aLn =

8<:pLn�� Hn p

Hn

�(1� Hn )if pLn >

Hn pHn

(1� Hn )

0 if pLn < Hn p

Hn

(1� Hn )

pLnaLn � wLn = 0

pHn aHn � wHn = pHn a

Ln � wLn (63)

Then substituting (31)-(32) in (63), and simplifying, yields the conclusions of the Proposition.

Proof of Proposition 5: The optimization problem is:

OBJ= Hn�mn(H)f(a

Hn )�RaHn

�+�1� Hn

� �mn(L)f(a

Ln)�RaLn � '

�aHn � aLn

��The �rst order conditions are

@OBJ

@aHn= Hn

�mn(H)f

0(aHn )�R���1� Hn

�'

@OBJ

@aLn= mn(L)f

0(aLn)�R + '

Separating is feasible for intermediate �n. To see this, note that the interior levels of invest-

ment in the low and high states (as provided by the �rst order conditions above) coincide

when,R� 'mn(L)

=1

mn(H)

�R +

�1� Hn Hn

�'

�=

R

�n (�h � �`) + �`

Thus, separating is feasible when,

Z(�n) ��n + �h

(1��)�n(1��)�n+�(1��n)

i+ �

>R

R� ' . (64)

The conclusions of the Proposition then follow from the foregoing.

Proof of Corollaries 3-4: Follows directly from Proposition 5.

30

Proof of Proposition 6: The optimization problem is:

OBJ = maxan(s);wn(s)

Z n(s;hn) [mn(s)f(an(s))�Ran(s)� wn(s)] ds

subject to,

wn(s) + 'an(s) = wn(s0) + 'an(s

0) for all s; s0 2 [sL; sH ] (65)

Since wages in (65) are determined by highest action an, let �an � maxs2S(an(s)) and let the

set of signals that correspond to the highest action be given by �S � fs 2 S : an(s) = �ang.

Then, it is optimal to set wn(s) = '(�an� an(s)) for all s 2 S (i.e., wn(s) = 0 for all s 2 �S ).

Thus, we can write OBJ as

OBJ = maxan(s);wn(s)

ZS� �S

n(s;hn) [mn(s)f(an(s))�Ran(s)� '(�an � an(s))] ds

+

Z�S n(s;hn) [mn(s)f(�an)�R�an] ds

Moreover, for any given �an optimality requires that

f 0(a�n(s)) =R� 'mn(s)

for s 2 S � �S

Since mn(s) is monotone in the signal s it is optimal to set �S = (�s(�an); sh) where �s(�an)

satis�es:

mn(�s(�an)) =R� 'f 0(�an)

.

Next, the level of �an is given by the solution to the problem (where �s(�an) and a�n(s) are given

by above):

�an 2 argmaxa

Z �s(�an)

sL n(s;hn) [m(s)f(a

�n(s))�Ra�n(s)� ' (�an � a�n(s))] ds

+

Z sH

�s(�an)

n(s;hn) [m(s)f(�an)�R�an] ds

31

The �rst order conditions for optimality are:

0 = �'Z �s(�an)

sL n(s;hn)ds+

Z sH

�s(�an)

n(s;hn) [m(s)f0(�an)�R] ds

The second order conditions for optimality are:

0 = �'�s0(�an) n(�s(�an);hn) +Z sH

�s(�an)

n(s;hn) [m(s)f00(�an)] ds

��s0(�an) n(�s(�an);hn) [m(�s(�an))f 0(�an)�R]

=

Z sH

�s(�an)

n(s;hn) [m(s)f00(�an)] ds < 0

The �rst order condition simpli�es to:

0 = �'Pr(sn < �s(�an)jhn)Pr(sn > �s(�an)jhn)

+ f 0(�an)E(snjhn; sn > �s(�an))�R.

By de�nition f 0(�an) =R�'

mn(�s(�an))and we can write the �rst order condition for an interior

solution as,

R + 'Pr(sn < �s(�an)jhn)Pr(sn > �s(�an)jhn)

= [R� '] E(snjhn; sn > �s(�an))mn(�s(�an))

.

Pooling occurs at the corner solution �an = apooln de�ned by:

[E(�j�n)(�h � �`) + �`] f 0(apooln )�R = 0,

and is optimal when

R

R� ' >[E(�j�n)(�h � �`) + �`] f 0(apooln )

mn(sL).

or

R

R� ' >(�h � �`)�n + �`

(�h � �`) pn(sL;�n) + �`

=[(�h � �`)�n + �`]

��g1(s

L)� g0(sL)��n + g0(s

L)�

(�hg1(sL)� �`g0(sL))�n + �`g0(sL)� �Z(�n).

32

Thus, lim�n!0+�Z(�n) = lim�n!1�

�Z(�n) = 1 < RR�' , and pooling is optimal at extreme

levels of expected productivity.

Proof of Corollary 5: Follows directly from the above analysis.

33

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