Exit option in hierarchical agency

Doyoung Kim a, Jacques Lawarrée b, Dongsoo Shin c

a Department of Business, University of Idaho, Moscow, ID 83844, USA; dkim@uidaho.edu

b Department of Economics, University of Washington, Seattle, WA 98195, USA and

ECARES, Brussels, Belgium; lawarree@u.washington.edu

c Department of Economics, Leavey School of Business, Santa Clara University, Santa Clara,

CA 95053, USA; dshin@scu.edu

May 25, 2004

Abstract

We explain why organizations that limit the voice of their agents can benefit from granting

them an exit option. We study a hierarchy with a principal, a productive supervisor and an agent.

Communication is imperfect in that only the supervisor can communicate with the principal,

while the agent has no direct voice to the principal. We show that the principal is better off if she

grants the agent the option to walk away from the contract. By doing so, the principal is

implicitly giving a “veto” power to the agent. This, in turn, restricts the manipulation of report by

the supervisor. Thus, the exit option can be interpreted as a remedy for limits on communication.

Our finding contrasts to the traditional result from the contract theory literature that the exit

option reduces the principal’s welfare, while protecting the agent. Our result is robust to the case

of collusion between the supervisor and the agent. We also examine the optimal exit option, i.e.,

whether exit should entail a payment to or from the agent.

JEL Classification: D82, L22

Key words: Contract, Exit option, Communication

2

1. Introduction

Casual observation of real world contracts reveals that while some contracts limit the exit

option by making it costly for the agent to quit, others grant the agent a relatively costless exit

option. Costly exit options are common in the entertainment industry. For example, musicians or

actors cannot easily walk away from the contract with their employers.1 Similarly, many private

contracts specify liability clauses that are essentially penalties in case of breaches. Relatively

costless exit options exist for many administrative staff or line workers who are able to quit

without many restrictions. This variety of exit options in contracts requires an analysis that

endogenizes the exit option. In this paper, we present a novel explanation why some

organizations give their workers an option to quit while others limit severely their exit option.

It is well established in the literature that granting an exit option, which is limiting the

liability of the agent, reduces the principal’s welfare while protecting the agent (e.g., Sappington,

1983). However, this result is based on the assumption that the agent has a direct communication

channel to the principal. If the agent’s message to the principal can be manipulated by a third

party, our analysis reveals that the principal is actually better off by granting an exit option to the

agent.

Our result explains why administrative staff and line workers are given exit options that allow

them to walk away from contracts easily while employees such as musicians and actors are not.

Giving an exit option in fact serves as a communication mechanism. Employees whose

performance is publicly observable and therefore easily communicated to their employers do not

require an exit option. In contrast, line-workers do not directly communicate with their employer

and their performance is often observable only by their close supervisors who could take credit

1 The costliness of an exit option was graphically illustrated by the musician Prince who appeared at the

1995 BRIT Awards ceremony with the word "slave" written across his forehead in protest against his

record company, Warner Brothers (http://www.memorabletv.com/musicworld/halloffame/prince.htm).

3

for their performance. When the employer only communicates with the supervisor, granting an

exit option to the worker serves as a communication mechanism that will alert the employer to the

supervisor’s misconduct.

Our model presents a three-tier hierarchy: a principal, a productive supervisor, and an agent.

Both the supervisor and the agent produce the output, but only the supervisor can communicate

directly with the principal. The agent has no direct voice to the principal. Mintzberg (1979)

describes this type of organization as “machine bureaucracy.” For instance, the supervisor is a

foreman or a mid-level manager who makes regular reports to the principal. There are many

reasons why the communication between the principal and the agent is limited. It can be the

result of costly information processing (see Radner, 1993 or Bolton and Dewatripont, 1994) or, as

argued in Friebel and Raith (2004), the result of an optimal organization design intended to

prevent conflicts between the supervisor and the agent over hiring decisions.2

In our model, since the supervisor is the only one reporting to the principal, he could try to

take credit for the agent’s performance by manipulating the report to the principal. The

supervisor’s manipulation of the report on the agent’s type can make the agent’s ex post payoff

negative. If given an exit option, the agent could then discard such a contract. Since the

supervisor knows this, his manipulation of the report on the agent’s type is now subject to

keeping the agent’s ex post payoff non-negative. The principal effectively gives a “veto” power

to the agent via the exit option, and this, in turn, restricts the supervisor’s manipulation of the

2 Friebel and Raith argue that the supervisor may fear that a more productive agent would replace him if the

latter can communicate directly with the principal. This fear can lead the supervisor to hire less competent

agents unless communication between the principal and the agent is limited. They show that it can be

optimal to prevent communication between the principal and the agent and force the agent to go through

the chain of command (i.e., the supervisor) when communicating with the principal. See also the model of

Laffont and Meleu (1997) where collusion makes it optimal to restrict the voice of one agent.

4

report. The principal thus protects herself by protecting the agent. The exit option remedies the

ills of limited communication.

The existing literature has overlooked the communication rationale for the exit option and

has, instead, relied on other reasons. For example, Jovanovic (1979) shows how the exit option

can increase matching quality due to the voluntary dismissal of unmatched agents. Polinsky and

Shavell (1979) show how the exit option can reduce the risk imposed onto risk averse agents.

Cooper and Ross (1985) explain how the exit option can alleviate the moral hazard by the

principal. Our paper complements the existing literature by presenting a new explanation for the

existence of exit option.

An application of our results can be seen in the concept of “noisy withdrawal,” which has

been recognized by the American Bar Association (1992). A lawyer who is bound by the duty of

confidentiality toward his client can quit when his services may otherwise be used by his client to

perpetrate fraud. While the lawyer cannot reveal the details of his client’s confidences, he may

withdraw and disaffirm documents prepared during his client’s representation. Exit becomes a

substitute for communication which, in this case, is limited by the duty of confidentiality; it

becomes a tool for a lawyer to alert outside parties without violating this duty.3

Our results also shed light on manufacturers’ generous return policies for consumers.

Manufacturers do not usually have direct communication channels with consumers. When

quality maintenance by a retailer is not easily observable, the retailer can misrepresent the quality

of a product or even sell a degraded product to consumers. In such cases, dissatisfied consumers

3 Recently, the Security and Exchange Commission has made “noisy withdrawal” part of the regulation to

implement the Sarbanes-Oxley Act. In its “Implementation of Standards of Professional Conduct for

Attorneys” (17 CFR Part 205), the Security and Exchange Commission requires an attorney to withdraw

from the representation of a client committing a material violation that is likely to cause substantial injury

to the financial interest of the investors.

5

can “exit” by returning the product for a full refund.4 The existing literature has identified

alternative justifications for return policies5, but our model proposes the limited communication

between the manufacturer and the consumer regarding the retailer’s unobservable quality

maintenance as a new explanation.

What the size of the exit option (sometimes referred as “goodbye payment”6) should be upon

the agent’s exit is another important question. Should the exit option entail payment to or from

the agent? We show that, in the optimal contract, the principal should make the agent’s payoff

equal to his reservation level if he chooses to quit. If the agent’s payoff is lower than the

reservation level upon exit, it only increases the scope of the supervisor’s manipulation of the

report on the agent’s type. If the agent’s payoff is higher than the reservation level, it only

increases the monetary transfer to the agent without further restricting the scope of the

supervisor’s manipulation.

One may wonder if the supervisor and the agent could jointly manipulate their report to

increase their joint payoffs. In other words, we need to verify whether our analysis is robust to

the possibility of collusion between the supervisor and agent. We show that it is indeed the case.

Unlike a traditional result in the literature on collusion (e.g., Tirole, 1986, Kofman and Lawarrée,

1993), we find that the supervisor’s individual incentives are not captured by the collective

incentives. The reason is that the supervisor is the only one to report and, therefore, he will

ignore the agent’s payoff when it is profitable to do so. The exit option still limits the

supervisor’s report manipulation under collusion.

4 Mann and Wissink (1990) also argue that a full refund with money back warranties is better than

replacement warranties in these cases.

5 Return policies can be also served as a risk reduction tool for risk-averse consumers (Wood, 2001) or a

signal for product quality as warranty does (Moorthy and Srinivasan, 1997).

6 See Almazan and Saurez (2003) or Bebchuk and Fried (2003) for a discussion in this issue.

6

Our work is closely related to the literature on limited liability. Although there have been

many studies that use the agent’s exit option as a modeling device, only a few papers have taken

it as the main issue. The seminal work that extensively analyzes the issue in an adverse selection

framework is Sappington (1983). The author shows that limiting an agent’s liability can bring

down the principal’s payoff from the first best level. In a moral hazard framework, Innes (1990)

adopts the concept to financial contracts. Dewatripont, Legros and Matthews (2003) extend the

analysis to renegotiable contracts. Lawarrée and Van Audenrode (1996) show that if the

principal’s output observation is imperfect, the agent’s exit option prevents the principal from

achieving the second best outcome under asymmetric information. Kim (1997) characterizes the

necessary and the sufficient conditions to implement the first best contract when the agent’s

liability is limited. Laux (2001) studies a model with multi-project to show that the principal can

relax the limited liability constraint by averaging out the liability of the agent between projects.

In the papers mentioned above, unlike ours, limiting the agent’s liability is exogenous and always

detrimental to the principal.

Finally, this paper is also related to the studies on exit and voice stemming from Hirschman’s

(1970) seminal work: dissatisfied employees tend to quit, especially if they cannot express their

discontent. Hirschman assumed that exit is always an option to the agent and studied the benefit

of adding voice. His work had a major influence in many disciplines and, indeed, it is now a

common assumption in the incentive literature that economic agents have an exit option. For

instance, Aghion and Tirole (1997) refer to the exit option as “the standard institution of letting

subordinates quit if they are unhappy with their superiors’ decision.” In this paper, we go one

step further to endogenize the exit option when the voice of the agent is limited.

The remainder of this paper is organized as follows. Section two presents the model. Section

three analyzes the optimal contract with and without the exit option, and discusses our findings.

Some extensions to the basic setup are discussed in section four. Section five concludes the

paper. All the proofs are relegated to the appendix.

7

2. Model

We present an adverse selection model with a three-tier hierarchy: a principal, a productive

supervisor, and an agent. All players are risk neutral. The principal has incomplete information

about the supervisor’s and the agent’s production costs – their types. The types are identically

and independently distributed and can be either L with low marginal cost βL and Prob(L) =πL, or

H with high marginal cost βH and Prob(H) =πH, where ∆β = βH –βL > 0 and πL + πH = 1. We

denote by i the supervisor’s type and by j the agent’s type where i, j ∈ {L, H}. The probability

distributions of the supervisor’s and agent’s types are common knowledge. The principal does

not know their types.

As in a standard adverse selection model, we assume that both the supervisor and the agent

know their own types before signing the contract. To ease our presentation, we also assume that

they learn each other’s type before they sign the contract. We will show later (section 4) that the

same results are derived under the alternative information structure where the supervisor and the

agent learn each other’s type only after signing the contract.

The supervisor reports both his own type and the agent’s type to the principal whereas the

agent has no voice. This exclusive communication channel captures the typical feature of better

access for the middle manager to the top of the hierarchy.

The supervisor and the agent produce the intermediate outputs qS and qA respectively for a

final output Q by Leontief technology.7 Thus, the intermediate outputs are complementary to

each other and qS = qA = Q in equilibrium. The principal observes the final output Q only. The

principal’s value function of the final output is V(Q). It is concave and twice differentiable on [0,

+∞), and satisfies the Inada conditions, limq→0V′(q) = ∞ and limq→∞V′(q) = 0.

7 For this type of technology, see Baron and Besanko (1992) and Laffont and Martimort (1998) among

others. The complementary production function implies that both the supervisor and the agent are

necessary to produce outputs.

8

The contract is a triplet )}ˆ,ˆ(),ˆ,ˆ(),ˆ,ˆ({ jiwjitjiQ , where i is the supervisor’s report on his

own type, j is the supervisor’s report on the agent’s type, and t and w are the monetary transfers

from the principal to the supervisor and the agent respectively. Since the principal cannot

observe the types of the supervisor and the agent, the principal’s contractual offers are contingent

on the supervisor’s report. For convenience, we denote ),ˆ,ˆ(),ˆ,ˆ( jitjiQ and )ˆ,ˆ( jiw by Qij, tij,

and wij respectively. The supervisor’s and the agent’s payoffs are SUij = tij – βiQij and AUij = wij –

βjQij respectively. Their reservation payoffs are normalized to zero.

The principal can grant the agent8 an exit option if she wishes to do so. Without an exit

option, the supervisor and agent get severely punished if they don’t produce according to the

contract. With an exit option, the agent can always discard the contract (i.e., quit) at any time. If

the exit option is exercised, then both the supervisor and agent get zero payoffs.9 Thus, the

contract becomes a “non-slavery contract.”

The timing of the game is summarized as follows.

1. Nature chooses the type of the supervisor and the agent, i and j. Each learns his own

and the other’s type.

2. The principal offers a contract {Qij, tij, wij} to the supervisor and the agent.

3. The supervisor and the agent accept/reject the contract. If either the supervisor or the

agent rejects the contract, the game ends.

4. The supervisor reports the types. The report is publicly observed.

5. The supervisor and the agent engage in production.

6. The principal observes the final output only.

7. Transfers occur.

8 The effect of an exit option for the supervisor is vacuous in our model since the contract must also satisfy

the individual rationality constraints: see footnote 10.

9 We discuss in section 4 the optimality of assuring the reservation payoff when the agent chooses to exit.

9

As a benchmark, we briefly review the optimal contract in the case of full information. When

the principal can observe both the supervisor’s and the agent’s type, her problem is to maximize

the expected payoff subject to the supervisor’s and agent’s individual rationality constraints.

These constraints induce them to sign the contract by guaranteeing a non-negative ex ante payoff.

There is no uncertainty in this setup, so the exit option is vacuous since the individual rationality

constraints also guarantee a non-negative ex post payoff. The contract under full information

gives the principal the first best outcome. The optimal output, *ijQ , is described by

.)(

,)()(

,)(

*

**

*

HHH

HLHLLH

LLL

2QV

QVQV

2QV

β

ββ

β

=′

+=′=′

=′

Both the supervisor and the agent receive no rent in any case. For the rest of this paper, we will

refer to these outputs as the benchmark for efficiency.

3. The optimality of granting an exit option

In this section, we look at the cases with and without an exit option for the agent. We then

compare these two cases and show that the principal prefers to grant the exit option to the agent.

Collusion will be introduced in the next section.

3.1. The principal’s problem with no exit option (Pn)

Without the exit option, the principal is not restricted to protect a truthful agent from ending

up with ex post negative payoff off-the-equilibrium path. The principal faces the following

optimization problem (Pn):

10

.,)(

,,)(

,,)(

,,)(

,,0)(

,,0)(..

])([)(

,

,

,

,

,,

jiQtQtIC

jiQtQtIC

jiQtQtIC

jiQtQtIC

jiQwIR

jiQtIRts

wtQVMaxP

ijHijHHHHHijHH

ijHijHLHHLijHL

ijLijLHLLHijLH

ijLijLLLLLijLL

ijjijA

ij

ijiijSij

i jijijijjiwtQ

n

∀−≥−

∀−≥−

∀−≥−

∀−≥−

∀≥−

∀≥−

−−∑∑

ββ

ββ

ββ

ββ

β

β

ππ

The principal maximizes her expected payoff subject to the individual rationality and

incentive compatibility constraints. The first constraints, (IRijS), are the individual rationality

constraints (hereafter IR) for the supervisor, while the next constraints, (IRijA), are the IRs for the

agent. The rest of the constraints are incentive compatibility constraints (hereafter IC) for the

supervisor that induces truthful type reporting. Recall that the supervisor reports both his type

and the agent’s type. Thus, the ICs prevent the supervisor from misreporting not only his type

but also the agent’s type. For example, the constraint (ICLL,ij) prevents the supervisor from any

misreport while both the supervisor and the agent are type L. There are 12 incentive

compatibility constraints for the supervisor. Since the agent has no communication channel to the

principal, no ICs need to be applied to the agent.

One difficulty for the principal is that she does not have a standard screening device to

prevent misreport about the agent’s type. Indeed, the supervisor produces the output using his

marginal cost, not the agent’s marginal cost. To induce a truthful report, the principal must

design the contract such that the supervisor’s payoff does not depend on the agent’s type. The

following equations reflect this idea.

tLL – βLQLL = tLH – βLQLH, (1)

tHL – βHQHL = tHH – βHQHH. (2)

11

Equation (1) is implied by (ICLL,LH) and (ICLH,LL). Equation (2) is implied by (ICHL,HH) and

(ICHH,HL). Therefore, those constraints are binding. They prevent the supervisor from lying about

the agent’s type.

As in a standard adverse selection problem, the principal gives the supervisor an information

rent if he is type L in order to prevent him from pretending to be type H, which is captured in

(ICLH,HH) and (ICLL,HL). Lemma 1 in the appendix proves that both of these IC constraints are

binding:

(ICLH,HH) tLH – βLQLH = tHH – βLQHH,

(ICLL,HL) tLL – βLQLL = tHL – βLQHL.

These constraints prevent the supervisor from lying about his own type. Together with (1) those

equations imply

tHL – βLQHL = tHH – βLQHH. (3)

Therefore, (ICLL,HH) and (ICLH,HL) are binding. These constraints prevent the supervisor from

simultaneously lying about his type and the agent’s type.

Knowing which constraints are binding, we can present the solution to the principal’s

problem (Pn) in the following proposition.

Proposition 1. Without an exit option, the optimal contract entails n

LLQ > nLHQ > n

HLQ = nHHQ ≡ n

HjQ

and nHj

nLj QSU β∆= , 0AUSU n

ijnHj == , ∀i, j , where n

ijQ s are characterized as

.)(

,)(

,)(

βππβ

ββ

β

∆+=′

+=′

=′

H

2L

HnHj

HLnLH

LnLL

2QV

QV

2QV

12

Proof. See Appendix. □

Clearly, from the result above, *

LLnLL QQ = , *

LHnLH QQ = , and *

HHnHH

nHL

nHj QQQQ <=≡ . As usual,

when both the supervisor and the agent are type L, “efficiency at the top” holds. An interesting

result is that there is “bunching at the bottom”: QHL = QHH. We can explain this result. The

supervisor can misreport not only his type but also the agent’s type. However, since the

supervisor produces his output using only his own marginal cost, he is not able to enjoy a benefit

from cost reduction by misreporting the agent’s type only. Suppose that the supervisor’s rent

increases with the output. Now the supervisor has an incentive to misreport the agent’s type if

such misreport increases the output and therefore his rent. Bunching at the bottom removes the

incentive to misreport.

In other words, the supervisor’s ability to manipulate the agent’s type can only affect the

output and therefore can only increase his total rent if the rent per unit of output is positive. The

supervisor can create this positive per-unit rent directly by misreporting his type, more

specifically by “overstating” his type (i.e., misreporting his own marginal cost as βH when it is βL).

To discipline this overstating incentive, the principal gives a rent when the supervisor is type L,

but not when he is type H. The amount of rent to the type L supervisor is proportional to the

output for the type H. The usual monotone result for efficiency, QHL > QHH, would create an

incentive for the supervisor to “understate” the agent’s type (i.e., misreporting the agent’s type as

βL when it is βH). By doing so, his rent would be proportional to QHL, rather than QHH. To reduce

the rent, the principal makes a downward distortion in QHL. However, QHL cannot be distorted

below QHH or the supervisor would have an incentive to overstate, rather than understate the

agent’s type. So in the optimal contract QHL=QHH. Therefore, the distortion below the first best

is introduced to reduce the rent from the supervisor’s manipulation of his own type while the

bunching prevents the supervisor’s manipulation of the agent’s type.

13

Another interesting result is that the optimal output is efficient when the supervisor’s type is

L and the agent’s type is H. The reason is as follows. As in a standard adverse selection model,

the principal distorts the output to reduce the information rent given to the supervisor when he has

an incentive to misreport his type. As usual, the type H supervisor does not have an incentive to

misreport his type as L. In other words, a report of a type L must be truthful. Furthermore, the

type L supervisor cannot enjoy a benefit from cost reduction by misreporting the agent’s type as

explained above. So, if the type L supervisor overstates the agent’s type, it must be the case that

the principal gives him a larger payoff when he reports the agent’s type as H. Knowing the

supervisor’s incentive, the principal will not give a larger payoff to the type L supervisor when he

reports the agent’s type as H. Then the type L supervisor has no incentive to overstate the agent’s

type. Thus, no distortion is needed in QLH. In fact, to avoid not only the incentive to overstate

but also the incentive to understate the agent’s type, the principal gives the same payoff to the

supervisor regardless of the agent’s type, as shown by equations (1) and (2). Therefore, the rent

given to the supervisor depends on his type, but not on the agent’s type.

3.2. The principal’s problem with the exit option (Pe)

In this subsection, we analyze the case where the principal faces a constraint on the agent’s

liability. As mentioned before, thanks to the exit option, the agent could discard the contract if he

anticipates that his ex post payoff will be negative after observing the supervisor’s report.

In the previous section, it is important to observe that all the ex-post equilibrium payoffs for

the agent are non-negative. So it seems that the exit option has no bite in this model. Since the

supervisor and the agent know their type when they sign the contract, there is no new information

provided by nature, therefore no uncertainty that could affect their payoffs. However, the limit on

the agent’s communication makes the exit option relevant. The ability of the agent to quit the

contractual relationship relaxes the supervisor’s incentive constraints and allows the principal to

offer a more profitable contract. If the supervisor lied about the agent’s type, the new contract

14

would induce a negative payoff for the agent who would then quit, thus removing the supervisor’s

incentive to misreport the agent’s type. In other words, the exit option has a bite off- the-

equilibrium path.

The principal’s optimization problem with the exit option, (Pe), becomes as follows.

∀≥−−

≥−

∀≥−−

≥−

∀≥−−

≥−

∀≥−−

≥−

∀≥−

∀≥−

−−∑∑

.0

,,0)(

,0

,,0)(

,0

,,0)(

,0

,,0)(

,,0)(

,,0)(..

])([)(

,

,

,

,

,,

otherwise

jiQwifQtQtIC

otherwise

jiQwifQtQtIC

otherwise

jiQwifQtQtIC

otherwise

jiQwifQtQtIC

jiQwIR

jiQtIRts

wtQVMaxP

ijHijijHij

HHHHHijHH

ijLijijHij

HLHHLijHL

ijHijijLij

LHLLHijLH

ijLijijLij

LLLLLijLL

ijjijA

ij

ijiijSij

i jijijijjiwtQ

e

βββ

βββ

βββ

βββ

β

β

ππ

The difference from the previous problem appears in the ICs. In (Pn), where the agent is not

allowed to quit after signing the contract, the supervisor does not consider the agent’s payoff at

all when he reports the types. This is possible because the agent cannot quit after the supervisor’s

report even if he knows that his ex post payoff will become negative.

Here in (Pe), when the supervisor reports the types, he must consider the agent’s payoff. The

reason is that the agent would rather quit if he anticipates that his ex post payoff is negative when

choosing the output level in line with the supervisor’s report. With a Leontief production

technology, once the agent quits, the supervisor cannot supply the output by himself. Since we

15

assume that the principal would then give a zero transfer to the supervisor,10 the latter can

misreport the types only if the agent’s ex post payoff remains non-negative. This idea is reflected

in the ICs. For example, in (ICLL,ij), the supervisor can misreport the types only if wij – βLQij ≥ 0.

The solution to the principal’s problem (Pe) is presented in the following proposition.

Proposition 2. With an exit option, the optimal contract entails e

LLQ > eLHQ > e

HLQ > eHHQ and

eHL

eLL QSU β∆= , e

HHeLH QSU β∆= , 0AUSU e

ijeHj == , ∀i, j, where e

ijQ s are characterized as

.)(

,)(

,)(

,)(

βππ

β

βππ

ββ

ββ

β

∆+=′

∆++=′

+=′

=′

H

LH

eHH

H

LHL

eHL

HLeLH

LeLL

2QV

QV

QV

2QV

Proof. See Appendix. □

From above, it appears that *

LLeLL QQ = , *

LHeLH QQ = , *

HLeHL QQ < and *

HHeHH QQ < . The outcome

in (Pe) has similarities with that in (Pn). For example, both eLLQ and e

LHQ are efficient as in (Pn),

and only the type L supervisor receives a rent. However, the output schedules in (Pn) and (Pe)

have a significant difference. The bunching at the bottom, a key feature in (Pn) is eliminated in

(Pe). In particular, .eHL

nHj

eHH QQQ << The intuition is as follows. Recall that, in the case of no

exit option, bunching at the bottom occurs because, if QHL > QHH, the supervisor has an incentive

to understate the agent’s type while overstating his own type. The agent’s option to exit makes

the supervisor unable to understate the agent’s type. If the supervisor did, the high-cost agent

10 Therefore, assuming that the supervisor is also protected by limited liability does not affect our results.

Assuming instead that the supervisor gets severely punished would only reinforce our reasoning.

16

would have to produce the output intended for the low-cost agent, which gives him a negative

payoff. The agent would quit making the supervisor’s payoff zero. The exit option alone can

discipline the supervisor’s manipulation of the report on the agent’s type. Thus, with the exit

option, an extra distortion in the form of bunching at the bottom is no longer needed. The

distortions in QHL and QHH are solely due to the supervisor’s incentive to overstate his own type.

Unlike the case of no exit option, the rent given to the supervisor can be made larger when

the agent is type L (due to the larger output). This difference in rent creates an incentive for the

type L supervisor to understate the agent’s type but, again, the exit option prevents such

misreporting. The rent no longer has to be the same regardless of the agent’s type.

It seems that an exit option is costly to the principal. However, this is not the case. The exit

option would be costly if it strengthened the agent’s individual rationality constraints, i.e., if the

exit option made the principal pay the agent more compared to the case of no exit option. But, in

our model, the agent always receives his reservation payoff regardless of the exit option. Thus,

the exit option is a costless device to discipline the supervisor’s manipulation of the report on the

agent’s type.11 Technically, less IC constraints are now binding in the principal’s problem. We

can summarize our main result of this paper in the following proposition.

Proposition 3. In a three-tier hierarchy with limited communication, the principal is better off

by granting an exit option to the agent.

Proof. See Appendix. □

It has been shown in the literature that granting an exit option to the agent constrains the

principal’s welfare. Proposition 3 presents a different result. The difference between our model

and a standard adverse selection model stems from a hierarchical structure where the agent cannot

11 Notice that it is only costless because the agent does not report to the principal.

17

communicate with the principal. With the exit option, the agent can quit if he anticipates an ex

post negative payoff after the supervisor’s report, making the supervisor’s report effectively

conditional on a non-negative ex post payoff for the agent. This implies that granting an exit

option eases the supervisor’s incentive compatibility constraints. For instance, (ICLH,HL) is

binding in (Pn) but relaxed in (Pe) because the RHS of the constraint becomes zero with the

agent’s exit option. By granting the exit option to the agent, the principal is giving him a “veto”

power. This, in turn, restricts the supervisor’s manipulation. Protecting the agent through the

exit option purely helps the principal since it enables her to adjust the incentive scheme without

any cost. One could interpret the agent’s ability to quit as a substitute for communication. Often,

in hierarchies, an employee quits to express his protest against a supervisor or a company policy.

Resigning is sometimes seen as the only credible communication channel for some employees.

4. Extensions

In this section, we discuss some extensions of the basic setup such as the optimal exit option,

the possibility of collusion between the supervisor and the agent, the possibility that the agent has

a voice, an alternative information structure, and an alternative production function.

4.1. Optimal exit option

We have shown that granting an exit option to the agent is optimal for the principal when

there are communicational limits between them. However, we have just assumed that the

principal does not pay a “goodbye” payment when the agent exercises the exit option. So a

question arises: what is the optimal exit option? In this section, we argue that no payment upon

exit, which makes the agent’s payoff equal to the reservation payoff, is in fact optimal. A

negative payment only increases the scope of the supervisor’s manipulation, and a positive

payment increases the monetary transfer to the agent.

18

To see this clearly, we assume that the agent’s reservation payoff is 0≥U , rather than zero.

The agent’s IRs become

.,)( jiUQwIR ijjij

Aij ∀≥− β

If the principal does not pay a goodbye payment when the agent quits, the RHS of the

supervisor’s ICs become

∀≥−−

.0

,,

otherwise

jiUQwifQt ijjijijiij ββ

To show the optimality of no goodbye payment, suppose first that, the principal imposes a

negative payment, which makes the agent’s payoff equal to ijxU − , where 0>ijx . This is the

case where the agent buys the exit option.12 It does not affect the agent’s IRs since the principal

must guarantee the reservation payoff in order to make the agent participate in the contract.

However, it does affect the RHS of the supervisor’s ICs:

∀−≥−−

.0

,,

otherwise

jixUQwifQt ijijjijijiij ββ

The supervisor now can manipulate the agent’s type even if the agent ends up receiving a payoff

less than the reservation payoff by doing so. It increases the scope of the supervisor’s

manipulation and therefore the rent given to him. Thus, comparing to the case of no payment, the

principal can never be better off by imposing a negative payment.

12 In equilibrium, the agent does not pay for the exit option since he does not exercise it.

19

Next, suppose that the principal pays a positive payment 0>ijx . This is the case where the

principal pays for the exit option. This time, because the agent can quit anytime, the agent’s IRs

are affected. They become

.,)( jixUQwIR ijijjij

Aij ∀+≥− β

To keep the agent in the organization, the principal has to pay the agent ijx on top of the

reservation payoff. It creates an extra cost to the principal, and therefore the principal would be

worse off unless it also reduces the scope of the supervisor’s manipulation, which is captured in

the RHS of the supervisor’s ICs as shown below:

∀+≥−−

.0

,,

otherwise

jixUQwifQt ijijjijijiij ββ

At first glance, it seems that the RHS of the supervisor’s ICs is different from that in the case of

no goodbye payment in the way that the scope of the supervisor’s manipulation decreases by ijx .

However, this is not the case. Indeed, there is no change in the RHS of the supervisor’s ICs since

ijw increases as well by ijx due to the binding IR constraints. Therefore, paying ijx upon exit is

beneficial only to the agent. Thus, we can conclude as follows.

Proposition 4. It is optimal to keep the agent’s payoff at his reservation level upon exit.

4.2. Robustness to collusion

When the supervisor benefits from misreporting, he might be able to compensate (bribe) the

agent who gets hurt by this misreporting. Also the agent might be able to bribe the supervisor to

induce him to overstate the agent’s type. Collusion is therefore an issue.

20

When collusion between the supervisor and the agent is possible, the supervisor considers not

only his individual payoff but also the collective payoff that includes the agent’s payoff. The

supervisor may want to misreport the agent’s and his own types for collective interest as well as

his individual interest. Thus, the principal now must include the following coalition incentive

constraints, hereafter CIC, (see, for example, Tirole, 1986 or Laffont and Martimort, 1997).

.,)(

,,)()()(

,,)()()(

,,)(

,

,

,

,

jiQ2wtQ2wtCIC

jiQwtQwtCIC

jiQwtQwtCIC

jiQ2wtQ2wtCIC

ijHijijHHHHHHHijHH

ijHLijijHLHLHLHLijHL

ijHLijijLHHLLHLHijLH

ijLijijLLLLLLLijLL

∀−+≥−+

∀+−+≥+−+

∀+−+≥+−+

∀−+≥−+

ββ

ββββ

ββββ

ββ

The CICs prevent the supervisor from misreporting for joint benefit. For example, in

(CICLL,ij), the LHS of the constraint is the joint payoff from the supervisor’s truthful report when

both players are type L, and the RHS is the joint payoff from misreporting by the supervisor.

In the previous sections, the exit option benefited the principal because it relaxed the

supervisor’s individual incentive constraints. However, the exit option may not relax the

coalition incentive constraints since the supervisor and the agent can exchange bribes. This is

particularly relevant since coalition incentives are usually stronger than individual incentives

because collective misreporting opportunities encompass individual misreporting opportunities.13

Therefore, there is a need to verify that our main result still holds.

To do so, we assume that the supervisor and the agent can sign a contract that specifies side-

transfers between them after signing the grand contract offered by the principal, but before

reporting their types to the principal.14 Without an exit option, the principal’s problem, (Pcn), is

(Pn) plus the CICs. The solution to (Pcn) is summarized in the following proposition.

13 See, for example, Tirole (1986) and Laffont and Martimort (1998).

14 The allocation of side transfers is not an issue from the principal’s point of view.

21

Proposition 5. Under collusion, without an exit option, the optimal contract entails cnLLQ > cn

LHQ > cnHLQ = cn

HHQ ≡ cnHjQ and cn

HjcnHL

cnLj QAUSU β∆== , cn

LHcnLL QAU β∆= , 0AUSU cn

iHcnHj == , ∀i, j,

where cnijQ s are characterized as

.)(

,)(

,)(

βππ

β

βππ

ββ

β

∆+=′

∆++=′

=′

H

LH

cnHj

H

LHL

cnLH

LcnLL

2QV

QV

2QV

Proof. See Appendix. □

The proposition shows that cn

LLQ = *LLQ , *

LHcnLH QQ < and cn

HjQ ≡ cnHLQ = cn

HHQ < *HHQ . As usual,

efficiency at the top holds. As in the case without collusion, there is also bunching at the

bottom: cnHLQ = cn

HHQ . This bunching occurs due to the same reason, i.e., the supervisor’s individual

incentive to misreport the agent’s type, as mentioned in section 3. But, here we need careful

explanation why the supervisor’s concern on joint interest does not overtake or nullify his

incentive to manipulate the agent’s type for his own individual interest. Intuitively it is because

the individual incentives of the supervisor can be stronger than the collective incentives. If, for

instance, the supervisor benefits from misreporting the agent’s type while the agent receives a

negative payoff, the collective incentive would tell the supervisor not to make such report while

the individual incentive would still induce him to do so. Therefore, bunching controls the

individual incentives of the supervisor to misreport the agent’s type.

However, the principal still has to deal with the collective incentive to overstate the agent’s

type. In this case, the agent could bribe the supervisor. To prevent this, we show the principal

must give away a joint rent. An interesting result here is that not only the type L supervisor but

also the type L agent receives a rent. Since the supervisor is the one reporting the types, collusion

is relevant only in the case where the agent bribes the supervisor, not the other way around. If the

22

supervisor overstates the agent’s type, then the agent enjoys a benefit from cost reduction on his

production. So the agent has an incentive to bribe the supervisor to overstate his type. To

discipline this incentive, the principal has to give a rent as usual. But, this rent could go to either

the supervisor or the agent. If the supervisor receives the rent, he has no collective incentive to

overstate the agent's type. If the agent receives the rent, he has no incentive to bribe the

supervisor. As long as the expected rents are the same, the principal would be indifferent

between two cases. However, this intuition is misleading as it “forgets” the supervisor’s

individual incentives. If the principal did give the rent to the supervisor, the type L supervisor

would understate the agent’s type for his own interest even when he reports his own type

truthfully (LH would be reported as LL). In other words, the rent to the supervisor can control his

collective incentive, but, in some cases, increase his individual incentive to misreport. Thus, the

principal chooses to give the rent to the agent, to prevent the agent from bribing the supervisor.

The supervisor enjoys a rent only because he can manipulate his type, not the agent’s type. Note

also that to minimize the rent given to the agent the principal distorts QLH below the first best

level.

Now consider the case where the principal grants an exit option to the agent. Her problem,

(Pce), is (Pe) plus CICs. The solution to (Pce) is summarized in the following proposition.

Proposition 6. Under collusion, the optimal contract with an exit option entails

ceLLQ > ce

LHQ = ceHLQ ≡ ce

MQ > ceHHQ and ce

HHceM

ceLL QQSU ββ ∆+∆= , ce

HHceHL

ceLH QSUSU β∆== , ce

HHSU =

0AU ceij = , ∀i, j, where ce

ijQ s are characterized as

.)(

,)(

,)(

βππ

βππ

β

βππ

ββ

β

∆

+∆+=′

∆++=′

=′

2

H

L

H

LH

ceHH

H

LHL

ceM

LceLL

22QV

2QV

2QV

23

Proof. See Appendix. □

From above, *** and , HH

ceHHHL

ceHL

ceLH

ceMLL

ceLL QQQQQQQQ <<=≡= . Also, .ce

McnHL

ceHH QQQ << As

expected, there is no bunching at the bottom. Again, the exit option serves as a device to

discipline the supervisor’s individual incentive to manipulate the agent’s type. The exit option

replaces bunching at the bottom.

Unlike the case of no exit option, there is bunching in the middle: ceLHQ = ce

HLQ . This results

exclusively from collusion. Since the exit option disciplines the supervisor’s individual incentive,

the principal now considers only the supervisor’s collective incentive in her contract offer. Then

it does not matter for the principal who is type L and type H.

In this case, however, there are two optimal transfer schemes. The rent to prevent the

overstating of the agent’s type ( HHQ β∆ ) can go either to the agent (as it did in proposition 5) or

to the supervisor (as we presented it in proposition 6) since, under collusion and with the exit

option, the principal no longer has to worry about the supervisor’s incentive to understate the

agent’s type (see the corollary in the appendix).

Now we can compare the cases with and without the exit option in the following proposition.

Proposition 7. Even if collusion between the supervisor and the agent is possible, the principal

is better off by granting an exit option to the agent.

Proof. See Appendix. □

Since collective incentives usually overtake individual incentives, it seems that the exit option

does not help the principal under collusion. However, we have shown that this is not the case as

some IC constraints stay binding. For instance, in (Pcn), (ICLH,LL) is binding while (CICLH,LL) is

24

slack. With an exit option, the principal can relax constraints such as (ICLH,LL), which becomes

slack in (Pce). Therefore, the exit option still helps the principal.

4.3. Alternative information structure

The timing of our model assumes that both the supervisor and the agent know each other’s

type when signing the contract. To verify the robustness of our result, we discuss an alternative

information structure: the supervisor and the agent know only their own type when they sign the

contract but find out each other’s type after signing the contract. 15

Since the other player’s type is unknown when signing the contract, the IR constraints in the

basic model should be replaced with the ones in expected terms as follows.

(IRi

S)' πL(tiL – βiQiL) + πH(tiH – βiQiH) ≥ 0 ∀i,

(IRjA)' πL(wLj – βjQLj) + πH(wHj – βiQHj) ≥ 0 ∀j.

In this setting, the principal has more degrees of freedom since he could give an ex-post

negative payoff to the agent and the supervisor. So granting an exit option by making ex-post

payoffs non-negative may be costly to her. Therefore, it is necessary to verify our result under

this alternative information structure. The following claim shows that our result still holds.

Claim. The principal’s payoff is the same in the optimal contract when the IRs in (Pn) and (Pcn)

are replaced with (IRiS)' and (IRj

A)'.

Proof. See Appendix. □

15 For example, in a corporation, a manager and his direct subordinate quickly find out each other’s abilities

once they start working together while the owner (the principal) usually knows less about them.

25

The claim shows that, without an exit option, the principal receives the same payoff with the

new IR constraints. Therefore granting the exit option will still benefit her. Intuitively, it is

because, to induce a truthful report on the agent’s type, the principal still has to give the

supervisor the same payoff independent of the agent’s type as shown in equations (1) and (2).

This makes the principal unable to give the supervisor ex-post negative payoffs. The

principal could give ex-post negative payoffs to the agent, but that would not improve the

principal’s payoff since all IRs for the agent are binding.16 Therefore, even if the supervisor and

the agent do not learn each other’s type prior to their participation, the principal’s payoff is

unaffected.

5. Conclusion

The exit option has usually been treated as exogenous in the contract theory literature. The

literature relies on the existence of some legal restrictions, such as bankruptcy law, corporate

investment regulation, and minimum wage law. The role of these exogenous constraints has been

to guarantee the agents an ex post reservation payoff, thus limiting the principal’s welfare.

In this paper, we allowed the principal to choose to grant an exit option to the agent if she

wants to do so. We have focused our analyses on a hierarchy where the communication channel

is limited to the supervisor. In this type of hierarchy, we showed that the principal becomes

better off by granting an exit option to the agent. By doing so, the principal gives the agent at the

low end of the hierarchy a right to discard the contract, which imposes a restriction on the

manipulation of information by the supervisor at the middle of the hierarchy. Thus, the exit

option can be seen as a remedy for communicational limits in the hierarchy. Our result still held

under the possibility of collusion between the supervisor and the agent. We also showed that it is

optimal for the principal to pay the agent his reservation payoff when he exercises the exit option.

16 In the appendix, we show that this claim is still true under collusion.

26

Acknowledgements

We thank Gorkem Celik, Mathias Dewatripont, Wouter Dessein, Fahad Khalil, Jean-Jacques

Laffont, Helen Popper, Luis Rayo, Bernard Sinclair-Desgagné, Bill Sundstrom, the seminar

participants at the Econometric Society Winter Meetings (2004), the anonymous referees and the

editor for helpful comments. Lawarrée thanks the support of the RRF at the University of

Washington.

Appendix

Proof of Proposition 1. We first find the binding constraints to solve for the optimal output

levels. Notice that LHS of (ICLL,LH) and the RHS of (ICLH,LL), and the RHS of (ICLL,LH) and the

LHS of (ICLH,LL) are identical. Due to this reciprocity, (ICLL,LH) and (ICLH,LL) are immediately

binding. Similarly, (ICHL,HH) and (ICHH,HL) are binding because of the reciprocity. We solve the

problem assuming that other four (ICHj,ij) are slack (i.e., (ICHL,LL), (ICHL,LH), (ICHH,LL), (ICHH,LH)).

We check later that the solution satisfies this assumption. Then, (IRSH j) are binding since

otherwise the principal can improve her payoff by decreasing tH j without violating any constraints.

The other (IRSL,j)are assumed to be slack. Also, (IRA

ij) are binding since otherwise the principal

can improve her payoff by decreasing wij without violating any constraints. Next we use lemma 1.

Lemma 1. Both (ICLH,HH) and (ICLL,HL) are binding in the optimal contract.

Proof. First, we show by contradiction that both (ICLH,HH) and (ICLL,HL) must be binding. First,

suppose neither IC is binding in the optimal contract. Then (ICLL,HH) and (ICLH,HL) are slack too

and the principal can improve her payoff by slightly decreasing tLH and tLL together without

violating any other constraints. Next, suppose only (ICLH,HH) is binding. With binding

constraints (ICLL,LH) and (ICLH,HH), we can express (ICLL,HL) as follows: tHH – βLQHH > tHL – βLQHL.

27

This inequality, together with binding (ICHL,HH), implies that the following must be satisfied in

the optimal contract.

QHL < QHH (a1)

Binding constraints (ICLL,LH), (ICHL,HH), and (ICLH,HH) and IRs for the supervisor and the agent of

type H give the following transfers: tHL = βHQHL, tHH = βHQHH, tLL = βLQLL + (βH – βL)QHH, tLH =

βLQLH + (βH –βL)QHH, and wLL = βLQLL, wLH = βHQLH, wHL

= βLQHL , wHH = βHQHH. After replacing

the transfers with their values in the objective function, the first order condition gives, V′(QHL) =

βL + βH and V′(QHH) = 2βH +(πL/ πΗ

2 ) (βH – βL). This, however, implies that QHL > QHH, which

contradicts (a1). Similarly, it can be shown that only (ICLL,HL) binding leads to a contradiction. It

implies that QHL > QHH , but the first order condition gives QHL < QHH. □

Therefore, the binding constraints in (Pn) are (IRS

H j), (IRAij), (ICLL,LH), (ICLL,HL), (ICLH,HH),

and (ICHL,HH). Binding (ICHL,HH), (ICLH,HH), (ICLL,LH) and (ICLL,HL) imply that QHL = QHH = QHj.

The transfers are obtained from the binding constraints: tLL = βLQLL + ∆β QHj, tLH = βLQLH + ∆β

QHj, tHL = βHQHj, tHH = βHQHj, wLL = βLQLL, wLH = βHQLH, wHL = βLQHj , wHH = βHQHj. After

replacing the transfers with their values in the objective function, we have

πL

2[V(QLL) – 2βLQLL – (βH – βL)QHj] + πLπH[V(QLH) – (βL + βH)QLH – (βH – βL)QHj]

+ πLπH[V(QHj) – (βL + βH)QHj] + πH2[V(QHj) – 2βHQHj]

The first order condition gives the optimal output schedule. The solution satisfies the assumption

made at the beginning and also satisfies all the constraints. The solution is unique since the

objective function is concave and the constraints are linear. □

28

Proof of proposition 2. We first solve the problem assuming that (ICHj,ij) ∀i, j, are slack. Then

(IRSH j) ∀j, are binding since otherwise the principal can improve her payoff by decreasing tH j

without violating any constraints. The other (IRSL,j) are assumed to be slack. Also, (IRA

i j) ∀i,j, are

binding otherwise the principal can improve her payoff by decreasing wij without violating any

other constraints. Thanks to the exit option, the RHS of (ICLH,LL) and (ICLH,HL) becomes zero.

Therefore, since the RHS of (ICLH,HH) is non-negative, (ICLH,HH) is stronger than (ICLH,LL) and

(ICLH,HL). Because of (IRSHH) binding, the RHS of (ICLH,HH) is positive and therefore (ICLH,HH)

must be binding. Otherwise, the principal could increase her payoff by decreasing tLH without

violating the constraints (recall that IRSLH is slack). Also, not all (ICLL,ij) can be slack since

otherwise the principal can increase her payoff by decreasing tLL without violating the constraints

(recall that (IRSLL) is slack). Next we use lemma 2 to show which (ICLL,ij) is binding.

Lemma 2. (ICLL,HL) is binding in the optimal contract.

Proof. Suppose to the contrary that (ICLL,HL) is slack. Binding (ICLH,HH) implies that both

(ICLL,LH) and (ICLL,HH) are binding. From binding constraints, we have tLL =βLQLL + ∆βQHH, tLH

=βLQLH + ∆βQHH, tHL

=βHQHL, tHH =βHQHH,, wLL =βLQLL, wLH

=βHQLH, wHL =βLQHL, wHH =βH QHH.

After substituting for these transfers in the objective function, we have the first order conditions

V′(QHL) = βL +βH and V′(QHH) = 2βH + (πL/πH

2)∆β. Thus, we have QHL > QHH. By plugging in the

transfers derived above, the RHS of (ICLL,HL) is bigger than its LHS which violates (ICLL,HL). □

Finally, the binding constraints in this problem are (IRSH j), (IRA

ij), (ICLL,HL) and (ICLH,HL) ∀i, j.

These binding constraints yield the following monetary transfers to the supervisor and the agent:

tLL =βL QLL

+ ∆βQHL, tLH =βLQLH + ∆βQHH, tHL =βHQHL, tHH =βHQHH, wLL =βLQLL, wLH =βHQLH,

wHL =βLQHL, wHH =βHQHH. Substituting for these transfers in the objective function gives the first

order conditions in the proposition, which show the solution of the principal’s problem. The

29

solution satisfies the assumption made at the beginning and also satisfies all the constraints. The

solution is unique since the objective function is concave and the constraints are linear. □

Proof of proposition 3. The solutions to (Pn) and (Pe) are different, but the solution to (Pn)

satisfies all the constraints in (Pe). □

Proof of proposition 5. The pair (ICLL,LH) and (ICLH,LL), the pair (ICHL,HH) and (ICHH,HL) and

the pair (CICLH,HL) and (CICHL,LH) are binding since the LHS (RHS) of one constraint is identical

to the RHS (LHS) of other. We solve the problem assuming the other four (ICHj,ij) are slack:

(ICHL,LL), (ICHL,LH), (ICHH,LH) and (ICHH,LL). We later check that the solution satisfies this

assumption. We focus our attention to (CICLL,LH), (CICLL,HH), (CICHL,LH) and (CICHL,HH), solve the

problem ignoring the rest of CICs and then check that the solution indeed satisfies them. Then

(IRSHH) and (IRS

HL) are binding since otherwise the principal can increase her payoff by

decreasing tHH and tHL without violating any constraints. Also, (IRALH) and (IRA

HH) must be

binding in the optimal contract. Otherwise, the principal can improve her payoff by decreasing

wiH without violating the other constraints. The constraints (IRALL) and (IRA

HL) are slack. Now, at

least one of (CICLL,LH) and (CICLL,HH) must be binding because otherwise the principal can

increase her payoff by decreasing wLL without violating other constraints. Using (ICLH,HH),

(IRALH) and (IRA

HH) imply that the RHS of (CICLL,LH) is larger than the RHS of (CICLL,HH) as long

as QLH > QHH. We solve the problem under this monotone condition and then check later that it is

indeed the case. Then (CICLL,LH) is binding. Next we use lemma 3.

Lemma 3. Both (ICLL,HL) and (ICLH,HH) are binding in the optimal contract.

Proof. We will show by contradiction that both (ICLL,HL) and (ICLH,HH) must be binding. First,

suppose that neither ICs is binding in the optimal contract. Then, (ICLL,HH) (ICLH,HL) are slack too.

The principal can improve her payoff by decreasing tLH and tLL together without violating any

30

other constraints. Next, suppose that only (ICLL,HL) is binding. Binding (ICLL,LH) implies that the

RHS of (ICLL,HL) is bigger than the RHS of (ICLH,HH), i.e., tHL – βLQHL > tHH – βLQHH. This

inequality, together with binding (ICHL,HH), implies that the following must be satisfied in the

optimal contract:

QHL > QHH. (a2)

Binding (ICLL,LH), (ICLL,HL), (ICHL,HH), (CICLL,LH), (CICHL,LH) and binding IRs give the

following transfers: tLL = βLQLL + ∆β QHL, tLH = βLQLH + ∆β QHL, tHL = βHQHL, tHH = βHQHH, wLL =

βLQLL + ∆β QLH, wLH = βHQLH, wHL = βLQHL + ∆β QHL , wHH = βHQHH. After substituting for the

transfers in the objective function, we have the following first order condition: V′(QHL) = 2βH

+(1/πH)∆β and V′(QHH) = 2βH . This, however, implies that QHL < QHH, which contradicts (a2).

Similarly, it can be shown that only (ICLH,HH) binding leads to a contradiction too. It implies that

QHL < QHH , but the first order condition gives QHL > QHH. □

Therefore, the binding constraints are (IRS

H j), (IRAij), (ICLL,LH), (ICLL,HL), (ICLH,LL), (ICLH,HH),

(ICHL,HH), (ICHH,HL), (CICLL,LH) and (CICHL,LH). Binding (ICHL,HH), (ICLH,HH), (ICLL,LH) and

(ICLL,HL) imply that QHL = QHH ≡ QH j. The following transfers are obtained from the binding

constraints: tLL = βLQLL + ∆β QH j, tLH = βLQLH + ∆βQH j, tHL

= βHQHj, tHH = βHQH j, wLL = βLQLL +

∆βQLH, wLH = βHQLH, wHL = βLQH j + ∆β QHj, wHH = βHQH j. After replacing the transfers with their

values in the objective function, we have the first order conditions in the proposition, which give

the optimal output schedule. The solution satisfies the assumption made above and also satisfies

all the constraints. The solution is unique since the objective function is concave and the

constraints are linear. □

31

Proof of proposition 6. We first solve the problem ignoring all ICs and then check that the

solution indeed satisfies them. In addition, we solve the problem assuming (CICHH,ij) ∀i, j,

(CICLH,LL), and (CICHL,LL) are slack and then check later that the solution satisfies this assumption.

Then (IRSHH) and (IRA

HH) are binding, otherwise the principal can increase her payoff by

decreasing tHH and wHH without violating other constraints. The constraints (CICLH,HL) and

(CICHL,LH) are binding since the LHS (RHS) of one constraint is identical to the RHS (LHS) of

other. Then at least one of (CICLH,HH) and (CICHL,HH) must be binding, since otherwise the

principal can increase her payoff by decreasing tLH + wLH and tHL + wHL together without violating

other constraints. Then binding (CICLH,HL) implies that both of (CICLH,HH) and (CICHL,HH) are

binding. Next, we use lemma 4.

Lemma 4. Both (CICLL,LH) and (CICLL,HL) are binding in the optimal contract.

Proof. We will show by contradiction that both (CICLL,LH) and (CICLL,HL) are binding. First,

suppose to the contrary that both (CICLL,LH) and (CICLL,HL) are slack. Then it must be case

(CICLL,HH) is binding otherwise the principal can increase her payoff by decreasing tLL+wLL

without violating any other constraints. Then the binding constraints give the following joint

transfers: tLL+wLL = 2βLQLL + 2∆βQHH, tLH+wLH = (βL +βH)QLH +∆β QHH, tHL+wHL = (βL +βH)QHL +

∆βQHH, tHH+wHH = 2βHQHH. After substituting for the transfers in the objective function, we have

the following first order condition: V′(QLH) = βL + βH and (1/πH)∆β and V′(QHH) = 2βH + ∆β [πL

(1+πH)/πH2]. This implies that QLH > QHH. Given this inequality, the above transfers violate

(CICLL,LH). Next, suppose that only (CICLL,LH) is binding. Then, using the slack (CICLL,HL) we

have tLH+wLH –2βLQLH > tHL+wHL–2βLQHL. Together with binding (CICLH,HL), it implies

QLH > QHL. (a3)

32

Binding (CICLL,LH), (CICLH,HH), (CICHL,HH) and binding IRs give the following joint transfers:

tLL+wLL = 2βLQLL + ∆βQLH + ∆βQHH, tLH+wLH = (βL +βH)QLH +∆β QHH, tHL+wHL = (βL +βH)QHL +

∆βQHH, tHH+wHH = 2βHQHH. After substituting for these joint transfers in the objective function,

we have the following first order conditions: V′(QLH) = βL +βH +(πL/πH )∆β, V′(QHL) = βL +βH. It

implies that QLH < QHL, which is a contradiction to (a3). Similarly, it can be shown that only

(CICLL,HL) binding leads to a contradiction too. It implies that QLH < QHL, but the first order

condition gives QLH > QHL. □

Using the lemma, finally binding constraints are (CICLL,LH), (CICLL,HL), (CICLH,HL), (CICLH,HH),

(CICHL,LH), (CICHL,HH), (IRSHH) and (IRA

HH). Binding (CICLL,LH), (CICLL,HL), (CICLH,HL) implies

that QLH = QHL ≡ QM . The binding constraints give the following joint transfers: tLL+wLL =

2βLQLL + ∆βQM + ∆βQHH, tLH+wLH = (βL +βH)QM +∆β QHH, tHL+wHL = (βL +βH)QM + ∆βQHH,

tHH+wHH = 2βHQHH. After replacing the transfers with their values in the objective function, we

have the first order conditions in the proposition, which give the optimal output schedule. The

solution satisfies the assumption made above and also satisfies all the constraints. The solution is

unique since the objective function is concave and the constraints are linear. □

Corrolary. Two sets of individual transfer schemes satisfy the constraints.

Proof. (ICLL,HL) and (IRSHL) imply that tLL ≥ βLQLL +∆βQM, and (ICLH,HH) and (IRS

HH) imply that

tLH ≥ βLQM +∆βQHH. Given these conditions, we have the following two individual transfer

schemes that satisfy the ignored ICs: (i) tLL =βL QLL +∆β QM+∆β QHH, tLH =βLQM +∆β QHH, tHL

=βHQM+∆β QHH, tHH =βHQHH, wLL =βLQLL, wLH =βHQM, wHL =βLQM, wHH =βHQHH, and (ii) tLL

=βL QLL +∆β QM, tLH =βLQM +∆β QHH, tHL =βHQM, tHH =βHQHH, wLL =βLQLL+∆β QHH, wLH

=βHQM, wHL =βLQM+∆β QHH, wHH =βHQHH. □

33

Proof of proposition 7. The solutions to (Pcn) and (Pce) are different, but the solution to (Pcn)

satisfies all the constraints in (Pce). □

Proof of the claim. Define (P0

n) and (P0cn) as the principal’s problem when the IR constraints in

(Pn) and (Pcn) respectively are replaced with (IRiS)' and (IRj

A)'. We prove that the principal’s

payoffs in (P0n) and (P0

cn) are at most the same as those in (Pn) and (Pcn) respectively. [i] Consider

first the case of (P0n). As in (Pn), we have tLL – βLQLL = tLH – βLQLH from (ICLL,LH) and (ICLH,LL),

and tHL – βHQHL = tHH – βHQHH from (ICHL,HH) and (ICHH,HL). From these, in order to satisfy (IRiS)',

we must have that tij – βiQij ≥ 0 ∀i, j. It implies that the expected IRs for the supervisor in (P0n)

become de facto the ex-post IRs in (Pn). Thus, there is no difference from (Pn) regarding the

supervisor’s IRs. However, unlike the case of the supervisor, the principal can make the agent’s

ex-post payoff negative using her leeway on wij as long as it satisfies (IRjA)'. But, it does not

improve the principal’s payoff. Since wij does not enter the ICs, the principal could not change the

supervisor’s incentive using her leeway on wij. In addition, due to the principal’s risk neutrality

in his payoff, it does not matter to her whether the agent’s IRs are in the form of expected term or

ex-post term as long as all the IRs are binding, which is also the case in (P0n) since wij does not

enter the ICs. Thus, the principal’s payoff can never be larger in (P0n) than in (Pn). [ii] Next,

consider the case of (P0cn). For the same reason as in (P0

n) explained above, we must have that tij

– βiQij ≥ 0 ∀i, j to satisfy (IRiS)'. The expected IRs for the supervisor in (P0

cn) become the ex-post

IRs, and therefore there is no difference from (Pcn) regarding the supervisor’s IRs. Since the IRs

and ICs are the same for the supervisor, the argument made in lemma 3 is also applicable. Thus

(ICLH,HH) is binding. Using the binding (ICLH,HH), (CICLH,HH) becomes

wLH – βHQLH ≥ wHH – βHQHH. (a4)

34

Also, using (ICHL,HH), (CICHH,HL) implies that

wHH – βHQHH ≥ wLH – βHQLH. (a5)

Therefore, wLH – βHQLH = wHH – βHQHH from (a4) and (a5). To satisfy (IRHA)', we must have that

wLH – βHQLH ≥ 0 and wHH – βHQHH ≥ 0. For the cases where the agent is type L, again using the

binding (ICLL,HL), (CICLL,HL) becomes

wLL – βLQLL ≥ wHL – βLQHL. (a6)

Using (ICHH,HL), (CICHL,HH) implies that

wHL – βLQHL ≥ wHH – βLQHH. (a7)

Since wHH – βHQHH ≥ 0 as shown above, (a7) implies that wHL – βLQHL ≥ 0, which in turn implies

that wLL – βLQLL > 0 from (a6). Therefore, to summarize, we must have that wij – βjQij ≥ 0 ∀i, j.

The expected IRs for the agent as well as for the supervisor in (P0cn) become ex-post IRs. Thus,

the principal’s payoff can not be larger in (P0cn) than in (Pcn). □

References

Aghion, P. and J. Tirole, 1997, Formal and real authority in organizations, Journal of Political

Economy 105, 1-29.

Almazan, A. and J. Suarez, 2003, Entrenchment and severance pay in optimal governance

structures, Journal of Finance 58, 519-547.

American Bar Association Standing Committee on Ethics and Professional Responsibility, 1992,

Withdrawal when a lawyer’s services will otherwise be used to perpetrate a fraud, Opinion

92-366, August 8, 1992.

35

Banerjee, A. and R. Somanathan, 2001, A simple model of voice, Quarterly Journal of Economics

116, 189-227.

Baron, D. and D. Besanko, 1992, Information, control, and organizational structure, Journal of

Economics and Management Strategy 1, 237- 275.

Bebchuk, L. and J. Fried, 2003, Executive compensation as an agency problem, Journal of

Economic Perspectives 17, 71-92.

Bolton, P. and M. Dewatripont, 1994, The firm as a communication network, Quarterly Journal of

Economics 109, 809-839.

Cooper, R. and T. Ross, 1985, Product warranties and double moral hazard, Rand Journal of

Economics 16, 103-113.

Dewatripont, M., P. Legros and S. Matthews, 2003, Moral hazard and capital structure dynamics,

forthcoming in the Journal of the European Economic Association.

Friebel, G. and M. Raith, 2004, Abuse of authority and hierarchical communication, Rand Journal

of Economics 35.

Hirschman, A., 1970, Exit, voice, and loyalty (Harvard University Press, Cambridge, MA).

Innes, R., 1990, Limited liability and incentive contracting with ex-ante action choices, Journal of

Economic Theory 52, 45-67.

Jovanovic, B., 1979, Job matching and the theory of turnover, Journal of Political Economy 87,

972-990.

Kim, S., 1997, Limited liability and bonus contracts, Journal of Economics and Management

Strategy 6, 899-913.

Kofman A. and J. Lawarrée, 1993, Collusion in hierarchical agency, Econometrica 61, 629-656.

Laffont, J.-J. and D. Martimort, 1997, Collusion under asymmetric information, Econometrica 65,

875-911.

Laffont, J.-J. and D. Martimort, 1998, Collusion and delegation, Rand Journal of Economics 29,

280-305.

36

Laffont, J.-J. and M. Meleu, 1997, Reciprocal supervision, collusion and organizational design,

Scandinavian Journal of Economics 99, 519-40.

Laux C., 2001, Limited liability and incentive contracting with multiple projects, Rand Journal of

Economics 32, 514-526.

Lawarrée, J. and M. Van Audenrode, 1996, Optimal contract, imperfect output observation, and

limited liability, Journal of Economic Theory 71, 514-531.

Mann, D. and J. Wissink, 1990, Money-back warranties vs. replacement warranties, American

Economic Review 80, 432-436.

Mintzberg, H., 1979, The Structuring of Organizations (Prentice-Hall, Englewood Cliffs, NJ).

Moorthy, S. and K. Srinivasan, 1995, Signaling quality with a money-back guarantee: The role of

transaction costs, Marketing Science 14, 442-466.

Polinsky, M, and S. Shavell, 1979, The optimal tradeoff between the probability and magnitude

of fines, American Economic Review 69, 880-891.

Radner, R., 1993, The organization of decentralized information processing, Econometrica 61,

1109-1146.

Tirole J., 1986, Hierarchies and bureaucracies: On the role of collusion in organizations, Journal

of Law, Economics and Organization 2, 181-214.

Sappington, D., 1983, Limited liability contracts between the principal and agent, Journal of

Economic Theory 29, 1-21.

Wood, S., 2001, Remote purchase environments: The influence of return policy leniency on two-

stage decision processes, Journal of Marketing Research 38, 157-169.