Becker (1968) a n d St igler (1970) in i t ia ted the l i t e ra ture on the e c o n o m i c s o f c r ime by showing h o w the m a r k e t for c r ime , as any o t h e r marke t , cou ld be analyzed us ing e c o n o m i c theory. S ince then , mos t papers have e x a m i n e d the behav io r o f the cr iminals for a given market and for a g iven (possibly op t imal ly chosen ) pub l ic d e t e r r e n c e policy, 1 n e g l e c t i n g the ef fec t the cr iminals can have o n the m a r k e t itself. A no t ab l e e x c e p t i o n is Sah (1991) w h o e x a m i n e d the p e r c e p t i o n s a n d choices o f po ten t i a l c r imina ls w h e n the i n f o r m a t i o n available to t h e m is part ly d e t e r m i n e d by the i r choices .
In this paper , a m o d e l o f the m a r k e t for c r ime is bu i l t in which the p h e n o m e n o n o f c r imina l -on-c r imina l v io lence is t aken in to account . This type o f v io lence r ep resen t s a s igni f icant share o f the v io len t acts c o m m i t t e d in m o d e r n societies. 2 I n d e e d , casual obse rva t ion reveals that b e i n g a c r imina l increases the l i ke l ihood o f e n c o u n t e r i n g v io lence . For example , in C a n a d a in 1991, 45% o f the h o m i c i d e vict ims had a c r imina l
I would like to thank Stephen Gordon, Nguyen Manh Hung, Sherwin Rosen, Bruce Shearer, Pascal St-Amour, and two anonymous referees for useful comments, and Etienne Saint-Pierre for research assistance. I am also grateful to the Social Sciences and Humanities Research Council of Canada (SSHRCC) for financial support. Errors are mine.
1A large literature has examined the optimal sanction issue. See the recent papers by Bebchuk and Kaplow (1992), Kaplow (1990, 1992), Malik (1990), and Shavell (1987, 1991, 1992).
ZOne only has to think about street gangs. On this, see Reiss and Roth (1993, Ch. 13).
International Review of Law and Economics 17:193-201, 1997 @ 1997 by Elsevier Science Inc. 655 Avenue of the Americas, New York, NY 10010
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194 The market for crime
record, S whereas only 9.4% of all Canadians had a cr iminal record. 4 The main po in t of this pape r is as follows. When an individual part ic ipates in the marke t for crime, there is a risk he will encoun te r violence (e.g., mee t ing rival criminals) . Hence , only the individuals that have a high tolerance for violence (i.e., that suffer less from being a victim of violence) will par t ic ipate in this market. It is assumed that the more an individual is to lerant of violence, the less he dislikes commit t ing a violent act. There- fore, because the level of violence is itself de t e rmine d by the degree of to lerance of those who are active in the market , mul t ip le equi l ibr ia can obtain. Indeed , it is possible that the economy settles down in a high-violence equi l ibr ium, a l though a low-violence equi l ibr ium is possible. Intuitively, a high-violence equi l ibr ium implies that those who become criminals are very to lerant of violence. Those persons in turn are the ones that cause a lot of violence, suppor t ing the equi l ibr ium. A low-violence equi l ibr ium is also possible: Individuals that are no t to lerant of violence en te r the market , and those are the ones that cause less violence, thus suppor t ing this o ther equi l ibr ium. It is shown that policies a imed at reduc ing violence (e.g., gun control) will have different impacts d e p e n d i n g on the equi l ibr ium in which the economy rests.
The rest of the pape r is divided in three sections. In Section II, a simple model of cr iminal choices and marke t equi l ibr ium is described, and there is a discussion on its implicat ions for policy. A mult iple equi l ibr ia example is p resen ted in Section III. The conclusion follows.
II. The Model
Consider an economy inhab i ted by N individuals each of whom decide whether to par t ic ipate in the marke t for cr ime or not. If an individual part ic ipates in this market , there is probabi l i ty 7r he will encoun te r violence at a level v, someth ing he dislikes. On the o ther hand, if he does not encoun te r violence (which happens with probabi l i ty 1 - ~r), he gets a benef i t b from his par t ic ipat ion. Individuals are he te rogeneous in their to lerance for violence ~b dis t r ibuted according to a dis t r ibut ion with density g(+) on [~b z, cb h]. The h igher the to lerance of an individual, the smaller his ~b, and the less he suffers from violence. An individual of type ~b who encounte rs violence v, suffers a loss of +v, where v is i n d e p e n d e n t of the individual 's type. Assuming that the individuals are risk neutral , 5 the expec ted utility E of an individual of type ~b is simply:
E(qb) = (1 - ~ r ) b - ~rqbv. (1)
Assuming that alternative utility is equal to zero, an individual will par t ic ipate 6 in the marke t if E(d~) ~> 0, i.e., if he is of type + ~< ~b, where (b is a critical po in t to be
:aSee Wright (1991), p. 13. 4According to Correct ional Services Canada (1991), p. 11. :'This assumption is only tbr simplicity. The point made in this pape r could go th rough with risk aversion. ~Note that the p rob lem of the individual considered here departs significantly fi'om the one considered in the
s tandard literature. In the s tandard model , the individual faces a public sanction swith some probability p de t e rmined by public spending. The indixidual thus participates if the sanction a n d / o r the probability of apprehens ion are sufficiently low. In the s tandard model , the marke t for c r ime is represen ted by some benefit, say b*, that the criminal obtains if he is not caught. This paper addresses the de te rmina t ion of this (net) benefi t f rom being active in the market . In the cur ren t analysis, b* = E(+) . Hence , the two problems could be integrated with the overall expected udlity U o f an individual of type ~b given a pair (s, p) equal to:
U(cb; s, p) = (1 p ) E ( ¢ , ) - p s - (1 - p)((1 "rr)b - "rrqbv) - p s .
N. MARCEAU 195
below. T h o s e of type ~b > qb do n o t pa r t i c ipa t e in the m a r k e t because d e t e r m i n e d E(qb) < 0 for them.
C o n s i d e r now the level o f v io lence v in the m a r k e t for cr ime. Assume that:
v = v* - ~xdO 6, (2)
whe re v* > 0 is an e x o g e n o u s cons t an t t ha t ref lects the ins t i tu t iona l e n v i r o n m e n t ( to be d iscussed be low) , c~ > 0 a n d 0 < [3 < 1 are posi t ive constants , a n d ~b is the average t o l e r a n c e o f a p a r t i c i p a n t in the m a r k e t for c r ime. As m e n t i o n e d above, it is a s sumed tha t the m o r e an ind iv idua l is t o l e r a n t of v io lence , the less he disl ikes c o m m i t t i n g a v io len t act. This is r e f l ec t ed in (2) by the fact tha t a h i g h e r average t o l e r ance (a lower ~b) leads to m o r e v io lence in the marke t . Because 0 < [3 < 1, the func t ion is a s s u m e d convex in &, m e a n i n g tha t c r imina l -on -c r imina l v io lence decreases at an inc reas ing ra te when average t o l e r a n c e decreases . 7 Because the la rges t poss ib le ~b is d0 h, a s sume tha t v* - ~xqb h6 > 0, t h e r e b y e n s u r i n g tha t v > 0 for any qb.
F r o m above, it is c lea r tha t the average t o l e r ance in the m a r k e t d e p e n d s on the cri t ical value ~b be low which an ind iv idua l will pa r t i c ipa te in the marke t :
f+ ' + g ( + ) d+
$(d;) - ' ( 3 )
f+g(+)d+ O f course , qb (~b) is i nc reas ing in its a r g u m e n t . H e n c e , the choices of the ind iv idua ls have an effect o n the level o f v io lence .
An e q u i l i b r i u m in this s imple m o d e l is a qb such that:
E ( $ ) = (a - - = 0 . ( 4 )
As qb does n o t e n t e r l inear ly in (4), mu l t i p l e equ i l ib r i a are possible . 8 Fo r e x a m p l e , t he re
The point m a d e here that the marke t is affected by the characteristics of those who decide to participate in it, wonld remain valid.
7This assumption may have some empirical validity. Consider, for example , the n u m b e r of serious assaults that occur red in Canadian penitentiaries in 1991. The distribution of the inmates in the penitentiaries is is follows: 15.4% of them are in m i n i m u m security penitentiaries, 55.6% are in m e d i u m security" ones, and 29% are in m a x i m u m security ones [Correctional SeI~ices Canada (1991), p. 22]. Assume that the Canadian system is able to separate the inmates according to their true tolerance of violence and thus, that the low-tolerance inmates are in m i n i m u m security penitentiaries, the medium-to lerance imnates are in m e d i u m security penitentiaries, and the high-tolerance inmates are in m a x i m u m security penitentiaries. .&ssuming that v is convex would then be correct, because of the 42 serious assaults that occur red in the Canadian penitentiaries in the first three trimesters of 1991, 9.5% occur red in m i n i m u m security penitentiaries, 54.8% occur red in m e d i u m security penitentiaries, and 35.7% occur red in m a x i m u m security penitentiaries [Correctional Services Canada (1992), p. 4]. Max imum security penitentiaries therefore have a larger share of violent acts than their share of imnates, whereas it is the reverse for m i n i m u m security penitentiaries.
8The approach above, with only one endogenous variable, is used for heuristic reasons. The following alternative, using two endogenous variables, is more adequate to examine the issne of the stability of the equilibria. Let d~ - A+(v) be given by the r ight-hand side of (1) when it equals 0, and let v - A,,(d)) be g)ven by (2), after substitution of (3). An equil ibr ium of this model is then a pair (d~ ~, v ~) snch that, simultaneously, (b ~ - A+(v") and v ~ Av(d)~). Then , in the (d), v) space, A+ is a decreasing and convex fnnction whereas A , is decreasing but of ambiguous curvature. An equil ibr ium corresponds to an intersection of the two curves and is stable if, at the point of intersection, A+ is s teeper than A,,.
196 The market for crime
FIG. 1. A n e c o n o m y w i t h two e q u i l i b r i a .
could be two equilibria9: one with a low qb co r re spond ing to a high-violence equi l ibr ium, and ano the r one with a high ~ co r respond ing to a low-violence equi l ibr ium. 1° Again, the intui t ion is as follows. A high-violence equi l ibr ium implies that those who become criminals are very to lerant of violence. Those persons in turn are the ones that cause a lot of violence, suppor t ing the equi l ibr ium. A low-violence equi l ibr ium can also arise because the individuals that are no t to lerant of violence en te r the market , and those are the ones that dislike commit t ing a violent act, thus suppor t ing this o the r equi l ibr ium. Graphically, we can plot equat ion (4) in the (d), E) space. This is done in Figure 1. The curve^depic ted in this f igure gives the expec ted utility of an individual with to lerance qb, this individual being the less tolerant individual in the market ('~ust in the market") . An equi l ibr ium is then a si tuation where such an individual jus t gets his al ternative utility, which is equal to zero. Thus, in equi l ibr ium, the curve must be crossing the x axis. In Figure 1, there are mul t ip le equi l ibr ia because the curve crosses the x axis twice. H
In this two-equilibria case, consider v* which reflects, as men t ioned above, the insti tutions of this economy. For example , it is natural to th ink of gun control as a reduc t ion of v*. 12 From (2) it is clear that, o the r things be ing equal, a lower v* implies a lower v, which will be refer red to as the direct effect. Note, however, that any ~ solving (4) is a funct ion of v*. This impact of v* on dp will be refer red to as the self-selection effect, i.e., the effect of v* on the decis ion of the potent ia l cr iminals to par t ic ipate or not in the marke t for crime. As will be seen below, the direct and the self-selection effects can work in opposi te direct ions, and the lat ter can domina te the former .
Consider th e impact of gun control on violence. First note that the impact o f v* on qb, the self-selection effect, is ambiguous:
9Of course, there could be m o r e than two equilibria. l °Note that when ~ goes down, average to lerance increases, implying an increase in the level of violence bu t a
reduc t ion in the number of violent acts. In the cu r ren t analysis, the level of violence and the n u m b e r of violent acts always move in opposi te directions. The mode l could be easily a m e n d e d to allow for m o v e m e n t s in the same direct ion. For example , "rr and N could also be affected by the choices of the individuals ra ther than be ing fixed.
11Note that the curve need not be U-shaped. In fact, it could have a lmost any shape, and it could cross the x axis once, twice, o r m o r e times.
a2Gun control (or any v io lence-reducing policy) could also be r ep resen ted as a decrease in a a n d / o r 13.
N. Mie.c~u 197
j/fL
FIG. 2. I m p a c t o f a n i n c r e a s e in v*,
= <>o. ( 5 )
To unders tand why the self-selection effect is ambiguous, we have to go back to Figure 1. First consider the equilibrium with a large ~b, labeled L because it is a low-violence equilibrium (because there are individuals in the market that are not tolerant of violence). Note that in this equilibrium, the expected utility of the individual '~ust in the market" is increasing in (b, i.e., OE($)/0$ > 0. The reason why it is so is that an increase in (b has a large reducing effect on violence through average tolerance (because violence is increasing and convex in average tolerance, and that at this equilibrium the average is relatively large). The reducing effect on violence through the average dominates the fact that the individual '~ust in the market" is less tolerant of violence and thus, this individual is better off when (~ increases. The reverse story is true at the equilibrium with a small ~), labeled H because it is a high-violence equilibrium. In this equilibrium, the expected utility of the individual '~ust in the market" is decreasing in +, i.e., aE($) /a$ < 0. There, an increase in • has a small reducing effect on violence through the average because the average is small in this equilibrium. The reduction in violence caused by the increase in d) is dominated by the fact that the individual '~ust in the market" is less tolerant of violence. This individual is therefore worse off when ~) increases.
The fact that the expected utility of the individual '~just in the market" can be increasing or decreasing in ~ explains why the self-selection effect is ambiguous, la For example, if OE(~)/a~ > 0, then the denominator of (5) is positive and &~/dv* > O. The intuition in this case is as follows. First note that in Figure 1 an increase in v* corresponds to a movement downward of the whole curve. Hence, consider Panel (a) of Figure 2, which depicts the movement of the curve in the port ion where it is increasing. As ment ioned above, in that port ion of the curve, changes in ~ lead to changes in the level of violence through the average that dominates (in terms of
131t is also possible to show, using the functions A+ and A,, defined supra in note 8, that the equilibrium is stable when the expected utility of the individual "just in the market" is decreasing (OE(~)/O~J < 0), whereas it is unstable if it is increasing (OE(~J)/O$ > 0).
198 The .market for crime
expected utility) the fact that the to lerance of the individual ' lust in the market" has changed. When v* increases, the utility of any individual '~just in the market" goes down, and in part icular , that of the individual with the initial equi l ibr ium ~b is now negative. To reestablish the equi l ibr ium, it is necessary to inove to a new + with a larger expec ted utility (such that E(~)) = 0); But in that region of the curve, the way to increase expec ted utility is to increase ~b: The equi l ibr ium therefore moves from po in t L to po in t L' . Thus, in that region, an increase in v* leads to a larger +.
For the case where OE(~) /O~ < 0, the d e n o m i n a t o r of (5) is negative and so, d~b/dv* < 0. Cons ider Panel (b) of F igure 2. When v* increases, the curve goes down and it is again necessary to move to a new + where expec ted utility is larger. As discussed above, when E(qb) is decreasing, the change in the to lerance of the individual '~just in the market" dominates the effect on violence th rough the average. Not surprisingly, it is therefore necessary to increase the level of to lerance to increase the expec ted utility. Hence, the equi l ibr ium moves f rom po in t H to po in t H' : A larger v* leads to a smaller $.
The change in the level of violence v after the in t roduc t ion of gun control is also ambiguous; i t depends on the equi l ibr ium in which the economy rests, and on the react ion of qb to a change in v*:
dv - v (6) - 1 - cx[3~)~-'~ ' d+ _
dv* dv* otf3(b f~- l dp' ~b - v
A change in v* affects violence v th rough the d i rec t and the self-selection effects. In the case where OE(d#)/O6 < 0, as in the high-violence equi l ibr ium, 14 the self-selection and the direct effects both tend to reduce v after a reduc t ion in v* (gun control) . Therefore , violence v decreases after the in t roduct ion of gun control.
I f 0 E ( 6 ) / 0 6 > 0, as in the low-violence equilibrium,~5 the self-selection and the d i rec t effects work in opposi te directions. For a decrease in v*, the d i rec t effect tends to decrease v but is domina t ed by the self-selection effect, a decrease in qb, which tends to increase v. Consequently, the level of violence v increases after the in t roduc t ion of gun control.
It is in teres t ing to note that the high-violence equi l ibr ium is stable, whereas the low-violence one is not. This could be in te rp re ted as an a rgumen t in favor of gun control because the economy is more likely to rest in the high-violence equi l ibr ium (where gun control is effective) than in the low-violence one. 16
14Note that such an equi l ib r ium is stable because the expec ted utility of the individual 'Just in the marke t" is
decreas ing in ~). ~5Such an equi l ibr ium is unstable.
J6Note that it may be possible to identify" the sort o f equi l ibr ium in which the economy rests by empirically
i m p l e m e n t i n g the model . To es t imate the model , the following p rocedure could be used. Assume that we have data at the state level for a few years (a panel) on the level o f violence (a me a su r e of v), the probabil i ty of violent acts (a
measu re o f '~ ) , per capita i n c o m e (a proxy for b), and o the r soc ioeconomic variables. It is then possible to const ruct the
critical po in t ~it for State i at t ime t as ~it = (I - ~it)bzJav,vit. Assume that the dis t r ibut ion of dO within a state is identical for all states and that it is invar iant t h rough t ime (i.e., there are no states or per iods in which individuals are systematically less or m o r e tolerant o f violence): for example , suppose that cb,t - N(~ , ~ ) . Condi t ional on this distr ibution, and us ing ~a, we can calculate that t runca ted m e a n ~it o f State i at t ime t. Then , hav ing recovered &a, we can es t imate equat ion (2), v~t = v* - ~ ~ + ezt, a "violence equat ion," and obta in es t imators for v* (a constant) ,
~, and [3. Note that we could have some s tandard identif icat ion p rob lems with this es t imat ion [because (2) is poorly ident if ied] . Also note that the es t imat ion is condi t ional on the postulated dis tr ibut ion of cb. However , us ing Bayesian data augm en ta t i on techniques, it would he possible to recover the pa ramete r s p, and cr 2. Finally, and mos t interestingly,
N. MARCEAU 199
2
0
-2
137437 / E(~) 2 8 . 8 1 7 ~ /
\ ! . . . . . . . . . ,o , , o , o
Fie . 3. N u m e r i c a l e x a m p l e w i t h two e q u i l i b r i a .
As an i l lus t ra t ion, I now p re s en t a two-equil ibria example in which v io lence will increase or decrease after the i n t r o d u c t i o n of g u n control , d e p e n d i n g o n whe the r the e c o n o m y is in a low-violence or h igh-violence equ i l ib r ium.
III . Example
Let the pa ramete r s of the above m o d e l take the fol lowing values: ~r = 0.2, b = 1, c¢ = 1, [3 = 1 /2 , ~b / = 1, a n d qb t, = 30. Also assume that qb is un i fo rmly d is t r ibuted . T h e n , for v* = 4, the cons t r a in t such that v > 0 for any ~b is satisfied. In this case, there are two ~b that solve the model . T h e first o n e is ~b = qb I = 1.37437, which co r re sponds to a h igh-violence (h igh- to lerance) e q u i l i b r i u m because v = v~ = 2.91042. The second o n e is a low-violence ( low-tolerance) e q u i l i b r i u m with qb = 62 = 28.8177 a n d v = v 2 = 0 .138804. T h e f u n c t i o n E(d~) d e f i n ed in (4) is p lo t ted in Figure 3. It can be seen that for the re levant r ange of qb, it crosses the x axis twice.
Cons ide r the impac t o f i n t r o d u c i n g legis la t ion l ike g u n con t ro l that decreases v* f rom 4 to 3.9999. In the h igh-violence equ i l ib r ium, ~b~ increases to 1.37443 a n d the level o f v io lence Vl decreases to 2.91031. G u n cont ro l therefore achieves its goal of r e d u c i n g violence. O n the o t h e r h a n d , if the e c o n o m y has sett led down in the low-violence equ i l ib r ium, the decrease in v* decreases ~b 2 to 28.816 so that the level o f v io lence increases to 0.138812. Thus , in a low-violence equ i l ib r ium, g u n cont ro l fails to r educe violence.
IV. Conclusion
In this paper , a s imple m o d e l o f the d e t e r m i n a t i o n of v io lence in the marke t for cr ime is p resen ted . T h e m a i n feature of the m o d e l is that po ten t ia l c r iminals choose the i r act ions tak ing in to a c c o u n t the level of v io lence in the market . Yet, this same vio lence is d e t e r m i n e d by the characterist ics of those who decide to par t ic ipate in the market . Even with this s imple s t ructure , the m o d e l genera tes mul t ip le equi l ibr ia . T h e m a i n message del ivered in this p a p e r is the re fore that appa ren t ly s imilar e c o n o m i e s may expe r i ence drastically d i f fe rent levels of violence, a7
p e r f o r m i n g such an es t imat ion would m a k e it possible to establish whe ther State i at t ime t is in a low- or high-violence
equi l ibr ium. 7Note that the po in t m a d e in this pape r also has consequences on the level o f violence in prisons. I f individuals
self-select to en t e r the m a r k e t for cr ime, then this initial self-selection m e a n s that only a cer ta in sort o f people can end
200 The market for crime
Furthermore, it is also shown that the success of policies like gun control, aimed at reducing violence, depends on the equilibrium in which the economy initially rests. TM
The success or failure of such policies depends on two effects: a standard direct effect and a self-selection effect, which reflect the fact that the only individuals that are willing to participate in the market for crime have some minimal tolerance of violence. It should be noted that the empirical literature on gun control is somewhat inconclu- sive. a9 For Canada, Mundt (1990) argues that the 1977 legislation on gun control had no impact on firearms violence, because despite the legislation the availability of firearms rose in the years after it. The empirical literature on other violence-reducing policies also seems to be inconclusive. Consider the case of stricter enforcement. In Reiss and Roth (1993, p. 6), it is reported that the average prison time served per violent crime in the United States roughly tripled between 1975 and 1989, whereas reported levels of serious violent crimes were stable a round the level of 2.9 million per year. Many other violence-reducing policies have been proposed. Reiss and Roth (1993, p. 22) make the following six recommendat ions: (a) intervening in the biological and psycho- social development of individuals' potentials for violent behavior; (b) modifying places, routine activities, and situations that promote violence; (c) maximizing the violence reduction effects of police interventions in illegal markets; (d) modifying the roles of commodities (firearms, alcohol, and other psychoactive drugs) in inhibiting or pro- moting violent events or their consequences; (e) intervening to reduce the potentials for violence in bias crime, gang activities, and community transitions; and (f) imple- ment ing a comprehensive initiative to reduce partner assault. In the analysis, a violence- reducing policy like gun control was the equivalent of a reduction in v*. However, some of the above recommendat ions are not the equivalent of a reduction in v*. For example, recommendat ions (a) and (f) seek to affect violence through a change in the distribution of tolerance. Others simply cannot be properly dealt with in the current model. Future work on these recommendat ions should prove interesting.
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up in jail. If the criminals caught by the authorities are a random draw from the pool of criminals active in the market, then violence in prison should reflect violence in the market for crime.
~SA theoretical argument according to which gun control does not necessarily reduce violence can also be found in Taylor (1995). Taylor studies gun control in a game theoretic framework, and he argues that by disarming victims gun control may actually encourage criminal predation. He concludes that to be effective, gun control should be targeted at criminals. Hence, add-on penalties and background checks are better policies than general gun bans or high taxes on guns.
J°See Kleck and Patterson (1993) and the papers listed in footnote 1 of Taylor (1995).
N. MARCEAU 201
KLECK, G., AND PATTERSON, E.B. (1993). The impact of gun control and gun ownership levels on violence rates. Journal of Quantitative Criminology. 9:249-287.
MALIK, A.S. (1990). Avoidance, screening, and opt imum enforcement . Rand Journal of Economics. 21: 341-353.
MUNDT, R.J. (1990). Gun control and rates of firearms violence in Canada and the United States. Canadian Journal of Criminology. 32:137-154.
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