Allowing the lesser of two evils:
bribery or extortion?
Draft November 27, 2006

FAHAD KHALIL†

JACQUES LAWARRÉE‡

SUNGHO YUN*

Abstract
Rewards to prevent enforcement agents from accepting bribes create incentives for extortion. We
present a model where a supervisor who can engage in bribery and extortion can still be useful inproviding incentive. We show that bribery may be allowed, but extortion is never tolerated in the
optimal design of organizations. Allowing extortion penalizes good behavior which increases
incentive cost; allowing bribery introduces the bribe as a penalty for bad behavior, which helps restore
incentives somewhat. As a key modeling insight, we point out the importance of the appropriate
notion of soft information. We demonstrate that the fight against corruption should be rooted in
making information hard.

JEL Classification:
D82, L23
Key words:
Monitoring, Corruption; Collusion, Bribery, Extortion; Framing.

†

Department of Economics, University of Washington, Seattle, WA 98195, khalil@u.washington.edu
Department of Economics, University of Washington, Seattle, WA 98195 and ECARES, Brussels
lawarree@u.wahington.edu* Department of Economics, Hanyang University, Ansan, Korea, uwyunsh@hanyang.ac.kr‡

1.

Introduction

In the design of optimal organizations, the fight against corruption by enforcement
officers relies on strong incentives to detect and report violations by agents. Suchincentives raise the specter of extortion since rewards to deter bribery may act as
inducements to engage in extortion. Consider the case of an enforcer whose role is todetect and report violations by an agent. Offering a reward to the enforcer for turning in
the agent will lower his incentive to accept a bribe from that agent. For instance, a driverunder the influence of alcohol may attempt to bribe a police officer to let him off thehook for a DUI conviction, but a corrupt officer will find it less profitable to accept abribe if he can collect a reward when turning in the drunk driver.1 Now consider the caseof an officer catching drivers who run red lights. Again, a reward would lower his
incentive to accept a bribe from a driver caught running the light, but the same reward
may invite a corrupt officer to claim that the driver ran the light when he did not.
Incentive to deter bribery may lead a corrupt officer to extort innocent drivers.
Notice the important difference between the nature of evidence in the DUI caseand the red light case, which turns out to be critical in studying the trade-off between
deterring bribery and inducing extortion. In the DUI case, a corrupt officer cannot claimthat a sober driver is drunk because hard evidence (such as a blood test) is required. In
the red light case however, the testimony of the officer may be enough to convict adriver.

We will say that the evidence is soft when the officer can manipulate theevidence (e.g., his testimony), either to help a guilty driver in exchange for a bribe or to
extort an innocent driver. Evidence that cannot be manipulated will be described as hard
evidence, but we allow for hard evidence to be concealed.2 The distinction between hardand soft evidence is key to analyzing the trade-off between bribery and extortion and it isrelevant to many other settings such as financial or tax audits.

1

The reward can be non-monetary such as good reputation, promotion, etc.
See, e.g., Tirole (1986). We will make the definitions of hard and soft information precise in our model
section.
2

1

The difference between bribery and extortion relies on the type of evidence
manipulation. 3 The enforcer can manipulate evidence in two different ways: (a) make afavorable report about the agent — this will be called bribery in this paper; (b) make an
unfavorable report about the agent — this will be called extortion in this paper. We also
use the generic term of corruption to describe bribery and extortion.
In this paper, we present a model which captures the trade-off between deterring
bribery and inducing extortion, and find two new results: (i) extortion should always be
deterred but bribery should not; (ii) bribery is deterred when information is hard but may
be allowed when information is soft.
The intuition for our result (i) is straightforward. Both bribery and extortion makeit more costly to provide incentive to an agent, but there is a critical difference between
bribery and extortion. Extortion penalizes the agent after “good behavior”, while bribery
penalizes the agent after “bad behavior”. Since bribery occurs when a violation isdetected, the bribe is a penalty for “bad behavior”, and helps somewhat in providing
incentive. This is in line with the less formal literature that suggests that bribes may havesome positive role to play but extortion does not.

Bribery can help “grease” theincentives in badly run organizations. It is also consistent with the fact that extortion ismainly a problem in less developed countries relying mostly on soft evidence, while indeveloped countries hard evidence is more common and it is mainly bribery that makes
the news4. We show that allowing some form of bribery can be a key part of the optimal
design of incentives in an organization.
The intuition for our result (ii) can be understood in light of the existing literature.
There is an extensive literature in economics dealing with bribery but our result that thethreat of extortion makes bribery optimal is new.5 Our focus is on the agency literaturethat followed the pioneering work by Tirole (1986, 1992) as opposed to the non-agency
3

Precise definitions are given later in the model section.
In the financial world for instance, making information hard can take various forms and be represented by
the use of institutions like lawyers, CPAs, auditors, bankruptcy courts, independent directors and legal
actions by the shareholders (see the survey paper by La Porta (2000)).
5
Several papers have shown that it may be optimal to allow bribery by putting restrictions on contracts.
For instance, Kofman and Lawarree (1996) (uncertain auditor type); Che (1995) and Mookherjee and Png(1995) (auditor moral hazard); Strausz (1997), Olsen and Torsvik (1998), Lambert-Mogiliansky (1998),
and Khalil and Lawarree (2006) (renegotiation and no-commitment).
42

literature (as reviewed in Bardhan (1997)). In the agency literature, there is no trade-off
between bribery and extortion (other than a few exceptions noted below), and a chiefreason is that this literature relies on hard information.
To see this, consider a standard moral hazard model with a supervisor whomonitors the agent’s performance ex post.

Suppose, as in Tirole (1986), that thesupervisor either finds hard evidence (positive or negative) or finds no conclusive
evidence. With hard evidence, the supervisor can hide information and pretend she has
found no conclusive evidence but she cannot forge evidence. So if the supervisor has noconclusive evidence, she has no discretion and no bribery or extortion can occur. If thesupervisor has incriminating evidence, the agent will want to bribe the supervisor toconceal it. However, this can be deterred without inducing a threat of extortion by
rewarding the supervisor only for producing incriminating evidence. Consequently, if
she has positive evidence about the agent and wants to threaten to extort by concealing it,
her threat is not credible. This is because she will not be rewarded if she reports noconclusive evidence. Therefore extortion is not an issue.
In our model also the information for the supervisor is hard. However, and this is
key, we assume that the information for the supervisor-agent coalition is soft, i.e., thesupervisor can forge evidence with the help of the agent. Now, the principal also has toreward the supervisor for not forging evidence, and not just for presenting incriminating
evidence.

The new reward goes to the supervisor when she reports no conclusive

evidence.

This reward makes extortion credible when the supervisor has positive

evidence and threatens to conceal it.

The trade-off between deterring bribery andextortion appears when information is soft, and we find that allowing bribery is optimal.
One important implication of our analysis is that the fight against bribery should
be rooted in making information hard.

Most of the literature following Tirole has

focused on the problem of bribery in models where extortion is not relevant, i.e., not acredible threat. 6

Other than special circumstances, noted in the footnote above, the6

For instance in Kessler (2000) and Vafai (2005), the information is hard. Baliga (1999) analyzes the caseof soft information but extortion does not increase the implementation costs because the mechanism of thegame allows the agent to quit when faced with the possibility of extortion. See also Faure Grimaud,Laffont and Martimort (2003) for a model of soft information with asymmetric information between the3literature largely finds that it is optimal to deter bribery. Therefore, we contribute to thisliterature by pointing out that if information is soft, the threat of extortion may make it
optimal to allow bribery. Our result suggests that the trade-off between bribery andextortion can be avoided by making information hard
One of our contributions is to develop a framework where a supervisor is usefuldespite the presence of extortion and bribery without having to assume the existence ofincorruptible enforcers.

In the recent literature, two prominent papers also feature

extortion but in different settings and with a different focus. Polinsky and Shavell (2001)
study an optimal law enforcement problem, while Hindriks et al. (1999) is a tax-evasion
model with a focus on the redistributive properties of the tax scheme.

To detercorruption, both papers rely on the availability of incorruptible external enforcement
agents and the penalties they can impose. Instead, we focus on internal mechanisms todeter bribery and extortion by developing an informational structure that makes asupervisor useful even though she can engage in bribery and extortion and incorruptible
external enforcers are absent.
The remaining sections of the paper are organized as follows. Section 2 outlines
the model. Section 3 presents the benchmark cases and shows that bribery-proof contract
is vulnerable to framing. Section 4 finds the optimal regime by comparing various
regimes with each other and characterizes the contract. Section 5 concludes the paper.

2.

The Setup

We present a standard principal/supervisor/agent hierarchy with a key new feature thatmakes extortion relevant. The principal (it) is the owner of a firm, the agent (he) is the
productive unit in the firm, and the supervisor (she) collects information for the principal.
The agent produces output x which depends on his level of effort, e ∈ {0, 1}. If the agent
works, that is, e = 1, he produces xH with probability π and xL with probability 1 – π,
where xH – xL = ∆x > 0, and π ∈ (0, 1). If he shirks, that is, e = 0, he produces xL withprobability one. While the level of output x is observed by all parties, the level of effort e
supervisor and the agent. In Kofman and Lawarree (1993) the information structure allows forging ofevidence but rules out extortion by assumption.

4

is private information of the agent. The agent’s disutility of effort in terms of money isgiven by ϕ . e, where ϕ > 0. The output belongs to the principal, who pays a transfer w to
the agent. We assume that the agent is risk averse with a separable utility function given
by, U(w, e) = u(w) – ϕ e, where u is concave, u(0) = 0, and satisfies Inada’s conditions (u′(0) = + ∞ and u′(+ ∞) = 0). The principal who is risk-neutral offers a take-it-or-leave-it
contract to the supervisor and the agent. We assume that ∆x is large enough that it isalways profitable to induce the agent to work, that is, exert e = 1. The principal’s
objective is to minimize its expected cost of inducing e = 1.
In the absence of a supervisor, the contract for the agent could only be based on x
and the wages would be wL when xL is produced and wH when xH is produced. The
optimal contract in the absence of a supervisor — the well-known second-best contract— requires that wsH = u −1 (ϕ / π ) and wsL = 0. In other words, the principal compensates
the agent only when there is definitive evidence that the agent worked, i.e., when xH isrealized. The agent does not obtain any rent.
The supervisor’s role is to collect information about the agent’s effort level and toreport it to the principal. Since xH can be realized only with e = 1, there is no reason to
use the supervisor following xH, and the principal will send the supervisor only when it
observes xL. Following Tirole (1986), we assume that the supervisor observes the truelevel of effort with probability p or obtains no conclusive evidence with probability 1 – p,where p ∈ (0, 1). The supervisor’s signal σ can take three values: σ ∈ {0, ∅, 1}, where ∅denotes that the supervisor does not have conclusive evidence about effort. Therefore,
the agent is given a wage wH following xH, and wr, following xL, where r is thesupervisor's report with r ∈ {0, ∅, 1}. We assume that the supervisor is costless but the
principal may want to pay her a wage s to deter corruption. The supervisor is risk neutral.
Without loss of generality, the wage to the supervisor depends only on her own reportand is denoted by sr. We assume that the agent’s and the supervisor’s reservation utilities
are zero, and that they are protected by limited liability such that wr ≥ 0 and sr ≥ 0. 7

7

Without limited liability, the first best could be reached since e = 0 is off the equilibrium path. When the
supervisor reports that e = 0, the principal can impose an infinite punishment on the agent, and also give alarge reward to the supervisor if she is corruptible.

5Supervision Technology and Corruption: key assumption
We assume that the supervisor is corrupt in the sense that she may not always report what
she has observed to the principal. She will report the truth only if it is in her interest to
do so. In this environment, we identify two types of corrupt behavior, which we define
below:
Definition 1. Bribery occurs when one party accepts a payment in return for
manipulating information in favor of the other party.
Definition 2. Extortion occurs when the supervisor obtains money from the agent bythreatening to falsify evidence that is favorable to the agent.

We say framing has

occurred if the attempt at extortion fails and the supervisor falsifies information that isfavorable to the agent.
We assume that the supervisor’s information is hard: she cannot fabricate theevidence by herself and the only way to manipulate information by herself is to suppress
it, i.e. if σ = e, she can only report r ∈ {e, ∅}, and if σ = ∅, the only possible report is r
= ∅. Thus, extortion involves threatening to suppress information favorable to the agent.
However, for extortion to be relevant in this framework, we need to make acritical assumption. We assume that the information for the coalition is soft: with theagent’s cooperation, the supervisor can make up evidence and report that the agent has
worked regardless of what she observed, i.e. it is possible to have r ∈ {0, ∅, 1} regardlessof σ. We refer to this as bribery.
It may seem counterintuitive that to make extortion by the supervisor relevant,information has to be soft for the coalition while it is hard for the supervisor. However,this assumption is critical because supervisory extortion would not be an issue if theinformation were only soft or hard. If the information were soft for the supervisor, thesupervisor would be useless. If the information were hard for both the supervisor and thecoalition, extortion would not be relevant. This is because a threat of extortion is credible
only if the supervisor is able to collect a reward by suppressing information. Since
evidence cannot be created, the supervisor has no discretion when σ = ∅, and there is noneed to reward the supervisor when σ = ∅.

6

Therefore, the threat of extortion bysuppressing evidence is vacuous in a model with hard information as it is the case inmany prominent models like Tirole (1986, 1992) or Kessler (2000).
Besides the standard assumption of enforceable side-contracts (see Tirole 1992),
we need to make two additional assumptions.

First, since bribery may occur inequilibrium, we need to be explicit in how side transfers are determined. We assume
they are determined according to the Nash bargaining solution. Second, we assume thatthe side-contract is signed after the supervisor finds out about the effort of the agent.
This timing is natural but, as we shall see, it also turns out to be important when
examining the occurrence of corruption. If the enforceable side-contract was signed
before the agent takes his effort, the bribery-proof principle would apply and bribery
would not occur in equilibrium. We will have more to say on this in section 4.
Bribery and extortion are accompanied by side-contracts between the supervisor
and the agent whereas framing is not. With bribery, the supervisor and agent jointly
manipulate information to maximize their joint surplus. With extortion (resp. framing),
the supervisor acts alone by threatening to suppress (resp. actually suppress) evidence
since she is acting against the agent’s interest. We require that extortion or framing besequentially rational; the supervisor's threat of suppressing information is credible only if
she receives a higher utility by suppressing evidence than by revealing it truthfully.
We summarize the model by presenting the timing of moves:(1) The principal offers a contract specifying the transfers to the agent as a function of
output and the supervisor’s report; and the transfers to the supervisor as a function of herreport.
(2) The agent and the supervisor accept/reject the contract.
(3) The agent decides whether to work (e = 1) or shirk (e = 0).
(4) Output x is realized. If the principal observes xL, it sends the supervisor. If it
observes xH, the game moves to (8).
(5) The supervisor and the agent observe the signal σ.
(6) The supervisor and the agent choose whether or not to make a side-contract.
(7) The supervisor makes a report r.
(8) Transfers are realized.

7

4. Optimal contract with a corrupt supervisor: trade-off between bribery
and extortion
If the supervisor were incorruptible, the optimal contract would specify that thesupervisor will not be paid any reward, sr = 0, for all r. The agent would only be
rewarded when there is definitive evidence of effort, i.e., if xH occurs or if xL occurs andthe supervisor finds evidence of work (r = 1); the agent will be paid zero otherwise. The
agent does not obtain any rent and he is equally compensated both when xH is realized
and when r = 1 with xL. i.e., wH = w1 > 0 = w∅ = w0 (see Appendix A for details of theincorruptible-supervisor contract). Compared with the second-best or no-supervisor case,the agent receives a positive wage more often, and therefore, his wage after xH is smaller
than under the second best. Given the effort e = 1, the agent obtains better insurance, andthat reduces the principal's expected wage payment relative to the second-best contract.
This contract, however, is vulnerable to bribery. The supervisor is not being
rewarded (sr = 0) since she is assumed to be truthful. If the supervisor is corruptible8, theagent will bribe the supervisor when she finds no-evidence or evidence of shirking, andhelp her fabricate evidence to give a report of work (r = 1) so that they can share thehigher wage collected by the agent (w1).
On first sight, this threat of bribery can be combated by introducing a reward forthe supervisor when she reports shirking (r = 0) or no-evidence (r =∅). If the reward isequal to w1 (i.e., s0 = s∅ = w1), there will be no incentive to bribe. The supervisor isturned into a bounty hunter as in, e.g., Tirole (1986) or Kofman and Lawarrée (1993).
However, in our framework, this would introduce a new problem of extortion by thesupervisor. To see this, note first that s1 = 0 since there is no perceived threat of a bribefrom the agent when σ = 1. Thus, when she has evidence of work, the supervisor will
have an incentive to suppress this evidence to obtain the reward s∅ > 0 rather than get s1

8

It is common knowledge that the supervisor is corruptible. For a dynamic model where the supervisor
privately knows her propensity for corruption, see Carillo (2000).

8

= 0.9 That is, we see the emergence of the trade-off that we alluded to in the introduction,
namely, strong incentives to deter bribes creates scope for a new kind of corruption,
namely extortion. As noted above, this trade-off would not appear if we had assumed
that information is hard as in many prominent models (e.g., Tirole 1986, 1992, Kessler
2000).10
Next we present the contract where the principal deters both bribery and extortion.
However, we also show later that this contract is not optimal.
The least-cost-corruption-proof (LCCP) contract: no bribery or extortion
It is not clear a priori whether it is optimal to deter all types of corruption. In
particular, we have already shown above that rewards for deterring bribery can encourage
extortion/framing, which means there is a trade-off in deterring different kinds ofcorruption. To study this trade-off, it is useful to characterize as a benchmark the leastcost-corruption-proof contract that deters both types of corrupt behavior. The LCCP
contract is also a critical step when we derive the optimal contract in the next section.
We show in Lemma 2 that the LCCP contract dominates any contract that allows
extortion to occur in equilibrium. The main implication of deterring both bribery andextortion is the principal loses much of the value of retaining a supervisor. It cannot fully
utilize the information provided by the supervisor to differentiate the agent’s payments
according to realized states. We show later that the LCCP contract is not optimal ingeneral, but it can be under specific conditions, e.g., if the agent had all the bargaining
power when negotiating the side-contract.
To prevent bribery the principal will have to ensure that the contract satisfies the
Coalition Incentive Compatibility (CIC) constraints.
(CICσ, r)Tσ ≥ Tr,

where Tσ = wσ + sσ, Tr = wr + sr, forσ, r ∈ {0, ∅, 1}.

9

Anticipating extortion the agent will refuse to put in high effort (his incentive constraint will be violated).
Note also that raising s1 to s∅ is not optimal since it would encourage the coalition to report r = 1 when σ =∅.
10There is a series of papers by Vafai (cited in Vafai (2005)) analyzing extortion under hard information.
To make extortion credible Vafai relies on the “prohibitive psychological or emotional cost” of not carryingout a threat and he shows that bribery can be deterred without cost.

9We have six (CIC) constraints and these can be satisfied only when T0 = T∅ = T1, i.e., theaggregate transfers in every state following xL must be the same. This can also be written
as:
w0 + s0 = w1 + s1,=>

s0 = w1 + s1 – w0(1)w∅ + s∅ = w1 + s1,=>s∅ = w1 + s1 – w∅

(2)

Since extortion/framing may occur only by suppressing evidence when σ ∈ {0, 1},
the principal will have to ensure that the contract satisfies two additional
extortion/framing deterring (EF) constraints to prevent extortion/framing. These can be
written as:(EF1)

s1 ≥ s∅,(EF0)

s0 ≥ s∅.

If one of the above constraints is not satisfied, the supervisor will choose to either extort
or frame the agent, whichever gives her a higher payoff. Note however that only (EF1) is
the relevant constraint for deterring extortion since it deters suppression of positive
evidence, whereas (EF0) deters suppression of negative information, where bribery is thepertinent issue. Therefore, we will ignore the (EF0) constraint and just verify ex post that
it is satisfied by our identified solutions in each case below. We also assume that theagent and the supervisor do not collude when they are indifferent between colluding or
not colluding, and the supervisor will not extort when she is indifferent.
A corruption-proof contract satisfies the (CIC) and (EF) constraints and detersboth bribery and extortion/framing. The agent’s participation and incentive constraints
and the supervisor’s participation constraint are the same as those in the incorruptible
supervisor case discussed above (see also Appendix A).11 Thus, the principal’s program
which prevents both bribery and extortion/framing, denoted by Po, can be written as
follows:

11

We can ignore the IR constraints as they are implied by the IC and the limited liability constraints.

10

Min

π(wH) + (1 – π) [p(w1 + s1) + (1 – p) (w∅ + s∅)]

s.t.

(IC)

πu(wH) + (1 – π) pu(w1) – π(1 – p) u(w∅) – pu(w0) ≥ ϕand(1), (2), (EF1), (EF0), wH ≥ 0, wr ≥ 0 and sr ≥ 0,
where r ∈ {0, ∅, 1}

The solution to this problem is called the least-cost-corruption-proof contract and it ischaracterized in the following lemma:
Lemma 1 The least-cost corruption-proof (LCCP) contract has the following features:(i) If p ≤ π, the contract is equivalent to the second-best or no-supervisor contract ofsection 3 with wr = sr = 0 for r ∈ {0, ∅, 1}, and wH = wHs.
(ii) If p > π, it is optimal to use the supervisor, and the contract to the agent satisfies:
wHo > w1o = w∅o > 0 = w0o ,whereu ′( w1o ) 1 − π=, π ( wHo ) + ( p − π )u ( w1o ) = ϕ ,u ′( wHo ) p − πi.e., the agent obtains an ex ante rent.
•

The supervisor's contract involves:

s1o = s ∅o = 0 < s 0o = w1o ,but the supervisor receives no ex ante rent.12•

The principal’s expected cost, denoted by Co, can be written as
Co = π(woH) + (1 – π)wo1.

Proof. See Appendix B.
There are two main findings from this lemma: (a) the threat of extortion restricts
the principal’s ability to use the supervisor’s information, and (b) the supervisor will be
used only if she is accurate enough. We explain these below in turn.
As we argued earlier, rewards for turning down bribes introduce incentive toextort/frame. In particular, a reward to the supervisor for reporting σ = ∅ truthfully
Since the agent does not shirk in equilibrium, the signal σ = 0 is off the equilibrium path, and thesupervisor’s rent is zero even though s0 > 0.

12

11

would encourage the supervisor to extort/frame when σ = 1. In the corruption-proof
contract, this incentive is avoided by reducing s∅ to zero, but then the (CIC) requires thatw∅ = w1. Therefore, it is no longer possible to only reward the agent after definitive
evidence of work, and the agent who shirks without being caught must also be treated asif he worked. The agent gets a high wage w1 (= w∅) with probability 1 – p even when heshirks since the supervisor is not perfectly accurate.
This implies that the supervisor may not be useful if she is not accurate enough,which is different from the case of the incorruptible supervisor where she is useful forany p > 0. If the agent works, he gets this payment with probability (1 – π)(p + (1 – p))= 1 – π. The net effect on the (IC) can be seen by setting w∅ = w1 and rearranging terms:

πu(wH) + (p – π)u(w1) = ϕ .
If p ≤ π, the agent is more likely to receive the transfer w1 when he shirks rather thanwhen he works, in which case it would be optimal to set w1 = 0. We have w1 = w∅ = w0
= 0, and the principal does not rely on the supervisor’s report at all, and we also have sr =0 for all r. Thus, the contract is equivalent to the second-best contract.
On the contrary, if p > π, paying a positive w1 is useful in providing incentive tothe agent since he is more likely to receive a positive transfer when he works. However,this is costly to the principal since it also pays a positive w∅ (= w1) and therefore it is
optimal to set wo1 < woH. The expected cost for the principal is smaller than under thesecond best, but higher than the case with an incorruptible supervisor.
Note that it is not the supervisor but the agent who benefits from the supervisor’s
ability to manipulate information under the corruption-proof contract. The reason is asfollows; the only way to prevent both bribery and extortion/framing is to give up theinformativeness of r = ∅ and treat it as if r = 1 in shaping the agent’s incentives. Thus
the supervisor cannot affect the agent’s payoff by misreporting that r = ∅ when σ = 1.
As a result, she cannot command any rent. The agent who is the potential victim, on thecontrary, obtains a higher utility than his reservation level. Otherwise the agent will shirk
and get w1 ( = w∅) with probability 1 – p.

12

Optimal Contract
In this section we characterize the optimal contract when the supervisor can engage inboth types of corruption. The principal has always the fall-back option of offering thesecond-best or no-supervisor contract and ignore the supervisor's report, but we knowthat the least-cost corruption-proof contract dominates this contract when p > π, i.e.,
when she is accurate enough. Therefore, the interesting question is whether it is possible
to improve upon the least-cost corruption-proof contract by allowing some type of
corruption.13
Since we allow for the possibility of corruption to occur in equilibrium, we haveto account for payoffs resulting from side contracts. We assume that when the agent andsupervisor engage in a side contract, their payoffs are determined by the Nash bargaining
solution. For example, if the agent bribes the supervisor to report work (r = 1) when
there is no evidence (σ = ∅), the coalition will get s1 + w1 which they will share. Thisimplies that the agent’s payoff when σ = ∅ and r = 1 is not w1, but rather the outcome
from Nash bargaining. Therefore, all the computations, and particularly the agent’s (IC)constraint, have to be derived using the relevant Nash bargaining payoffs. They arepresented in detail in the appendix and we only outline the main intuition here in the text.
We first prove in the following lemma that extortion will never be allowed.

Lemma 2: Any contract that induces e = 1, but violates (EF1) is strictly dominated by theleast-cost corruption-proof contract.
Proof: See Appendix C.

The intuition for never allowing extortion is that it appears as a penalty after theagent has done the right thing, i.e., exerted effort. Thus extortion makes it difficult forthe principal to reward the agent for his effort and increases the cost of providing
13

Note that if it is possible to improve on the corruption-proof contract, it will be optimal to use thesupervisor even when p < π , but for high enough p.

13

incentive. Technically (see Appendix C), this is seen from the outcome of the Nashbargaining between the agent and supervisor when (EF1) is violated. If (EF1) is violated,i.e., if the threat to report ∅ when σ = 1 is credible, we show that the agent gets the same
payoff from the Nash bargaining whether the state is ∅ or 1. Therefore, the supervisor's
report is not useful in distinguishing between these states and the agent has less incentive
to provide effort. As shown in our lemma 1, the least-cost corruption-proof contract does
not distinguish between ∅ and 1 either but it is less costly to the principal since thesupervisor is not rewarded (s1 = s∅ = 0). Therefore the least-cost corruption-proof
contract dominates any contract that induces extortion.
We can now present our main result showing that allowing some bribery is indeed
optimal, but allowing extortion is not, which is a novel result in the literature.
Proposition It is optimal to use the supervisor if p > π. If the agent does not have all thebargaining power, the optimal contract induces bribery when the signal σ = ∅, but
deters extortion and framing, and the optimal contract will have the following features:•

w*H > w*1 > 0 = w*∅ = w*0; when σ = ∅, the agent obtains kw*1 > 0, where k < 1
and k depends on the agent's relative bargaining power.14•

s1* = s∅* = 0 < s0* = w1* ; the supervisor obtains (1 - k) w*1 > 0 when σ = ∅.

•The principal’s expected cost, denoted by C*, is given by
C* = π(w*H) + (1 – π)w*1.

Proof: See Appendix D.
To see why bribery may help, note from our lemma 1 that the only way to deter
all corruption is by not utilizing every piece of information provided by the supervisor.
In particular, the principal can no longer pay the agent only after definitive evidence ofwork. The agent receives the same compensation when the signal is ∅ and 1 even though
the supervisor reports truthfully. This raises the cost of providing incentive to the agent
since a shirking agent will also obtain a positive compensation when the signal isinconclusive about the true effort.

A way to restore some variation in the agent's

In the Appendix, we define w1∅ as the agent’s payoff in state σ = ∅ as a result of Nash bargaining andreporting r = 1, and thus k = w1∅ / w*1.

14

14

compensation between the states ∅ and 1 is by allowing bribery to occur in state ∅.
Suppose a bribe from the agent leads the supervisor to overstate performance in state ∅and report 1. Then the principal will make the same aggregate transfer in both states ∅and 1, but the agent's payoff in state ∅ is lowered since he has to pay a bribe to thesupervisor, and this lowers the cost of inducing high effort.15
This captures nicely an intuition often mentioned in the applied literature, thatallowing bribery can create markets that improves incentives. Here, the principal relieson the supervisor to extract a bribe from the agent and lower the agent's payoff in state ∅,when it cannot directly do so in fear of encouraging extortion.

The latter is also

consistent with the widely held belief that extortion is always counter productive since it
penalizes agents when they have obeyed rules or done what they are supposed to.
Extortion punishes the agent when he has done the “right thing”, while bribery occurs if
the agent shirks or violates rules.
Supervisor’s and agent’s bargaining power
From Tirole (1986) we know that when bribery is deterred, the bargaining powerof the coalition members does not matter. The principal competes with the agent for thesupervisor’s report and the reward given to the supervisor must exceed any viable offerfrom the agent. In our model, since the principal lets bribery occur in equilibrium, thebargaining power is relevant.

We show that the principal is better off when the

supervisor has relatively more bargaining power. The reason is that the supervisor can
extract a larger bribe from the agent who will find bribery less profitable. Consider acase where the agent has little bargaining power. When the supervisor receives a signal∅, the agent will want to bribe the supervisor to report r = 1 hoping to receive the larger

wage w1 > w∅ = 0. However, after bribing the supervisor, the agent will only collect asmall fraction of w1. In other words, an agent with little bargaining power receives little
15

Polinsky and Shavell (2001) find that, depending on parameter values, it may be optimal to allow
extortion/framing and deter bribery. Their model is very different from ours and relies on incorruptible
external enforcers to detect corruption. More specifically, the principal can choose different probabilitiesof detecting bribery, framing, and extortion, and also choose different levels of sanctions for each offence.
They also introduce another parameter θ that determines how likely an innocent agent will be in a position
to be framed. The relative values of these parameters may make it optimal to deter bribery and allow
extortion/framing. For instance if the parameter θ is very small, then allowing extortion/faming is not very
costly, and the principal should focus on deterring bribery.

15

benefit from bribing because the bribe acts as an effective penalty. Consequently it iseasier for the principal to provide incentive to the agent and therefore the principal isbetter off when the supervisor has more bargaining power.
Perhaps more interestingly, when the agent has close to zero bargaining power,
the principal’s payoff is similar to its payoffs when extortion is not an issue. In otherwords, when the supervisor has all the bargaining power, it is as if extortion was not
relevant. To see this, let us envision a case where the supervisor can engage in bribery
but cannot extort by assumption. We begin by recalling the incorruptible supervisor
contract that is vulnerable to bribery. When the supervisor finds negative or no evidence
about the agent’s effort, the agent will bribe her to report r = 1. To prevent bribery, thesupervisor should be rewarded for reporting negative or no evidence, i.e., s0 = s∅ = w1.
Raising s∅ would raise the specter of extortion, but remember that we have ruled out
extortion by assumption for the sake of the argument. It can be shown that the newcontract that deters bribery but ignores extortion – call it ωb – is characterized by wH > w1
= s∅ = s0 > 0 = w∅= w0 = s1. The principal’s payoff from this contract is identical to its
payoff in the optimal contract of our proposition when the agent’s bargaining power isclose to zero.16 We provide the argument below.
In both ωb and the optimal contract, the principal has to incur the payment w1, either as
reward or wage, in each of the states σ = ∅ or 1. Thus, the principal would be indifferent
between the two contracts if the w1 were identical. We argue next that this is indeed thecase when the agent has no bargaining power. Note that in both contracts the agent gets
w1 when σ = 1, but whenσ = ∅, the agent gets zero in ωb or his coalition share of w1 in
the optimal contract. Since the latter is close to zero if the supervisor has almost all thebargaining power, the incentive cost of the principal to satisfy the agent’s (IC) constraint
is identical to that under ωb. Therefore, the wage w1 (and wH for that matter) is identicalin the two cases, and so is the principal’s payoff. To deter extortion, the principal has toallow the agent to obtain a part of w1 in the side contract with the supervisor, but thisdoes not raise the cost of providing incentive when the agent has no bargaining power.
Remark: The principal’s payoff increases as the agent’s bargaining power decreases,
and the threat of extortion poses no additional cost if the agent has (almost) no
bargaining power.
16

Note that when the agent’s bargaining power is zero exactly, the agent is indifferent between bribing or
not. If we assume that he does not, the principal receives the same payoff as in the incorruptible supervisor
contract.

16

5.

Conclusion

A key intuition that has not played much of a role in the literature on bribery inhierarchies is that rewards to enforcement agents to turn down bribes may also encouragethem to engage in extortion. Part of the problem is in finding an appropriate model inwhich a supervisor or enforcement agent remains useful even though they can engage inextortion.

Tirole (1986) showed that a corruptible supervisor can still be useful.

However, his model and much of the subsequent literature did not feature the effect ofextortion since extortion was not a credible threat in these models. By introducing anappropriate notion of soft information, we are able to present a model of extortion inwhich the supervisor remains useful.
We show that the effect of the trade-off implied by the above intuition may imply
that allowing bribery is optimal due to the threat of extortion. In many theoretical modelsof bribery, the principle of collusion proofness applies and bribery does not appear inequilibrium. We show that this result depends on the softness of information. When
information is soft, there is a trade-off between bribery and extortion and collusion orbribery appears in equilibrium. If information is hard, there is no such trade-off andbribery does not occur in equilibrium. Our results suggest that organizations that must
rely on soft information may also need to allow bribery. By making its information“harder” an organization will suffer less from corruption. Making information harder canbe costly.

For instance, speeding tickets should rely on sophisticated cameras or

shareholders ought to be able to appeal auditing reports to reliable and incorruptible
experts. Developing countries with less resources and technological abilities, and weaklegal environment also have less capability to make information hard and, therefore, weshould expect that bribery to be a more pervasive problem. Again the reason is that they
do not have the ability to rely on hard information. The fight against corruption should
therefore focus on the reliance on hard evidence.

17

One implication of bribery occurring in equilibrium is to validate in a model the
popular notion that bribery can be useful to “grease the wheels” in inefficientorganizations. However, it must be kept in mind that this is a second-best result. For
example, bribery is optimal in our model because it allows the principal to cause avariation in the agent’s payoffs when direct payments from the principal would only haveresulted in introducing extortion, which is a worse problem. Extortion penalizes an agent
after “good” behavior, while bribery at least imposes some penalty for “bad” behavior.
Finally, our result that allowing bribery may be optimal depends on the fact that
we do not allow corrupt behavior to be detected ex post. For example, if there wereincorruptible enforcement agents available to detect and sanction corrupt behavior, it
would be possible to eliminate bribery in equilibrium. However, it is well known thatpolicing the police is not an easy task, and incorruptible enforcement agents may bescarce and expensive in many contexts.

18Appendix A

Incorruptible Supervisor

Suppose the supervisor always reports truthfully what he has observed. The agent’s
participation and incentive constraints are as follows:(IR)

πu(wH) + (1 – π) [pu(w1) + (1 – p) u(w∅)] – ϕ ≥ 0,(IC)

πu(wH) + (1 – π) [pu(w1) + (1 – p) u(w∅)] – ϕ ≥ pu(w0) + (1 – p) u(w∅,)πu(wH) + (1 – π) pu(w1) – π(1 – p) u(w∅) – pu(w0) ≥ ϕ .

or,Given limited liability, and since zero effort entails zero cost, the incentive constraint will
imply that the participation constraint is satisfied in each of the cases we consider. The
supervisor's participation constraint is also satisfied due to limited liability. Thus, we will
ignore both the agent's and the supervisor's participation constraints from now on.
The principal’s program when the supervisor is truthful, Pt, can be written as
follows:
Min

π(wH) + (1 – π) [p(w1 + s1) + (1 – p) (w∅ + s∅)]s.t.

(IC), wH ≥ 0, wr ≥ 0 and sr ≥ 0, where r ∈ {0, ∅, 1}.

The principal’s problem has the following Lagrangian:L = π(wH) + (1 – π) [p(w1 + s1) + (1 – p) (w∅ + s∅)]– λ [πu(wH) + (1 – π) pu(w1) – π(1 – p) u(w∅) – pu(w0) – ϕ]with the additional non-negativity constraints where λ ≥ 0 is the Lagrange multiplier.
The Kuhn-Tucker conditions for minimization are:
∂L
∂L
∂L

∂L

∂wH
∂w1
∂w∅

∂w0

⎛
⎝

⎞
⎠

(a1)

w1 ⎜ ∂L

⎛
⎝

⎞
= 0,
∂w1 ⎟⎠

(a2)

⎛
⎝

⎞
= 0,
∂w∅ ⎟⎠

(a3)

⎛
⎝

⎞
= 0,
∂w0 ⎟⎠(a4)= π – λπ u′ (wH) ≥ 0;

wH ⎜ ∂L ∂w ⎟ = 0,

= (1 – π) p – λ(1 – π) p u′ (w1) ≥ 0;
= (1 – π) (1 – p) + λ π(1 – p) u′ (w∅) ≥ 0;w∅ ⎜ ∂L= λ p u′ (w0) ≥ 0;

w0 ⎜ ∂L

19

H

∂L
∂L

∂s1
∂s∅

= (1 – π) p ≥ 0;

s1 ⎜ ∂L

⎛
⎝

⎞
= 0,
∂s1 ⎠⎟(a5)= (1 – π) (1 – p) ≥ 0;s∅ ⎜ ∂L

⎛
⎝

⎞
= 0,
∂s∅ ⎟⎠(a6)

plus the complementary slackness conditions for the constraints.
From (a3), (a5) and (a6), we have w∅ = 0, s1 = 0 and s∅ = 0. Since s0 does notenter the Lagrangian, it can be any non-negative number and the principal’s expected costis independent of s0.
Now suppose that λ = 0. From (a1) and (a2), we have wH = w1 = 0, which violates
the constraint (IC). The assumption that λ = 0 leads to a contradiction. Hence λ > 0 and(IC) is binding. Now (a4) implies that w0 = 0.
The result of λ > 0 also implies that wH = w1 > 0. First we argue that both wages
are positive and then show that they are equal. If ∂L ∂w > 0, then wH =0 and 1-λu′H(0)>0, but then (a2) implies that 1-λu′ (w1)>0 since w1 ≥ 0 and u″ < 0. This would
imply that w1 = 0, but having both wH = 0 and w1 = 0 violates (IC). So we must have∂L∂wH

= 0 and therefore wH > 0. Likewise, λ > 0 implies that w1 > 0. Therefore, wehave ∂L ∂w = 0 and ∂L
H∂w1L= 0. which leads to λ = 1()

wH = w1 in (IC), we have w = w1 = u −1 ϕ π + (1 − π ) p ;H

Appendix B

u '( wH )= 1u '( w1L ). Finally, usingw∅ = w0 = 0. .

Proof of Lemma 1

In the problem Po of section 4, we will first ignore the constraint (EF0) and verify laterthat it is satisfied by the optimal contract. Using (2) to replace s« everywhere, we canrewrite (EF1) as (EF1b) and state the principal’s problem as follows:
Min πwH + (1 – π) (w1 + s1),

20

s.t.
(IC)

π u(wH) + (1 – π) pu(w1) – π(1 – p) u(w∅) – pu(w0) ≥ ϕ,

(EF1b)w« ≥ w1,(1)

s0 = w1 + s1 – w0,
and the non-negativity constraints.

Note that once we ignore (EF0), the variable s0 does not appear anywhere else in the
problem except in (1). Therefore, we are free to choose s0 to satisfy this constraint (1) aslong as s0 ≥ 0. We can now set up the following Lagrangian for this problem:
L = π(wH) + (1 – π) (w1 + s1)– δ1 [πu(wH) + (1 – π) pu(w1) – π(1 – p) u(w∅) – pu(w0) – ϕ]– δ2 (w∅ – w1),with the additional non-negativity constraints.
The Kuhn-Tucker conditions for minimization are:
∂L

∂L
∂L
∂L
∂L

= π – δ1π u′ (wH) ≥ 0;

wH ( ∂L ∂w ) = 0,(b1)

= (1 – π) – δ1 (1 – π) p u′ (w1) + δ2 ≥ 0;

w1 ( ∂L) = 0,(b2)= δ1 π(1 – p) u′ (w∅) – δ2 ≥ 0;w∅ ( ∂L) = 0,(b3)= δ1 pu′ (w0) ≥ 0;

w0 ( ∂L) = 0,(b4)= (1 – π) ≥ 0;

s1 ( ∂L) = 0,

(b5),

∂wH

∂w1
∂w∅
∂w0
∂s1

H

∂w1

∂w∅

∂w0

∂s1

plus the complementary slackness conditions for the constraints.
From (b5), we have s1= 0 since (1 – π) > 0. This result, (EF1), and limited liability
imply that s∅ = 0. Thus, we have w1 = w∅ from (2).
Now suppose that δ1 = 0. From (b1) and (b2), we have wH = w1 = 0, which
violates the constraint (IC). The assumption that δ1 = 0 leads to a contradiction. Henceδ1 > 0, (IC) is binding.

21

The result of δ1 > 0 also implies that wH > 0 because condition (b1) is violated if
we assume that wH = 0 and thus u′ (wH) = ∞. Therefore, we have ∂L ∂w = 0 and δ1 =H

1/u′(wH).
Now (b4) implies that w0 = 0, which leads to w1 = s0 from (1).
Since we showed above that w1 = w∅, then using condition (b2) and (b3), we havethe following condition
∂L

∂w1

+ ∂L

∂w∅

= (1 – π) – δ1 (p – π) u′ (w1) ≥ 0

There are two cases to be considered: (i) p ≤ π and (ii) p > π. When (i) p ≤ π, ∂L
∂L

∂w∅

∂w1

+

is always strictly positive, which means w1 = w∅ = 0 since at least one of themmust be zero. From (IC), we have wH = u −1 (ϕ / π ) . The contract becomes equivalent tothe case when the supervisor is not available.
When (ii) p > π, ∂L
∂L

∂w∅

∂w1

+ ∂L

∂w∅

must be zero. If we assume that ∂L> 0, then we have w1 = w∅ =0. However, this implies that ∂Lsince u′ (w1) = ∞, which is a contradiction. By solving ∂L

∂w1

+ ∂L

∂w∅

∂w1

+ ∂L

∂w1

∂w∅

+

<0

= 0, we have thefollowing;u ′( w1 ) 1 − π=.
u ′( wH ) p − π

The above equation gives us values of wH and w1 = w∅ with binding (IC). Finally, s0 =w1 is given by (1) and note that the ignored constraint (EF0) is satisfied in each case.
Appendix C

É

Proof of Lemma 2

We proceed in steps. First, we show that the agent receives the same payoff from Nashbargaining for σ œ {«, 1} if the constraint (EF1) is violated, but the supervisor earns an
ex ante rent. We then show that there exists a corruption-proof contract that achieves thesame cost but is more costly than the least-cost corruption-proof contract. This proves22the claim. [Note that the least-cost corruption-proof contract is strictly better since it also
pays the agent the same wage for σ œ {«, 1} but the supervisor earns no ex ante rent.](i) If (EF1) is violated, i.e., s1 < s∅, then the agent gets identical payoffs for σ = ∅ or σ =1; the same is true for the supervisor.
Define Tk: Tk = wk + sk for k = {0, ∅, 1}, and define m by Tm = max {T0, T∅, T1}. Thendefine wrσ and srσ as the agent and the supervisor’s respective payoffs (from Nashbargaining where relevant) when the signal is σ and the supervisor reports r.
(a) If Tm = T∅: Given s1 < s∅, the supervisor will report r = ∅ when σ = {∅, 1}, and theagent will not find it profitable to bribe the supervisor into announcing r = 1. Therefore,
payoffs will be: wm1 = wm∅ = w∅ ; sm1 = sm∅ = s∅.
(b) If Tm > T∅: The supervisor reports r = m and the coalition receives Tm for σ = {∅, 1}.
Their payoffs are given by Nash bargaining. Since only the supervisor reports, the threat
point is r = ∅ for σ œ {∅, 1} since s1 < s∅. The bargaining problem is given bymax ( u ( w) − u ( w∅ ) ) ( s − s∅ )
α

1−α

w, s

s.t.

w + s = Tm ,where α œ (0, 1) is the agent’s bargaining power. The solution is denoted by wmσ and smσ
for σ œ {∅, 1}. Since the bargaining set and the threat point remain unchanged whether
σ = ∅ or 1, their respective payoffs must also remain unchanged. They are: wm1 = wm∅;
sm1 = sm∅ > 0 since s∅ > s1 ≥ 0.
Therefore, from (a) and (b), we have proved that wm1 = wm∅ regardless of m.
(ii) Expected cost of any contract that induces e = 1 but violates (EF1).
Consider the contract denoted by {wˆ H , wˆ r , sˆr } that induces e = 1, but violates (EF1),sˆ∅ > sˆ1 . Then the expected cost is:23π ( wˆ H ) + (1 – π) (Tl m ) where Tl m = max {Tl 0 , Tl φ , Tl 1 },and {wˆ H , wˆ r , sˆr } satisfy the (IC) constraint:(IC)

π u( wˆ H ) + (1 – π){p u( wˆ m1 ) + (1 - p) u( wˆ m∅ )} - ϕ ≥ p u( wˆ m 0 ) + (1 - p) u( wˆ m∅ ).

Define Wˆm = wˆ m1 = wˆ m∅ , Sˆm = sˆm1 = sˆm∅ and simplify (IC):17(IC)l m ) – ϕ ≥ p u( wˆ )π u( wˆ H ) + (p – π) u( Wm0

Note that S m > 0 since the supervisor receives at least ŝ∅ from Nash bargaining
and sˆ∅ > sˆ1 ≥ 0 .
(iii) Implement e = 1 with a (constructed) corruption-proof contract {w′H , wr′ , sr′ } that has
the same expected cost as {wˆ H , wˆ r , sˆr } .
l m , w′ = 0, s′ = s′ = S m ,Construct {w′H , wr′ , sr′ } by defining: w′H = wˆ H , w1′ = w∅′ = W
01∅
and s0′ = Tl m .
Check that {w′H , wr′ , sr′ } is indeed corruption-proof and implements e = 1:
(CIC) is satisfied since

wk′ + sk′ = Tl m ,k ∈ {0, ∅, 1},(EFk) is satisfied sincesk′ ≥ s∅′k ∈ {0, 1}, and(IC) is satisfied since

w'k must satisfy (IC) given that wˆ k satisfies (IC) where k ∈{H, m0, m∅, m1} and given that w0′ ≤ wˆ m∅ .
Finally, note that {w′H , wr′ , sr′ } is not the least-cost corruption-proof contract since S m > 0,
whereas in least-cost corruption-proof contract s10 = s∅0 = 0. Therefore, the least-cost
opportunity-proof contract strictly dominates both {w′H , wr′ , sr′ } and {wˆ H , wˆ r , sˆr } . É

24

Appendix D

Proof of the Proposition

The agent-supervisor coalition will choose the report to maximize their joint payoff,which will be Tm. Note that since we do not impose (CIC) constraints bribery may
potentially occur. Then the objective function becomes

π wH + (1 – π) Tm
From lemma 2 we know that the (EF1) must be satisfied:(EF1) s1 ≥ s«.
The (IC) constraint is:
π u( wH ) + (1 – π) p u( wm1 ) – π (1 – p) u( wm∅ ) – p u( wm 0 ) – ϕ ≥ 0,
where wrσ denotes the agents payoff from Nash bargaining when the report is r and thesignal is σ. We ignore the constraint (EF0) for now and verify later that it is indeed
satisfied by the optimal contract.
We consider three cases depending on whether m = 1, ∅, or 0 respectively, and show thatcase I is optimal.
Case I: Tm = T1
Min π wH + (1 – π) T1(IC)

π u( wH ) + (1 – π) p u(w1) – π (1 – p) u(w1∅) – p u(w10) – ϕ ≥ 0(EF1) s1 ≥ sφ
We make some observations to simplify the optimization problem.
(a) Note that wm1 = w1 because s1 ≥ s∅ and Tm = T1. The Nash Bargaining Solution (NBS)
implies that s11 = s1, and w11 = w1.
(b) T0 = T1 and w0 = 0: To see this, note that w0 and s0 only appear in (IC) through w10.
By setting s0 = T1 and w0 = 0 the principal can make w10 = 0 and this does not cost the17

Note that s0 could be larger or smaller than s∅ – both cases are captured in wˆ m 0 .
25

principal anything since s0 does not appear in the objective function. Given that s0 = T1
and w0 = 0, T0 = T1.
Since s0 = T1, we have s0 ≥ s∅, and (EF0) is satisfied.
(c) w∅ = 0: To see this, note that w∅ does not appear in objective function and enters only
the (IC) through w1∅ via the threat-point payoff of the agent in the Nash bargaining
problem. The Nash bargaining problem that determines w1∅ and s1∅ is given bymax ( u ( w) − u ( w∅ ) ) ( s − s∅ )
α

1−α

w, s

s.t.

w + s = w1 + s1

It can be shown that a decrease in w∅ decreases w1∅. Therefore, from the (IC) w∅ = 0.
(d) s∅ = s1: To see this note that s∅ does not appear in objective function and enters only
the (IC) through w1∅ via the threat-point payoff of the supervisor. It can also be shownthat an increase in s∅ reduces w1∅. Therefore, from the (IC) the principal can raise s∅
until (EF1) binds and thus s∅ = s1.
(e) s1 = 0: In the Nash bargaining problem, s = s1 + w1 – w. Since s∅ = s1, the bargaining
problem becomes max (u(w))α (w1 – w)1-α, which is independent of s1. Therefore, s1 canbe reduced to zero to minimize the objective function.
Given (a), (b), (c), (d), (e) and the binding (IC) constraint, we can write the Lagrangian as
follows:
L = π wH + (1 – π) w1 - λ [ π u( wH ) + (1 – π) p u(w1) – π (1 – p) u(w1∅) – ϕ]
∂L

∂L

∂w H

∂w1

= π – λ π u′(wH) = 0(c1)= (1 – π) – λ[(1 – π) p u′(w1) – π (1 – p) u′(w1∅)

26

dw1∅
]=0
dw1

(c2)

From (c1)u′(wH) =From (c2)u′(w1) =1

λ,1 π (1 − p )dw+u′(w1∅) 1∅ .
λ p (1 − π ) pdw1

Since the bargaining set becomes bigger as w1 increases, it can be shown that

dw1∅
> 0,
dw1

and therefore u′(wH) < u′(w1), which implies wH > w1.
The solution is such that wH > w1 > 0 = s1 = s∅ = w∅ = w0 and s0 = w1 = T1. Note that the(CIC) is violated when σ = ∅ – the coalition is strictly better off by reporting r = 1 or r
= 0.
Case 2: Tm = T∅
Min π wH + (1 – π) T∅(IC)

π u( wH ) + (1 – π) p u(w∅1) – π (1 – p) u(w∅) – p u(w∅0) – ϕ ≥ 0(EF1)

s1 ≥ s∅

We make some observations to simplify the optimization problem.
(a) w∅ ≥ w1: To see this, note that T∅ ≥ T1 and s1 ≥ s∅.
(b) s0 = T∅ and w0 = 0: To see this note that s0 and w0 only appear in (IC) through w∅0.
By setting s0 = T∅ and w0 = 0, the principal can make w∅0 = w0 = 0 since s0 does notappear in the objective function. Given s0 = T∅ and w0 = 0, we have T0 = T∅. Note alsothat (EF0) is satisfied since s0 = T∅ ≥ s∅.
(c) w1 = w∅: To see this, note that w1 only appears in (IC) through w∅1 via the threat point
payoff of the agent. Therefore the principal can increase w∅1 and relax the (IC) by
increasing w1. Since w∅ ≥ w1 from (a), w1 will be increased until w1 = w∅.

27

(d) s1 = s∅: To see this, note that s1 only enters (IC) through w∅1. The principal canincrease w∅1 by reducing s1 since s1 is the threat-point payoff of the supervisor. It canalso be shown that a decrease in s1 reduces w∅1. Therefore, from the (IC), the principal
can reduce s1 until (EF1) binds and thus s1 = s∅.
(e) w∅1 = w∅ = w1: To see this, note that s1 = s∅, w1 = w∅ and T1 = T∅.
(f) s∅ = 0: given that w∅0 = 0, s∅ only appears in the objective function and therefore canbe reduced to zero.
Also, since T∅ = T1 = w1, we can rewrite the minimization problem as
Min π wH + (1 – π) w1(IC)

π u( wH ) + (p – π) u(w1) – ϕ ≥ 0

And the Lagrangian is:
L = π wH + (1 – π) w1 + λ [ π u( wH ) + (p – π) u( w1 ) – ϕ].
The FOCs give the optimal wH and w1 for case II:
∂L
∂L

∂wH
∂w1

= π – λ π u′(wH) = 0(c3)= (1 – π) – λ (p – π) u′(w1) = 0(c4)

Therefore, we have shown that the optimal contract under case II is the least-costcorruption-proof contract.
Case 3: Tm = T0
Min π wH + (1 – π) T0(IC)

π u( wH ) + (1 – π) p u(w01) – π (1 – p) u(w0∅) – p u( w0 ) – ϕ ≥ 0(EF1)

s1 ≥ s∅

We make a few observations to simplify the optimization problem.

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(a) s0 = T0 and w0 = 0: To see this, note that in the NBS w01 and w0∅ are not affected by
the distribution of T0 between s0 and w0 as long as w0 + s0 remains the same. Note that by
reducing w0, (IC) can be relaxed and the objective function reduced. Therefore theprincipal sets w0 = 0 and s0 = T0. Note that (EF0) is also satisfied since s0 = T0 = Tm ≥ s∅.
(b) s1 = s∅ and w1 + s1 = T0: To see this, note that s1 and w1 only affect w01. By
decreasing s1 and increasing w1, w01 can be increased and (IC) relaxed. Therefore, s1 is
reduced until (EF1) binds, and thus s1 = s∅. And w1 is increased until w1 + s1 = T0 since
T0 is Tm.
(c) s∅ = w∅ = 0: To see this, note that in the Nash bargaining problem s = w1 + s1 – w
since T1 = T0. Since s1 = s∅, the Nash bargaining problem that determines w0∅ becomesmax [u ( w) − u ( w∅ )] ( w1 − w)1−ααwwhich is independent of s∅. Therefore, s∅ is reduced to zero to relax the (IC) since (EF1)binds from (b). Reducing s∅ allows the principal to reduce s1 and increases w01 to relax
the (IC). From the NBS w0∅ is reduced by decreasing w¯ to zero and therefore relaxing
the (IC). Finally, since s1 = s∅ = 0, w1 = T0.
We have proved that the optimization problem and thus the solution for case III is
identical to case I. Therefore to find the optimal solution, we only need to compare cases
I and II which we do now.
(Case I) Min π wH + (1 – π) w1(IC)

π u(wH) + (1 – π) p u(w1) – π (1 – p) u(w1∅) – ϕ = 0(Case II) Min π wH + (1 – π) w1(IC)

π u(wH) + (p – π) u(w1) – ϕ = 0

Since Nash bargaining implies w1∅ < w1 for α <1, the lowest expected cost under case II
can be achieved under case I with a slack (IC). Therefore, the optimal contract under29

case I results in a smaller expected cost than case II. We have proved that case I is
optimal, and it will induce bribery when σ = ∅.

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