Further Notes on Information, Corruption, And Optimal Law Enforcement

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Abstract We consider two important notes on optimal law enforcement with cor-ruption. First, we analyze the role of asymmetric information on the emergence of collusion between criminals and enforcers. Second, our paper proposes that theoptimal criminal sanction for the underlying offense is not necessarily maximal. Weachieve this result by coupling the criminal sanction for the underlying offense with acriminal sanction for corruption, both imposed on offenders. A higher criminal sanc-

tion for the underlying offense implies that the government must spend more resourcesto detect and punish corruption (since the likelihood of collusion increases). Thus, thegovernment could reduce this sanction, save on detection, and increase the criminalsanction for corruption (in order to offset the negative effect on deterrence).

Keywords Fine Æ Probability of detection and punishment Æ Corruption ÆInformation

JEL Classification K4

1 Introduction

Corruption has been an important issue in the economic literature of lawenforcement. In their seminal article, Becker and Stigler (1974) argued that it

We are grateful to Mitch Polinsky and two anonymous referees for helpful suggestions. The usualdisclaimers apply.

N. Garoupa (&)

Faculdade de Economia, Universidade Nova de Lisboa, Campus de Campolide,1099-032 Lisbon, Portugale-mail: [email protected]

M. JellalThe Institute of Economic Analysis and Prospective Studies, Al Akhawayn University,PO. Box 104, Avenue Hassan II, Ifrane, Moroccoe-mail: [email protected]

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Eur J Law Econ (2007) 23:59–69DOI 10.1007/s10657-007-9002-1

Further notes on information, corruption, and optimal

law enforcement

Nuno Garoupa

Mohamed Jellal

Published online: 15 May 2007 Springer Science+Business Media, LLC 2007


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might be advantageous to extend private enforcement to the criminal law andother areas where the law is now enforced publicly. Their principal argument wasthat public enforcement creates incentives to bribery which undermine deter-rence. If law enforcement were privatized, however, competitive private enforcers

could be rewarded with the fines offenders paid and enforcers would have noincentive to take bribes. In the last few years scholars have begun to pay moreattention to corruption in corruption.1 A central conclusion of this literature isthat corruption is usually socially undesirable, because it dilutes deterrence. As aconsequence, it is usually optimal to expend resources to detect and penalizecorruption.

The first point of our paper is to consider the role of asymmetric information onthe emergence of collusion between criminals and enforcers, in the frameworkproposed by Bowles and Garoupa (1997) and Chang et al. (2000), and to sameextent by Polinsky and Shavell (2001). Indeed, particularly in the case of casual

corruption, the hypothesis of asymmetric information about private costs (oppor-tunity costs) of enforcers engaging in collusion seems very plausible. The asymmetryof information leads the enforcers to overestimate this cost in order to increase theircorruption rent. Therefore, asymmetric information might eventually deter corrup-tion and deter bargaining between the two parties. We show otherwise, and discussthe shortcomings of the solutions proposed by Bowles and Garoupa (1997) andPolinsky and Shavell (2001). In particular, we provide a comprehensive character-ization of incentives and derive an endogenous likelihood of corruption underasymmetric information.2

Our second point is to consider a collusion-proof solution as in the regulatoryliterature (Laffont and Tirole 1993). We show that the optimal criminal sanction forthe underlying offense3 is not necessarily maximal as in Bowles and Garoupa (1997)and Polinsky and Shavell (2001).4 We achieve this result by coupling the criminalsanction for the underlying offense with a criminal sanction for corruption, bothimposed on offenders. A higher criminal sanction for the underlying offense impliesthat the government must spend more resources to detect and punish corruption(since the likelihood of collusion increases). Thus, the government could reduce thissanction, save on detection, and increase the criminal sanction for corruption (inorder to offset the negative effect on deterrence).5

1 Bowles and Garoupa (1997), Chang et al. (2000), and Polinsky and Shavell (2001) model cor-ruption under a regime of public enforcement. Marjit and Shi (1998) and Garoupa and Klerman(2001) model corruption under a regime of private enforcement, and Garoupa and Klerman (2004)discuss non-monetary sanctions when there is collusion between offenders and enforcers. AlsoHarris and Ravin (1978), Besley and McLaren (1993), Flatters and MacLeod (1995), and Mook-herjee and Png (1995) discuss compensation of enforcers in a context of corruption.2 For example, Laffont and Martimort (1997) look at collusion under asymmetric information, butnot in a specific context of enforcement.3 The underlying offense is the offense detected by the enforcer and hence creates the oppor-tunity for corruption. Examples could be traffic violations, drug traffic offenses or moneylaundering.4 Bowles and Garoupa (1997) already have a brief extension to harmful corruption where they arguethat the maximal fine might not be optimal. They do not however relate the argument to the nowstandard corruption proof solution of Laffont and Tirole (1993).5 Although the nature of the result is independent of asymmetry of information, the magnitudedepends on the incentives generated by the asymmetry of information concerning private costs.

60 N. Garoupa, M. Jellal

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We differ from Chang et al. (2000), because we do not incorporate social norms.They show that a less than maximal sanction could be optimal due to the ‘snow-balling’ effect of social norms; raising fines could be counterproductive in deterringcrime if collusion is widespread. Our result instead could be seen as an application of

the marginal deterrence principle. If detecting the underlying offense is moreexpensive than detecting corruption, deterring the underlying offense is relativelymore important than deterring corruption. Thus, the optimal criminal sanction forthe underlying offense is maximal. However, if detecting the underlying offense isless expensive than detecting corruption, deterring corruption is relatively moreimportant than deterring the underlying offense. Thus, the optimal criminal sanctionfor the underlying offense is less than maximal, whereas the optimal sanction forcorruption is higher than otherwise.6

The paper goes as follows: In Sect. 2, we introduce the basic model from Bowlesand Garoupa (1997) and Polinsky and Shavell (2001). In Sect. 3, we analyze the

bribing game under asymmetric information; in section four, we discuss optimaldeterrence in the context of a corruption proof solution. Final remarks are addressedin section five. Proofs of results are in appendix at the end of the paper.

2 Basic model

Each risk neutral individual decides whether or not to become an offender. Once heacts as a criminal, he may be detected or not (the probability of detection and

punishment of an offender is p). If detected, he starts a bargaining process over thebribe with the police officer. If such process is successful (it will be with probability1–r , where r is the rate of honesty across enforcers), he pays a bribe R to thecorrupted police officer. After the bribery has occurred, the corrupted police officermay be detected (the probability of detection and punishment of an enforcer is q),and if so, both enforcer and offender are punished.

There are four possible states of nature: (a) an offender is not detected, (b) anoffender is detected and does not collude with the enforcer (he pays the sanction forthe underlying offense, that is, a fine F ), (c) an offender is detected and colludes withthe enforcer; they are detected by the government (an offender pays a fine F plus a

penalty T for corruption, and the enforcer pays a fine S), (d) an offender is detectedand colludes with the enforcer; they are not detected by the government.

The prospective gain from crime to the offender is b, and the expected utility of each potential offender who commits the act is given by

U ¼ ð1 pÞb þ pfr ðb F Þ þ ð1 r Þqðb R F T Þ þ ð1 r Þð1 qÞðb RÞg

¼b p½r þ ð1 r ÞqF pð1 r ÞqT pð1 r ÞR: ð1Þ

The expected utility of an enforcer who accepts the bribe is:

V ¼ ð1 qÞR þ qðR S wÞ

¼ R qðS þ wÞ ð2Þ

6 Notice that on the side of the enforcer the marginal deterrence problem is less interesting sincethere is only one offense, i.e., accepting a bribe (corruption).

Optimal law enforcement with corruption 61

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where w includes, for example, psychological costs and opportunity costs borne by apolice officer when she is bribed and shamed for exposition (hence, we could say thatit is the cost of shame).

3 Bribing game under asymmetric information

In Bowles and Garoupa (1997), the costs represented by w are public information inthe moment of the bargaining. In our model, w is private information to enforcers.We relax the assumption that w is observable by criminals and suppose that it isknown only to the officer when the bargaining takes place. Indeed, particularly in thecase of casual corruption, the hypothesis of asymmetric information about privatecosts of enforcers engaging in collusion seems very plausible.

The asymmetry of information of course leads the enforcers to overestimate this

cost in order to increase their corruption rent. Therefore, asymmetric informationmight eventually deter corruption and break bargaining between the two parties.The purpose of this section is to analyze the extent asymmetric information affectscorruption.

The private cost w is a continuous parameter that belongs to the closed interval I ¼ ½w; w , where w[w . Hereafter w denotes a realization of a random variable withcumulative distribution L(w) and the corresponding positive probability densityfunction l (w). By making use of the incentive contract theory and the revelationprinciple as developed by Laffont and Tirole (1993) we can show the following

result:

Proposition 1 Corruption occurs with the following probability: one if w w0 andzero if w[w0. The endogenous number of corrupted officers is given by1 r ¼ Lðw0Þ, where

ð1 qÞF ¼ qðS þ T þ w0 þ Lðw0Þ=l ðw0ÞÞ

Contrary to Bowles and Garoupa (1997) and Polinsky and Shavell (2001), we

have an endogenous likelihood of corruption under asymmetric information. Ahigher fine for the underlying offense seems to reduce the rate of honesty sincecriminals are willing to pay higher bribes, making officers more willing to accept abribe. Punishment of corruption deters bribing by increasing the rate of honestyacross offenders, either in the form of more severe punishment for corrupted officersand corrupter offenders, or in the form of higher likelihood of punishment.

The optimal bribe under asymmetric information is given by qðS þ w0Þ , for allw w0 . This is an important corollary because it implies that the amount offered asbribe does not depend on the type of enforcers w 2 I . The bribe is determined at themarginal level (by type w0). The informational rent for an enforcer type w w0 is a

proportion of the difference between her own private costs and the critical level forbribing given by qðw0 wÞ , for all w w0.

Having determined the optimal bribe contract in presence of corrupted officersand asymmetric information, we analyze the incentive individuals have to committhe underlying offense. We rewrite (1) as:


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L ¼

Z 1 pF

ðb hÞ gðbÞ db c1F =ðF þ S þ T þ wÞ c0 p þ k0ðF F T Þ þ k1ðS SÞ

ð5Þ

where hk0; k1i are the associated multipliers.

Proposition 2 Corruption-proof law enforcement is characterized by

(i) F ðh pF Þ gð pF Þ ¼ c0,(ii) S ¼ S,(iii) F ¼ F and T = 0 if c0 p c1~q,(iv) F ¼ c0 pF =ðc1~qÞ and T ¼ F F if c0 p\c1~q.

The collusion proof solution implies that it is optimal to impose a maximal fineon officers while the optimal fine on criminals may be lower than maximal. Thisresults contrasts with Polinsky and Shavell (2001) since they have shown that allfines should be maximal when controlling corruption. The rationale for this resultis obtained when enforcement costs are higher for corruption than for theunderlying offense. The fine imposed on criminals should be less than maximalbecause a higher sanction makes bribes more likely and thus more costly todeter. However, note that the fine should not be zero since then no one wouldbe punished and everyone would commit the underlying offense. The sanctionfor corruption imposed on criminals is positive and determined by the wealth

constraint.When enforcement costs are higher for the underlying offense than for corrup-

tion, the fine imposed on criminals should be maximal and, by the wealth constraint,the fine for corruption imposed on offenders is zero. It is more important to detercrime than corruption because crime detection is more expensive than corruptiondetection.

5 Conclusion

This paper presents a simple model to evaluate the alternative enforcement policiesin presence of corruption with asymmetry of information. We have shown that theoptimal criminal sanction for the underlying offense is not necessarily maximal in acorruption-proof solution.

There are two main public policy implications from these notes on optimal lawenforcement with corruption. First, if deterring corruption is a priority for thegovernment, then the fine for the underlying offense should not be maximal incontrast with Bowles and Garoupa (1997) and Polinsky and Shavell (2001) be-cause some marginal cost should be borne by an offender who prefers to bribe

the prosecutor rather than pay his dues. Second, by creating more shame for onlya subset of corrupted enforcers, the government is able to reduce corruption andhence also deter the underlying offense more effectively. The rationale is thatthere is asymmetry of information, and hence the relevant indicator for offendersis the distribution of costs from shame, not the exact cost for a particular en-forcer. By concentrating shame on a small group of enforcers (but enough to


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change the individual at the margin), the government may achieve generaldeterrence.

Appendix: Proofs

Proof of Proposition 1

We make the following (usual) assumption about the distribution L(.); it satisfies themonotone hazard rate property:

@

@ w

LðwÞ

l ðwÞ

0

In this setup, an offender designs a compensation (bribe) structure that maximizeshis expected utility while guaranteeing the enforcer at least her reservation utility.8

The proof consists in maximizing the expected utility of an offender subject to thestandard participation and incentive compatibility constraints by which the enforceris willing to participate and reveal her private costs.

From Laffont and Tirole (1993), it is well known that, without loss of generality,one can restrict the search to the class of mechanisms that induces a truthful reve-lation of the enforcer’s cost parameter w. In our context, it is easy to see that anyoptimal mechanism M that induces a truthful reporting can be represented as the

following M ¼ hrðwÞ; RðwÞi I , where r (w) is the probability of offering a bribe of amount R(w) for the corrupted enforcer of type w 2 I .Given a mechanism M , let the level of utility achieved by the officer of type w if

she reports type ~w be Vðw; ~wÞ ¼ rð~wÞ½Rð~wÞ qðS þ wÞ, where VðwÞ ¼ Vðw;wÞ de-notes truthful reporting of costs.

The incentive compatibility constraint (IC) to guarantee truthful reporting is givenby Vðw;wÞ Vðw; ~wÞ, for all w; ~w 2 I . The individual rationality constraint (IR) isVðwÞ 0, for all w 2 I .

In this context the problem for the offender is to maximize his expected utilitysubject to (IC) and (IR). Once detected by an officer, the expected utility of a

criminal is:

U c ¼rðb R qF qT Þ þ ð1 rÞðb F Þ

¼b F rðR þ qT ð1 qÞF Þ

where the bribe is accepted with probability r.The criminal solves the following program:

8 The alternative sequential model would be for the enforcer to propose a menu of contracts to anoffender. Another possibility is to consider a simultaneous (bribing) game that, given the linearity of the menu of contracts, is analytically isomorphic to the game we present. Given the need forconsistent beliefs and the nature of the asymmetry of information, any other model would be morecomplex. See Inderst (2002). Furthermore, we are in the context of casual corruption where it makesmore sense to expect offenders to make a first move.


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maxðr;RÞ

Z I

½b F rðwÞðRðwÞ þ qT ð1 qÞF Þ dLðwÞ

subject to (IC) and (IR).

We are now able to solve the problem of side payments (bribing) with asymmetricinformation. To find the optimal solution, we begin by characterizing the class of bribing contracts that satisfies the incentive constraint in order to implement M in adominant strategy.

Define V cðwÞ such that:

V cðwÞ ¼ max~w2 I

frð~wÞ½Rð~wÞ qðw þ SÞg

Then V cðwÞ is an upper envelope of a linear function in w, it is convex and we havealmost everywhere (using the envelope theorem):

V 0cðwÞ ¼ qrðwÞ

V 00c ðwÞ ¼ qr0ðwÞ 0

only if r0ðwÞ 0 for all w 2 I .By integration of V 0cðwÞ such that V cð

wÞ ¼ 0 , we obtain

V cðwÞ ¼ qZ ww

rð xÞ d x

The following lemma follows:

Lemma 1 The bribe contract satisfies the incentive constraint if and only if

(i) V cðwÞ ¼ qR ww rð xÞ d x and

(ii) r0ðwÞ 0 for all w 2 I .

The rent from corruption V cðwÞ is the informational rent left to an officer of type w by the criminal. Indeed because of asymmetric information about privatecosts of enforcer, the criminal is forced to give up a costly rent to the officer. Theinformational rent is used to discipline the enforcer into revealing her trueprivate cost of being bribed. From the first lemma, we can remark that theinformational rent is decreasing in w. Hence to be willing to reveal her cost, alower w type must be rewarded with a more substantial rent than a higher wtype. Furthermore, from the monotonicity assumption such that r0ðwÞ 0 , anofficer with low private cost is characterized by an increased probability of beingoffered a side contract (i.e., a bribe). Note also that the informational rentincreases with r(.).

In order to find the components of the optimal bribing contract M , we mustdetermine the expected utility of the criminal. From the definition of V cðwÞ we have:

rðwÞRðwÞ ¼ qðw þ SÞrðwÞ þ V cðwÞ


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where the left-hand-side is the expected bribe, and the right-hand-side is the sum of the expected cost of being bribed plus the informational rent.

The expected utility of an offender is:

U ¼Z

I ½b F rðwÞðRðwÞ þ qT ð1 qÞF Þ dLðwÞ

¼

Z I

½b F þ rðwÞð1 qÞF rðwÞqðw þ S þ T Þ V cðwÞ dLðwÞ

with

V cðwÞ ¼ q

Z ww

rð xÞ d x

as showed in lemma one.Then, after integrating by parts, we derive:

U ¼

Z I

½b F þ rðwÞð1 qÞF rðwÞqðw þ S þ T Þ rðwÞqLðwÞ=l ðwÞ dLðwÞ

The first part of the proposition follows easily from maximizing U in r. Theoptimal r* (w) is determined by the first-order condition:

@ U =@ rðwÞ ¼ ð1 qÞF qðS þ T þ w þ LðwÞ=l ðwÞÞ

Then r* = 1 if ð1 qÞF [qðS þ T þ w þ LðwÞ=l ðwÞÞ. And r ¼ 0 if ð1 qÞF \qðS þ T þ w þ LðwÞ=l ðwÞÞ. Let w0 be given by

ð1 qÞF ¼ qðS þ T þ w0Þ þ qLðw0Þ=l ðw0Þ

Therefore, from assumption one (on monotone hazard rate of L(.)), it follows thatrðwÞ ¼ 1 holds only if w w0 given that r

0ðwÞ 0 . Conversely rðwÞ ¼ 0 whenw[w0. The second part of the proposition is a corollary.

Also notice that from lemma one we have:

V cðwÞ ¼ q

Z w0w

rð xÞ d x þ q

Z ww0

rð xÞ d x ¼ qðw0 wÞ

By construction, we have:

RðwÞ ¼ qðS þ wÞ þ V cðwÞ

where the bribe (when occurs with probability one) equals expected cost plus

informational rent. Then, for w w0,

RðwÞ ¼ qðS þ wÞ þ qðw0 wÞ ¼ qðS þ w0Þ


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Proof of Proposition 2

The first-order conditions are:

L p ¼ F ðh pF Þ gð pF Þ c0 ¼ 0LS ¼ c1F =ðF þ S þ T þ wÞ

2 k1 ¼ 0

LF ¼ pðh pF Þ gð pF Þ c1ðS þ T þ wÞ=ðF þ S þ T þ wÞ2 k0 ¼ 0

LT ¼ c1F =ðF þ S þ T þ wÞ2 k0 ¼ 0

Second-order conditions are assumed to be satisfied. From the first-order condi-tion with respect to S, we have k1[0 and S ¼ S.

From the first-order condition with respect to p, we can write:

ðh pF Þ gð pF Þ ¼ c0=F

From the first-order condition with respect to T , we can write:

k0 ¼ c1F =ðF þ S þ wÞ2

We can rearrange the remaining first-order condition as:

LF ¼ pc0=F c1=ðF þ S þ wÞ ¼ 0

The optimal solution depends on the following situations:

(a) c0 p c1~q. We should have F ¼ F and T = 0.(b) c0 p\c1~q. We should have F ¼ pc0 F =ðc1~qÞ and T ¼ F F .

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Further Notes on Information, Corruption, And Optimal Law Enforcement

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