ESSAYS IN INFORMATION ECONOMICS

By

John Andrew Withers

A DISSERTATION

Submitted to

Michigan State University

in partial fulﬁllment of the requirements

for the degree of

Economics — Doctor of Philosophy

2018

ABSTRACT

ESSAYS IN INFORMATION ECONOMICS

By

John Andrew Withers

The ﬁrst chapter of this dissertation studies a repeated interaction between a regulator

and a regulated ﬁrm. In each period, the ﬁrm completes a project for the regulator, and the

regulator observes the project’s cost. The ﬁrm’s intrinsic cost level is a component of the

project’s cost. Thus, the regulator gathers information about the ﬁrm’s intrinsic cost level

by observing the project’s cost. This information is valuable to the regulator; the more she

knows about the ﬁrm’s intrinsic cost level, which is ﬁxed over time, the more eﬃcient is the

outcome of their interaction in each period.

An important feature of the interaction is that the project’s cost is stochastic; that is,

the ﬁrm has imperfect control over the project’s cost. The ﬁrm determines the expected

project cost by choosing its eﬀort, but a noise term determines the cost realization. The

ﬁrst chapter demonstrates that the regulator’s ﬁrst period contract choice determines how

much she learns about the ﬁrm’s intrinsic cost level. The main contribution of the ﬁrst

chapter is to show that, given a reasonable assumption about the distribution of noise, the

low cost ﬁrm’s ﬁrst period eﬀort is lower than his second period eﬀort. This result aligns

with anecdotal, experimental and empirical evidence of the ratchet eﬀect.

The second chapter examines an interaction that is similar to the ﬁrst chapter, with one

important diﬀerence: the agent’s productivity, which is akin to his intrinsic cost level in

the ﬁrst chapter, is positively correlated over time, rather than ﬁxed. Unlike the standard

ratchet eﬀect literature, the low productivity agent has an incentive to reveal information to

the principal. If the high ability agent is not too much more productive than the low ability

agent, or if the high productivity agent is suﬃciently likely ex-ante, the optimal ﬁrst period

contract restricts what the principal learns about the agent’s ﬁrst period type.

The third chapter considers a two period contracting problem between one principal, one

agent, and an outside labor market. In the ﬁrst period, the principal hires the agent to exert

unveriﬁable eﬀort on a project that may either succeed or fail. Eﬀort can be high or low. In

the second period, the labor market makes the agent a wage oﬀer if the project is successful.

The principal has the opportunity to match the outside oﬀer, or let the agent leave the ﬁrm.

When the agent leaves the ﬁrm, the principal incurs a cost of replacing the agent.

The agent is “self motivated.” That is, the expected value of the outside oﬀer is high

enough that the agent prefers high eﬀort to low eﬀort in the absence of an incentive wage.

When the cost of replacing the agent exceeds a certain threshold, the principal prefers low

eﬀort to high eﬀort, even though the agent is self motivated.

ACKNOWLEDGMENTS

I am grateful for the countless hours that my adviser, Thomas Jeitschko, spent discussing

this dissertation with me over the past four years. Without his encouragement, guidance

and mentorship, it would not have been possible for me to ﬁnish. I would also like to thank

my other committee members, Mike Conlin, Arijit Mukherjee, and Adam Candeub, for their

helpful comments and advice.

I would like to thank Lori Jean Nichols for her support and expert administrative help

during my time as a graduate student at Michigan State University. I would also like to

thank Todd Elder for helping guide me through the daunting task of ﬁnding a job.

Most importantly, I would like to thank parents, my sister and my ﬁanc´e for their unwa-

vering love and support.

iv

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Chapter 1 Dynamic Regulation with Stochastic Costs: Signal Dampen-
ing, Experimentation and the Ratchet Eﬀect . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1
1.2 Model
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Second period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 First period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1
Signal dampening . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.2 Experimentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Equilibrium ratchet eﬀect
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Noisy Observable Outcomes

Chapter 2 Repeated Short-Term Contracting with Correlated Types and
. . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
2.2 Model
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Second period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 First period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1
Signal dampening . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Experimentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3 Total ﬁrst period transfer: signal dampening or experimentation? . .
2.5 Equilibrium rent preservation . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 3 Task Assignment Under Moral Hazard, with Eﬀort-Dependent
Human Capital and Outcome-Dependent Outside Options . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
3.2 Model
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Existence of countervailing incentives, Region 1 . . . . . . . . . . . .
3.3.2 Countervailing incentives, Regions 2 and 3 . . . . . . . . . . . . . . .
3.3.3 Analysis of contracting for a ﬁxed cost of eﬀort
. . . . . . . . . . . .
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
7
12
19
25
28
30
35

37
37
42
44
53
57
59
62
64
69

72
72
76
81
84
90
93
99

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

v

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

vi

LIST OF TABLES

Table 3.1: Expected revenues and expected costs when e = eH . . . . . . . . . . . .

Table 3.2: Expected revenues and expected costs when e = eL . . . . . . . . . . . . .

85

86

vii

LIST OF FIGURES

Figure 1.1: The probability density of costs depends on the agent’s eﬀort choice . . .

26

Figure 1.2:

If c0
over time.

1 lies in the shaded region, the low cost ﬁrm has his eﬀort increased
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

Figure 2.1: Second period beliefs . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

Figure 2.2: First period contract favors rent preservation for all α . . . . . . . . . . .

68

Figure 2.3: First period contract favors learning for some α . . . . . . . . . . . . . .
Figure 3.1: Game tree and payoﬀs (Ue, ue), e ∈ {L, H} . . . . . . . . . . . . . . . . .
Figure 3.2: The agent is self motivated for α ≥ α(ψ). . . . . . . . . . . . . . . . . . .

70

79

85

Figure 3.3: Countervailing incentives, Region 1 . . . . . . . . . . . . . . . . . . . . .

89

Figure 3.4: Countervailing incentives, pH + pL > 1 . . . . . . . . . . . . . . . . . . .

93

viii

Chapter 1

Dynamic Regulation with Stochastic

Costs: Signal Dampening,

Experimentation and the Ratchet

Eﬀect

1.1

Introduction

In regulated industries, ﬁrms and regulators have long-term relationships with one another.

The rules and procedures that govern these relationships are revised over time. When the

regulator cannot commit at the outset of the relationship to how these rules and procedures

will be updated in the future, the ratchet eﬀect arises.

In repeated principal-agent interactions, the ratchet eﬀect describes the agent’s response

to the principal’s inability to commit to long term contracts. The principal learns about the

agent’s ability, or the economic environment, by observing his performance. The principal

then adjusts the agent’s compensation in the future based on what she learns from this

observation. The more the principal learns about the agent, the more rent she is able to

extract. To obscure the principal’s learning process, the agent restricts his performance, or

1

reduces his eﬀort. This allows the agent to avoid more stringent incentives in the future.

Take, for example, a regulated monopoly that provides electricity to consumers. Period-

ically, the regulator will undertake a rate case to evaluate whether current electricity prices

oﬀer the utility a fair return on capital. During the rate case, the regulator observes the

utility’s operating expenses, along with other measures such as the ﬁrm’s rate base (capital),

taxes and depreciation expenses. Based oﬀ these measures, the regulator determines the

revenue that the ﬁrm needs to earn to recoup operating expenses and make a fair return for

their investors. This revenue target in turn determines the prices that the utility can charge

consumers.

During this process, the regulator learns about the ﬁrm’s eﬃciency by observing the

ﬁrm’s operating expenses. The regulator expects that a ﬁrm with high operating expenses

in the current rate cycle will have high operating expenses again in the next rate cycle, and is

thus more willing to give a generous reimbursement. Therefore, the ﬁrm has little incentive

reduce operating costs, since a better performance today implies a less generous revenue

requirement in the next rate cycle.

Some of the earliest anecdotal evidence of the ratchet eﬀect comes from studies of piece

rate factory workers (see Matthewson (1931), Roy (1952), Montgomery (1979) and Clawson

(1980)). Matthewson (1931) documented that piece-rate workers understood that a good

performance today ultimately made them worse oﬀ in the long run. To see this, suppose

a worker produces more units of output in the current period than in the previous pay

period. Since the worker is paid per unit, the worker earns more in the current period

than in the previous period. Workers learned, however, that the factory manager’s response

to this improved performance was to reduce the worker’s piece rate. Thus, the worker

had to keep producing a high output just to earn as much take home pay as they did before

2

revealing favorable information about their productive ability. In response to this behavior by

factory managers, Matthewson documented that workers “never worked at anything like full

capacity.” Berliner (1957) documented that factory managers in the Soviet Union responded

similarly to incentive systems based on output targets.

The anecdotal evidence discussed above suggests that agents restrict their performance

(i.e., reduce eﬀort) when the principal bases their future compensation on information that

she gathers about them. Recent empirical evidence supports this notion. Macartney (2016)

adapts the theoretical model of Weitzman (1980) to examine if teacher value-added schemes

induce dynamic eﬀort distortions among teachers in North Carolina. Teachers in a given

school receive a bonus in the current year if the school-wide average on a standardized test

is above a pre-speciﬁed target. The key feature of these schemes is that the target score is

a function of the school’s average standardized test score in the previous year. Clearly, the

higher is the school’s average test score this year, the more diﬃcult it will be for teachers

to exceed next year’s target and receive a bonus. Macartney exploits diﬀerences in grade

composition across schools to show that teachers respond to the value-added schemes by

reducing their eﬀort on improving their students standardized test scores.

In the kind of repeated interactions described by Matthewson (1931) and Macartney

(2016), agents with high ability have the strongest incentive to reduce eﬀort in the present

to maintain information rents in the future. Charness, Kuhn and Villeval (2011) use an

experimental design to study the eﬀects of labor market competition on the ratchet eﬀect.

As a baseline case, they examine a two-period relationship between one ﬁrm and one worker.

In this baseline case, roughly 60 percent of the experimental subjects who are designated as

having high ability reduce their eﬀort in the ﬁrst period so that they can maintain a second

period information rent. In a related experimental paper, Cardella and Depew (2018) study

3

the impact of evaluating performance at the individual versus group level on the ratchet

eﬀect. The authors ﬁnd that workers suppress eﬀort when evaluated individually.

In most theoretical models of the ratchet eﬀect, the good agent’s eﬀort does not evolve

as one would expect based on the anecdotal, empirical, and experimental evidence discussed

above. For example, Laﬀont and Tirole (1987) examine a two-period interaction between a

regulator and a regulated ﬁrm in which the ﬁrm completes a project for the regulator. The

observable outcome is the project’s cost. The project cost depends on the ﬁrm’s intrinsic

cost level, which is the ﬁrm’s private information. The regulator cannot commit, in the ﬁrst

period, to the second period incentive scheme.

In this setting, the low-cost ﬁrm exerts the ﬁrst best level of eﬀort in the ﬁrst and second

period unless he places a large enough weight on the second period contract. One reason the

low-cost ﬁrm’s eﬀort in Laﬀont and Tirole (1987) does not evolve in a manner that ﬁts with

received evidence is because the ﬁrm is assumed to have perfect control over the observable

outcome. That is, the only way for the low cost agent to hide his private information is to

mimic (pool with) the high cost ﬁrm.

Contrast this with the case in which the agent does not have perfect control over the

observable outcome (i.e., the relationship between the agent’s actions and the project’s out-

come is stochastic). In the framework of Laﬀont and Tirole (1987), this can be achieved

by assuming project costs depend on an additive, zero-mean noise term. Laﬀont and Tirole

(1986) and Laﬀont and Tirole (1993) show that an additive, zero-mean noise term has no

impact on incentives in a static setting.1

In a dynamic setting, however, noise plays the crucial role of slowing the principal’s

learning process. Jeitschko, Mirman and Salgueiro (2002) and Jeitschko and Mirman (2002)

1This assumes that both the ﬁrm and the regulator are risk neutral.

4

study two-period interactions in which an agent produces output for a principal. Output in

each period depends on the agent’s eﬀort, his inherent productivity, and a zero-mean noise

term. The agent’s productivity is his private information and can take one of two values. In

each period, the agent’s compensation depends only on observed output.

In this setting, the agent’s eﬀort choice determines the expected output level. In equilib-

rium, the agent chooses his eﬀort so that his expected output is equal to an output target

proposed by the principal. Therefore, the principal’s choice of equilibrium output targets

determines the distribution of output for each type of agent in each period. For this reason,

the principal’s second period beliefs are a function of her ﬁrst period contract choice.

Jeitschko et al. (2002) and Jeitschko and Mirman (2002) show that two opposing in-

centives determine the ﬁrst period output targets. First, the principal can design the ﬁrst

period contract to increase what she learns about the agent’s private information. By doing

so, she increases her expected second period payoﬀ. Second, the principal can design the

ﬁrst period contract to decrease what she learns about the agent’s private information. By

doing so, she decreases the ﬁrst period transfer to the high productivity agent.

This paper examines a two-period model of regulation. In each period, a ﬁrm completes

a project for a regulator. The observable outcome is the project’s cost. The project’s cost

is stochastic, and the principal uses the cost observation to update her beliefs about the

ﬁrm’s type. We show that when the noisy component of the project’s cost follows a general

distribution, the low-cost agent has his eﬀort increased over time. Therefore, we present

a theoretical model whose predictions match with anecdotal, empirical and experimental

evidence of the ratchet eﬀect.2

2Jeitschko et al. (2002) assume the noise follows a uniform distribution and show that it is optimal for
the high ability agent to exert less than the ﬁrst best eﬀort in the ﬁrst period, which in turn implies that
his ﬁrst period eﬀort is lower than his second period eﬀort. Jeitschko and Mirman (2002) examine a similar
setting in which the distribution of noise is general and are unable to determine how the high ability agent’s

5

This paper is related to two strands of dynamic principal-agent literature. First, this

paper is related to theoretical models of the ratchet eﬀect. The ratchet eﬀect has most

famously been studied in the context of regulation and procurement (Freixas, Guesnerie and

Tirole (1985), Laﬀont and Tirole (1987), Laﬀont and Tirole (1988) and Laﬀont and Tirole

(1993)). It has also been studied in settings such as piece-rate incentive contracts (Gibbons

(1987)), optimal income taxation (Dillen and Lundholm (1996)), and government corruption

(Choi and Thum (2003)). These papers diﬀer from the current paper in that the agent is

assumed to have perfect control over the observable outcome.

This paper is also related to a growing dynamic mechanism design literature. Athey and

Segal (2013) and Pavan, Segal and Toikka (2014) derive eﬃcient and revenue maximizing

dynamic mechanisms, respectively, when the principal can commit to future mechanisms and

the agent’s private information changes over time (for a survey of dynamic mechanism design

when the principal can commit to future incentive schemes, see Bergemann and Valimaki

(2017)). Because the principal is assumed to commit to future mechanisms, the ratchet eﬀect

problem does not arise.

The dynamic mechanism design literature most closely related to this paper studies

dynamic mechanisms in which the principal has limited commitment power. First, Skreta

(2015) studies a two period model in which a seller cannot commit not to re-sell an indivisible

good if the ﬁrst period mechanism fails to allocate the good to one of several buyers. Deb and

Said (2015) study a sequential screening problem that builds oﬀ of Courty and Li (2000).

The seller can commit in the ﬁrst period to the terms of consumption of a good in the

second period, but cannot commit to the selling mechanism oﬀered in the second period.

The principal in both Skreta (2015) and Deb and Said (2015) is concerned with maximizing

eﬀort evolves over time.

6

revenue, while the principal in our paper maximizes welfare. Additionally, consumption only

occurs once in each paper; in either the ﬁrst or second period in Skreta (2015), and at the

end of the second period in Deb and Said (2015). In our paper, the agent completes a task

for the principal in each period. The principal gathers information about the agent from the

outcome of the ﬁrst period project, and uses this information to increase the eﬃciency of

the second period interaction.

Finally, Gerardi and Maestri (2017) study an inﬁnitely repeated principal-agent interac-

tion. The principal is uninformed about the agent’s private cost characteristic, which may be

high or low. The agent produces a good of observable and veriﬁable quality for the principal.

Depending on the principal’s prior beliefs and the discount factor, the principal learns the

agent’s type immediately, over time, or never at all. Because Gerardi and Maestri (2017)

study a pure adverse selection setting, there are no direct comparisons between our paper

and theirs about how the low cost agent’s eﬀort evolves over time.

1.2 Model

Consider a two period interaction between a welfare-maximizing regulator (she) and a regu-

lated ﬁrm (he). In each period, the regulator oﬀers the ﬁrm a contract to complete a project

that has gross-beneﬁt S.

In return for completing the project each period, the regulator

reimburses the ﬁrm for the project’s cost, ct, and pays the ﬁrm an additional transfer, tt(ct).

The additional transfer is a function of the project’s realized cost in each period, and incen-

tivizes cost-reducing eﬀort. The project’s cost in each period depends on the ﬁrm’s intrinsic

cost parameter, β, its unobservable eﬀort, et, and a homoskedastic, zero mean noise term,

7

εt:

ct = β − et + εt,

t = 1, 2.

(1.1)

The random variable εt is assumed to be distributed over the entire real line according to the

distribution function G(ε) with associated density g(ε). The density satisﬁes the monotone

likelihood ratio property. While the full support assumption is analytically convenient, it

raises two issues that bear mention.

The ﬁrst issue is that the low cost ﬁrm’s eﬀort from mimicking the high cost type may be

negative in the second period. This occurs when the ﬁrst period cost realization is suﬃciently

low. A common assumption in static models is that the regulator’s prior belief that the ﬁrm

has low costs is small enough that this situation does not arise. However, in this dynamic-

stochastic setting, the regulator’s second period beliefs are endogenous, and depend on the

ﬁrst period cost realization. Thus, the analysis allows for negative eﬀort levels. Second, the

full support assumption implies that negative cost realizations are possible. While unrealistic,

the possibility of negative costs does not aﬀect the results of this paper.

It is important to note that εt is unobservable both ex-ante and ex-post. Thus, while the

regulator is able to observe total cost ct in each period, she cannot determine the individual

impacts of the ﬁrm’s type, its eﬀort, and noise. This captures the intuition that the ﬁrm

does not have perfect control over the project’s cost. The ﬁrm aﬀects the distribution of

costs by exerting eﬀort, but the project’s cost depend on factors outside of the ﬁrm’s control.

Another interpretation of noise is that of an “accounting error.” Given the complexity of

accounting rules, and constraints on her time, the regulator may not able to perfectly discern

which costs should and shouldn’t be reimbursed after observing the ﬁrm’s income statement

or other supporting documents.

8

The ﬁrm’s type can be either β or ¯β, with 0 < β < ¯β, and remains constant over the

course of the interaction. Throughout, type β is referred to as the “low cost type” or “low

cost ﬁrm,” and type ¯β as the “high cost type” or “high cost ﬁrm.” The ﬁrm’s type is its

private information; the regulator’s prior belief that the ﬁrm is the low cost type is given by

ρ. The ﬁrm experiences a disutility of eﬀort that can be expressed in monetary terms by



ψ(et) =

γ
2

e2
t ,

0,

et > 0,
et ≤ 0,

(1.2)

(1.3)

where γ > 0. Thus, the ﬁrm’s per period utility is given by

Ut = tt(ct) − ψ(et).

Although project costs are stochastic, the ﬁrm’s eﬀort is not; in each period, the ﬁrm chooses

his eﬀort before the realization of εt.

The regulator’s objective in each period is to maximize expected welfare, which is the

sum of taxpayer surplus and the ﬁrm’s utility. In each period, welfare is given by

Wt = S − (1 + λ)(cid:0)ct + tt(ct)(cid:1) + Ut.

(1.4)

Taxpayers enjoy beneﬁt S from the project, compensate the ﬁrm for its costs ct, and pay

out the incentive fee tt(ct). Since the cost reimbursement and incentive transfer are raised

via distortionary taxation, one dollar paid to the ﬁrm costs taxpayers $(1 + λ), where λ > 0

denotes the shadow cost of public funds.

The solution concept used is that of a perfect Bayesian equilibrium. In each period, the

9

regulator designs an incentive scheme to maximize expected welfare. The incentive scheme

depends on the regulator’s beliefs about the ﬁrm’s type. In the ﬁrst period, the regulator

considers the impacts of the ﬁrst period contract on expected second period welfare.

At the beginning of the second period, the regulator observes the ﬁrst period project cost,

and updates her beliefs about the ﬁrm’s type using Bayes’ rule. Contracts are short term;

thus, when designing the second period contract, the regulator cannot commit to ignore any

information she learns about the ﬁrm’s type from observing the realized ﬁrst period project

cost.

The ﬁrm chooses whether to participate or not in each period. If the ﬁrm chooses to

participate, he chooses his eﬀort to maximize his expected utility given the transfer designed

by the regulator. In the ﬁrst period, he considers the impact that his actions have on the

regulator’s second period beliefs, and thus his expected second period payoﬀs.

In the analysis to follow, the regulator’s problem in each period is to maximize expected

welfare by choosing a cost target for each type of ﬁrm. These targets serve two purposes.

First, whatever cost the ﬁrm decides to target determines the ﬁrm’s eﬀort. To see this, recall

that eﬀort is chosen before the realization of εt. Thus, the ﬁrm simply chooses its eﬀort such
that its expected cost, E[ct] = β − et, is equal to its chosen cost target.

Second, for a given type of ﬁrm, the cost target serves as the mean of the distribution

of project costs in each period. Since the incentive transfer is a function of project costs,

the expected transfer in each period depends on the cost target. Thus, at the beginning of

each period the regulator chooses cost targets that, in expectation, form an incentive feasible

menu.

Framing the regulator’s problem as a choice of cost target for each type of ﬁrm is with-

out loss of generality as long as there exists an incentive transfer, based solely on realized

10

costs, that satisﬁes the three following properties in expectation. First, the high cost ﬁrm’s

expected utility from targeting ct must be equal to his outside option of zero. Second, the

low cost ﬁrm’s expected utility from targeting ct must be equal to his expected utility from
targeting ct. Third, the ﬁrm’s expected utility from targeting ct /∈ {ct, ct} is lower than his

expected utility from targeting either ct or ct.

When these three properties are satisﬁed, the high cost ﬁrm’s participation constraint and

the low cost ﬁrm’s incentive constraint are satisﬁed in expectation in each period. Further,

neither ﬁrm has an incentive to target a cost level other than the cost target designed

for him by the regulator. The paper proceeds by assuming that there exists a transfer

based on observed costs, tt(ct), such that the expected transfer, E[tt(ct)], satisﬁes the three

aforementioned properties.

Caillaud, Guesnerie and Rey (1992), Picard (1987) and Melumad and Reichelstein (1989)

study the existence of such reward schedules when the agent’s type space is continuous. When

the agent’s type may only take on two values, there are fewer constraints placed on the reward

schedule. However, the lower envelope of the high and low cost agent’s indiﬀerence curves

is kinked, which implies that it may not be possible to implement the high cost ﬁrm’s exact

cost target. However, one can implement a cost target that is arbitrarily close (see Jeitschko

and Mirman (2002)).

Throughout the paper, the focus is on deriving an equilibrium that is “separating in

actions.” Because cost observations are noisy, and this uncertainty is not resolved ex-post,

the regulator is not able to determine with certainty the ﬁrm’s type by on observing the

cost realization. That is, even when the ﬁrst period contract is designed in a way that the

low cost ﬁrm and high cost ﬁrm target distinct cost levels, the regulator does not have full

information about the ﬁrm’s type in the second period. Thus, the equilibrium is separating

11

in actions when the regulator designs distinct targets for each type of ﬁrm, and each type of

ﬁrm targets the expected cost designed for for him by the regulator. This means in period

t = 1, 2, the low cost ﬁrm targets ct, and the high cost ﬁrm targets ct.

1.3 Second period

Since the model is solved using backward induction, the analysis begins with the second

period. Suppose that the ﬁrst period contract is separating in actions. At the beginning of

the second period, the regulator observes the ﬁrst period cost realization and updates her

beliefs about the ﬁrm’s type using Bayes’ rule. Therefore, her second period belief that the

ﬁrm is the low cost type is given by

ρ2 :=

ρg(c1 − c1)

ρg(c1 − c1) + (1 − ρ)g(c1 − c1)

.

(1.5)

Consider the numerator of (1.5). The regulator’s prior belief that the ﬁrm has low costs

is given by ρ. In the ﬁrst period, the low cost ﬁrm targets c1; when the ﬁrm targets c1,

the ﬁrst period cost realization is c1 = c1 + ε1. Since g(ε1) represents the density of noise
in the ﬁrst period, g(c1 − c1) is the probability density of ﬁrst period costs when the agent
targets c1. Thus, g(c1 − c1) gives the value of the probability density function when the cost

realization is c1 and the agent targets c1.

Similarly, the probability density of costs when the agent targets c1 is given by g(c1− c1).
Since noise has full support on the real line, both g(c1 − c1) and g(c1 − c1) are strictly

positive on the entire real line. Thus, the principal never believes to be fully informed about

the agent’s type in the second period. That is, because of the full support assumption,

12

ρ2 ∈ (0, 1).

With beliefs given in (1.5), the regulator’s problem is to choose expected costs c2 and ¯c2

to maximize expected welfare, subject to incentive and participation constraints (which are

derived below):

(cid:90)

(cid:104)

S − (1 + λ)(cid:0)c2 + t2(c2)(cid:1) + t2(c2) − γ
(cid:90)

(β − c2)2(cid:105)
(cid:104)
S − (1 + λ)(cid:0)c2 + t2(c2)(cid:1) + t2(c2) − γ

2

max
c2, c2

ρ2

R

+ (1 − ρ2)

R

g(c2 − c2)dc2

( ¯β − ¯c2)2(cid:105)

2

g(c2 − ¯c2)dc2. (1.6)

Because the second period game is static, and both the regulator and the ﬁrm are risk

neutral, zero-mean noise has no impact on incentives. Thus, the binding constraints on the

regulator’s problem are the low cost type’s incentive compatibility constraint and the high

cost ﬁrm’s participation constraint.3

First, consider the low cost type’s incentive compatibility constraint. The optimal second

period cost targets make the low cost ﬁrm’s expected utility from targeting c2 equal to his

expected utility from targeting c2. When the low cost ﬁrm targets c2, he chooses his eﬀort
in the second period such that e2 = β − c2, and thus his private cost of eﬀort is equal to

(cid:0)β − c2

(cid:1)2.

γ
2

When the low cost ﬁrm chooses his eﬀort in this manner, it is easy to see that

E[c2] = E(cid:2)β − β + c2 + ε2

(cid:3) = c2.

(1.7)

Therefore, the second period project cost can be written as c2 = c2 + ε2, which implies
that the density of second period costs is given by g(c2 − c2). Therefore, the low cost ﬁrm’s
3The low cost ﬁrm’s incentive constraint depends on whether the low cost type’s eﬀort from mimicking

the high cost type is positive or negative. This issue is addressed shortly.

13

expected second period utility from targeting c2 is given by

(β − c2)2(cid:105)

g(c2 − c2)dc2 = t2 − γ
2

(cid:0)β − c2

(cid:1)2 ,

(1.8)

E [U2 | c2] :=

t2(c2) − γ
2

R

where t2 :=(cid:82)R t2(c2) · g(c2 − c2)dc2.

Similarly, when the low cost type targets ¯c2, his eﬀort is given by ¯e2 − ∆β = β − ¯c2, and
the density of second period costs is given by g(c2 − ¯c2). Thus, his expected utility from

(cid:90)

(cid:104)

(cid:90)

(cid:104)

R

targeting ¯c2 is

(β − ¯c2)2(cid:105)

(cid:0)β − ¯c2

(cid:1)2 ,

(1.9)

E [U2 | ¯c2] :=

t2(c2) − γ
2

g(c2 − ¯c2)dc2 = ¯t2 − γ
2

where ¯t2 :=(cid:82)R t2(c2) · g(c2 − ¯c2)dc2. The low cost ﬁrm’s incentive compatibility constraint

makes him indiﬀerent, in expectation, between targeting c2 and ¯c2:

E [U2 | c2] = E [U2 | ¯c2] =⇒ t2 − γ
2

(cid:0)β − c2

(cid:1)2 = ¯t2 − γ

(cid:0)β − ¯c2

(cid:1)2 .

(1.10)

2

The second period game is designed to extract all expected rent form the high cost type.
When the high cost type targets ¯c2, his cost of eﬀort is ¯e2 = ¯β − ¯c2, and the density of
expected costs is given by g(c2 − ¯c2). Thus, the high cost type’s expected second period rent

is given by

E(cid:2)U 2 | ¯c2

(cid:3) :=

(cid:90)

R

(cid:104)
t2(c2) − γ
2

( ¯β − ¯c2)2(cid:105)

g(c2 − ¯c2)dc2 = ¯t2 − γ
2

(cid:0) ¯β − ¯c2

(cid:1)2 .

(1.11)

14

Therefore, the high cost type’s participation constraint is given by

E(cid:2)U 2 | ¯c2

(cid:3) = 0 =⇒ ¯t2 − γ

2

(cid:0) ¯β − ¯c2

(cid:1)2 = 0.

(1.12)

Simplifying the objective function in (1.6) and using (1.10) and (1.12) to substitute for

the expected transfers leaves the following unconstrained problem:

(cid:104)

S − ρ2

max
c2, ¯c2

(1 + λ)

c2 +

− (1 − ρ2)(1 + λ)

¯c2 +

(cid:16)

(cid:16)

(cid:16) γ

γ
2

(β − c2)2(cid:17)
( ¯β − ¯c2)2(cid:17)

γ
2

,

+ λ

( ¯β − ¯c2)2 − γ
2

2

(β − ¯c2)2(cid:17)(cid:105)

(1.13)

where γ

2 ( ¯β − ¯c2)2 − γ

2 (β − ¯c2)2 is the low cost ﬁrm’s expected information rent.

The ﬁrst order conditions of this problem imply the following equilibrium eﬀorts and cost

targets:

and

e2 = β − c2 =

1
γ

,

¯e2 = ¯β − ¯c2 =

1
γ

− ρ2
1 − ρ2

λ

1 + λ

∆β.

(1.14)

(1.15)

Thus, the low cost type exerts the ﬁrst best eﬀort in the second period, and the high cost

type’s eﬀort is distorted away from the ﬁrst best according to the principal’s second period

beliefs. Notice that the eﬀort levels given in (1.14) and (1.15) correspond to the standard

static game in which beliefs are given by ρ2. This illustrates than in a static setting, additive

noise has no impact on incentives when the regulator and ﬁrm are risk neutral.

One concern in this model is that the low cost ﬁrm’s eﬀort from mimicking the high cost

15

ﬁrm,

¯e2 − ∆β = β − ¯c2 =

− 1 + λ − ρ2
(1 − ρ2)(1 + λ)

1
γ

∆β,

(1.16)

can be less than zero for values of ρ2 close to one. “Negative eﬀort” captures any measures

taken to increase the project’s cost. To understand why the low cost type might have to

increase the project’s cost to mimic the high cost type, recall that the expected cost for the

high cost type is equal to its type minus its cost reducing eﬀort. When the ﬁrst period cost

observation is low, this leads the regulator to believe that she is very likely to be contracting

with the low cost type in the second period. In response, she reduces the eﬀort of the high

cost type in order to extract rent from the low cost type. When this eﬀort is small enough
(i.e. when ρ2 is close to one), ¯c2 = ¯β − ¯e2 > β.

This possibility is usually assumed away in static models. However, as ε has full support

on the real line, it must be considered in this setting. Since g satisﬁes the monotone likeli-

hood ratio property, the principal’s posterior belief that the ﬁrm has low costs is monotone

decreasing in ﬁrst period cost realizations. Therefore, there exists a unique value of ρ2,

deﬁned

ρ0
2 := ρ2(c0

1) =

(1 + λ)(1 − γ∆β)

1 + λ − γ∆β

< 1,

(1.17)

such that for every c1 ≤ c0

1, the low cost type’s eﬀort from mimicking the high cost type is

negative.

Since the ﬁrm cannot experience a dis-utility from negative eﬀort (that is, ψ(et) = 0
when et ≤ 0), the low cost type’s second period incentive compatibility constraint is written

t2 − γ
2

(β − c2)2 = ¯t2.

(1.18)

16

The high cost ﬁrm’s participation constraint remains unchanged. Together, this implies that
the regulator’s unconstrained problem when c1 ≤ c0

1 is given by

(cid:104)

(cid:16)

S − ρ2

max
c2, ¯c2

(1 + λ)

c2 +

− (1 − ρ2)(1 + λ)

¯c2 +

(cid:0) ¯β − ¯c2

(cid:1)2(cid:105)

(β − c2)2(cid:17)
( ¯β − ¯c2)2(cid:17)

γ
2

,

+ λ

γ
2

γ
2

(cid:16)

(1.19)

(cid:0) ¯β − ¯c2

(cid:1)2.

where the low cost ﬁrm’s expected information rent is now given by γ
2

The ﬁrst order condition for this problem with respect to ¯c2 implies the following equi-

librium eﬀort for the high cost type (the low cost type still exerts the ﬁrst best eﬀort):

2 = ¯β − ¯c2 =
¯e0

(1 − ρ2)(1 + λ)

1 + λ − ρ2

.

1
γ

(1.20)

The following proposition summarizes the second period game:

Proposition 1.1. When c1 > c0

1, the regulator’s problem is given by (1.13), while for c1 ≤
c0
1, the regulator’s problem is given by (1.19). The ﬁrst order conditions of (1.13) and (1.19)

with respect to c2 and ¯c2 imply that the low cost ﬁrm’s equilibrium expected rent is given by

U2(ρ2) =

(¯e2 − ∆β)2 =: u2,

(¯e2)2 − γ
2
2)2 =: u0
2,

(¯e0

γ
2
γ
2

if

if

c1 > c0
1,
c1 ≤ c0
1,

(1.21)

where ¯e2 is given in (1.15), ¯e2 − ∆β in (1.16), and ¯e0

2 in (1.20). Similarly, equilibrium

17



expected second period welfare is given by

S − ρ2

S − ρ2

(cid:104)
(cid:104)

W2(ρ2) =

(cid:16)
(cid:16)

(cid:17)
(cid:17)

(1 + λ)

(1 + λ)

β − 1

2γ

β − 1

2γ

+ λu2

+ λu0
2

(cid:105) − (1 − ρ2)(1 + λ)
(cid:16) ¯β − ¯e2 + γ
(cid:105) − (1 − ρ2)(1 + λ)(cid:0) ¯β − ¯e0

2 + γ

2 (¯e2)2(cid:17)
2)2(cid:1) =: w0

2 (¯e0

=: w2,

2,
(1.22)

when c1 is greater than c0

1 and less than c0

1, respectively.

Regardless of the size of c1, the second period game exhibits the classic rent extrac-

tion/eﬃciency trade-oﬀ present in static adverse selection models:



dU 2(ρ2)

dρ2

=

du2
d¯e2
du0
2
d¯e0
2

d¯e2
dρ2
d¯e0
2
dρ2

=

=

−1

λ

1 + λ

(1 − ρ2)2
−λ(1 + λ)2

γ∆β2 < 0,
1 − ρ2

γ

(1 + λ − ρ2)3 < 0,

if

if

c1 > c0
1,
c1 ≤ c0
1.

(1.23)

This is an important consideration for the regulator in the ﬁrst period, since ρ2 is a function

of c1 and ¯c1.

To see how second period beliefs, and thus second period welfare, depend on the ﬁrst

period contract, consider ˜c1 = c1 + x, for some ﬁxed value x. From (1.5), the closer together

are c1 and ¯c1, the closer together are the values of g(˜c1) and ¯g(˜c1). The closer together are

g(˜c1) and ¯g(˜c1), the closer ρ2 is to the prior, ρ; indeed, if c1 = ¯c1, then g(˜c1) = ¯g(˜c1) for all

x, and the posterior is equal to the prior. Conversely, the further apart are c1 and ¯c1, the

smaller is ¯g(˜c1) relative to g(˜c1), and the closer the posterior is to one.

Thus, the distance between ﬁrst period cost targets directly inﬂuences how much the

regulator updates her prior, given a ﬁrst period cost realization. The further apart are the

ﬁrst period cost targets, the more accurate are the regulator’s second period beliefs; the

more accurate are the regulator’s second period beliefs, the closer second period welfare is

18

to the ﬁrst-best. However, this information comes at a cost. Since the low cost ﬁrm’s second

period rent is decreasing in ρ2, spreading the cost targets apart decreases (in expectation)

the low cost ﬁrm’s rent from targeting c1, and increases his rent from targeting ¯c1 in the ﬁrst

period. This increases the low cost type’s ﬁrst period transfer. Thus, the regulator faces a

tradeoﬀ between increasing the expected second period welfare or preserving the low cost

ﬁrm’s expected second period rent.

1.4 First period

The second period beliefs, ρ2, serve as the link between the ﬁrst and second period contracts.

When choosing the ﬁrst period cost targets, the regulator considers not only the impact that

they have on ﬁrst period welfare, but what impact they have on expected second period

welfare as well. The regulator’s ﬁrst period problem is to maximize the expectation of ﬁrst

and (discounted) second period welfare, subject to incentive compatibility and participation

constraints, which are derived below:

max
c1, ¯c1

(cid:90)

S − ρ

R
− (1 − ρ)

(cid:104)
(cid:90)
(1 + λ) (c1 + t1(c1)) + t1(c1) − γ
(cid:104)
2
(1 + λ) (c1 + t1(c1)) + t1(c1) − γ
2

(cid:1)2(cid:105)
(cid:0)β − c1
(cid:0) ¯β − ¯c1

R

(cid:1)2(cid:105)

g(c1 − c1)dc1

g(c1 − ¯c1)

(1.24)

where W2(ρ2) is given in (1.22), and

+ δE[W2(ρ2)],

(cid:90)

R

E[W2(ρ2)] =

W2(ρ2) [ρg(c1 − c1) + (1 − ρ)g(c1 − ¯c1)] dc1.

(1.25)

19

A well known issue in dynamic games is that the ﬁrst period payment to the low cost ﬁrm

may be so large that the high cost type’s incentive compatibility constraint binds (the so-

called “take the money and run” strategy). For now, consider the low cost ﬁrm’s incentive

compatibility constraint and the high cost ﬁrm’s participation constraint.4 The low cost

ﬁrm’s incentive constraint requires that his expected utility from targeting c1 equal his

expected utility from targeting c1. That is,

(cid:104)
(cid:104)

(cid:90)
(cid:90)

R

R

E[U1| c1] :=

=

t1(c1) − γ
2
t1(c1) − γ
2

(β − c1)2 + δU2(ρ2)
(β − ¯c1)2 + δU2(ρ2)

(cid:105)
(cid:105)

gdc1

¯gdc1 =: E[ U2| ¯c2],

(1.26)

where g := g(c1 − c1) and g := g(c1 − c1). The left hand side of (1.26) is the low cost ﬁrm’s
expected utility when he targets c1 in the ﬁrst period. He exerts eﬀort e1 = β − c1, and

receives an expected ﬁrst period transfer and expected second period rent, where expectations

are taken over the real line according to the density g. If the low cost ﬁrm instead chooses to
target ¯c1, he experiences a disutility from eﬀort ¯e1 − ∆β = β − ¯c1, and receives an expected

ﬁrst period transfer and expected second period rent. These expectations are taken according

to the density ¯g.

From the perspective of the high cost ﬁrm, the ﬁrst period game is essentially static

since the second period game extracts all the rent from the high cost type. Therefore, the

high cost ﬁrm’s participation constraint requires that his expected ﬁrst period utility from

targeting c1 be equal to his outside option of zero:

(cid:90)

(cid:12)(cid:12) ¯c1] :=

(cid:104)
t1(c1) − γ
2

R

E[U 1

(cid:0) ¯β − ¯c1

(cid:1)2(cid:105)

¯gdc1 = 0.

(1.27)

4In suﬃciently noisy environments, the high cost ﬁrm’s incentive constraint is slack. See Appendix A.

20

By deﬁning t1 and ¯t1 analogously to t2 and ¯t2, one can simplify (1.26) and (1.27) and

solve for the low cost ﬁrm’s expected ﬁrst period transfer:

t1 =

(β − c1)2 +

γ
2

γ
2

( ¯β − ¯c1)2 − γ
2

(β − ¯c1)2 + δ

R

(cid:90)

U2(ρ2)(¯g − g)dc1.

(1.28)

The ﬁrst three terms on the right hand side of (1.28) comprise the familiar static transfer;

the low cost ﬁrm must be compensated for the cost of its eﬀort, and also for the ability to

“hide behind” the high cost ﬁrm.

In dynamic games, there is an additional component of the low cost ﬁrm’s ﬁrst period

transfer. Because the density of noise, g, satisﬁes the monotone likelihood ratio property,

the distribution of costs induced by targeting ¯c1 ﬁrst order stochastically dominates the

distribution induced by targeting c1. Therefore, the low cost ﬁrm enjoys a higher expected

second period rent when he targets ¯c1 than he does when he targets c1.5 The ﬁrst period

transfer must compensate him for this opportunity cost to induce him to target c1.

In a deterministic setting, unless the the ﬁrm cares little about the future (i.e., the ﬁrm

heavily discount future payoﬀs), this additional component of the low cost ﬁrm’s ﬁrst period

transfer can make it impossible to induce a separating equilibrium. To see this, recall that

in a deterministic setting, the ﬁrm has perfect control over the project’s cost. Suppose

the regulator’s contract speciﬁes that the high and low cost ﬁrms complete the project at

diﬀerent cost levels.

If the ﬁrm accepts such a contract, his actions perfectly reveal his

type to the regulator; information revelation in a deterministic separating equilibrium is an

“all-or-nothing” proposition.

Thus, when the the low cost ﬁrm follows the equilibrium in the ﬁrst period, the regulator

5That is, because g satisﬁes the monotone likelihood ratio property,(cid:82)R U2(ρ2)(¯g − g)dc1 > 0.

21

believes with probability one that she is contracting with the low cost type in the second

period, and he is held to his reservation utility. Further, when the low cost ﬁrm takes out-of-

equilibrium actions in the ﬁrst period and mimics the high cost ﬁrm, at the beginning of the

second period the regulator believes the ﬁrm to be the high cost type. In this case the low

cost ﬁrm enjoys his highest possible second period information rent, U 2(0). To induce him

to target c1, the principal must increase the low cost ﬁrm’s ﬁrst period transfer by δU 2(0).

This rationale changes in a stochastic setting. First, simply by following the equilibrium

and targeting c1 in the ﬁrst period, the low cost ﬁrm enjoys expected second period rent

(cid:90)

R

U2(ρ2)gdc1 > 0.

(1.29)

Second, the low cost ﬁrm’s gains from mimicking the high cost ﬁrm are diminished. Suppose

the low cost ﬁrm deviates and targets ¯c1 in the second period. The corresponding density of

ﬁrst period costs is g, so that the low cost ﬁrm’s expected second period rent from targeting

c1 is

(cid:90)

R

(cid:90)

R

U2(ρ2)¯gdc1 <

U2(0)¯gdc1 = U2(0).

(1.30)

Therefore, the additional component of the low cost ﬁrm’s ﬁrst period transfer is smaller in

a stochastic setting than it is in a deterministic environment.

To proceed with the principal’s ﬁrst period problem, consider the following assumption:

Assumption 1.1. The single crossing property holds in the ﬁrst period. That is,

γ(cid:0) ¯β − c(cid:1) ≥ γ(cid:0)β − c(cid:1) + δ

(cid:90)

=⇒ γ∆β ≥ δ

dU2
dρ2

dρ2
dc1

R

(cid:90)

g(c1 − c)dc1

dU2
dρ2

dρ2
R
dc1
g(c1 − c)dc1.

(1.31)

22

The single crossing assumption guarantees a regular ﬁrst period problem by ensuring that

the high cost type’s marginal cost of decreasing the cost target c is higher than the low cost

type’s marginal cost of decreasing the cost target for every c. From (1.31), this condition is

satisﬁed when

dρ2
dc1

is small, i.e. when the posterior beliefs are not too sensitive to changes

in ﬁrst period cost. Since the magnitude of

dρ2
dc1

depends on the slope of the density, and

the slope of the density goes to zero when the variance is large, this condition is satisﬁed

in suﬃciently noisy environments. The single crossing condition is also more likely to be

satisﬁed when the diﬀerence between the low and high cost ﬁrm’s intrinsic cost levels, ∆β,

is large.

Proposition 1.2. The regulator’s full ﬁrst period problem is given by

(cid:20)

S − ρ

max
c1, ¯c1

(1 + λ)

c1 +

− (1 − ρ)(1 + λ)

¯c1 +

(cid:16)

(cid:18) γ

(β − c1)2(cid:17)
( ¯β − ¯c1)2(cid:17)

γ
2

+ λ

γ
2

(cid:16)

( ¯β − ¯c1)2 − γ
2

(β − ¯c1)2 + δ

2

(cid:19)(cid:21)

U2(ρ2)(¯g − g)dc1

(cid:90)

R

+ δE[W2(ρ2)],

(1.32)

where E[W2(ρ2)] is given in (1.25). The ﬁrst order conditions imply the following ﬁrst period

eﬀorts (and cost targets):

e1 = β − c1 =

1
γ

+

δ

γρ(1 + λ)

d
dc1

(cid:20)

ρλ

(cid:90)

R

U2(ρ2)(¯g − g)dc1 − E[W2]

(cid:21)

,

(1.33)

and

¯e1 = ¯β − ¯c1 =

−

1
γ

ρλ

(1 − ρ)(1 + λ)

∆β

+

δ

γ(1 − ρ)(1 + λ)

d
d¯c1

(cid:90)

R

(cid:20)

ρλ

23

U2(ρ2)(¯g − g)dc1 − E[W2]

(cid:21)

.

(1.34)

If the regulator were able to commit to the ﬁrst and second period cost targets at the

outset of her relationship with the ﬁrm, she would implement the same contract in each

period. In periods one and two, the low cost agent exerts the ﬁrst best level of eﬀort,

ec = e∗ =

1
γ

.

The high cost ﬁrm’s eﬀort distortion remains the same in periods one and two:

¯ec =

−

1
γ

ρλ

(1 − ρ)(1 + λ)

∆β.

(1.35)

(1.36)

Comparing (1.35) to (1.33) and (1.36) to (1.34), one can see that each type of ﬁrm’s eﬀort

is distorted away from the commitment optimum. Whether the low cost ﬁrm exerts more or

less eﬀort than in the commitment optimum depends on how the additional component of

the low cost ﬁrm’s ﬁrst period transfer and expected second period welfare change with the

low cost ﬁrm’s ﬁrst period cost target.

In particular, if

d
dc1

(cid:20)

(cid:90)

U2(ρ2)(¯g − g)dc1 − E[W2]

ρλ

R

(cid:21)

< 0,

(1.37)

the low cost ﬁrm exerts less eﬀort in the ﬁrst period than he does in the second period. To

see this, recall that the second period game is static. In a static game, the low cost ﬁrm

exerts the ﬁrst best eﬀort. The low cost ﬁrm also exerts the ﬁrst best eﬀort in every period

when the principal can commit. Therefore, if the low cost ﬁrm’s ﬁrst period eﬀort, given in

(1.33), is less than the commitment eﬀort given in (1.35), then his ﬁrst period eﬀort is lower

than his eﬀort in the second period.

This case is of particular interest in light of the discussion of the ratchet eﬀect in the

24

introduction. If e1 < e2, then the theoretical predictions of this paper match with anecdotal,

experimental and empirical evidence which shows that high ability agents decrease their eﬀort

at the beginning of their relationship with a principal to maintain information rents in the

future.

1.4.1 Signal dampening

Recall that the low cost ﬁrm’s expected second period rent is higher when he targets c1

that it is when he targets c1. The additional component of the low cost type’s ﬁrst period

transfer,

δ

U2(ρ2)(¯g − g)dc1,

(1.38)

(cid:90)

R

compensates him for this diﬀerence in expected second period rents. Without this additional

component, the principal cannot induce the low cost ﬁrm to target c1. Clearly, the larger is

(1.38), the larger is the low cost ﬁrm’s ﬁrst period transfer, given in (1.28). This subsection

demonstrates that the principal can decrease (1.38), and thus decrease the low cost ﬁrm’s

expected ﬁrst period transfer, by reducing the distance between the ﬁrst period cost targets.

The intuition for this argument is simple. Because the density of noise satisﬁes the

monotone likelihood ratio property, the principal’s belief that the ﬁrm is the low cost type

is monotone decreasing in the ﬁrst period cost realization. That is, the higher is the ﬁrst

period cost, the lower is the principal’s second period belief that the ﬁrm is the low cost

type.

The lower is the principal’s belief that the ﬁrm is the low cost type, the more eﬀort the

high cost ﬁrm exerts in the second period. The more eﬀort that the high cost ﬁrm exerts, the

higher is the low cost ﬁrm’s information rent. Thus, the less the principal’s second period

25

beliefs change depending on which cost level the ﬁrm targets, the lower is the low cost ﬁrm’s

incentive to mimic the high cost ﬁrm. To see this, consider Figure 1.1.

g(c1)

g

c1

¯c1

¯g

c1

Figure 1.1: The probability density of costs depends on the agent’s eﬀort choice

When the ﬁrm targets c1, the density of ﬁrst period costs is given by g in Figure 1.1.

Similarly, when the ﬁrm targets c1, the density of ﬁrst period costs is g. The closer together

are c1 and c1, the closer together are the values of g and g for any given ﬁrst period cost

realization. The closer together are the values of g and g, the closer second period beliefs,

given in (1.5), are to the prior, ρ.

The less the regulator updates her beliefs for any given ﬁrst period cost realization, the

closer is the low cost ﬁrm’s expected second period rent from targeting c1 compared to when

he deviates and targets ¯c1. This decreases the low cost type’s incentives to mimic the high

cost type in the ﬁrst period, which reduces the low cost type’s ﬁrst period transfer. and thus

alleviates the ﬁrst period incentive problem.

The following proposition formalizes this logic by, for the time being, abstracting from

the impacts of the ﬁrst period contract on expected second period welfare. The proof makes

use of the connection between eﬀort and cost targets; an increased cost target implies a

decrease in eﬀort, and vice-versa. The proof formalizes the intuition that the regulator can

decrease the low cost ﬁrm’s ﬁrst period transfer by decreasing the distance between ¯c1 and

26

(cid:21)

(cid:20)

ρλ

(cid:20)

ρλ

(cid:90)

R

(cid:90)

R

c1. To do this, the proof shows that the ﬁrst period transfer is decreasing in c1 and increasing

in ¯c1. This equilibrium transfer eﬀect decreases (increases) the low cost (high cost) type’s

equilibrium ﬁrst period eﬀort.

Proposition 1.3. The eﬀect of the dynamic portion of the low cost ﬁrm’s ﬁrst period transfer

is to decrease (increase) the low cost (high cost) ﬁrm’s ﬁrst period eﬀort. That is,

and

d
dc1

d
d¯c1

U2(ρ2)(¯g − g)dc1

U2(ρ2)(¯g − g)dc1

< 0,

(cid:21)

> 0.

(1.39)

(1.40)

The proof of Proposition 1.3, which is found in Appendix A, establishes that even though

the regulator cannot commit to ignore information she learns about the ﬁrm when designing

the second period contract, in a stochastic environment the regulator can commit to learn less

via her choice of ﬁrst period cost targets. Doing so preserves the low cost ﬁrm’s equilibrium

expected second period rent and decreases his gains from deviation, which in turn decreases

his ﬁrst period transfer, alleviating the dynamic incentive problem.

Tying cost targets to eﬀorts also allows a discussion of how the ratchet eﬀect behaves in a

stochastic setting versus a deterministic one. In a deterministic separating equilibrium, the

high cost type has his eﬀort decreased over time, while the low cost type always exerts the

ﬁrst best eﬀort.6 As Proposition 1.3 shows, and as the above intuition argues, in a stochastic

setting the regulator distorts the eﬀorts of both types of ﬁrm in the ﬁrst period, as opposed

to just the high cost ﬁrm. In particular, to decrease the low cost ﬁrm’s ﬁrst period transfer,

the principal decreases the low cost type’s eﬀort, and increases the high cost type’s eﬀort,

6See Laﬀont and Tirole (1993).

27

relative to the commitment optimum.

1.4.2 Experimentation

Proposition 1.3 establishes that the regulator has an incentive to restrict how much informa-

tion she gathers about the ﬁrm. However, an opposing incentive exist as well. The more the

regulator learns about the ﬁrm’s type by observing the ﬁrst period project cost, the better

she can tailor the second period contract to the ﬁrm’s type. The stronger is the regulator’s

belief that the ﬁrm is the low cost type (i.e., the closer ρ2 is to one), the lower is the high cost

agent’s eﬀort. This extracts rent from the low cost ﬁrm in the second period. The stronger

is the regulator’s belief that the ﬁrm is the high cost type (i.e., the closer ρ2 is to zero), the

higher is the high cost type’s cost-reducing eﬀort.

Thus, the better is the principal’s information in the second period, the more accurate

is the high cost ﬁrm’s eﬀort distortion in the second period. This improves expected second

period welfare by either inducing more cost-reducing eﬀort from the high cost ﬁrm or ex-

tracting more rent from the low cost ﬁrm. The following lemma establishes that information

about the ﬁrm’s type is valuable to the regulator in the second period.7

Lemma 1.1. Information is valuable. That is, expected second period welfare is convex in

second period beliefs:

d2W2(ρ2)

dρ2
2

> 0.

(1.41)

The proof of Lemma 1.1 is a straightforward envelope theorem argument, and is relegated

to Appendix A. Given that information is valuable, one can show that the regulator increases

expected second period welfare, E[W2(ρ2)], by increasing the distance between ﬁrst period

7Information is valuable in the sense of Blackwell (1951).

28

cost targets.

To see the intuition for this result, return attention to Figure 1.1. As the distance between

ﬁrst period cost targets grows, so does the diﬀerence between the value of g and g for any

given ﬁrst period cost realization. The further apart are the values of g and g, the more the

regulator updates her prior beliefs for any given ﬁrst period cost realization.

Thus, the information asymmetry between the regulator and the ﬁrm in the second period

diminishes with the distance between ﬁrst period cost targets. Since welfare distortions in the

second period arise because of asymmetric information, an increase in the distance between

c1 and ¯c1 increases expected second period welfare.

This incentive to manipulate ﬁrst period cost targets to increase how much the principal

learns about the agent’s type can be interpreted in terms of equilibrium ﬁrst period eﬀorts.

As the following proposition shows, the principal increases expected second period welfare

by increasing the low cost ﬁrm’s eﬀort, and decreasing the high cost ﬁrm’s eﬀort, relative to

the commitment optimum.

Proposition 1.4. The eﬀect of expected second period welfare is to increase (decrease) the

low cost (high cost) ﬁrm’s ﬁrst period eﬀort. That is,

and

dE[W2(ρ2)]

dc1

< 0,

dE[W2(ρ2)]

d¯c1

> 0

(1.42)

(1.43)

The proof of Proposition 1.4 (found in Appendix A) establishes that the principal in-

creases expected second welfare by increasing the distance between the ﬁrst period cost

29

targets. Since the game ends after the second period interaction, the only welfare distortions

in the second period arise because of the presence of asymmetric information (i.e. there are

no dynamic considerations as there are in the ﬁrst period). Thus, any measures the regulator

can take to decrease the information asymmetry in the ﬁrst period increase expected second

period welfare.

1.5 Equilibrium ratchet eﬀect

The analysis has shown that two opposing incentives determine the optimal ﬁrst period

contract. To decrease the low cost ﬁrm’s ﬁrst period transfer, the regulator must decrease

the distance between the ﬁrst period cost targets, and restrict how much she learns about

the ﬁrm’s type. To increase expected second period welfare, the regulator must increase the

distance between ﬁrst period cost targets, and increase how much she learns about the ﬁrm’s

type.

To determine the combined eﬀect of these competing incentives on the ﬁrst period cost

targets, consider the following re-formulation of the regulator’s ﬁrst period problem:

(cid:104)

S − ρ

max
c1, ¯c1

(1 + λ)

c1 +

− (1 − ρ)(1 + λ)

(cid:16)

(cid:16)

γ
2

(β − c1)2(cid:17)
( ¯β − ¯c1)2(cid:17)

γ
2

+ λ

(cid:16) γ
(cid:20)
( ¯β − ¯c1)2 − γ
2

2

(β − ¯c1)2(cid:17)(cid:105)
(cid:18)

ρwF B + (1 − ρ)

wF B − 1 + λ
2γ

(cid:19)(cid:21)

(cid:90) c0
(cid:90) ∞

1
−∞

c0
1

+ δ

+ δ

+ δ

¯c1 +

(1 − ρ)(1 + λ)¯e0

(cid:110)
2 − (1 + λ − ρ)
(cid:110)
(1 − ρ)(1 + λ)¯e2 − (1 + λ − ρ)
γ
2

γ
2

2)2(cid:111)

(¯e0

¯gdc1

¯e2
2 + ρλ

γ
2

(cid:16)

(cid:17)

(¯e2 − ∆β)2(cid:111)
(cid:16) ¯β − 1
(cid:17)

2γ

Note that wF B = S − (1 + λ)

β − 1

2γ

and wF B = S − (1 + λ)

are the ﬁrst best

¯gdc1.

(1.44)

30

welfare for the low and high cost ﬁrm, respectively.

In (1.44), the expected transfers have already been substituted using the low cost ﬁrm’s

incentive constraint and the high cost ﬁrm’s participation constraint. The second period

welfare distortions (how much rent to leave the low cost ﬁrm and how much eﬀort to induce

in the high cost ﬁrm) are captured by the two integrals. Recall that the high cost ﬁrm’s

second period eﬀort determines how much rent is left to the low cost ﬁrm. Now, deﬁne

A := (1 − ρ)(1 + λ)¯e0

2 − (1 + λ − ρ)

γ
2

(¯e0

2)2,

and

B := (1 − ρ)(1 + λ)¯e2 − (1 + λ − ρ)

γ
2

¯e2
2 + ρλ

(¯e2 − ∆β)2.

γ
2

(1.45)

(1.46)

The ﬁrst order conditions of this problem imply the following eﬀort levels for the low and

high cost ﬁrm:

e1 = β − c1 =

−

1
γ

δ

ρ(1 + λ)γ

d
dc1

(cid:34)(cid:90) c0

1

−∞ A ¯g dc1 +

(cid:35)

B ¯g dc1

,

(1.47)

(cid:90) ∞

c0
1

¯e1 = ¯β − ¯c1 =

1
γ
−

−

ρλ

(1 − ρ)(1 + λ)

∆β

δ

(1 − ρ)(1 + λ)γ

d
d¯c1

(cid:34)(cid:90) c0

1

−∞ A ¯g dc1 +

(cid:35)

B ¯g dc1

.

(1.48)

(cid:90) ∞

c0
1

Again, the equilibrium eﬀorts in (1.47) and (1.48) are distorted relative to the commit-

ment optimum targets in (1.35) and (1.36). The overall eﬀect of the ﬁrst period contract is

31

to restrict how much the regulator learns about the ﬁrm’s type if ¯ec < ¯e1 < e1 < ec, and to

increase learning if ¯e1 < ¯ec < ec < e1.

When the distribution of noise is uniform, the overall eﬀect of the ﬁrst period contract is

to decrease the distance between performance targets, relative to the commitment optimum,

and restrict learning. This implies that the high-ability agent has its eﬀort increased over the

course of his interaction with the principal.8 However, this result depends on the distribution

of noise being uniform. Here, this result is extended to show that when the distribution of

noise is general, the net eﬀect of the two competing incentives is to restrict learning; that is,

the low cost ﬁrm has his eﬀort increased over the course of his interaction with the regulator.

This result that an agent with favorable private information increases his eﬀort over

time is appealing because it ﬁts with anecdotal, experimental, and empirical evidence of

the ratchet eﬀect. Anecdotal evidence of piece-rate factory workers documented that skilled

workers learned to restrict their output in order to avoid either an increase in their output

quotas or a decrease in their piece rates.9 In experimental settings that study two-period

principal agent interactions, high ability workers restrict their output (reduce their eﬀort)

in the ﬁrst period to maintain a second period information rent.10 Empirical studies of the

ratchet eﬀect show that teachers reduce their eﬀort on improving student’s standardized test

scores when their compensation in the future depends on their student’s scores today.11

With this discussion on the relevance of the ratchet eﬀect in mind, consider the following

proposition:

8For the uniform noise case, see Jeitschko et al. (2002) and for the general noise case, see Jeitschko and

Mirman (2002).

9See Matthewson (1931), Clawson (1980), Montgomery (1979) and Roy (1952).
10See Charness et al. (2011) and Cardella and Depew (2018).
11See Macartney (2016).

32

Proposition 1.5. The Ratchet Eﬀect: If the low cost ﬁrm’s second period eﬀort from
mimicking the high cost ﬁrm is positive for all ﬁrst period cost realizations c1 ≥ c1, then the

low cost ﬁrm has his eﬀort increased over the course of the relationship with the regulator.

That is,

d
dc1

d
d¯c1

(cid:34)(cid:90) c0
(cid:34)(cid:90) c0

1

1

(cid:35)
(cid:35)

(cid:90) ∞
(cid:90) ∞

c0
1

c0
1

−∞ A ¯g dc1 +

B ¯g dc1

> 0,

−∞ A ¯g dc1 +

B ¯g dc1

< 0.

(1.49)

(1.50)

The proof of Proposition 1.5 is given in Appendix A. The important implication of

Proposition 1.5 is that the low cost ﬁrm’s ﬁrst period eﬀort, given in (1.47), is less than his

eﬀort when the regulator can commit, (1.35). Since the low cost ﬁrm exerts the ﬁrst best

eﬀort in the ﬁrst period when the principal can commit, and he exerts the ﬁrst best eﬀort

in the second period regardless of the principal’s commitment powers, this implies that the

low cost ﬁrm’s eﬀort increases over time.

Since the low cost ﬁrm’s ﬁrst period eﬀort is less than in the commitment optimum and

the high cost ﬁrm’s eﬀort is greater than in the commitment optimum, the ﬁrst period cost

targets are closer together than the commitment optimum targets. Therefore, the optimal

ﬁrst period contract favors reducing the ﬁrst period transfer to the low cost ﬁrm at the

expense of having worse information about the ﬁrm’s type in the second period.

Proposition 1.5 requires that the low cost ﬁrm’s eﬀort from mimicking the high cost ﬁrm

in the second period be positive for all ﬁrst period cost realizations greater than the low

cost ﬁrm’s ﬁrst period cost target. Recall from the discussion of the second period game

that there exists a unique ﬁrst period cost realization, c0

1, such that for all c1 ≤ c0

1, the low

cost ﬁrm’s eﬀort from mimicking the high cost ﬁrm in the second period is negative, and for

33

g(c1)

c1

ˆc1

g

¯c1

¯g

c1

Figure 1.2: If c0

1 lies in the shaded region, the low cost ﬁrm has his eﬀort increased over time.

all c1 > c0

1 the low cost ﬁrm exerts positive eﬀort to mimic the high cost ﬁrm. Therefore,

Proposition 1.5 requires that c0

1 be less than or equal to the low cost ﬁrm’s ﬁrst period cost

target.

Figure 1.2 illustrates the restriction that Proposition 1.5 places on c0

1, which we consider

to be natural. Suppose that c0

1 > c1. This implies that for some cost realizations greater

than the low cost ﬁrm’s cost target, ρ2 is close enough to one that the high cost ﬁrm’s second

period eﬀort is close to zero. When the high cost ﬁrm’s eﬀort is close to zero, the low cost

ﬁrm has to increase costs above its intrinsic cost level, β, to mimic the high cost ﬁrm.

Under the conditions outlined in Proposition 1.5, the value of information is decreased in

a repeated relationship; not only is the regulator content to have imperfect information in the

second period, but she chooses to learn less than she could by implementing the commitment

optimum. This is because the beneﬁt of better information in the second period does not

outweigh the concomitant increase in the low cost type’s expected ﬁrst period transfer.

34

1.6 Conclusion

In this two-period model of regulation, the regulator and the ﬁrm contract over the comple-

tion of a socially valuable project. The ﬁrm has private information about its intrinsic cost

level, which can be high or low, and has imperfect control over the project’s ﬁnal cost (costs

are stochastic). In this setting, the regulator determines how much information she gathers

about the ﬁrm’s type via her choice of ﬁrst period cost targets.

The regulator can gather more information about the ﬁrm by increasing the distance

between ﬁrst period cost targets. The better the regulator’s information is about the ﬁrm’s

type in the second period, the higher is expected second period welfare. Conversely, the

regulator gathers less information about the ﬁrm by decreasing the distance between ﬁrst

period cost targets. The less the regulator learns about the ﬁrm’s type, the higher is the

low cost ﬁrm’s equilibrium expected second period rent, and the lower is its beneﬁt from

mimicking the high cost ﬁrm. Thus, by decreasing the distance between ﬁrst period cost

targets, the regulator decreases the low cost ﬁrm’s ﬁrst period transfer.

Given a natural restriction on the regulator’s second period beliefs, the net eﬀect of the

ﬁrst period contract is to decrease the distance between the ﬁrst period cost targets. Thus,

the regulator’s desire to reduce the ﬁrst period transfer is stronger than her desire to improve

expected second period welfare.

This implies that the low cost type exerts less than the ﬁrst-best eﬀort in the ﬁrst pe-

riod, and has his eﬀort ratcheted up over the course of his interaction with the regulator.

Anecdotal, experimental and empirical evidence of the ratchet eﬀect suggests that agents

with favorable private information preserve their future information rents by taking actions

to keep this information private. Thus, the prediction that the low cost ﬁrm increases his

35

eﬀort over time aligns closely with observed repeated principal-agent interactions.

36

Chapter 2

Repeated Short-Term Contracting

with Correlated Types and Noisy

Observable Outcomes

2.1

Introduction

In many principal-agent interactions, the agent performs the same task for the principal

over time. A car salesman sells cars year after year and a science teacher covers the same

material, at the same grade level, year after year. In procurement, a ﬁrm provides the same

good or service to a government agency over time.

In these various settings, the agent’s performance on a given task is positively impacted

by both his inherent skill and the amount of eﬀort he exerts. No matter how talented

and diligent the agent is, however, factors outside his control can impact his performance.

Suppose, for example, that the science teacher is judged on his student’s standardized test

scores. Clearly, he cannot control how much sleep his students get the night before the test

or whether some students are sick on the day of the exam. Therefore, the student’s test

scores are a noisy indicator of the teacher’s eﬀort and ability.

Contracts between principal and an agent in which compensation is based (at least in

37

part) on a noisy signal of performance are commonplace. In some states (see [31]), teachers

receive a bonus if the average of their student’s standardized test scores exceed some pre-

speciﬁed benchmark. Similarly, a car salesman may receive a bonus if his monthly sales

exceed a quota. Often, procurement contracts call for cost sharing between the ﬁrm and

government agency for cost overruns.

Jeitschko et al. (2002) and Jeitschko and Withers (2018) study two-period principal-agent

interactions in which the agent’s reward depends solely on a noisy, observable outcome. In

both papers, the principal is unable to commit to long term contracts. Therefore, the

principal updates her beliefs about the agent’s type after observing the noisy outcome, and

uses this information when designing the second period contract. The key insight from these

papers is that the principal’s ﬁrst period contract choice impacts how much information

she gathers about the agent’s private characteristic. The principal can design the ﬁrst

period contract to increase how much she learns about the agent’s private information, which

increases her expected second period payoﬀ, or she can design the ﬁrst period contract to

reduce how much she learns about the agent’s private information, which reduces the good

agent’s ﬁrst period transfer.

Jeitschko and Withers (2018) shows that the optimal ﬁrst period contract favors reducing

the good agent’s ﬁrst period transfer at the expense of reducing the principal’s expected

second period payoﬀ, regardless of the distribution of noise.1 One key feature of these

models, however, is that the agent’s private information is ﬁxed over time; if the agent has

high ability at the beginning of the relationship, he is guaranteed to have high ability at the

end of the relationship as well.

1It is only assumed that the density satisﬁes the monotone likelihood ratio property. This extends the

results of Jeitschko et al. (2002) and Jeitschko and Mirman (2002).

38

In real-world settings, however, it may not be realistic to think about the agent’s type

as ﬁxed. Consumers’ preferences change over time, a taxpayer’s ability to earn income

changes over time, and ﬁrms that have industry leading production technology today can be

surpassed in the future if a rival makes an innovation.

This paper considers a two period model in which an agent produces output for a prin-

cipal in each period. The agent’s inherent productivity (type) is positively correlated across

periods. The principal observes a noisy signal of the agent’s performance at the end of the

ﬁrst period. From this signal, the principal learns something about what the agent’s type

was in the ﬁrst period, and she uses this information when designing the second period con-

tract. As in Jeitschko et al. (2002), Jeitschko and Mirman (2002) and Jeitschko and Withers

(2018), the principal has competing incentives to increase her expected second period payoﬀ

and reduce the good agent’s ﬁrst period transfer. Unlike these papers, however, a third

incentive is introduced: with some positive probability, the agent with low ability in the

ﬁrst period will have high ability in the second period. Thus, the agent with low ability in

the ﬁrst period earns earns an expected second period rent. Therefore, the principal must

consider the impact of the ﬁrst period contract on the low productivity agent’s ﬁrst period

transfer.

The following results hold regardless of the degree of positive correlation. First, the

principal reduces the high productivity agent’s ﬁrst period payment by designing the ﬁrst

period contract to restrict how much she learns about the agent’s ﬁrst period type. Second,

the principal increases her expected second period payoﬀ, and decreases the low productivity

agent’s ﬁrst period transfer, by designing the ﬁrst period contract to increase how much she

learns about the agent’s ﬁrst period type. Third, the principal reduces the total expected ﬁrst

period transfer (the prior-weighted sum of the high and low productivity agent’s ﬁrst period

39

transfers) by designing the ﬁrst period contract to restrict how much she learns about the

agent’s private information. Lastly, suﬃcient conditions are given for the optimal ﬁrst period

contract to favor the reduction of the total expected ﬁrst period transfer at the expense of

having worse information in the second period.

In addition to the literature on principal-agent contracting when the contractible outcome

is stochastic, this paper is related to three strands of dynamic principal agent literature.

First, this paper is closely related to theoretcial studies of the ratchet eﬀect. The ratchet

eﬀect arises in dynamic pricipal agent interactions in which the principal cannot commit to

future incentive schemes. The intuition behind the ratchet eﬀect is that the agent can avoid

more demanding incentives in the future by reducing his eﬀort in the present. In Weitzman

(1980), the ratchet eﬀect arises because the agent’s present-day performance target depends

explicitly on his past output history. In Freixas, Guesnerie and Tirole (1985), Laﬀont and

Tirole (1987) and Laﬀont and Tirole (1988), asymmetric information drives the ratchet

eﬀect. As in the current paper, the agent’s performance on a project is a function of his

inherent ability and his performance-enhancing eﬀort. However, these papers assume that

the project’s outcome is perfectly determined by the agent’s choice of eﬀort. For this reason,

the dynamic predictions of these papers diﬀer markedly from the predictions of the dynamic-

stochastic literature (see Jeitschko and Withers (2018) for further discussion).

Second, this paper is also related to dynamic principal-agent models in which the agent’s

type is stochastic. One of the ﬁrst such papers is Baron and Besanko (1984), which studies

a multi-period relationship between a regulator and a ﬁrm, in which the ﬁrm’s private cost

characteristics may change over time. Laﬀont and Tirole (1996) examine market based and

regulatory solutions for investing in pollution reducing technologies when a ﬁrm’s “valuation

for polluting” changes over time. Battaglini (2005) studies an inﬁnite-horizon pricing prob-

40

lem in which the consumer’s preferences evolve according to a Markov process. Battaglini

(2007) considers the optimal renegotiation-proof contract in a two period model of procure-

ment in which the agent’s type is positively correlated over time. Battaglini and Coate

(2008) study optimal income taxation in which an individual’s income generating abilities

may change over time.

Lastly, this paper is related to a growing dynamic mechanism design literature. Recently,

Athey and Segal (2013) and Pavan et al. (2014) study eﬃcient and revenue maximizing

dynamic mechanisms, respectively, when the agent’s private information is allowed to change

over time. An important diﬀerence between these papers and the current paper is that the

principal has the power to commit to future mechanisms (for a survey on dynamic mechanism

design when the principal can commit, see Bergemann and Valimaki (2017)). Skreta (2015)

and Gerardi and Maestri (2017) study dynamic mechanisms in which the principal has limited

commitment powers, but the agent’s private information is ﬁxed.

Most closely related to the current paper is Deb and Said (2015). The authors study

a monopolist that faces two cohorts of buyers; consumption occurs only at the end of the

second period, and the principal cannot diﬀerentiate between second cohort buyers and ﬁrst

cohort buyers who did not agree to a contract in the ﬁrst period. While the principal can

commit to a contract in the ﬁrst period that speciﬁes the terms of consumption in period

two, she cannot commit in the ﬁrst period to the contract that will be oﬀered in the second

period. The setting is dynamic in the sense that the preferences of buyers in the ﬁrst cohort

may change between the ﬁrst and second periods.2 The principal ﬁnds it optimal to induce

some subset of ﬁrst period buyers to delay their purchase until the second period.

While the principal has limited commitment powers and the agent’s private information

2Deb and Said (2015) build on the work of Courty and Li (2000).

41

may change over time, the nature of the principal’s problem in Deb and Said (2015) is

quite diﬀerent from the principal’s problem in the current paper. In the current paper, the

principal interacts with the same agent in each period, and the agent produces the same good

for the principal in each period. As in models of the ratchet eﬀect, the principal’s concern is

to determine how much information she gathers about the agent’s private characteristic via

her ﬁrst period contract choice.

2.2 Model

The agent produces output for the principal in time periods one and two. The agent’s output

in period t is given by

yt = θtet + εt,

t = 1, 2,

(2.1)

and depends on the agent’s ability, θt, his eﬀort, et, and a zero mean noise term, εt. Eﬀort is
positive (et ∈ R+), so the agent improves his expected output in each period by increasing
his eﬀort. The agent’s productivity can be low or high (θt ∈ {θ, ¯θ} with 0 < θ < θ), where

type θ is the high productivity or high ability type, and type θ is the low productivity or
low ability type. The noise term εt is assumed to be distributed uniformly on [−η, η]. Prior

to her interaction with the agent, the principal believes that the ﬁrm has high productivity
in the ﬁrst period with probability ρ ∈ (0, 1).

The agent’s type is positively correlated over time. Thus, with probability α ∈ [1/2, 1]

the agent’s type in the second period is the same as his type in the ﬁrst period, and with
probability 1− α, the agent’s type switches between the ﬁrst and second periods. Therefore,

P (θ2 = θ(cid:12)(cid:12) θ1 = θ) = P (θ2 = θ| θ1 = θ) = α, and P (θ2 = θ| θ1 = θ) = P (θ2 = θ(cid:12)(cid:12) θ1 = θ) =

42

1 − α.3 The special case of α = 1 captures the case in which the agent’s type is ﬁxed, while

α = 1/2 capture the cases in which the agent’s ﬁrst and second period types are uncorrelated.

The agent’s utility in each period, ut, is the diﬀerence between the transfer he is paid by

the principal, rt(yt), and his private monetary cost of eﬀort, e2

t (i.e., ut = rt(yt)− e2

t ). Thus,

the agent’s expected utility is

E[ut] = E[rt(yt) − e2
t ].

(2.2)

Notice that the transfer paid to the agent, rt(yt), is a function only of observed output.

Speciﬁcally, this transfer cannot be based on a message from the agent to the principal.

This assumption is maintained in order to focus on the impact that imperfect observability

has on dynamic incentive problems.4

The principal’s payoﬀ in each period, vt = yt − rt, is simply the diﬀerence between the

monetary value the principal places on the agent’s output and the transfer paid to the agent.

Thus, her expected payoﬀ in each period is

E[vt] = E[yt − rt(yt)].

(2.3)

The timing of the game is as follows. First, the agent learns his type, θ1. The principal

proposes a payment function, r1(y1), that maps from observed output to rewards. If the agent

rejects the contract, he receives his outside option of zero. If the agent accepts the contract,

he chooses his eﬀort e1. After his eﬀort has been chosen, ε1 is realized.5 The realization of

ε1 determines y1, which in turn determines the agent’s reward and the principal and agent’s

3This approach to modeling transition probabilities is borrowed from Battaglini (2007).
4For a discussion on when it is optimal to base contracts on an additional message from the agent to the

principal, see Melumad and Reichelstein (1989).

5In each period, eﬀort is chosen before the realization of εt; thus, eﬀort is deterministic.

43

ﬁrst period payoﬀs.

At the beginning of the second period, the agent observes his second period type, θ2. The

principal observes the ﬁrst period output realization, y1, and updates her beliefs about the

agent’s ﬁrst period type using Bayes’ rule. She uses her updated beliefs about the agent’s

ﬁrst period type, along with the transition probability α, to form her second period beliefs:

ρ2 := P (θ2 = θ) = α · P (θ1 = θ(cid:12)(cid:12) Y1 = y1) + (1 − α) · P (θ1 = θ| Y1 = y1).

(2.4)

Again, the principal oﬀers a reward schedule, r2(y2), that maps from second period output

realizations to rewards, and the agent accepts or rejects this contract. If the agent accepts

the contract, he chooses his eﬀort, and then ε2 is realized. The principal observes the second

period output realization and the principal and agent obtain their second period payoﬀs. At

the end of the second period, the relationship ends. Note that in the following analysis, all

proofs are relegated to Appendix B.

2.3 Second period

Suppose the ﬁrst period equilibrium is such that each type of agent chooses its eﬀort to reach

a distinct expected output; that is, suppose that the agent with high productivity chooses

his eﬀort in the ﬁrst period such that E[y1] = y1, while the agent with low productivity

in the ﬁrst period chooses his eﬀort such that E[y1] = y1, and y1 > y1. Thus, if the

agent has high productivity in the ﬁrst period, the set of equilibrium output realizations is
y1 ∈ [y1 − η, y1 + η], while if the agent has low productivity in the ﬁrst period, the set of
equilibrium output realizations is y1 ∈ [y1 − η, y1 + η].

44

ρl = 1 − α

ρm = ρα + (1 − ρ)(1 − α)

ρh = α

y1 − η

¯y1 − η

y1 + η

¯y1 + η

y1

Figure 2.1: Second period beliefs

If the agent targets y1 in the ﬁrst period, the smallest possible ﬁrst period output realiza-
tion is y1− η. Therefore, when the principal observes output realizations y1 < y1− η, her be-

lief that the ﬁrm had low productivity in the ﬁrst period is equal to one. That is, following an
output realization y1 < y1 − η, the principal’s beliefs about the agent’s ﬁrst period type, up-

dated using Bayes’ rule, are as follows: P (θ1 = θ| Y1 = y1) = 1 and P (θ1 = θ(cid:12)(cid:12) Y1 = y1) = 0.

Given these beliefs about the agent’s ﬁrst period type, the principal’s second period belief

that the agent has high productivity is given by

ρ2 = α · 0 + 1 · (1 − α) = 1 − α =: ρl.

(2.5)

Similarly, if the agent targets y1 in the ﬁrst period, the largest possible output realization

is y1 + η. Therefore, when the principal observes output realizations y1 > y1 + η, her belief,

updated using Bayes’ rule, that the agent had high productivity in the ﬁrst period is equal

to one. Given such beliefs about the agent’s ﬁrst period type, the principal’s second period

belief that the agent has high productivity is given by

ρ2 = α · 1 + (1 − α) · 0 = α =: ρh.

(2.6)

When the ﬁrst period output realization is greater than ¯y1 − η, but less than y1 + η,

the principal is unsure of the agent’s ﬁrst period type. Such output could have been the

result of equilibrium behavior by either type of agent in the ﬁrst period. Because noise is

45

distributed uniformly on [−η, η], the value of the probability density function is equal to 1

2η

for every potential output realization. Therefore, when the principal observes ﬁrst period

output realizations y1 ∈ [¯y1− η, y1 + η], she updates her beliefs about the agent’s ﬁrst period

type using Bayes’ rule as follows:

P (θ1 = θ(cid:12)(cid:12) Y1 = y1) =

ρ · 1

2η

ρ · 1

2η + (1 − ρ) · 1

2η

= ρ.

(2.7)

Similarly, P (θ1 = θ| Y1 = y1) = 1 − ρ. Thus, for intermediate output realizations, the

principal’s second period belief that the ﬁrm is the high productivity type is given by

ρ2 = ρα + (1 − ρ)(1 − α) =: ρm.

(2.8)

In summary, the principal’s second period beliefs that the ﬁrm is the high productivity type

are given by:



ρ2 =

1 − α =: ρl,
ρα + (1 − ρ)(1 − α) =: ρm,

α =: ρh,

if

if

if

y1 < ¯y1 − η,
y1 ∈ [¯y1 − η, y1 + η],
y1 > y1 + η.

(2.9)

Consider the second period game in which the principal has generic belief ρ2 that the

agent is the high productivity type in the second period. Given these beliefs, the principal’s

problem is to design a reward schedule that maximizes her expected second period payoﬀ,

subject to a set of participation and incentive constraints. This reward schedule, r2(y2), is

based solely on observed second period output.

Rather than focus on deriving this reward schedule, however, the ﬁrst and second period

46

analysis focuses on the principal’s choice of proﬁt maximizing output targets.

In period

t = 1, 2, the high productivity agent’s output target is given by yt, and the low productivity

agent’s output target by yt. As long as there exists a reward schedule that satisﬁes the three

following properties in each period, re-framing the principal’s problem in such a manner is

without loss of generality.

First, the reward schedule must make the high productivity type’s expected utility from

targeting yt equal to his expected utility from targeting yt. Second, it must make the low

productivity agent’s expected utility from targeting yt equal to his outside option of zero.

Lastly, the reward schedule must be such that the high productivity agent’s expected utility

from targeting yt /∈ {yt, yt} is strictly lower than his expected utility from targeting yt or
yt, and the low productivity ﬁrm’s expected utility from targeting yt is greater than his

expected utility from targeting any other period-t output level.

As Jeitschko et al. (2002) show, one can explicitly derive such a reward schedule when

noise follows a uniform distribution. The reward schedule resembles a base pay/bonus in-

centive scheme. The low productivity agent receives a transfer equal to his cost of eﬀort

for all output realizations in [yt − η, yt + η]; this ensures that his participation constraint

is satisﬁed in expectation. The high productivity agent receives a bonus if his output ex-

ceeds some cutoﬀ; the cutoﬀ is chosen to satisfy, in expectation, the high productivity types

incentive constraint. Since noise is uniform (i.e., has bounded support), the principal can

prevent the low productivity agent from shirking by severely punishing output realizations

less than yt − η. The high productivity agent does not wish to reduce his eﬀort, as his

expected transfer decreases faster than his disutility of eﬀort. It is shown in the Appendix

that such a reward schedule exists when the agent’s type is positively correlated.

Given the existence of a reward schedule that implements output targets yt and yt in

47

each period, consider the high productivity agent’s second period incentive constraint. The

agent’s expected output is simply the product of his type and his eﬀort. Thus, the high
productivity agent’s expected second period output is E[y2] = θ · e2.6 Therefore, to target

output level y2, the high productivity agent chooses his eﬀort to equal y2/θ. When the high

productivity agent chooses his eﬀort in this manner, the expected transfer is

r2(y2) · 1
2η

dy2 =: r2,

(2.10)

(cid:90) y2+η

y2−η

(cid:90) y2+η

y2−η

where r2(y2) is the second period reward schedule. Thus, the high productivity agent’s ex-
pected utility is simply r2 − (y2/θ)2.

When the high productivity agent chooses his eﬀort so that e2 = y2/θ, he targets y2.

When he targets y2, his expected transfer is

r2(y2) · 1
2η

dy2 =: r2,

(2.11)

and his expected utility is r2 − (y2/θ)2. Through a similar line of reasoning, the low pro-
ductivity ﬁrm’s expected utility from targeting y2 is r2 − (y2/θ)2.

Incentive compatibility for the high productivity type requires that his expected utility

from targeting y2 be equal to his expected utility from targeting y2. Individual rationality

for the low productivity type requires that his expected utility from targeting y2 be equal

to his outside option of zero. The principal’s second period problem is to maximize her

expected payoﬀ, subject to the high productivity agent’s incentive constraint and the low

6From (2.1), E[y2] = E[θ2 · e2 + ε2] = θ2 · e2.

48

productivity agent’s participation constraint:

max
y2, y2

s.t.

(cid:104)
y2 − r2
+ (1 − ρ2)
(cid:19)2
(cid:18) y2

= r2 −

(cid:105)

(IC2)

θ

(cid:104)

(cid:105)
y2 − r2
(cid:19)2
(cid:19)2

(cid:18) y2
(cid:18) y2

θ

θ

ρ2

r2 −

r2 −

= 0.

(IR2)

(2.12)

After using the incentive and participation constraints to substitute out for the expected

transfers, one derives the second period equilibrium output targets for the high and low

productivity agent, respectively:

and

where C(ρ2) =

rent is given by

y2 =

2

θ
2

,

y2 = C(ρ2)

θ2
2

,

(2.13)

(2.14)

1−ρ2
1−ρ2Θ2 , and Θ = θ/θ. The high productivity agent’s expected information

(cid:104)

1 − Θ2(cid:105)

u2(ρ2) = C2(ρ2)

θ2
4

.

(2.15)

Observing (2.14) and (2.15), one can see that this second period game exhibits the classic

rent-extraction/eﬃciency trade-oﬀ that characterizes static asymmetric information games.

As the principal’s belief that the agent has high productivity approaches one, the low pro-

ductivity type has his output target, and thus his eﬀort, reduced to zero (C(ρ2) goes to zero

as ρ2 goes to one). By reducing the low productivity agent’s eﬀort as second period beliefs

49

go to one, the principal extracts rent from the high productivity worker:

(cid:32)

2(cid:2)1 − ρ2Θ2(cid:3)(cid:33)2
θ(cid:2)1 − Θ2(cid:3)

du2(ρ2)

dρ2

= −

< 0.

(2.16)

This trade-oﬀ has important implications for the ﬁrst period problem. To see these

implications, consider the following lemma:

Lemma 2.1. The principal’s belief that the agent is the high productivity type in the second
period is increasing in ﬁrst period output. That is, ρl ≤ ρm ≤ ρh.

The proof of Lemma 2.1 is straightforward, given that α ≥ 1/2. The important implica-

tion of Lemma 2.1 is summarized in Corollary 2.1 below:

Corollary 2.1. The high productivity agent’s second period rent is decreasing in ﬁrst period

output. That is, u2(ρl) > u2(ρm) > u2(ρh).

The proof of Corollary 2.1 follows directly from Lemma 2.1, and the fact that

du2(ρ2)

dρ2

< 0.

To see the importance of Corollary 2.1 in the ﬁrst period, consider the high productivity

agent’s ﬁrst period eﬀort choice. By targeting y1 in the ﬁrst period, the high productivity
type’s set of possible ﬁrst period output realizations is [y1 − η, y1 + η]. With probability
y1−y1

, he has a favorable output shock, and the principal learns that the he has high ability

2η

in the ﬁrst period. Her second belief that the worker has high productivity is given by ρh.
With probability 1− y1−y1

, the ﬁrst period high productivity type has an unfavorable output

2η

realization. In this case, the principal’s belief that the agent has high productivity in the

second period is given by ρm. The worker only obtains a second period information rent if

he remains the high productivity type in the second period. Therefore, the ﬁrst-period high

50

productivity type’s expected second period rent from targeting y1 is given by

E1[u2(ρ2)| y1] := α

· u2(ρm) +

· u2(ρh)

.

(2.17)

(cid:20)(cid:18)
1 − y1 − y1

(cid:19)

2η

(cid:18) y1 − y1

(cid:19)

2η

(cid:21)

Suppose instead the high productivity agent targets y1 in the ﬁrst period. If he experi-

ences a negative output shock, the principal believes that he was the low productivity agent

in the ﬁrst period. Conditional on remaining the high productivity type in the second period,

he enjoys expected second period rent u2(ρl). If he experiences a favorable output shock

when targeting y1, his expected second period rent is u2(ρm). From the perspective of the

ﬁrst period, his expected second period rent from targeting y1 is

E1[ u2(ρ2)| y1] := α

· u2(ρm) +

(cid:20)(cid:18)
1 − y1 − y1

(cid:19)

2η

(cid:18) y1 − y1

(cid:19)

(cid:21)
· u2(ρl)

.

(2.18)

2η

Using Corollary 2.1, one can easily verify that the high productivity agent’s expected

second period rent from targeting y1 is larger than his expected second period rent when

targeting y1. Thus, the high productivity agent beneﬁts from mimicking the low productivity

type in the ﬁrst period. As in the standard ratchet eﬀect literature, the principal must

increase his ﬁrst period transfer by E[u2(ρ2)| y1] − E[u2(ρ2)| y1] to induce him to target y1

in the ﬁrst period.

Unlike the standard ratchet eﬀect literature in which the agent’s type is ﬁxed, the agent

with low productivity in the ﬁrst period has an expected second period rent, since with
probability 1 − α he has high ability in the second period. Suppose the agent who has low

productivity in the ﬁrst period chooses to target y1 in the ﬁrst period. With probability
1 − y1−y1

, he has a favorable output shock, and the principal has beliefs ρm that the agent

2η

51

is the high productivity type in the second period. With the opposite probability, he has an

unfavorable output shock, and the principal learns that the agent has low productivity in

the ﬁrst period. Therefore, the low productivity agent’s expected second period rent from

targeting y1 in the ﬁrst period is

E1[u2(ρ2)| y1] := (1 − α)

(cid:20)(cid:18)

(cid:19)

1 − y1 − y1

2η

· u2(ρm) +

(cid:18) y1 − y1

(cid:19)

2η

(cid:21)

· u2(ρl)

.

(2.19)

When the low productivity agent targets y1 in the ﬁrst period, his expected second period

rent is

E1[u2(ρ2)| y1] := (1 − α)

(cid:20)(cid:18)
1 − y1 − y1

2η

(cid:19)

· u2(ρm) +

(cid:18) y1 − y1

(cid:19)

2η

(cid:21)
· u2(ρh)

.

(2.20)

It is clear to see that the low productivity agent prefers a ﬁrst period equilibrium that

increases the probability that the principal learns the agent’s ﬁrst period type. The intuition

is straightforward; the low productivity agent induces a more favorable distribution of second

period output by targeting y1 than if he were to mimic the high productivity agent in the ﬁrst
period. The further apart are y1 and y1, the bigger is the diﬀerence between E1[u2(ρ2)| y1]
and E1[u2(ρ2)| y1]. Therefore, the principal can reduce the low productivity agent’s ﬁrst

period transfer by increasing the distance between ﬁrst period output targets.

Lastly, consider the impacts of the ﬁrst period contract on the principal’s second period

payoﬀ, which is given by

(cid:34)

v2(ρ2) = ρ2

2

θ
4

− C2(ρ2)

θ2
4

1 − Θ2(cid:105)(cid:35)
(cid:104)

+ (1 − ρ2)

(cid:34)

C(ρ2)

θ2
2

(cid:20)

1 − C(ρ2)

2

(cid:21)(cid:35)

.

(2.21)

By increasing the distance between output targets, the principal increases the probability

52

that she learns the agent’s ﬁrst period type. This reduces her uncertainty regarding the

agent’s second period type, which allows her to reduce the good agent’s expected second

period rent or reduce the bad agent’s eﬀort distortions. Therefore, the principal increases

her expected second period payoﬀ by increasing the distance between ﬁrst period output

targets.

2.4 First period

As in the second period, the principal’s ﬁrst period problem is to choose output targets y1

and y1 to maximize the sum of her ﬁrst and (discounted) second period expected payoﬀs,

[y1 − r1(y1)]

dy1 + (1 − ρ)

1
2η

[y1 − r1(y1)]

1
2η

dy1 + δE[v2(ρ2)],

(2.22)

(cid:90) y1+η

y1−η

ρ

(cid:90) y1+η

y1−η

where

(cid:18) y1 − y1

(cid:19)

2η

E[v2(ρ2)] =

[ρv2(ρh) + (1 − ρ)v2(ρl) − v2(ρm)] + v2(ρm),

(2.23)

subject to incentive compatibility and participation constraints, which are derived below.

A well-known problem that arises in dynamic games of asymmetric information is that

the low productivity agent’s incentive constraint may bind in the ﬁrst period. As discussed

in the previous section, the principal must increase the high productivity agent’s ﬁrst period

transfer to induce him to target y1 rather than y1. When output is deterministic and the

agent’s type is ﬁxed, the low productivity type is tempted to mimic the high productivity

agent in the ﬁrst period. By doing so, he receives the high productivity agent’s large ﬁrst

period transfer, and can walk away from the relationship in the second period.

53

In the current setting, two factors alleviate this dynamic incentive problem. First, output

is noisy, which implies that the principal’s learning process is slowed. For intermediate output

realizations, the principal is unsure of the agent’s ﬁrst period type. Additionally, the low

cost agent receives an expected second period rent, and this rent is higher when targeting y1

than if he were to target y1. Thus, the dynamic incentive problem is alleviated relative to

the deterministic case, and even the stochastic case in which the agent’s type is ﬁxed over

time. The ﬁrst period problem proceeds by assuming the low productivity ﬁrm’s ﬁrst period

incentive constraint is slack. Once the equilibrium output targets are derived, it is veriﬁed

in Appendix B that for η large enough, this assumption holds.

First, consider the high productivity type’s incentive compatibility constraint. When he

chooses his eﬀort so that e1 = y1/θ, his expected ﬁrst period reward is

r1 :=

r1(y1)

1
2η

dy1.

(2.24)

Therefore, his expected ﬁrst period utility is r1 − (y1/θ)2. His expected second period rent

from targeting y1 is given by (2.17). Suppose instead he chooses his eﬀort so that e1 = y1/θ.

In this case, his expected ﬁrst period reward is

(cid:90) y1+η

y1−η

(cid:90) y1+η

y1−η

r1 :=

r1(y1)

1
2η

dy1,

(2.25)

and his ﬁrst period utility is r1− (y1/θ)2. His expected second period rent from targeting y1

is given in (2.18). Therefore, incentive compatibility for the high productivity type requires

that

r1 −

(cid:18) y1

(cid:19)2

θ

(cid:18) y1

(cid:19)2

θ

+ δE[u2(ρ2)| y1] = r1 −

54

+ δE[u2(ρ2)| y1].

(2.26)

From this incentive compatibility constraint, we can derive the high productivity ﬁrm’s

expected ﬁrst period transfer:

(cid:18) y1

(cid:19)2

θ

r1 =

+ r1 −

(cid:18) y1

(cid:19)2

θ

(cid:18) y1 − y1

(cid:19)

2η

+ δα

(u2(ρl) − u2(ρh)) .

(2.27)

Next, consider the high cost ﬁrm’s participation constraint. By choosing his eﬀort so

that e1 = y1/θ, the low cost ﬁrm’s expected ﬁrst period utility is r1 − (y1/θ)2. His expected
second period rent from targeting y1 is given in (2.19). Therefore, the low productivity ﬁrm’s

participation constraint is given by

(cid:18) y1

(cid:19)2

θ

r1 −

+ δ(1 − α)

(cid:20)(cid:18) y1 − y1

2η

(cid:19)

u2(1 − α) +

(cid:18)
1 − y1 − y1

2η

(cid:19)

(cid:21)

u2(ρ2
2)

= 0.

(2.28)

The principal’s ﬁrst period problem is as follows:

ρ [y1 − r1] + (1 − ρ)

max
y1, y1

(cid:105)
(cid:104)
y1 − r1
(cid:19)2
(cid:18) y1
(cid:19)2
(cid:20)(cid:18)y1 − y1
(cid:19)2 − δ(1 − α)

+ r1 −

+ δα

θ

+ δE[v2(ρ2)]

(cid:19)

(cid:18) y1 − y1
(cid:19)

2η

(cid:18) y1
(cid:18) y1

θ

θ

s.t.

r1 =

r1 =

(cid:19)
(cid:18)
(u2(1 − α) − u2(α))
1 − y1 − y1

2η
u2(1 − α) +

(cid:21)

u2(ρ2
2)

.

(2.29)

2η

After using the incentive and participation constraints to eliminate the expected ﬁrst period

transfer from the principal’s problem, one obtains the following equilibrium ﬁrst period

output targets for the high and low productivity agent, respectively:

y1 =

2

θ
2

2

θ
2

1
ρ

δ
2η

+

· A,

(2.30)

55

and

where

y1 = C(ρ)

−

θ2
2

1

1 − ρΘ2

θ2
2

δ
2η

· A,

(2.31)

A := ρv2(ρh) + (1 − ρ)v2(ρl) − v2(ρm)

− (ρα(u2(ρl) − u2(ρh)) − (1 − α)(u2(ρl) − u2(ρm))) .

(2.32)

One of the main results from the stochastic contracting literature in which the agent’s

type is ﬁxed is that it is optimal for the principal to learn less about the agent’s type than

she could by setting the ﬁrst period performance targets equal to the commitment optimum

targets.7 Suppose the principal is able to commit to the second period incentive scheme at

the beginning of the interaction with the agent. The ﬁrst period commitment output targets

for the high and low productivity agent, respectively, are given by

and

yc
1 :=

2

θ
2

yc
1 := C(ρ)

θ2
2

.

(2.33)

(2.34)

Therefore, if A < 0, it is optimal for the the principal to learn less about the agent’s ﬁrst

period type than she could by choosing the commitment output targets.8 Before exploring

the relationship between the equilibrium ﬁrst period output targets and the commitment

output targets, we examine separately the three eﬀects that determine their relationship:

7See Jeitschko et al. (2002) and Jeitschko and Withers (2018).
8If A < 0, then yc

1 ≤ y1 ≤ y1 ≤ yc
1.

56

the principal’s desire to reduce the high productivity ﬁrm’s ﬁrst period transfer, her desire

to reduce the low productivity ﬁrm’s ﬁrst period transfer, and her desire to increase her

expected second period payoﬀ.

2.4.1 Signal dampening

The principal’s choice of ﬁrst period output targets determines the probability with which

she learns the agent’s ﬁrst period type. When the output targets are close together, this

probability is small. When the probability that the principal learns the agent’s ﬁrst period

type is small, the high productivity agent has little incentive to mimic the low productivity

agent in the ﬁrst period. Therefore, the principal reduces the high productivity agent’s ﬁrst

period transfer by setting the ﬁrst period output targets close together.

To see this, note that when the agent targets y1 in the ﬁrst period, the principal’ second

period belief that the agent has high productivity is either ρm or ρh. The closer together are

the ﬁrst period cost targets, the more likely it is that the principal’s second period beliefs are

given by ρm. Since u2(ρm) > u2(ρh), bringing the ﬁrst period output targets closer together
increases E1[u2(ρ2)| y1].

Similarly, if the high productivity agent targets y1 in the ﬁrst period, the principal’s

second period belief that the agent has high productivity is either ρm or ρl. The closer

together are the ﬁrst period output targets, the more likely it is that the principal’s second

period beliefs are given by ρm. Since u2(ρm) < u2(ρl), bringing the ﬁrst period output
targets closer together decreases E1[u2(ρ2)| y1]. Thus, by bringing the ﬁrst period output
targets closer together, the principal decreases the diﬀerence between E1[u2(ρ2)| y1] and
E1[u2(ρ2)| y1]. By reducing the diﬀerence in these expected utilities, the principal reduces

the high productivity agent’s ﬁrst period transfer.

57

Proposition 1 below formalizes this argument. To make the statement of Proposition 1

more clear, solve for r1 in (2.28) and substitute it into (2.27). The resulting expression for

the high productivity agent’s ﬁrst period expected transfer can be decomposed as follows:

where

and

r1 = rS

1 + rD
1 ,

(cid:18) y1

(cid:19)2

θ

(cid:19)2 −

(cid:18) y1

θ

(cid:18) y1

(cid:19)2

θ

,

+

rS
1 :=

(cid:18) y1 − y1

2η

(cid:19)(cid:2)α(u2(ρl) − u2(ρh)) − (1 − α)(u2(ρl) − u2(ρm))(cid:3)

rD
1 := δ

− δ(1 − α)u2(ρm).

(2.35)

(2.36)

(2.37)

The “dynamic” portion of this expected transfer, rD

1 , represents the diﬀerence in ex-

pected second period rent that the high productivity agent receives from targeting y1 versus

targeting y1, less the amount that the principal is able to extract from the low productivity

agent. If rD

1 is increasing in the distance between ﬁrst period output targets, then the prin-

cipal can decrease the high productivity ﬁrm’s expected ﬁrst period transfer by bringing the

ﬁrst period output targets closer together.

Proposition 2.1. The principal can decrease the high productivity ﬁrm’s ﬁrst period transfer

by decreasing the distance between y1 and y1. That is,

d rD
1

d(y1 − y1)

> 0.

58

(2.38)

It is important to note that Proposition 2.1 is true for any degree of positive correlation

between the agent’s ﬁrst and second period types. As discussed above, the beneﬁt of reducing

the distance between ﬁrst period output targets is to decrease the high productivity agent’s

incentive to mimic the low ability agent in the ﬁrst period. The cost of reducing the distance

between ﬁrst period output targets is that the principal is able to extract less rent from the

low ability agent in the ﬁrst period.9

For the purposes of reducing the high productivity agent’s expected ﬁrst period transfer,

the beneﬁt of reducing the distance between output targets outweighs the costs for all levels

of positive correlation for two reasons. First, since the low ability agent’s incentive constraint

is slack, it matters less to the principal to extract rent from the low ability type than it does

from the high ability type. Second, since types are positively correlated, an agent with high

ability in the ﬁrst period is more likely to receive a second period rent than an agent with

low ability in the ﬁrst period.

2.4.2 Experimentation

Just as the principal can decrease the probability of learning the agent’s ﬁrst period type

by bringing the ﬁrst period output targets closer together, the opposite is true as well. By

spreading y1 and y1 further apart, the principal increases the probability that she learns the

agent’s ﬁrst period productivity parameter.

Because the agent’s type is correlated over time, the principal does not have complete

information about the his second period type even when she learns the his ﬁrst period type.

Nevertheless, the principal beneﬁts in two ways from learning the agent’s ﬁrst period type.

First, the principal can either extract more rent from the high productivity type or induce

9This will be discussed in more detail in the following section.

59

the low productivity type to exert more eﬀort. Second, she is able to reduce the ﬁrst period

expected transfer to the low productivity agent.

First, consider the impact of the distance between ﬁrst period output targets on the low

productivity agent’s ﬁrst period expected transfer. Recall that the low productivity agent

chooses his eﬀort to target y1 (his incentive constraint is slack). The low productivity agent

prefers a lower y1 for two reasons. First, a lower output target requires less eﬀort to achieve.

Second, the further y1 is from y1, the larger is the low productivity agent’s expected second

period rent.

To see this, recall that the low productivity agent receives no second period rent if he

remains the low productivity type. Unlike the high productivity agent, therefore, he has no

incentive to conceal his ﬁrst period private information. When he targets y1, the principal’s

second period belief that the agent has high productivity is either ρl or ρm, depending on
whether the ﬁrst period output realization is less than or greater than y1 − η. Conditional

on becoming the high productivity type in the second period, the low productivity agent

prefers a ﬁrst period equilibrium that places more weight on u2(ρl) as opposed to u2(ρm).

That is, he beneﬁts from a ﬁrst period equilibrium that increases the probability that the

principal learns his ﬁrst period type.

Proposition 2.2 formalizes this logic. To simplify the statement of the proposition, let

where

r1 = rS

1 + rD
1 ,

(cid:18) y1

(cid:19)2

θ

,

rS
1 :=

60

(2.39)

(2.40)

and

(cid:18) y1 − y1

(cid:19)

2η

1 := −δ(1 − α)
rD

(u2(ρl) − u2(ρm)) − δ(1 − α)u2(ρm).

(2.41)

Proposition 2.2. The principal can decrease the low productivity ﬁrm’s ﬁrst period transfer

by moving the ﬁrst period output targets further apart. That is,

d rD
1

d(y1 − y1)

< 0.

(2.42)

The intuition of the result is clear; the portion of the low productivity agent’s ﬁrst period

transfer that depends on the distance between ﬁrst period output targets is decreasing in the

distance between those targets. Therefore, the principal decreases the expected ﬁrst period

transfer to the low productivity agent by increasing the distance between y1 and y1. The

economic intuition for this result is discussed in the argument preceding the statement of

Proposition 2.2.

Next, consider the eﬀect of the distance between ﬁrst period output targets on the prin-

cipal’s expected second period payoﬀ. Lemma 2.2 establishes that information about the

agent’s ﬁrst period type is valuable to the principal; the more the principal knows about the

agent’s ﬁrst period type, the better she can balance the rent extraction/eﬃciency tradeoﬀ in

the second period. Given that information is valuable, Proposition 2.3 establishes that the

principal acquires better information about the agent’s ﬁrst period type, and thus increases

her expected second period payoﬀ, by increasing the distance between ﬁrst period output

targets.

Lemma 2.2. Information is valuable to the principal. That is, the principal’s second period

61

expected payoﬀ is convex in second period beliefs (see Blackwell (1951)):

d2v2(ρ2)

dρ2
2

> 0.

(2.43)

Given Lemma 2.2, it is easy to show that the principal increases her expected second

period payoﬀ by increasing the distance between ﬁrst period output targets.

Proposition 2.3. The principal increases her expected second period payoﬀ, E[v2(ρ2)], by

increasing the distance between ﬁrst period output targets. That is,

dE[v2(ρ2)]
d(y1 − y1)

> 0.

(2.44)

Proposition 2.2 and Proposition 2.3 establish that two incentives drive the principal to

learn the agent’s ﬁrst period private information. Like the stochastic contracting literature in

which the agent’s type is ﬁxed, the principal increases her expected second period payoﬀ by

learning more about the agent’s ﬁrst period private information. Unlike the aforementioned

literature, however, the principal has an incentive to learn more to decrease the ﬁrst period

transfer to the low productivity agent. These two incentives combine with the incentive

to decrease the high productivity agent’s ﬁrst period transfer to determine the ﬁrst period

output targets.

2.4.3 Total ﬁrst period transfer: signal dampening or experimen-

tation?

Both the high and low productivity ﬁrm’s ﬁrst period transfers depend on the distance be-

tween ﬁrst period output targets. Proposition 2.1 demonstrates that the principal decreases

62

the high productivity ﬁrm’s transfer by learning less about the agent’s ﬁrst period type,

while Proposition 2.2 demonstrates that the principal decreases the low productivity agent’s

transfer by learning more about the agent’s ﬁrst period type.

In the principal’s ﬁrst period problem, the high and low productivity ﬁrm’s ﬁrst period

transfers are weighted by the principal’s prior beliefs about the agent’s type. Deﬁne the

“total expected ﬁrst period transfer,” E[r1], as follows:

E[r1] := ρr1 + (1 − ρ)r1,

(2.45)

where r1 and r1 are given in (2.35) and (2.39), respectively. One component of the total

expected transfer, r1, is increasing in the distance between output targets, while the other

component, r1, is decreasing in the distance between output targets. This raises the question

of how E[r1] depends on the distance between ﬁrst period output targets.

To see why this question is interesting, consider the relationship between the optimal ﬁrst

period output targets, given in (2.30) and (2.31), and the commitment output targets, given

in (2.33) and (2.34). This relationship depends on the principal’s incentives to decrease

the high and low productivity agent’s ﬁrst period transfers and her incentive to increase

her expected second period payoﬀ. Proposition 2.3 establishes that to increase her expected

second period payoﬀ, the principal increases the distance between ﬁrst period output targets.

This result does not depend on the value of the correlation parameter, the level of the prior,

or the ratio of the intrinsic productivity levels.

This implies that if, for some values of the primitives, the principal decreases E[r1] by

increasing the distance between output targets, then the optimal ﬁrst period output targets

lie outside the commitment optimum output targets. This result would stand in contrast to

63

the stochastic contracting literature in which the agent’s type is ﬁxed; when the agent’s type

is ﬁxed, a robust result is that the optimal performance targets reveal less information about

the agent’s type to the principal than she would gather by setting the ﬁrst period output

targets equal to the commitment level. The following proposition, however, shows that this

is never the case:

Proposition 2.4. The principal can decrease the total expected ﬁrst period transfer by de-

creasing the distance between the ﬁrst period output targets. That is,

dE[r1]
d(y1 − y1)

> 0.

(2.46)

An interesting takeaway from Proposition 2.4 is that, regardless of her prior beliefs, ρ, the

principal decreases the total expected transfer by designing the ﬁrst period output targets

to decrease the the ﬁrst period payment to the high productivity agent. Put another way,

if the principal’s objective is to reduce the total expected ﬁrst period transfer, it is never

in her interest to design the ﬁrst period output targets to increase the probability that she

learns the agent’s ﬁrst period type, no matter how unlikely the high productivity agent is

ex-ante.

2.5 Equilibrium rent preservation

With the exception of subsection 2.4.3, the three incentives that determine the distance be-

tween the ﬁrst period output targets have been considered in isolation. First, the principal

decreases the high productivity agent’s ﬁrst period transfer by decreasing the distance be-

tween ﬁrst period output targets. By decreasing the distance between ﬁrst period output

64

targets, she decreases the probability that she learns the agent’s ﬁrst period type. This

reduces the high ability agent’s incentive to mimic the low ability agent in the ﬁrst period,

and reduces the high ability agent’s ﬁrst period transfer.

Second, the principal decreases the low ability agent’s ﬁrst period transfer by increasing

the distance between ﬁrst period output targets. By increasing the distance between ﬁrst

period output targets, she increases the probability that she learns the agent’s ﬁrst period

type. The low ability agent beneﬁts when the principal learns his ﬁrst period type; when

the principal believes the agent has low ability in the ﬁrst period, she designs the second

period contract to induce more eﬀort from the low ability agent. The higher is the low

ability agent’s eﬀort in the second period, the higher is the high ability agent’s information

rent. Since the low ability agent receives no second period rent if his type remains low in

the second period, he maximizes his expected second period rent by revealing his ﬁrst period

type to the principal.

In this case, if his type switches between periods, he receives the

highest possible second period rent.

Third, the principal increases her expected second period payoﬀ by increasing the distance

between ﬁrst period output targets. When the principal believes she is fully informed about

the agent’s ﬁrst period type, she strikes a better balance in the second period between

inducing eﬀort in the low ability agent and extracting rent from the high ability agent.

If she believes that the agent’s type is low in the ﬁrst period, the second period contract

induces more eﬀort in the low ability type and thus leaves a higher rent for the high ability

agent. When she believes that the agent’s type is high in the ﬁrst period, the second period

contract calls for a lower eﬀort in the low ability agent, which extracts rent from the high

ability agent. Thus, the better is the principal’s information about the agent’s ﬁrst period

type, the more appropriately she can distort the low ability agent’s second period eﬀort.

65

From the perspective of the ﬁrst period, therefore, her expected second period payoﬀ is

increasing in the probability that she learns the agent’s ﬁrst period type.

The combined eﬀect of these three incentives determine how likely it is that the principal

learns the agent’s ﬁrst period type. To give a frame of reference for the following discussion,

we will say that the ﬁrst period contract favors reducing the high ability agent’s ﬁrst period

transfer if the optimal ﬁrst period output targets, given in (2.30) and (2.31), lie within the

commitment output targets given in (2.33) and (2.34). That is, if yc

1 ≤ y1 ≤ y1 ≤ yc

1, the

principal learns less about the agent’s ﬁrst period type than she could by setting the ﬁrst

period output targets equal to the commitment optimum output targets.

Conversely, the ﬁrst period contract favors reducing the low ability agent’s ﬁrst period

transfer and increasing the principal’s expected second period payoﬀ if y1 ≤ yc

1 ≤ yc

1 ≤ y1.

In this case, the probability that the principal learns the agent’s ﬁrst period type is higher

than if the principal set the ﬁrst period output targets equal to the commitment optimum.

The following proposition shows that as long as either the high and low productivity

agent do not diﬀer too much in their ability, or the high productivity agent is not too

unlikely ex-ante, the optimal ﬁrst period contract favors reducing the upfront payment to

the high ability agent.

Proposition 2.5. If the high ability agent is not too much more productive than the low

ability agent, or if the principal’s prior belief that the agent is the high productivity type

is not too low, then the overall impact of the ﬁrst period contract is to reduce the distance

between ﬁrst period output targets, relative to the commitment optimum, for every level of
positive correlation. Speciﬁcally, if Θ2 ≥ 1/2 or ρ ≥ 1/3, then

1 ≤ y1 ≤ y1 ≤ yc
yc
1,

66

(2.47)

for every α ∈ [1/2, 1].

The interpretation of Proposition 2.5 is straightforward. First, consider the suﬃcient
condition on the ratio of the agent’s types, Θ2 ≥ 1/2. Since Θ = θ/θ, Proposition 2.5 states

that as long as the low ability agent is at least 71 percent as productive as the high ability

agent, the principal ﬁnds it optimal to design the ﬁrst period contract to reduce the high

productivity agent’s ﬁrst period transfer.

Recall that the principal reduces the high productivity agent’s ﬁrst period transfer by

reducing the distance between ﬁrst period output targets. Doing so reduces the diﬀerence in

the low productivity agent’s expected second period rent given that he targets y1, and his

expected second period rent given that he targets y1. This in turn reduces his ﬁrst period

transfer.

The cost of bringing the output targets closer together is that the principal reduces

the probability that she learns the agent’s ﬁrst period type. First, this increases the low

productivity agent’s ﬁrst period transfer. However, Proposition 2.4 shows that the beneﬁt

of reducing the high productivity agent’s ﬁrst period transfer always outweighs the beneﬁt

of reducing the low productivity agent’s ﬁrst period transfer.

Second, when the principal is unsure of the agent’s ﬁrst period type, the low productivity

agent exerts less eﬀort in the second period than when the principal believes she is fully

informed about the agent’s ﬁrst period type. The low productivity agent’s eﬀort distortion,
however, is decreasing in Θ. Proposition 2.5 shows that as long as Θ2 ≥ 1/2, the loss in

the principal’s expected second period payoﬀ due to the increased probability of low eﬀort

from the low productivity agent is outweighed by the beneﬁt of decreasing the ﬁrst period

transfer to the high productivity agent.

Similar reasoning explains the suﬃcient condition on the principal’s beliefs that the agent

67

is the high productivity type at the beginning of the interaction, ρ ≥ 1/3. When the high

productivity type is likely enough, the beneﬁt of designing the ﬁrst period contract to reduce

the high productivity agent’s ﬁrst period transfer outweighs the second period reduction in

the principal’s payoﬀ implied by the low productivity agent’s reduced second period eﬀort.

The following example illustrates how the optimal ﬁrst period tradeoﬀ between rent

preservation and learning depends on the degree of positive correlation. In Figure 2.2, it is
assumed that θ = 1, δ = 1, and that noise is distributed uniformly on [−1, 1]. The principal’s

prior belief that the agent is the high productivity type is given by ρ = .4. The low ability

agent is 80 percent as productive as the high ability agent (i.e. Θ = .8 ). The horizontal axis

measures the degree of positive correlation, and the vertical axis measures how much closer

together the ﬁrst period output targets are than the commitment output targets. One can

clearly see that the diﬀerence between ﬁrst period output targets is not monotone in α; y1

and y1 are closest together when α = .93, and as α approaches one-half, they approach the

commitment optimum targets.

Figure 2.2: First period contract favors rent preservation for all α

68

When the suﬃcient conditions outlined in Proposition 2.5 do not hold, it is possible that

the optimal ﬁrst period output targets lie outside the commitment output targets. In the

following example, assume once again that θ = 1, δ = 1, and noise is distributed uniformly
on [−1, 1]. Suppose, however, that the high productivity type is unlikely ex-ante (ρ = .1),

and that the low ability agent is only 50 percent as productive as the high ability agent

(Θ = .5).

From Figure 2.3, one can see that for α between one-half and approximately three-fourths,

the ﬁrst period output targets lie outside the commitment output targets.10 Thus, for smaller

degrees of positive correlation, the optimal ﬁrst period contract increases the probability that

the principal learns the agent’s ﬁst period type relative to what she would learn under the

commitment optimum. When the high productivity type is unlikely and the diﬀerence in the

agent’s productivity parameters is large, the beneﬁt of reducing the low productivity agent’s

ﬁrst period transfer, coupled with the beneﬁt of better information in the second period,

outweighs the beneﬁt of reducing the high productivity agent’s ﬁrst period transfer. Still,

this beneﬁt only holds when the agent’s type is weakly positively correlated.

2.6 Conclusion

This paper examines a repeated relationship between a principal and an agent. The agent

produces output that the principal values. Two key features of this relationship are that

the agent has imperfect control over output (output is stochastic), and the agent’s private

information may change over time (the agent’s type is positively correlated).

The principal determines the probability that she learns the agent’s ﬁrst period type via

10That is, y1 < yc

1 < yc

1 < y1.

69

Figure 2.3: First period contract favors learning for some α

her choice of ﬁrst period output targets. As long as the high productivity type is not too

unlikely ex-ante, or as long as the diﬀerence in the agent’s ability levels is not too large, the

optimal ﬁrst period contract favors reducing the high productivity ﬁrm’s ﬁrst period transfer.

This is achieved by reducing the distance between ﬁrst period output targets, relative to the

commitment optimum.

The low productivity agent’s ﬁrst period transfer depends on the ﬁrst period output

targets; this feature does not arise in stochastic contracting models in which the agent’s type

is ﬁxed over time. The low productivity agent’s expected second period rent is increasing in

the probability that the principal learns the agent’s ﬁrst period type. Therefore, to decrease

the low productivity ﬁrm’s ﬁrst period transfer, the principal increases the distance between

the ﬁrst period output targets.

Both the high and low productivity agent’s expected second period rents depend on the

ﬁrst period contract. Therefore, the question arises whether the principal reduces the prior-

weighted sum of the high and low productivity agent’s ﬁrst period transfers by increasing or

70

decreasing the distance between ﬁrst period cost targets. Regardless of the principal’s prior

beliefs about the agent’s type, this total expected transfer (prior-weighted sum) is reduced

by decreasing the distance between output targets.

71

Chapter 3

Task Assignment Under Moral

Hazard, with Eﬀort-Dependent

Human Capital and

Outcome-Dependent Outside Options

3.1

Introduction

Firms want to develop their employees’ skills. They want them to be better decision makers,

better marketers, better at managing and motivating subordinates, and better product de-

velopers. One common way to develop skills is through training. Another potential avenue

for developing human capital is learning by doing; speciﬁcally, the ﬁrm can assign workers to

complete tasks, or implement projects. If the experience of implementing the project endows

the worker with skills, this practice will beneﬁt him and the ﬁrm in the future. The next

time he undertakes the same task or one that is closely related to it, he will be better at it

than he is today.

The above method of human capital acquisition has been discussed extensively, beginning

with the work of Becker (1962). This paper examines a similar framework in which a principal

72

is interested in developing her employees’ human capital, but with the added supposition

that the amount of human capital the agent develops depends positively on the amount of

eﬀort that he exerts on the task. If eﬀort is costly to the agent, then the principal will have

to induce the agent to exert eﬀort in order to develop more than the minimum level of human

capital.

Making this assumption allows us to examine human capital acquisition within the frame-

work of the moral hazard literature. Following Mirrlees (1976) and Holmstrom (1979), when

eﬀort is unobservable, the principal will have to tie compensation to an observable and

veriﬁable outcome in order to induce the desired level of eﬀort from the agent.

In the absence of an outside labor market, nothing substantial changes between the

classical moral hazard problem and the model augmented with human capital acquisition. If

human capital confers some beneﬁt to the principal, then the model augmented with human

capital acquisition only increases the cost of eﬀort at which the principal is indiﬀerent between

high and low eﬀort.

However, the existence of an outside labor market may alter the agent’s disutility of

eﬀort. If outside ﬁrms view a successful project as a signal that the employee worked hard,

and developed valuable human capital, they may try to bid the agent away from his current

ﬁrm when he has a success. Since exerting eﬀort makes it more likely that the project will

be successful, the increase in the expected value of the outside option from exerting high

eﬀort may outweigh the agent’s increased cost of eﬀort. If this is the case, then ex ante he

prefers high eﬀort to low eﬀort, and we say the agent is self-motivated.

The introduction of the outside labor market borrows from the literature that discusses

the theory of wage and promotion dynamics inside ﬁrms. Speciﬁcally, it is most closely

related to Waldman (1984). Waldman considers an asymmetric learning model in which

73

the agent’s ability is initially unknown to the entire economy. The principal can assign the

worker to one of two jobs: one that depends on his ability, and one that does not.

In equilibrium, the principal assigns all workers to the ability-independent job in the ﬁrst

period. After the ﬁrst period production process, the principal learns the agent’s ability

perfectly. The outside market does not, but does observe the ﬁrst period ﬁrm’s decision to

promote the agent to the ability-dependent job or keep him in the same job. The outside

market uses this signal to update its beliefs about the agent’s ability.

Due to the outside market’s bidding behavior, which is inﬂuenced by the asymmetric

information between the principal and the outside market, the principal promotes an inef-

ﬁciently small number of agent-types in the second period. The agent must be productive

enough in the new, ability dependent job, to oﬀset the higher wage that the principal must

pay him in order to keep him from leaving to work for the outside labor market.

In this paper, the principal and the outside market are not interested in learning about

ability. They are instead interested in whether the agent has developed human capital. Like

in Waldman (1984), the agent’s ﬁrst period employer has an informational advantage over

the outside market. The principal observes the agent’s eﬀort, and so knows whether the

agent developed human capital. The outside market is again left to infer the agent’s level of

human capital from a signal; in this paper, the signal is the success or failure of the project.

Unlike Waldman (1984), however, the market in this paper does not update expectations

about the agent’s eﬀort choice using the project outcome. The market sends an outside

oﬀer if and only if the agent is successful. One can interpret this as the market over-valuing

success; in this sense, success is costly to the ﬁrst period employer. The ﬁrm wants to keep

workers who exert eﬀort and develop human capital, but they do so only if his beneﬁt to the

ﬁrm outweighs the increased cost of matching the outside oﬀer.

74

In Waldman (1984), the principal’s important strategic decision is whether to promote

the agent or not after the ﬁrst period is over. In this paper, the principal’s strategic decision

comes before the ﬁrst period begins. She must decide whether the human capital that the

agent will develop by exerting eﬀort is valuable enough to oﬀset the increased wage she would

have to pay him, were he successful.

Consider the following real world example to motivate this paper. First, consider an

assistant district attorney who develops litigation skills by taking cases to trial. The district

attorney’s oﬃce certainly values their employee’s litigation skills, but should value their

prosecutorial discretion more. That is, the district attorney’s oﬃce wants a case to go to

trial (or plea agreement) if the evidence warrants it. The assistant district attorney’s chances

of employment with a criminal defense ﬁrm, however, may be increasing in the number of

visible successes that he has in court. Thus, there may arise cases in which the evidence

against the defendant does not warrant a charge from the district attorney’s point of view,

but the assistant district attorney may nevertheless feel that the chances of a conviction are

high.

Throughout the analysis, we restrict attention to the case in which the agent is self-

motivated. Because success is costly to the principal, we examine the circumstances under

which the principal prefers to induce low eﬀort, when the agent is self-motivated. We say

the contracting problem suﬀers from countervailing incentives when both of these conditions

are met.

75

3.2 Model

Consider a two-period model, with one principal (she) and one agent (he). Both the principal

and the agent are risk neutral, but the agent faces limited liability. In the ﬁrst period, the

principal decides whether to delegate completion of a project to the agent. If the agent exerts

high (low) eﬀort, the project succeeds with probability pH (pL), and fails with probability
1 − pH (1 − pL). When the project is successful, the principal realizes a proﬁt πS, and with
failure, πF . We assume πS ≥ πF > 0. The project’s outcome is observable and veriﬁable.
The agent incurs an eﬀort cost c(eH ) = ψ if he exerts eﬀort, where ψ ∈ [0,∞). There is
no cost of low eﬀort (c(eL) = 0). We assume that the principal can observe e ∈ {eH , eL},

but eﬀort is not veriﬁable, and so contracts must be written on the project’s outcome alone.

We augment the model with human capital acquisition and an outside labor market. If

the agent exerts high (low) eﬀort when implementing the project, he gains human capital

v(eH ) = v (v(eL) = 0). We can think about this as learning by doing, with the added

assumption that the amount of human capital developed depends on how much eﬀort the

agent exerts while undertaking the project. The key point is that if the agent shirks in

Period 1, he does not improve or acquire any skills that the principal values, whereas if he

exerts high eﬀort, he develops these skills whether the project succeeds or fails.

The principal and the outside labor market value the skills that the agent develops by

exerting eﬀort. However, they may have diﬀerent valuations for his newly developed skills.

At the end of the ﬁrst period, the outside market observes the outcome of the project. If

the project is a success, the outside market sends an oﬀer of α to the agent. The labor

market is a “black box” in the sense that the agent gets an outside oﬀer at the beginning

of the second period when he is successful, regardless of whether he actually exerted eﬀort

76

and gained human capital. Similarly, he gets no outside oﬀer when he exerts high eﬀort but

fails.

We allow α ∈ [0, ∞). When 0 ≤ α < v, outside ﬁrms value his human capital less

than the inside ﬁrm, in which case the human capital developed is more ﬁrm speciﬁc. When

α = 0, we have the classic case of purely ﬁrm speciﬁc human capital. However, we allow
α ≥ v to capture the idea that the skills a worker develops may be more useful or productive

outside his current ﬁrm.

To see an example in which the principal and an outside labor market may have diﬀerent

valuations for the human capital developed on a task, recall the assistant district attorney

who develops litigation experience by taking cases to trial. The district attorney only values

the assistant’s litigation experience on a given case if the evidence warrants going to trial.

A private criminal defense ﬁrm, however, may value the litigation experience regardless of

the merit of the case.

The principal observes the agent’s outside oﬀer, and decides whether to match it or let

him leave. The principal only obtains the human capital payoﬀ if she retains the agent, and

the agent exerted eﬀort in the ﬁrst period. If she lets the agent leave, she faces a cost c of

replacing the agent, whether or not the agent exerted eﬀort in the ﬁrst period.

The outside oﬀer has two important eﬀects on the principal’s decision making. First,

when α is large enough, the agent is too expensive to keep, regardless of ﬁrst period eﬀort.

Second, the outside oﬀer will create the possibility that the agent is “self motivated.” We

will use this phrase to mean that in the absence of a wage, the agent prefers to exert high

eﬀort rather than low eﬀort (Under the usual contracting problem under moral hazard, the

“status-quo” level of eﬀort is low eﬀort, since eﬀort is costly). When α is large enough, the

agent prefers to exert eﬀort, and increase his probability of getting an outside oﬀer, even

77

when eﬀort is costly. Notice that when the agent’s cost of eﬀort is small, the outside oﬀer

does not have to be large in order for the agent to be self motivated.

Self motivation will plays an important role in the analysis to follow. We concentrate

on the contracting problem between the principal and the agent, when the agent is self

motivated. We characterize parameter restrictions that ensure the existence of countervailing

incentives, which occur when when the agent is self motivated, but proﬁt maximization

dictates that the principal induce low eﬀort.

The timing of the game is as follows: In Period 1, the principal oﬀers the agent a contract

w = (wS, wF ). The agent chooses his eﬀort, and at the end of Period 1, the principal realizes
project payoﬀ πo, and pays the agent his outcome contingent wage, wo, where o ∈ {S, F}.

At the end of Period 1, outside ﬁrms observe whether the agent successfully implemented

the project. If he did, then he gets a wage oﬀer α at the beginning of Period 2. The principal

observes α, and has the opportunity to match the outside wage oﬀer. If she chooses to match,

then at the end of Period 2, she realizes the second period human capital payoﬀ, v, less the
outside oﬀer. Her second period payoﬀ is then U = v − α. If she chooses not to match, she

does not get v, and incurs cost c > 0 of replacing the agent.

The principal’s payoﬀ depends on whether she matches the outside oﬀer and whether the

agent develops human capital:

U = πo − wo + v(e) − α,

(3.1)

where v(e) = v if the agent exerted high eﬀort, and zero otherwise. Likewise, if the project

78

fails, α = 0. The agent’s payoﬀ is as follows:

u = wo − c(e) + α,

(3.2)

where c(e) = ψ if the agent exerts eﬀort, and zero otherwise. Again, if the project fails,

α = 0. Figure 3.1 illustrates the timing of the game, the principal’s second period strategy

choices, and the principal and agent’s outcome contingent payoﬀs.

e L

e

H

A

1

1

−

p

L

p

L

−

p

H

p

H

P

P

(πF − wf , wf )

a v e

L e

t

L e

Match

(πS − ws − c, ws + α)

(πS − ws − α, ws + α)

(πF − wF + v, wF − ψ)

a v e

L e

t

L e

Match

(πS − wS − c, wS + α − ψ)

(πS − wS + v − α, wS + α − ψ)

Figure 3.1: Game tree and payoﬀs (Ue, ue), e ∈ {L, H}

When the agent exerts low eﬀort he does not develop human capital. The project succeeds
with probability pL, and fails with probability 1− pL. The principal’s payoﬀ from low eﬀort,

when she chooses “Let Leave” and “Match” in the second period, are

L = pL(πS − ws − c) + (1 − pL)(πF − wf )
U L

(3.3)

79

and

M = pL(πS − ws − α) + (1 − pL)(πF − wf ),
U L

(3.4)

respectively. If she matches, she must pay the value of the outside oﬀer to the agent to retain

him. If she lets him leave, she incurs cost c of replacing the agent. Comparing (3.3) and

(3.4), we can see that when the agent exerts low eﬀort, the principal will match an outside

oﬀer when

α < c.

(3.5)

When the agent exerts eﬀort, he develops human capital. The project succeeds with
probability pH , and fails with probability 1 − pH . The principal’s payoﬀ from high eﬀort,

when she chooses “Let Leave” and “Match” in the second period are

L = pH (πS − wS − c) + (1 − pH )(πF − wF + v)
U H

and

M = pH (πS − wS + v − α) + (1 − pH )(πF − wF + v),
U H

(3.6)

(3.7)

respectively. Comparing (3.6) and (3.7), we can see that when the agent exerts eﬀort, the

principal will play “Match” if and only if

α < c + v.

(3.8)

80

3.3 Analysis

The principal’s decision of which eﬀort level to implement depends on the size of the outside

oﬀer that the agent receives when he is successful, the agent’s cost of exerting high eﬀort,

and the diﬀerence in expected project payoﬀs when the agent exerts high and low eﬀort. To

implement high eﬀort, the principal must satisfy the following incentive and participation

constraints:

pH wS + (1 − pH )wF + pH α − ψ ≥ pLwS + (1 − pL)wF + pLα

pH wS + (1 − pH )wF + pH α − ψ ≥ 0.

(3.9)

(3.10)

We derive the condition for self-motivation from the agent’s incentive constraint, (3.9). When

the agent is self-motivated, (3.9) is satisﬁed in the absence of a wage (i.e. when wS = wF = 0):

pH α − ψ ≥ pLα.

(3.11)

From (3.11) follows the deﬁnition of a self motivated agent:

Deﬁnition 3.1. The agent is self motivated when the outside oﬀer is large enough that the

agent prefers to exert high eﬀort in the absence of a wage. That is, when

⇒ α ≥ ψ
∆p

.

(3.12)

This is the opposite of the canonical contracting problem under moral hazard, in which

eﬀort is costly and the agent must be oﬀered an incentive wage to exert eﬀort. In such a case,

the principal rewards success and punishes failure when she wants the agent to exert eﬀort,

81

since success is more likely when the agent exerts eﬀort. If she wants the agent to shirk, she

oﬀers a wage that is the same whether the project succeeds or fails, and just satisﬁes the

agent’s participation constraint. When the agent is self-motivated, she must punish success

and reward failure if she wants the agent to shirk, while she can oﬀer a ﬂat wage if she wants

the agent to exert eﬀort. The agent’s incentive and participation constraints to implement

low eﬀort are as follows:

pLwS + (1 − pL)wF + pLα ≥ pH wS + (1 − pH )wF + pH α − ψ

pLwS + (1 − pL)wF + pLα ≥ 0.

(3.13)

(3.14)

If both (3.9) and (3.10) hold with equality, the optimal (unlimited liability) wage schedule

to induce high eﬀort is given by1

(1 − pL)

∆p

ψ − α,

wS =

wF =

−pL
∆p

ψ.

(3.15)

(3.16)

Since the agent has limited liability, however, the principal must pay the agent a non-

negative wage. When the agent is self motivated, (3.15) and (3.16) are negative. Therefore,

the principal implements the following wage schedule to induce high eﬀort:

w∗
S = w∗

F = 0.

(3.17)

Substituting these wages into (3.7) and (3.6), the principal’s equilibrium proﬁt when the

1This is also the optimal unlimited liability wage schedule to induce low eﬀort.

82

agent exerts eﬀort and she matches the outside oﬀer is

M = pH (πS + v − α) + (1 − pH )(πF + v),
U H

(3.18)

and when she lets the agent leave,

L = pH (πS − c) + (1 − pH )(πF + v).
U H

(3.19)

To induce low eﬀort, the best the principal can do is set the wage the agent receives upon

a success equal to zero, and make the agent’s incentive constraint (3.13) bind. This results

in the following wage schedule:

ws = 0,
wf = α − ψ
∆p

.

(3.20)

Substituting these wages into (3.4) and (3.3) the equilibrium proﬁt for the principal when

the agent exerts low eﬀort and she matches the outside oﬀer is

M = pL(πS − α) + (1 − pL)
U L

πF − α +

ψ
∆p

(cid:18)

(cid:19)

,

(3.21)

(3.22)

and when she lets the agent walk,

L = pL(πS − c) + (1 − pL)
U L

(cid:18)

πF − α +

ψ
∆p

(cid:19)

.

In what follows, we break the analysis up based on the size of α. In Region 1 of Figure

3.2, α is small enough so that principal matches an outside oﬀer whether the agent exerts

83

high or low eﬀort; that is, Region 1 is deﬁned by α < c. In Region 2, the principal matches

an outside oﬀer if the agent exerts high eﬀort, but not if the agent exerts low eﬀort; that is,
Region 2 is deﬁned by α ∈ [c, c+v]. In Region 3, the principal lets the agent walk, regardless

of ﬁrst period eﬀort; that is, Region 3 is deﬁned by α > c + v.

We study the principal’s decision making in each region. Speciﬁcally, we are interested

in situations in which the principal induces a self-motivated agent to shirk. We call this

occurrence countervailing incentives:

Deﬁnition 3.2. The contracting problem between the principal and the agent suﬀers from

countervailing incentives when the agent is self motivated, but the principal’s proﬁt is

maximized by inducing the agent to shirk.

A necessary condition for countervailing incentives to arise is that the agent is self mo-

tivated. Therefore, in the analysis to follow, we restrict attention in each Region to outside
oﬀers α ≥ ψ

∆p. In Figure 3.2, the function α(ψ) gives, for every cost of eﬀort ψ, the outside

oﬀer that just makes the agent self-motivated. Therefore, for α < α(ψ), the agent with cost
of eﬀort ψ must be incentivized to exert high eﬀort. For α ≥ α(ψ), the agent with cost ψ is

self motivated, and the principal must reward failure to induce the agent to exert low eﬀort.

In each Region, the analysis to follow is restricted to outside oﬀers large enough that the

agent is self motivated (that is, outside oﬀers that lie above α(ψ).

3.3.1 Existence of countervailing incentives, Region 1

In this sub-section, characterize the existence of countervailing incentives in Region 1. Recall

that Region 1 captures all outside oﬀers that are less than the cost of replacing the agent,

α < c.

84

α

Region 3

Region 2

Region 1

α(ψ) := ψ
∆p

c + v

c

0
Figure 3.2: The agent is self motivated for α ≥ α(ψ).

ψ

Given that α < c, the largest cost of eﬀort that the agent can possess and still face

countervailing incentives in Region 1 is

ψc := c · ∆p.

(3.23)

Note that ψc is the cost of eﬀort at which α(ψ) = c. If the agent has cost of eﬀort ψc, the

smallest outside oﬀer which makes him self-motivated is α = c. To begin the analysis of

countervailing incentives, consider Table 3.1, which gives the principal’s expected revenues

(row 1) and expected costs (row 2) when the agent exerts high eﬀort.

Period 1

pH πS + (1 − pH )πF

0

Period 2

v

pH α

E[R]
E[C]

Table 3.1: Expected revenues and expected costs when e = eH

In Period 1, the project either succeeds or fails. Because the agent is self motivated, the

85

principal does not have to pay the agent to exert high eﬀort (see (3.17)). Therefore, expected

costs in the ﬁrst period are equal to zero. In Period 2, the principal obtains the beneﬁt of

the human capital payoﬀ, v, because the agent exerted high eﬀort. With probability pH , the

agent receives an outside oﬀer α. Since this outside oﬀer is less than the cost of replacing the

agent, the principal decides to match the outside oﬀer and retain the agent. Thus, expected

costs in Period 2 are pH α.

Period 1

pLπS + (1 − pL)πF
(1 − pL)(α − ψ
∆p)

Period 2

0

pLα

E[R]
E[C]

Table 3.2: Expected revenues and expected costs when e = eL

When the agent exerts low eﬀort (see Table 3.2), both ﬁrst and second period expected

revenues decrease. First period expected revenue decreases because the project is less likely

to be successful, and second period expected revenue decreases because the agent does not

develop human capital when he exerts low eﬀort. Expected costs increase in the ﬁrst period

and decrease in the second period, relative to when the agent exerts high eﬀort. The principal

has to reward failure in order to incentivize low eﬀort; therefore, when the project fails, the
principal pays the agent wf = α − ψ/∆p. However, she is less likely to have to match an

outside oﬀer, which decreases expected second period costs from pH α to pLα.

Expected revenues are lower when the agent exerts low eﬀort than when the agent exerts

high eﬀort. Therefore, it is necessary for expected costs under low eﬀort to be lower than

expected costs under high eﬀort for the principal’s expected proﬁts under low eﬀort to be

higher than her expected proﬁts under high eﬀort. The left hand side of the ﬁrst inequality

in (3.24) is the principal’s expected cost of inducing low eﬀort, and the right hand side is

86

the expected cost when the agent exerts high eﬀort:

pLα + (1 − pL)(α − ψ
∆p
⇒ α <

) < pH α

1 − pL
1 − pH

ψ
∆p

.

(3.24)

Since pH > pL, we know that the probability of failure conditional on low eﬀort is larger

than the probability of failure conditional on high eﬀort. Thus, the cost of inducing low eﬀort

is lower than the cost of inducing high eﬀort when the agent is “not too self motivated,”

that is, when

α ∈

(cid:20) ψ

∆p

1 − pL
1 − pH

,

ψ
∆p

(cid:21)

.

(3.25)

Recall that countervailing incentives exist when the agent is self-motivated and the princi-

pal’s expected proﬁts from low eﬀort are higher than her expected proﬁts from low eﬀort.

Therefore, countervailing incentives will exist for some sub-set of

(cid:104) ψ

∆p ,

(cid:105)

.

1−pL
1−pH

ψ
∆p

From (3.18) and (3.21), the principal’s expected proﬁts under low eﬀort are higher than

her expected proﬁts under high eﬀort when

U H
M < U L

M ⇒ α <

1 − pL
1 − pH

ψ
∆p

− ∆p∆π + v
1 − pH

=: α1(ψ).

(3.26)

We refer to α1(ψ) as the principal’s Region 1 decision rule. This decision rule reﬂects the

principal’s proﬁt maximizing allocation of eﬀort. For every value of ψ, it gives the outside

oﬀer that makes the principal indiﬀerent between high and low eﬀort. For α > α1(ψ), the

principal induces high eﬀort, and for α < α1(ψ), the principal induces low eﬀort.

For countervailing incentives exist in Region 1, it must be the case that the principal

87

prefers low eﬀort, but the agent is self motivated. Therefore, it must be the case that

1 − pL
1 − pH

<

ψ
∆p
⇒ ψ > ∆p∆π + v =: ˆψ.

ψ
∆p

− ∆p∆π + v
1 − pH

(3.27)

The above condition is necessary for countervailing incentives to exist in Region 1, but

it is not suﬃcient. Cost of eﬀort ˆψ := ∆p∆π + v is the smallest cost of eﬀort for which it

is possible that an outside oﬀer makes the principal indiﬀerent between high and low eﬀort

proﬁts, and the agent self motivated. Put diﬀerently, for every cost of eﬀort less than ˆψ,

whenever the outside oﬀer is low enough to make low eﬀort proﬁts optimal, the agent is not

self motivated.

Consider further the meaning of the necessary condition in (3.27). When the agent exerts

high eﬀort, expected project revenues increase by ∆p∆π. Further, the agent develops human

capital, which is valuable to the principal. Therefore, what (3.27) shows is that the agent’s

cost of eﬀort must be high large enough to outweigh this increase in expected revenues. This

reasoning also explains why (3.27) is not suﬃcient for countervailing incentives to exist in

Region 1. As discussed above, the largest cost of eﬀort in Region 1 is ψc. Therefore, for

countervailing incentives to exist in Region 1, it must be the case that ˆψ < ψc. The following

proposition provides a suﬃcient condition for this to be the case.

Proposition 3.1. If the cost associated with replacing the agent is large enough, then coun-

tervailing incentives exist in Region 1. That is, if

c ≥ ∆p∆π + v

∆p

,

88

(3.28)

then countervailing incentives exist in Region 1.

Proof. Suppose c ≥ ∆p∆π+v

∆p

. Then

∆pc ≥ ∆p∆π + v
⇒ ψc ≥ ˆψ.

(3.29)

Notice that ˆψ represents the intersection between α1(ψ) and the principal’s Region 1 decision

rule. Since ˆψ < ψc, this intersection occurs in Region 1. Since this intersection occurs in
Region 1, for ψ ∈ [ ˆψ, ψc] there exist outside oﬀers α < c for which the agent is self motivated

and the principal induces low eﬀort (see Figure 3.3).

α(ψ)

α1(ψ)

α(ψ) = ψ
∆p

c

ψ

Region 1

0

ˆψ

ψc

Figure 3.3: Countervailing incentives, Region 1

Proposition 3.1 shows that the cost of replacing the agent must be large enough for

countervailing incentives to occur in Region 1. This is true even though the principal never

89

incurs the cost of replacing the agent for outside oﬀer α < c. The larger is c, however, the

larger are the outside oﬀers contained in Region 1. When c is large enough, the cost of eﬀort

that makes the principal indiﬀerent between high and low eﬀort, ˆψ, is less than the largest

cost of eﬀort in Region 1, ψc.

This is illustrated in Figure 3.3. The principal induces the agent to shirk in the shaded
region. In this shaded area, the agent with cost of eﬀort ψ ∈ [ ˆψ, ψc] is self motivated, because

the outside oﬀer is larger than α(ψ). Further, the principal’s proﬁt maximizing eﬀort choice

is low eﬀort because α < α1(ψ).

When countervailing incentives exist, it is not because high eﬀort wage payments are

prohibitively expensive; the agent is self motivated, so it is costless for the principal to

induce high eﬀort. Recall that countervailing incentives exist only if the agent is not too self

motivated (see (3.25)). When the agent is not too self motivated, the cost of inducing low

eﬀort is small. When the cost of replacing the agent is large, the outside oﬀers in Region

1 are large. Therefore, countervailing incentives exist in Region 1 when the outside oﬀer is

close to the cost of replacing the agent (i.e. when α is close to c) and the cost of inducing

low eﬀort is small (i.e. the agent is not too self motivated).

3.3.2 Countervailing incentives, Regions 2 and 3

In Region 2, the principal matches the outside oﬀer only if the agent exerted eﬀort and

developed human capital. If the agent was successful despite shirking, she lets the agent

leave in the second period. The principal’s proﬁts from high and low eﬀort are given by

90

(3.18) and (3.22), respectively.2 Therefore, the principal induces low eﬀort if

(cid:20)1 − pL

∆p

(cid:21)

U H
M < U L

L =⇒ α >

1

1 − pH − pL

ψ − ˆψ − pLc

=: α2(ψ).

(3.30)

Above, α2(ψ) represents the principal’s Region 2 decision rule; the principal implements low
eﬀort when α ≥ α2(ψ), and high eﬀort when α < α2(ψ).

In Region 3, the outside oﬀer is so large that the principal lets the agent leave, regardless

of his ﬁrst period eﬀort choice. Therefore, the principal’s proﬁts from high and low eﬀort

are given by (3.19) and (3.22), respectively. The principal induces low eﬀort if

U H
L < U L

L =⇒ α <

ψ
∆p

+ (ψc − ˆψ) + pH v =: α3(ψ).

(3.31)

Once again, we refer to α3(ψ) as the principal’s Region 3 decision rule. The principal
implements low eﬀort if α ≤ α3(ψ), and implements high eﬀort if α > α3(ψ).

We state without formal proof that if countervailing incentives exist in Region 1, they

exist in Regions 2 and 3. The intuition is easy to see from the principal’s decision rule

in Region 3. Recall that countervailing incentives exist in Region 1 if ψc > ˆψ. From

(3.31), if ψc > ˆψ, then α3(ψ) lies above α(ψ).
If α(ψ) < α3(ψ), then for outside oﬀers
α ∈ [α(ψ), α3(ψ)], the agent is self motivated and the principal induces low eﬀort. Therefore,

if countervailing incentives exist in Region 1, they exist in Region 3.

The principal’s behavior can be explained in a similar manner to her behavior in Region 1.

The wage-cost of inducing high eﬀort is zero, but on the chance that the agent is successful,

the principal must pay cost c to replace the agent. When the agent is not too self motivated,

2It is assumed that the average probability of success is greater than one half. That is, 1 < pH + pL.

This ensures the decision rule in Region 2 is downward sloping.

91

the cost of inducing low eﬀort is small. By inducing low eﬀort, the principal reduces the

probability that she has to replace the agent. Thus, when c is large enough, the principal

beneﬁts from inducing the agent that is not too self motivated to shirk.

In Region 2, the intuition is less clear. Notice that the principal’s decision rule in Region
2 is negatively sloped, since 1 − pH − pL < 0. The reason we study this case is because it

creates a non-monotonicity in the agent’s proﬁt maximizing allocation of eﬀort, which we

examine in more detail below.

For the purpose of showing that countervailing incentives exist in Region 2, we can make

an argument similar to the one for the existence of countervailing incentives in Region 1. If

countervailing incentives exist in Region 1, we can show that the smallest cost of eﬀort in

Region 2 for which countervailing incentives can exist, which we will denote ˇψ, is less than

the largest cost of eﬀort in Region 2, which we denote ψc+v.

Notice that ˇψ is the Region 2 equivalent of ˆψ. For every cost of eﬀort less that ˇψ, the

principal will induce high eﬀort in Region 2. Similarly, ψc+v is the Region 2 equivalent of

ψc. When the agent has cost of eﬀort ψc+v, the smallest outside oﬀer which makes him
self motivated is α = c + v, which is the upper bound on Region 2. For ψ ∈ ( ˇψ, ψc+v),

countervailing incentives occur when

α ∈(cid:16)

max{α3(ψ), c}, c + v

(cid:17)

.

(3.32)

The intuition is the same as in Regions 1 and 3. When the agent is not too self motivated

in Region 2, the principal is better oﬀ inducing low eﬀort.

92

3.3.3 Analysis of contracting for a ﬁxed cost of eﬀort

Now that we have characterized the existence of countervailing incentives in Regions 1, 2,

and 3, and discussed the intuition for why the principal induces a self motivated agent to

shirk, we examine the contracting game between the principal and the agent when the agent’s
cost of eﬀort is ¯ψ ∈ ( ˆψ, ˜ψ). 3 We restrict ourselves to these costs of eﬀort because we are

interested in the non-monotonicity in the principal’s proﬁt maximizing allocation of eﬀort
when 1 − pH − pL < 0.4

α(ψ)

α3(ψ)

c + v

α(ψ) = ψ
∆p

c

ψ

0

ˆψ = ˇψ

¯ψ

˜ψ

Figure 3.4: Countervailing incentives, pH + pL > 1

In the shaded areas of Figure 3.4, the principal induces the agent with cost of eﬀort ¯ψ

to shirk, while in the non-shaded areas, he is allowed to exert eﬀort. As the outside oﬀer

increases, the principal’s proﬁt maximizing choice of eﬀort changes three times. We will

3 ˜ψ is the cost of eﬀort at which α1(ψ) is equal to c
4For graphical simplicity, we assume that ˇψ = ˆψ, however, they need not be equal.

93

examine why the optimal allocation of eﬀort, and thus the decision of whether to allow human
capital acquisition or not, is so sensitive to the size of the outside oﬀer when 1−pH −pL < 0.

We brieﬂy discussed the principal’s proﬁt maximizing eﬀort choice when the outside oﬀer

is in Region 1. Here, we will re-visit that process for ﬁxed cost of eﬀort ¯ψ. We can show that

the principal is actually better oﬀ when the agent fails, regardless of the ﬁrst period eﬀort

choice. Recall that the principal’s payoﬀ, when the agent exerts high eﬀort and fails, is

M|F = πF + v.
U H

When the agent exerts high eﬀort and succeeds, her payoﬀ is

Notice that

M|S = πS + v − α
U H

M|F − U H
U H

M|S = α − ∆π,

(3.33)

(3.34)

(3.35)

which is positive since the agent is self motivated and ¯ψ > ˆψ.

Similarly, the principal is better oﬀ when the agent fails after exerting low eﬀort:

M|F − U L
U L

M|S = (πF − (α − ¯ψ

∆p

)) − (πS − α) =

¯ψ
∆p

− ∆π,

(3.36)

which is also positive, since ¯ψ > ˆψ.

Further, we know that U H

M|S − U L

M|S > 0, since the only diﬀerence between the two

payoﬀs is that the agent develops valuable human capital when he exerts eﬀort, and does

not develop human capital when he shirks. We can also show that U H

M|F − U L

M|F > 0, since

94

the principal must pay to induce low eﬀort, and gets no human capital payoﬀ. Lastly, we

can determine that U L

M|F − U H

M|S > 0, since ¯ψ > ˆψ.

This last ordering may seem counter intuitive; the principal is better oﬀ when she pays

the agent to exert low eﬀort, gain no human capital, and the project fails, than she is when

the agent exerts high eﬀort “for free,” gains human capital, and the project succeeds. This

has to do with the structure of the wage to induce low eﬀort, and the fact that the principal

must match the outside oﬀer once the agent succeeds.

We are left with the following general ranking of the principal’s high and low eﬀort proﬁts

in Region 1, conditional on outcome:

M|S < U H
U L

M|S < U L

M|F < U H

M|F.

(3.37)

The principal is best oﬀ, regardless of ﬁrst period eﬀort choice, when the agent fails. Given

our assumptions on pH and pL, we can now explain the principal’s decision making in Region

1.

We can more clearly see the reasoning if we rearrange the principal’s decision rule. Recall

that in Region 1, the principal’s high and low eﬀort payoﬀs are given by (3.18) and (3.21),

respectively. To determine why the principal induces low eﬀort in an agent who is just
self motivated, and switches to high eﬀort as the outside oﬀer passes α1(·), consider the

expression:

(cid:104)

M − U H
U L

M =

(cid:105) −(cid:104)

∆p(πF − (πS − α))

(cid:105)

.

(1 − pL)(α − ¯ψ

∆p

) + v

(3.38)

The ﬁrst term in brackets in (3.38) is the principal’s beneﬁt of inducing low eﬀort, relative

95

to high eﬀort. We showed in (3.37) that the principal is better oﬀ when the agent fails than

when he succeeds, regardless of eﬀort choice. This is illustrated again here; under low eﬀort,
the project fails with probability (1 − pL), and under high eﬀort, it fails with probability
(1 − pH ). Then the probability of getting the project payoﬀ from failure is increased by
pH − pL when the principal induces low eﬀort, and the probability of getting the payoﬀ from
success, less the cost of matching the outside oﬀer, is decreased by pH − pL. This beneﬁts
the principal because when the cost of eﬀort is greater than ˆψ, if the outside oﬀer is large

enough to make the agent self-motivated, the project payoﬀ from failure is higher than the

project payoﬀ from success, less the cost of matching the outside oﬀer.

The second term in brackets in (3.38) is the principal’s cost of inducing low eﬀort. The

ﬁrst term is the wage cost of inducing low eﬀort, and the second term is the opportunity cost

of not inducing human capital acquisition. Notice that the beneﬁt of inducing low eﬀort,
relative to high eﬀort, is increasing in α at rate pH − pL, while the cost of inducing low eﬀort
is increasing at rate 1 − pL > pH − pL. Thus, eventually, the cost of inducing low eﬀort
outweighs the beneﬁt. This occurs exactly when α = α1( ¯ψ).

As α increases past α1( ¯ψ) in Region 1, the principal maintains high eﬀort. As α crosses

c, the principal will no longer match the outside oﬀer if the agent exerted low eﬀort. Thus,

the principal’s payoﬀ is still given by (3.18) if the agent exerted eﬀort in the ﬁrst period, but

is now given by (3.22) if the agent shirked.

We analyze Region 2 similarly to how we analyzed Region 1. We can obtain a similar

general ordering of payoﬀs in Region 2, conditional on outcome, as we did in Region 1. The

conditional payoﬀs U H

M|S and U H

M|F are unchanged, so we still have U H

M|F > U H

M|S, for the

96

same reasons as before. The principal’s conditional payoﬀs from low eﬀort are now given by:

and

L|S = πS − c
U L

L|F = πF − (α − ¯ψ
U L

∆p

).

(3.39)

(3.40)

The principal’s payoﬀ from low eﬀort, conditional on failure, is still higher than her payoﬀ

from high eﬀort, conditional on success:

L|F − U L
U L

L|S =

¯ψ
∆p

− (α − c) − (∆π) > 0.

(3.41)

This is because, in Region 2, α − c < c + v − c = v, and ¯ψ > ˆψ. Further, we know that

M|S − U L
U H

L|S = c + v − α > 0,

(3.42)

since α < c + v. Recall that in Region 1, U H

M|S − U L

M|S = v does not depend on the outside

oﬀer. In Region 2, proﬁts from success in the high eﬀort state approach proﬁts from success

in the low eﬀort state as α approaches c + v. This has an important eﬀect on the rate at

which the beneﬁt from inducing low eﬀort increases in α.

Additionally, since U L

L|F is the same expression as U L

M|F > U L
L|F
M|S, for the same reasons as in Region 1. Then we are left with a general

M|F , we know that U H

and U L

L|F > U H

ordering similar to the one we had in Region 1:

L|S < U H
U L

M|S < U L

L|F < U H

M|F.

(3.43)

97

Again, the principal is best oﬀ, regardless of eﬀort choice, when the agent fails. We can now

explain the principal’s decision making in Region 2.

We can rearrange the principal’s decision rule to see how the principal’s beneﬁts and

costs from inducing low eﬀort change as α increases inside Region 2:

(cid:104)
∆pπF − (pH (πS − α) − pL(πS − c))

(cid:105) −(cid:2)(1 − pL)(α − ¯ψ

L − U H
U L

M =

(cid:105)

.

) + v

(3.44)

∆p

The ﬁrst term in brackets in (3.44) is the principal’s beneﬁt from inducing low eﬀort relative

to high eﬀort. Again, the fact that the principal is better oﬀ when the agent fails is reﬂected

here. The principal’s probability of getting the project payoﬀ from failure increases by
pH − pL when she induces low eﬀort instead of high eﬀort. The principal’s probability of

getting the payoﬀ from success decreases, but so does her cost of matching the outside oﬀer.

When the agent exerts low eﬀort and succeeds, the principal lets him leave, and pays c < α

to replace him.

The second term in brackets is the principal’s cost of inducing low eﬀort, relative to high

eﬀort, and is the same expression as in Region 1. From (3.44), we see that the principal’s

beneﬁt from inducing low eﬀort relative to high eﬀort is increasing at rate pH in α, while
her cost is increasing at rate 1 − pL. Since 1 − pH − pL < 0 ⇒ pH > 1 − pL, her beneﬁt of

inducing low eﬀort relative to high eﬀort is increasing more rapidly in α than her cost. Once

α > α2(ψ), she will induce low eﬀort.

As α increases, the principal’s payoﬀ from high eﬀort, conditional on success, approaches
the principal’s payoﬀ from low eﬀort conditional on success. Since pH − pL is large enough

that countervailing incentives exist, the principal is better oﬀ shifting away from high eﬀort,

even though her best possible payoﬀ is U H

M|F . This payoﬀ is suﬃciently unlikely that the

98

principal prefers low eﬀort as α increases.

As α enters Region 3, the cost of matching the outside oﬀer becomes too expensive even

if the agent exerted high eﬀort, and developed human capital. The principal’s payoﬀs from

high and low eﬀort, conditional on success, are identical:

L|S = U H
U L

L |S = πS − c.

(3.45)

α <

L|F > U L

A general ranking is now more diﬃcult to give. We have that U L

L|S as long as
∆p + c − ∆π. However, as long as α < α3(ψ), the previous inequality is satisﬁed, and
¯ψ
L|S holds. In Region 3, the principal’s decision rule can be re-written as follows:

L|F > U L
U L

L − U H
U L

L = ∆p(πF − (πS − c)) − ((1 − pL)(α − ¯ψ

∆p

) + (1 − pH )v).

(3.46)

From (3.46) we can see that the principal’s beneﬁt of inducing low eﬀort, relative to high

eﬀort, does not depend on α. Her cost of inducing low eﬀort is decreased by pH v, since if

the agent exerts eﬀort and succeeds, the principal chooses not to match the outside oﬀer,

and does not obtain the human capital beneﬁt. Her cost of inducing low eﬀort is once again
decreasing in α at rate 1− pL. Once α > α3(ψ), inducing low eﬀort is too expensive relative

to letting the agent exert eﬀort.

3.4 Conclusion

We analyze a contracting game between one principal and one agent, when the principal

may hire the agent to exert high or low eﬀort on a project. If the project succeeds, the agent

gets an outside oﬀer, which the principal must match to retain the agent. If the agent exerts

99

high eﬀort, he develops valuable human capital, regardless of the project’s outcome.

The agent is self motivated for outside oﬀers large enough, which implies that the principal

will have to reward failure in order to get the agent to shirk. We show that when the diﬀerence

between the probability of success given high and low eﬀort is large enough, the agent can

have a high enough cost of eﬀort that the principal may ﬁnd it proﬁtable to induce low eﬀort,

instead of letting him exert low eﬀort like he prefers.

When the average probability of success on the project is greater than one-half, this

behavior can create an interesting non-monotonicity in the principal’s proﬁt maximizing

cost of eﬀort. The principal is best oﬀ when the agent fails, so as long as incentivizing low

eﬀort is not too expensive, she will do so.

100

APPENDICES

101

Appendix A

High cost type’s ﬁrst period incentive constraint

Given the expression for the low and high cost ﬁrm’s equilibrium eﬀorts, one can verify that

the high cost ﬁrm’s incentive constraint is satisﬁed in suﬃciently noisy environments. Since

the high cost type’s participation constraint binds in expectation, it is suﬃcient to check

(A.1)

(A.2)

(cid:21)

.

(A.3)

that

(cid:0) ¯β − c1

(cid:1)2 ≤ 0.

t1 − γ
2

Substituting for t1 from (1.28) and simplifying, this requires

(cid:90)

δ

U 2(ρ2)(¯g − g)dc1 ≤ ¯c1 − c1.

γ∆β

R

Now, from (1.33) and (1.34),1

¯c1 − c1 =

1 + λ − ρ

(1 − ρ)(1 + λ)

∆β

δ

+

γρ(1 − ρ)(1 + λ)
(cid:21)

(cid:90)
R U 2(ρ2)(¯g − g)dc1 − E[W2]

1And using the fact that

(cid:20)

d
d¯c1

ρλ

U 2(ρ2)(¯g − g)dc1 − E[W2]
(cid:21)
(cid:90)
R U 2(ρ2)(¯g − g)dc1 − E[W2]

(cid:90)

(cid:20)

ρλ

d
dc1

R

(cid:20)

ρλ

= − d
dc1

102

Thus, the high cost ﬁrm’s incentive constraint is satisﬁed when

1 + λ − ρ

(1 − ρ)(1 + λ)

∆β ≥

δ

γρ(1 − ρ)(1 + λ)
−

γ(1 − ρ)(1 + λ)

δλ

E[W2]

(cid:90)

d
dc1
d
dc1

R
U 2(ρ2)(¯g − g)dc1.

(cid:90)

+ δ

R

U 2(ρ2)(¯g − g)dc1

(A.4)

From Proposition 4,

d
dc1

is suﬃciently large,

E[W2] < 0. Therefore, it must be checked that when the variance

(cid:90)

R

U 2(ρ2)(¯g − g)dc1 + δ

(cid:90)

R

U 2(ρ2)(¯g − g)dc1 ≈ 0.

(A.5)

−

δλ

γ(1 − ρ)(1 + λ)

d
dc1

From Proposition 3,

(cid:90)

d
dc1

R

U 2(ρ2)(¯g − g)dc1 =

(cid:90) ∞

c0
1

du2
dρ2

k

(cid:104)
g(cid:48)¯g2 − g2¯g(cid:48)(cid:105)

(cid:90) c0

1
−∞

du0
2
dρ2

k

(cid:104)

g(cid:48)¯g2 − g2¯g(cid:48)(cid:105)

dc1 +

dc1.

(A.6)

As the variance of ﬁrst period cost increases, the slope of the density goes to zero. As the

(cid:82)R U 2(ρ2)(¯g − g)dc1.
Turning attention to(cid:82)R U 2(ρ2)(¯g − g)dc1, integration by parts yields

slope of the density goes to zero, so too does d
dc1

U 2(ρ2)(¯g − g)dc1 = −

1
−∞

du0
2
dρ2

dρ2
dc1

(cid:34)(cid:90) c0

(cid:2) ¯G − G(cid:3) dc1 +

(cid:90) ∞

c0
1

du2
dρ2

dρ2
dc1

(cid:2) ¯G − G(cid:3) dc1

(cid:35)

. (A.7)

(cid:90)

R

Since

dρ2
dc1

=

ρ(1−ρ)[g(cid:48)¯g−g¯g(cid:48)]

D2

goes to zero as the slope of the density goes to zero, this term

is close to zero when the variance is large. Thus, the high cost type’s incentive constraint is

satisﬁed in noisy enough environments.

103

Proof of Proposition 1.3

Proof. Consider the expression for the low cost type’s ﬁrst period eﬀort given by (1.33).

Abstracting from the eﬀect of the ﬁrst period contract on expected second period welfare,

the low cost type’s equilibrium ﬁrst period eﬀort is less than in a deterministic separating

equilibrium (that is, less than 1

γ , the ﬁrst best) when (1.39) is true. To show that (1.39)

holds, consider

(cid:90)

d
dc1

R

U2(ρ2)(¯g − g)dc1 =
(cid:90) c0

=

1
−∞

du0
2
dρ2

d
dc1

dρ2
dc1

1

(cid:34)(cid:90) c0
−∞ u0
(¯g − g) + u0

(cid:90) ∞
2(¯g − g)dc1 +
(cid:90) ∞

2g(cid:48)dc1 +

c0
1

(cid:35)

u2(¯g − g)dc1

du2
dρ2

(¯g − g) + u2g(cid:48)dc1.

(A.8)

dρ2
dc1

c0
1

Integrate the second term under each integral on the right hand side of (A.8) by parts. Doing

so yields

(cid:90) c0

1
−∞

du0
2
dρ2

(cid:20)dρ2

dc1

¯g −

(cid:18) dρ2

dc1

+

dρ2
dc1

(cid:19)
(cid:21)
(cid:90) ∞

g

+

c0
1

(cid:12)(cid:12)(cid:12)c0
(cid:20)dρ2

1
−∞
¯g −

dc1

dc1 + u0
2g

du2
dρ2

(cid:18) dρ2

dc1

(cid:19)

(cid:21)

g

dc1 + u2g(cid:12)(cid:12)∞

c0
1

+

dρ2
dc1

.

(A.9)

Now,

and

(A.10)

(A.11)

−ρ(1 − ρ)g(cid:48)¯g

D2

,

dρ2
dc1

=

ρ(1 − ρ)[g(cid:48)¯g − g¯g(cid:48)]

D2

,

dρ2
dc1

=

104

where D = ρg + (1 − ρ)¯g. Thus,

dρ2
dc1

+

dρ2
dc1

=

−ρ(1 − ρ)g¯g(cid:48)

D2

.

(A.12)

Further,

u0
2g

(cid:12)(cid:12)(cid:12)c0
−∞ + u2g(cid:12)(cid:12)∞

1

c0
1

(cid:104)

= g(c0
1)

(cid:105)

2(ρ0
u0

2) − u2(ρ0
2)

.

(A.13)

When ρ2 = ρ0

2, it is easily veriﬁed that

u0
2(ρ0

2) =

γ
2

∆β2 = u2(ρ0
2).

(A.14)

After substituting for the relevant terms and simplifying, (A.9) becomes

(cid:90) c0

1
−∞

du0
2
dρ2

k

(cid:104)
g(cid:48)¯g2 − g2¯g(cid:48)(cid:105)

dc1 +

(cid:90) ∞

c0
1

du2
dρ2

k

(cid:104)
g(cid:48)¯g2 − g2¯g(cid:48)(cid:105)

dc1,

(A.15)

where k =

Because

−ρ(1−ρ)

.

D2
du2
dρ2

< 0 and

du0
2
dρ2

< 0, to show that

g(cid:48)¯g2 − g2¯g(cid:48) < 0,

∀ c1,

(A.16)

it is suﬃcient to show that the above integrals are negative over their respective limits of

integration. This follows from the monotone likelihood ratio property (see, e.g., the proof of

105

Theorem 2 in Jeitschko and Mirman (2002)). Thus,

d
dc1

ρλ

R

U2(ρ2)(¯g − g)dc1

(cid:34)(cid:90) c0

= ρλ

1
−∞

du0
2
dρ2

k

(cid:104)
g(cid:48)¯g2 − g2¯g(cid:48)(cid:105)

dc1 +

(cid:90) ∞

c0
1

du2
dρ2

k

(cid:104)
g(cid:48)¯g2 − g2¯g(cid:48)(cid:105)

(cid:35)

dc1

< 0,

(A.17)

and the low cost ﬁrm’s ﬁrst period eﬀort is decreased. A similar proof shows that

(cid:90)

(cid:20)(cid:90)

d
d¯c1

R

U2(ρ2)(¯g − g)dc1

= − d
dc1

U2(ρ2)(¯g − g)dc1

> 0.

(A.18)

(cid:21)

(cid:20)(cid:90)

R

(cid:21)

Thus, the eﬀect of the dynamic portion of the low cost ﬁrm’s ﬁrst period transfer is to

decrease the distance between cost targets, and reduce how much the regulator updates her

prior for any given cost realization.

Proof of Lemma 1.1

Proof. From the perspective of the second period, expected second period welfare is given

by (1.22). When c1 > c0

1, welfare can be expressed as

w2 = argmax

e2, ¯e2

S − ρ2

(cid:16)

(e2)2(cid:17)
(cid:16)
(¯e2)2(cid:1).
− (1 − ρ2)(1 + λ)(cid:0) ¯β − ¯e2 +

β − e2 +

(1 + λ)

γ
2

+ λu2

(cid:17)

(A.19)

γ
2

106

By the envelope theorem,

dw2
dρ2

= −(1 + λ)(cid:0)β − e2(ρ2) +
= (1 + λ)(cid:0)∆β +

(e2(ρ2))2(cid:1) − λu2(¯e2(ρ2)) + (1 + λ)(cid:0) ¯β − ¯e2(ρ2) +

(cid:1) − λu2(¯e2(ρ2)) − (1 + λ)(cid:0)¯e2(ρ2) − γ

(¯e2(ρ2))2(cid:1).

γ
2

(¯e2(ρ2))2(cid:1)

γ
2

(A.20)

1
2γ

2

Thus,

d2w2
dρ2
2

= −λ

du2
d¯e2

d¯e2
dρ2

− (1 + λ)(1 − γ¯e2(ρ2))

d¯e2
dρ2

> 0,

(A.21)

d¯e2
dρ2

du2
d¯e2

> 0 and

since
implies (1− γ¯e2(ρ2)) > 0. Because (1− γ¯e0
is identical for w0

2. Thus, information is valuable.

2) > 0 and

< 0, and the high cost type’s eﬀort is less than the ﬁrst best, which

du0
2
d¯e0
2

> 0 and

d¯e0
2
dρ2

< 0 as well, the proof

Proof of Proposition 1.4

Proof. From the perspective of the ﬁrst period,

(cid:90) c0
−∞ w0

1

2

(cid:2)ρg + (1 − ρ)¯g(cid:3) dc1 +

(cid:90) ∞

c0
1

(cid:2)ρg + (1 − ρ)¯g(cid:3) dc1.

w2

E[W2(ρ2)] =

First, consider

dE[W2(ρ2)]

dc1

(cid:90) c0

1
−∞

=

dw0
2
dρ2

dρ2
dc1

(cid:2)ρg + (1 − ρ)¯g(cid:3) − w0

(cid:90) ∞

c0
1

+

2ρg(cid:48)dc1
dρ2
dc1

dw2
dρ2

(cid:2)ρg + (1 − ρ)¯g(cid:3) − w2ρg(cid:48)dc1.

107

(A.22)

(A.23)

Integrate the second term under each integral by parts. Doing so yields

(cid:90) c0

1
−∞

dw0
2
dρ2

(cid:20)(cid:18) dρ2

dc1

+

(cid:19)
(cid:90) ∞

dρ2
dc1

+

c0
1

Now,

− w0

2ρg

(cid:21)

(cid:12)(cid:12)(cid:12)c0

(1 − ρ)¯g

dρ2
dc1

(cid:20)(cid:18)dρ2

dc1

ρg +

dw2
dρ2

+

dρ2
dc1

(cid:19)
dc1 − w0

ρg +

2ρg

dρ2
dc1

(cid:21)

1
−∞
(1 − ρ)¯g

dc1 − w2ρg(cid:12)(cid:12)∞

c0
1

.

(A.24)

(cid:12)(cid:12)(cid:12)c0
−∞ − w2ρg(cid:12)(cid:12)∞

1

(cid:12)(cid:12)(cid:12)c0

1

+ w2ρg(cid:12)(cid:12)c0

1

= 0.

(A.25)

,

dρ2
dc1

=

ρ(1−ρ)[g(cid:48)¯g−g¯g(cid:48)]

D2

, and

dρ2
dc1

+

dρ2
dc1

=

2ρg

= − w0

c0
1
−ρ(1−ρ)g(cid:48)¯g

D2

From the proof of Proposition 3,
−ρ(1−ρ)g¯g(cid:48)

.

D2

dρ2
dc1

=

Substituting the above into (A.24) yields

(cid:90) c0

1
−∞

dw0
2
dρ2

(cid:34)−ρ(1 − ρ)g¯g(cid:48)

D2

(cid:35)
(cid:34)−ρ(1 − ρ)g¯g(cid:48)

(1 − ρ)¯g

ρg − ρ(1 − ρ)g(cid:48)¯g
(cid:90) ∞

D2

dc1

dw2
dρ2

+

c0
1

ρg − ρ(1 − ρ)g(cid:48)¯g

D2

D2

(cid:35)

(1 − ρ)¯g

dc1

(A.26)

(cid:34)(cid:90) c0

1
−∞

dw0
2
dρ2

= −

2(1 − ρ)¯g(cid:48)dc1 +
(cid:34)(cid:90) ∞
ρ2

−

c0
1

(cid:90) c0

1
−∞

(cid:35)
(1 − ρ2)2ρg(cid:48)dc1
(cid:90) ∞

dw0
2
dρ2
2(1 − ρ)¯g(cid:48)dc1 +
ρ2

dw2
dρ2

dw2
dρ2

c0
1

(cid:35)

.

(A.27)

(1 − ρ2)2ρg(cid:48)dc1

108

Using the fact that (1 − ρ2)2 = 1 − ρ2 − ρ2(1 − ρ2), re-write (A.27) as

(cid:90) c0

1
−∞

−

(cid:2)ρ2(1 − ρ)¯g(cid:48) − ρ(1 − ρ2)g(cid:48)(cid:3) dc1 −
(cid:90) ∞

(cid:90) c0
(cid:2)ρ2(1 − ρ)¯g(cid:48) − ρ(1 − ρ2)g(cid:48)(cid:3) dc1 −

1
−∞

ρ2

ρ2

dw0
2
dρ2
−

dw2
dρ2

c0
1

dw0
2
dρ2

(1 − ρ2)ρg(cid:48)dc1
(cid:90) ∞

dw2
dρ2

c0
1

(1 − ρ2)ρg(cid:48)dc1.

Since ρ2 =

ρg

D and 1 − ρ2 = (1−ρ)¯g
D ,

ρ2(1 − ρ)¯g − ρ(1 − ρ2)g(cid:48) =

ρ(1 − ρ)

D

[¯g(cid:48)g − ¯gg(cid:48)] = −dρ2
dc1

D.

(A.28)

(A.29)

Thus, (A.28) becomes

dw0
2
dρ2

dρ2
dc1

ρ2Ddc1 −

(cid:90) c0

1
−∞

dw0
2
dρ2

+

(1 − ρ2)ρg(cid:48)dc1
(cid:90) ∞

dw2
dρ2

dρ2
dc1

c0
1

ρ2Ddc1 −

(cid:90) ∞

c0
1

dw2
dρ2

(1 − ρ2)ρg(cid:48)dc1.

(A.30)

(1 − ρ2)ρg(cid:48)dc1

dw0
2
dρ2

=

dw0
2
dρ2

(1 − ρ2)ρg

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)c0

1

−∞

(cid:32)

(cid:90) c0

1
−∞

−

d2w0
2
dρ2
2

dρ2
dc1

(1 − ρ2) − dw0
2
dρ2

dρ2
dc1

(cid:33)

ρgdc1,

(A.32)

109

(cid:90) c0

1
−∞

(cid:90) c0

1
−∞

(cid:90) c0

1
−∞

Once again, use the fact that Dρ2 = ρg, and (A.30) becomes

(cid:90) c0

1
−∞

dw0
2
dρ2

dρ2
dc1

ρgdc1 −

dw0
2
dρ2

(1 − ρ2)ρg(cid:48)dc1
(cid:90) ∞

dw2
dρ2

dρ2
dc1

+

c0
1

(cid:90) ∞

c0
1

dw2
dρ2

ρgdc1 −

(1 − ρ2)ρg(cid:48)dc1.

(A.31)

Integrate the second and fourth integrals in (A.31) by parts:

and

(cid:90) ∞

c0
1

(1 − ρ2)ρg(cid:48)dc1

dw2
dρ2

=

dw2
dρ2

(1 − ρ2)ρg

(cid:12)(cid:12)(cid:12)(cid:12)∞

c0
1

−

(cid:32)

(cid:90) ∞

c0
1

d2w2
dρ2
2

dρ2
dc1

(1 − ρ2) − dw2
dρ2

dρ2
dc1

(cid:33)

ρgdc1.

(A.33)

Substituting back in to (A.31) yields

(cid:90) c0

1
−∞

dE[W2(ρ2)]

dc1

=

d2w0
2
dρ2
2

(1 − ρ2)

dρ2
dc1

ρgdc1 +

(cid:90) ∞

c0
1

(1 − ρ2)

d2w2
dρ2
2
+ (1 − ρ2)ρg

dρ2
dc1

(cid:16) dw2

dρ2

ρgdc1

− dw0
2
dρ2

(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)c0

1

.

(A.34)

Since

dρ2
dc1

< 0 by the monotone likelihood ratio property, by Lemma 1 the integrals are

negative for all c1. It is left to show that, when evaluated at c0
1,

Lemma 1 gives the expression for

, and a similar argument yields

= (1 + λ)(cid:0)∆β +

1
2γ

dw0
2
dρ2

(cid:1) − λu0

2(ρ2) − (1 + λ)(cid:0)¯e0

2(ρ2) − γ
2

2(ρ2))2(cid:1).

(¯e0

dw2
dρ2

− dw0
2
dρ2

= 0.

dw2
dρ2

(cid:105)

Thus, when evaluated at c0
1,

(cid:104)

dw2
dρ2

− dw0
2
dρ2

= λ

u0
2(ρ0

2) − u2(ρ0
2)

+ (1 + λ)

(cid:104)

¯e0
2(ρ0

2) − ¯e2(ρ0

2) +

2))2(cid:105)

(¯e0

2(ρ0

γ
2

(¯e2(ρ0

2))2 − γ
2

110

(A.35)

(A.36)

.

(A.37)

From Proposition 1, u0

2) − u2(ρ0

2(ρ0

2) = 0. Further,

Thus,

and

dE[W2(ρ2)]

dc1

(cid:90) c0

1
−∞

=

d2w0
2
dρ2
2

(1 − ρ2)

dρ2
dc1

ρgdc1 +

(cid:90) ∞

c0
1

A similar proof shows that

(cid:34)(cid:90) c0

dE[W2(ρ2)]

d¯c1

= −

1
−∞

d2w0
2
dρ2
2

(1 − ρ2)

dρ2
dc1

ρgdc1 +

2(ρ0
¯e0

2) = ∆β = ¯e2(ρ0
2).

dw2
dρ2

− dw0
2
dρ2

= 0,

(A.38)

(A.39)

d2w2
dρ2
2

(1 − ρ2)

dρ2
dc1

ρgdc1 < 0.

(A.40)

(cid:90) ∞

c0
1

d2w2
dρ2
2

(1 − ρ2)

dρ2
dc1

(cid:35)

ρgdc1

> 0.

(A.41)

Thus, the eﬀect of expected second period welfare is to increase the distance between the

ﬁrst period cost targets.

Proof of Proposition 1.5

Proof. To prove Proposition 1.5, consider

(cid:90) ∞

c0
1

(cid:35)

(cid:90) c0
−∞ A(cid:48) d¯e0

2
dρ2

1

(cid:90) ∞

c0
1

dρ2
dc1

¯gdc1 +

B(cid:48) d¯e2
dρ2

dρ2
dc1

¯gdc1,

(A.42)

−∞ A ¯g dc1 +

B ¯g dc1

=

(cid:34)(cid:90) c0

1

d
dc1

where

A(cid:48) = (1 − ρ)(1 + λ) − (1 + λ − ρ)γ¯e0
2,

(A.43)

111

and

First, focus on

Since

B(cid:48) = (1 − ρ)(1 + λ)(1 − γ¯e2) − ρλγ∆β.

(cid:90) c0
−∞ A(cid:48) d¯e0

2
dρ2

1

dρ2
dc1

¯gdc1.

γ¯e0

2 =

(1 − ρ2)(1 + λ)

1 + λ − ρ2

,

(A.44)

(A.45)

(A.46)

it follows that

A(cid:48) =

1 + λ

1 + λ − ρ2

(cid:2)(1 − ρ)(1 + λ − ρ2) − (1 − ρ2)(1 + λ − ρ)(cid:3) =

λ(1 + λ)(ρ2 − ρ)

1 + λ − ρ2

.

(A.47)

Since c1 ≤ c0

1 < ˆc1 (see Figure 1), ρ2 > ρ. Thus , A(cid:48) > 0.

Further, for every c1 ≤ c1, g is increasing, so g(cid:48) ≥ 0 (see Figure 1); thus,

Therefore, since

d¯e0
2
dρ2

< 0 for all c1, and since c0

1 ≤ c1,

for c1 ∈(cid:0)−∞, c0

1

(cid:3).

Now, return attention to

−ρ(1 − ρ)g(cid:48)¯g

D2

dρ2
dc1

=

< 0.

(cid:90) c0
−∞ A(cid:48) d¯e0

2
dρ2

1

dρ2
dc1

¯gdc1 > 0

(cid:90) ∞

c0
1

B(cid:48) d¯e2
dρ2

dρ2
dc1

¯gdc1.

112

(A.48)

(A.49)

(A.50)

Using the deﬁnition of

dρ2
dc1

, (A.50) can be re-written

(cid:90) ∞

c0
1

−ρ
1 − ρ

B(cid:48) d¯e2
dρ2

(1 − ρ2)2g(cid:48)dc1.

(A.51)

Integrating by parts yields

(cid:12)(cid:12)(cid:12)(cid:12)∞

c0
1

B(cid:48) d¯e2
(cid:34)
(cid:90) ∞
dρ2

−

c0
1

(1 − ρ2)2g

−(1 − ρ)(1 + λ)γ

(cid:18) d¯e2

(cid:19)2

dρ2

(cid:34)

(1 − ρ2) + B(cid:48)

d2¯e2
dρ2
2

(1 − ρ2) − 2

d¯e2
dρ2

(cid:35)(cid:35)

(1−ρ2)

dρ2
dc1

gdc1.

(A.52)

Notice that

(1 − ρ2) − 2

d2¯e2
dρ2
2

d¯e2
dρ2

=

−2λ∆β(1 − ρ2)
(1 − ρ2)3(1 + λ)

−

−2λ∆β

(1 − ρ2)2(1 + λ)

= 0.

(A.53)

Thus, (A.51) becomes

(cid:34)

−ρ
1 − ρ

−B(cid:48) d¯e2
dρ2

(1 − ρ2)2g

(cid:12)(cid:12)(cid:12)(cid:12)c0

1

(cid:90) ∞

c0
1

+

(cid:18) d¯e2

(cid:19)2

dρ2

(1 − ρ2)2 dρ2
dc1

(cid:35)

gdc1

.

(A.54)

(1 − ρ)(1 + λ)γ

First, notice that

(cid:90) ∞

c0
1

−ρ
1 − ρ

(1 − ρ)(1 + λ)γ

(cid:18) d¯e2

(cid:19)2

dρ2

(1 − ρ2)2 dρ2
dc1

gdc1 > 0,

(A.55)

since

dρ2
dc1

< 0 for all c1, and every other term under the integral in (A.55) is positive. Now,

113

consider

Since

(A.56) can be simpliﬁed to

ρ
1 − ρ

B(cid:48) d¯e2
dρ2

(1 − ρ2)2g

(cid:12)(cid:12)(cid:12)(cid:12)c0

1

.

d¯e2
dρ2

=

−λ∆β

(1 − ρ2)2(1 + λ)

,

−ρλ∆β

(1 − ρ)(1 + λ)

B(cid:48)(c0

1)g(c0
1).

When evaluated at c1 = c0

1, ¯e2 = ∆β. Thus,

B(cid:48)(c0

1) = (1 − ρ)(1 + λ)

(cid:20)
1 − 1 + λ − ρ
(1 − ρ)(1 + λ)

(A.56)

(A.57)

(A.58)

(cid:21)

γ∆β

,

(A.59)

and (A.56) further simpliﬁes to

−ρλ∆β

(cid:20)

1 − 1 + λ − ρ
(1 − ρ)(1 + λ)

γ∆β

(cid:21)

g(c0

1).

(A.60)

Clearly, the term in brackets in (A.60) is less than one. It is also equal to γ(¯e2(ρ) − ∆β),
where ¯e2(ρ) − ∆β is the low cost ﬁrm’s eﬀort from mimicking the high cost ﬁrm in a static

game in which the regulator’s beliefs are given by ρ. This is assumed to be positive; thus,

the expression given in (A.60) is negative. However, the terms multiplying g(c0

1) are small,

and if g(c0

1) ≈ 0, the term in (A.60) can be ignored in signing the ﬁrst order condition.

114

Thus,

(cid:34)(cid:90) c0

1

d
dc1

−∞ A ¯g dc1 +

(cid:90) ∞

(cid:35)
(cid:90) ∞

c0
1

B ¯g dc1

c0
1
− ρ
1 − ρ

1

≈

(cid:90) c0
−∞ A(cid:48) d¯e0
2
dρ2
(1 − ρ)(1 + λ)γ

¯gdc1

dρ2
dc1

(cid:18) d¯e2

(cid:19)2

dρ2

(1 − ρ2)2 dρ2
dc1

gdc1 > 0,

(A.61)

and the desired result is obtained. A similar proof shows that

(cid:35)

(cid:34)(cid:90) c0

1

(cid:35)

B ¯g dc1

.

(A.62)

(cid:90) ∞

c0
1

(cid:34)(cid:90) c0

1

d
d¯c1

(cid:90) ∞

c0
1

−∞ A ¯g dc1 +

B ¯g dc1

= − d
dc1

−∞ A ¯g dc1 +

Thus, the low cost (high cost) ﬁrm’s eﬀort in the ﬁrst period is below (above) the commitment

optimum, and his eﬀort is increased (decreased) over the course of the interaction with the

regulator.

115

Appendix B

Existence of a reward schedule that implements output

targets in each period

There exists ˆy1 such that the following is an equilibrium reward schedule:



r1(y1) =

(cid:18) y1
(cid:18) y1

θ

θ

(cid:19)2 − δE1[u2(ρ2)| y1],
(cid:19)2 − δE1[u2(ρ2)| y1]+

(cid:34) y2

1

P (y1 ≥ ˆy1)

1 − y2

1

2 + δα

θ

if

y1 ∈ [y1 − η, ˆy1)

(cid:19)

(cid:35)
(u2(ρl) − u2(ρh))

,

if

y1 ∈ [ˆy1, y1 + η]

(cid:18)y1 − y1

2η

To see this, note that the principal punishes output realizations less than y1 − η. Therefore,

the low productivity agent does not shirk. To see that the high productivity agent is strictly

worse oﬀ from targeting y1 /∈ {y1, y1}, note that due to the uniform noise assumption,

the probability that he receives the bonus increases linearly in his choice of output target.

Because his disutility from eﬀort is convex, however, the increase in his expected transfer

from a slight increase in eﬀort is outweighed by the increase in his disutility from eﬀort.

This reward schedule and the second period reward schedule r2(y2) are a straightforward

extension of Proposition 4 in [24]. Therefore, see [24] for further discussion.

116

The low ability agent’s ﬁrst period incentive constraint

is slack

Since the low ability agent’s expected utility from targeting y1 is equal to his outside option

of zero, it suﬃces to show that there exists eta large enough such that

(cid:18) y1

(cid:19)2 ≤ 0.

θ

r1 −

(B.1)

(cid:32)

(cid:33)2(cid:104)

δ
2η

θ2
2

After some algebra, one can show that this is equivalent to

(cid:16)

α

C2(1 − α) − C2(α)

(cid:17) − (1 − α)

(cid:16)

(cid:17)(cid:105)

C2(1 − α) − C2(ρm)

− 1
ρ

2

θ
2

δ
2η

A +

1

1 − ρΘ2

θ2
2

δ
2η

2

A ≤ θ
2

+ C(ρ)

θ2
2

.

(B.2)

The right hand side does not depend on η, and the left hand side goes to zero as η grows.

Proof of Proposition 2.1

Proof. First, note that

(cid:2)α(u2(ρl) − u2(ρh)) − (1 − α)(u2(ρl) − u2(ρm))(cid:3).

d rD
1

d(y1 − y1)

=

δ
2η

Suﬃcient for this term to be positive is

α
1 − α

>

u2(ρl) − u2(ρm)
u2(ρl) − u2(ρh)

.

117

(B.3)

(B.4)

By Corollary 1, the right hand side of (B.4) is always less than one, and the left hand side
is greater than or equal to one for all α ∈ [ 1

2 , 1]. Therefore,

d rD
1
d(y1−y1) > 0 as desired.

Proof of Proposition 2.2

Proof. From (2.41),

d rD
1

d(y1 − y1)

= −δ(1 − α)

2η

(u2(ρl) − u2(ρm)) .

(B.5)

By Corollary 1, u2(ρl) − u2(ρm) > 0, and thus

d rD
1
d(y1−y1) < 0, as desired.

Proof of Lemma 2.2

The principal’s expected second period payoﬀ can be expressed as follows:

(cid:34)

(cid:18) y2

(cid:19)2 −

(cid:18) y2

(cid:19)2

+

θ

(cid:19)2(cid:35)

(cid:18) y2

θ

v2 = argmax
y2, ¯y2

ρ2

y2 −

θ

(cid:34)

(cid:19)2(cid:35)

.

(B.6)

(cid:18) y2

θ

+ (1 − ρ2)

y2 −

By the envelope theorem,

(cid:18) y2(ρ2)

θ

(cid:19)2 −

(cid:18) y2(ρ2)

(cid:19)2

θ

+

(cid:18) y2(ρ2)

θ

(cid:34)

(cid:19)2 −

y2(ρ2) −

(cid:18) y2(ρ2)

(cid:19)2(cid:35)

θ

.

(B.7)

= y2(ρ2) −

dv2
dρ2

In the second period, the optimal output target for the high productivity ﬁrm does not

depend on second period beliefs (y2 = θ

2

/2), and the low productivity worker’s optimal

118

output target as a function of second period beliefs is given by (2.14). Thus,

dv2
dρ2

=

2

θ
4

− C2(ρ2)

θ2
4

(cid:104)

1 − Θ2(cid:105) − C(ρ2)

(cid:20)

θ2
2

1 − C(ρ2)

2

(cid:21)

,

(B.8)

and

d2v2
dρ2
2

= −2C(ρ2)C(cid:48)(ρ2)
= −2C(ρ2)C(cid:48)(ρ2)

θ2
4
θ2
4

(cid:20)
(cid:104)
1 − Θ2(cid:105) − θ2
(cid:104)
1 − Θ2(cid:105) − C(cid:48)(ρ2)

2

C(cid:48)(ρ2)
θ2
2

(cid:18)

1 − C(ρ2)

2

(cid:19)

(cid:21)

− C(ρ2)

C(cid:48)(ρ2)

2

[1 − C(ρ2)] > 0,

(B.9)

since C(cid:48)(ρ2) < 0, and 1 − C(ρ2) > 0. Thus, the principal’s expected second period payoﬀ is

convex in second period beliefs.

Proof of Proposition 2.3

Proof. With E[v2(ρ2)] given in (2.23),

dE[v2(ρ2)]
d(y1 − y1)

=

1
2η

[ρv2(ρh) + (1 − ρ)v2(ρl) − v2(ρm)] .

(B.10)

Suﬃcient for this to be positive is

ρv2(ρh) + (1 − ρ)v2(ρl) ≥ v2(ρm)).

(B.11)

119

Using the deﬁnition of ρh, ρm and ρl,

ρv2(ρh) + (1 − ρ)v2(ρl) = ρv2(α) + (1 − ρ)v2(1 − α)
≥ v2(ρα + (1 − ρ)(1 − α))

= v2(ρm),

(B.12)

where the inequality holds due to Lemma 2.2. Hence the desired result.

Proof of Proposition 2.4

Proof. First, notice that

(cid:16)

(cid:16)
(cid:17)
rS
1 + rD
1
(cid:18) y1 − y1
(cid:19)(cid:16)
1 + (1 − ρ)rS

(cid:17)
+ (1 − ρ)
rS
1 + rD
1
1 − δ(1 − α)u2(ρ2)
ρα(u2(1 − α) − u2(α)) − (1 − α)(u2(1 − α) − u2(ρ2
2))

2η

(cid:17)

.

(B.13)

ρr1 + (1 − ρ)r1 = ρ

= ρrS

+ δ

Therefore,

dE[r1]
d(y1 − y1)

= ρα(u2(ρl) − u2(ρh)) − (1 − α)(u2(ρl) − u2(ρm)),

(B.14)

and suﬃcient for

dE[r1]
d(y1−y1) > 0 is

ρα (u2(ρl) − u2(ρh)) − (1 − α) (u2(ρl) − u2(ρm)) ≥ 0.

(B.15)

120

second period rent

u2(ρ2) = C2(ρ2)

θ2
4

where C(ρ2) =

1−ρ2
1−ρ2Θ2 , and Θ = θ

θ

. Thus,

(cid:104)

1 − Θ2(cid:105)

(cid:104)
1 − Θ2(cid:105)(cid:16)
1 − Θ2(cid:105)(cid:16)

(cid:104)

and

u2(ρl) − u2(ρh) =

u2(ρl) − u2(ρm) =

θ2
4

θ2
4

Thus, we wish to show that

,

(B.16)

(cid:17)

(cid:17)

C2(1 − α) − C2(α)

,

(B.17)

C2(1 − α) − C2(ρ2
2)

.

(B.18)

Recall that for generic second period beliefs ρ2, the high productivity agent receives expected

(cid:16)

C2(1 − α) − C2(α)

(cid:17)

(cid:16)

> (1 − α)

(cid:17)

C2(1 − α) − C2(ρ2
2)

.

(B.19)

ρα

Note that

C2(1 − α) − C2(α) = (C(1 − α) − C(α)) · (C(1 − α) + C(α)) ,

(B.20)

and

C2(1 − α) − C2(ρ2

2) =

(cid:16)

(cid:17) ·(cid:16)

(cid:17)

.

C(1 − α) + C(ρ2
2)

C(1 − α) − C(ρ2
2)

One can show that

C(1 − α) − C(α) =

(1 − Θ2)(2α − 1)

(1 − (1 − α)Θ2)(1 − αΘ2)

(B.21)

(B.22)

121

and

C(1 − α) − C(ρ2

2) =

ρ(1 − Θ2)(2α − 1)
(1 − (1 − α)Θ2)(1 − ρ2

2Θ2)

,

Thus, (B.19) is equivalent to

α

1 − αΘ2 (C(1 − α) + C(α)) ≥ 1 − α
1 − ρ2

2Θ2

(cid:16)

C(1 − α) + C(ρ2
2)

This statement is true if

(cid:34)

C(1 − α)

α

1 − αΘ2 − 1 − α
1 − ρ2

2Θ2

(cid:35)

(cid:34)

+ (1 − α)

(B.23)

(B.24)

≥ 0.

(B.25)

(cid:17)

.

(cid:35)

(1 − αΘ2)2 − 1 − ρ2
(1 − ρ2

α

2

2Θ2)2

First, notice that

α

1 − αΘ2 ≥ 1 − α

1 − αΘ2 ≥ 1 − α
1 − ρ2

2Θ2

.

(B.26)

Both inequalities inequality hold since since α ≥ 1/2 (for the second inequality, note that
α ≥ 1/2 implies that α ≥ ρ2
(α ≥ 1/2 ⇒ α ≥ 1 − ρ2

2, which in turn implies that 1 − αΘ2 ≤ 1 − ρ2

2Θ2). Similar logic

2) shows that

α

(1 − αΘ2)2 ≥

1 − ρ2
(1 − ρ2

2

2Θ2)2

.

(B.27)

Thus,

ρα(u2(1 − α) − u2(α)) − (1 − α)(u2(1 − α) − u2(ρ2

2)) ≥ 0

(B.28)

for every α ∈ [1/2, 1], and

dE[r1]
d(y1−y1) > 0 as desired. The principal can decrease the total

expected transfer by decreasing the distance between the ﬁrst period cost targets.

122

Proof of Proposition 2.5

Proof. First, consider the high productivity agent’s ﬁrst period output target. Suﬃcient for
y1 ≤ yc for all α is to show that A < 0 for all α, where A is given in (2.32). The ﬁrst task

is to show that

A = B ·(cid:16)

where

(1 − ρ)(2 − C(α)) − (1 − ραΘ2)(C(1 − α) + C(ρ2
2))

(cid:17)

,

(B.29)

B :=

ρ(1 − Θ2)2(2α − 1)2
2Θ2)(1 − (1 − α)Θ2)(1 − αΘ2)

(1 − ρ2

.

(B.30)

To see this, note that for generic second period beliefs ρ2, the principal’s second period

payoﬀ, given in (2.21), can be re-written

v2(ρ2) = ρ2

2

θ
4

+ (1 − ρ2)C(ρ2)

θ2
4

.

(B.31)

Thus,

ρv2(α) + (1 − ρ)v2(1 − α) − v2(ρ2
2)

(cid:104)
(cid:104)

=

=

θ2
4
θ2
4

ρ(1 − α)C(α) + (1 − ρ)αC(1 − α) − (1 − ρ2
α(1 − ρ)

(cid:17) − ρ(1 − α)

C(1 − α) − C(ρ2
2)

(cid:16)

(cid:105)
(cid:17)(cid:105)
2)C(ρ2
2)
2) − C(α)

C(ρ2

(cid:16)

.

(B.32)

123

Next, consider

ρα[u2(1 − α) − u2(α)] − (1 − α)[u2(1 − α) − u2(ρ2
2)]

(cid:16)

1 − Θ2(cid:17)(cid:104)

=

θ2
4

ρα(C2(1 − α) − C2(α)) − (1 − α)(C2(1 − α) − C2(ρ2
2))

Ignoring θ2

4 , which factors out, A becomes

(cid:16)

α(1 − ρ)

C(1 − α) − C(ρ2
2)

−(cid:16)

1 − Θ2(cid:17)(cid:104)

(cid:17) − ρ(1 − α)

(cid:16)

(cid:17)
2) − C(α)

C(ρ2

ρα(C2(1 − α) − C2(α)) − (1 − α)(C2(1 − α) − C2(ρ2
2))

(cid:105)

(cid:105)

.

(B.33)

.

(B.34)

Consider the term in brackets on the second line of (B.34):

(cid:16)

= ρ

ρα(C2(1 − α) − C2(α)) − (1 − α)(C2(1 − α) − C2(ρ2
2))

(cid:16)

= ρ

α(C2(1 − α) − C2(α))) − (1 − α)(C2(1 − α) − C2(ρ2
2)

− (1 − ρ)(1 − α)(C2(1 − α) − C2(ρ2
2)

(cid:17)

(cid:17)

α(C2(1 − α) − C2(ρ2

2)) + α(C2(ρ2

2) − C2(α)) − (1 − α)(C2(1 − α) − C2(ρ2
2))

(cid:16)

(cid:17)
− (1 − ρ)(1 − α)(C2(1 − α) − C2(ρ2
2)
2) − C2(α)

C2(1 − α) − C2(ρ2
2)

C2(ρ2

(cid:17)

(cid:16)

+ ρα

= ρ(2α − 1)

(cid:16)

C2(1 − α) − C2(ρ2
2)

(cid:17)

.

(B.35)

− (1 − ρ)(1 − α)

124

Therefore, (B.34) is equivalent to

(cid:16)

α(1 − ρ)

C(1 − α) − C(ρ2
2)
− (1 − Θ2)ρ(2α − 1)

(cid:16)

(cid:17) − ρ(1 − α)
(cid:16)

C(ρ2
C2(1 − α) − C2(ρ2
2)

(cid:17)
(cid:17) − (1 − Θ2)ρα
2) − C(α)
(cid:16)

(cid:16)

(cid:17)
(cid:17)
2) − C2(α)

C2(ρ2

+ (1 − Θ)2(1 − ρ)(1 − α)

C2(1 − α) − C2(ρ2
2)

.

Next, use the fact that x2 − y2 = (x − y)(x + y) combined with the fact that

C(1 − α) − C(ρ2

2) =

ρ(1 − Θ2)(2α − 1)
(1 − (1 − α)Θ2)(1 − ρ2

2Θ2)

and

and (B.36) becomes

C(ρ2

2) − C(α) =

(1 − ρ)(1 − Θ2)(2α − 1)
(1 − αΘ2)(1 − ρ2
2Θ2)

,

(B.36)

(B.37)

(B.38)

ρ(1 − ρ)(1 − Θ2)(2α − 1)

(1 − ρ2

2Θ2)

(cid:16)

(C(1 − α) − C(α))
− ρ2(1 − Θ2)2(2α − 1)2
(1 − (1 − α)Θ2)(1 − ρ2
2Θ2)
− ρ(1 − ρ)(1 − Θ2)(2α − 1)
ρ(1 − ρ)(1 − Θ2)(2α − 1)

(cid:18) α(1 − Θ2)
(cid:18)(1 − α)(1 − Θ2)

(1 − ρ2

2Θ2)

1 − αΘ2 (C(ρ2

C(1 − α) + C(ρ2
2)

(cid:17)

(cid:19)

2) + C(α))

(1 − ρ2

2Θ2)

1 − (1 − α)Θ2)

(C(1 − α) + C(ρ2
2))

(cid:19)

.

(B.39)

(B.40)

+

Notice that

α(1 − Θ2)
1 − αΘ2 = 1 − C(α)

125

and

(1 − α)(1 − Θ2)
1 − (1 − α)Θ2)

= 1 − C(1 − α).

(B.41)

Therefore, the expression in (B.39) becomes

ρ(1 − ρ)(1 − Θ2)(2α − 1)

(1 − ρ2

2Θ2)

ρ(1 − ρ)(1 − Θ2)(2α − 1)

+

(1 − ρ2

2Θ2)

(C(1 − α) − C(α))
− ρ2(1 − Θ2)2(2α − 1)2
(1 − (1 − α)Θ2)(1 − ρ2
2Θ2)
(C(1 − α) − C(α))

(cid:16)
(cid:16)

(cid:17)

C(1 − α) + C(ρ2
2)

1 − C(α) − C(1 − α) − C(ρ2
2)

Combining terms with like denominators, this is equivalent to

ρ(1 − ρ)(1 − Θ2)(2α − 1)

(1 − ρ2

2Θ2)

(cid:16)

(C(1 − α) − C(α))

2 − C(α) − C(1 − α) − C(ρ2
2)

− ρ2(1 − Θ2)2(2α − 1)2
(1 − (1 − α)Θ2)(1 − ρ2
2Θ2)

C(1 − α) + C(ρ2
2)

.

(B.42)

.

(B.43)

(cid:17)

Next, use the fact that

C(1 − α) − C(α) =

(1 − Θ2)(2α − 1)

(1 − (1 − α)Θ2)(1 − αΘ2)

,

(B.44)

and (B.43) becomes

ρ(1 − ρ)(1 − Θ2)2(2α − 1)2
2Θ2)(1 − (1 − α)Θ2)(1 − αΘ2)

(1 − ρ2

2 − C(α) − C(1 − α) − C(ρ2
2)

− ρ2(1 − Θ2)2(2α − 1)2
(1 − (1 − α)Θ2)(1 − ρ2
2Θ2)

C(1 − α) + C(ρ2
2)

.

(B.45)

(cid:17)

(cid:16)

126

(cid:17)

(cid:17)

(cid:17)

(cid:16)

(cid:16)

Deﬁne

B :=

ρ(1 − Θ2)2(2α − 1)2
2Θ2)(1 − (1 − α)Θ2)(1 − αΘ2)

(1 − ρ2

,

(1 − ρ)(2 − C(α) − C(1 − α) − C(ρ2

2)) − ρ(1 − αΘ2)(C(1 − α) + C(ρ2
2))

(B.46)

,

(B.47)

(cid:17)

and (B.45) becomes

B ·(cid:16)

which simpliﬁes to

B ·(cid:16)
A = B ·(cid:16)

(1 − ρ)(2 − C(α)) − (1 − ραΘ2)(C(1 − α) + C(ρ2
2))

.

(B.48)

(cid:17)

Thus,

(1 − ρ)(2 − C(α)) − (1 − ραΘ2)(C(1 − α) + C(ρ2
2))

(cid:17)

.

(B.49)

With this result in hand, notice that B ≥ 0 for all α ∈ [1/2, 1], ρ ∈ (0, 1), and Θ2 ∈ (0, 1).

Therefore, it is left to show that if Θ2 ≥ 1/2 or ρ ≥ 1/3, then

(1 − ρ)(2 − C(α)) − (1 − ραΘ2)(C(1 − α) + C(ρ2

2)) ≤ 0

(B.50)

for all α ∈ [1/2, 1]. Deﬁne

f (α) := (1 − ρ)(2 − C(α)) − (1 − ραΘ2)(C(1 − α) + C(ρ2

2)).

(B.51)

127

Notice that

(cid:18)

1 +

(cid:19)

1 − ρ
1 − ρΘ2

f (1) =(1 − ρ)2 − (1 − ρΘ2)

= − ρ(1 − Θ2) < 0,

(B.52)

2 = ρ when α = 1. Further, when α = 1/2, C(α) =

since C(1) = 0, C(0) = 1, and ρ2
C(1 − α) = C(ρ2

2) = 1

2−Θ2 . Thus,

(cid:18)1

(cid:19)

2

f

(cid:18)

(cid:19)

(cid:19)

(cid:19)(cid:18) 2

2 − Θ2

1 − ρΘ2
2

(cid:18)

−

.

2 − 1

=(1 − ρ)
2 − Θ2
1 − 2Θ2 − 3ρ(1 − Θ2)

=

2 − Θ2

(B.53)

(B.54)

It is clear that f (1/2) < 0 if Θ2 ≥ 1/2. Notice that f (1/2) < 0 is also true if

ρ ≥ 1 − 2Θ2
3(1 − Θ2)

.

The right hand side of (B.54) is decreasing in Θ2, and as Θ2 goes to zero, the right hand
side of (B.54) goes to 1/3. Thus, ρ ≥ 1/3 ⇒ f (1/2) < 0. Therefore, if f (α) is convex on
[1/2, 1], then f (α) < 0 for all α ∈ [1/2, 1]. From (B.51), one can verify that

f(cid:48)(cid:48)(α) = − (1 − ρ)C(cid:48)(cid:48)(α) + 2ρΘ2(cid:16)

=

2(1 − ρ)Θ2(1 − Θ2)

(1 − αΘ2)3
2(1 − ραΘ2)Θ2(1 − Θ2)

+

(1 − (1 − α)Θ2)3

+

(cid:17) − (1 − ραΘ2)

(cid:16)

C(cid:48)(1 − α) + C(cid:48)(ρ2
2)
2ρ(1 − 2ρ)Θ2(1 − Θ2)
2ρΘ2(1 − Θ2)
(1 − (1 − α)Θ2)2 +

(1 − ρ2
2)2

2(1 − ραΘ2)(1 − 2ρ)2Θ2(1 − Θ2)

(cid:17)

C(cid:48)(cid:48)(1 − α) + C(cid:48)(cid:48)(ρ2
2)

+

(1 − ρ2

2Θ2)3

,

(B.55)

where the diﬀerentiation is with respect to α. Examining (B.55), it is clear that f(cid:48)(cid:48)(α) > 0

128

for ρ ≤ 1/2. Now, suppose ρ > 1/2. Notice that

2ρ(1 − 2ρ)Θ2(1 − Θ2)

(1 − ρ2
2)2

+

Now,

2(1 − ραΘ2)(1 − 2ρ)2Θ2(1 − Θ2)

(1 − ρ2

=

2Θ2)3
2(1 − 2ρ)(1 − ρ)(1 − ρΘ2)Θ2(1 − Θ2)

(1 − ρ2
2)3

.

(B.56)

2(1 − ρ)Θ2(1 − Θ2)

(1 − αΘ2)3

− 2(2ρ − 1)(1 − ρ)(1 − ρΘ2)Θ2(1 − Θ2)

(cid:104)

(1 − ρ2
2)3
2Θ2)3 − (1 − αΘ2)3(2ρ − 1)(1 − ρΘ2)

(1 − ρ2

(cid:105)

.

(B.57)

2(1 − ρ)Θ2(1 − Θ2)
(1 − αΘ2)3(1 − ρ2
2Θ2)3

=

Notice that

(1 − ρ2

2Θ2)3 ≥ (1 − αΘ2)3 > (1 − αΘ2)3(2ρ − 1)(1 − ρΘ2),

(B.58)

where the ﬁrst inequality is true since α ≥ ρ2
2 for all α ∈ [1/2, 1], and the second inequality
holds because (2ρ − 1)(1 − ρΘ2) < 1. Therefore, f(cid:48)(cid:48)(α) > 0 for all ρ (f (α) is convex). Thus,
if Θ2 ≥ 1/2 or ρ ≥ 1/3, then A < 0 for all α ∈ [1/2, 1], which implies that y1 ≤ yc and
y1 ≥ yc. The principal learns less about the agent’s ﬁrst-period private information than

she would by setting the ﬁrst period output targets equal to the commitment optimum.

129

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