A simple model of optimal hate crime legislation.

Gan, Li ; Williams, Roberton C., III ; Wiseman, Thomas 等

I. INTRODUCTION

Reported "hate crimes" have increased dramatically in the United States in recent years; the number of reported incidents rose from 4,588 in 1991 to 7,160 in 2005. (1) This rise, along with the attention paid by popular media to high profile cases, such as the murders of James Byrd in Texas and Matthew Shepard in Wyoming, has produced unprecedented public interest in the issue and potential remedies. As of 1999, 41 states had passed "hate crime" or "bias crime" laws: laws that create or enhance penalties for crimes motivated at least in part by the victim's race, religion, or other protected category (see Grattet and Jenness 2001). (2) In essence, these laws mandate stronger punishment based on the motivation behind the crime. Debate over these laws centers on the question of whether this additional punishment is justified. (3)

In this paper, we present a formal model of law enforcement in the presence of hate crimes, and use that model to investigate the conditions under which it is optimal to impose harsher penalties for hate crimes--or, more generally, to exert greater public effort to prevent hate crimes--than for otherwise similar crimes not motivated by hatred. We also examine the implications of these conditions for other aspects of hate crime policy: for example, should government policy encourage effort by members of targeted groups to make themselves less likely to be victims of hate crime?

Our model is similar in nature to Becker's (1968) economic model of crime. (4) In this model, potential criminals derive some benefit from committing a crime, but weigh that against their expected cost--both in terms of effort and expected punishment--when deciding whether or not to commit a crime. Law enforcement and efforts by potential targets to protect themselves from crime can increase the expected cost to potential criminals, and thus reduce crime rates.

Hate crime has attracted substantial attention in the fields of law, sociology, psychology, and other social sciences, but little within the field of economics. Jefferson and Pryor (1999), Medoff (1999), and Gale, Heath, and Ressler (2002) provide empirical analyses of hate crime data. (5) The latter two papers also briefly present theories to explain hate crimes. Glaeser (2005) provides a more extensive economic theory of hatred.

To our knowledge, though, the study of Dharmapala and Garoupa (2004) is the only prior work that addresses the normative issue of optimal hate crime policy. While the general structure of their model is similar to ours, the rationale for hate crime laws is different. In their model, effort by one potential victim to avoid crime causes criminals to seek out other victims, thus increasing the probability that others will be attacked. Without this "displacement" effect, the optimal punishment for hate crimes in their model would be the same as for other crimes. In contrast, our explanations do not rely on displacement. (Indeed, our model assumes that no displacement occurs.)

While there is widespread agreement on the defining characteristic that distinguishes hate crime from other crime--the motivation for the crime--there is much less agreement on the effects of hate crime and on exactly why such crimes should be punished more harshly. To accommodate this diversity of opinion, we consider five possible ways in which hate crimes may differ from other crimes and thus warrant greater punishment. First, society may put a lower weight on the utility of hate criminals than on the utility of other criminals. For example, the utility that a mugger gets from stolen money would count in calculating social welfare, but the pleasure that the perpetrator of a hate crime gets from inflicting suffering is invalid and should not be counted. (6) Second, hate crime may harm other people in addition to the direct victim. In particular, other members of the targeted group may suffer disutility from sympathy for the victim or fear that they themselves will be attacked.

Third, efforts by potential targets to avoid being victims of hate crime may generate a negative externality. For example, they might try to hide their identities, thus resulting in a loss of diversity to society. Fourth, hate crime might be more difficult for victims to avoid. A person can reduce his chance of being mugged by avoiding visible displays of wealth, but cannot change his skin color to avoid race-based hate crime. Fifth, there may be fewer potential targets of hate crime than for other crimes, leading to a higher probability that any particular potential target will in fact be a victim. While the primary focus of this paper is on hate crimes, it is worth noting that the first four of these differences also apply to acts of terrorism, and thus many of our conclusions will also apply to anti-terrorism policy as well.

We show that each of these differences may justify greater public effort to prevent hate crimes than to prevent other crimes, even if the direct harm to the victim is the same. However, in most of these cases, whether the optimal public effort is greater or smaller depends on the complementarity or substitutability between public and private effort--a point that prior hate crime research has not noted. (7)

We also show that these differences between hate crime and other crimes affect the socially optimal level of private effort relative to the individually optimal level. Thus, if the government can encourage or discourage private effort through means other than just the overall level of public effort--for example, if it is possible to subsidize private effort or to choose types of public effort that are more complementary to private effort--then it will be optimal for hate crime policy to differ from policy toward other crimes on those dimensions as well as on the overall level of public effort. This finding suggests that some hate crimes warrant more punishment effort than others. For example, if there is a negative externality from segregation, it may be optimal to have a harsher penalty for a hate crime committed against a black family living in an otherwise all-white neighborhood than for an otherwise-identical hate crime committed against a black family living in a black neighborhood, in order to provide better incentives for integrating neighborhoods.

In the next section, we present a simple model that incorporates these differences and demonstrates their implications for the optimal design of hate crime legislation. The final section contains conclusions and a discussion of additional factors that fall outside the scope of the model.

II. THE MODEL

In this section, we first provide a baseline model of crime and law enforcement that applies to any crime, and then extend that model to capture the ways in which hate crime may differ from other crimes.

A. The Baseline Model

Three types of agents interact with each other: potential criminals (who form a continuum of mass [N.sup.C]), potential victims (mass [N.sup.V] > [N.sup.C]), and the government. All potential victims are assumed to be identical. If a crime is committed against a particular potential victim, that victim suffers a utility loss of [DELTA]U [member of] (0, 1). The protection level for potential victim i, P([a.sub.i], g), is a function both of individual effort to avoid crime ([a.sub.i] [greater than or equal to] 0) and of the level of government effort to catch and punish criminals (g [greater than or equal to] 0). The function P is assumed to be bounded between 0 and 1, and to be increasing and strictly concave in each of its arguments: [P.sub.1] > 0, [P.sub.2] > 0, [P.sub.11] < 0, and [P.sub.22] < 0. That is, exerting more effort of either type raises the level of protection, but the marginal effect falls as the level of effort rises. For now, we make no assumptions on the degree of complementarity or substitutability between individual and government effort (that is, on the sign and magnitude of the cross-partial derivative [P.sub.12]). As shown later in this paper, this parameter has important implications for hate crime policy. For simplicity, we assume that all third- and higher order derivatives of P are zero--that is, P is quadratic.

The cost of individual effort [C.sup.V] ([a.sub.i]) increases quadratically. That is, [C.sup.V.sub.1] > 0, [C.sup.V.sub.11] > 0, and all higher order derivatives are zero. The probability that individual i is victimized depends on his protection level P([a.sub.i], g) and on the ratio of the mass of potential criminals to the mass of potential victims [N.sup.C]/[N.sup.V]. In particular, we assume that the probability is given by [N.sup.C][1 - P([a.sub.i], g)]/[N.sup.V]. If each potential victim has the same protection level P, then the total number of crimes committed is given by [N.sup.C][1 - P]. (8)

Given government effort g, each potential victim maximizes expected utility by solving the problem

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We assume that [C.sup.V.sub.1](0) is sufficiently small to ensure an interior solution to Equation (1). The necessary and sufficient condition for this unique solution a* is given by

(2) [DELTA]U [N.sup.C]/[N.sup.V][P.sub.1](a*, g) -- [C.sup.V.sub.1](a*) = 0.

Each individual chooses the same level of effort a(g), which varies with government effort g, so the average protection level is equal to P(a(g), g). Implicit differentiation of Equation (2) yields the derivative

(3) [a.sub.1](g) = [P.sub.12](a(g), g)[N.sup.C]/[N.sup.V][DELTA]U/ [C.sup.V.sub.11](a(g)) - [P.sub.11](a(g), g)[N.sup.C]/[N.sup.V][DELTA]U

Because the denominator of Equation (3) is positive, the sign of [a.sub.1] (g) is the same as the sign of the cross-partial derivative [P.sup.12]. If individual and government efforts are complementary ([P.sub.12] > 0), then the optimal level of individual effort increases with the level of government effort. If, on the other hand, government effort acts as a substitute for individual effort ([P.sub.12] < 0), then individuals exert less effort as government effort rises. Effort to avoid crime and effort to catch and punish criminals might be strongly complementary if, for example, avoidance effort takes the form of increased alertness and watchfulness for suspicious activity. In that case, the higher probability of witnesses greatly increases the effectiveness both of police effort to catch criminals and of prosecutorial effort to convict them. On the other hand, if potential victims seek to avoid crime by secluding themselves at home, increased avoidance effort might hinder government effort to catch and punish criminals, and so public and private efforts are substitutes (see Ben-Shahar and Harel 1995).

The total effect of a marginal increase in government effort (g) on the average protection level is [P.sub.2](a(g), g) + [a.sub.1](g)[P.sub.1](a(g), g). We assume that this effect is positive--individual and government efforts are not such strong substitutes [[a.sub.1](g) is not so negative] that increasing g actually results in more crime.

Potential criminals are the second type of agent. Each criminal i chooses to commit one crime if the benefit [B.sub.i] from doing so exceeds the expected cost [C.sub.C]. The benefit from crime for each criminal is drawn independently from a uniform distribution on the interval [0, 1]. The cost of committing a crime is given by the protection level of potential victims, P, and thus is the same for all potential criminals. (9) The mass of crimes committed, then, is [N.sup.C][1 - P], as specified earlier.

The third and last type of agent is the government. The government faces a cost of effort in catching and punishing criminals (which we sometimes refer to just as punishment effort) given by the increasing and convex function [C.sup.G](g). (We assume that the government's cost depends neither on the mass of crimes committed nor on the size of the group of potential victims. In Section III, we discuss the effects of relaxing that restriction.) The government chooses its effort level to maximize expected societal utility, which is the sum of potential victims' and criminals' utility, less the cost of government effort. (10) Thus, the government's problem is given by

(4)[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In selecting its effort level g, the government must weigh the benefits of increasing g against the costs. A higher government effort level g affects the welfare of the potential victims in two ways. First, there is the direct effect of a lower crime rate resulting from the increased cost of committing a crime P(a(g), g). Second, increasing g changes the equilibrium level of avoidance effort by the victims a(g), and thus leads to a different cost of effort [C.sup.V] (a(g)). That second effect may either increase the cost for victims (if a and g are complements) or lower it (if they are substitutes). On the other hand, the social cost of raising government effort g includes both the direct cost to the government [C.sup.G](g) and the cost to potential criminals due to the higher cost of committing crimes. The first-order condition for an interior solution to the government's problem is given by

(5a) [N.sup.C] x [DELTA]AU[[P.sub.1](a(g), g)[a.sub.1](g) + [P.sub.2](a(g), g)] -[N.sup.V][a.sub.1] (g)[C.sup.V.sub.1] (a(g)) + NC. [[P.sub.1](a(g), g)[a.sub.1] (g) + [P.sub.2](a(g), g)] x [-1 + P(a(g), g)] - [C.sup.G.sub.1](g) = 0.

Substituting in the first-order condition for potential victims ]Equation (2)] and suppressing the arguments for the sake of clarity yields Equation (5b):

(5b) [N.sup.C][P.sub.2]{[DELTA]U - 1 + P} +[N.sup.C][a.sub.1][P.sub.1]{- 1 + P} - [C.sup.G.sub.1] = 0.

We assume that the cost function [C.sup.G](g) is sufficiently convex that the government's objective function [in Equation (4)] is concave. In that case, Equation (5b) is both necessary and sufficient for an interior solution to the government's optimization.

Finally, it is useful to consider the socially optimal level of avoidance effort by potential victims, in order to help interpret later results concerning the government's optimal policy. Furthermore, while in this model the government can affect the level of avoidance effort only by changing the level of punishment effort, in a richer model, the government might have more influence. For example, the government might be able to tax or subsidize some types of avoidance activities, or might choose among different methods of law enforcement that vary in their degree of complementarity or substitutability with avoidance effort by potential victims. The socially optimal level of private effort a** is the level that maximizes social utility [from Equation (4)]. The first-order condition is given by

(6) [N.sup.C][P.sub.1](a**, g){[DELTA]U - 1 + P(a**, g)} - [N.sup.V][C.sup.V.sub.1](a**) = 0.

The notation a** is used to distinguish the socially optimal level from the level that is optimal from the perspective of a potential victim, a*. Note that a**< a*: that is, the level of avoidance effort chosen by potential victims exceeds the socially optimal level. This occurs because potential victims do not take into account the utility of potential criminals, who are made worse off by greater avoidance effort. (11)

Absent any cost of effort by the government and potential victims, the optimal level of crime would equate the marginal cost ([DELTA]U x [N.sup.C]) to the victims of the additional crime resulting from lowering the expected cost P to the benefit (1 x [N.sup.C][1- P]) that the mass of criminals committing crimes get from the lower cost. That is, the optimal expected cost P would satisfy [DELTA]U - 1 + P = 0. Because the value of [DELTA]U lies between 0 and 1, that optimal P is also between 0 and 1. When government and victim costs are included, we make two assumptions on the optimal level of government effort g*. First, we assume that g* is strictly positive, so that the unique solution is characterized by Equation (5b). That is, there is scope for government action after the private effort by the potential victims is undertaken. Second, we assume that g* is such that [DELTA]U - l + P(a(g*), g*) >0. That is, the equilibrium level of crime at the optimal level of government effort is higher if effort is costly than if effort is costless. By making those two assumptions, we restrict attention to what we consider the interesting, realistic case.

B. Hate Crimes

We consider five possible ways in which hate crime might differ from other crime in the context of our model, each of which could imply a higher optimal level of government effort to prevent hate crime than to prevent other crimes. In each case, we hold the rest of the model fixed.

The first is that the utility of those who commit hate crimes might get a lower weight in the government's objective function than the utility of those who commit other crimes. One argument that would support this approach is that some sources of utility should not count toward social welfare. As stated by a survey respondent cited in Iganski (2001, p. 632), "it is somehow more odious to harm someone for no other reason than because of who they are, not because they have something that you want." Thus, while the utility that a mugger gets from the money that he steals would count, the utility that the perpetrator of a hate crime gets from inflicting suffering would not count.

Glaeser's (2005) theory of hatred provides an alternative argument. He notes that one effective strategy for fighting hatred is to publicize images of violent hate-motivated attacks on minorities, which leads people to "hate the haters." In this case, those who hate the haters would derive utility from making potential hate criminals worse off, which would reduce the weight on the potential criminals' utility (or even lead to a negative weight, for a sufficiently large number of people with a sufficiently strong level of hate against the haters). Under either argument, the relative weight on criminals' utility would be lower. We model this by multiplying the term for the criminals' utility ([N.sup.C][[florin].sup.1.sub.p] B - p]) in the government's objective function in Equation (4) by the variable [delta], where [delta] < 1 represents the relative weight on criminals' utility.

Second, hate crime might generate a negative externality. This could occur in several ways. Members of the targeted group other than the direct victim may also suffer disutility from the crime, because they feel threatened or feel sympathy for the victim. A report by the U.S. Department of Justice says that, "A hate crime victimizes not only the immediate target but every member of the group that the immediate target represents" (U.S. Department of Justice, Bureau of Justice Assistance 1997, p. x). Other members of society may feel ashamed that such crimes took place. Hate crimes may incite retaliatory attacks by members of the targeted group, making the victims of those attacks worse off. The Department of Justice report goes on to note that, "Apart from their psychological impacts, violent hate crimes can create tides of retaliation and counter-retaliation." Recall that the number of crimes is ([N.sup.C][1 - P(a(g), g)]). We model the negative externality by adding the term f([N.sup.C][1- P(a(g), g)]) to the government's objective function in Equation (4), where f(x) is a negative and decreasing function. (12)

Third, avoidance effort by potential victims may generate a negative externality. One way to avoid being a victim of hate crime is to hide one's identity: Jews may try to pass as non-Jews, or gays may remain closeted, for example. If society values diversity, efforts like this to hide one's identity will create a negative externality. (13) There will be a similar negative externality if such efforts interfere with victims' ability to form profitable social networks, as in Dharmapala and Garoupa (2004). These efforts may also increase hatred. In Glaeser's (2005) model, contact with members of a minority group makes it more costly to hate that group, and thus reduces the level of hatred. But that effect could be reduced or eliminated if minority individuals hide their identities or avoid contact with those who hate them. (14) And hate crimes often target those who are fighting for minority rights (e.g., civil rights workers and politically active blacks in the American South during the 1960s). We model these cases by adding the term e(a(g)) to the government's objective function in Equation (4), where e(x) is a negative and decreasing function that represents this negative externality. (15)

Fourth, it may be more difficult to avoid being the victim of a hate crime than to avoid being the victim of other crimes. McDevitt et al. (2001, p. 706) say that unlike victims of other crimes, "... bias crime victims expressed feelings of frustration when asked how to prevent or reduce such crimes in the future. They generally did not indicate that their actions had done anything to provoke or exacerbate a situation." It is relatively easy to avoid visible displays of wealth and thus reduce one's chances of being mugged, but it is impossible to change one's skin color to avoid race-based hate crime, for example. We model this as a uniform increase in the marginal cost of avoidance effort by potential victims, [C.sup.V.sub.1], to [C.sup.V.sub.1] + [epsilon]. The second derivative, [C.sup.V.sub.11], is unchanged. (16)

Fifth, because hate crimes typically target minorities, the number of potential victims may be smaller for a hate crime than for other crimes, thus raising the probability that any particular individual in that group will be a crime victim. We model this by having a smaller mass of potential victims [N.sup.V].

Note that we do not explicitly consider the case in which the direct harm to the victim ([DELTA]U) differs between hate crimes and other crimes. The implications of that case are obvious: if hate crime does more harm to the victim than other crime, then, all else the same, this difference will imply a higher optimal punishment.

Nor do we explicitly consider the case in which the benefit to the criminal ([B.sub.i]) differs between hate crimes and other crimes. A higher benefit to the criminal would increase the level of crime, but would also increase the optimal level of crime. The net result could be either an increase or a decrease in the optimal level of punishment effort.

III. RESULTS

Here we analyze the optimal government effort to catch and punish criminals (g) for each of the five potential differences between hate crime and other crimes, as described in the previous section. We show that in either of the first two cases (discounting criminals' utility, negative externality from crime), the optimal punishment effort is higher for a hate crime than for other crime, all else equal. For the other three cases, the effect depends on the degree of complementarity or substitutability between victim and government effort. In the third case (negative externality from avoidance effort), optimal government effort is higher for a hate crime if individual and government efforts are substitutes, but lower if they are complements. In the fifth case (smaller mass of potential victims), the result is just the opposite: optimal government effort is lower for a hate crime than for an equivalent non-hate crime if individual and government efforts are substitutes and higher if they are complements. In the fourth case (higher marginal cost of avoidance effort), the result depends not just on whether individual and government efforts are substitutes, but on how substitutable they are: optimal government effort is higher for hate crime only if they are sufficiently strong substitutes. If they are complements or are sufficiently weak substitutes, the optimal government effort will be lower for hate crime than for other crime.

Formally, let g* denote the optimal government effort in the baseline model, and let [g.sup.H1], [g.sup.H2], [g.sup.H3], [g.sup.H4], and [g.sup.H5], respectively, denote the optimal levels for the five ways in which hate crime may differ from other crimes, taken one at a time. Our first result pertains to the first case, in which there is a lower weight in the government's objective function on the utility of potential hate criminals. One of the social costs of higher government effort is that it lowers the utility of potential criminals. If their utility gets a lower weight, that effectively lowers the social cost of government effort, and thus increases the optimal level of government effort. (17)

RESULT 1. All else equal, if the utility of potential hate criminals gets a lower weight in the government's objective function than the utility of other potential criminals, the optimal punishment effort is higher for hate crime than for other crimes.

Proof: In the baseline model, Equation (5b) is satisfied at g*. Discounting the criminals' utility, [N.sup.C][[florin].sup.1.sub.p(a(g),g)] B- P(a(g), g)], in the government objective function in Equation (4) by multiplying it by [delta] means that the term [N.sup.C][[P.sub.1][a.sub.1] + [P.sub.2]]{-1 + P} will be multiplied by [delta] in the derivative in Equation (5a). That term is strictly negative, as long as public and private efforts are not such strong substitutes that higher public effort reduces private effort by enough to increase the crime rate, a case that we have ruled out as unrealistic. Thus, multiplying that term by [delta] (which is less than 1) drives the value of the equation above zero. Because the objective function is concave, then, [g.sup.H1] must exceed g* to restore optimality.

For the case in which there is a negative externality from hate crime, the benefit of reducing crime is greater, because less crime implies both less harm to victims and less harm from the externality. Thus, the optimal level of government effort is higher.

RESULT 2. All else equal, if there is a negative externality from hate crime, the optimal level of punishment effort is higher for hate crime than for other crimes.

Proof: Adding a negative externality f([N.sup.C] [1 - P(a (g), g)]) to the government's objective function in Equation (4) means adding the term --[f.sub.1]([N.sup.C][1 - P])([P.sub.1][a.sub.1] + [P.sub.2]) to the derivative on the left-hand side of Equation (5b). That term is positive (again, as long as an increase in g leads to a decrease in crime). So, as in the previous proof, [g.sup.H2] must be greater than g* for the government's first-order condition to be satisfied.

Note that in both of these first two cases, the effect is different from that of simply increasing the harm to victims ([DELTA]U). In the baseline model, potential victims choose a level of avoidance effort (a*) that exceeds the socially optimal level (a**), because they do not take into account the effect on potential criminals' utility. That effect is still present if [DELTA]U increases. On the other hand, lowering the weight on criminals' utility reduces that effect--if the weight is zero, then a* and a** are equal. Similarly, introducing a negative externality from crime creates a positive externality from avoidance effort that offsets the negative external effect on criminals. In either case, the socially optimal level of avoidance effort increases.

For the first two definitions of hate crime, then, the increase in the optimal level of punishment effort relative to a baseline crime is different from what it would be for an equivalent rise in the harm to victims. If public and private efforts are substitutes ([P.sub.12] < 0), then there is less incentive for the government to discourage private effort by increasing public effort, and so the optimal increase is smaller. Conversely, if public and private efforts are complements ([P.sub.12] > 0), then the optimal level of punishment effort will increase by more in either of these first two cases than for a corresponding increase in the harm to victims.

Next, consider the third possible difference between hate crime and other crimes: a negative externality from avoidance effort. Now, the effect on optimal punishment effort is more complicated. Recall that if government and individual efforts are substitutes ([P.sub.l2] < 0), then the equilibrium level of individual effort a(g) decreases with g. In that case, the externality increases the marginal benefit of government effort, because a higher g leads to lower victims' effort. On the other hand, if [P.sub.12] > 0, then [a.sub.1](g) > 0, and the externality decreases the marginal benefit of government effort. Thus, a hate crime in this case requires a higher level of punishment effort in the case of substitutes, and a lower effort in the case of complements. That result is formalized below.

RESULT 3. All else equal, if there is a negative externality from hate crime avoidance effort, then the optimal punishment effort will be higher for hate crime than for other crime if punishment effort and avoidance efforts are substitutes, and will be lower if they are complements.

Proof: Adding a negative externality e(a(g)) to the government's objective function in Equation (4) means adding the term el (a(g))[a.sub.1] (g) to the derivative on the left-hand side of Equation (5a). According to Equation (3), if [P.sub.12] < 0, then so is [a.sub.1](g), and the new term is positive. As before, then, [g.sup.H2] must be greater than g* to restore equality. Analogously, if [P.sup.12] > 0, then the new term is negative, and [g.sup.H2] is less than g*.

Note that this assumption about hate crimes leads to the opposite effect on the socially optimal level of avoidance effort a** from the first two cases: it is even further below the level that potential victims choose a* than it is in the baseline model. Thus, it is optimal for the government to do more to discourage avoidance effort, either through changing the level of punishment effort or through other means.

Now consider the fourth possible difference between hate crimes and other crimes: hate crimes are associated with a higher marginal cost of avoidance effort by potential victims. In that case, each government effort level g is associated with a lower effort level and a higher marginal cost of effort in equilibrium for a hate crime relative to a regular crime. Suppose that [P.sup.12] > 0, so that victim and government efforts are complementary, implying that a(g) increases with g. Then a hate crime has three effects, all of which act to lower the optimal government effort. First, the marginal cost of individual effort (which is increasing in g) is higher. Second, the total marginal effect of g on the protection level P, [P.sub.1][a.sub.1] + [P.sub.2] falls (as is shown in the proof). Finally, a lower level of victim effort means a greater mass of criminals committing crimes. (18) Thus, the societal cost of increasing P (the expected cost of committing a crime) increases, because there are more criminals incurring that cost. If [P.sub.12] > 0, then, [g.sup.H4] < g*

When individual and government efforts are substitutes ([P.sub.12] < 0), two of the three effects change sign. Individual effort a(g) now is decreasing in g, so the higher marginal cost of individual effort makes government effort more attractive. Similarly, the overall marginal effect of g on P rises with the fall in a. On the other hand, there is still a greater mass of active criminals, which reduces the welfare gain from raising g. If the degree of substitutability is high enough, then the first two effects outweigh the third, and [g.sup.H4] > g*. Those two findings are shown in Result 4.

RESULT 4. All else equal, if avoidance effort is more expensive for hate crimes than for other

crimes, the optimal level of punishment effort is lower .lot" hate crimes than for other crimes unless punishment and avoidance efforts are sufficiently strong substitutes.

Proof: See Appendix.

Finally, consider the fifth way in which hate crime may differ from other crimes: the mass of potential victims [N.sup.V] is smaller. That change does not directly affect the number of crimes committed, which depends only on the mass of criminals [N.sup.C] and the average protection level. But because those crimes are concentrated on a smaller number of potential victims, each potential victim has a greater incentive to exert private effort to avoid crime, and thus the equilibrium level of such effort is higher. This has two effects. First, if government and private efforts are substitutes, then this higher level of private effort reduces the productivity of government effort. Conversely, if they are complements, then government effort is more productive. Second, the fewer victims who are incurring the cost of individual effort, the less benefit there will be from decreasing the equilibrium level of such effort: even though each individual's effort level will be more sensitive to government effort, the smaller number of victims dominates that effect, and thus the total cost of private effort will be less sensitive to the level of government effort. That is, reducing the number of potential victims makes increasing g less attractive in the case of substitutes, and more attractive in the case of complements. Thus, both effects--both the change in productivity of government effort and the influence of government effort on the total cost of private effort--imply a lower optimal level of government action if it is a substitute for effort by the victims, and a higher optimal level if it is a complement.

RESULT 5. All else equal, if there are fewer potential hate crime victims than potential victims of other crimes, the optimal punishment effort is lower for hate crimes than for other crimes when punishment and avoidance efforts are substitutes, and higher when they are complements.

Proof: See Appendix.

Note that we have assumed that the cost of government effort depends neither on the volume of crimes committed [N.sup.C][I - P], nor on the number of potential victims [N.sup.V]. The consequence of relaxing that assumption and supposing instead that government costs increase with the volume of crime is to raise the optimal level of government effort g both in the baseline model and for each definition of hate crime. There are no qualitative effects on any of our results, however. If we allow the cost of government effort to rise with the mass of the potential victims, there is a qualitative change only for this last case, in which there is a smaller pool of potential hate crime victims than of potential victims of other crimes. In this case, having the cost of government effort depending on the number of victims implies a lower cost of effort to prevent hate crimes, and thus a higher optimal level of government effort.

IV. CONCLUSION

We have presented a simple model of the effects of hate crime legislation. It shows that even if the harm to the direct victim of hate crime is the same as the harm from non-hate-motivated crime, other differences may lead to a higher optimal punishment for hate crime. However, the implications of these other differences are not always straightforward. In several of the cases we consider, the optimal level of public effort to prevent hate crime could be greater than or less than the optimal effort for other crime, depending on the complementarity or substitutability between public and private effort. Even for the cases in which the optimal punishment for hate crimes is unambiguously higher than for other crimes, we find important and previously unrecognized implications for policy that encourages or discourages private effort to prevent hate crime.

While this paper has focused on hate crime, the implications of this model extend to any other crimes that have similar effects. One notable example is terrorism, which shares many of the characteristics of hate crime: society puts little weight (or even a negative weight) on the welfare of terrorists, terrorism creates negative externalities (such as fear and the possibility of discrimination against Arab Americans), and private effort to avoid terrorism (for example, by avoiding air travel) is very difficult. Thus, our results should apply equally well to laws that require higher penalties for terrorism or for policies that encourage or discourage private effort to prevent terrorism.

There are several important aspects of this problem that are beyond the scope of our analysis, and thus represent potential directions for future research. First, we have focused only on efficiency issues, and have ignored equity issues. Protected groups are often economically disadvantaged, which could also provide an equity argument for additional protection--although this argument depends on the identity of the victim, whereas hate crime laws are typically based on the motivation for the crime, not the victim's identity.

Second, we have considered the normative issue of optimal hate crime policy, but not the positive political economy issue of why hate crime policy has developed in the way that it has, although the issues are clearly linked. It would be interesting to research the factors that explain states' decisions to enact hate crime laws.

Third, and perhaps most importantly, further empirical research on the effects of hate crime would be very valuable. The psychology and sociology literatures include substantial research on the harm to those who are direct victims of hate crime relative to the harm to victims of other crimes, although this literature is far from conclusive (see McDevitt et al. 2002). And for the other effects modeled in this paper, there exists only anecdotal evidence. That evidence suggests, though (particularly the damage that hate crimes inflict on the broader community rather than just the direct victim), that these effects exist and are important. Empirically measuring these effects would be difficult, but very valuable.

APPENDIX: PROOFS OF RESULTS 4 AND 5

Proof of Result 4

Let the marginal cost of avoidance effort be [C.sup.V.sub.1] + [epsilon], First, we rewrite Equation (2) to find the new level of individual effort:

(A1) [DELTA]U [N.sup.C]/[N.sup.V] [P.sub.1]([a.sup.*],g) - [C.sup.V.sub.I]:([a.sup.*]) - [epsilon] = 0.

Thus, the partial derivative of a(g, [epsilon]) with respect to [epsilon] is given by

(A2) [N.sup.C][[P.sub.2] + [a.sub.1][P.sub.1]]{[DELTA]U - 1 + P} - [N.sup.V][a.sub.1]([C.sup.V.sub.1] + [epsilon]) - [C.sup.G.sub.1] = 0.

where the arguments are suppressed for clarity. Note that [a.sub.2](g, [epsilon]) is strictly negative. Similarly, we rewrite Equation (5b) to get the first-order condition for the optimal government effort level [g.sup.*] ([epsilon]):

(A3) [N.sup.C][[P.sub.2] + [a.sub.1][P.sub.1]]{[DELTA]U - 1 + P} - [N.sup.V][a.sub.1]([C.sup.V.sub.1] + [epsilon]) - [C.sup.G.sub.1] = 0.

Note that [a.sub.1](g, [epsilon]) is still given by Equation (3), and that it does not vary with a, g, or [epsilon]. The effect of a marginal increase in [epsilon] on the left-hand side of Equation

(A.3) is given by [N.sup.C][{[DELTA]U - 1 + P}([P.sub.12] + [a.sub.1][P.sub.11]) + ([P.sub.2] + [a.sub.1][P.sub.1])[P.sub.1]][a.sub.2](g, [epsilon]) - [N.sup.V][A.sub.1][[C.sup.V.sub.11][a.sub.2](g. [epsilon]) + 1].

Substituting using first Equation (A.2) and then Equation (3) yields

(A.4a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If [P.sub.12] > 0, then [a.sub.1] > 0, and each of the three terms in the square brackets in Equation (A.4a) is greater than zero. Because [a.sub.2](g, [epsilon]) is strictly negative, then, the value of Equation (A.4a) is negative, so an increase in [epsilon] lowers the value of Equation (A.3) below zero. To restore equality, g must fall. (Remember that the government's objective function is concave in g.)

If [P.sub.12] = 0, then so does [a.sub.1], and Equation (A.4a) reduces to

(A.4b) [N.sup.C][[P.sub.1][P.sub.2]][a.sub.2](g, [epsilon]),

which again is strictly negative.

Because the square-bracketed term in Equation (A.4a) is increasing without bound in [P.sub.12], for low enough values of [P.sub.12] Equation (A.4a) is positive. In that case. an increase in [epsilon] implies that g must rise.

Thus. if government punishment effort and victims' avoidance effort are strong enough substitutes in raising the expected cost of committing a crime, then an increase in the marginal cost of victims' effort leads to an increase in the optimal level of government effort. Otherwise, it leads to a decrease.

Proof of Result 5 Let [a.sub.1N](g, [N.sup.V]) denote the partial derivative of [a.sub.1](g, [N.sup.V]) with respect to [N.sup.V]. Implicit differentiation of Equation (3) yields

(A.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note that the sign of [a.sub.1N](g, [N.sup.V]) is the opposite of the sign of [P.sub.12]. Note also that

(A.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which has the same sign as [P.sub.12].

The marginal effect of an increase in [N.sup.V] on the left-hand side of Equation (5b) is given by

(A.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In Equation (A.7), both of the terms in square brackets are positive. If [P.sub.12] > 0, then [a.sub.1N] < 0 and [N.sup.V] [a.sub.1N] + [a.sub.1] > 0, so the value of Equation (A.7) is negative. Thus, an increase in [N.sup.V] lowers the left-hand side of Equation (5b) below zero, and government effort g must fall to restore equality. Analogously, if [P.sub.12] < 0, then an increase in [N.sup.V] implies a rise in g.

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(1.) See Federal Bureau of Investigation (1995, 2005). Of course, the rise could be due to increased reporting rather than higher rates of such crime, but in either case, it indicates increased attention to the problem.

(2.) The set of protected categories varies across different states' laws; as of 1999, nearly all included race, religion. and national origin, while gender, sexual orientation, and disability were protected in roughly half of the states with hate crime laws. A handful of states also include such categories as political affiliation, age, marital status, involvement in civil rights, or service in the armed forces.

(3.) Jacobs and Potter (1998, p. 147) articulate the argument against hate crime laws: "We do not believe that crimes motivated by hate invariably are morally worse or lead to more severe consequences for the victims than the same criminal act prompted by other motivations."

(4.) Freeman (1999) and Polinsky and Shavell (2000) provide surveys of the extensive literature related to the Becker model.

(5.) Jefferson and Pryor (1999) find that sociological and economic factors are not good predictors of the presence of hate groups across locations. Medoff (1999) finds that market wages, the value of time, mean age, and law-enforcement activity predict hateful activity, but urbanization, occupational status, and social mobility do not. Gale, Heath, and Ressler (2002) find that hate crime rates are positively correlated with unemploynrent rates, abuse rates, and parity of income between blacks and whites, and negatively correlated with law-enforcement expenditures.

(6.) The phenomenon of "hating the haters" noted by Glaeser (2005) would also lead to a lower (or even negative) weight on the utility of hate criminals.

(7.) Many studies in law and economics have shown that the optimal level of government effort to prevent crime depends strongly on the degree of complementarity or substitutability between government and private effort (see, e.g., Ben-Shahar and Harel 1995). Some of our results are similar to the results of those studies in that they stem from distortions in potential victims' decisions about how much effort to exert to avoid being victimized. However, the sources of those distortions in this paper are quite different from those in prior work.

(8.) Note that individual avoidance effort has no external effect on the probability that others are victimized: such effort actually prevents crime, rather than merely displacing it onto other potential victims. As noted earlier, this is a key difference between our model and Dharmapala and Garoupa (2004).

(9.) For simplicity, we assume that there is no uncertainty in the cost of committing a crime. This assumption should not affect the results, because this certain cost could represent the certainty equivalent of an uncertain cost.

(10.) Glaeser and Sacerdote (2003) find that patterns of homicide sentencing are inconsistent with the predictions of this type of optimal law-enforcement model, and posit that this is caused by a taste for vengeance. Our analysis is primarily normative, and thus we ignore such issues, but they could be important in a positive analysis.

(11.) Note that this effect could be reversed if there is a positive externality from avoidance effort--if, for example, avoidance effort by one potential victim also benefits other potential victims. Ayres and Levitt (1998) note that the Lojack system to prevent car theft has just such a positive externality. This system sends a radio signal to help police track a stolen car. Because the radio transmitter on the car is difficult for prospective thieves to discover, the presence of Lojack on some cars reduces car thefts even for cars without the system. In contrast, a highly visible alarm system might encourage a potential thief to steal a different car, thus creating a negative externality. For simplicity, we rule out such externalities from avoidance effort in this baseline model.

(12.) On the other hand, widespread hatred of the targeted group would imply a positive externality from hate crime, because other people who share the perpetrator's hatred will also derive utility from the harm he inflicts on the victim, or a hate crime might generate a smaller negative externality than other crimes, because people outside the target group do not feel directly threatened. This could lead to a lower optimal penalty or even a reward for crimes against a sufficiently hated minority--a very troubling conclusion.

(13.) However, this could also create a positive externality if visible minorities generate disutility--again, a troubling conclusion. For example, those who are homophobic may prefer to have gays be forced to stay in the closet. This, along with the argument in the previous footnote, may explain why support is much weaker for hate crime laws that include sexual orientation among the protected categories than for those that do not (see Johnson and Byers 2003).

(14.) McDevitt et al. (2002) find that roughly 25% of hate crimes reported in Boston targeted minority households that had recently moved into a previously all-white block, with the apparent goal of convincing the outsider to move to a different neighborhood.

(15.) Again, it is possible that this effect works in the opposite direction. For example, to avoid the risk of mugging entirely might require never leaving the house.

(16.) This change can also be interpreted as a fall in the marginal productivity of avoidance effort for hate crime victims, which would have an identical effect.

(17.) More generally, the government might maximize a social welfare function other than the (weighted) sum of

individuals' utilities. Discounting utility achieved through crime in any such social welfare function would have an effect similar to the one described in Result 1.

(18.) This last effect has less impact when the utility of criminals is discounted, as in Result 1.

LI GAN, ROBERTON C. WILLIAMS III and THOMAS WISEMAN *

* The authors thank Valerie Bencivenga, Jeff Ely, Don Fullerton, Dan Hamermesh, Preston McAfee, Gerald Oettinger, Steve Trejo, Abraham Wickelgren, seminar participants at the University of Texas, and Tim grennan and two anonymous referees for their helpful suggestions.

Gan: Associate Professor. Department of Economics, Texas A&M University, College Station, TX 77843. Phone 979-862-1667, Fax 979-847-8757, E-mail gau@econmail.tamu.edu

Williams: Associate Professor, Department of Agricultural and Resource Economics, University of Maryland, Symons Hall, College Park, MD 20742. Phone 202-5079729, Fax 301-314-9091, E-mail robwilliams@mail.utexas.edu

Wiseman: Associate Professor, Department of Economics, University of Texas, 1 University Station C3100, Austin, TX 78712. Phone 512-475-8516, Fax 512-471-3510, E-mail wiseman@eco.utexas.edu

doi: 10.1111/j.1465-7295.2009.00281.x