Éric Langlais
email: eric.langlais@u-paris10.fr.z

Marie Obidzinski
email: marie.obidzinski@u-paris2.fr.

Eric Langlais

Public law enforcers and political competition

Keywords: public law enforcement, deterrence, monetary sanctions, electoral competition. JEL classi…cation codes : D72, D73, H1, K14, K23, K4

In this paper, we analyze how political competition a¤ects the design of public law enforcement policies. Assuming that the cost of enforcement is linear, criminals'type is uniformally distributed, and society's wealth is large enough, the article arrives at two main conclusions: 1) electoral competition entails no loss of e¢ ciency at equilibrium for both minor and major o¤enses (e.g. minor o¤enses are not enforced, while major ones are fully deterred); 2) di¤erent distortions arises at equilibrium for the intermediate o¤enses: enforcement expenditures for small o¤enses are lower than the optimal level, such that the issue of under-deterrence is exacerbated; in contrast, for larger o¤enses, enforcement measures are higher, and there is more deterrence than what e¢ ciency requires. We show that these results also holds under more general assumptions (convex costs of enforcement, a general cdf of illegal bene…ts, a lower society's wealth), excepted that full deterrence of major o¤enses is not achievable.

Introduction

Political views are often at stake when the public law enforcer chooses the level of sanction, or the means given to the police. In real life, a public law enforcer might be a chief of police, a judge, a chief attorney, a regulator, a city mayor, etc. In many cases, a realistic view of the law enforcer is to consider that her/his decisions regarding the level of sanction and the means given to detection, apprehension and conviction are in ‡uenced by the political market, and as such, by citizens'preferences through the electoral competition.

Enforcement expenditures may increase under the threat of (re)election, an intuition which seems to be supported by data. A large body of empirical literature has indeed documented the in ‡uence of election on the probability of detection, apprehension and conviction. [START_REF] Levitt | Using Electoral Cycles in Police Hiring to Estimate the E¤ect of Police on Crime[END_REF] shows that the number of sworn o¢ cers signi…cantly increases during election years (and remains stable meanwhile). Berdejó and Yuchtman (2012) present evidence that elected judges (in the State of Washington) tend to respond to political pressure by increasing the severity of their judgment: sentences are around 10% longer at the end of a judge's political cycle than at the beginning. [START_REF] Dyke | Electoral Cycles in the Administration of Criminal Justice[END_REF] shows that the district attorneys are less likely to dismiss cases in the election year. Prosecutors running for elections tend also to take more cases to trial rather than plea bargain [START_REF] Mccannon | Prosecutor Elections, Mistakes, and Appeals[END_REF].

By contrast, some legal rules are weakly enforced, if ever. Examples are numerous. Many Parisian cyclists do not abide to tra¢ c law. Many citizens throw their cigarettes butts on the street. The …nes provided for by the law are respectively 90 euros (running red lights) and 68 euros (for throwing the cigarettes), but are scarcely applied. We can also think to illegal downloading, which is considered as largely under-deterred. Furthermore, up to the latest presidential elections, French citizens expected that the new president would award a large amnesty for infractions to speed limits, parking regulation, running red and so on. [START_REF] Makowsky | Political Economy at Any Speed: What Determines Tra¢ cs Citations?[END_REF] shows that the probability of getting a ticket (rather than a warning) for excessive speed and the size of the …ne is negatively a¤ected by the fact to reside in the town (compare to live and vote out of the town). In other words, policemen tend to favor local constituents.

These observations question1 the link between elections and public law enforcement policies. The purpose of our paper is to understand why and how deterrence policies might be a¤ected when the law enforcer is elected as compared to the standard case of a benevolent dictator. Mainly, our purpose is to characterize the distortions in enforcement policies chosen by elected enforcers, both in terms of intensity/severity, and in terms of adequacy to the gravity of o¤enses (harm).

Our main results show that political competition entails no loss of e¢ ciency both at the top and the bottom of the distribution of o¤enses, at least to the extend that society's wealth is large enough and enforcement measures develops with constant marginal cost. For minor o¤enses, the outcome of the political market is the laissez-faire which is the optimal policy; to the opposite, major o¤enses are fully deterred in a political equilibrium, which prooves also to be e¢ cient. More precisely, we show that when o¤enses entail an external harm to society which is lower (larger) than a threshold (that depends on the level of the marginal cost of enforcement measures), a benevolent enforcer and an elected enforcer will agree on the policy to enforce: for both, the best policy consists in no enforcement measures and zero deterrence (maximal enforcement measures, and full deterrence), since enforcement measures are …nanced by taxes and entail a marginal cost which is too high (small) when addressed to minor o¤enses. In contrast, for intermediate o¤enses (i.e. excuding the smallest and the largest), we show that electoral competition promotes ine¢ cient enforcement policies since the preferences of the majority of voters depart from the preferences of the benevolent enforcer. The distortions that appear can be classi…ed according to the size of the external harm, along two areas (let say: small and large). First, for small o¤enses for which the external harm is not negligible (compared to the marginal cost of enforcement) but limited, a "weak enforcement" equilibrium emerges from the electoral process: citizens vote for enforcement's expenditures which are lower than the optimal ones, and the proportion of undeterred o¤enders is larger. Second, for large o¤enses having a higher external harm, a "strong enforcement" equilibrium prevails where citizens vote for high enforcement's expenditures; in this case, the policy reaches a level of deterrence which is higher than the optimal one. In this perspective, our paper addresses the important point of the enforcers motivations and objectives, an issue that has been rarely addressed in the literature stemming from [START_REF] Becker | Crime and Punishment: an Economic Approach[END_REF]. A …rst exception is [START_REF] Friedman | Why Not Hang Them All: The Virtues of Ine¢ cient Punishment[END_REF], who argues that the public law enforcer is merely self interest (as any other agent), and observes that the literature about law enforcement considers criminals as highly sophisticated and rational individuals, while the State is usually considered as a simple "proxy" (benevolent automate) or "a wise, benevolent and wholly altruistic organization"2 . However, as Friedman emphasized, societies do not generally choose the most e¢ cient way to enforce law in practice. One explanation lays with the objectives of law enforcers; they wish to maximize their own rents rather than the social welfare, thus departing from the socially optimal solution of the literature. Therefore, Constitutions should impose costly punishment (such as prison) in order to avoid excessive punishment. In relation with the seminal work of Friedman, Garoupa and Klerman (2002) 3 analyze the issue of law enforcement and deterrence in the realm of the rent seeking model of government. They assume that the objective function of the enforcer is to maximize the revenues minus the harm to the government and the cost of law enforcement. 4 Wickelgren (2003) also build from the point made by [START_REF] Friedman | Why Not Hang Them All: The Virtues of Ine¢ cient Punishment[END_REF] and justi…es the use of costly forms of sanctions (prison rather than corporal punishments), in a model with two enforcing authorities acting sequentially. However, in Wickelgren's work both levels of the enforcement system share a similar objective, i.e. maximizing a (weighted) social welfare function.

Our paper discusses a point similar to [START_REF] Friedman | Why Not Hang Them All: The Virtues of Ine¢ cient Punishment[END_REF], [START_REF] Garoupa | Optimal Law Enforcement with a Rent-Seeking Government[END_REF] and [START_REF] Wickelgren | Justifying Imprisonment: On the Optimality of excessively Costly Punishment[END_REF], although it adopts an alternative view. Speci…cally, rather than assuming that the enforcer's preferences are exogenously …xed, we consider here a case where the objective function of the elected public law enforcer is endogenous, resulting of the electoral game. Our aim is to discuss in a simple framework whether/how political competition may (or not) promote the toughness of enforcement policies in the various domains of administrative law and penal law.

The paper is organized as follows. Section 2 sets the general framework and recall the results obtained in the standard beckerian approach relying on a benevolent planner in order to de…ne our benchmark. Section 3 analyzes the case where the public enforcer is elected with a simple model of Downsian electoral competition. Section 4 shows how these results extend under more general assumptions (a lower level of society's wealth, convex costs of enforcement, a general cdf for illegal bene…ts). Section 5 concludes.

Model and assumptions

We introduce here our basic framework, which elaborates on the model of law enforcement à la Becker5 . Let us consider a population of risk neutral individuals, the size of the population being normalized to 1. Each individual considers the opportunity to engage in the legal activity (and earns 0), or to engage in the illegal activity which yields a bene…t b that varies in the population. Public authorities do not observe the type b, but only know that bene…ts are distributed according to the uniform law on [0; 1]. The uniform distribution assumption makes easier the exposition of results.6 The (external) loss/harm to the rest of the society in case of an o¤ense is h, whatever the private bene…t for the o¤ender.

Monitoring the illegal activity entails a cost for public authorities, equal to m(p), where for the sake of simplicity p is the probability of control (encompassing apprehension and conviction for an illegal behavior). The enforcement cost function writes as m(p) = m:p, with m > 0 8p 2 [0; 1]. The paper highlight …rst this case with a constant marginal cost of enforcement, which has been often exempli…ed in the literature [START_REF] Garoupa | Optimal magnitude and probability of …nes[END_REF][START_REF] Garoupa | Optimal Law Enforcement with a Rent-Seeking Government[END_REF]. We will discuss alternative assumptions in the last section of the paper.

Also, we assume that this enforcement cost is …nanced through a lump sum tax t plus the …ne f levied on the detected (with probability p) o¤enders. The maximal …ne is the legal wealth of the population w, i.e. f 2 [0; w]. For the moment, we make the next assumption:

Assumption 1. w > 1 > m.
The management costs (associated with the monetary penalty), as usual in the literature, are assumed to be negligible. Throughout the paper, we will consider only balance-budget policies. The public budget constraint writes as:

m(p) = t + qpf (1)
where q is the proportion of non abiding people. As usual in the literature, we will show that the proportion of o¤enders equals q = 1 b , with b the deterrence threshold.

O¤enders and law abiding citizens

We assume that an o¤ense hurts citizens through a pure external term a¤ecting individuals' utility level, de…ned as qh. Note that both the criminals and the honest people su¤er from the externalities imposed by o¤enses. 7The population of citizens is distributed along the value of the potential illegal bene…t b 2 [0; 1], but only those citizens who become o¤enders ("criminals") will e¤ectively retain their b, whereas those who abide the law ("honest") will forgive their b. Let us denote the utility level of a risk neutral citizen b who considers the opportunity to become an o¤ender, as:

u c = w + b t pf qh
whereas when he considers the opportunity to be law abiding, he obtains a utility level written as:

u h = w t qh
Hence, if u c > u h the citizen b becomes an o¤ender, i.e. he decides to undertake the illegal activity if the illegal bene…t he receives from doing it is higher than the expected punishment: b pf ; and if u c < u h , he is law abiding. As usual, the marginal o¤ender b = pf is de…ned by u c = u h . Given that b is uniformly distributed on [0; 1], we obtain that q = 1 b = 1 pf .

The benchmark: benevolent enforcer

Let us remind the choice made by a benevolent law enforcer: he determines both the level of …ne f u and the probability of detection p u by maximizing the social welfare function S:

S = Z b 0 u h db + Z 1 b u c db = w t + Z 1 b (b pf h)db
and substituting with (1) yields:

S = w + Z 1 pf (b h)db mp (2)
which is the standard formulation considered in the literature. The …rst (integral) term of S correspond to the expected private bene…t associated with the illegal activity. The last one is the cost of monitoring for public authorities.

The …ne paid by the o¤ender when arrested is a mere transfer (the expected probability of paying the …ne, is equal to the expected probability of collecting it).

We will denote the pair (p u ; f u ) as the optimal enforcement policy. De…ning:

h 1 = m w and h 2 = h 1 + 1, we have:
Proposition 1 The optimal enforcement policy (p u ; f u ) may be one of the three following solutions: i) If h < h 1 , then p u = 0, i.e. no enforcement expenditures, and the policy is associated with zero deterrence. ii) If h 1 < h < h 2 , then p u = (h h 1 ) 1 w > 0; f u = w, and the policy is associated with under deterrence. iii) If h > h 2 , then p u = 1 w ; f u = w, and the policy associated with full deterrence.

Proof. The derivatives of S with respect to f and p are given by:

@S @f = (h pf )p @S @p = (h pf )f m
We have:

@S @p jp=0 = hf m i) Thus, if hw m < 0 , h < m w , then @S @p jp=0
< 0 and it must be that p = 0, and the choice of f is of no matter. ii) On the other hand, if hw m > 0, it is not optimal to choose f = h=p < w since this would imply @S @p < 0 for any p > 0. Hence, it must be that f u = w. Note that for the optimal …ne, we also have:

@S @p jp=1=w = (h 1)w m
Thus, as long as h < h 1 +1, there exists an interior solution where p u satis…es @S @p = 0 or:

(h p u w)w = m (3) implying h p u w > 0. Solving for p u yields p u = (h h 1 ) 1 w > 0. iii) Finally, when h > h 1 + 1, @S @p jp=1=w
> 0 such that the optimal policy is (p u = 1 w ; w), and full deterrence is achieved (since p u w = 1).

Proposition 1 is depicted in the next graph: It reminds us that the optimal enforcement policy for minor o¤enses (h < h 1 ) is the laissez-faire. In contrast, for intermediate o¤enses (h 1 < h < h 2 ), the best policy consists in enforcement expenditures mixed with a maximal …ne, and under deterrence occurs. Finally, for the largest o¤enses (h > h 2 ), maximal enforcement measures allowing complete deterrence are optimal.8

Law enforcement under political competition

In this section, we depart from the usual assumption that the enforcer is benevolent. Instead, we assume that he is elected; for that purpose, we introduce a simple model of electoral competition in the vein of the Downsian model (see [START_REF] Downs | An Economic Theory of Democracy[END_REF]. Assume there exist two candidates i = 1; 2 representative of two political parties, competing for national (presidential or legislative) or local (municipal) elections. Competing for elections here is alike a rent seeking contest, where V , the exogenous rent obtained in case of victory is attached to holding o¢ ces, ministries and so on.

The objective of politician i is to maximize the expected value of the rent i V , where i is the probability that he wins the elections. To this end, candidate i proposes to electors an electoral platform (f i ; p i ). We consider the (simple) majority rule for voting. All citizens are electors and do participate: each voter simply votes for the candidate whose platform allows him to reach the highest utility level, and if he is indi¤erent, he tosses a coin to decide for whom he votes.

The electoral competition game between the candidates and the citizens/voters is as follows: after Nature moves at stage 0 (choosing the type of citizens, not observable for politicians), the electoral competition begins at stage 1, which is a simultaneous move (non cooperative) game between the candidates, where they both choose and announce their platforms (f 1 ; p 1 ), (f 2 ; p 2 ), both satisfying the balanced budget constraint (1); at stage 2, elections take place, and citizens simultaneously choose between the two candidates9 ; at stage 3, the elected candidate implements his policy 10 -it becomes a law; at stage 4, citizens choose to abide or not the law; at stage 5, law is enforced.

In the next paragraphs, we solve for the equilibrium. To this aim, we specifically highlight two main stages: stage 2, where citizens vote (3.1), and stage 1 where candidates choose their policies (3.2).

Analysis of citizens'best policies

Solving backward, it comes that at stage 4, any policy (p; f ) that is implemented after the elections will induce a screening of citizens between those who abide the law, and those who do not. The analysis of paragraph 2.2 still holds, and it is straightforward that the deterrence threshold at equilibrium is b = pf .

At stage 2, each citizen depending on his type b consistently takes into account that in the future, either he will comply or not to the law, and vote for his preferred policy which maximizes his utility over the whole game.

Best decision of a citizen who will comply

For a citizen who anticipates he will comply to the law, the policy he votes for at stage 2 is: (f h ; p h ) = arg max (f;p) fu h under (1)g. Substituting (1) in u h leads to:

u h = w m:p + (1 pf ) (pf h)
We obtain:

Proposition 2 Law abiding citizens vote for a policy (p h ; f h ) which is one of the following solutions:

i) If h < h 2 , then p h = (1 + h h 1 ) 1 2w < 1; f h = w. ii) If h > h 2 , then p h = 1 w ; f h = w.
Proof. The derivatives of u h with respect to f and p are:

@u h @f = (1 + h 2pf ) p @u h @p = (1 + h 2pf )f m
We have:

@u h @p jp=0 = (1 + h)f m
Note that at f = w, @u h @p jp=0 cannot be negative under assumption 1. 11 Furthermore, it is not rational for an individual to choose f < w (i.e. such that @u h @f = 0) since this would imply that @u h @p = m < 0 for any p > 0. Thus, for the maximal …ne f h = w, we obtain: @u h @p jp=1=w = (h 1)w m and as long as h < h 2 , p h is the solution to @u h @p = 0, or:

1 + h 2p h w = m w (4)
which implies h p h w ? 0. Solving for p h yields p h = (1 + h h 1 ) 1 2w > 0. On the other hand, when h > h 2 , @u h @p p=1=w > 0 meaning that the best policy is (p h = 1 w ; w) and full deterrence is achieved (since p h w = 1).

Note that compliant citizens prefer a positive probability of detection and conviction p h > 0, even for minor o¤enses that wouldn't be worth to deter from the point of view of e¢ ciency (i.e. for h < h 1 , p u = 0). On the other hand, for larger o¤enses, they prefer enforcement measures higher as well as lower than the optimal ones. More precisely, one can show that:

1 1 Assuming that w < m, would imply that @u h @p jp=0 > 0 only for h > m w 1. However, this new threshold is not relevant for the equilibrium analysis.

Proposition 3 Law abiding citizens vote for a policy with enforcement measures that are never lower than the optimal ones.

Proof. Straightforward from the comparison of p h and p u , since:

i) if h < h 2 , then p h > p u ; ii) if h > h 2 , then p h = 1 w = p u .
The results of propositions 2 and 3 are represented in the next graph: The basic force explaining these results is a rent seeking argument 12 : to what extent the best policy preferred by law abiding people is mainly …nanced thanks to the …nes levied on non compliant citizens? The argument runs as follows.

h h 1 p h 2 1/w 0 P h p u
A benevolent government would not …nd e¢ cient to enforce all o¤enses, excepted when the social bene…t (in terms of avoiding the external cost) retained from doing so is larger than the cost borne in apprehending and convicting o¤enders. Thus, minor o¤enses (h < h 1 ) representing a social harm small enough compared to the marginal cost of enforcement are not enforced and not deterred by a benevolent government. In contrast, under political competition, citizens who expect to be law abiding demand that any o¤ense (the size being small or large) be enforced in proportion to the social harm they in ‡ict to society. The reason is that although all citizens do pay taxes, only those who do not comply the law are facing the burden of …nes. Thus for "honest" citizens, enforcing the minor o¤enses yields a private bene…t (net of the external cost of o¤enses) with two components: the higher rate to which the …ne is collected, on the one hand, and on the other hand the lower tax pressure required to …nance the policy. In other words, the cost of enforcement measures associated with a small but positive probability of control required by honest citizens to deter minor o¤enses is very easy to …nance, thanks to a low tax and a large rate to which the …ne be paid. Indeed, for compliant individuals the expected sanction is neither a cost, nor a mere social transfer, and the policy is …nanced mainly by the population of the non compliant people (i.e. less tax, more expected …ne).

By the same token, it is easy to understand that for intermediate o¤enses (h 1 < h < h 2 ) which would be worth to deter from a social point of view, law abiding citizens require a policy entailing more enforcement expenditures and reaching a higher level of deterrence than would be required by e¢ ciency. The private bene…ts they retain is still larger than the social bene…t (the …ne is a mere transfer for the benevolent enforcer, having no bene…t and no cost) associated with their deterrence. In this area of intermediate values for the external harm, a medium size for the probability of control represents a reasonable enforcement cost, which is still easy to …nance with taxes and a high rate of recovering the …ne.

But for major o¤enses (h > h 2 ), law abiding citizens will choose enforcement expenditures which are the optimal ones.

Best decision of a citizen who will not comply

For a citizen who anticipates he will not abide the law, let us denote the stage 2 preferred policy as: (f c ; p c ) = arg max (f;p) fu c under (1)g. Substituting (1) in u c yields:

u c = w + b m(p) pf + (1 pf ) (pf h)
Let us de…ne h 3 = h 1 + 2. We have now:

Proposition 4 Not complying citizens vote for a policy (p c ; f c ) which may be one of the following solutions:

i) If h < h 1 , then p c = 0. ii) But if h 1 < h < h 3 , then p c = (h h 1 ) 1 2w ; f c = w. iii) If h > h 3 , then p c = 1 w ; f c = w.
Proof. We have:

@u c @f = (h 2pf )p @u c @p = (h 2pf )f m
We obtain:

@u c @p jp=0 = hf m
Thus, if hw m < 0 , h < h 1 , then @uc @p jp=0

< 0 and it must be that p = 0. On the other hand, if h > h 1 , then it is not rational to choose f 6 = w (such that @uc @f = 0), since this would also imply that @uc @p = m 0 (p) < 0 for any p > 0. Thus it must be that f c = w, implying that:

@u c @p jp=1=w = (h 2)w m
As long as h < h 3 , p c is de…ned as the solution to @uc @p = 0, or:

h 2p c w = m w (5)
Solving for p c yields p c = (h h 1 ) 1 2w > 0, such that h p c w > 0. Finally, when h > h 3 , @uc @p p=1=w > 0, the best policy is p c = 1 w , and full deterrence is achieved.

The results of propositions 2 and 3 are represented in the next graph: Notice that for minor o¤enses (h h 1 ), non compliant citizens prefer the laissez-faire which is the e¢ cient policy in this area. Moreover, for the largest o¤enses (h > h 3 ), they also prefer full deterrence which is still e¢ cient. But for intermediate levels of harm (h 1 < h < h 3 ), o¤enders always prefer a level of enforcement expenditures lower than the e¢ cient one (since p c = pu 2 ). The underlying intuition is that -in contrast both to the law abiding citizens for whom the …ne paid by criminals is a bene…t -o¤enders have to consider the additional cost (over the tax) represented by the …nes they pay, when choosing their best policy. As the level of enforcement measures is raised, the probability of getting caught and …ned increases, and the burden of the policy is mainly borne by the population of o¤enders rather than the whole population of taxpayers.

Finally, a straightforward result is also that the deterrence level is at least as high in an equilibrium where (p h ; f h ) is chosen, as compared to an equilibrium where (p c ; f c ) arises, since: p h p c .

Equilibria

Now, we turn to the stage of electoral competition, where candidates announce their policy, and characterize the (subgame perfect) equilibrium.

At stage 1, candidates propose a policy for which the number of voters is maximized, anticipating that when implemented, this policy will induce a screening of citizens. We show in the next propositions that the political equilibrium that emerges depends on the size of the external cost of o¤ense h we consider. We start with the range of minor o¤enses (h < h 1 ).

Proposition 5 Assume that h < h 1 . The unique equilibrium is such that both candidates announce the laissez-faire policy: p c = 0. 13 Proof. We have to compare the proportion of citizens voting for the policy (p h ; w), which is p h w =1 2 (1 + h h 1 ), to the proportion voting for (p c = 0; f ) given by: 1 p c f = 1. The result is straightforward since p h w < 1 2 on the domain where h < h 1 .

In words, electoral competition creates no distortion in the area of minor o¤enses, where the external cost to society is small enough: the equilibrium policy emerging from elections is the optimal policy that a benevolent enforcer would choose, with no enforcement expenditure and zero deterrence.

Let us consider now the case of larger o¤enses (but less than major o¤enses: h < h 2 ).

Proposition 6 Assume that h 1 < h < h 2 . The equilibrium may be one of the following: i) If h < h 1 + 1 2 , then both candidates announce the policy (p c = (h h 1 ) 1 2w ; w). ii) If h 1 + 1 2 < h < h 2 , then both candidates announce the policy (p h = (1 + h h 1 ) 1 2w ; w).

Proof. Consider the domain of o¤enses h 1 < h < h 2 . It is easy to verify that the proportion of citizens voting for (p h ; w) satis…es now p h w =1 2 (1 + h h 1 ) > 1 2 since h > h 1 . On the other hand, the proportion voting for (p c ; w) is 1 p c w = 1 1 2 (h h 1 ) and satis…es 1 p c w > 1 2 as long as h < h 2 , or 1 p c w < 1 2 as long as h > h 2 . Thus in the range h 1 < h < h 2 : either p h w > 1 p c w and thus both candidates maximize their chances to win the election soon as they propose (p h ; w); or p h w < 1 p c w and thus both candidates maximize their chances to win the election when they propose (p c ; w). 14 Note that substituting for p h and p c , the condition p h w < 1 p c w writes equivalently as h < h 1 + 1 2 , and vice versa (p h w > 1 p c w () h > h 1 + 1 2 ). Moreover, it is easy to verify that (p u = 1 w (h h 1 ); w) is not an equilibrium, since: -on the range m w + 1 2 < h < h 2 , (p u = 1 w (h h 1 ); w) does not destroy (p h ; w), since:

p u w = h h 1 < p h w = 1 2 (1 + h h 1 ) , h < h 2 , which holds; -on the range h < m w + 1 2 , (p u = 1 w (h h 1 ); w) does not destroy (p c ; w), p u w = h h 1 < 1 p c w = 1 1 2 (h h 1 ) , h < h 1 + 2 3
, which also holds.

In the case of illegal acts for which society su¤ers from external costs with intermediate values, (h 1 < h < h 2 ), we …nd that political competition may lead to a weak or strong enforcement equilibrium. For moderate levels of harm (h < h 1 + 1 2 ), the maximal …ne is associated with enforcement's expenditures which are lower than the optimal ones (p c < p u ) under a "weak enforcement equilibrium". This means that where the weak enforcement prevails, the electoral competition leads to less deterrence than at the optimum. In contrast, enforcement expenditures are higher than the e¢ cient level, when the "strong enforcement equilibrium" emerges (h 1 + 1 2 < h < h 2 ), and a higher level of deterrence is obtained.

Last, we come to the range of major o¤enses.

Proposition 7 Assume that h > h 2 . The unique equilibrium is such that both candidates announce the policy (p = 1 w ; w).

Proof. When h > h 2 , the proportion of citizens voting for (p h = 1 w ; w) is 1. On the other hand, as long as h < h 3 , the proportion voting for (p c = 1 2w (h h 1 ) ; w) is 1 p c w < 1 2 ; while for h > h 3 , the proportion voting for (p c = 1 w ; w) is 0; hence the result.

Figure 4 shows the optimal probability of detection (p u ), the probability preferred by law abiding citizens (p h ) and the probability pro¤ered by not compliant citizens (p c ), as a function of h. Finally, the bold lines represents the probability of detection emerging at the voting equilibrium.

For minor o¤enses (below h 1 ), there is no deterrence at equilibrium as shown in proposition 5. This result is e¢ cient, since p u = 0 for all h < h 1 . The intuition is that the harm is so small relative to the marginal cost of enforcement that it is not worth spending some money on deterrence. To the opposite, the major o¤enses (above h 2 ) are always and fully deterred, the rational being the reverse. For intermediate o¤enses (between h 1 and h 2 ), the characteristics of the equilibrium may be of two opposites kinds. The …rst one occurs for moderately harmful acts (between h 1 and h 1 +1 2 ), where the "weak enforcement" equilibrium prevails as shown in proposition 6; there is less deterrence than at optimum, such that the issue of under-deterrence is aggravated. A majority of citizens decide to not abide the law. Then, enforcement expenditures are lower than the social welfare maximizing one, and the proportion of o¤enders exceeds the social welfare maximizing one.

The second kind, associated with more deterrence than at optimum, occurs in the range of more harmful acts (between h 1 + 1 2 and h 2 ) where the "strong enforcement" equilibrium emerges; the resulting probability of detection is higher than the social welfare maximizing. In such a case, a majority decides to abide the law. Enforcement expenditures are higher than the e¢ cient level, and consequently the proportion of o¤enders is lower than what would require e¢ ciency.

To sum up, electoral competition in our set up yields zero distortion both at the top (major o¤enses) and the bottom (minor o¤enses) of the distribution of social harms. Distortions only occur in the range of intermediate harms, and typically correspond to under (over) enforcement of small (large) o¤enses. 15 Note that the comparative statics of the model are very simple, and depend mainly on the marginal cost of enforcement's expenditures relatively to society's wealth ( m w ). All else equal, the lower m w : (1) the higher p h and p c , (2) the higher (smaller) the number of voters for p h (p c ). On the other hand, a change in m w also modi…es the thresholds of harms associated with the di¤erent equilibria: as m w decreases, then h 1 , h 1 + 1=2 and h 2 shift to the left. This means that some of the (initially) minor o¤enses are now deterred with a positive probability, while some of the (initially) major o¤enses become under-deterred. Similarly, in the range of intermediate o¤enses, some of the low o¤enses being initially under-deterred, become now over-deterred and so on.

Extensions

We consider here some simple developments of our analysis. An important discussion is related to assumption 1. In a sense we have assumed that citizens were rich enough, i.e. their personal wealth were larger than the highest illegal bene…t (w > 1). A straightforward implication of such an assumption is that it allows full deterrence of major o¤enses (those requiring maximal enforcement expenditures, p =1 w ). Relaxing this assumption, it is easy to verify that full deterrence is never obtained (see paragraph 4.1). On the other hand, the mix of the uniform distribution for the illegal bene…t, and of a constant marginal cost for enforcement expenditures, have the main expositional interest to allow us to fully characterize the equilibria. We will relax both in paragraph 4.2.

Enforcement, wealth, and political competition

Let us substitute assumption 1 with the next one:

Assumption 2. 1 > w > 3 4 > m.
In such a case, we have to introduce new thresholds for the external cost of crime (see appendix 1):

ĥ2 = h 1 + w; ĥ3 = h 1 + 2w; h 4 = h 1 + 2w 1 It is straightforward to verify that 1 > w > m implies: h 4 < ĥ2 < ĥ3 , while w > 3 4 gives: h 1 < h 1 + 1 2 < h 4 < h 1 + 1 < ĥ3 .
As a result, 16 proposition 5 still holds, while propositions 6 and 7 are substituted with the next one: Proposition 8 A/ Assume that h 1 < h < h 4 , the equilibrium may be one of the following: i) If h 1 < h < h 1 + 1 2 , then both candidates announce the policy (p c = (h h 1 ) 1 2w ; w). ii) If h 1 + 1 2 < h < h 4 , then both candidates announce the policy (p h = (1 + h h 1 ) 1 2w ; w). B/ Assume h > h 4 , the unique equilibrium is such that both candidates announce the policy (p = 1; w) and incomplet deterrence occurs.

Proof. See appendix 1.

Thus this case where 1 > w > 3 4 is qualitatively very similar to the former one, the exception being that some of the larger o¤enses (but not the major, i.e. only for h 4 < h < h 2 ) are drastically deterred with maximal enforcement expenditures, although incomplet deterrence occurs.

To end up, it is straightforward to verify that when w < 1 2 , we obtain h 4 < h 1 , and thus the equilibrium is as follows:

Proposition 9 i) If h < h 1 , both candidates announce the policy (p c = 0; w). ii) If h 1 < h < ĥ3 , both candidates announce the policy (p c = (h h 1 ) 1 2w ; w). iii) If h > ĥ3
, both candidates announce the policy (p = 1; w) and incomplet deterrence occurs.

The reason is that when w < 1 2 the number of citizens voting for (p h ; w) (whether we have p h = (1 + h h 1 ) 1 2w or p h = 1) satis…es p h w < 1 2 since h 4 < h 1 , while the number voting for (p c = (h h 1 ) 1 2w ; w) satis…es 1 p c w 1 w > 1 2 .

More general distributions and technologies

Let us assume that b is distributed according to a general, continuous law represented by a density g > 0 at any b and a cumulative function G de…ned on [0; 1]. Wlog, we will assume that 1 G g is decreasing on [0; 1]. Regarding the monitoring costs associated with the control of illegal activities, we will assume the following conditions hold: 8p 2 [0; 1], m 0 > 0, m 00 > 0, and m 0 (1) ! 1. This mean that the enforcement activity runs with decreasing returns to scale. Our main results are summarized in the …nal proposition:

Proposition 10 Assume that b follows a continuous probability distribution represented by a density g > 0 at any b and a cumulative function G de…ned on [0; 1], and that 8p 2 [0; 1], m 0 > 0, m 00 > 0, with m 0 (1) ! 1. The political equilibrium may be one of the following solutions: i) If h < m 0 (0) wg(0) , then both candidates propose the laissez-faire policy, (p c = 0). ii) If h > m 0 (0) wg(0) , there exists a threshold ĥ > m 0 (0) wg(0) such that both candidates propose the policy (p c < 1; w) if m 0 (0) wg(0) < h < ĥ; in contrast, both candidates propose the policy (p h < 1; w) if ĥ < h.

Proof. See appendix 2.

In appendix 2, we also show that the optimal enforcement expenditures are characterized as follows: for h < m 0 (0) wg(0) , then p u = 0; but, for h > m 0 (0) wg(0) then p u < 1 and is larger than the ones for which o¤enders vote, but lower than those chosen by law abiding citizens: p c < p u < p h . In words, in a strong (weak) enforcement equilibrium, there are more (less) deterrence than at the optimum.

Concluding remarks

The central issue of our paper is the relationships between law enforcers'objectives and the public enforcement of law. Our analysis is based on the assumption that citizens vote before deciding whether or not to abide the law. Depending on the level of harmfulness of the act and the marginal cost of enforcement, either a "strong" or a "weak" law enforcement equilibrium can emerge. When the "strong enforcement" equilibrium emerges, the preferences of o¤enders (and thus, crime bene…ts) are no longer taken into account -in a sense, criminals' preferences are not representative of social preferences, the majority of citizens that emerges in a political equilibrium being law abiding. But, it cannot be ignored that a "weak enforcement" equilibrium might also emerge, in which the criminals'preferences become representative of social preferences.

The paper contributes to the debate concerning the limits of the beckerian approach, and mainly the early criticisms that focused on the inclusion of crime bene…ts in the social welfare function [START_REF] Lewin | The Social Value of Crime?[END_REF]Trumbull, 1990, Dau-Schmidt, 1990). According to [START_REF] Lewin | The Social Value of Crime?[END_REF], including criminal bene…ts in the social welfare function lowers the deterrence threshold. [START_REF] Dau-Schmidt | An Economic Analysis of the Criminal Law as a Preference-Shaping Policy[END_REF] also argues that it is morally shocking to include criminal bene…ts in the social welfare function. Our paper re-conciliates in a way the two positions, by establishing a clear distinction between what is socially optimal (the beckerian approach) and what should emerge from a political process (deterrence under election). We show that what emerges from the political process does not maximize social welfare (the social welfare is lower under democracy than in the implausible utilitarian social planner).

Our paper also provides an argument to explain casual observations regarding existing similarities or to the contrary di¤erences, between countries relative to citizens' attitude and public policies orientation in the area of crime deterrence. On the one hand, the issue of criminality became a main concern in electoral campaigns for more than a decade in most European countries; on the other hand, there is an ongoing debate about the non criminalization/legalization of some o¤enses, such as drug consumption (except in relation with international tra¢ cs and criminal networks) or illegal downloading. In the …rst case, the growing place of crime deterrence in electoral campaigns can be seen as a consequence of the election strategies of the politicians, anticipating the "strong enforcement" equilibrium for major crimes. In the second case, some o¤enses such as illegal downloading might be considered as involving minor harm relatively to their private bene…ts (the evaluation of those depending on cultures).

The intriguing empirical result of Lin ( 2007) is partly explain by our paper. [START_REF] Lin | Does Democracy Increase Crime ? The Evidence from International Data[END_REF] attempted to verify empirically whether di¤erences arise in criminal law enforcement policies (in particular …ghting minor and major crimes) according to the level and quality of democracy. Using an index of political liberty from the comparative freedom survey to distinguish "low democracies" from "high democracies", he shows that countries characterized by a higher level of democracy tend to punish major crimes relatively more severely as compared to countries with a lower level of democracy, the reverse being true for minor crimes. More precisely, the deterrence of homicides is quite strong 17 and the homicide rate lower in high democracy by comparison with low democracy. On the contrary, it seems that democracy has a negative impact on less serious crimes such as burglary, robbery, car theft. However, no explanation of such an empirical result has been yet provided. When the harm generated by an o¤ense is small (large) relative to the marginal cost of enforcement, a "weak (strong) enforcement" equilibrium should emerged, provided that the o¤enders (respectively, honest citizens) represent the majority. It is possible that the harm generated by car theft is quite low relative to the marginal cost of detecting, apprehending and convicting the o¤enders, therefore leading to a relatively weak enforcement. The reverse being true for homicide.

A limitation of the model is the assumption regarding the commitment of elected law enforcers to enforce their electoral platform. Here, we deal with preelection politics, and assume that electoral promises are binding and enforceable. A signi…cant extension would be to study the case where politicians could decide not to implement the announced policy despite reelection concerns. 18 We also abstract from the existence of lobbying activities that yield other kinds of imperfections on the political market. We leave for future research the analysis of public enforcement when partisan pressures exist, which will allow to study the e¤ects of di¤erent assumptions departing from the one of a benevolent enforcer.

To complete the picture, two extensions might develop. First, the interplay of the voting model with social norms (Acemoglu, Jackson 2015) can be worth been investigated. For instance, some moral or social reasons can impede people to do illegal acts, even if law is in practice not enforced (for instance, illegal downloading or throwing cigarettes butts in France). Second, another signi…cant extension of the paper might be to consider the case of error. [START_REF] Mccannon | Prosecutor Elections, Mistakes, and Appeals[END_REF] shows that, in addition of taking more case to trial (rather than plea bargain) during reelection campaign, prosecutors face a decreased probability of having the conviction being upheld by the appellate court. An interesting point would be investigate the relation between election and accuracy in conviction.

1 + h 2p h w = m w
Solving for p h yields p h = (1 + h h 1 ) 1 2w > 0, which implies h p h w ? 0. On the other hand, If h > h 4 , then @u h @p jp=1 > 0 which implies p h = 1.

Not abiding people. When w < 1, the analysis of the best policy chosen by people not abiding law changes as follows (given that f = w is still optimal); we have: @u c @p jp=0 = hw m @u c @p jp=1 = (h 2w)w m Thus, h < h 1 implies p c = 0. Moreover, as long as h 1 < h < ĥ3 m w + 2w, p c is de…ned as the solution to @uc @p = 0, or:

h 2p c w = m w (6)
Solving for p c yields p c = (h h 1 ) 1 2w > 0, such that h p c w > 0. But If h > ĥ3 , then @uc @p jp=1 > 0 and p c = 1.

It is straightforward to verify that assuming 1 > w > 3 4 (> m) implies:

h 1 < h 1 + 1 2 < h 4 < ĥ2 < h 1 + 1 < ĥ3 .
Proof of proposition 9. First note that for h < h 1 , there exists a proportion p h w = 1 2 (1 + h h 1 ) < 1 2 of voters for (p h ; w) and a proportion 1 p c w = 1 of voters for (p c = 0; w). Hence, for h < h 1 , the equilibrium is such that both candidates announce (p c = 0; w) (i.e. proposition 5 still holds).

i) Let us consider the domain of o¤enses h 1 < h < h 1 + 1.

-On the one hand, the proportion of law abiding citizens voting for (p h < 1; w) is p h w = 1 2 (1 + h h 1 ) > 1 2 (when h < h 4 ); or the proportion of law abiding citizens voting for (p h = 1; w) is p h w = w > 1 2 (when h > h 4 ). -On the other hand, the proportion of o¤enders voting for (p c < 1; w) is 1 p c w = 1 1 2 (h h 1 ) and satis…es 1 p c w > 1 2 as long as h < h 1 + 1. Thus it can be veri…ed that: a) when h 1 < h < h 4 (< h 1 + 1): the condition p h w = 1 2 (1 + h h 1 ) < 1 p c w = 1 1 2 (h h 1 ) still writes equivalently as h < m w + 1 2 , and vice versa (p h w > 1 p c w () h > m w + 1 2 ). This shows that the equilibrium is such that both candidates announce: i) (p c < 1; w) if h 1 < h < h 1 + 1 2 ; or ii) (p h < 1; w) if h < h 1 + 1 2 < h < h 4 .

@u h @f = [(1 G(pf )) g(pf )(pf h)] p @u h @p = [(1 G(pf )) g(pf )(pf h)] f m 0 (p)

and we have: @u h @p jp=0 = (1 + g(0)h)f m 0 (0) i) Thus, if (1 + g(0)h)w m 0 (0) < 0 , h < m 0 (0) wg(0) 1 g(0) , then @u h @p jp=0 < 0 and the solution is p h = 0 whatever f is. ii) On the other hand, if h > m 0 (0) wg(0) 1 g(0) , then it is not rational to choose f 6 = w such that @u h @f = 0 since this would imply that @u h @p = m 0 (p). Thus, f h = w such that @u h @f jp h w > 0, and p h is de…ned according to:

h p h w + 1 G g jp h w = m 0 (p h ) g(p h w)w (8)
which implies h p h w ? 0. To sum up, we have :

Lemma A. The policy preferred by honest citizens (p h ; f h ) may be one of the two following solutions: i) Assume h < h 1 1 g(0) ; then the policy is p h = 0, and is associated with zero deterrence. ii) Assume h > h 1 1 g(0) ; then the policy is p h > 0; f h = w, and is associated with either over or under deterrence: p h w ? h.

On the other hand, a citizen who anticipates to become a criminal at stage 4 votes for a policy (f c ; p c ) = arg max (f;p) fu c under (1)g. Substituting (1) in u c yields:

u c = w + b m(p) pf + (1 G(pf )) (pf h)
We have now:

@u c @f = [ G(pf ) g(pf )(pf h)] p @u c @p = [ G(pf ) g(pf )(pf h)] f m 0 (p)
and thus we obtain: @u c @p j0 = g(0)(h)f m 0 (0)

LHS term is decreasing in p, while each RHS is increasing in p. The result is straightforward given that h pw G g jpw < h pw < h pw + 1 G g jpw .

Figure 1 :

 1 Figure 1: The optimal enforcement policy

Figure 2 :

 2 Figure 2: The best enforcement policy of law abiding citizens

Figure 3 :

 3 Figure 3: The best enforcement policy of not complying citizens

Figure 4 :

 4 Figure 4: The equilibrium enforcement policy

By de…nition, empirical studies do not allow for normative conclusions. Some signi…cant issues remains such as the relation between the level of enforcement expenditure under the in ‡uence of election and e¢ ciency. Furthermore, these empirical papers focus on harmful crimes such as homicide, rather than dealing with a larger scope of crimes (such as regulatory crimes, parking …nes and so on).

See Friedman, 1999, p. 262.

See also[START_REF] Gradstein | Rent Seeking and the Provision of Public Goods[END_REF], for an analysis of the impact of a rent seeking government for the provision of public goods.Dittman (2004) addresses the case where the government puts some weight on the residual budget of the prosecution policy.

This issue of self-interest has been raised for judges'decision making notably by[START_REF] Epstein | The Independence of Judges: The Uses and Limitations of Public Choice Theory[END_REF] and[START_REF] Posner | What Do Judges and Justices Maximize? (The Same Thing Everybody Else Does)[END_REF]. Examining the case of ordinary tenured judges,[START_REF] Posner | What Do Judges and Justices Maximize? (The Same Thing Everybody Else Does)[END_REF] examines the case where their utility is a¤ected by income, leisure, and judicial voting.

See the surveys by[START_REF] Garoupa | The Theory of Optimal Law Enforcement[END_REF] and[START_REF] Polinsky | The Economic Theory of Public Enforcement of Law[END_REF].

See the last section for a more general assumption.

Examples of such o¤enses are numerous (see, for instance,[START_REF] Polinsky | The optimal trade-o¤ between the probability and magnitude of …nes[END_REF]: polluting the air (while non respecting a regulatory standard), speeding or double parking, car theft, throwing cigarettes butts, drug consumption, etc. Each of these o¤ense imposes a cost to the rest of the society.

This last result re ‡ects that the marginal cost of enforcement is constant. This case vanishes when one relaxes this assumption, assuming rather than m 0 (p) > 0, m 00 (p) > 0 with for example m 0 (1) ! 1; as a result pu < 1 for any h > h 1 . See[START_REF] Garoupa | Optimal magnitude and probability of …nes[END_REF], and the last section of this paper.

Citizens are dynamically consistent players here. Each citizen votes, anticipating his future behavior, i.e. whether he will behave as honest people or criminals.1 0 i.e., we assume that candidates commit to their own electoral platform -without specifying the reasons explaining neither why those platforms are credible announcements, nor how they become a law. These (obviously important) issues are beyond the scope of the paper.

2[START_REF] Garoupa | Optimal Law Enforcement with a Rent-Seeking Government[END_REF] have analyzed the the enforcement policy for a rent-seeking government, and found it displays over enforcement of some minor o¤enses combined with under enforcement of some major o¤enses, compared to the optimal policy. However, note that proposition

does not give the equilibrium policy, but characterizes only the best response function of abiding citizens. Our equilibrium analysis (propositions 5,6,7) will show that, although electoral competition is consistent with the existence of rent-seeking behaviors (by both candidates and voters), the properties of the equilibrium are very di¤erent than those found byGaroupa and Klerman.

3 Note that at equilibrium, each candidate wins with probability 1

(this remark applies to all propositions).

4 It is obvious that when p h w = 1 pcw, both candidates maximize their chances to win the election when they propose either (w; p h ) or (w; pc) indi¤erently. We let aside this kind of situation for the moment, but we will discuss the occurrence of multiple and asymmetric equilibria later on.

5 The enforcement strategy choosen by a rent-seeking government in[START_REF] Garoupa | Optimal Law Enforcement with a Rent-Seeking Government[END_REF] is roughly speaking the opposite: over enforcement of some minor o¤enses combined with under enforcement of some major o¤enses; moreover, it displays some distortions both at the top and the bottom of the distribution of o¤enses.

6 The analysis of the optimal policy is changed in two ways, compared to proposition 1: in part ii) h

is replaced with ĥ2 ; in part iii), pu = 1 and partial deterrence occurs; see appendix 1.

7 According to multiple criteria: average prison length, average clearance rates. 1 8 For example, up to the latest presidential elections, French citizens could anticipate that the new president would award a large amnesty for infractions to speed limits, parking regulation, running red and so on.

APPENDIX 1

Benevolent enforcers. When w < 1, the analysis of the optimal policy is changed as follows (given that f = w is still optimal); we have: @S @p jp=0 = hw m @S @p jp=1 = (h w)w m Thus, h < h 1 implies p u = 0. Moreover, as long as h 1 < h < h 1 + w ĥ2 , the solution corresponds to a p u satisfying @S @p = 0 or:

(h p u w)w = m implying under deterrence: h p u w > 0. Solving for p u yields p u = (h h 1 ) 1 w > 0.

In contrast, when h > ĥ2 , the optimal policy is (p u = 1; w), and incomplet deterrence occurs (p u w < 1).

Law abiding citizens. When w < 1, the analysis of the best policy chosen by law abiding people changes as follows (given that f = w is still optimal); we have:

Thus, as long as

. This implies that when h 4 < h < h 1 + 1, the equilibrium is such that both canidates announce (p h = 1; w).

ii) When h > h 1 +1, the proportion of citizens voting for (p h = 1; w) is w > 1 2 . On the other hand, when h < ĥ3 the proportion voting for (p c = 1 2w (h h 1 ) ; w) is 1 p c w < 1 2 ; while for h > ĥ3 , the proportion voting for (p c = 1; w) is 1 w < 1 2 . Hence, when h > h 1 +1, the equilibrium is such that both candidates announce (p h = 1; w).

APPENDIX 2

In this appendix, we extend our main results to more general environments. Let us assume that b is distributed according to a general, continuous law represented by a density g > 0 at any b and a cumulative function G de…ned on [0; 1]. Wlog, we will assume that 1 G g is decreasing on [0; 1]. Regarding the monitoring costs associated with the control of illegal activities, we will assume the following conditions hold: 8p 2 [0; 1], m 0 > 0, m 00 > 0, and m 0 (1) ! 1.

In this case, a benevolent enforcer [START_REF] Garoupa | Optimal magnitude and probability of …nes[END_REF], would choose p u = 0 for any h < h 1 = m 0 (0) wg(0) ; otherwise, the optimal policy is (p u ; f u = w) where p u < 1 (given that m 0 (1) ! 1) is the solution to:

with h p u w > 0.

The analysis of the electoral game is now developed with these new assumptions.

At stage 4, any policy (p; f ) will induce a screening of citizens between those who abide the law, and those who become criminals. Using the analysis of paragraph 2.1, it is straightforward that the deterrence threshold at equilibrium is b = pf .

At stage 2, a citizen who anticipates to behave honestly at stage 4 votes for a policy (f h ; p h ) = arg max (f;p) fu h under (1)g. Substituting (1) in u h leads to:

The derivatives of u h with respect to f and p are given by: i) Thus, if g(0)hw m 0 (0) < 0 , h < m 0 (0) wg(0) , then @uc @p j0 < 0 and it must be that p = 0. ii) On the other hand, if h > h 1 , then it is not rational to choose f 6 = w such that @uc @f = 0, since this would also imply that @uc @p = m 0 (p) < 0; thus it must be that f c = w satisfying @uc @f jpcw > 0, and p c is de…ned by:

such that h p c w > 0. To summarize:

Lemma B. The policy preferred by criminals (p c ; f c ) may be one of the two following solutions: i) Assume h < h 1 ; then the policy is p c = 0 and is associated with zero deterrence. ii) Assume h > h 1 ; then the policy is p c > 0; f c = w, and is associated with under deterrence: p c w < h.

Equilibrium. Now, we turn to the initial stage of the game. We …rst consider the equilibrium associated with a small external cost, i.e. h < h 1 .

Once more, it is easy to show that the unique equilibrium is such that both candidates announce the laissez-faire policy: p c = 0. To see this, note …rst that when h < h 1 1 g(0) , both honest citizens and criminals prefer the laissezfaire. Hence, an equilibrium cannot exist except when both candidates announce p = 0. Assume now that h 1 1 g(0) < h < h 1 ; we have to compare the proportion of citizens voting for (p h ; w): G(p h w), to the proportion voting for (p c = 0; f ): 1 G(p c f ) = 1. The result is straightforward.

Consider now that the external cost is large, i.e. h > h 1 . Let us compare the proportion of citizens voting for (p h ; w): G(p h w), to the proportion voting for (p c ; w): 1 G(p c w). Either: G(p h w) > 1 G(p c w) and thus both candidates maximize their chances to win the election soon as they propose (w; p h ); or: G(p h w) < 1 G(p c w) and thus both candidates maximize their chances to win the election when they propose (w; p c ). Note that using (8) and ( 9), we can write equivalently:

De…ne ĥ as the value of the external harm for which 1

since the RHS in (10) decreases in h, and the RHS in (11) increases in h; as a result, the unique equilibrium is such that both candidates announce the policy (p c ; w). ii) In contrast if h > ĥ, then the unique equilibrium is such that both candidates announce the policy (p h ; w).

Finally, using (7), ( 8) and ( 9) which de…ne as an interior solution respectively p u ; p h ; p c , it can be verify that p c < p u < p h . By second order condition, each