A CRIMINAL ORGANIZATION AND THE OPTIMAL LAW
JUIN-JEN CHANG, HUEI-CHUNG LU, and MINGSHEN CHEN*
This article develops a simple but general criminal decision framework in which individual crime and organized crime are coexisting alternatives to a potential of-fender. It enables us to endogenize the size of a criminal organization and explore interactive relationships among sizes of criminal organization, the crime rate, and the government’s law enforcement strategies. We show that the method adopted to allocate the criminal organization’s payoffs and the extra benefit provided by the criminal organization play crucial roles in an individual’s decision to commit a crime and the way in which he or she commits that crime. The two factors also jointly de-termine the market structure for crime and the optimal law enforcement strategy to be adopted by a government. (JEL K4)
Organized criminal activities have been regarded as major economic and social issues since the early 1900s. Although there has been, as pointed out by Fiorentini and Peltzman (1995), enormous research interest in the eco-nomics of criminal activities, economists have done relatively little work on issues specifically related to the economics of organized crime. We only see that much of the work has been done by following Becker’s (1968) framework where the target is the individual agent’s allocative choice between legal and illegal
activities in the face of different deterrence arrangements. Among the few exceptions, a criminal organization is often thought of as a monopolistic firm, and the theory of monopoly is predominantly used to analyze organized crimes. Schelling (1967), Buchanan (1973), Backhaus (1979), Gambetta and Reuter (1995), and more recently Garoupa (2000) investigate the optimal public policy toward organized crimes through a welfare comparison between a monopoly (organized criminals) and competitive supplies (individ-ual criminals). Given that criminal activities are seen as ‘‘social bads,’’ the monopolistic market is considered to be better than a per-fectly competitive one, because its ‘‘output of crime’’ is smaller.1
The monopolistic model implies that po-tential offenders have no other criminal choices but are forced to join the criminal
organization if they decide to commit
a crime. Clearly, it is less than exhaustive in terms of describing agents’ choices in rela-tion to criminal behavior. In fact, the criminal *We are grateful to an anonymous referee who
pro-vided with many insightful comments and suggestions in relation to a previous version of this article. Any errors or shortcomings, however, remain our own responsibility. We also thank seminar participants at Institute of Eco-nomics, Academia Sinica, and Tunghai University for their helpful discussions.
Chang: Associate Research Fellow, Institute of Econom-ics, Academia Sinica, and Associate Professor, De-partment of Economics, Fu-Jen Catholic University, Nankang, Taipei 115, Taiwan. Phone 886-2-27822791 ext. 532, Fax 886-2-27853946, E-mail email@example.com
Lu: Professor, Department of Economics, Fu-Jen Catho-lic University, No. 510, Chung Cheng Road, Hsin Chuang, Taipei 242, Taiwan. Phone 886-2-29052718, Fax 886-2-22093475, E-mail firstname.lastname@example.org. edu.tw
Chen: Associate Professor, Department of Finance, Na-tional Taiwan University, No. 1, Section 4, Roosevelt Road, Taipei 106, Taiwan. Phone 886-2-33661088, Fax 886-2-23626744, E-mail chen0811@management. ntu.edu.tw
1. Criminologists sometimes include terrorist groups as criminal organizations; therefore, some studies view the criminal organization as a state, within which members undertake their own legal and illegal activities. Following this line of research, Grossman (1995) and Posner (1998) model the Mafia as a competitor of the state in the pro-vision of public goods and services. Skaperdas and Syro-poulos (1995) argue that the relationship between the state and a criminal organization is not completely antagonis-tic. It is often symbiotic, a live-and-let-live arrangement.
(ISSN 0095-2583) doi:10.1093/ei/cbi046
Vol. 43, No. 3, July 2005, 661–675 Western Economic Association International 661
organization may not be monopolistic after all, as argued by Dick (1995) and others.2 Criminologists Johnson (1962), Rubin (1973), and Reiss (1986) observe that most criminal organizations are endemic in some particular areas and/or restrictive in some particular illegal businesses. These criminal activities, such as bootlegging, drug dealing, pimping, and prostitution, are often less organized and would be difficult to successfully monop-olize. Dick (1995) also points out that pros-titution, smuggling, fencing, and narcotics importation often involve substantial com-petition among downstream suppliers and can hardly be regarded as monopolistic businesses. In other similar studies, Klein and Crawford (1967), Morash (1983), and Sarnecki (1986) conclude that most youths who engage in delinquency are at most street gangsters and cannot be seen as members of a highly structured organization.
In reality, the determination of the market structure for crime should be endogenous, which has notable implications for the optimal crime enforcement policies and crime itself. To recover the conventionally neglected facts and provide a more complete picture regarding or-ganized crime, this article develops a simple but more general model. In this framework, individual and organized crimes are coexisting alternatives to a potential offender, and join-ing a criminal organization is seen as only one of the agent’s rational choices. In deciding whether to join a criminal organization, the methods used to allocate the criminal organi-zation’s payoffs and the extra benefits pro-vided by the criminal organization are given serious consideration. They both play crucial roles in determining the sizes of the criminal organizations as well as in influencing the in-teractive relationship between the market structure for crime and the government’s op-timal law enforcement.
This study contributes to the existing liter-ature and provides new insights in three respects. First of all, by allowing the agent to choose rationally between engaging in indi-vidual or organized crime, we not only endogenize the size of the criminal organiza-tion but also explore the factors that
deter-mine the type (high or low criminal ability) of offenders that enter the criminal organiza-tion. A new theoretical attempt is made in terms of providing an explanation to the ob-servation of Anderson (1995): There are con-siderable variations in personal qualities and values across different criminal organizations. This issue has not previously been addressed in the existing literature.
Second, this article sheds light on the role of the distribution arrangements of the Mafia’s criminal payoff in the determination of the equilibrium market structure for crime. Schelling (1960), Tullock (1974), Grossman (1991), Hirshleifer (1991), Skaperdas (1992), and Polo (1995) have expended much effort in answering such questions as: How does a criminal organization divide its en-dowment among its members? How do the members of the organization engage in rent-seeking by investing their efforts in militia and productive activities? This work, how-ever, does not follow this line of research; rather, it seeks to understand how criminal payoff arrangements determine the size of criminal organizations and the market struc-ture for crime.
Two types of payoff sharing schemes, namely, uniform and ability-adherent sharing schemes, are employed to examine the determi-nation of the market structure for crime. Doing this has two advantages. From a micro-per-spective, our analysis shows that the different payoff arrangements of a criminal organiza-tion will attract the entry of members that differ in quality in terms of criminal skill. More pre-cisely, if the Mafia’s payoff is divided uniformly among its members, the Mafia will only be able to attract offenders with relatively low criminal skill to join the organization. Due to the free-rider problem, the offenders with relatively higher ability will refuse to join the Mafia to avoid their distinctly high contributions from being shared by others. However, if the crimi-nal organization’s method of sharing is in accordance with its members’ abilities, the high-ability offenders could be organized by the Mafia. From a macro-perspective, we show that the sharing arrangement within the crim-inal organization plays a dominant role in the determination of the market structure for crime and, consequently, influences the optimal en-forcement strategy of a government. Our find-ings benefit the formulation of public policies in relation to organized crime.
2. Dick (1995), following Williamson’s (1979, 1985) approach, propounds that transaction costs, rather than monopoly power, primarily determine the market struc-ture for crime.
Third, one of the salient features of criminal organizations is that they engage in activities to enhance their influence, improve their busi-nesses, and even reduce the effectiveness of deterrence against them (Abadinsky 1994). Thus, it is believed that there exists an extra benefit for members from joining a criminal organization that is not available to individual criminals. In this article, we model such an ex-tra benefit for the Mafia’s members and show that it is another important factor influencing the optimal law enforcement. When the Mafia creates more of these extra benefits for its members, the equilibrium crime rate will rise in response. When there is no extra benefit, the existence of a monopolistic Mafia will result in a reduction in the government’s enforcement and, consequently, will become a welfare-improving mechanism within the society, as in the case of the classical view, as espoused by Buchanan (1973) and Garoupa (2000). However, when a positive extra benefit is taken into account, the existence of a monop-olistic Mafia is no longer welfare-improving.
The remainder of this article is organized as follows. In section II we describe the scenario by extending Garoupa’s (2000) model with the inclusion of the criminal choices in a sequential game. Section III provides equilibria for the three parties, that is, potential offenders, the organization, and the government, in a bench-mark model in which the criminal organization adopts a uniform-sharing payoff scheme to distribute its criminal rents. Section IV consid-ers related discussions. In particular, we extend the analysis to an ability-adherent sharing scheme and show that the properties of equilib-ria in an ability-adherent sharing scheme are different from those in a uniform-sharing pay-off scheme. Finally, we conclude in section V.
II. THE SCENARIO
The analysis is performed within the frame-work of a three-stage game. This game con-sists of three protagonists: the authorities, the criminal organization (the Mafia), and po-tential offenders. For simplicity, all agents are assumed to be risk-neutral. In a way that departs from the existing literature, the crim-inal market in our model is not restricted to being perfectly competitive or monopolistic. We emphasize that criminals are in a self-selection market in which all agents make rational decisions regarding participating in organized crime (denoted by O), engaging in individual crime (denoted by I), or engaging in no crime at all (denoted by N).
Figure 1 serves as a supplementary tool to assist us in our exposition. Let us denote f1and f2as the sanctions imposed on the organized and individual criminal offenses, respectively. In the first stage, the government and the Mafia formulate their optimal policies by simultaneous moves, that is, a Cournot-Nash game. The policy maker chooses sanctions f1, f2, and a detection probability p to maximize social welfare W. The sanctions are restricted by a maximum level F, i.e., f1, f2 F, which can be thought of as the offenders’ maximum wealth. In addition, we assume that the crim-inal organization provides extra benefits (or protection) to its members and, at the same time, extracts criminal rents from them. In other words, a member has to pay entry fees y in exchange for the Mafia’s extra benefits.3 FIGURE 1
The Three-Stage Game Tree
3. The traditional literature, for example Garoupa (2000), specifies that in a monopolistic model potential offenders have to buy a license from the Mafia to be able to commit the offense; that is, the Mafia regulates entry into the criminal market.
Following Garoupa (2000), the criminal syn-dicate will choose y to maximize the total rev-enue, P, extracted from its members.
A potential offender engages in a two-stage decision (during stages 2 and 3). Given the pol-icies formed by the government and the Mafia in stage 1, a potential offender first decides whether to commit a crime (denoted by C) in the second stage. If he decides to commit a crime, then the offender in turn decides whether to become an organized criminal (O) or an individual criminal (I) in the third stage. An offender obtains b (criminal rent) from committing a crime. The b is private in-formation known only to the offender (and to the criminal organization if he joins the Mafia) and is not revealed to the authorities. In this article, b also reflects the offender’s quality in terms of criminal skills, which vary across potential offenders. For simplicity, b is uniformly distributed over [0,1].
A potential offender’s net payoff corre-sponding to the ith choice is denoted by ui, where i¼ N, O, I. Thus we have
uN ¼ 0; if he chooses N ;
uO¼ B pf1þ e y; if he chooses O;
uI ¼ b pf2; if he chooses I :
The term B is a member’s payoff received from participation in organized crime, which comes from the Mafia’s payoff shared by all members. The extra benefit e is defined as what a member would obtain by joining a criminal organization, which exceeds what he would obtain when committing the crime alone. As pointed out by Abadinsky (1994), the extra benefit may take the form of greater influence, better business, or better protec-tion. We interpret it concretely as either an increase in income or a reduction in expected sanctions (i.e., the expected sanction of choosing O is [pf1 e]) for the Mafia’s mem-bers.4 It will play a prominent role in our analysis.
The member’s payoff B crucially depends on the incentive sharing schemes within the criminal organization. In this article we deal with this problem by introducing two differ-ent sharing arrangemdiffer-ents, namely, uniform and ability-adherent sharing schemes. We begin our analysis with a uniform sharing scheme in the benchmark model and later, in section IV, we will extensively discuss an incentive scheme that adheres to criminals’ skills (i.e., in which members receive payoffs based on their performance in the conduct of their criminal activities). In the uniform sharing scheme, the Mafia’s payoff is di-vided equally among its members and, conse-quently, every member receives an average criminal benefit, that is, B ¼ b. The average payoff b is endogenous and dependent on the members’ average criminal skills within the organization.
Ideally, the extra benefit might depend on whether the criminal organization can recruit high-quality members and on how the orga-nization runs its business. However, to focus our point, we shed light on the former effect and abstract the latter effect from the anal-ysis. Thus, it is convincing that the higher the average quality of the Mafia’s members is, the greater will be the extra benefits e that the Mafia can create. Therefore, we specify e ¼ e(b) ¼ ab, 0 < a < 1, for convenience, and it also allows us to yield a simpler solution.5
III. THE MODEL
Following a backward induction, we solve this three-stage game by starting with the criminal choices of a potential offender.
The Potential Offender’s Decision
In the third stage, given that he enters the criminal market, an individual evaluates the benefits and costs of joining the Mafia to de-cide whether to become an organized criminal. From (1), the ‘‘self-selection constraint’’ re-veals that a potential offender will participate 4. The Mafia may try to recruit individual offenders
forcibly through the imposition of violence or harassment. The term e can also be viewed as a benefit from joining the Mafia, stemming from the escape from the Mafia’s harass-ment. When all agents are risk-neutral, such a benefit as joining the Mafia is equivalent to the disutility from refus-ing to join the Mafia (i.e., engagrefus-ing in individual crime) in our model.
5. In some circumstances, the extra benefit may de-pend on the size of the criminal organization. Such a spec-ification will yield similar results in the article and merely complicate the model without adding too much insight. A mathematical deduction is available on request.
in organized crime as long as uO> uI, or equiv-alently bþ e y pf1¼ bð1 þ aÞ y pf1 >b pf2: ð2Þ
Otherwise, he will be an individual criminal. In the second stage, potential offenders de-cide whether to commit an offense. According to (1), the ‘‘participation constraints’’ indicate that committing a crime is worthwhile if uI>0 or if uO>0, that is, if
bð1 þ aÞ pf1 y > 0 or
if b pf2>0:
On the contrary, committing a crime is not worthwhile if
bð1 þ aÞ pf1 y 0 and
if b pf2 0:
Based on the self-selection constraint in (2) and the participation constraints in (3) and (4), we propose the following proposition. PROPOSITION 1. Under the uniform sharing scheme, potential offenders with relatively high criminal skills will tend to commit crimes alone, whereas those with relatively low potential crimi-nal rents will choose not to enter the crimicrimi-nal mar-ket. The potential offenders who have medium abilities will choose to be organized by the Mafia. Proof. By a backward induction, we first deal with the offender’s decision regarding whether to join the criminal organization, given that he enters the criminal market. Under the uniform sharing arrangement, other things being equal, the self-selection constraint (2) indi-cates that low-ability offenders (with lower criminal benefits b) are more likely to join the Mafia and receive more criminal payoffs than they would have received when
commit-ting crimes alone.6However, under a uniform sharing scheme, the high-ability offenders tend not to participate in organized crime, be-cause their distinctly high criminal rents will be shared by other free riders with lower criminal rents. Due to the free-rider problem, the offenders with a higher b will choose I, rather than O.7Accordingly, as shown in Figure 2, the region I (individual criminals) should be on the right-hand side of O (organized criminals) according to the criminal rent distribution.
We turn to the potential offender’s decision regarding whether to commit crimes (i.e., choose C) or not (i.e., choose N). In (4) the participation constraint b pf2indicates that potential offenders with very low criminal skills will not commit individual crimes. In ad-dition, the average payoff for Mafia members b is dependent on the qualities of the offenders that are organized. Given a government’s en-forcement pf1and the Mafia’s fee y, if mostly low-ability offenders join the Mafia, then their participation will result in a very low level of the Mafia’s average payoff b, thus making it more likely that their own net benefit will be negative (i.e., b[1þ a] y þ pf1in equation (4) is true). Thus, it will not be worthwhile for low-ability potential offenders to participate in organized crime, and they will remain as law-abiding citizens. Therefore, Figure 2 indicates that the region N (commits no FIGURE 2
The Equilibrium Margins in [N, O, I]
6. Once those low-ability offenders choose O, the av-erage shared payoff for Mafia members will decrease ac-cordingly. Given that e > 0, as we will see in next subsection, under certain conditions, some offenders will still be willing to join the Mafia as long as the net profit from joining is positive.
7. If the Mafia provides an unduly large extra benefit to its members (or if the difference f2 f1is positive and
extremely large), then offenders with higher criminal rents will also join the criminal organization. However, in such a situation, individual criminals will eventually disappear. To ensure that O and I coexist in equilibrium, we rule out these possibilities in our analysis. In general, as we will see in the next section, given the free-rider problem, [N, O, I] will be a unique equilibrium.
crime) will be on the left-hand side of O and I within the criminal rent distribution.
Based on these deductions, under a uniform sharing arrangement, people with the lowest criminal rent (or skill) will not commit crime at all (N), people with medium criminal rent will become organized criminals (O), and those who have the highest criminal rent will become individual criminals (I). This consti-tutes an [N, O, I] equilibrium as described in
Figure 2. Q.E.D.
In what follows we will prove the existence of the [N, O, I] equilibrium, and show that the type [N, O, I] is a unique equilibrium under the Mafia’s uniform sharing arrangement. The Solution of the Equilibrium [N, O, I]
We denote RO, RI, and RNas the propor-tions of citizens choosing O, I, and N, re-spectively, in the community. There are two critical levels of potential criminal rent, namely, bl and bu (where bl bu), that de-termine whether an individual will commit crime and the way he commits the crime. According to Proposition 1, citizens with criminal rents 0 b bl will not enter the criminal market at all. For those who decide to enter the criminal market, offenders with criminal rents bl< b buwill join a criminal organization, and those who have a relatively higher criminal ability bu< b 1 will decide to commit crime alone. Thus, we can define the average payoff of a criminal organization
b, and the equilibrium proportions RN, RO, and RI as follows: b¼ ðbu bl bdb. ð bu bl db¼ ðblþ buÞ=2; ð5Þ RN¼ ðbl 0 db¼ bl; ð6Þ RO¼ ðbu bl db¼ bu bl; ð7Þ RI ¼ ð1 bu db¼ 1 bu: ð8Þ
The marginal potential offender who has criminal rent bl is indifferent in terms of choosing between joining a criminal organi-zation and committing no crime. Thus, from
the participation constraints and (5), we have:
0¼ ð1 þ aÞðblþ buÞ=2 y pf1:
By analogy, the potential offenders who have a criminal rent bu are indifferent in terms of choosing between I and O. That is,
ð1 þ aÞðblþ buÞ=2 y pf1¼ bu pf2:
We now define the regions of N, O, and I as XN, XO, and XI, respectively. Thus, we estab-lish Proposition 2 to describe the equilibrium [N, O, I] as follows.
PROPOSITION 2. Under the [N, O, I] equi-librium, the criminal ability distribution of three types of agents N, O, and I, are, respectively:
XN[½0; 2ðy þ pf1Þ=ð1 þ aÞ pf2;
XO[ð2½y þ pf1=ð1 þ aÞ pf2;pf2;
Proof. From (9) and (10), the equilibrium crit-ical values bland bu, respectively, are given by:
bl ¼ 2ðy þ pf1Þ=ð1 þ aÞ pf2 and
bu ¼ pf2:
According to (11), we immediately have the equilibrium regions of N, O, and I. Q.E.D. From (11), we also learn that the existence of the equilibrium [N, O, I] should satisfy the condition 0 < bl< bu<1, that is, (1þ a)pf2/ 2 pf1< y < (1þ a)pf2 pf1and pf2<1. When critical rents bland buare determined, the average criminal rent can also be solved from (5) and is given by b¼ (pf1þ y)/ (1 þ a).
We can be assured of the existence of this equilibrium. Proposition 2 indicates that the offenders with criminal rents b2 XOwill join a criminal organization. Let the lower bound of XObe 2(yþ pf1)/(1þ a) pf2þ e, where erepresents an infinitesimal positive value, that is, e / 0þ. Accordingly, the average criminal rent b is (pf1þ y)/(1 þ a) þ e/2. Substituting this into (1), the payoff for potential offenders choosing O, uO¼ (1 þ a)e/2 > 0, which satisfies the participation constraint. If an offender with an upper-boundary criminal rent pf2chooses I, then his payoff is b pf2¼ pf2 pf2¼ 0, which
is less than (1þ a)e/2. This implies that he must be a Mafia member.
Similarly, it follows from Proposition 2 that the offenders with b 2 XI choose to commit crimes alone. By letting the lower bound of X1 be pf2 þ e, from (1), an offender with b ¼ pf2 þ e, will obtain a benefit uI ¼ b pf2¼ e if he chooses I, and he will receive a ben-efit (1þ a)e/2 if he chooses O. Because 0 < a < 1, we have e > (1þ a)e/2, implying that this offender will rationally choose I. Therefore, we verify the existence of the equilibrium [N, O, I] and ensure its credibility.8
The Optimal Law Enforcement and the Equilibrium Crime Rate
Given that the government and the Mafia play a Cournot-Nash game, this subsection will analyze these two parties’ behaviors and in turn determine the Mafia’s optimal extrac-tion, the government’s optimal law enforce-ment, and the equilibrium crime rate. Mafia. In our model the criminal organiza-tion is structured as a pure franchise that pro-vides benefits B¼ e ¼ (1 þ a) b and receives an entry fee y. The way the criminal organiza-tion runs the business on the basis of the extra benefit, such as by giving bribes to officials, is abstracted from the model. In particular, we do not analyze how the criminal organization pays these amounts to arrive at an optimal level of the extra benefit a. For the sake of compar-ison with the existing literature, we treat a as a parameter. Though simple, such a specifica-tion allows us to capture a number of scenarios and focus attention on the relationship between theindividual’srationalchoice,themarketstruc-ture for crime, and the optimal enforcement.
Given the government’s enforcement pol-icy, the Mafia’s optimization problem is to maximize its total profit, that is,
max y P¼ ðbu bl ydb Z ¼ yðbu blÞ Z; ð12Þ
where Z is the Mafia’s fixed operation cost.9 By substituting equilibrium values bl and bu from (11) into (12), we obtain the first-order condition @P/@y ¼ 2[pf2 (2y þ pf1)/ (1þ a)] ¼ 0, which yields the Mafia’s optimal extraction:
yRF¼ ½ð1 þ aÞpf2 pf1=2:
Equation (13) conveys very intuitive results whereby the Mafia’s optimal extraction yRF increases with the extra benefit a and decreases with the expected sanction of organized crime pf1. By contrast, when the expected sanction of the alternative choice I, pf2, increases, the Mafia will be able to extract more rent from its members.
Government. Let us denote h1 and h2 as the average harm to a society resulting from an organized and an individual criminal offense, respectively. It is plausible to specify h1 h2>0. It is also important to take into account the attendant externality generated by the Mafia. For this, we assume that producing the extra benefit e will generate a negative externality be to a society, where b reflects the degree of such an externality. Accordingly, the aggregate surplus stemming from O can be described by Ðbu
blðbþ ae h1 beÞdb:
For analytical convenience, we further define h [ b a (and 0 < h < 1), which measures the degree of the net externality of producing e.
Furthermore, it is evident that the surplus stemming from I isÐb1
uðb h2Þdb: To add these
two surpluses, the social welfare W is given by:Ðbu
bl ðb h1 heÞdb þ
buðb h2Þdb CðpÞ:
The term C stands for the cost of detection and conviction, which is an increasing function of p, that is, @C/@p > 0. For simplicity, we as-sume C(p)¼ cp with c > 0 in the analysis that follows.
Given the Mafia’s extraction y, the authori-ties choose the optimal enforcement p, f1, and f2 so as to maximize the social welfare W, that is,
8. The analysis can be easily reduced to two possible special cases. First, following Figure 2, the equilibrium [N, I] exists if bl bu>0 and bu<1, implying that the
conditions y (1 þ a)pf2 pf1and pf2<1 must be
sat-isfied. Second, if 0 < bl< buand bu 1, the equilibrium
[N, O] will emerge. The corresponding conditions of exis-tence are (1þ a)/2 pf1< y <(1þ a) pf1and pf2 1.
9. If Z is related to the extra benefit and takes the form Z ¼ z e with z > 0, our main results are not altered.
max p;f1;f2 W ¼ ðbu bl ðb h1 heÞdb þ ð1 bu ðb h2Þdb Cð pÞ ¼ ½1 ð1 ahÞb2 l ahb2u=2 ðbu blÞh1 ð1 buÞh2 Cð pÞ: ð14Þ
We recall that in the analysis we should con-fine the sanctions to the limitation f1, f2,F, due to the problem of limited liability. In addition, we further assume that f2 ¼ f and f1¼ f þ Df, where Df 0, reflecting the fact that an organized criminal may incur higher penalties than an individual criminal. With this assumption, (11) becomes
bl¼ ½2y þ ð1 aÞpf þ 2p Df
=ð1 þ aÞ and bu¼ pf;
and, accordingly, the resulting equations yield the following comparative statics:
@bl=@p¼ ½ð1 þ aÞf þ 2Df =ð1 þ aÞ; @bl=@f ¼ ð1 aÞpð1 þ aÞ; @bl=@Df ¼ 2p=ð1 þ aÞ; @bu=@p¼ f ; @bu=@f ¼ p; and @bu=@Df ¼ 0:
Differentiating W with respect to p, f, and Df, we have @W =@p¼ ½h1 ð1 ahÞbl½ð1 aÞf þ 2Df =ð1 þ aÞ ½ahbuþ ðh1 h2Þ f c; ð15aÞ @W =@f ¼ pfð1 aÞ½h1 ð1 ahÞbl =ð1 þ aÞ ½ahbu þ ðh1 h2Þg; ð15bÞ @W =@Df ¼ 2p½h1 ð1 ahÞbl =ð1 þ aÞ: ð15cÞ
Due to (1 a)f þ 2Df > 0, we should specify h1 (1 ah)bl >0 in (15a) to guarantee an
interior solution for p (i.e., @W/@p¼ 0). Given that, from (15c) we have @W/@Df > 0, imply-ing that the government would like to impose a higher penalty on organized criminals than individual criminals. That is, as a best policy, the government will set Df as the maximal level, that is, Df [ f1 f2¼ F f. By substitut-ing Df¼ F f into (15a) and letting the result-ing equation be 0 (i.e., @W/@p ¼ 0), the optimal p satisfies:
h1 ð1 ahÞbl
¼ ð1 þ aÞ½c þ ðahbuþ h1 h2Þf
=½2F ð1 þ aÞf : ð15dÞ
Putting (15b) and (15d) together thus immedi-ately yields
@W =@f ¼ p½2ðF f Þ
ðahbuþ h1 h2Þ þ ð1 aÞc
=½2F ð1 þ aÞf : ð16Þ
In (16), if f¼ F, then @W/@f > 0 is true. In ad-dition, from (15b) we also have @2W/@f2 ¼ p [(1 a)(1 ah)/(1 þ a) (@bl/@f) þ ah (@bu/@f)] < 0, meaning that the W func-tion is concave in f. With this understand-ing, the optimal f will be bound at F and hence Df ¼ F F ¼ 0. In other words, the government’s best policy is to set the highest sanction, that is, fRF
1 ¼ f2RF¼ F:
1 ¼ f2RF ¼ F; from (11#) and (15a),
the government’s optimal law enforcement is given by ðpf ÞRF¼ ð1 þ aÞ½2ah1þ ð1 þ aÞh2 2ð1 ahÞð1 aÞy =ð1 þ aÞ ð1 þ aÞc=F =ð4a2hþ a2 2a þ 1Þ: ð17Þ
Equation (17) indicates that the optimal law enforcement is negatively related to the level of the Mafia’s entry fee y. In the [N, O, I] equi-librium, a higher entry fee discourages the po-tential offenders from committing any crime, and thus the government can save on the law enforcement budget given a tolerable crime rate. That is to say, the government can regard the Mafia’s entry fee as a substitute for law enforcement in maximizing social welfare.
The two reaction functions (13) and (17) determine the equilibrium law enforcement and the Mafia’s extraction simultaneously:
p*f *¼ ð1 þ aÞ½2ah1 þ ð1 þ aÞh2 ð1 þ aÞc=F =ða3hþ 3a2h a þ 1Þ; y*¼ ap*f *=2 ¼ að1 þ aÞ½2ah1 þ ð1 þ aÞh2 ð1 þ aÞc=F =2ða3hþ 3a2h a þ 1Þ: ð18Þ
Substituting (18) into (11) yields the equilib-rium critical values bland buas follows:
bu*¼ p*f * and b*l ¼ p*f *=ð1 þ aÞ:
PROPOSITION 3. Under the uniform sharing scheme, (i) if the Mafia cannot provide any ex-tra benefit to its members (a¼ e ¼ 0), the crim-inal market turns out to be perfectly competitive and all offenders are individual criminals; (ii) if p*f * /1, the criminal market will be charac-terized by monopoly.
Proof. It follows from (19) that if a¼ e ¼ 0, in equilibrium the critical value bl will atrophy to bu. By referring to Figure 2, this implies that no one will choose O and all offenders will commit individual crimes. Thus, the criminal market becomes perfectly com-petitive. By contrast, when p*f * / 1, from (11) we learn that the critical values reduce to bu* / 1 and b* / 1/(1 þ a). These indicatel that RI shrinks to zero and RO¼ bu* bl* ¼
a/(1þ a). Q.E.D.
The economic intuition behind the result of Proposition 3 is straightforward. As a¼ e ¼ 0, relative to I, the only benefit of O is in the shared average payoff b. However, any of-fender with a relatively higher ability will choose I to avoid sharing his distinctly higher criminal rent with those who have lower rents. The free-rider problem thus chokes off any in-centive for joining the Mafia. By contrast, if the government raises its law enforcement to a very high level, that is, p*f* / 1 (or even larger than 1), the high intensity of law en-forcement will eliminate all possibility of
indi-vidual crimes.10However, because the Mafia’s extra benefit could take the form of a reduction in terms of the government’s detection, all offenders will in such circumstances choose O and take shelter in the Mafia. The criminal market will therefore be characterized by mo-nopoly.11
The Equilibrium Crime Rate. Let us define the equilibrium crime rate as R [ ROþ RI. By referring to Figure 2, the equilibrium crime rate is
R¼ 1 bl*¼ 1 ½2ah1
þ ð1 þ aÞh2 ð1 þ aÞc=F
=ða3hþ 3a2h a þ 1Þ:
According to (19) and (20), we then have the following.
PROPOSITION 4. If the Mafia creates a larger extra benefit for its members, the equi-librium crime rate will rise in response. How-ever, a higher Mafia extra benefit may not always be helpful in recruiting more members and increasing the size of the criminal organiza-tion after all.
Proof. See Appendix A.
From (20), the result @R/@a¼(@bl*/@a)> 0 indicates that a rise in the Mafia’s extra ben-efit will cause some citizens who initially abide by the law to join the Mafia and, as a consequence, the overall crime rate will rise.
This, however, does not imply that the
number of the Mafia’s members RO ¼
bu* bl* (see ) will also be increased by a. Intuitively speaking, on the one hand, @bl*/@a < 0 reveals that following a rise in a, some potential offenders with relatively low b will be enticed to join the Mafia. However, on the other hand, by referring to (11) and (19), the result @bu*/@a ¼ @(p*f*)/@a 5., 0
10. Generally speaking, pf would be larger than 1 (the upper bound of criminal benefit) only when the social harm resulting from an offense is very large. If pf were large enough, then all criminal behavior would be elimi-nated.
11. Under such a case when the Mafia provides more extra benefits to its members (a larger a), then the size of the monopolistic criminal organization will grow.
indicates that a higher a will have a mixed effect in terms of attracting high-ability offenders. A higher a may also generate an incentive effect in terms of attracting high-ability offenders to the criminal organi-zation. Nevertheless, when the Mafia pro-vides more benefits to its members, at the same time, it will also extract more from them (inferred from ). Because the gov-ernment regards the Mafia’s entry fee as a substitute for law enforcement, the optimal law enforcement will decrease (inferred from ). This will encourage offenders to choose I rather than O. Due to this negative effect, a higher Mafia extra benefit may not always help in terms of recruiting more offenders and increasing the size of the criminal orga-nization. This result potentially contributes to an interesting implication whereby the size of the Mafia is ambiguously related to the crime rate.
Propositions 3 and 4 also contribute an im-portant implication to the existing literature. For ease of comparison with the existing liter-ature, we assume that (1) an individual criminal does the same harm to society as an organized criminal, that is, h1¼ h2¼ h; and (2) the extra benefit e is a pure ‘‘social transfer’’ and hence h¼ 0 (i.e., a¼ b) (for example, an increase in the extra benefit via a reduction in the government’s deterrence would mean an increase in social harm). Given those assumptions, if the Mafia’s extra benefit is absent (a¼ 0), the equilibrium crime rate (20) and law enforcement (19) are re-duced to those of Garoupa (2000). As such, as argued by Garoupa (2000), the existence of the Mafia will thus become welfare-improving to society as a whole. However, it follows from (19) and (20) that when a > 0, because all offenders can choose the criminal patterns that best suit them under the self-selection mecha-nism, the society will have a higher crime rate under self-selection market than under monop-oly. Moreover, a monopolistic crime market does not necessarily result in a reduction of the optimal law enforcement. As a result, Prop-ositions 3 and 4 sharply contradict the tradi-tional viewpoints of Buchanan (1973) and Garoupa (2000).12
IV. DISCUSSIONS AND EXTENSIONS The benchmark framework is well worth extending to tackle related issues. In this sec-tion we consider two extensions.
An Ability-Adherent Sharing Scheme
In section III it is found that, under a uniform sharing scheme, [N, O, I] is the unique equilib-rium in the self-selection criminal market. This equilibrium indicates that only relatively low-ability criminals join a criminal organization, whereas high-ability criminals commit crimes alone. Perhaps this gives us the impression that a criminal organization might merely be formed by low-end criminals or by ‘‘a group of street gangsters.’’ However, this equilibrium does not seem to properly capture the typical profiles of criminal organizations that we see in real life, mostly because we assume the exis-tence of a uniform sharing scheme. In this sub-section we consider another sharing scheme that adheres to members’ abilities and, accord-ingly, demonstrates the existence of a distinctive equilibrium, namely, the [N, I, O] equilibrium. This [N, I, O] equilibrium may fit into some of the familiar profiles we often read about in the literature regarding criminal organizations. One point should be noted here. Although our central concern is to verify the existence of the [N, I, O] equilibrium, we do not, however, intend to exclude the possibility of the existence of other equilibrium, such as [N, O, I], in the ability-adherent sharing scheme.13
In the ability-adherent sharing scheme, the Mafia’s payoff is distributed according to members’ abilities. We denote R(b) as a shar-ing function, with the payoff positively relat-ing to members’ personal criminal rents within the Mafia. Thus, we can specify a member’s payoff as R(b)b(1 þ a) y pf1, where Ð b2XOR(b)db¼ Ð b2XOdb and R#(b) > 0. 14
12. See Chang et al. (2002) (an earlier version of this article) for a more complete discussion.
13. In fact, some characteristics of the [N, O, I] equi-librium under an ability-adherent sharing scheme are sim-ilar to those in the uniform sharing case. See the working paper of Chang et al. (2002) for the details.
14. To make a meaningful comparison with a ‘‘uni-form’’ sharing scheme, we use the average benefit b, rather than the total benefitÐb2XObdb to describe a member’s
distributed benefit. When R(b) ¼ 1 "b, the ability-adherent sharing arrangement is reduced to the uni-form-sharing one. We first denote S(b) as the ‘‘actual’’ sharing proportion out of the Mafia’s total payoff, and then Ðb2XOS(b)db ¼ 1. By so doing, we obtain S(b)
Ð b2XObdb¼ R(b) b, where R(b) [ S(b) Ð b2XOdb and b [Ðb2XObdb/ Ð b2XOdb. That is, Ð b2XOR(b)db [ Ð b2XOdb.
Assume that there are two critical values bl and bu(where bl< bu) such that
b2 ½0; bl 0 no crime;
b2 ðbl;bu0 individual crime;
b2 ðbu;10 organized crime:
Given (21), the average benefit to the Mafia’s members is b¼ (1 þ bu)/2. Accordingly, we can find the critical values bland bu, which satisfy both the self-selection and participation con-straints as follows: ð1 þ aÞRðbuÞð1 þ buÞ=2 y pf1 ¼ bu pf2; ð22aÞ bl pf2¼ 0: ð22bÞ
Equation (22a) can be further rewritten as: Uðbu;aÞ ¼ y þ p Df ;
where U(bu; a) [ (1þ a)R(bu)(1þ bu)/2 bu. Equation (23) indicates that, compared with I, the relative benefit of choosing O is U(b; a) and its relative cost is yþ p Df.
Based on (21)–(23), we have the following proposition,
PROPOSITION 5. Under the ability-adherent sharing arrangement, if the self-selection constraint U(b; a) > 0"b 2 (bu, 1] and @U/ @bjb¼bu > 0, then the equilibrium [N, I, O] exists.
Proof. See Appendix B.
Proposition 5 points out that the [N, I, O] equilibrium exists as long as the following conditions hold. First, for potential offenders with b > bu, the relative benefit of joining a criminal organization compared to commit-ting crime alone is positive (i.e., U(b; a) > 0, "b 2 (bu, 1]). To satisfy this, the Mafia must provide a sufficiently large extra benefit to its members. Second, the sharing scheme is de-signed such that the relative benefit from join-ing a criminal organization increases with an offender’s criminal ability (i.e., @U/@bjb¼bu>0). A simple example is useful to verify the existence of the equilibrium [N, I, O]. Let us denote S(b) as a member’s ‘‘actual’’ sharing proportion out of the Mafia’s total
crimi-nal payoff and assume that SðbÞ ¼ b2=
2db ¼ 3b2=ð1 b3
uÞ: Accordingly, we
can further calculate the sharing functions RðbÞ ¼ SðbÞÐb2XOdb¼ 3b2=½1 þ b
uþ b2u and
b2XOR(b)db ¼ 1 bu. From (23) with this sharing function, we can obtain
Uðbu;aÞ ¼ bu þ 3ð1 þ aÞb2 uð1 þ buÞ =½2ð1 þ buþ b2uÞ > 0 if a >½2 buð1 þ buÞ =ð3buð1 þ buÞ; ð24Þ @U=@bjb¼bu¼ ½1 þ ð1 þ 3aÞbu þ 3ð1 þ 2aÞb2u þ ð1 þ 3aÞb3 u þ ð1 þ 3aÞb4 u=2 =½ð1 þ buþ b2uÞ 2 > 0: ð25Þ
Equations (24) and (25) satisfy the require-ments stated in Proposition 5 and verify the existence of the equilibrium [N, I, O]. Law Enforcement Distortion of the Mafia
In this extension we consider a situation where the Mafia is able to command some influ-ence in weakening criminal deterrinflu-ence. One way of seeing this is that the influence on law en-forcement is exerted through the corruption of the law enforcers or police officers.15To model this distortion of the Mafia, we simply specify that the extra benefit e provided by the Mafia increases with pf1and takes the following form:
e¼ a bþ spf1; with 0 < s < 1:
According to (26), the payoff of an organized criminal changes to:
uO¼ b pf1þ e y
¼ b ð1 sÞpf1þ a b y:
We learn from (27) that the effective law en-forcement (1 s)pf1could be reduced by the Mafia and s measures the degree of such an enforcement distortion. Note that, under this specification, a b will only be viewed as ex-tra income from joining the Mafia.
15. We are grateful to an anonymous referee for bring-ing this point to our attention.
Because the modification of the extra ben-efit e does not alter the equilibrium character-istic of [N, O, I], according to (27), the critical values of the lower and upper bounds in (11#) can be changed to:
bl¼ ½2y þ ð1 a 2sÞpf
þ 2ð1 sÞp Df =ð1 þ aÞ and bu¼ pf :
The government, subject to 0 < f1, f2 F, chooses p, f, and Df to maximize the social wel-fare W reported in (15). Solving this optimiza-tion problem yields the following first-order conditions @W =@p¼ ½h1 ð1 ahÞbl ½ð1 a 2sÞf þ 2Df =ð1 þ aÞ ½ahbu þ ðh1 h2Þf c; ð28aÞ @W =@f ¼ pfð1 a 2sÞ ½h1 ð1 ahÞbl=ð1 þ aÞ ½ahbuþ ðh1 h2Þg; ð28bÞ @W =@Df ¼ 2ð1 sÞp½h1 ð1 ahÞbl=ð1 þ aÞ: ð28cÞ
To obtain an interior solution for p (i.e., @W/ @p¼ 0), we should assume [h1 (1 ah)bl] [(1 a 2s)f þ 2Df] > 0. In the relevant studies, there is a common assumption whereby the social harm stemming from a crime exceeds the criminal rent, that is, h1 (1 ah)bl>0 in our model.
Following a similar approach to that in the last section, from (28a)–(28c) we can easily obtain the optimal sanctions f1RF¼ f2RF¼ F, and, as a consequence, the government’s optimal law enforcement is given by
ðpf ÞRF¼ ð1 þ aÞ½2ða þ sÞh1 þ ð1 þ aÞh2 2ð1 ahÞ ð1 a 2sÞy=ð1 þ aÞ ð1 þ aÞc=F =½ð1 a 2sÞ2ð1 ahÞ þ ð1 þ aÞ2ah: ð29Þ
From (29), we immediately have @ðpf ÞRF=@s5,.0;
implying that, in the presence of the enforce-ment distortion s, increasing the severity of law enforcement is no longer an effective pol-icy in the reduction of crimes and in the improvement of social welfare. As the enforce-ment distortion s increases, some individuals who choose N initially will be allured to be or-ganized by the Mafia (i.e., it follows from equation (11$) that bl falls as s increases) and, consequently, organized crime will in-crease. To deter organized crime and reduce the social harm arising there from, the author-ities at first would like to increase the law en-forcement pf. The stronger law enen-forcement is, however, the greater will be the enforcement distortion generated by the Mafia. To con-strain the law enforcement distortion, the gov-ernment then may decrease pf. Because the law enforcement is congested, the government will be in a dilemma in terms of deterring crime and maximizing social welfare. This result is akin to that in Chang et al. (2000), who incor-porate police corruption and the social-norm mechanism into the Becker (1968) model and use it to address the dilemma of law enforce-ment facing a governenforce-ment.
The Mafia’s optimization problem is simi-lar to that of (12). Nevertheless, to incorporate the corruption cost into the Mafia’s objective function, we modify the specification of oper-ating costs as Z¼ Z0þ Z1 spf1, where Z0and Z1are constant and positive. Given that, by solving the Mafia’s optimization problem, we obtain the optimal extraction as
yRF¼ ½ð1 þ aÞpf2 ð1 sÞpf1=2:
This indicates that a higher s will allow the Mafia to extract more rent from its members.
Two reaction functions in (13#) and (29) determine the equilibrium law enforcement and the Mafia’s extraction simultaneously:
p*f *¼ ð1 þ aÞ½2ða þ sÞh1 þ ð1 þ aÞh2 ð1 þ aÞc=F =½ð1 ahÞð1 a 2sÞð1 sÞ þ ð1 þ aÞ2ah; y*¼ ða þ sÞp*f *=2: ð30Þ
Moreover, substituting (30) into (11$) yields the equilibrium critical values:
bl* ¼ ð1 sÞp*f *=ð1 þ aÞ and
bu* ¼ p*f *:
These results allow us to establish Proposi-tion 6 as follows.
PROPOSITION 6. In the presence of the enforcement distortion of the Mafia, under theﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ(sufficient) condition, s > 1
ð1 þ aÞ2ah=½2ð1 ahÞ q
an increase in the enforcement distortion s will result in a higher rate of crime in the society as a whole. Proof. From (30) and (31), it is easy to derive:
@R=@s¼ f½1 þ ð1 sÞCs=C ½2ða þ sÞh1 þ ð1 þ aÞh2 ð1 þ aÞc=F þ 2ð1 sÞh1g=C; ð32Þ where C [ (1 ah)(1 a 2s)(1 s) þ (1 þ a)2ah >0 and Cs[ @C/@s ¼ (1 ah)(3 þ a þ 4s). Given the (sufficient) condition s > 1
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 þ aÞ2ah=½2ð1 ahÞ q
, C þ (1
s)Cs>0 is true, and, as a result, @R/@s > 0. Q.E.D. In (32) there are three strengths in govern-ing the impact of a rise in s on the aggregate crime rate. First, as s rises, given a particular entry fee y, some individuals who choose N initially will be allured to choose O. Under this situation, the government may resort to stronger law enforcement to deter the growth of organized crime. This will give rise to a de-terrence effect in terms of decreasing the crime rate. However, the later second and third effects refer to opposing impacts on the crime rate. The second effect is that as already emphasized, in the presence of s the discipline effect of the government’s enforcement is plagued by the distortion of the Mafia. To constrain the enforcement distortion, the gov-ernment should let pf decrease. Crimes thus may grow, accordingly. Third, we learn from (14#) that the Mafia will extract more rents from its members because the Mafia can pro-vide better protection against the criminal penalty imposed on organized criminals. Given that the government’s enforcement
and the Mafia’s entry fee are substitutes, in re-sponse to a higher y, the government will de-crease law enforcement pf and tolerate more crimes. Clearly, if the last two effects suffi-ciently outweigh the first effect, the equilib-rium crime rate rises in response to a higher degree of enforcement distortion s. The suffi-cient condition indicates that intuitively, the outcome is more likely to become true under the circumstance where corruption is initially overwhelming (i.e. the initial s is greater).
In a way that departs from the traditional framework, this article has developed a more general model in which individual and orga-nized crime are coexisting alternatives from which a potential offender can choose. Based on this framework, we are able to analyze the interactive relationships among individuals’ choices of crimes, the market structure for crimes, the government’s law enforcement strategies, and the aggregate crime rate.
The article has considered two different sharing schemes—the uniform and ability-adherent sharing schemes—for Mafia mem-bers. Under the uniform-sharing scheme, if the Mafia cannot provide any extra benefit for its members, then the so-called free-rider problem will choke off any incentive to join the Mafia. As a result, the criminal market will turn out to be perfectly competitive, and all offenders will become individual criminals. Even if it is able to provide pos-itive extra benefits to its members, the Mafia will only entice offenders with relatively low criminal skills to join the organization. The potential offenders who are highly skilled will tend to commit crimes alone. The mar-ket structure for crimes will thus exhibit the [N, O, I] equilibrium. Given this equilibrium, we have shown that when the Mafia creates a larger extra benefit for its members, the equilibrium crime rate will rise in response. Furthermore, in the presence of a positive Mafia extra, the existence of the Mafia may not be welfare-improving.
Under the ability-adherent sharing scheme, the criminal market may end up with a very different market structure. By extending the benchmark model, we have proved the exis-tence of a distinctive equilibrium, namely, the [N, I, O] equilibrium. This implies that
the Mafia can entice potential offenders who have high criminal skills to join the organiza-tion, as the Mafia’s payoff is distributed according to members’ abilities. Given the dif-ferent sharing schemes within criminal organ-izations, the article potentially provides an explanation for Anderson’s observation that there are considerable variations in personal qualities and values across different criminal organizations.
Some assumptions in this article are still debatable and should be extensively discussed in future research. First, under the ability-adherent sharing scheme, section IV focused on the existence of the [N, I, O] equilibrium, whereas we have not tackled the relevant issues concerned with the crime rate and the government’s optimal law enforcement under different [N, I, O] and [N, O, I] equi-libria. It is interesting to investigate these issues and compare the consequences under two distinctive equilibria. For a preliminary discussion, one may refer to the working pa-per by Chang et al. (2002). Second, we have described the interaction between the govern-ment and the Mafia by a Cournot-Nash game. However, it may be a case where the government and the Mafia play a Stackelberg game instead. Along this line, it may be also interesting to extend the model to compares the equilibrium solutions under different games (including Cournot-Nash and Stackel-berg games) and different market structures (including perfect competition, monopoly, and self-selection markets). For the relevant discussions, readers is also referred to our working paper. Third, in this article the Mafia’s extra benefits have been endogenized and are related to the members’ average abil-ity within the organization (in the bench-mark) and the government’s law enforcement (in the extension). Nevertheless, here we do not formally take account of how the crimi-nal organization runs its business. Garoupa (2001) sets up a principal-agent model to an-alyze how the principal (the leader of the criminal organization) provides extra infor-mation to his members. By means of a mo-nopolistic competition model, Kugler et al. (2003) address the issue of how the criminal organizations compete with each other in re-lation to the crime and corruption. Following this line of research, it would also be worth-while in our future research to engage in a rel-evant extension. To seriously deal with this
issue, more sophisticated analysis concerned with the interaction between a government and the Mafia is needed.
APPENDIX A: PROOF OF PROPOSITION 4 Define W [ a3h
a þ 1 and hence Wa[ @W/
hþ 6ah 1. To guarantee R > 0, we should also restrict W > 0. With this restriction, from (20), we have
@R=@a¼ f½W=ð1 aÞ þ Wa½ð1 þ aÞh2
2ah1 ð1 þ aÞc=F
þ 2Wðh1 h2þ c=FÞ=ð1 aÞg=W2:
In (A1) when we restrict the law enforcement p*f* > 0, 2ah1þ (1 þ a)h2 (1 þ a)c/F > 0 must be met.
Further-more, because a and h are less than 1, then W=ð1 aÞ þ Wa¼ 2ahð3 a2Þ=ð1 aÞ > 0:
These conditions allow us to conclude @R/@a > 0, meaning that, as a increases, the equilibrium crime rate will rise as a response. In addition, from (19) we have
RO¼ bu* bl*
¼ a½2ah1þ ð1 þ aÞh2 ð1 þ aÞc=F=W:
According to (A2), the comparative static concerning R with respect to a is given by
@RO=@a¼ fðW aWaÞ
½2ah1þ ð1 þ aÞh2 ð1 þ aÞc=F
aW½2h1 h2þ c=Fg=W2 .5,0
APPENDIX B: PROOF OF PROPOSITION 5 Consider two potential offenders with criminal rents bu
and buþ e, where e / 0þ. According to self-selection
con-straint and (21), the existence of [N, I, O] must satisfy: ð1 þ aÞRðbuÞð1 þ buÞ=2 y pf1 bu pf2
0Uðbu;aÞ y þ p Df and
ð1 þ aÞRðbuþ eÞð1 þ buþ eÞ=2 y pf1>bu pf2þ e
0Uðbuþ e; aÞ > y þ p Df :
These conditions indicate that U(bu þ e; a) > U(bu; a),
implying that @U/@bjb¼bu>0. Because yþ p Df > 0,
we also need the condition U(b; a) > 0,"b 2 (bu, 1]. That
is, the Mafia must provide a large enough a to ensure the existence of the equilibrium [N, I, O]. If the extra benefit effect is so small that U(b; a) < 0"b, then we cannot find a critical value bu2 (pf1, 1]. Q.E.D.
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