ELSEVIER
The Self-Reinforcing Nature of Crime
JOEL SCHRAG
Department of Economics, Emo~y University
E-mail: [email protected]
SUZANNE SCOTCHMER
Graduate School of Public Policy, University of California, Berkeley
E-mail: [email protected]
Crime is n o t evenly distributed in the population. T h e r e are various explanations for
this fact, each of which plays a role in the public debate a b o u t solutions. We p o i n t out
that attempts to trace crime to exogenous factors in the e n v i r o n m e n t or in the moral
character of citizens may be d o o m e d to disappointment. Crime is a mutually reinforcing
activity, a n d therefore e n f o r c e m e n t parameters or aspects of moral character do n o t
u n i q u e l y d e t e r m i n e the crime rate. T h e u n e v e n distribution of crime may simply be an
expression of multiple equilibria, a n d there is n o obvious way to make the system
coordinate o n low-crime equilibria rather than high-crime equilibria. © 1997 by
Elsevier Science Inc.
I. Introduction
Crime is n o t distributed evenly in the population. Instead, it is c o n c e n t r a t e d a m o n g the
young, the poor, a n d certain ethnic groups. For example, 17.5% of the people arrested
in the U n i t e d States in 1991 for c o m m i t t i n g property crimes were between the ages of
20 a n d 24, whereas only 10.8% were in the more n u m e r o u s g r o u p of citizens between
the ages of 30 a n d 34. African Americans a c c o u n t e d for one-third of adults arrested for
property crimes in 1991 [Bureau of Justice statistics (1993)], b u t comprise only a b o u t
12% of the population. Crime rates also vary widely between countries. In the U n i t e d
States, there were 4,697 thefts per 100,000 people in 1988, whereas in J a p a n there were
1,138 [Kurian (1991)].
T h e r e are various explanations for the u n e v e n distribution of crime, each of which
plays a role in the public debate a b o u t solutions. O n e explanation is that citizens with
little to lose c a n n o t be deterred by the p u n i s h m e n t s available in a h u m a n e society, a n d
We thankPhilipCookand Ann Wittefor usefuldiscussion.For financialsupportwe thank the NSFunder grant 90
119 10, the John M. Olin Foundationthrough grants to BoaltSchool of Lawand the YaleLaw School, and the Alfred
P. SloanFoundation.
InternationalReviewof Law and Economics17:325-335,1997
@1997by ElsevierScienceInc.
655 Avenueof the Americas,NewYork, NY10010
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Self-reinforcing nature of crime
the high-crime citizens are the disadvantaged. T h e solution is t h e n to improve o p p o r tunities. A m o r e sinister e x p l a n a t i o n m i g h t b e that some p e o p l e have a "taste" for crime
that is transferred between generations. T h e solution is then education. These arguments might partially explain the distribution of crime, b u t they seem incomplete. H e r e
we argue that the uneven distribution of crime can be partly e x p l a i n e d by its selfreinforcing nature: Citizens who live in high-crime c o m m u n i t i e s will find crime m o r e
attractive.
T h e self-reinforcing n a t u r e of crime is a direct c o n s e q u e n c e of citizens' incentives,
which are d e t e r m i n e d in part by the criminal justice system. We c o n s i d e r two alternative
hypotheses a b o u t the policy that the criminal justice system pursues. First, we suppose
that the authorities can c o m m i t to a particular probability o f a p p r e h e n d i n g criminals,
a n d therefore the e n f o r c e m e n t b u d g e t is e n d o g e n o u s to the crime rate. Next, we
suppose that the authorities must c o m m i t to a particular e n f o r c e m e n t budget, a n d
therefore the crime rate a n d probabilities o f a p p r e h e n s i o n are e n d o g e n o u s to the
budget. In b o t h cases there are multiple equilibria with different crime rates, a n d
individual citizens have a greater incentive to c o m m i t crime when the crime rate is high.
T h e intuition b e h i n d this result d e p e n d s on which of the two policies the authorities
pursue. If probabilities o f a p p r e h e n s i o n are fixed, crime is self-reinforcing if many
p e o p l e have the o p p o r t u n i t y to c o m m i t a particular crime. W h e n the crime rate is high,
a citizen realizes that if he foregoes a crime opportunity, it will p r o b a b l y be taken by
s o m e o n e else. If the police identify suspects on the basis of who h a d an o p p o r t u n i t y to
c o m m i t the crime a n d sometimes make e r r o n e o u s arrests, then the i n n o c e n t citizen
may find himself accused a n d p u n i s h e d w h e t h e r o r n o t he is guilty. Knowing this, he has
less incentive to forego the crime.
T h e a r g u m e n t that p e o p l e c o m m i t crimes because they anticipate arrest even if they
are i n n o c e n t may seem farfetched. Nevertheless, the intuition b e h i n d o u r result reveals
an i d e a that is m o r e general than o u r particular model. To see this, it is i m p o r t a n t to
recognize that d e t e r r e n c e d e p e n d s on the difference between the e x p e c t e d punishm e n t costs that a citizen faces when he is guilty a n d the costs he faces when b e is
innocent. O u r m o d e l identifies how multiple equilibria arise when this difference in
e x p e c t e d p u n i s h m e n t costs d e p e n d s on the level o f crime in the community. Punishm e n t costs may d e p e n d on the level o f crime because p e o p l e share crime opportunities,
b u t there are o t h e r plausible explanations for why this relationship exists.
Suppose, for example, that p e o p l e who live in high-crime c o m m u n i t i e s value j a i l time
positively. In his study o f u r b a n gangs Sanchez-Jankowski (1991, p. 116) points out that:
Nearly every Chicano gang member anticipates that he will be incarcerated at some
point in his gang career, and incarceration has been elevated to a position of high
status. A gang member who has not been arrested, convicted, and incarcerated has
not been an active and honorable member.
If the p u n i s h m e n t that the criminal justice system metes o u t b e c o m e s a b a d g e o f h o n o r
in a high-crime community, then this penalty loses its punitive value a n d deters less
crime. Thus, multiple equilibria will arise if the stigma attached to j a i l t i m e - - a n d
therefore the true cost o f the p u n i s h m e n t - - a n d the level o f crime in society are b o t h
d e t e r m i n e d e n d o g e n o u s l y by the equilibrium.
U r b a n gang behavior provides a n o t h e r e x p l a n a t i o n for why the difference in exp e c t e d p u n i s h m e n t costs d e p e n d s on the level of crime in the community. Many o f a
typical u r b a n gang's criminal enterprises are collective efforts that raise funds to
s u p p o r t the gang's c o n t i n u e d survival. [Sanchez-Jankowski (1991), p. 113] Like all
J. SCHRAGAND S. SCOTCHMER
327
teams [ H o l m s t r o m (1982)], gangs m u s t c o n t e n d with m o r a l hazard o n the part o f their
m e m b e r s , who may try to take advantage o f the social network that the gang offers
without b e a r i n g any o f the risk associated with the crimes that s u p p o r t the gang's
activities. To r e d u c e a m e m b e r ' s t e m p t a t i o n to shirk the g a n g may sanction his decision
n o t to participate in g r o u p crimes:
The leadership of the gang also encourages members to invest their time in
collective economic [criminal] projects by nurturing the social pressures to participate. Here, the leadership reinforces the ideology of group commitment, that
every member must give of himself to the brothers of his gang. Those who give are
honorable, and should be admired and respected by the entire membership, while
those who do not are an abomination and should be penalized. [Sanchez-Jankowski
(1991), p. 114]
A g a n g m e m b e r may quite rationally fear that his g a n g will punish him if he chooses n o t
to c o m m i t a particular crime, whereas a p e r s o n who is n o t p a r t o f a g a n g does n o t
usually anticipate that his friends will penalize him if he obeys the law. T h e r e f o r e , an
individual who lives in a high-crime c o m m u n i t y where gangs flourish may perceive a
smaller difference between his e x p e c t e d p u n i s h m e n t costs when guilty a n d when
i n n o c e n t than s o m e o n e who lives in a gang-free n e i g h b o r h o o d .
Now suppose that the e n f o r c e m e n t b u d g e t is fixed, a n d t h e r e f o r e the probabilities o f
a p p r e h e n s i o n are e n d o g e n o u s to the crime rate. In this case multiple equilibria arise
because high crime rates force the police a n d prosecutors to screen suspects less
t h o r o u g h l y than if the e n f o r c e m e n t b u d g e t were less stretched. Because stretching the
b u d g e t leads to mistakes, the probability o f arresting the p e r p e t r a t o r goes down,
whereas the probability o f arresting an i n n o c e n t citizen c o u l d go up. These effects
increase the incentive for crime, a n d t h e r e f o r e high crime a n d r a n d o m n e s s in indictmerits are mutually reinforcing. This p h e n o m e n o n lies at the h e a r t o f the empirical
literature o f the 1970s that tried to u n d e r s t a n d how e n f o r c e m e n t p a r a m e t e r s such as the
probability o f arrest affect crime rates, given that crime rates also affect the probability
o f arrest. See Cook (1980) for a b r o a d discussion o f these issues, a n d Schmidt a n d Witte
(1984) for estimates o f how e n f o r c e m e n t affects crime a n d vice versa.
Finally, we p o i n t out how a j u r y reinforces the citizen's incentive to c o m m i t crime in
the high-crime equilibrium. If a case goes to trial, the j u r y will evaluate the specific
evidence linking the d e f e n d a n t to the crime a n d will d e c i d e w h e t h e r the evidence
warrants conviction. However, the evidence c a n n o t usually distinguish guilt from innocence with certainty, a n d t h e n the j u r y must d e c i d e guilt or i n n o c e n c e based on the ex
post probabilities that it infers from the specific evidence a n d from what it knows o f the
i n d i c t m e n t process. W h e n the e n f o r c e m e n t b u d g e t is fixed a n d the crime rate is high
r a n d o m n e s s in indictments will shift the j u r y ' s p r i o r beliefs toward innocence. This shift
will m a k e the j u r y m o r e likely to acquit, which u n d e r a reasonable hypothesis u n d e r mines d e t e r r e n c e . Thus the j u r y c o m p o u n d s the r e i n f o r c e m e n t to crime already provided by a s t r e t c h e d e n f o r c e m e n t b u d g e t a n d f u r t h e r increases the range o f crime rates
that are consistent in equilibrium with a given e n f o r c e m e n t budget.
This a r g u m e n t hinges on two key assumptions. First, we assume that a j u r o r ' s p r i o r
belief a b o u t a d e f e n d a n t ' s guilt d e p e n d s on how the j u r o r believes the criminal justice
system operates. Second, we assume that a j u r o r ' s p r i o r belief a b o u t the d e f e n d a n t ' s
guilt influences what evidence the j u r o r will n e e d to convict the d e f e n d a n t . If these
assumptions are satisfied, then j u r i e s that are i m p a n e l e d in jurisdictions where the
328
Self-reinforcing nature of crime
police have a p o o r reputation for matching suspects and crimes should be reluctant to
convict defendants.
W h e t h e r these assumptions are in fact satisfied is an empirical question, t h o u g h one
that is apparently difficult to answer. Recent high-profile criminal trials, such as the first
trial of Los Angeles police officers accused of brutality in the beating of motorist Rodney
King and the O.J. Simpson m u r d e r trial, seem to support the idea that people who live
in different communities enter the jury box with different beliefs about how the
criminal justice system works. Many observers of the criminal justice system argued that
the decision to move the first Rodney King trial out of ethnically diverse Los Angeles
and to largely white Simi Valley played an important role in the acquittal of the police
officers.1 As Loyola Law School Professor Laurie Levenson put it, "Frankly, the people
in Simi Valley worship the police," [Margolick (1992a)]. After his client was acquitted,
J o h n D. Barnett, attorney for Rodney King defendant T h e o d o r e Briseno, stated that "I
wouldn't say the case was won at that point, but if [the change of venue] h a d n ' t been
granted, the case would have been lost, no question," [Margolick (1992b)].
But do jurors' different prior beliefs influence the conclusions that they draw from
the evidence? Some suggestive evidence exists in the literature on the effect of a jury's
racial composition on its decision making. A key finding of this literature, summarized
in J o h n s o n (1985), is that racially mixed juries tend to come to different conclusions
than all-white or predominately white juries, particularly when the defendant is black.
This may occur because, as Hans and Vidmar (1986, p. 50) note, "[w]hat may appear
to white jurors as a black defendant's implausible story may ring true to black jurors with
a greater knowledge of the context and norms of black experience."
The disproportionate concentration of crime in some segments of the population
might in fact be traceable to differences in propensities to commit crime that arise from
exogenous differences a m o n g citizens. However, the message of this paper is that
unequal crime rates in different segments of the population might alternatively represent different equilibria. Crime rates can differ even t h o u g h citizens in different
subpopulations are exogenously identical and face the same enforcement budget.
There is no guarantee that increasing the enforcement budget will reduce crime.
The model below builds on Becket's (1968) classic study in that crime and crime
enforcement are modeled as rational activities. Becker's model does not find the
multiplicity we explore in Section I because he did not consider the mutual interdep e n d e n c e of citizens' crime decisions when their crime opportunities overlap. Becker
also did not explore the fact that for a fixed enforcement budget the probabilities of
apprehension and conviction d e p e n d on the crime rate and vice versa. However,
Blumstein and Nagin (1978) explored the related p h e n o m e n o n that with a fixed prison
capacity sentences must be reduced in length if they are increased in number, and that
this can u n d e r m i n e enforcement. Finally, most closely related to this work is Rasmusen
(1996). In his model, employers make inferences about potential employees' criminality on the basis of their criminal records. He assumes that employers are willing to pay
a higher wage to workers who are not criminals. When the crime rate is low most
workers who are conviction-free are not criminals, and hence employers are willing to
pay a high wage, which makes it optimal for most people to forego crime. W h e n the
crime rate is high, on the other hand, employers are less sure that someone who is
1See citations given in (1993), O u t of the frying pan or into the fire? Race and choice of venue after Rodney King.
Harvard Law Review 106:705-722.
J.
SCHRAG AND S. SCOTCHMER
329
conviction-free is n o t a criminal, a n d t h e r e f o r e they offer a lower wage, which in turn
makes crime m o r e attractive. Thus, t h e r e can be multiple equilibria with different crime
rates. W h e r e a s Rasmusen focuses o n social stigma as a potential source o f multiplicity,
we show how the criminal justice institutions a n d the i n t e r d e p e n d e n c e o f citizens'
choice p r o b l e m s can also l e a d to multiple equilibria.
In Section II we study the self-reinforcing n a t u r e o f crime with e x o g e n o u s probabilities o f arrest a n d conviction. In Section III we analyze the multiplicity that arises
t h r o u g h e n d o g e n o u s probabilities o f arrest when the b u d g e t is fixed, a n d in Section IV
we show how the j u r y can f u r t h e r reinforce high crime rates t h r o u g h unwillingness to
convict d e f e n d a n t s when they know that the police make mistakes.
II. Incentives for Crime
We suppose that citizens face "opportunities" for crime. We distinguish between crime
o p p o r t u n i t i e s that are "exclusive" a n d only available to a single citizen a n d those that
are "nonexclusive" a n d available to m a n y potential criminals. A n e x a m p l e o f the f o r m e r
is the e m b e z z l e m e n t o f c o m p a n y funds by a chief financial officer (CFO); only the CFO
can wire funds to a Swiss b a n k a c c o u n t a n d cover h e r tracks. A nonexclusive crime is the
r o b b e r y o f the c o r n e r convenience store; a n y o n e in the n e i g h b o r h o o d c o u l d take such
an opportunity. We assume that crime o p p o r t u n i t i e s have benefits m / > 0 distributed
a c c o r d i n g to the density function c(m) a n d the cumulative distribution function C(m).
If a crime is in a citizen's " o p p o r t u n i t y set," he must d e c i d e w h e t h e r o r n o t to d o it.
T h e citizen makes his decision by c o m p a r i n g the benefit from c o m m i t t i n g the crime
with the e x p e c t e d increase in his p u n i s h m e n t costs. T h e latter d e p e n d s o n the different
probabilities that he will be investigated, indicted, a n d convicted if guilty a n d if innocent.
We assume that i f a citizen commits a particular crime, t h e n the police arrest him with
probability q. If he does n o t c o m m i t a crime o f value m in his o p p o r t u n i t y set, then
s o m e o n e else commits the crime with probability p. (m), 1 i> ix (m) /> 0. If s o m e o n e else
commits a crime in a citizen's o p p o r t u n i t y set, we assume the i n n o c e n t p e r s o n will b e
arrested with probability p. Thus, even if the citizen foregoes a crime in his o p p o r t u n i t y
set, he nevertheless b e c o m e s a suspect for that crime with probability p,(m)p. T h e
probability o,(m)p d e p e n d s o n the n a t u r e o f the crime, as well as o n the equilibrium
b e h a v i o r o f citizens. If the crime is the "exclusive" o p p o r t u n i t y o f a particular citizen,
t h e n IX(m) = 0. If, as we assume, citizens share crime opportunities, then it is possible
that p, (m) > 0. S h a r e d crime o p p o r t u n i t i e s lead to multiple equilibria in some o f which
high crime rates will be self-reinforcing.
After a citizen is i n d i c t e d the j u r y assesses the evidence against him. 2 T h e strength o f
evidence against the citizen, e, is a r a n d o m variable distributed o n an interval [0, ~)
a c c o r d i n g to probability density functions f ( ~ i) a n d f ( ~ g ) , where i means the citizen was
i n n o c e n t o f the crime for which he was charged, a n d g m e a n s that he was guilty. A low
value e c o r r e s p o n d s to weak evidence that matches the d e f e n d a n t to the crime. This
i n t e r p r e t a t i o n o f evidence reflects the fact that an individual may g e n e r a t e circumstanZFor simplicity we a s s u m e h e r e that all the suspects arrested g o to trial. In a previous p a p e r (1994) we showed that
t h e r e is n o loss in a s s u m i n g this. I f the p r o s e c u t o r has n o better e v i d e n c e than the police, t h e n h e has n o basis o n which
to distinguish a m o n g cases. I f h e has a signal o f the evidence, e, that will be p r e s e n t e d at trial a n d only prosecutes cases
h e expects to win, t h e n the suspects w h o are convicted are the s a m e o n e s as if all cases went to trial. This is because
we a s s u m e in b o t h cases that the j u r y knows the decision process o f the p r o s e c u t o r a n d u p d a t e s its likelihood o f
i n n o c e n c e a n d guilt accordingly. See S c h r a g a n d S c o t c h m e r (1994) for elaboration.
Self-reinforcing nature of crime
330
tial evidence despite his i n n o c e n c e . We assume that f( e[g) / f( el i) is increasing3 a n d
therefore that F(t~i) > F(e]g) for all e, where F(e]i) a n d F(ehg) are the cumulative
distribution functions.
We suppose for simplicity that each citizen has the same value M > 0 of e n g a g i n g in
law-abiding activities, so M is the opportunity cost of crime. M is d e t e r m i n e d by the
citizen's tastes a n d the value of his time spent in law-abiding activities. A citizen will take
a crime opportunity with value m if the benefit, m - M, exceeds the expected increase
in p u n i s h m e n t costs, which we represent by the function:
K(e, P, q) -~ {q[1 - F(elg)] - 911 - F(eli)]}S
where p = Ix(m)p, S is the monetized value of the sentence that a citizen will receive if
convicted, [1 - F(~ x)] is the probability of b e i n g convicted when the jury's standard of
evidence (the m i n i m u m evidence necessary for conviction) is e a n d the citizen takes
action x ~ {i[nnocent], g[uilty]}. The citizen will take all crime opportunities that
satisfy:
•(m,
e, p, q) -~ m -
M-
K(e, 0, q) >1 0
(1)
We now take the e n f o r c e m e n t parameters p, q, a n d e as fixed, a n d we show that when
crime opportunities are nonexclusive there are multiple equilibria a m o n g the citizens,
each equilibrium having a different crime rate. Thus the level of e n f o r c e m e n t does n o t
uniquely d e t e r m i n e the crime rate. In general each citizen's strategy would be a cutoff
m / such that he would c o m m i t any crime with value m > m i a n d forego any crime with
value m < m i. However, to show that there are multiple equilibria it is sufficient to
describe symmetric Nash equilibria in which each citizen has the same cutoff. If the
c o m m o n cutoff is m*, then any crime with m > m* will be taken with probability 1, a n d
any crime with m < m* will be taken with probability 0. Thus we can define symmetric
Nash equilibrium as follows: A symmetric Nash equilibrium is a strategy m* such that for
each citizen i, ~ ( m, e, 0, q) ~ < 0 f o r m < ~ m* a n d w ( m, e, p, q) i> 0 f o r m/> m*.
I n the first inequality we have e n t e r e d p = 0 to reflect tx(m) = 0 (ifa particular citizen
does n o t take a crime with value m < m*, t h e n n o other citizen will take it either), a n d
in the second inequality we have e n t e r e d 9 = P to reflect Ix(m) = 1 (if a particular
citizen does n o t take a crime with value m > m*, then it will be taken by s o m e o n e else
with probability one).
PROPOSITION 1: Let m I a n d r e 2 satisfy'rr ( m 2, e, O, q) = 0 and w ( m x, e, p, q) = 0. Then any
m* such that ml < m* < m 2 is a symmetric Nash equilibrium. I f C ( m l ) < C(m2), there are
symmetric Nash equilibria with different crime rates.
PROOF: Because ~r(m, e, 9, q) is increasing in p and m, it follows that m 2 > mx, and each
m* between m I and m 2 satisfies the definition of equilibrium. The total crime rate is 1 - C(m*),
and thus the crime rates for m* = ml and m* = m 2 are different.
We illustrate Proposition 1 in Figure 1.
The equilibrium m* = mI is the o n e that maximizes the total utility of citizens in their
capacity as potential criminals (not a c c o u n t i n g for losses in their capacity as potential
victims). However, even if citizens wanted to coordinate o n that equilibrium, they could
SThis is the familiar m o n o t o n e likelihood ratio property,
J.
SCHRAGAND S. SCOTCHMER
331
rc(m , e,p,q)
rc(m ,e,O,q)
0
~
possible
m*
possiblem*
J
ml
...................
/ ..........
....
in
m2
FIG. 1. I l l u s t r a t i o n o f P r o p o s i t i o n I.
not necessarily do S O . 4 If a citizen faces a crime opportunity that no other citizen would
take, then with high probability he can avoid any possibility of prosecution by foregoing
the crime opportunity. This is not true if the crime he considers would be done by
someone else if not by him. In that case he might be wrongfully prosecuted and
convicted anyway, and this prospect decreases his incentive to forego the opportunity.
In this way crime is mutually reinforcing, and that is the content of Proposition 1. We
cannot predict the crime rate from the enforcement policy because there are multiple
equilibria in which crime is self-reinforcing to different extents. It follows that even if
the judicial system treats all citizens alike in the sense that all citizens face the same
probabilities of arrest and conviction conditional on their actions, we might observe
different crime rates in different communities.
HI. Reinforcement through Resource Constraints in Enforcement
In the previous section we held fixed the probabilities of arrest and conviction, and we
showed that there are multiple equilibria among the citizens. In these equilibria crime
is self-reinforcing to different degrees. However, even if we focus on a particular
4Akerlof and YeUen (1993) study the behavior of criminal "gangs," which permit explicit coordination. Their focus
is nn the ability of the gang to undermine enforcement by gaining the complicity of the surrounding community. In
our context a gang would be a natural coordinating body to steer the cidzens to the high-crime equilibrium rather than
to any other equilibrium. Once there, the equilibrium would be self-reinforcing. In a related paper [Schrag and
Scotchmer (1994)] we used a similar model to focus on the strategic interaction between the citizens and the jury. To
concentrate on that interaction we did not discuss the multiplicity of citizens' replies that we discuss here. Instead we
assumed that the citizens coordinate on the equilibrium m* = mI that maximizes their joint payoff.
332
Self-reinforcing nature of crime
selection of equilibrium given the probabilities of arrest and conviction, e.g., the one
that maximizes the crime rate, various levels of criminal activity can be consistent with
a given enforcement budget. In this section we explore this in more detail. In the next
section we carry this line of reasoning one step further and show how reinforcement of
high crime rates can be exacerbated even further by a jury that rationally accounts for
probabilities of arrest.
To show how the endogeneity of the parameters q and p further exacerbate the
multiplicity problem, we will assume that, conditional on the e n f o r c e m e n t parameters
q and p, the citizens play the equilibrium m* that maximizes the crime rate; i.e., the m*
fnl.
If the police face a higher crime rate with a fixed enforcement budget, they can
devote less effort to solving each case. With greater demands on their time the police
can devote less effort to screening the citizens that had opportunity to do each crime,
and this will introduce an element of randomness into their selection of a suspect. In
addition there might be fewer arrests as a proportion of crimes. Randomness in arrests
would mean that q becomes lower and p becomes higher. But if there are fewer arrests,
p might also become lower. The assumption in the following proposition means intuitively that the stretched budget introduces randomness in screening, so that if both q
and p are lower in the high-crime equilibrium, then the reduction in p is dominated by
the reduction in q. The hypotheses imply that K(e, PH, qH) < K(e, PL, qL), and the
following proposition follows immediately.
PROPOSITION 2: Suppose that qH/PH < qL/PI., that qH < q~., and that m*, m~, respectively,
solve ~ ( m * , e, PH, qH) = 0 and ~(m*, e, PL, qI) = 0. Then m * < m*.
Thus the endogeneity of probabilities of arrest due to a fixed enforcement budget
increase the n u m b e r of equilibria even beyond the multiplicity identified in Section II.
It remains true, of course, that the whole set of equilibria could be moved toward lower
crime rates if the enforcement budget were increased.
IV. The Role of the Jury in Reinforcing High Crime Rates
The task of the jury is to dispense expostjustice. So far we have taken the jury's standard
of evidence for conviction as fixed, but in fact the jury must choose the standard of
evidence. Because the evidence cannot usually distinguish between guilt and innocence
with certainty, the jury will presumably choose a standard of evidence that reflects its
prior belief in the defendant's guilt or innocence. We assume that the jury consists of
rational Bayesian updaters who fully understand the system of criminal justice in which
they operate. Although this is a strong assumption, there is no obvious alternative. If we
open the door to irrational beliefs, almost any behavior can be justified.
As is now customary in the literature [e.g., Rubinfeld and Sappington (1987); Schrag
and Scotchmer (1994)], we assume that the jury pursues ex post justice by minimizing
the expected cost of errors. That is, it balances the possibility of convicting the innocent
(a Type I error) against the possibility of acquitting the guilty (a Type II error). The
jury's objective function is:
minimizee~0[1 - F(e Ii)]e(i]/) L ' + F( elg) e ( g[ l) L H
(5)
where f(il/) and f ( ~ / ) represent the jury's prior probabilities of innocence and guilt,
conditional on the defendant having been indicted but before seeing the evidence e.
J. SCH~G AND S. SCOTCHMER
333
(/represents "Indictment.") L I and L n are, respectively, the costs of Type I and Type II
errors. The jury's best reply, say e*, satisfies
>
e(illO L"
f(e*[g)
j:(e*li )
-
~(g~I) L tl
o = e*
as
0 < e * < ~
e:~ ~
<
(6)
oo
The likelihood ratio ~ (i[/)/2 (g~/) can be understood as follows. According to Bayes'
Rule:
£(i11)
prob(/~ i)prob(i)
e ( I n i)
e(g~/)
prob(/]g)prob(g)
f ( I n g)
The probability of indictment, conditional on being guilty is prob(~g) = q, whereas
the probability that a r a n d o m citizen is guilty is prob(g) = 1/N.
The probability of indictment, conditional on innocence, prob (~ i), must account for
the probability that the defendant had this crime in his opportunity set and for the
probability that he did not. The police might make mistakes in identifying which
citizens had opportunity for the crime as well as in choosing suspects from a m o n g that
group. We have ignored the possibility of such mistakes in our previous discussion
because such mistakes have no bearing on the citizen's decisions about crime. If a crime
is not available to a particular criminal, then he cannot commit it, and the prospect of
being arrested for it does not further deter him, However, the possibility that citizens
could be arrested for crimes outside their opportunity sets has an indirect bearing on
deterrence because it affects the jury's assessment of the relative likelihood that the
defendant is i n n o c e n t or guilty.
If n people other than the guilty party have opportunity for each crime, the probability that a r a n d o m i n n o c e n t citizen had the opportunity at h a n d is n~ (N - 1 ), and the
probability that the r a n d o m innocent citizen did not have the opportunity is (N - n 1 ) / ( N - 1 ). We wiU let p' denote the probability that a particular citizen is arrested for
a particular crime that is n o t in his opportunity set. ~ Thus prob(~i) = p n / ( N - 1) +
p ' ( N - n - 1 ) / ( N - 1). Because the probability that a r a n d o m citizen is innocent is
prob(/) = ( N - 1)/N, we have:
?.(ilI)_ pn + p'(N-
e ( gl l)
q
1 - n)
(7)
In the following proposition, the hypothesis L ~ ~ L u means that the jury's subjective
cost of wrongful conviction is at least as great as its subjective cost of wrongful acquittal.
This hypothesis seems to reflect the jury's mandate, which is to protect the innocent. As
in Proposition 2, the hypotheses qu < qL and qt-z/P~ < qlfPt~ reflect the idea that in the
subpopulation or in the equilibrium labeled H the police are enforcing more randomly
due to a stretched enforcement budget. The hypothesis that P~4 ~> P~ means that in a
high-crime environment the overburdened police are more likely to arrest a citizen who
~Of course, p' depends on N, with dp'/dN< 0; when the population is larger there is a smaller probability that any
particular citizen without opportunity for the crime will nevertheless he a suspect, We suppress this dependence
because we take N to be exogenous.
334
Self-reinforcing nature of crime
d i d n ' t even have opportunity for the crime. However, if the stretched budget means
that arrests are made in a smaller fraction of cases, then all the probabilities of arrest p,
p', and q might decrease with the crime rate. The result requires that the likelihood
ratio f (i[/)/e (g~/) is higher in the H than the L equilibrium, i.e., that when the budget
is overburdened the defendant at trial is relatively more likely to be innocent. This can
hold even if P/4 < PL and p ~ / < pj~ provided qH is substantially smaller than qL.
PROPOSITION3: Suppose that L I ~ L II. Suppose there are two equilibria with different probabilities
of arrest and different crime rates, labeled H (for high crime rate) and L (for low crime rate). If
qH/PH ~< qL/PL, qH ~< qL, and P~I ~> P[, and one of the inequalities is strict, then the standards
of evidence e* and e* that satisfy equation (6) have the property that e* > e *. Further, the m *
a n d m * thatsatisfyqr(m*,e*,pH, qH) = O a n d ~r(ml.,
* eL,
* PL, qL) = 0 have the property that
m * < m*.
PROOF: The hypotheses ensure that the likelihood ratio in equation (6) is smaller in the H
equilibrium than in the L equilibrium. By the monotone likelihood ratio property, namely, that
f(e*lg)/f(e*[i) is increasing, it foUows that e* > e*. By the monotone likelihood ratio property,
K(e, PL, qL) is strictly quasi-concave in e and maximized wheref(e[g) /f(eli) = PL/qL. It follows
from the hypotheses and from equation (6) that e* is larger than this maximizer. Because e * >
e* > argrnax K(e, PL, qL), by quasi-concavity K(e*, PL, q~) < K(e*, PL, qL)" But because
K(e, PH, qH) < K(e, PL, qL) f oralle, it follows thatK(e*, PH, qn) < K(e*, PL, qL) < K(e*,
PL, qL)" Thus, because ~ ( m * , e*, PH, qH) = 0 = "or(m*, e*, PL, qL), m * < m*.
The intuitive m e a n i n g of e* > e* is that in the equilibrium with higher crime the jury
will hold defendants to a higher standard of evidence because it starts with a higher
prior belief that the defendant is innocent. This belief arises from the randomness in
indictment. If the police are stuck "bringing in the usual suspects" because the enforcem e n t budget is overstretched, the jury will be more willing to acquit. Their willingness
to acquit reinforces the high crime rate, as implied by m * < m*.
V. Conclusion
We return briefly to our motivating question: Why is crime unevenly distributed in the
population? The population might divide naturally into communities, each with opportunity for a different set of crimes. Because the different communities have different
criminal opportunities, they could have different equilibria, and some could have
higher crime rates than others. Furthermore, in some communities "exclusive" rather
than nonexclusive crime opportunities may be more c o m m o n . The requirements for
self-reinforcement described in Section II are that crime opportunities are not exclusive
and that the police and prosecutor make mistakes when they arrest and indict. The case
of street gangs provide an example where these hypotheses are likely to be met. A gang
m e m b e r may be reinforced in his life of crime because his fellow gang members are
willing to take any opportunity that he himself passes up. Because the police may have
difficulty identifying the party who was responsible for any given crime, a particular
gang m e m b e r may be arrested and jailed even if he is innocent, thus increasing his
incentive to commit the crime in the first place. Contrast this with embezzlement. If
there is only one accountant with access to c o m p a n y funds, then he can avoid any
possibility of prosecution by simply not stealing them.
Sections III and IV discussed reinforcement to crime that comes from stretching the
enforcement budget. This type of reinforcement can arise whether crimes are exclusive
or nonexclusive. Stretching the enforcement budget plausibly makes the police less
J. SCHRAGAND S. SCOTCHMER
335
a c c u r a t e in i d e n t i f y i n g p e r p e t r a t o r s . E v e n with a f i x e d s t a n d a r d o f e v i d e n c e at trial, s u c h
r a n d o m n e s s w o u l d r e i n f o r c e a h i g h c r i m e rate, b u t in fact t h e j u r y will f u r t h e r r e i n f o r c e
t h e h i g h c r i m e rate by its r e l u c t a n c e to convict. Its r e l u c t a n c e to c o n v i c t rationally
reflects r a n d o m n e s s in i n d i c t m e n t s b e c a u s e it starts its d e l i b e r a t i o n with less c o n f i d e n c e
in t h e d e f e n d a n t ' s guilt.
T h e d i s p r o p o r t i o n a t e distribution o f c r i m e p r e s u m a b l y has m a n y causes. A p r i m a r y o n e
m i g h t b e that citizens in d i f f e r e n t age g r o u p s o r s u b p o p u l a t i o n s have d i f f e r e n t o p p o r t u n i t y
costs o f crime. However, the p o i n t o f this p a p e r is that e v e n h o l d i n g the e x o g e n o u s factors
fixed, t h e r e are m u l t i p l e equilibria with d i f f e r e n t c r i m e rates, a n d t h e r e f o r e e x o g e n o u s
factors are p r o b a b l y n o t a c o m p l e t e e x p l a n a t i o n o f the u n e v e n distribution o f crime. A
s u p p l e m e n t a r y e x p l a n a t i o n c o u l d simply he that the u n e v e n distribution o f c r i m e is an
a c c i d e n t o f history. W e d o n o t n e e d to d i s e n t a n g l e the c o m p l i c a t e d forces o f history that
m i g h t have led a particular s u b p o p u l a t i o n o r n a t i o n to tolerate c r i m e to observe that o n c e
this has h a p p e n e d the criminal justice system a n d b e h a v i o r o f o t h e r citizens r e i n f o r c e each
citizen's c r i m i n a l choices. O u r message can be s e e n as e i t h e r optimistic o r pessimistic. It is
pessimistic in the sense that we offer n o g u i d a n c e o n h o w to e n s u r e that society coordinates o n low-crime equilibria. But it is optimistic in the sense that by c h a n g i n g to a n o t h e r
e q u i l i b r i u m c r i m e c o u l d be r e d u c e d w i t h o u t increasing e n f o r c e m e n t costs.
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