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Rewards versus Imprisonment

Rewards versus Imprisonment Abstract This article considers the possibility of simultaneously reducing crime, prison sentences, and the tax burden of financing the criminal justice system by introducing rewards, which operate by increasing quality of life outside of prison. Specifically, it proposes a procedure wherein a part of the imprisonment budget is redirected towards financing rewards. The feasibility of this procedure depends on how effectively the marginal imprisonment sentence reduces crime, the crime rate, the effectiveness of rewards, and how accurately the government can direct rewards towards individuals who are most responsive to such policies. A related welfare analysis reveals an advantage of rewards: they operate by transferring or creating wealth, whereas imprisonment destroys wealth. Thus, the conditions under which rewards are optimal are broader than those under which they can be used to jointly reduce crime, sentences, and taxes. With an exogenous [resp. endogenous] budget for law enforcement, it is optimal to use rewards when the imprisonment elasticity of crime is small [resp. the marginal cost of public funds is not high]. These conditions hold, implying that using rewards is optimal, in numerical examples generated by using estimates for key values from the empirical literature. 1. Introduction There is widespread concern over the way imprisonment is used within the U.S. criminal justice system. This has caused scholars, critics, and commentators to use phrases like “mass incarceration” and “overincarceration” to describe their opinion that in the United States too many people are being incarcerated and/or that imprisonment sentences are too long. Even from a purely consequentialist perspective, economists have noted that imprisonment is used to a point where there are small returns from increased sentences in the form of crime reduction, i.e., the sentence elasticity of crime is quite small.1 This, combined with the high costs of imprisonment, including estimates going up to |$\$81,000$| per year per inmate in California,2 has caused many people to question the effectiveness of imprisonment and to seek alternatives to using lengthy imprisonment sentences to combat crime. In this article, I consider an alternative that has not received much attention in the theoretical economics of crime literature, namely the use of rewards .3 I consider as rewards any investment that increases the quality of life outside of prison. These can be in the form of cash payments for refraining from crime but may also include less obvious but valuable benefits, such as public health insurance;4 housing for the homeless;5 vocational and educational programs;6 nuisance abatement;7 and many others,8 which can have crime reducing effects as noted in existing empirical studies. Thus, rewards operate by improving people’s lives (outside of prison), and hence increasing the opportunity cost of committing crime in the standard Beckerian economic model of criminal behavior (Becker 1968). Conversely, punishment operates by reducing the value of committing crime. The effect of both policy instruments is to increase the gap between the legal and illegal options and thereby deter the commission of crimes.9 Despite this functional equivalence, economic theories of crime, starting with Becker (1968), have focused on punishment and have almost universally ignored rewards with a handful of important exceptions.10 Becker’s initial analysis of “Crime and Punishment” has since been extended in countlessly many directions.11 Yet rewards have not even been incorporated into standard crime and deterrence models as an alternative to imprisonment and monetary fines. This would be a serious deficiency in the economics of law enforcement literature, if rewards may serve as a viable tool in complementing punishment devices, and in particular imprisonment, in achieving deterrence. Thus, my objective here is to question whether rewards can, in fact, perform this function, and in the process to propose a simple way to incorporate rewards into the standard model of crime and deterrence. A natural way to begin this investigation is to point out the obvious disadvantage of rewards. As Wittman (1984) notes, given a low (detected) crime rate, the number of people who would be recipients of rewards is much larger than the number of people who would be convicted and punished. Thus, financing rewards would be difficult. A similar point comes to the fore in Demougin and Schwager (2000), who consider the possibility of using wealth redistributions from the employed to the unemployed as a means to reduce crime.12 Thus, in their setting, redistribution becomes an inferior option to increasing the probability of enforcement when the proportion of unemployed individuals—which is exogenously given in their setting—is high. Based on these simple observations, one may be concerned that administering a criminal justice system wherein rewards are used to deter people would require the imposition of a very large tax burden on society. This problem may appear insurmountable until one notes that rewards are meant to be used as an alternative to longer imprisonment sentences, which also generate large tax burdens to society. Stated differently, both rewards, and the imprisonment system generate a large tax burden to society. Therefore, the important question is not whether rewards generate a large tax burden, but whether the marginal tax dollar generates greater deterrence via rewards or via imprisonment. Once the question is posed in this manner, the advantage of rewards versus imprisonment becomes apparent: the ineffectiveness of imprisonment in achieving deterrence. As discussed in the empirical literature, the imprisonment elasticity of crime is rather low, with recent estimates around |$0.13$| (Lee and McCrary 2017). Thus, what needs to be assessed is whether the large-recipient-pool-problem of rewards is greater than the ineffectiveness problem of imprisonment. To make this assessment, one can consider the following thought experiment, which consists of two steps. First, suppose that the state reduces sentences imposed on convicts by a certain percentage. This naturally results in a reduction in deterrence, but, reduces the tax burden by |$X$| dollars.13 Second, suppose that the state announces that it will provide rewards to each individual who is not convicted by splitting the |$X$| dollars saved through the reduction in sentences. This will naturally lead to an increase in deterrence. If the reduction in deterrence due to reduced sentences is smaller than the increase in deterrence caused by the prospect of receiving rewards, then employing rewards will lead to a reduction in crime. It is also worth noting that, if it is possible to reduce crime through rewards in this manner, then it will also be possible to jointly reduce crime, sentences, and the tax burden. To see this, note that since rewards financed by exactly |$\$X$| leads to a reduction in crime, one can choose a slightly smaller amount |$\$Y<\$X$|⁠, which still leads to a reduction in crime. Thus, the state may use |$\$Y$| to finance rewards, and can return |$\$(X-Y)$| back to tax payers to jointly reduce crime, sentences, and the monetary burden imposed on taxpayers. The analysis reveals that rewards can be employed as described above, only if the monetary equivalent of the disutility imposed on a convict by investing an additional dollar towards prolonging his sentence is much smaller than the utility he would gain from rewards financed by a dollar. Thus, the relevant condition, which appears strong, depends on factors which are very hard to measure,14 such as the marginal disutility of prison; the monetary cost of imprisonment; the marginal utility generated by a dollar spent on rewards; the crime rate; and the probability of detection. This makes it hard to intuitively interpret these conditions, and, thus, I seek more intuitive and weaker sufficient conditions under which rewards can be put to beneficial use. For this purpose, I pursue two separate approaches. First, I question under what circumstances using rewards may enhance social welfare, defined as either the maximization of aggregate utility or as the aggregate utility of noncriminals.15 Second, I question whether rewards can be used to jointly reduce crime, sentences and taxes—as described in the above example—when the government can target either subsets of the population with high crime rates, or judgment proof offenders, with rewards. The analysis of the first question reveals that rewards enhance welfare under broader conditions than those under which rewards can be used to jointly reduce crime, sentences and taxes—as described above. The intuition behind this result relates to a very simple advantage enjoyed by rewards compared to imprisonment: rewards operate not by causing pain, displeasure, or disutility on to others, but, to the contrary, by enhancing the well-being of individuals by conferring benefits onto them. Thus, each dollar used to finance rewards rather than imprisonment enhances welfare further by causing an increase in the well-being of nonconvicts, and also by reducing the disutility of each convict when criminals’ utilities are included in the welfare function. An additional benefit of conducting a welfare analysis is that it reveals that, given a fixed budget allocated to the criminal justice system, rewards enhance welfare, if the sanction elasticity of crime is sufficiently small. In other words, the analysis produces a critical sanction elasticity of crime, which is a function of key variables in the analysis, such that any elasticity below this value implies that the use of rewards enhances welfare. Investigating this critical value reveals a stricter, but more intuitive sufficient condition: rewards are optimal as long as the ratio between total imprisonment costs and the total costs of crime (including imprisonment costs) is greater than the imprisonment elasticity of crime. This result can be interpreted as suggesting that there is an upper bound to using imprisonment, above which it is socially desirable to switch to using rewards. Moreover, this upper bound is decreasing in how effectively rewards can be used to incentivize people to stay away from crime. This approach, of using the stricter but more intuitive threshold for elasticities, has the advantage of switching the focus from specific valuations of marginal disutilities from prison towards values about which there exists several empirical estimates, namely the aggregate criminal responsiveness in society and the total costs of crime and imprisonment. Using recent estimates of imprisonment elasticities from the literature (Levitt 1998; Helland and Tabarrok 2007; Iyengar 2008; Lee and McCrary 2017) and estimates of the aggregate costs of crime and imprisonment (Bureau of Justice Statistics 2012; Chalfin 2015; Wagner and Rabuy, 2017) as examples suggests that substituting some imprisonment for some rewards in the United States is likely to enhance welfare. A welfare analysis can also be conducted where the budget allocated towards the criminal justice system is not fixed but is chosen optimally. The analysis of this problem relates the optimality of using rewards to the marginal cost of public funding, which refers to distortionary losses caused in the process of raising taxes. There is extensive literature on this topic containing both theoretical arguments as to how optimal tax policies can generate marginal costs of public funding which equal one (i.e., the cost of raising |$\$1$| equals |$\$1$|⁠, see, e.g., Jacobs (2018)) as well as empirical efforts to evaluate the marginal cost of public funds in various countries. Unsurprisingly, the analysis reveals that if the marginal cost of public funding is, in fact, one, then rewards are always optimal, since they can be used to enhance deterrence without generating any social costs. On the other hand, when the marginal cost of public funds exceeds one, a trade-off may emerge between the costs of securing funds for financing rewards and the comparative benefits of rewards over imprisonment due to its utility conferring, rather than destructive, nature (i.e., the fact that it operates by increasing nonconvicts’ utilities rather than reducing convicts’ utilities). When such trade-offs are present, one can calculate an upper bound on the marginal cost of public funding, such that a value below this bound implies that it is optimal to complement imprisonment with rewards. The analysis in Section 2 contains examples of what this upper bound would be, as a function of the imprisonment rate, by using an estimate for the marginal disutiltiy of imprisonment from Abrams and Rohlfs (2011). The exercise reveals that these upper bounds are larger than recent estimates of the marginal cost of public funds reported (e.g., 1.05–1.1 in Bjertnæs (2018)) suggesting that the use of rewards is optimal in these examples. The second question pertains to whether rewards can be used to serve the triple function of reducing crime, sentences, and taxes, if they can be used to target a specific subpopulation. As noted previously, the primary disadvantage of rewards is that they must be conferred to a very large number of individuals, and this is likely to reduce the deterrent effect of the per-person reward. If, however, rewards could be selectively provided to individuals, this disadvantage of rewards could be alleviated. In particular, rewards do not serve a deterrence function when they are provided to individuals who would not have committed crime, even if they were not offered rewards. These individuals make up a large proportion of the general population. Thus, if the government could use rewards exclusively to incentivize subpopulations with crime rates higher than the general population, it could reduce the proportion of rewards which do not serve a deterrence effect. Although one can identify many high crime subpopulations, targeting some of these groups may generate other costs or concerns. If, for instance, target-groups were defined based on behavior, the availability of targeted-rewards could cause moral hazard problems.16 Moreover, defining groups based on certain exogenous characteristics of individuals may also cause concern and be politically infeasible, if they perpetuate existing inequalities.17 Thus, it is worth recalling that the objective of this article is to identify advantages and disadvantages within the criminal justice system, and that when thinking about actual policies to be implemented, one must, of course, bear in mind many different factors that are traditionally excluded from theoretical analyses of crime. Nevertheless, one can think of groups that can be targeted without generating the types of concerns mentioned above. The juvenile and young adult population, for instance, has a much higher crime rate than the general population. Thus, implementing an age-based targeting scheme can increase the feasibility of using rewards.18 One can make the group even smaller and focus on subgroups among juveniles and young adults, who are statistically more likely to commit crimes. Empirical evidence suggests, for instance, that people who have grown up in a father-absent home have higher crime and incarceration rates (Harper and McLanahan 2004).19 Therefore, targeting rewards to these subpopulations can reduce crime in addition to generating many other social benefits which are excluded from my analysis. These possibilities are fomalized in section 3.1. Subsequently, I question whether targeting offenders based on their ability to pay monetary fines can similarly improve the effectiveness of rewards. This requires analyzing the case where, in addition to imprisonment, the government uses monetary sanctions to deter criminal activity. In this case, when introducing rewards, in addition to reducing sentences as outlined above, the government may increase monetary fines to keep the crime rate of wealthy individuals, who are able to pay larger fines, constant. This leads to two important effects: first, it is unnecessary to allocate any of the cost savings from imprisonment to the wealthy (i.e., nonjudgment proof), since their criminal behavior can be kept constant through relevant adjustments in monetary fines; and, second, the increased monetary fines received by wealthy offenders can be used to increase the amount that is devoted towards the rewards receivable by judgment proof offenders. These two additional considerations, formalized in Section 3.2.1, increase the effectiveness of targeted rewards. However, targeting based on ability to pay may be different in important respects from targeting based on exogenous characteristics, because it can cause moral hazard problems by distorting some individuals’ work incentives. To account for this possibility, I consider an extension in Section 3.2.2 where individuals first decide whether to work and subsequently decide on whether to commit crime. The analysis reveals two important insights. First, it reveals that the work propensity of marginal individuals is (weakly) increasing with their criminal propensity, and this can cause rewards to have very limited distortion effects on work incentives. Second, even when these distortions are sizeable, cost savings from using rewards in the criminal justice system are large enough to eliminate moral hazard problems altogether through hypothetical work incentives under conditions very similar to those I identify in the model without moral hazard considerations. Overall, this article identifies important factors that affect the (in-) feasibility of using rewards as a cost effective crime reduction tool. Whether one can, in fact, use rewards to achieve the goals described above is a matter of current debate. One recent contribution to legal scholarship, for instance, claims that rewards can, in fact, be successfully used to reduce crime in a cost-effective manner (Galle 2021). This view stands in contrast to prior scholarship which is skeptical of rewards based on a focus on the disadvantages of rewards described above.20 This article contributes to this debate by formalizing important considerations which bear on the feasibility of using rewards in a cost-effective manner. It does so by identifying a series of sufficient conditions under which rewards can be put to good use, and by discussing how rewards would need to be used to achieve these sufficient conditions. My analysis represents a step towards understanding and scrutinizing a tool that has been largely ignored in the theoretical economics of crime literature. The models presented in the next two sections propose a starting point with the hope of providing the theoretical foundations for understanding the appropriate role for rewards in the criminal justice system. The limitations of the model, the abstractions it contains, how it relates to the existing literature, and various additional considerations are discussed in Section 4. Section 5 concludes, and an Appendix in the end contains proofs of some of the propositions. 2. Model I consider the standard law enforcement model (see, e.g., Polinsky and Shavell 2007), where a continuum of potential risk-neutral offenders differ from each other with respect to their criminal benefits and opportunity costs of committing crime. This is formalized by assuming that a person who commits crime increases his well-being by an amount of |$b$|⁠, which will henceforth be called his criminal benefit, and which differs from person to person. The cumulative distribution function |$F(b)$| with support |$[0,\infty)$| describes the proportion of individuals with criminal benefits |$b$| that equal or are smaller than |$b$|⁠, with |$f=F^{\prime}$|⁠. To deter the commission of crimes, the government employs an enforcement mechanism which detects the commission of an offense with probability |$p$|⁠, and imprisons detected offenders for a length of time which causes them to suffer disutility |$v(s)$| with |$v^{\prime}>0,$| where |$s$| is the monetary cost of imprisoning the person for the amount of time that causes this disutility. Following Polinsky and Shavell (1984), I assume that the cost of imprisonment per person is proportional to the sentence such that |$s$| also represents the length of the sentence measured in the appropriate unit of time. Absent rewards, which are defined next, potential offenders commit crime if their criminal benefits exceed |$pv(s)$|⁠. In addition to imprisoning people the government may use rewards, which may either be monetary transfers or programs designed to increase potential offenders’ benefits from pursuing legal options. The government chooses the amount to be spent on rewards per recipient, which is denoted |$r$|⁠. The receipt of a reward in the form of a government program generates utility of |$u(r)$| for the recipient. Thus, a person who is eligible for a reward of |$r$| commits crime only if:21 $$\begin{equation} b^{r}(s,r)\equiv p(v(s)+u(r))<b. \end{equation}$$(1) I assume that |$u^{\prime}(r)\geq1$| for all |$r$|⁠, since, in the worst case, the government can offer monetary transfers as rewards instead of administering programs to meet the needs of the recipient whenever the marginal utility from the program falls below one. The government may choose to offer rewards to the entire population (as in Section 2), or it may choose to announce a subset of the population who is eligible to participate in the program (as in Section 3). Sections 2 and 3 are organized to answer questions related to the use of rewards. Section 2.1 analyzes the benchmark case where rewards must be used nonselectively and asks under what circumstance one can introduce rewards to simultaneously reduce crime, imprisonment sentences, and the tax burden generated by the criminal justice system. Section 2.2 conducts welfare analyses, assuming again that rewards are used non-selectively, both when the budget for the criminal justice system is fixed, and when it is a choice variable. Section 3 returns to the question asked in Section 2.1 but considers cases where the rewards can be used in a targeted manner. Throughout the analysis, the arguments of functions are often omitted to abbreviate notation when doing so causes no ambiguity. 2.1 Nontargeted Use of rewards When the government commits to a policy of making the entire population eligible for receiving rewards the crime rate, |$\theta$|⁠, is given by $$\begin{align} \theta(s,r)=1-F(b^{r}(s,r)) \end{align}$$(2) and the monetary cost of financing the criminal justice system is $$\begin{equation} T(s,r)=p\theta s+(1-p\theta)r\text{.} \end{equation}$$(3) This is because out of |$\theta$| criminals |$p\theta$| are caught and imprisoned, whereas |$(1-p)\theta$| evade detection and receive rewards along with the population of size |$1-\theta$| who refrain from committing crime. A regime which makes no use of rewards and employs an imprisonment sentence of |$\overline{s}>0$| leads to a crime rate of |$\theta(\overline{s},0)$| and a tax burden of |$T(\overline{s},0)$|⁠. To investigate whether the imposition of rewards can lead to superior results compared to the regime |$(\overline{s} ,0)$|⁠, one can consider regimes that lead to the same degree of deterrence as |$(\overline{s},0)$|⁠, which consist of a sentence |$s<\overline{s}$| along with a reward |$\overline{r}(s)>0$| characterized by $$\begin{equation} b^{r}(s,\overline{r}(s))=b^{r}(\overline{s},0)\text{ for all }s\in \lbrack0,\overline{s}). \end{equation}$$(4) The definition of |$b^{r}$| in (1) immediately reveals such |$\overline {r}(s)$| exists and that $$\begin{equation} \frac{d\overline{r}}{ds}=-\frac{v^{\prime}(s)}{u^{\prime}(\overline{r}(s))}. \end{equation}$$(5) Here, |$\frac{d\overline{r}}{ds}$| can be interpreted as being analogous to the marginal rate of technical substitution between rewards and imprisonment in the “production” of deterrence. Thus, |$\overline{r}(s)$| can be depicted as an isodeterrence curve with a slope of |$\frac{d\overline{r}}{ds}$| as in Figure 1, above. The same figure depicts a second function, namely $$\begin{equation} r^{T}(s)\equiv\frac{T(\overline{s},0)}{1-p\theta(\overline{s},0)} -\frac{p\theta(\overline{s},0)}{1-p\theta(\overline{s},0)}s \end{equation}$$(6) which represents a hypothetical tax-constraint outlining the feasible set of rewards and sentences that can be financed with the tax budget used in the initial regime |$(\overline{s},0)$|⁠, if, somehow, deterrence could be held constant at the level it is in the regime |$(\overline{s},0)$|⁠. Figure 1 Open in new tabDownload slide Isodeterrence and hypothetical tax-constraint. Figure 1 Open in new tabDownload slide Isodeterrence and hypothetical tax-constraint. Although this tax constraint is hypothetical (since not all |$s,r$| combinations below |$r^{T}$| lead to the same level of deterrence), it is useful to graphically explain the result reported in proposition 1, below, and it also provides the background necessary to develop a similar graphical representation (see Figure 2) in explaining the value of targeted rewards in Section 3, below. An important property of this tax constraint is that any point below |$r^{T}$| which generates a crime rate of |$\theta(\overline{s},0)$| (if there exist any) leads to a lower tax burden than |$T(\overline{s},0)$|⁠, because it uses lower rewards for the same number of nonconvicts (i.e., |$1-p\theta(\overline{s},0)$|⁠), shorter sentences for the same number of convicts (i.e., |$p\theta(\overline{s},0)$|⁠), or both, compared to some regime on |$r^{T}$| which leads to a tax burden of |$T(\overline{s},0)$| when the crime rate is |$\theta(\overline{s},0)$|⁠. Figure 2 Open in new tabDownload slide Expansion of the hypothetical tax-constraint. Figure 2 Open in new tabDownload slide Expansion of the hypothetical tax-constraint. In Figure 1, point |$I$| is depicted in the lower right corner and corresponds to an initial regime where rewards are not used and sentences equal |$\overline{s}$|⁠. Moving along the isodeterrence curve from point |$I$| to point |$A$|⁠, by definition, holds deterrence constant. Thus, point |$A=(s^{\prime },\overline{r}(s^{\prime}))$| generates a lower tax burden than |$T(\overline {s},0)$|⁠, since it leads to the same crime rate as the regime |$(\overline {s},0)$|⁠, and lies below |$r^{T}$|⁠. This implies that slightly increasing rewards above |$\overline{r}(s^{\prime})$| and keeping sentences at |$s^{\prime}$| (i.e., moving from point |$A$| to |$B$| in Figure 1) increases deterrence above the initial level while maintaining some of the tax savings generated by point |$A$|⁠. Thus, in the situation depicted in figure 1,22 one can introduce rewards to reduce sentences from |$\overline{s}$| to |$s^{\prime}$|⁠, while lowering the tax burden, and reducing crime at the same time. Of course, this observation is not specific to a single example but extends to all cases where |$r^{T}$| declines faster than |$\overline{r}(s)$| around the initial point |$(\overline {s},0)$|⁠. The intuition behind this result is explained after it is formalized and proven via proposition 1, below. Proposition 1 One can reduce crime, imprisonment sentences, and the tax burden generated by any regime which relies solely on imprisonment (with a sentence of |$\overline{s}>0$|⁠) by introducing rewards if $$\begin{equation} \frac{u^{\prime}(0)}{1-p\theta(\overline{s},0)}>\frac{v^{\prime}(\overline {s})}{p\theta(\overline{s},0)}. \end{equation}$$(7) Proof The tax burden generated by any regime |$(s,\overline{r}(s))$| is given by |$T(s,\overline{r}(s))$|⁠. Differentiating |$T(s,\overline{r}(s))$| with respect to |$s$| reveals that $$\begin{equation} \frac{\partial T}{\partial s}+\frac{\partial T}{\partial r}\frac{d\overline {r}}{ds}=p\theta-\frac{v^{\prime}(s)}{u^{\prime}(\overline{r}(s))} (1-p\theta)\geq0\text{ iff} \end{equation}$$(8) $$\begin{equation} p\theta\geq\frac{v^{\prime}(s)}{u^{\prime}(\overline{r}(s))+v^{\prime}(s)}. \end{equation}$$(9) The inequality in (9) reveals that whenever (7) holds there exists a regime |$(s^{\prime},r^{\prime})$| with |$s^{\prime}<\overline{s}$| and |$r^{\prime}>0$| such that |$\theta (\overline{s},0)=\theta(s^{\prime},r^{\prime})$| and |$T(s^{\prime},r^{\prime })<T(\overline{s},0)$|⁠. Thus, there exists |$\varepsilon>0$| such that |$T(s^{\prime},r^{\prime}+\varepsilon)<T(\overline{s},0)$| and |$\theta (\overline{s},0)>\theta(s^{\prime},r^{\prime}+\varepsilon)$|⁠. □ The condition identified by proposition 1, and graphically depicted in Figure 1, is relatively intuitive. The right-hand side (hereafter, RHS) of (7) describes the marginal deterrence effect—divided by |$f(b^{r}(\bar{s},0))$| to simplify the expressions—of increasing spending on imprisonment: any increase in the budget allocated towards imprisonment would have to be split equally to increase the punishment of |$p\theta$| convicts, and would cause a marginal deterrence effect of |$v^{\prime}(\overline{s} )f(b^{r}(\bar{s},0))$|⁠. Similarly, the left-hand side (hereafter, LHS) of (7) describes how much deterrence can be enhanced by increasing the reward budget: the increase in the budget would have to be split equally by |$1-p\theta$| people, and would cause a marginal deterrence effect of |$u^{\prime}(0)f(b^{r}(\bar{s},0))$|⁠. An important and simple observation is that, given all other parameters, if the crime rate is sufficiently low the condition in (7) does not hold. Conversely, the inequality in (7) is more likely to hold when the imprisonment rate is high. This is due to two interrelated reasons. First, convicts do not receive rewards, and, thus, a lower population of convicts increases the per person rewards receivable by nonconvicts. Second, the budget savings from reducing sentences is proportional to the imprisonment rate. Thus, a large imprisonment rate implies greater budget savings from reduced sentences which are used to finance rewards. Expanding on this point, one can note that the numerators of the expressions on the two sides of (7) reflect the primary advantage of rewards while the denominators highlight the primary disadvantages of using rewards. Given long imprisonment sentences, the marginal deterrence impact of increasing sentences (i.e., |$v^{\prime}(\overline{s})$|⁠) is likely to be smaller than |$1$|⁠, which is the lower bound for the marginal impact of rewards (i.e., |$u^{\prime}(0)$|⁠). On the other hand, rewards are less cost effective, because they are provided to all non-punished individuals (i.e., |$1-p\theta$| people), whereas punishment is reserved for a much smaller subpopulation (i.e., |$p\theta$| people). Therefore, whether crime and taxes can be reduced by the use of nonselective rewards depends on how ineffective prison is and how large the convicted criminal population is. This observation naturally leads one to question whether rewards can be used in a selective manner to reduce the denominator in the LHS of (7) and thereby make rewards a more effective tool. This question is analyzed in Section 3, after a further advantage of rewards is formalized through welfare analyses. 2.2 Welfare Analysis As noted in the Section 1, rewards operate by enhancing the well-being of people who are not convicted, as opposed to reducing the well-being of convicts. This causes rewards to enjoy an advantage over imprisonment, which can be formalized through a welfare analysis. For this purpose, I consider a conventional welfare function which aggregates the sum of all utilities, but, I point out that the implications of the analysis remain unchanged when one excludes criminals’ utilities from the welfare function in footnotes. As part of this analysis, first, I consider a welfare maximization problem subject to an exogenously determined tax constraint. Subsequently, I consider a similar maximization problem when taxes are chosen to maximize welfare. In both cases, to focus on the trade-off between rewards and imprisonment, I consider an exogenously given probability of detection |$\left( p>0\right) $|⁠, which could be interpreted as the welfare maximizing probability of detection when |$r=0$|⁠. As noted in the literature, the standard Beckerian result (that the maximum sanction along with a low probability of detection is always optimal) does not hold when |$v^{\prime\prime}<0$| (Polinsky and Shavell 1999). Thus, focusing on a fixed |$p$| is a harmless simplification in identifying sufficient conditions for the optimality of using rewards, since the case where |$p$| is endogenously determined can only broaden the conditions under which it is optimal to use rewards. This is because altering the probability of detection to the optimal level (instead of keeping it constant at |$p$|⁠) cannot reduce welfare.23 2.2.1 Welfare maximization subject to an exogenously given tax constraint The sum of all utilities generated by the commission of crimes, the imposition of punishment, and the conveying of rewards is given by: $$\begin{equation} W(s,r)=\int_{b^{r}}^{\infty}\left( b-h\right) f(b)db+(1-p\theta)u(r)-p\theta v(s), \end{equation}$$(10) where |$h$| denotes the expected harm inflicted through the commission of each crime, which is assumed to be high enough that any feasible regime leads to underdeterrence. When the government’s objective is to maximize this function subject to a tax constraint (⁠|$\overline{T}$|⁠), the relevant maximization problem can be formalized as $$\begin{equation} \underset{s,r}{\max}W(s,r)\text{ such that }T(s,r)=p\theta s+(1-p\theta )r\leq\overline{T}. \end{equation}$$(11) An inspection of this problem reveals the following result.24 Proposition 2 (i) Using rewards is optimal under broader conditions than those identified in Proposition 1. (ii) Moreover, the use of rewards is optimal if the imprisonment elasticity of crime (denoted |$\varepsilon_{s}$|⁠) is small enough, and a sufficient condition for the optimality of rewards is met when this elasticity is smaller than the ratio between total imprisonment costs and the total costs of crime including imprisonment costs, i.e., |$\varepsilon_{s} \leq\frac{\theta ps}{\theta\lbrack ps+h]}$|⁠. Proof See Appendix. □ Proposition 2 reveals conditions under which rewards are optimal. In particular it relates the optimality conditions to the sanction elasticity of crime rates: rewards are optimal, if the ratio of total imprisonment costs to total costs of crime, inclusive of imprisonment costs, is greater than the imprisonment elasticity of crime. Although it is hard to get precise estimates for these values, for purposes of a quick comparison, one can consider the estimates provided in prior studies. For example, Chalfin (2015) reviews the literature on estimating the total cost of crime, and reports that in 2012 the total harm from crime in the United States can be estimated around $\$$ 310 billion. For the total cost of imprisonment in the United States, there are two studies which report differing numbers, one from the Bureau of Justice Statistics (henceforth BJS) (BJS, 2012) and a second from the Prison Policy Initiative (henceforth PPI) (Wagner and Rabuy 2017). They report different numbers, among other reasons, because the BJS only includes the cost of operating correctional facilities, whereas the PPI study includes additional expenses, e.g., criminal court expenditures and bail fees. Thus, the BJS reports a cost of around $\$$81 billion (BJS, 2012) whereas the PPI study reports a cost of around $\$$182 billion. Using these numbers yields an estimate for the ratio on the RHS of the inequality expressed in Proposition 2 of about 21|$\%$| and 37|$\%$|⁠, respectively. Both of these numbers are significantly greater than |$0.07$|⁠, |$0.1$|⁠, and |$0.13$|⁠, which are the elasticities reported in Helland and Tabarrok (2007), Iyengar (2008), and Lee and McCrary (2017), respectively. However, both numbers are smaller than the elasticity of |$0.4$| reported in Levitt (1998). Thus, even rewards in the form of cash transfers appear optimal except when one assumes the highest elasticities estimated for criminal responsiveness by Levitt (1998). Of course, one should bear in mind that the above exercises are meant to illustrate the relevant dynamics in the model by using estimates from the literature. They abstract from many considerations, some of which are listed in Section 4. The primary purpose here is to illustrate that when one attempts to maximize welfare as opposed to achieving the triple goals of reducing crime, sentences, and tax burdens, the conditions under which rewards can be used in a socially desirable manner is broadened significantly. It is also worth bearing in mind that the above calculations attempt at finding intuitive sufficient conditions, meaning that if one could assess weaker conditions reported in the proof of Proposition 2 (see, expression (A.3) in the Appendix, below), one would find that rewards are more likely to be optimal than the assessment above would lead one to think. This analysis focuses on whether the use of rewards is socially desirable, given any exogenously determined tax constraint which must be used towards either imprisonment or rewards. Thus, the analysis does not question whether the resources devoted towards the administration of the criminal justice system may be above or below the optimum amount. It is possible, of course, that the government may have devoted above the optimal amount of resources towards fighting crime, and this may cause the imprisonment elasticity of crime to be low. Given this possibility, one may question whether there may ever be room for using rewards, if the optimal amount of resources are devoted towards the criminal justice system. This issue is explored, next. 2.2.2 Welfare maximization with optimal budget allocation The collection of taxes may generate welfare losses due to distortions they cause in the economy. Thus, increasing the size of the budget allocated towards the criminal justice system by a dollar may cause a loss that is greater than a dollar. To incorporate this possibility, when the taxes to be collected are endogenously determined, I consider a social welfare function of the form $$\begin{equation} \widehat{W}(s,r)=\int_{b^{r}}^{\infty}\left( b-h\right) f(b)db+(1-p\theta )u(r)-p\theta v(s)-k[p\theta s+(1-p\theta)r], \end{equation}$$(12) where |$k\geq1$| represents the marginal cost of public funds, and, as previously noted, |$p\theta s+(1-p\theta)r$| is the budget required to administer a system with sentence |$s$| and rewards of |$r$|⁠. Thus, it follows that the impact of imprisonment on welfare is $$\begin{equation} \frac{\partial\widehat{W}}{\partial s}=-\frac{\partial\theta}{\partial s}[h+kp(s-r)]-p\theta v^{\prime}(s)-kp\theta. \end{equation}$$(13) On the other hand, the marginal impact of rewards on welfare is $$\begin{align} \frac{\partial\widehat{W}}{\partial r} & =-\frac{\partial\theta}{\partial r}[h+kp(s-r)]+(1-p\theta)u^{\prime}(r)-k(1-p\theta)\\ & =-\frac{\partial\theta}{\partial s}\frac{u^{\prime}(r)}{v^{\prime} (s)}[h+kp(s-r)]+(1-p\theta)u^{\prime}(r)-k(1-p\theta),\nonumber \end{align}$$(14) where the second line of (14) is obtained by noting |$\frac {\partial\theta}{\partial r}=\frac{\partial\theta}{\partial s}\frac{u^{\prime }(r)}{v^{\prime}(s)}$|⁠. The expressions in (13) and (14) can be used to discuss the comparative advantages and disadvantages of rewards versus imprisonment. The second terms in (13) and (14) reflect the fact that rewards operate by enhancing rather than reducing the well-being of people, which is one of the primary advantages of rewards. On the other hand, increasing the reward receivable by each person who is not convicted by a dollar requires more funding than investing a dollar to prolong the sentence of each convict, as reflected by the last terms in (13) and (14). Finally, an increase in the reward provided to each recipient causes a greater deterrent effect compared to a similar increase in the amount spent towards financing an increase in each convict’s sentence as long as |$v^{\prime}(s)<1$|⁠. This is reflected by the first terms in (13) and (14). These observations reveal that the desirability of using rewards hinges on three key values: the marginal cost of public funding (⁠|$k$|⁠), the comparative deterrence effects of monetary values versus additional time in prison (i.e., |$u^{\prime }(r)$| versus |$v^{\prime}(s)$|⁠), and the incarceration rate (i.e., |$p\theta$|⁠). The next proposition identifies a precise relationship between these key values, which, when met, implies that using rewards is optimal.25 Proposition 3 Suppose that some deterrence is optimal, and that the optimal sentence is finite. (i) Then, it is optimal to use rewards if $$\begin{equation} \frac{1-p\theta}{u^{\prime}(0)}-\frac{p\theta}{v^{\prime}(s^{\ast})}<\frac {1}{k}, \end{equation}$$(15) where |$s^{\ast}$| is the optimal imprisonment sentence. (ii) This implies that using rewards is optimal if the marginal cost of public funding is sufficiently small. Proof (i) Some degree of deterrence requires that either |$r^{\ast}>0$| or |$s^{\ast }>0$|⁠. Thus, |$r^{\ast}=0$| is possible only if |$s^{\ast}>0$| characterized by the first order condition: $$\begin{equation} -\frac{\partial\theta}{\partial s}[h+kps^{\ast}]=p\theta\lbrack k+v^{\prime }(s^{\ast})] \end{equation}$$(16) and $$\begin{equation} \frac{\partial\widehat{W}(s^{\ast},0)}{\partial r}=-\frac{\partial\theta }{\partial s}\frac{u^{\prime}(0)}{v^{\prime}(s^{\ast})}[h+kps^{\ast }]+(1-p\theta)u^{\prime}(0)-k(1-p\theta)\leq0. \end{equation}$$(17) But, plugging (16) into (17) reveals that this impossible if $$\begin{equation} \frac{u^{\prime}(0)}{v^{\prime}(s^{\ast})}[kp\theta+p\theta v^{\prime} (s^{\ast})]+(1-p\theta)u^{\prime}(0)-k(1-p\theta)>0 \end{equation}$$(18) or, equivalently: $$\begin{equation} \frac{1-p\theta}{u^{\prime}(0)}-\frac{p\theta}{v^{\prime}(s^{\ast})}<\frac {1}{k} \end{equation}$$(19) which is the stated condition. (ii) The condition holds whenever |$k=1$| since |$u^{\prime}(0)\geq1$|⁠, and |$\theta(s,r)>0$| for all |$s$|⁠. □ As noted in part (ii) of Proposition 3, an implication of the condition specified is that if securing public funding generates very little distortions in the economy, then using rewards is certainly optimal. Quite importantly, many previous economic analyses of law enforcement implicitly invoke a similar assumption by supposing that the use of fines generates no direct costs or benefits. Thus, under similar assumptions where monetary fines and rewards are both costless transfers, rewards are trivially optimal. When |$k>1$|⁠, the analysis is more complicated. Unfortunately, without specifying precise numbers, it is not possible to pinpoint a concrete value for the marginal cost of public funding which makes rewards optimal. However, for interpretation purposes a conservative upper-bound for |$k$| as a function of the conviction rate can be calculated by using an estimate of |$v^{\prime}$| from Abrams and Rohlfs (2011), which finds that “the typical defendant in our [Philadelphia] sample would be willing to pay roughly $\$$1,000 for 90 d[ays] of freedom.” The cost of imprisonment per year in 2018 was at least |$\$42,000$| in Philadelphia according to the Philadelphia District Attorney’s office.26 These numbers imply a |$v^{\prime}$| which is smaller than |$0.1$|⁠. This number is unlikely to increase much if one adjusted the Abrams and Rohlfs (2011) estimate for inflation while also incorporating the fact that they were looking at the willingness to pay to avoid imprisonment for the first, rather than last, 90 days of incarceration, which would imply a smaller |$v^{\prime}$| given diminishing marginal disutility from imprisonment. Thus, one can calculate |$\overline{k}(p\theta)$|⁠, an upper bound on |$k$| which satisfies (19) using |$0.1$| as a conservative upper bound on |$v^{\prime}(s^{\ast})$| as follows $$\begin{equation} \overline{k}(p\theta)\equiv\frac{1}{1-1.1p\theta}. \end{equation}$$(20) To give an idea of the magnitudes of these upper bounds, Table 1, below, provides calculations of |$\overline{k}(p\theta)$| for different levels of conviction rates. The numbers reported in Table 1 range between around |$1.12$| and |$1.49$| and are higher than the recent estimates provided in Bjertnæs (2018), which range between 1.05 and 1.1. This suggests that using rewards is optimal in these numerical examples. Table 1. Estimates of |$\overline{k}$| using |$v^{\prime }(s^{\ast})<0.1$| |$p\theta=$| . |$1\%$| . |$2\%$| . |$3\%$| . |$\overline{k}(p\theta)=$| |$1.\,123\,6$| |$1.\,282\,1$| |$1.\,492\,5$| |$p\theta=$| . |$1\%$| . |$2\%$| . |$3\%$| . |$\overline{k}(p\theta)=$| |$1.\,123\,6$| |$1.\,282\,1$| |$1.\,492\,5$| Open in new tab Table 1. Estimates of |$\overline{k}$| using |$v^{\prime }(s^{\ast})<0.1$| |$p\theta=$| . |$1\%$| . |$2\%$| . |$3\%$| . |$\overline{k}(p\theta)=$| |$1.\,123\,6$| |$1.\,282\,1$| |$1.\,492\,5$| |$p\theta=$| . |$1\%$| . |$2\%$| . |$3\%$| . |$\overline{k}(p\theta)=$| |$1.\,123\,6$| |$1.\,282\,1$| |$1.\,492\,5$| Open in new tab 3. Targeted Use of Rewards As noted in the discussion following Proposition 1, the primary disadvantage of using nontargeted rewards is that the pool of reward recipients is large compared to the subpopulation being punished. Thus, as highlighted via (7), reducing sentences, taxes, and crime simultaneously is more likely to be possible when the imprisonment rate is high. A simple corollary of this observation is that if rewards can be used selectively to target only high-crime groups, they would be more likely to achieve these three goals simultaneously. However, when imprisonment sentences are restricted to be nondiscriminatory, i.e., equal across high-crime and low-crime groups, targeting high-crime groups with rewards while simultaneously reducing sentences for the entire population causes an increase in the low-crime group’s crime rate. In this case, it becomes necessary to target the group with the higher criminal responsiveness per rewardee, which may or may not coincide with the high-crime group. Another possible way of increasing the effectiveness of rewards is to target them exclusively towards individuals who are judgment-proof, which would allow one to reduce sentences across the entire population while increasing monetary sanctions to keep deterrence constant. The latter type of targeting can, of course, lead to moral hazard problems. This section discusses both cases, starting with the targeting of high-crime groups, first. 3.1 Rewards Targeting High-Crime Groups To discuss the possibility of targeting high-crime groups, I partition the population into two groups |$i\in\{H,L\}$|⁠, and let |$F_{i}$| denote the cumulative criminal benefit distributions conditional on being in group |$i\in\{H,L\}$|⁠, and let |$f_{i}$| denote the corresponding probability density functions. Here, the labels |$H$| and |$L$| may refer to high- and low-crime groups, respectively, when the government intends to use rewards targeting high crime groups. However, as explained below, the government may wish to target groups based on a characteristic other than crime rates, in which case these labels will refer to the ranking (as high and low) of that characteristic. Given this notation the unconditional distribution of criminal benefits is $$\begin{equation} F=\eta_{H}F_{H}+\eta_{L}F_{L}, \end{equation}$$(21) where |$\eta_{i\in\{H,L\}}$| with |$\eta_{H}+\eta_{L}=1\,\ $|represents the relative size of group |$i$|⁠. Thus, the crime rates within each group are $$\begin{equation} \theta_{i}=1-F_{i}(b_{i}^{r})\text{ for }i\in\{H,L\} \end{equation}$$(22) and the crime rate among the overall population is $$\begin{equation} \theta=1-\eta_{H}F_{H}(b_{H}^{r})-\eta_{L}F_{L}(b_{L}^{r}) \end{equation}$$(23) In (23), the subscript for |$b^{r}$| is used to indicate that the criminal benefit thresholds may differ across groups, since the two groups may be targeted with different enforcement schemes. Specifically, I consider two cases: one where both the sentence and the rewards can vary across groups, and a second case where sentences must be equal across the two groups but the government may nevertheless use different rewards across the two groups. 3.1.1 Targeting when sentences can vary across groups When both |$s$| and |$r$| can be based on group membership, the government can simply choose to target a group with a high crime rate to use rewards more effectively. This would not be possible when sentences are required to be non-discriminatory across groups. However, there are certain categories of offenders who can, in fact, be punished or treated differently in the criminal justice system based on their group membership. In the United States, the best example is presumably juveniles who can be subjected to very different treatments than adults. Thus, suppose the government partitions the population into two groups such that |$\theta_{H}>\theta_{L}$| in the initial punishment scheme which does not utilize rewards and punishes group |$H$| with a sentence of |$\bar{s}$|⁠. This sentence may, but need not, differ from the initial sentence applicable to group |$L$|⁠. In this case, proposition 1 can be applied simply by replacing the entire population with the high crime group, which reveals the following. Corollary 1 One can reduce crime, imprisonment sentences, and the tax burden generated by any regime that relies solely on imprisonment (involving a sentence |$\bar {s}>0$|⁠) by introducing rewards if $$\begin{equation} \frac{u^{\prime}(0)}{1-p\theta_{H}(\bar{s},0)}>\frac{v^{\prime}(\bar{s} )}{p\theta_{H}(\bar{s},0)}. \end{equation}$$(24) The primary difference between (24) and its nontargeted analog in (7) is that it involves the crime rate of the high criminal propensity group as opposed to the crime rate for the whole population. As noted previously, the condition in Proposition 1 (and therefore Corollary 1) is more likely to hold when the crime rate is higher. Thus, using rewards in a socially desirable manner is more likely to be feasible when the government can use different sentences as well as different rewards across the two groups. 3.1.2 Targeting with non-discriminatory sentences When the government can use targeted rewards but is constrained to impose the same sentences across groups, it can no longer leave the policy applicable to the low crime group unchanged while targeting the high crime group. Thus, to target one group with rewards, it will have to reduce the sentences applicable to the entire population. This type of policy will naturally have the effect of reducing the deterrence of the untargeted group. However, this reduction in deterrence can potentially be off-set by an increase in the deterrence of individuals in the targeted group, if the rewards directed towards them more than off-set the impact of reduced sentences. This result is more likely to be observed when the targeted group has a large ratio of marginal offenders to non-convicts (i.e., |$\frac{f_{i}}{1-p\theta_{i}}$|⁠), since this ratio is proportional to the per-dollar deterrence effect of rewards (i.e., |$\frac{f_{i}u^{\prime}(r)}{1-p\theta_{i}}$|⁠). In particular, when the per-dollar deterrence effect of rewards within the targeted group is greater than the per-dollar deterrence effect of imprisonment within the entire population, the targeting scheme described above can be successfully implemented. This is noted via the following corollary, whose proof is relegated to the Appendix since it follows steps very similar to those used in proving Proposition 1. Corollary 2 (i) One can reduce crime, imprisonment sentences, and the tax burden generated by any regime that relies solely on imprisonment (involving a sentence |$\bar{s}>0$|⁠) by using rewards that target group |$i$| if $$\begin{equation} u^{\prime}(0)\frac{f_{i}(b^{r})}{1-p\theta_{i}(\bar{s},0)}>v^{\prime}(\bar {s})\frac{f(b^{r})}{p\theta(\bar{s},0)} \end{equation}$$(25) (ii) Let |$H$| and |$L$| be selected such that |$\frac{f_{H}(b^{r})}{1-p\theta _{H}}>\frac{f_{L}(b^{r})}{1-p\theta_{L}}$|⁠, then when |$i=H$| (25) represents a weaker condition than (7). Proof See Appendix. □ A few points related to corollary 2 are worth highlighting. First, the condition identified for successfully using targeted rewards is independent of the relative size of the group being targeted. This is because, given any population and initial enforcement scheme, the smaller the size of the group being targeted, the smaller are the rewards needed to incentivize the group as a whole, and therefore the smaller are the sentence reductions required to finance those incentives. Second, unlike previously identified targeting mechanisms, it is not necessarily the case that the high-crime group ought to be targeted. Instead, the group with the higher marginal offender to nonconvict ratio ought to be targeted. This group naturally corresponds to the one with the higher crime rate whenever that group is at least as responsive to sanctions as the low crime group. This case is consistent with assumptions made in the theoretical literature (e.g., Bjerk 2007; Mungan 2018) as well as empirical findings regarding some high-crime groups (e.g., Levitt 1998). Finally, targeting groups with higher than average marginal offender to non-convict ratios always expands the conditions under which rewards can be successfully used to reduce crime, sentences, and taxes simultaneously. 3.2 Rewards Targeting Judgment Proof Offenders The optimal enforcement regime where people can be punished through imprisonment as well as monetary fines, but without rewards, is analyzed in Garoupa and Mungan (2019). This framework is discussed in further detail in Section 4, below, and results in a punishment scheme where some offenders are judgment proof and some are not. To incorporate the possibility of using rewards to target judgment proof offenders, I consider an initial punishment scheme where offenders are punished not only through imprisonment sentences (of |$\overline{s}$|⁠), but also monetary fines, whose magnitude is given by |$\underline{m}$|⁠. Following Polinsky and Shavell (1984), I assume that there are two groups with low and high wealth: |$\lambda$| proportion of individuals are judgment proof, i.e., they possess disposable wealth |$w<\underline{m}$|⁠, and the wealth of the remaining population (of proportion |$1-\lambda$|⁠) exceeds |$\underline{m}$|⁠, and, thus, these individuals are capable of paying higher fines. In the initial punishment scheme, there are no rewards, and, thus, judgment proof offenders commit crimes if $$\begin{equation} b>p(v(\overline{s})+w), \end{equation}$$(26) whereas wealthy individuals commit crimes if $$\begin{equation} b>p(v(\overline{s})+\underline{m}) \end{equation}$$(27) The question to be answered is whether the government can reduce crime, sentences and the tax burden of the criminal justice system, jointly, by providing targeted rewards to judgment proof offenders alone. First, I answer this question assuming that the proportion of judgment proof individuals is exogenous, i.e., unaffected by the enforcement scheme employed by the government. Subsequently, I consider the case where the enforcement scheme employed by the government may cause moral hazard problems by distorting the incentives of individuals to engage in productive activities to increase their wealth. 3.2.1 Exogenously determined groups In this subsection, I assume that individuals are either judgment proof or not, and this is unaffected by the enforcement scheme in place. Since, nonjudgment proof offenders are capable of paying additional fines, the government can keep their level of deterrence constant by simultaneously increasing the fine above |$\underline{m}$| and reducing the sentence below |$\overline{s}$|⁠. In particular, one can easily define the monetary fine that achieves this, as follows: $$\begin{equation} \widehat{m}(s)\equiv\underline{m}+(v(\overline{s})-v(s)). \end{equation}$$(28) On the other hand, the rewards that must be used in order to keep the deterrence of judgment proof offenders constant is given by |$\overline{r}(s)$|⁠, whose property is described via (5), and is depicted in Figure 2, above, as an isodeterrence curve. In calculating the tax burden, one thing that needs to be noted is that the collection of fines reduces the tax burden by allowing the criminal justice system to partially self-finance. Bearing this in mind, the tax burden can be calculated as follows $$\begin{equation} T=\lambda\lbrack p\theta_{l}(s-w)+(1-p\theta_{l})r]+(1-\lambda)p\theta _{h}(s-m), \end{equation}$$(29) where |$\theta_{l}$| and |$\theta_{h}$| refer to the crime rates among judgment proof people and wealthy people, respectively.27 Following an approach similar to that in Section 2.1, one can conceive of a hypothetical tax constraint, which corresponds to the rewards |$\widehat{r} ^{T}(s)$| that could be directed towards judgment-proof offenders by using the same tax budget as in regime |$(\overline{s},0,\underline{m})$|⁠, by altering |$s$| while simultaneously adjusting fines according to |$\widehat{m}(s)$|⁠, if deterrence could, somehow, be kept constant. Using (29) reveals that |$\widehat{r}^{T}(s)$| and its derivative are given by $$\begin{equation} \begin{array} &\widehat{r}^{T}(s)\equiv \frac{T(\overline{s},0,\underline{m})} {\lambda(1-p\overline{\theta}_{l})}& - \frac{p\overline{\theta}_{l} }{1-p\overline{\theta}_{l}}(s-w) & - \frac{(1-\lambda)p\overline{\theta }_{h}}{\lambda(1-p\overline{\theta}_{l})}& + \frac{(1-\lambda )p\overline{\theta}_{h}}{\lambda(1-p\overline{\theta}_{l})}\widehat{m}(s)\\ & & & & & & & \\ {\textit{Savings from:}} & \textit{(I) sentence } & \textit{(II) sentence} & \textit{(III) extra fines}\\ & \textit{reductions } & \textit{reductions} & \textit{collected}\\ & \textit{for JP} & \textit{for non-JP} & \textit{from non-JP}\\ & & & & & & \\ \frac{d\widehat{r}^{T}(s)}{ds}= & - \overbrace{\frac{p\overline {\theta}_{l}}{1-p\overline{\theta}_{l}}} & - \overbrace{\frac{1-\lambda }{\lambda}\frac{p\overline{\theta}_{h}}{1-p\overline{\theta}_{l}}} & - \overbrace{\frac{1-\lambda}{\lambda}\frac{p\overline{\theta}_{h} }{1-p\overline{\theta}_{l}}v^{\prime}(s)} \end{array}, \end{equation}$$(30) where |$\overline{\theta}_{i\in\{h,l\}}$| are used to refer to the crime levels in the initial regime to abbreviate the expression. The derivative of |$\widehat{r}^{T}(s)$| is split into components (marked (I), (II), and (III)) to explain how rewards that target the judgment proof population expands the conditions under which they can be used to reduce crime, sentences, and taxes, relative to the condition identified in (9). These different components are illustrated in Figure 2, below, and are subsequently explained in further detail. First, because the deterrence of nonjudgment proof offenders are kept constant by increasing fines, the sole objective of rewards is to keep the deterrence of judgment proof offenders constant. Thus, the relevant crime rate in component (I) is the crime rate among judgment proof offenders, i.e., |$\theta_{l}$|⁠, which is empirically higher than the overall crime rate given by |$\theta=\lambda\theta_{l}+(1-\lambda)\theta_{h}$|⁠, which I assume throughout the analysis. Thus, the tax savings from a sentence reduction for judgment proof offenders (as captured by component (I)) is greater than its analog in (9). Second, the nonjudgment proof population’s sentences are reduced, and, yet, their deterrence is held constant through an increase in fines. Thus, the tax savings obtained from their sentence reductions are not directed towards financing rewards for their own group but are directed towards the financing of rewards that go to the judgment proof population. This generates additional savings which were not previously available, and this is reflected via (II). Finally, the additional fines collected from nonjudgment proof offenders to keep their deterrence constant are also directed towards the financing of rewards for the judgment proof population, and this is reflected through component (III). As previously noted, any point below |$\widehat{r}^{T}(s)$| (if there are any) that leads to the same level of crime for judgment proof offenders would result in tax savings. Thus, the outward expansion of |$\widehat{r}^{T}(s)$| relative to its analog in (6) corresponds to a broadening of the conditions under which rewards can be used to reduce crime, taxes, and sentences. This result is formalized in Proposition 4, below. Proposition 4 (i) When the government can exclusively target judgment proof offenders with rewards, it can reduce crime, imprisonment sentences, and tax burdens generated by a regime which relies on imprisonment (with a sentence of |$\overline{s}>0$|⁠) and fines (of size |$\underline{m}$|⁠) by introducing rewards, if $$\begin{equation} \frac{u^{\prime}(0)}{1-p\theta_{l}(\overline{s},0,\underline{m})} +\frac{1-\lambda}{\lambda}\frac{\theta_{h}(\overline{s},0,\underline{m} )}{\theta_{l}(\overline{s},0,\underline{m})}\frac{u^{\prime}(0)[1+v^{\prime }(\overline{s})]}{1-p\theta_{l}(\overline{s},0,\underline{m})}>\frac {v^{\prime}(\overline{s})}{p\theta_{l}(\overline{s},0,\underline{m})}\text{;} \end{equation}$$(31) and, (ii) this condition holds when the proportion of judgment proof offenders is sufficiently small. Proof See Appendix. □ The condition reported in Proposition 4 is obviously a weaker one compared to that in Proposition 1, due to reasons explained via components (I)–(III) in (30). Specifically, the second term in the LHS of (31) is positive and is not present in (7) (due to the emergence of components (II) and (III)), and the crime rate that appears in the denominator of the remaining terms is |$\theta_{l}$| rather than |$\theta$|⁠, which also makes the condition weaker (due to the emergence of component (I)). The extent to which the condition is broadened depends inversely to the proportion of judgment proof individuals, since more people benefiting from the reward system reduces the per-person impact of these rewards. The implication of this is stated in part (ii) of Proposition 4: when the proportion of judgment proof offenders is sufficiently small, rewards can certainly be used to jointly reduce crime, sentences and the tax burden. It is worth briefly noting that the objective here is to identify the simple dynamics which lead to benefits from targeted rewards. The implementation of the simple reward program presented may, of course, generate moral hazard problems by reducing the gains from not being judgment proof. The next subsection formalizes this possibility by introducing a first period in which potential offenders choose whether or not to work to increase their disposable wealth. 3.2.2 Endogenously determined groups To incorporate possible moral hazard problems, I consider a simple two period extension. In the first period people decide whether or not to engage in efforts to increase their wealth by an amount |$a>0$|⁠, and decide whether to engage in criminal acts in the last period. I call productive acts that increase one’s wealth by |$a$| “working” to abbreviate descriptions. As in Section 3.2.1, I consider two groups (⁠|$l$| and |$h$|⁠) with differing wealth with |$w_{h}>$||$\underline{m}+a$| and |$w_{l}<\underline{m}$|⁠. A person in the latter group who works increases his wealth above the monetary sanction in place, i.e., |$a>\underline{m}-w_{l}$|⁠, in which case he is no longer judgment proof, which is what gives rise to moral hazard problems. Working imposes costs of |$c$| which differs across individuals. This change is easily captured by letting |$f_{i\in\{l,h\}}(b,c)$| denote the joint distribution of |$b$| and |$c$| rather than the distribution of only |$b$| as in the previous sections. As in the previous analysis, the initial punishment scheme consists of the monetary fine |$\underline{m}$| and sentence |$\bar{s}$|⁠. The question to be answered is whether introducing rewards that target judgment proof offenders while slightly adjusting the punishment scheme can lead to an improvement in outcomes. These types of rewards can naturally reduce low wealth individuals’ returns from work, since by working a person can lose his eligibility to receive rewards. To focus on this problem, I restrict attention to schemes which involve monetary fines |$m\in\lbrack\underline{m},w_{l}+a)$|⁠, such that this disincentivizing effect is in fact present. Moral hazard problems can be formalized by ascertaining individuals’ behavior through backward induction. Specifically, in the last period, a judgment-proof person with wealth |$w_{l}$| commits crime if $$\begin{equation} b^{J}(s,r)\equiv p(v(s)+u(r)+w_{l})<b \end{equation}$$(32) since he is entitled to receive rewards of |$u(r)$| if not convicted. On the other hand, an individual who has wealth |$w>m$| commits crime if $$\begin{equation} b^{N}(s,m)\equiv p(v(s)+m)<b. \end{equation}$$(33) Thus, $$\begin{equation} b^{N}(\bar{s},\underline{m})>b^{J}(\bar{s},0), \end{equation}$$(34) and this ranking is preserved for small deviations away from the initial enforcement scheme (i.e., |$u(r)<m-w_{l}$|⁠), which is what the analysis below focuses on. Given these best responses, a person’s expected pay-off from working in the first period can be expressed as: $$\begin{equation} \Pi_{i}^{1}\equiv \begin{array} [c]{lll} w_{i}+a-c & {\rm if} & b\leq b^{N}(s,m)\\ w_{i}+a-c+b-p(m+v(s)) & {\rm if} & b>b^{N}(s,m) \end{array} \ \ \text{ for }i\in\{l,h\}. \end{equation}$$(35) On the other hand, the expected pay-off associated with not working for a person in group |$i\in\{h,l\}$| is $$\begin{align} \Pi_{l}^{0} & \equiv \begin{array} [c]{lll} w_{l}+u(r) & {\rm if} & b\leq b^{J}(s,r)\\ (1-p)(w_{l}+u(r))+b-pv(s) & {\rm if} & b>b^{J}(s,r) \end{array} \ \ \text{; and} \end{align}$$(36) $$\begin{align} \Pi_{h}^{0} & \equiv \begin{array} [c]{lll} w_{h} & {\rm if} & b\leq b^{N}(s,m)\\ w_{h}+b-p(m+v(s)) & {\rm if} & b>b^{N}(s,m) \end{array}. \end{align}$$(37) Thus, a person in group |$h$| chooses to work if |$c<a$|⁠,28 since |$\Pi_{h}^{1}-\Pi_{h}^{0}=a-c$| for all |$b$|⁠. On the other hand, a person in group |$l$| chooses to work if |$\Pi_{l}^{1}>\Pi_{l}^{0}$|⁠, which can be written more explicitly as $$\begin{equation} \begin{array} [c]{lll} c^{L}(r)\equiv a-u(r)>c & {\rm if} & b\leq b^{J}\\ c^{I}(b,s,r)\equiv a-b+p(w_{l}+v(s))-(1-p)u(r)>c & {\rm if} & b\in(b^{J},b^{N} ]\\ c^{H}(r,m)\equiv a-(1-p)u(r)-p(m-w_{l})>c & {\rm if} & b>b^{N} \end{array} \ \ , \end{equation}$$(38) where the superscripts on these thresholds refer to the low, intermediate, and high criminal benefits, in which they are obtained, respectively. Inspecting (32) immediately reveals that rewards for judgment proof offenders increases the critical criminal benefit for these individuals and this contributes to reducing crime. However, this comes at the cost of reducing the work incentives of individuals in group |$l$|⁠, which is apparent from (38). Thus, the introduction of rewards can give rise to a trade-off between deterrence and reduced work incentives. Prior to seeking ways to compare these two effects, it is worth making some observations regarding the possible magnitude of moral hazard problems. First, note that when individuals receive their marginal productivity from work, the work incentive distortions of people who are not interested in crime (i.e., people with |$b<b^{J}$|⁠) have negligible welfare impacts. This is because, as noted via (38), absent rewards, these individuals’ returns from work equal their working costs, which implies that their working decision has no impact on welfare.29 Therefore, any negative welfare consequences that arise due to work incentive distortions caused by the introduction of rewards are limited to those generated by the changed incentives of individuals with greater returns from crime (i.e., with benefits |$b\geq b^{J}$|⁠). This limits the potential negative welfare impacts that might be borne from moral hazard problems. Second, and relatedly, as (38) reveals, moral hazard problems caused by rewards come about by changing the incentives of individuals who are on the margin with respect to their working decisions. Thus, the importance of moral hazard problems are positively related to the size of the population who are on these margins. Quite importantly, work incentives (i.e., |$c$|⁠) and criminal benefits (⁠|$b$|⁠) within the population can be correlated in a way that causes few people to be on the work–incentive margins listed in (38). This would be the case, for instance, when people who are least inclined to work (i.e., people with large |$c$|⁠) also have the largest benefits from crime (⁠|$b$|⁠) and vice versa such that |$b$| and |$c$| are positively correlated within the population. One could then expect few individuals to be marginal with respect to their work efforts, since the work-incentive margins listed in (38) are weakly decreasing in |$b$| (i.e., |$f(b,c)$| is small for |$b\geq b^{J}(\bar{s},0)$| and |$c\leq a$|⁠). These observations may perhaps help in explaining why moral hazard effects in this context have not been interpreted as representing a significant obstacle (Galle 2021). Nevertheless, the second observation above relies on empirical facts which are hard to quantify and describe. Thus, it can be useful to identify more conservative, but easier to interpret, conditions under which moral hazard problems, even if sizeable, might be small compared to deterrence benefits that could be obtained from introducing rewards. Unfortunately, this trade-off, too, hinges on similar assessments of how many people are on work-margins versus criminal activity margins, in addition to the relative costs of moral hazard problems versus benefits from reduced crime. The former comparison involves difficult empirical assessments, and the latter may necessitate making debatable value judgments. These problems can be avoided by noting that the changes in enforcement schemes of interest lead to reductions in the tax burden. Thus, if this reduction in the tax burden can be used to off-set the reduced work incentives problem, one can conclude that the gains from introducing rewards more than off-set their costs. This is because the additional proceeds from introducing rewards can be (although need not be) used to fix incentive distortions that they cause. To identify the conditions under which rewards can indeed be introduced in this manner, I consider an adjusted mechanism which distributes part of the savings to workers with wealth |$w_{l}+a$| in the amount of |$n$| per-worker.30 The remaining components of the mechanism are as in the previous subsection. Using this mechanism, it can be verified that a departure away from an initial scheme (where |$s=\bar{s}$|⁠, |$m=\underline{m}$|⁠, |$n=0,$| and |$r=0$|⁠) to a new scheme with a sentence |$s$| below |$\bar{s}$|⁠, and $$\begin{align} \hat{r}(s) & =u^{-1}(v(\bar{s})-v(s))\\ \hat{m}(s) & =\underline{m}+u(\hat{r}(s))\text{, and}\nonumber\\ \hat{n}(s) & =u(\hat{r}(s))\nonumber \end{align}$$(39) leaves the incentives of all types completely unchanged. This can be verified by adjusting and evaluating the critical benefits and costs listed in (32), (33), and (38) under a scheme that satisfies the conditions in (39). The important question, of course, is how the move from the initial scheme to the new scheme described by (39) affects taxes. If the move increases taxes, then it follows that the moral hazard problems created by the targeting of judgment proof offenders cannot be eliminated by the proposed mechanism without increasing the cost of public funding. However, if the taxes necessary are reduced by such moves, then it follows that any moral hazard problems caused by the introduction of rewards are small enough to be off-set by the tax savings obtained from the introduction of rewards. Which outcome is obtained can be ascertained by writing the taxes necessary to implement this mechanism as follows $$\begin{align} T(s) & = \lambda\big\{ (1-\omega)[(1-p\theta_{l}^{0})\hat{r}(s)+p\theta_{l}^{0}(s-w_{l})]+\omega\lbrack\hat{n}(s) \nonumber\\ & \quad{} +p\theta_{l}^{1}(s-\hat{m}(s))]\big\} +(1-\lambda)p\theta_{h}(s-\hat{m}(s)), \end{align}$$(40) where |$\omega$| is the proportion of individuals in group |$l$| who choose to work, and |$\theta_{l}^{0}$| and |$\theta_{l}^{1}$| are the crime rates among people who work and who do not work in group |$l$|⁠, which can of course differ from each other. Since, all schemes described by (39) hold the work as well as crime incentives of all individuals constant, these values are the same as they are in the initial enforcement scheme. Differentiating |$T$| with respect to sentences reveals that one can introduce rewards and simultaneously eliminate moral hazard problems through payments to workers while still reducing the tax burden, if $$\begin{equation} \begin{array} [c]{ll} \frac{dT(\bar{s})}{ds}= & \lambda\left\{ (1-\omega)[p\theta_{l} ^{0}-(1-p\theta_{l}^{0})\frac{v^{\prime}(\bar{s})}{u^{\prime}(0)} ]+\omega\lbrack p\theta_{l}^{1}-(1-p\theta_{l}^{1})v^{\prime}(\bar {s})]\right\} \\ & +(1-\lambda)p\theta_{h}(1+v^{\prime}(\bar{s}))>0. \end{array} \ \ \ \ \end{equation}$$(41) The next proposition summarizes the implications of this observation by re-expressing (41) to allow a better comparison with the analogous condition obtained when moral hazard problems are not incorporated. Proposition 5 (i) When the government can exclusively target judgment proof offenders with rewards, it can reduce crime, imprisonment sentences, and tax burdens generated by a regime which relies on imprisonment and fines without reducing the work incentives of individuals if $$\begin{align} &\frac{u^{\prime}(0)}{1-p\theta_{l}^{0}} \left[1-\omega+\omega\frac{\theta _{l}^{1}}{\theta_{l}^{0}}\right]\nonumber\\ &\quad{} +\frac{1-\lambda}{\lambda}\frac{\theta_{h} }{\theta_{l}^{0}}\frac{u^{\prime}(0)(1+v^{\prime}(\bar{s}))}{1-p\theta_{l} ^{0}}>\frac{v^{\prime}(\bar{s})}{p\theta_{l}^{0}}\left[ 1-\omega+\omega u^{\prime}(0)\frac{1-p\theta_{l}^{1}}{1-p\theta_{l}^{0}}\right], \end{align}$$(42) and (ii) this condition holds when the proportion of judgment proof offenders is sufficiently small. Proof The discussion preceding the proposition explains how one can obtain the same crime rate and work incentives by employing the scheme described via (39). When (42) holds, employing this scheme through a slight reduction in |$s$| below |$\bar{s}$| leads to tax savings. These savings can be used to slightly adjust the scheme to reduce crime via |$r$| and |$n$|⁠. □ A comparison between (31) and (42) reveals that the two conditions with and without moral hazard problems in the targeting of judgment proof offenders is quite similar. In fact, when the difference in the crime rates among workers and nonworkers within group |$l$| are negligible, and they both equal |$\theta_{l}$|⁠, the condition becomes $$\begin{equation} \frac{u^{\prime}(0)}{1-p\theta_{l}}+\frac{1-\lambda}{\lambda}\frac{\theta_{h} }{\theta_{l}}\frac{u^{\prime}(0)(1+v^{\prime}(\bar{s}))}{1-p\theta_{l}} >\frac{v^{\prime}(\bar{s})}{p\theta_{l}}\left[ 1-\omega+\omega u^{\prime }(0)\right.]. \end{equation}$$(43) This condition differs from (31) only in that the term multiplying |$\frac{v^{\prime}(\bar{s})}{p\theta_{l}}$| is a convex combination of |$1$| and |$u^{\prime}(0)$| rather than simply |$1$|⁠. Thus, the RHS is greater than its analog in (31) only when |$u^{\prime}(0)>1$|⁠, which of course, makes even untargeted uses of rewards more likely to succeed. The rationale behind this result can be ascertained by inspecting the first line of (41). The effect on the tax revenue from introducing rewards according to (39) is very similar across workers and nonworkers in group |$l$|⁠. First, they cause a reduction in the imprisonment financing required within both groups (proportional to their respective crime rates of |$p\theta_{l}^{i\in\{0,1\}}$|⁠). Second, they cause an increase in the rewards receivable by nonworkers (proportional to |$(1-p\theta_{l}^{0} )\frac{v^{\prime}(\bar{s})}{u^{\prime}(0)}$|⁠). Third, they cause an increase in the amount of worker compensation that is made out to workers (proportional to |$v^{\prime}(\bar{s})$|⁠). Fourth, the burden of these worker compensations are partially off-set by the increased fines that workers are able to pay (proportional to |$p\theta_{l}^{1}v^{\prime}(\bar{s})$|⁠). Thus, the impacts on the tax burden caused through workers and nonworkers in group |$l$| are quite similar, and they differ only as explained, above. 4. Discussion The models presented here follow the tradition in the theoretical economics of criminal behavior literature by focusing on the deterrent effects of sanctions. It questions how sanction schemes must be designed to balance deterrence benefits against other costs, such as imprisonment costs and disutilities to offenders. In this regard, it can be interpreted as extending the optimal deterrence literature (see e.g., Polinsky and Shavell (2007) for a review of basic models) which includes articles that incorporate imprisonment costs (Polinsky and Shavell 1984; Shavell 1985; Polinsky 2006). Analyzing the simple framework provided here highlights some of the disadvantages and advantages of rewards compared to imprisonment. The primary disadvantage of rewards is that, unless they can be used in a targeted manner, they must be distributed to a very large population. This property of rewards reduces its deterrent effect, given that the administration of the criminal justice system must rely on a limited budget. On the other hand, one of its comparative advantages stems from the diminished deterrent effect of imprisonment which makes the marginal deterrent effect of a dollar spent on a person via rewards stronger than the marginal deterrent effect of a dollar spent on imprisoning a person for a longer time. A second advantage of rewards is that they operate by increasing people’s utilities through, at worst, wealth transfers, and at best, wealth creation, whereas imprisonment is a tool that operates through wealth destruction. The analysis reveals that the benefits of using rewards are likely to dominate their costs, if imprisonment sentences are very long to begin with. Most of my analysis focuses on a comparison of the benefits and costs of rewards versus imprisonment. However, as Section 3.2 demonstrates, it is possible to extend the analysis to a framework where monetary fines can be used as a third tool. The analysis in that section focuses on an initial regime where some offenders are judgment proof, whereas others can afford to pay greater monetary fines. A natural question to ask is why the government would not increase the fine further, prior to resorting to imprisonment. One answer is that the government must choose a probability of detection that is applicable to all individuals, and therefore fines that optimally deter wealthy individuals, by definition, underdeter judgment proof individuals. Thus, it becomes optimal to impose some degree of imprisonment to reduce the underdeterrence of people who are judgment proof. As noted in Polinsky (2006), this problem can be avoided, if the government can either implement wealth-based sentences, or if it can offer people a choice between two sanction schemes, such that the wealthy choose to pay high fines and a shorter imprisonment term. As noted in Garoupa and Mungan (2019), this solution can become impracticable when it is politically or constitutionally impermissible to have wealth-dependent imprisonment sentences. Under those circumstances, we show, the optimal scheme involves one that is similar to the initial scheme considered in section 3.2, where some individuals are judgment proof and some are not. If, instead, the initial scheme were—contrary to what is observed in reality—one where all individuals are subjected to fines which equal their wealth, the targeted sanctions considered in Section 3.2 could not be used, and one would have to revert to using the solutions discussed in Sections 2 and 3.1. Similarly, most of the analysis focuses on the case where the probability of detection, |$p$|⁠, is fixed. That this assumption is harmless in identifying sufficient conditions is explained in note 23, above, and the accompanying text. However, the incorporation of rewards has an important implication regarding the relative value of the certainty versus the severity of punishment, a topic discussed extensively in the criminology and economics literatures. As explained in this article, the value of rewards is greater when the detected crime rate (i.e. |$p\theta$|⁠) is greater, because this increases the deterrent effect of the budget devoted towards rewards by reducing the number of reward recipients. Thus, unlike the severity of punishment, the certainty of punishment serves an additional function of enhancing the benefits from rewards. Therefore, the presence of rewards is likely to add to the list of reasons as to why the certainty of punishment is more effective than the severity of punishment, a point that is also made in Dari-Mattiacci and Raskolnikov (2020). (For further analysis of this issue, see Polinsky and Shavell (1999) and Mungan and Klick (2014).) It is worth noting that the analysis abstracts from several important issues to focus on the properties of rewards listed above. A few of these issues are discussed here to highlight avenues for future research. Most importantly, the analysis does not incorporate the incapacitative effects of imprisonment, whose analysis may reveal important insights regarding rewards. Previous analyses of incapacitation (Shavell (1987); Miceli (2010); Miceli (2012); Shavell (2015)) reveal the difficulty of incorporating both deterrence and incapacitation in a unified model of criminal decision making. Nevertheless, an intuitive conjecture is that the presence of incapacitative benefits from imprisonment may reduce the attractiveness of substituting some imprisonment time with rewards, since this would remove part of the incapcitative gains. Whether this conjecture is likely to hold is complicated by two important factors. First, a well-known observation in the criminology literature is that the prevalence of offending is related to age in an inverse-U shaped manner.31 This would suggest that average marginal incapacitative benefits are decreasing with the imprisonment sentence and can be quite small given long sentences. Second, and perhaps more importantly, in addition to providing incapacitative benefits, incarceration may give rise to criminogenic effects (Mueller-Smith 2015), i.e., contribute to higher recidivism rates.32 This consideration cuts against the increased incapacitative benefits of imprisonment, and could imply that the marginal benefits from using rewards and shortening imprisonment sentences are, in fact, greater than those presented in this article. Thus, it is not clear, a priori, how the incorporation of incapacitative and criminogenic effects of imprisonment may impact the desirability of rewards compared to punishment, which suggests the need for additional research on this issue. There are also some obvious considerations, which presumably increase the attractiveness of rewards, that were not formally incorporated into the analysis, because this would require making ad hoc assumptions regarding the relative importance of these considerations. Perhaps most importantly, when a person is convicted, there are costs borne by relatives and other close ones of the convict, which were not included in the analysis. The existence of these costs obviously tilt the analysis in favor of rewards. Similarly, the analysis ignores possible benefits or costs to third parties arising from deviations from what the public may deem to be the fair punishment for a crime (these considerations can be incorporated in framework similar to that in Polinsky and Shavell 2000). If, as most of the commentary seems to suggest, current sentences are perceived as being longer than what the public perceives to be the fair punishment, rewards will enjoy an additional benefit over imprisonment. The analysis also does not focus on the possibility of rewards being used in a way to permanently alter recipients’ opportunity costs from committing crime. Programs designed to broaden the skill set of individuals who are at risk of committing crime can have long lasting effects by increasing the legal earning potential of recipients. These can be incorporated into the current model by assuming that |$u^{\prime}$| is a number that is greater than |$1$|⁠, or, in a richer setting with dynamic effects, they can be incorporated by assuming that such programs reduce the recipient’s future |$b$|⁠, i.e., his relative benefits from crime. Another consideration which has predictable implications relates to people’s risk attitudes. As discussed in the literature (e.g., Block and Lind 1975; Polinsky and Shavell 1999; Mungan and Klick 2014; Mungan and Klick 2016), people may exhibit risk-seeking attitudes with respect to prison sentences and yet exhibit risk-averse preferences with respect to monetary outcomes. If true, this would again cut in favor of using rewards more often. This is because rewards increase the monetary and certain benefit to be had by refraining from crime whereas punishment only probabilistically increases the sentence over which people have risk-seeking tendencies. The analysis also does not incorporate the possibility of repeat offenses, and thus abstracts from issues related to recidivism. One question that becomes particularly important is whether a potential reward recipient loses his eligibility to receive rewards upon being convicted. In standard models analyzing the punishment of repeat offenders, the impact of rewards considered herein is similar to the impact of rewards which are not conditional on having a clean criminal record.33 This is because an unconditional reward system of this type affects the incentives of first time as well as repeat offenders in a similar fashion, and thus has effects that are similar to those explained herein. However, if the reward eligibility were made conditional on having a clean record, a familiar trade-off between general and specific deterrence would arise (Mungan, 2017b). Rewards would increase the incentives of people with clean records to comply with the laws (as explained in this article). However, ex-convicts would not be eligible for rewards, and if faced with lower imprisonment sentences, they would be more inclined to commit crime compared to a regime wherein there are longer sentences and no rewards. How this trade-off ought to be addressed is a complicated question, which can hinge, among other things, on the relative sizes of marginal first time and marginal repeat offenders in equilibrium. Nevertheless, a very simple observation is that allowing the reward size to depend on one’s criminal history (just as sentences are dependent on criminal history) can only weakly increase the attractiveness of rewards compared to the case where reward size is independent of criminal history. Thus, extending the current analysis to a setting where repeat offenses are possible may lead to the identification of additional mechanisms in which rewards can be used more effectively than explained here. The ideas presented here also bring to mind prior studies analyzing the impact of inequality and redistribution on crime, and how various redistribution methods compare to punishment mechanisms in achieving deterrence (e.g. Eaton and White (1991); Johnsen (1986); Tabbach (2012)). As explained in these studies, redistribution, like punishment, can achieve deterrence by increasing what a criminal has to lose upon being convicted, and this function is equally served by the rewards considered herein. Additionally, redistribution may also enhance deterrence further by reducing what a criminal has to gain through committing property crimes, because redistribution causes a reduction in the assets of potential victims. Moreover, as noted in Eaton and White (1991), punishment, unlike redistribution, can cause criminals to take costly actions to reduce the negative consequences of convictions (an issue also explored in the avoidance literature following Malik (1990)), which is an additional source of inefficiency. These latter two aspects of redistribution were not considered here, because the types of rewards envisioned in the article do not increase the tax burden placed on individuals in society, but instead are financed through reductions in imprisonment sentences (and, only in Section 3.2, through increases in the fines payable by wealthy offenders). Thus, the type of rewards considered here are not expected to cause additional deterrence effects by reducing the assets of potential victims. Similarly, because rewards are receivable only if a person does not commit crime, they are likely to have similar incentive effects as punishment on the avoidance activity of criminals. Thus, although rewards can be financed through large wealth redistributions, the instant analysis has not focused on this possibility to emphasize how this tool compares to imprisonment in achieving deterrence. 5. Conclusion Rewards and imprisonment can be used to perform similar deterrence functions. Despite this, there exists very little theoretical work considering the possibility of using rewards in the criminal setting. This is perhaps because, as highlighted in the analysis in this article, for rewards to be viable substitutes for imprisonment in achieving deterrence, imprisonment sentences would need to be significantly long. However, given the figures reported in the empirical literature regarding the ineffectiveness of imprisonment in enhancing deterrence, many jurisdictions in the United States may have already passed beyond the sentence threshold above which using rewards is a realistic alternative for achieving deterrence. The primary objective of this article is to highlight this possibility and to provide a theoretical framework that can be extended to incorporate further considerations relevant to the social (un)desirability of rewards. A. Appendix Proof of Proposition 2 (i) When (7) holds, it trivially follows that one can enhance welfare through rewards, e.g., by reducing sentences and increasing rewards above zero to hold deterrence constant. When (7) does not hold, part (ii) shows that when the imprisonment elasticity of crime, |$\varepsilon_{s}\equiv-\frac{\partial\theta}{\partial s}\frac{\bar{s}}{\theta }$|⁠, is smaller than a critical level using rewards is optimal. Thus, using rewards is optimal under broader conditions than (7). (ii) First, note that |$\frac{\partial T(\overline{s},0)}{\partial s} =\frac{\partial\theta}{\partial s}p\bar{s}+p\theta>0$|⁠, if |$\varepsilon_{s}<1$|⁠, which holds when |$\varepsilon_{s}<\frac{\theta ps}{\theta\lbrack ps+h]}$|⁠. Therefore, consider the case where |$\frac{\partial T(\overline{s},0)}{\partial s}>0$|⁠. Next, note that if |$\frac{\partial T(\overline{s},0)}{\partial r}\leq 0$|⁠, welfare can be increased simply by introducing rewards. When this condition does not hold, we have that |$\frac{\partial T(\overline{s} ,0)}{\partial s},\frac{\partial T(\overline{s},0)}{\partial r}>0$|⁠, and rewards are optimal if |$\frac{\frac{\partial W(\overline{s},0)}{\partial s}} {\frac{\partial T(\overline{s},0)}{\partial s}}<\frac{\frac{\partial W(\overline{s},0)}{\partial r}}{\frac{\partial T(\overline{s},0)}{\partial r} }$|⁠. Differentiating the welfare function and the tax burden with respect |$r$| and |$s$| reveals that $$\begin{align} \frac{\frac{\partial W}{\partial s}}{\frac{\partial T}{\partial s}} & =\frac{\frac{\partial\theta}{\partial s}(b^{r}-h)-p\frac{\partial\theta }{\partial s}[u(r)+v(s)]-p\theta v^{\prime}(s)}{\frac{\partial\theta}{\partial s}p(s-r)+p\theta}\text{; and}\\ \frac{\frac{\partial W}{\partial r}}{\frac{\partial T}{\partial r}} & =\frac{\frac{\partial\theta}{\partial r}(b^{r}-h)-p\frac{\partial\theta }{\partial r}[u(r)+v(s)]+(1-p\theta)u^{\prime}(r)}{\frac{\partial\theta }{\partial r}p(s-r)+(1-p\theta)}.\nonumber \end{align}$$(A.1) Evaluating these expressions at |$r=0$| and |$s=\bar{s}$|⁠, rearranging terms, and noting that |$\frac{\partial\theta}{\partial r}=\frac{\partial\theta}{\partial s}\frac{u^{\prime}(r)}{v^{\prime}(s)}$| reveals that |$\frac{\frac{\partial W(\overline{s},0)}{\partial s}}{\frac{\partial T(\overline{s},0)}{\partial s} }<\frac{\frac{\partial W(\overline{s},0)}{\partial r}}{\frac{\partial T(\overline{s},0)}{\partial r}}$| if $$\begin{equation} -\frac{\partial\theta}{\partial s}\left[ h\left[ (1-p\theta)-\frac {u^{\prime}(0)}{v^{\prime}(\overline{s})}p\theta\right] +u^{\prime }(0)p\overline{s}\right] <\left[ u^{\prime}(0)+v^{\prime}(\overline {s})\right] (1-p\theta)p\theta. \end{equation}$$(A.2) Multiplying both sides of this inequality by |$\frac{\overline{s}}{\theta}$|⁠, and noting that |$\varepsilon_{s}=-\frac{\partial\theta}{\partial s}\frac {\bar{s}}{\theta}$| reveals that this inequality is equivalent to $$\begin{equation} \overline{\varepsilon}_{s}\equiv\frac{\left[ 1+\frac{v^{\prime}(\overline {s})}{u^{\prime}(0)}\right] (1-p\theta)}{\frac{h}{p\overline{s}}\left[ \frac{1-p\theta}{u^{\prime}(0)}-\frac{p\theta}{v^{\prime}(\overline{s} )}\right] +1}>\varepsilon_{s} \end{equation}$$(A.3) Thus, |$r>0$| is optimal whenever |$\varepsilon_{s}<\overline{\varepsilon}_{s}$|⁠. When (7) does not hold it follows that |$v^{\prime }(\overline{s})\geq\frac{u^{\prime}(0)p\theta}{1-p\theta}$|⁠, in which case |$\overline{\varepsilon}_{s}>\frac{1}{\frac{h}{u^{\prime}(0)p\overline{s}}+1}$|⁠. This implies that a sufficient condition for the optimality of rewards is $$\begin{equation} \frac{p\overline{s}u^{\prime}(0)}{h+p\overline{s}u^{\prime}(0)}\geq \varepsilon_{s} \end{equation}$$(A.4) which holds when $$\begin{equation} \frac{\text{Total imprisonment costs}}{\text{Total criminal harm plus imprisonment costs}}=\frac{\theta p\overline{s}}{\theta (h+p\overline{s})}\geq\varepsilon_{s} \end{equation}$$(A.5) since |$u^{\prime}(0)\geq1$|⁠. □ Proof of Corollary 2 Part (i): When the government targets group |$i$| the crime rate is $$\begin{equation} \theta(s,r)=\eta_{i}\theta_{i}(s,r)+\eta_{-i}\theta_{-i}(s,0) \end{equation}$$(A.6) and the taxes necessary to implement this enforcement scheme are given by $$\begin{equation} T(s,r)=p\theta(s,r)s+\eta_{i}[1-p\theta_{i}(s,r)]r. \end{equation}$$(A.7) This is because |$p\theta(s,r)$| people are imprisoned, but only non-convicts in group |$i$| receive rewards (who have a measure of |$\eta_{i}[1-p\theta _{i}(s,r)]$|⁠). Letting |$r^{T}(s)$| denote the per-person reward that can be distributed to group |$H$| while keeping the tax burden unchanged at |$T(\bar{s},0)$|⁠, one can calculate the increase in the rewards available to this group as a result of reductions in the sentence as follows: $$\begin{equation} \frac{dr^{T}(s)}{ds}=-\frac{T_{s}(s,r)}{T_{r}(s,r)}=-\frac{p[\frac {\partial\theta(s,r)}{\partial s}s+\theta(s,r)]-\eta_{i}pr\frac{\partial \theta_{i}(s,r)}{\partial s}}{\eta_{i}\left( \frac{\partial\theta_{i} (s,r)}{\partial r}p[s-r]+[1-p\theta_{i}(s,r)]\right) }. \end{equation}$$(A.8) Crime is reduced as a result of reducing |$s$| below |$\bar{s}$| while increasing |$r$| above |$0$| according to (A.8) if $$\begin{equation} -\eta_{i}\frac{\partial\theta_{i}(\bar{s},0)}{\partial r}\frac{dr^{T}(\bar {s})}{ds}<\frac{\partial\theta(\bar{s},0)}{\partial s}. \end{equation}$$(A.9) Plugging (A.8) into the LHS of (A.9) reveals that this condition is equivalent to $$\begin{equation} \frac{f_{i}(b^{r})}{1-p\theta_{i}}u^{\prime}(0)>\frac{f(b^{r})}{p\theta}\text{ }v^{\prime}(\bar{s}) \end{equation}$$(A.10) which is the condition stated in the corollary. Part (ii): A simple manipulation of the condition |$\frac{f_{H}(b^{r} )}{1-p\theta_{H}}>\frac{f_{L}(b^{r})}{1-p\theta_{L}}$| reveals that it is equivalent to |$\frac{f_{H}(b^{r})}{1-p\theta_{H}}>\frac{f(b^{r})}{1-p\theta}$|⁠. Thus, (7) implies (25). □ Proof of Proposition 4 (i) The tax burden generated by all regimes that lead to the same level of deterrence as the initial regime can be expressed as $$\begin{equation} T(s,\overline{r}(s),\widehat{m}(s))=\lambda\lbrack p\theta_{l}(s-w)+(1-p\theta _{l})\overline{r}(s)]+(1-\lambda)p\theta_{h}(s-\widehat{m}(s)). \end{equation}$$(A.11) It follows that $$\begin{equation} \frac{\partial T}{\partial s}+\frac{\partial T}{\partial r}\frac{d\overline {r}}{ds}+\frac{\partial T}{\partial m}\frac{d\widehat{m}}{ds}=\lambda\left[ p\theta_{l}-(1-p\theta_{l})\frac{v^{\prime}(s)}{u^{\prime}(r(s))}\right] +(1-\lambda)p\theta_{h}[1+v^{\prime}(s)]. \end{equation}$$(A.12) This expression, evaluated at |$\overline{s}$|⁠, is positive if: $$\begin{equation} \frac{u^{\prime}(0)}{1-p\theta_{l}}+\frac{1-\lambda}{\lambda}\frac{\theta_{h} }{\theta_{l}}\frac{u^{\prime}(0)[1+v^{\prime}(\overline{s})]}{1-p\theta_{l} }>\frac{v^{\prime}(\overline{s})}{p\theta_{l}} \end{equation}$$(A.13) which is the condition expressed in (31). Due to the same reason noted in the proof of Proposition 1, it follows that when this inequality holds, the government can reduce crime, imprisonment sentences, and tax burdens, by introducing rewards. (ii) Follows immediately from the way |$\lambda$| enters the second term in (31 ). □ Acknowledgement This article replaces the working paper “Positive Sanctions versus Imprisonment” (Mungan 2019). I thank two anonymous referees for their careful review of this article and their useful comments and suggestions. I also thank participants of the 29th Annual Meeting of the American Law and Economics Association, the 36th Annual Meeting of the European Association of Law and Economics, the 2019 Public Choice Society Meetings, and the 2019 Southern Economic Association Annual Meeting for valuable comments and suggestions. Footnotes 1. See, e.g., Lee and McCrary (2017) finding sentence elasticities of crime not exceeding |$0.13\,$| and Chalfin and McCrary (2017) for a review of the existing literature. 2. This number is taken from a report by the Legislative Analyst’s Office which was last updated in January 2019 and last accessed in October 2021 at the following URL: https://lao.ca.gov/policyareas/cj/6_cj_inmatecost. 3. Wittman (1984), explained below, Dari-Mattiacci and De Geest (2009), explained in note 10, below, De Geest and Dari-Mattiacci (2013), and Dari-Mattiacci and Raskolnikov (2019) referred to in note 9, below, consider asymmetries between sticks and carrots in different settings. It is also worth noting that there are similarities between the dynamics I explore here and those that emerge in studies investigating the impact of foreign aid on terrorism. Some studies in this field posit that foreign aid, especially when used in a targeted manner, can be an effective tool (and an alternative to military intervention) in reducing terrorism in the aid-recipient country. See, e.g., Azam and Thelen (2008, 2010) and Bandyopadhyay et al. (2011). 4. See, e.g., Wen et al. (2020), He and Barkowski (2020), Aslim et al. (2020), Vogler (2020), all finding crime reducing effects associated with Medicaid expansions. Relatedly, Bondurant et al. (2018) find that the presence of substance abuse treatment centers have a local crime reducing effect. 5. See, e.g., Cohen (2020) finding a crime reducing effect associated with certain housing assistance programs. 6. Many empirical studies report crime reducing effects of educational programs, see, e.g., Garces et al. (2002) for a prominent example, and others referenced in Welsh (2011). Some of these programs target children based on exogenous categories. Welsh (2011) notes, for instance, that “[p]reschool intellectual enrichment programs are generally targeted on the risk factors of low intelligence and attainment” and that these programs have been found in subsequent evaluations to reduce offending. 7. See, e.g., Mazerolle et al. (1998), where nuisance abatement measures are found to reduce observed drug selling relative to more traditional policing measures. 8. For a lengthier discussion of publicly provided benefits that may have crime reducing effects, see, e.g., Galle (2021) and Welsh (2011). 9. It is important to note that this substitutability between carrots and sticks does not extend to cases wherein actors can decide not to be subject to the regulations through which rewards and sanctions are imposed (Dari-Mattiacci and Raskolnikov (2019)). The current analysis focuses on criminal behavior wherein the agent does not have this option. 10. The most important exception is Demougin and Schwager (2000) mentioned, below. Dari-Mattiacci and De Geest (2009) also consider rewards in noncriminal settings and ask whether sticks and carrots may be used in a way that generate different incentives structures. However, their mechanism is not widely implementable in a criminal setting, because it requires specifying punishment schemes that are a function of the behavior profile of all individuals, which would make it administratively very costly to implement in the criminal setting, and it could violate the Equal Protection Clause, because it requires specifying unequal punishments for different violators in some cases. There is also some empirical literature, focusing on a different kind of carrot, which is not a direct choice variable available to the government, namely employment opportunities. See, e.g., Corman and Mocan (2005) and other references reviewed in Chalfin and McCrary (2017). 11. See, e.g., Garoupa (1997) and Polinsky and Shavell (2007) for an account of some of the primary extensions. 12. Unlike the instant article, Demougin and Schwager (2000) consider redistribution as an alternative to more policing, as opposed to the length of imprisonment. Their approach is also quite different in that it considers a redistribution mechanism as opposed to rewards conferred upon avoiding conviction; assumes that wealth transfers are dissipated if the recipient is subsequently convicted; and the set of potential criminals is observable by the state. This last assumption is particularly problematic in the current context. As will become clear in the next sections, the inability to observe the at-risk population becomes an important obstacle in the way of using rewards in a socially beneficial manner. 13. Here, |$\$X$| is assumed to be the reduction in imprisonment costs at the new equilibrium level of deterrence after both policy changes (including the change described in the second step) have been implemented. 14. See, e.g. Mastrobuoni and Rivers (2019) and Abrams and Rohlfs (2011) on empirical estimates of the disutility from imprisonment. Even with these innovative empirical studies, it is difficult to ascertain the marginal disutility from the last day (or any other unit of time) of prison, which is smaller than the average disutility from one day of prison when the disutility from prison rises less than proportionally as discussed in Polinsky and Shavell (1999). 15. I consider two separate welfare functions for this analysis, because there is a disagreement that dates back to the first articles proposing economic analyses of criminal behavior. Becker includes criminals’ benefits in the social welfare function (Becker 1968) and Stigler (1970) questions whether this approach is defensible. Recently, Curry and Doyle (2016) provided an explanation as to why maximizing a utilitarian social welfare function may be equivalent to minimizing criminal harm in some contexts. 16. For instance, although the crime rate among ex-offenders is higher than the general population, targeting ex-offenders can generate incentives to commit crime in the first place. 17. It is not hard to imagine, for instance, that a system which makes rewards available only to males (based on the fact that the crime rate is much higher among males than females) would lead to concerns that it would contribute to existing gender gaps. 18. Existing educational programs, some of which target children with specific needs, can be viewed as targeted rewards. 19. Some may argue that this type of targeted policy may incentivize a higher rate of father-absent families. However, these effects may be small due to reasons similar to those explained in the targeting of judgment proof offenders explained below. 20. See, e.g., Wittman (1984), and others referenced in Galle (2021). 21. I assume that indifferent individuals refrain from committing crime. 22. One may be curious about whether |$\overline{r}(s)$| is necessarily convex, as depicted in Figure 1, although this is not a necessary property for the derivation of any of the results. A quick look at |$\overline{r}$|’s derivative reveals that it is convex as long as |$v^{\prime\prime},u^{\prime\prime}<0$|⁠. 23. This argument can be made more explicit through the use of symbols. Suppose the function to be maximized is |$W(s,r,p)$|⁠. If |$\widehat{p}$| and |$\widehat{s}$| denote the optimal choices of |$p$| and |$s$| subject to the constraint that |$r=0$|⁠, and if |$r^{\prime}>0$| and |$s^{\prime}$| improve welfare relative to this solution subject to the constraint that |$p=\widehat{p}$|⁠, then it follows that $$ W(s^{\ast},r^{\ast},p^{\ast})\geq W(s^{\prime},r^{\prime},\widehat{p} )>W(\widehat{s},0,\widehat{p}), $$ where |$\ast$| indicate optimal levels when all policy variables are endogenously determined. Thus, it follows that |$r^{\ast}\neq0$|⁠, since otherwise it would follow that |$W(s^{\ast},r^{\ast},p^{\ast})=W(\widehat{s} ,0,\widehat{p})$|⁠, which is a contradiction with the above ranking. 24. When criminals’ utilities are excluded from the welfare function, assuming that criminal harm is uniformly distributed across the population, the welfare function reads $$ W(s,r)=(1-\theta)[u(r)-\theta h]. $$ Thus, the maximization problem becomes $$ \underset{s,r}{\max}(1-\theta)[u(r)-\theta h]\text{ st. }T(s,r)=p\theta s+(1-p\theta)r\leq\overline{T}. $$ An analysis of this problem reveals a bound on the imprisonment elasticity of crime of $$ \widetilde{\varepsilon}_{s}\equiv\frac{1}{\frac{h}{ps}\frac{1-2\theta }{1-\theta}(\frac{1-p\theta}{u^{\prime}(0)}-\frac{p\theta}{v^{\prime}(s)})+1} $$ such that—whenever the denominator of |$\widetilde{\varepsilon}_{s}$| is positive—an elasticity lower than |$\widetilde{\varepsilon}_{s}$| implies that welfare can be enhanced by the use of rewards. This implies the same results reported in Proposition 2. 25. When criminals’ utilities are excluded from the welfare function, assuming the tax burden is uniformly distributed across the population, the welfare function instead reads $$ \widehat{W}(s,r)=(1-\theta)\left[ [u(r)-\theta h]-k[p\theta s+(1-p\theta )r]\right.] $$ Repeating steps that are very similar to those in the proof of Proposition 3 to investigate this function reveals the same results reported in Proposition 3. 26. See, e.g., https://www.nbcphiladelphia.com/news/local/Philadelphia-Prosecutors-Must-Include-Cost-of-Prison-Time-During-Sentencing-476987333.html, last accessed in October 2021. 27. The subscripts |$l$| and |$h$| are used to distinguish these rates from |$\theta_{L}$| and |$\theta_{H}$| used in Section 3.1. 28. I assume that indifferent people choose not to work. 29. This dynamic is very similar to that underlying the optimality of underdeterrence in simple Beckerian law enforcement models, see, e.g., Polinsky and Shavell (2007). 30. Another approach is to calculate the moral hazard losses generated by rewards and compare them to tax savings. This approach generates similar results, but the conditions obtained are less intuitive because they relate to the density of individuals on work margins. 31. See, e.g., Loeber and Farrington (2014) discussing the age-crime curve. 32. Mueller-Smith notes that “the empirical results indicate that incarceration generates net increases in the frequency and severity of recidivism ... A cost–benefit exercise finds that substantial general deterrence effects are necessary to justify incarceration in the marginal population.” 33. There is a broad literature analyzing the punishment of repeat offenders. 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Rewards versus Imprisonment

American Law and Economics Review , Volume Advance Article – Nov 9, 2021

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Oxford University Press
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Copyright © 2021 American Law and Economics Association
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1465-7252
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1465-7260
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10.1093/aler/ahab011
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Abstract

Abstract This article considers the possibility of simultaneously reducing crime, prison sentences, and the tax burden of financing the criminal justice system by introducing rewards, which operate by increasing quality of life outside of prison. Specifically, it proposes a procedure wherein a part of the imprisonment budget is redirected towards financing rewards. The feasibility of this procedure depends on how effectively the marginal imprisonment sentence reduces crime, the crime rate, the effectiveness of rewards, and how accurately the government can direct rewards towards individuals who are most responsive to such policies. A related welfare analysis reveals an advantage of rewards: they operate by transferring or creating wealth, whereas imprisonment destroys wealth. Thus, the conditions under which rewards are optimal are broader than those under which they can be used to jointly reduce crime, sentences, and taxes. With an exogenous [resp. endogenous] budget for law enforcement, it is optimal to use rewards when the imprisonment elasticity of crime is small [resp. the marginal cost of public funds is not high]. These conditions hold, implying that using rewards is optimal, in numerical examples generated by using estimates for key values from the empirical literature. 1. Introduction There is widespread concern over the way imprisonment is used within the U.S. criminal justice system. This has caused scholars, critics, and commentators to use phrases like “mass incarceration” and “overincarceration” to describe their opinion that in the United States too many people are being incarcerated and/or that imprisonment sentences are too long. Even from a purely consequentialist perspective, economists have noted that imprisonment is used to a point where there are small returns from increased sentences in the form of crime reduction, i.e., the sentence elasticity of crime is quite small.1 This, combined with the high costs of imprisonment, including estimates going up to |$\$81,000$| per year per inmate in California,2 has caused many people to question the effectiveness of imprisonment and to seek alternatives to using lengthy imprisonment sentences to combat crime. In this article, I consider an alternative that has not received much attention in the theoretical economics of crime literature, namely the use of rewards .3 I consider as rewards any investment that increases the quality of life outside of prison. These can be in the form of cash payments for refraining from crime but may also include less obvious but valuable benefits, such as public health insurance;4 housing for the homeless;5 vocational and educational programs;6 nuisance abatement;7 and many others,8 which can have crime reducing effects as noted in existing empirical studies. Thus, rewards operate by improving people’s lives (outside of prison), and hence increasing the opportunity cost of committing crime in the standard Beckerian economic model of criminal behavior (Becker 1968). Conversely, punishment operates by reducing the value of committing crime. The effect of both policy instruments is to increase the gap between the legal and illegal options and thereby deter the commission of crimes.9 Despite this functional equivalence, economic theories of crime, starting with Becker (1968), have focused on punishment and have almost universally ignored rewards with a handful of important exceptions.10 Becker’s initial analysis of “Crime and Punishment” has since been extended in countlessly many directions.11 Yet rewards have not even been incorporated into standard crime and deterrence models as an alternative to imprisonment and monetary fines. This would be a serious deficiency in the economics of law enforcement literature, if rewards may serve as a viable tool in complementing punishment devices, and in particular imprisonment, in achieving deterrence. Thus, my objective here is to question whether rewards can, in fact, perform this function, and in the process to propose a simple way to incorporate rewards into the standard model of crime and deterrence. A natural way to begin this investigation is to point out the obvious disadvantage of rewards. As Wittman (1984) notes, given a low (detected) crime rate, the number of people who would be recipients of rewards is much larger than the number of people who would be convicted and punished. Thus, financing rewards would be difficult. A similar point comes to the fore in Demougin and Schwager (2000), who consider the possibility of using wealth redistributions from the employed to the unemployed as a means to reduce crime.12 Thus, in their setting, redistribution becomes an inferior option to increasing the probability of enforcement when the proportion of unemployed individuals—which is exogenously given in their setting—is high. Based on these simple observations, one may be concerned that administering a criminal justice system wherein rewards are used to deter people would require the imposition of a very large tax burden on society. This problem may appear insurmountable until one notes that rewards are meant to be used as an alternative to longer imprisonment sentences, which also generate large tax burdens to society. Stated differently, both rewards, and the imprisonment system generate a large tax burden to society. Therefore, the important question is not whether rewards generate a large tax burden, but whether the marginal tax dollar generates greater deterrence via rewards or via imprisonment. Once the question is posed in this manner, the advantage of rewards versus imprisonment becomes apparent: the ineffectiveness of imprisonment in achieving deterrence. As discussed in the empirical literature, the imprisonment elasticity of crime is rather low, with recent estimates around |$0.13$| (Lee and McCrary 2017). Thus, what needs to be assessed is whether the large-recipient-pool-problem of rewards is greater than the ineffectiveness problem of imprisonment. To make this assessment, one can consider the following thought experiment, which consists of two steps. First, suppose that the state reduces sentences imposed on convicts by a certain percentage. This naturally results in a reduction in deterrence, but, reduces the tax burden by |$X$| dollars.13 Second, suppose that the state announces that it will provide rewards to each individual who is not convicted by splitting the |$X$| dollars saved through the reduction in sentences. This will naturally lead to an increase in deterrence. If the reduction in deterrence due to reduced sentences is smaller than the increase in deterrence caused by the prospect of receiving rewards, then employing rewards will lead to a reduction in crime. It is also worth noting that, if it is possible to reduce crime through rewards in this manner, then it will also be possible to jointly reduce crime, sentences, and the tax burden. To see this, note that since rewards financed by exactly |$\$X$| leads to a reduction in crime, one can choose a slightly smaller amount |$\$Y<\$X$|⁠, which still leads to a reduction in crime. Thus, the state may use |$\$Y$| to finance rewards, and can return |$\$(X-Y)$| back to tax payers to jointly reduce crime, sentences, and the monetary burden imposed on taxpayers. The analysis reveals that rewards can be employed as described above, only if the monetary equivalent of the disutility imposed on a convict by investing an additional dollar towards prolonging his sentence is much smaller than the utility he would gain from rewards financed by a dollar. Thus, the relevant condition, which appears strong, depends on factors which are very hard to measure,14 such as the marginal disutility of prison; the monetary cost of imprisonment; the marginal utility generated by a dollar spent on rewards; the crime rate; and the probability of detection. This makes it hard to intuitively interpret these conditions, and, thus, I seek more intuitive and weaker sufficient conditions under which rewards can be put to beneficial use. For this purpose, I pursue two separate approaches. First, I question under what circumstances using rewards may enhance social welfare, defined as either the maximization of aggregate utility or as the aggregate utility of noncriminals.15 Second, I question whether rewards can be used to jointly reduce crime, sentences and taxes—as described in the above example—when the government can target either subsets of the population with high crime rates, or judgment proof offenders, with rewards. The analysis of the first question reveals that rewards enhance welfare under broader conditions than those under which rewards can be used to jointly reduce crime, sentences and taxes—as described above. The intuition behind this result relates to a very simple advantage enjoyed by rewards compared to imprisonment: rewards operate not by causing pain, displeasure, or disutility on to others, but, to the contrary, by enhancing the well-being of individuals by conferring benefits onto them. Thus, each dollar used to finance rewards rather than imprisonment enhances welfare further by causing an increase in the well-being of nonconvicts, and also by reducing the disutility of each convict when criminals’ utilities are included in the welfare function. An additional benefit of conducting a welfare analysis is that it reveals that, given a fixed budget allocated to the criminal justice system, rewards enhance welfare, if the sanction elasticity of crime is sufficiently small. In other words, the analysis produces a critical sanction elasticity of crime, which is a function of key variables in the analysis, such that any elasticity below this value implies that the use of rewards enhances welfare. Investigating this critical value reveals a stricter, but more intuitive sufficient condition: rewards are optimal as long as the ratio between total imprisonment costs and the total costs of crime (including imprisonment costs) is greater than the imprisonment elasticity of crime. This result can be interpreted as suggesting that there is an upper bound to using imprisonment, above which it is socially desirable to switch to using rewards. Moreover, this upper bound is decreasing in how effectively rewards can be used to incentivize people to stay away from crime. This approach, of using the stricter but more intuitive threshold for elasticities, has the advantage of switching the focus from specific valuations of marginal disutilities from prison towards values about which there exists several empirical estimates, namely the aggregate criminal responsiveness in society and the total costs of crime and imprisonment. Using recent estimates of imprisonment elasticities from the literature (Levitt 1998; Helland and Tabarrok 2007; Iyengar 2008; Lee and McCrary 2017) and estimates of the aggregate costs of crime and imprisonment (Bureau of Justice Statistics 2012; Chalfin 2015; Wagner and Rabuy, 2017) as examples suggests that substituting some imprisonment for some rewards in the United States is likely to enhance welfare. A welfare analysis can also be conducted where the budget allocated towards the criminal justice system is not fixed but is chosen optimally. The analysis of this problem relates the optimality of using rewards to the marginal cost of public funding, which refers to distortionary losses caused in the process of raising taxes. There is extensive literature on this topic containing both theoretical arguments as to how optimal tax policies can generate marginal costs of public funding which equal one (i.e., the cost of raising |$\$1$| equals |$\$1$|⁠, see, e.g., Jacobs (2018)) as well as empirical efforts to evaluate the marginal cost of public funds in various countries. Unsurprisingly, the analysis reveals that if the marginal cost of public funding is, in fact, one, then rewards are always optimal, since they can be used to enhance deterrence without generating any social costs. On the other hand, when the marginal cost of public funds exceeds one, a trade-off may emerge between the costs of securing funds for financing rewards and the comparative benefits of rewards over imprisonment due to its utility conferring, rather than destructive, nature (i.e., the fact that it operates by increasing nonconvicts’ utilities rather than reducing convicts’ utilities). When such trade-offs are present, one can calculate an upper bound on the marginal cost of public funding, such that a value below this bound implies that it is optimal to complement imprisonment with rewards. The analysis in Section 2 contains examples of what this upper bound would be, as a function of the imprisonment rate, by using an estimate for the marginal disutiltiy of imprisonment from Abrams and Rohlfs (2011). The exercise reveals that these upper bounds are larger than recent estimates of the marginal cost of public funds reported (e.g., 1.05–1.1 in Bjertnæs (2018)) suggesting that the use of rewards is optimal in these examples. The second question pertains to whether rewards can be used to serve the triple function of reducing crime, sentences, and taxes, if they can be used to target a specific subpopulation. As noted previously, the primary disadvantage of rewards is that they must be conferred to a very large number of individuals, and this is likely to reduce the deterrent effect of the per-person reward. If, however, rewards could be selectively provided to individuals, this disadvantage of rewards could be alleviated. In particular, rewards do not serve a deterrence function when they are provided to individuals who would not have committed crime, even if they were not offered rewards. These individuals make up a large proportion of the general population. Thus, if the government could use rewards exclusively to incentivize subpopulations with crime rates higher than the general population, it could reduce the proportion of rewards which do not serve a deterrence effect. Although one can identify many high crime subpopulations, targeting some of these groups may generate other costs or concerns. If, for instance, target-groups were defined based on behavior, the availability of targeted-rewards could cause moral hazard problems.16 Moreover, defining groups based on certain exogenous characteristics of individuals may also cause concern and be politically infeasible, if they perpetuate existing inequalities.17 Thus, it is worth recalling that the objective of this article is to identify advantages and disadvantages within the criminal justice system, and that when thinking about actual policies to be implemented, one must, of course, bear in mind many different factors that are traditionally excluded from theoretical analyses of crime. Nevertheless, one can think of groups that can be targeted without generating the types of concerns mentioned above. The juvenile and young adult population, for instance, has a much higher crime rate than the general population. Thus, implementing an age-based targeting scheme can increase the feasibility of using rewards.18 One can make the group even smaller and focus on subgroups among juveniles and young adults, who are statistically more likely to commit crimes. Empirical evidence suggests, for instance, that people who have grown up in a father-absent home have higher crime and incarceration rates (Harper and McLanahan 2004).19 Therefore, targeting rewards to these subpopulations can reduce crime in addition to generating many other social benefits which are excluded from my analysis. These possibilities are fomalized in section 3.1. Subsequently, I question whether targeting offenders based on their ability to pay monetary fines can similarly improve the effectiveness of rewards. This requires analyzing the case where, in addition to imprisonment, the government uses monetary sanctions to deter criminal activity. In this case, when introducing rewards, in addition to reducing sentences as outlined above, the government may increase monetary fines to keep the crime rate of wealthy individuals, who are able to pay larger fines, constant. This leads to two important effects: first, it is unnecessary to allocate any of the cost savings from imprisonment to the wealthy (i.e., nonjudgment proof), since their criminal behavior can be kept constant through relevant adjustments in monetary fines; and, second, the increased monetary fines received by wealthy offenders can be used to increase the amount that is devoted towards the rewards receivable by judgment proof offenders. These two additional considerations, formalized in Section 3.2.1, increase the effectiveness of targeted rewards. However, targeting based on ability to pay may be different in important respects from targeting based on exogenous characteristics, because it can cause moral hazard problems by distorting some individuals’ work incentives. To account for this possibility, I consider an extension in Section 3.2.2 where individuals first decide whether to work and subsequently decide on whether to commit crime. The analysis reveals two important insights. First, it reveals that the work propensity of marginal individuals is (weakly) increasing with their criminal propensity, and this can cause rewards to have very limited distortion effects on work incentives. Second, even when these distortions are sizeable, cost savings from using rewards in the criminal justice system are large enough to eliminate moral hazard problems altogether through hypothetical work incentives under conditions very similar to those I identify in the model without moral hazard considerations. Overall, this article identifies important factors that affect the (in-) feasibility of using rewards as a cost effective crime reduction tool. Whether one can, in fact, use rewards to achieve the goals described above is a matter of current debate. One recent contribution to legal scholarship, for instance, claims that rewards can, in fact, be successfully used to reduce crime in a cost-effective manner (Galle 2021). This view stands in contrast to prior scholarship which is skeptical of rewards based on a focus on the disadvantages of rewards described above.20 This article contributes to this debate by formalizing important considerations which bear on the feasibility of using rewards in a cost-effective manner. It does so by identifying a series of sufficient conditions under which rewards can be put to good use, and by discussing how rewards would need to be used to achieve these sufficient conditions. My analysis represents a step towards understanding and scrutinizing a tool that has been largely ignored in the theoretical economics of crime literature. The models presented in the next two sections propose a starting point with the hope of providing the theoretical foundations for understanding the appropriate role for rewards in the criminal justice system. The limitations of the model, the abstractions it contains, how it relates to the existing literature, and various additional considerations are discussed in Section 4. Section 5 concludes, and an Appendix in the end contains proofs of some of the propositions. 2. Model I consider the standard law enforcement model (see, e.g., Polinsky and Shavell 2007), where a continuum of potential risk-neutral offenders differ from each other with respect to their criminal benefits and opportunity costs of committing crime. This is formalized by assuming that a person who commits crime increases his well-being by an amount of |$b$|⁠, which will henceforth be called his criminal benefit, and which differs from person to person. The cumulative distribution function |$F(b)$| with support |$[0,\infty)$| describes the proportion of individuals with criminal benefits |$b$| that equal or are smaller than |$b$|⁠, with |$f=F^{\prime}$|⁠. To deter the commission of crimes, the government employs an enforcement mechanism which detects the commission of an offense with probability |$p$|⁠, and imprisons detected offenders for a length of time which causes them to suffer disutility |$v(s)$| with |$v^{\prime}>0,$| where |$s$| is the monetary cost of imprisoning the person for the amount of time that causes this disutility. Following Polinsky and Shavell (1984), I assume that the cost of imprisonment per person is proportional to the sentence such that |$s$| also represents the length of the sentence measured in the appropriate unit of time. Absent rewards, which are defined next, potential offenders commit crime if their criminal benefits exceed |$pv(s)$|⁠. In addition to imprisoning people the government may use rewards, which may either be monetary transfers or programs designed to increase potential offenders’ benefits from pursuing legal options. The government chooses the amount to be spent on rewards per recipient, which is denoted |$r$|⁠. The receipt of a reward in the form of a government program generates utility of |$u(r)$| for the recipient. Thus, a person who is eligible for a reward of |$r$| commits crime only if:21 $$\begin{equation} b^{r}(s,r)\equiv p(v(s)+u(r))<b. \end{equation}$$(1) I assume that |$u^{\prime}(r)\geq1$| for all |$r$|⁠, since, in the worst case, the government can offer monetary transfers as rewards instead of administering programs to meet the needs of the recipient whenever the marginal utility from the program falls below one. The government may choose to offer rewards to the entire population (as in Section 2), or it may choose to announce a subset of the population who is eligible to participate in the program (as in Section 3). Sections 2 and 3 are organized to answer questions related to the use of rewards. Section 2.1 analyzes the benchmark case where rewards must be used nonselectively and asks under what circumstance one can introduce rewards to simultaneously reduce crime, imprisonment sentences, and the tax burden generated by the criminal justice system. Section 2.2 conducts welfare analyses, assuming again that rewards are used non-selectively, both when the budget for the criminal justice system is fixed, and when it is a choice variable. Section 3 returns to the question asked in Section 2.1 but considers cases where the rewards can be used in a targeted manner. Throughout the analysis, the arguments of functions are often omitted to abbreviate notation when doing so causes no ambiguity. 2.1 Nontargeted Use of rewards When the government commits to a policy of making the entire population eligible for receiving rewards the crime rate, |$\theta$|⁠, is given by $$\begin{align} \theta(s,r)=1-F(b^{r}(s,r)) \end{align}$$(2) and the monetary cost of financing the criminal justice system is $$\begin{equation} T(s,r)=p\theta s+(1-p\theta)r\text{.} \end{equation}$$(3) This is because out of |$\theta$| criminals |$p\theta$| are caught and imprisoned, whereas |$(1-p)\theta$| evade detection and receive rewards along with the population of size |$1-\theta$| who refrain from committing crime. A regime which makes no use of rewards and employs an imprisonment sentence of |$\overline{s}>0$| leads to a crime rate of |$\theta(\overline{s},0)$| and a tax burden of |$T(\overline{s},0)$|⁠. To investigate whether the imposition of rewards can lead to superior results compared to the regime |$(\overline{s} ,0)$|⁠, one can consider regimes that lead to the same degree of deterrence as |$(\overline{s},0)$|⁠, which consist of a sentence |$s<\overline{s}$| along with a reward |$\overline{r}(s)>0$| characterized by $$\begin{equation} b^{r}(s,\overline{r}(s))=b^{r}(\overline{s},0)\text{ for all }s\in \lbrack0,\overline{s}). \end{equation}$$(4) The definition of |$b^{r}$| in (1) immediately reveals such |$\overline {r}(s)$| exists and that $$\begin{equation} \frac{d\overline{r}}{ds}=-\frac{v^{\prime}(s)}{u^{\prime}(\overline{r}(s))}. \end{equation}$$(5) Here, |$\frac{d\overline{r}}{ds}$| can be interpreted as being analogous to the marginal rate of technical substitution between rewards and imprisonment in the “production” of deterrence. Thus, |$\overline{r}(s)$| can be depicted as an isodeterrence curve with a slope of |$\frac{d\overline{r}}{ds}$| as in Figure 1, above. The same figure depicts a second function, namely $$\begin{equation} r^{T}(s)\equiv\frac{T(\overline{s},0)}{1-p\theta(\overline{s},0)} -\frac{p\theta(\overline{s},0)}{1-p\theta(\overline{s},0)}s \end{equation}$$(6) which represents a hypothetical tax-constraint outlining the feasible set of rewards and sentences that can be financed with the tax budget used in the initial regime |$(\overline{s},0)$|⁠, if, somehow, deterrence could be held constant at the level it is in the regime |$(\overline{s},0)$|⁠. Figure 1 Open in new tabDownload slide Isodeterrence and hypothetical tax-constraint. Figure 1 Open in new tabDownload slide Isodeterrence and hypothetical tax-constraint. Although this tax constraint is hypothetical (since not all |$s,r$| combinations below |$r^{T}$| lead to the same level of deterrence), it is useful to graphically explain the result reported in proposition 1, below, and it also provides the background necessary to develop a similar graphical representation (see Figure 2) in explaining the value of targeted rewards in Section 3, below. An important property of this tax constraint is that any point below |$r^{T}$| which generates a crime rate of |$\theta(\overline{s},0)$| (if there exist any) leads to a lower tax burden than |$T(\overline{s},0)$|⁠, because it uses lower rewards for the same number of nonconvicts (i.e., |$1-p\theta(\overline{s},0)$|⁠), shorter sentences for the same number of convicts (i.e., |$p\theta(\overline{s},0)$|⁠), or both, compared to some regime on |$r^{T}$| which leads to a tax burden of |$T(\overline{s},0)$| when the crime rate is |$\theta(\overline{s},0)$|⁠. Figure 2 Open in new tabDownload slide Expansion of the hypothetical tax-constraint. Figure 2 Open in new tabDownload slide Expansion of the hypothetical tax-constraint. In Figure 1, point |$I$| is depicted in the lower right corner and corresponds to an initial regime where rewards are not used and sentences equal |$\overline{s}$|⁠. Moving along the isodeterrence curve from point |$I$| to point |$A$|⁠, by definition, holds deterrence constant. Thus, point |$A=(s^{\prime },\overline{r}(s^{\prime}))$| generates a lower tax burden than |$T(\overline {s},0)$|⁠, since it leads to the same crime rate as the regime |$(\overline {s},0)$|⁠, and lies below |$r^{T}$|⁠. This implies that slightly increasing rewards above |$\overline{r}(s^{\prime})$| and keeping sentences at |$s^{\prime}$| (i.e., moving from point |$A$| to |$B$| in Figure 1) increases deterrence above the initial level while maintaining some of the tax savings generated by point |$A$|⁠. Thus, in the situation depicted in figure 1,22 one can introduce rewards to reduce sentences from |$\overline{s}$| to |$s^{\prime}$|⁠, while lowering the tax burden, and reducing crime at the same time. Of course, this observation is not specific to a single example but extends to all cases where |$r^{T}$| declines faster than |$\overline{r}(s)$| around the initial point |$(\overline {s},0)$|⁠. The intuition behind this result is explained after it is formalized and proven via proposition 1, below. Proposition 1 One can reduce crime, imprisonment sentences, and the tax burden generated by any regime which relies solely on imprisonment (with a sentence of |$\overline{s}>0$|⁠) by introducing rewards if $$\begin{equation} \frac{u^{\prime}(0)}{1-p\theta(\overline{s},0)}>\frac{v^{\prime}(\overline {s})}{p\theta(\overline{s},0)}. \end{equation}$$(7) Proof The tax burden generated by any regime |$(s,\overline{r}(s))$| is given by |$T(s,\overline{r}(s))$|⁠. Differentiating |$T(s,\overline{r}(s))$| with respect to |$s$| reveals that $$\begin{equation} \frac{\partial T}{\partial s}+\frac{\partial T}{\partial r}\frac{d\overline {r}}{ds}=p\theta-\frac{v^{\prime}(s)}{u^{\prime}(\overline{r}(s))} (1-p\theta)\geq0\text{ iff} \end{equation}$$(8) $$\begin{equation} p\theta\geq\frac{v^{\prime}(s)}{u^{\prime}(\overline{r}(s))+v^{\prime}(s)}. \end{equation}$$(9) The inequality in (9) reveals that whenever (7) holds there exists a regime |$(s^{\prime},r^{\prime})$| with |$s^{\prime}<\overline{s}$| and |$r^{\prime}>0$| such that |$\theta (\overline{s},0)=\theta(s^{\prime},r^{\prime})$| and |$T(s^{\prime},r^{\prime })<T(\overline{s},0)$|⁠. Thus, there exists |$\varepsilon>0$| such that |$T(s^{\prime},r^{\prime}+\varepsilon)<T(\overline{s},0)$| and |$\theta (\overline{s},0)>\theta(s^{\prime},r^{\prime}+\varepsilon)$|⁠. □ The condition identified by proposition 1, and graphically depicted in Figure 1, is relatively intuitive. The right-hand side (hereafter, RHS) of (7) describes the marginal deterrence effect—divided by |$f(b^{r}(\bar{s},0))$| to simplify the expressions—of increasing spending on imprisonment: any increase in the budget allocated towards imprisonment would have to be split equally to increase the punishment of |$p\theta$| convicts, and would cause a marginal deterrence effect of |$v^{\prime}(\overline{s} )f(b^{r}(\bar{s},0))$|⁠. Similarly, the left-hand side (hereafter, LHS) of (7) describes how much deterrence can be enhanced by increasing the reward budget: the increase in the budget would have to be split equally by |$1-p\theta$| people, and would cause a marginal deterrence effect of |$u^{\prime}(0)f(b^{r}(\bar{s},0))$|⁠. An important and simple observation is that, given all other parameters, if the crime rate is sufficiently low the condition in (7) does not hold. Conversely, the inequality in (7) is more likely to hold when the imprisonment rate is high. This is due to two interrelated reasons. First, convicts do not receive rewards, and, thus, a lower population of convicts increases the per person rewards receivable by nonconvicts. Second, the budget savings from reducing sentences is proportional to the imprisonment rate. Thus, a large imprisonment rate implies greater budget savings from reduced sentences which are used to finance rewards. Expanding on this point, one can note that the numerators of the expressions on the two sides of (7) reflect the primary advantage of rewards while the denominators highlight the primary disadvantages of using rewards. Given long imprisonment sentences, the marginal deterrence impact of increasing sentences (i.e., |$v^{\prime}(\overline{s})$|⁠) is likely to be smaller than |$1$|⁠, which is the lower bound for the marginal impact of rewards (i.e., |$u^{\prime}(0)$|⁠). On the other hand, rewards are less cost effective, because they are provided to all non-punished individuals (i.e., |$1-p\theta$| people), whereas punishment is reserved for a much smaller subpopulation (i.e., |$p\theta$| people). Therefore, whether crime and taxes can be reduced by the use of nonselective rewards depends on how ineffective prison is and how large the convicted criminal population is. This observation naturally leads one to question whether rewards can be used in a selective manner to reduce the denominator in the LHS of (7) and thereby make rewards a more effective tool. This question is analyzed in Section 3, after a further advantage of rewards is formalized through welfare analyses. 2.2 Welfare Analysis As noted in the Section 1, rewards operate by enhancing the well-being of people who are not convicted, as opposed to reducing the well-being of convicts. This causes rewards to enjoy an advantage over imprisonment, which can be formalized through a welfare analysis. For this purpose, I consider a conventional welfare function which aggregates the sum of all utilities, but, I point out that the implications of the analysis remain unchanged when one excludes criminals’ utilities from the welfare function in footnotes. As part of this analysis, first, I consider a welfare maximization problem subject to an exogenously determined tax constraint. Subsequently, I consider a similar maximization problem when taxes are chosen to maximize welfare. In both cases, to focus on the trade-off between rewards and imprisonment, I consider an exogenously given probability of detection |$\left( p>0\right) $|⁠, which could be interpreted as the welfare maximizing probability of detection when |$r=0$|⁠. As noted in the literature, the standard Beckerian result (that the maximum sanction along with a low probability of detection is always optimal) does not hold when |$v^{\prime\prime}<0$| (Polinsky and Shavell 1999). Thus, focusing on a fixed |$p$| is a harmless simplification in identifying sufficient conditions for the optimality of using rewards, since the case where |$p$| is endogenously determined can only broaden the conditions under which it is optimal to use rewards. This is because altering the probability of detection to the optimal level (instead of keeping it constant at |$p$|⁠) cannot reduce welfare.23 2.2.1 Welfare maximization subject to an exogenously given tax constraint The sum of all utilities generated by the commission of crimes, the imposition of punishment, and the conveying of rewards is given by: $$\begin{equation} W(s,r)=\int_{b^{r}}^{\infty}\left( b-h\right) f(b)db+(1-p\theta)u(r)-p\theta v(s), \end{equation}$$(10) where |$h$| denotes the expected harm inflicted through the commission of each crime, which is assumed to be high enough that any feasible regime leads to underdeterrence. When the government’s objective is to maximize this function subject to a tax constraint (⁠|$\overline{T}$|⁠), the relevant maximization problem can be formalized as $$\begin{equation} \underset{s,r}{\max}W(s,r)\text{ such that }T(s,r)=p\theta s+(1-p\theta )r\leq\overline{T}. \end{equation}$$(11) An inspection of this problem reveals the following result.24 Proposition 2 (i) Using rewards is optimal under broader conditions than those identified in Proposition 1. (ii) Moreover, the use of rewards is optimal if the imprisonment elasticity of crime (denoted |$\varepsilon_{s}$|⁠) is small enough, and a sufficient condition for the optimality of rewards is met when this elasticity is smaller than the ratio between total imprisonment costs and the total costs of crime including imprisonment costs, i.e., |$\varepsilon_{s} \leq\frac{\theta ps}{\theta\lbrack ps+h]}$|⁠. Proof See Appendix. □ Proposition 2 reveals conditions under which rewards are optimal. In particular it relates the optimality conditions to the sanction elasticity of crime rates: rewards are optimal, if the ratio of total imprisonment costs to total costs of crime, inclusive of imprisonment costs, is greater than the imprisonment elasticity of crime. Although it is hard to get precise estimates for these values, for purposes of a quick comparison, one can consider the estimates provided in prior studies. For example, Chalfin (2015) reviews the literature on estimating the total cost of crime, and reports that in 2012 the total harm from crime in the United States can be estimated around $\$$ 310 billion. For the total cost of imprisonment in the United States, there are two studies which report differing numbers, one from the Bureau of Justice Statistics (henceforth BJS) (BJS, 2012) and a second from the Prison Policy Initiative (henceforth PPI) (Wagner and Rabuy 2017). They report different numbers, among other reasons, because the BJS only includes the cost of operating correctional facilities, whereas the PPI study includes additional expenses, e.g., criminal court expenditures and bail fees. Thus, the BJS reports a cost of around $\$$81 billion (BJS, 2012) whereas the PPI study reports a cost of around $\$$182 billion. Using these numbers yields an estimate for the ratio on the RHS of the inequality expressed in Proposition 2 of about 21|$\%$| and 37|$\%$|⁠, respectively. Both of these numbers are significantly greater than |$0.07$|⁠, |$0.1$|⁠, and |$0.13$|⁠, which are the elasticities reported in Helland and Tabarrok (2007), Iyengar (2008), and Lee and McCrary (2017), respectively. However, both numbers are smaller than the elasticity of |$0.4$| reported in Levitt (1998). Thus, even rewards in the form of cash transfers appear optimal except when one assumes the highest elasticities estimated for criminal responsiveness by Levitt (1998). Of course, one should bear in mind that the above exercises are meant to illustrate the relevant dynamics in the model by using estimates from the literature. They abstract from many considerations, some of which are listed in Section 4. The primary purpose here is to illustrate that when one attempts to maximize welfare as opposed to achieving the triple goals of reducing crime, sentences, and tax burdens, the conditions under which rewards can be used in a socially desirable manner is broadened significantly. It is also worth bearing in mind that the above calculations attempt at finding intuitive sufficient conditions, meaning that if one could assess weaker conditions reported in the proof of Proposition 2 (see, expression (A.3) in the Appendix, below), one would find that rewards are more likely to be optimal than the assessment above would lead one to think. This analysis focuses on whether the use of rewards is socially desirable, given any exogenously determined tax constraint which must be used towards either imprisonment or rewards. Thus, the analysis does not question whether the resources devoted towards the administration of the criminal justice system may be above or below the optimum amount. It is possible, of course, that the government may have devoted above the optimal amount of resources towards fighting crime, and this may cause the imprisonment elasticity of crime to be low. Given this possibility, one may question whether there may ever be room for using rewards, if the optimal amount of resources are devoted towards the criminal justice system. This issue is explored, next. 2.2.2 Welfare maximization with optimal budget allocation The collection of taxes may generate welfare losses due to distortions they cause in the economy. Thus, increasing the size of the budget allocated towards the criminal justice system by a dollar may cause a loss that is greater than a dollar. To incorporate this possibility, when the taxes to be collected are endogenously determined, I consider a social welfare function of the form $$\begin{equation} \widehat{W}(s,r)=\int_{b^{r}}^{\infty}\left( b-h\right) f(b)db+(1-p\theta )u(r)-p\theta v(s)-k[p\theta s+(1-p\theta)r], \end{equation}$$(12) where |$k\geq1$| represents the marginal cost of public funds, and, as previously noted, |$p\theta s+(1-p\theta)r$| is the budget required to administer a system with sentence |$s$| and rewards of |$r$|⁠. Thus, it follows that the impact of imprisonment on welfare is $$\begin{equation} \frac{\partial\widehat{W}}{\partial s}=-\frac{\partial\theta}{\partial s}[h+kp(s-r)]-p\theta v^{\prime}(s)-kp\theta. \end{equation}$$(13) On the other hand, the marginal impact of rewards on welfare is $$\begin{align} \frac{\partial\widehat{W}}{\partial r} & =-\frac{\partial\theta}{\partial r}[h+kp(s-r)]+(1-p\theta)u^{\prime}(r)-k(1-p\theta)\\ & =-\frac{\partial\theta}{\partial s}\frac{u^{\prime}(r)}{v^{\prime} (s)}[h+kp(s-r)]+(1-p\theta)u^{\prime}(r)-k(1-p\theta),\nonumber \end{align}$$(14) where the second line of (14) is obtained by noting |$\frac {\partial\theta}{\partial r}=\frac{\partial\theta}{\partial s}\frac{u^{\prime }(r)}{v^{\prime}(s)}$|⁠. The expressions in (13) and (14) can be used to discuss the comparative advantages and disadvantages of rewards versus imprisonment. The second terms in (13) and (14) reflect the fact that rewards operate by enhancing rather than reducing the well-being of people, which is one of the primary advantages of rewards. On the other hand, increasing the reward receivable by each person who is not convicted by a dollar requires more funding than investing a dollar to prolong the sentence of each convict, as reflected by the last terms in (13) and (14). Finally, an increase in the reward provided to each recipient causes a greater deterrent effect compared to a similar increase in the amount spent towards financing an increase in each convict’s sentence as long as |$v^{\prime}(s)<1$|⁠. This is reflected by the first terms in (13) and (14). These observations reveal that the desirability of using rewards hinges on three key values: the marginal cost of public funding (⁠|$k$|⁠), the comparative deterrence effects of monetary values versus additional time in prison (i.e., |$u^{\prime }(r)$| versus |$v^{\prime}(s)$|⁠), and the incarceration rate (i.e., |$p\theta$|⁠). The next proposition identifies a precise relationship between these key values, which, when met, implies that using rewards is optimal.25 Proposition 3 Suppose that some deterrence is optimal, and that the optimal sentence is finite. (i) Then, it is optimal to use rewards if $$\begin{equation} \frac{1-p\theta}{u^{\prime}(0)}-\frac{p\theta}{v^{\prime}(s^{\ast})}<\frac {1}{k}, \end{equation}$$(15) where |$s^{\ast}$| is the optimal imprisonment sentence. (ii) This implies that using rewards is optimal if the marginal cost of public funding is sufficiently small. Proof (i) Some degree of deterrence requires that either |$r^{\ast}>0$| or |$s^{\ast }>0$|⁠. Thus, |$r^{\ast}=0$| is possible only if |$s^{\ast}>0$| characterized by the first order condition: $$\begin{equation} -\frac{\partial\theta}{\partial s}[h+kps^{\ast}]=p\theta\lbrack k+v^{\prime }(s^{\ast})] \end{equation}$$(16) and $$\begin{equation} \frac{\partial\widehat{W}(s^{\ast},0)}{\partial r}=-\frac{\partial\theta }{\partial s}\frac{u^{\prime}(0)}{v^{\prime}(s^{\ast})}[h+kps^{\ast }]+(1-p\theta)u^{\prime}(0)-k(1-p\theta)\leq0. \end{equation}$$(17) But, plugging (16) into (17) reveals that this impossible if $$\begin{equation} \frac{u^{\prime}(0)}{v^{\prime}(s^{\ast})}[kp\theta+p\theta v^{\prime} (s^{\ast})]+(1-p\theta)u^{\prime}(0)-k(1-p\theta)>0 \end{equation}$$(18) or, equivalently: $$\begin{equation} \frac{1-p\theta}{u^{\prime}(0)}-\frac{p\theta}{v^{\prime}(s^{\ast})}<\frac {1}{k} \end{equation}$$(19) which is the stated condition. (ii) The condition holds whenever |$k=1$| since |$u^{\prime}(0)\geq1$|⁠, and |$\theta(s,r)>0$| for all |$s$|⁠. □ As noted in part (ii) of Proposition 3, an implication of the condition specified is that if securing public funding generates very little distortions in the economy, then using rewards is certainly optimal. Quite importantly, many previous economic analyses of law enforcement implicitly invoke a similar assumption by supposing that the use of fines generates no direct costs or benefits. Thus, under similar assumptions where monetary fines and rewards are both costless transfers, rewards are trivially optimal. When |$k>1$|⁠, the analysis is more complicated. Unfortunately, without specifying precise numbers, it is not possible to pinpoint a concrete value for the marginal cost of public funding which makes rewards optimal. However, for interpretation purposes a conservative upper-bound for |$k$| as a function of the conviction rate can be calculated by using an estimate of |$v^{\prime}$| from Abrams and Rohlfs (2011), which finds that “the typical defendant in our [Philadelphia] sample would be willing to pay roughly $\$$1,000 for 90 d[ays] of freedom.” The cost of imprisonment per year in 2018 was at least |$\$42,000$| in Philadelphia according to the Philadelphia District Attorney’s office.26 These numbers imply a |$v^{\prime}$| which is smaller than |$0.1$|⁠. This number is unlikely to increase much if one adjusted the Abrams and Rohlfs (2011) estimate for inflation while also incorporating the fact that they were looking at the willingness to pay to avoid imprisonment for the first, rather than last, 90 days of incarceration, which would imply a smaller |$v^{\prime}$| given diminishing marginal disutility from imprisonment. Thus, one can calculate |$\overline{k}(p\theta)$|⁠, an upper bound on |$k$| which satisfies (19) using |$0.1$| as a conservative upper bound on |$v^{\prime}(s^{\ast})$| as follows $$\begin{equation} \overline{k}(p\theta)\equiv\frac{1}{1-1.1p\theta}. \end{equation}$$(20) To give an idea of the magnitudes of these upper bounds, Table 1, below, provides calculations of |$\overline{k}(p\theta)$| for different levels of conviction rates. The numbers reported in Table 1 range between around |$1.12$| and |$1.49$| and are higher than the recent estimates provided in Bjertnæs (2018), which range between 1.05 and 1.1. This suggests that using rewards is optimal in these numerical examples. Table 1. Estimates of |$\overline{k}$| using |$v^{\prime }(s^{\ast})<0.1$| |$p\theta=$| . |$1\%$| . |$2\%$| . |$3\%$| . |$\overline{k}(p\theta)=$| |$1.\,123\,6$| |$1.\,282\,1$| |$1.\,492\,5$| |$p\theta=$| . |$1\%$| . |$2\%$| . |$3\%$| . |$\overline{k}(p\theta)=$| |$1.\,123\,6$| |$1.\,282\,1$| |$1.\,492\,5$| Open in new tab Table 1. Estimates of |$\overline{k}$| using |$v^{\prime }(s^{\ast})<0.1$| |$p\theta=$| . |$1\%$| . |$2\%$| . |$3\%$| . |$\overline{k}(p\theta)=$| |$1.\,123\,6$| |$1.\,282\,1$| |$1.\,492\,5$| |$p\theta=$| . |$1\%$| . |$2\%$| . |$3\%$| . |$\overline{k}(p\theta)=$| |$1.\,123\,6$| |$1.\,282\,1$| |$1.\,492\,5$| Open in new tab 3. Targeted Use of Rewards As noted in the discussion following Proposition 1, the primary disadvantage of using nontargeted rewards is that the pool of reward recipients is large compared to the subpopulation being punished. Thus, as highlighted via (7), reducing sentences, taxes, and crime simultaneously is more likely to be possible when the imprisonment rate is high. A simple corollary of this observation is that if rewards can be used selectively to target only high-crime groups, they would be more likely to achieve these three goals simultaneously. However, when imprisonment sentences are restricted to be nondiscriminatory, i.e., equal across high-crime and low-crime groups, targeting high-crime groups with rewards while simultaneously reducing sentences for the entire population causes an increase in the low-crime group’s crime rate. In this case, it becomes necessary to target the group with the higher criminal responsiveness per rewardee, which may or may not coincide with the high-crime group. Another possible way of increasing the effectiveness of rewards is to target them exclusively towards individuals who are judgment-proof, which would allow one to reduce sentences across the entire population while increasing monetary sanctions to keep deterrence constant. The latter type of targeting can, of course, lead to moral hazard problems. This section discusses both cases, starting with the targeting of high-crime groups, first. 3.1 Rewards Targeting High-Crime Groups To discuss the possibility of targeting high-crime groups, I partition the population into two groups |$i\in\{H,L\}$|⁠, and let |$F_{i}$| denote the cumulative criminal benefit distributions conditional on being in group |$i\in\{H,L\}$|⁠, and let |$f_{i}$| denote the corresponding probability density functions. Here, the labels |$H$| and |$L$| may refer to high- and low-crime groups, respectively, when the government intends to use rewards targeting high crime groups. However, as explained below, the government may wish to target groups based on a characteristic other than crime rates, in which case these labels will refer to the ranking (as high and low) of that characteristic. Given this notation the unconditional distribution of criminal benefits is $$\begin{equation} F=\eta_{H}F_{H}+\eta_{L}F_{L}, \end{equation}$$(21) where |$\eta_{i\in\{H,L\}}$| with |$\eta_{H}+\eta_{L}=1\,\ $|represents the relative size of group |$i$|⁠. Thus, the crime rates within each group are $$\begin{equation} \theta_{i}=1-F_{i}(b_{i}^{r})\text{ for }i\in\{H,L\} \end{equation}$$(22) and the crime rate among the overall population is $$\begin{equation} \theta=1-\eta_{H}F_{H}(b_{H}^{r})-\eta_{L}F_{L}(b_{L}^{r}) \end{equation}$$(23) In (23), the subscript for |$b^{r}$| is used to indicate that the criminal benefit thresholds may differ across groups, since the two groups may be targeted with different enforcement schemes. Specifically, I consider two cases: one where both the sentence and the rewards can vary across groups, and a second case where sentences must be equal across the two groups but the government may nevertheless use different rewards across the two groups. 3.1.1 Targeting when sentences can vary across groups When both |$s$| and |$r$| can be based on group membership, the government can simply choose to target a group with a high crime rate to use rewards more effectively. This would not be possible when sentences are required to be non-discriminatory across groups. However, there are certain categories of offenders who can, in fact, be punished or treated differently in the criminal justice system based on their group membership. In the United States, the best example is presumably juveniles who can be subjected to very different treatments than adults. Thus, suppose the government partitions the population into two groups such that |$\theta_{H}>\theta_{L}$| in the initial punishment scheme which does not utilize rewards and punishes group |$H$| with a sentence of |$\bar{s}$|⁠. This sentence may, but need not, differ from the initial sentence applicable to group |$L$|⁠. In this case, proposition 1 can be applied simply by replacing the entire population with the high crime group, which reveals the following. Corollary 1 One can reduce crime, imprisonment sentences, and the tax burden generated by any regime that relies solely on imprisonment (involving a sentence |$\bar {s}>0$|⁠) by introducing rewards if $$\begin{equation} \frac{u^{\prime}(0)}{1-p\theta_{H}(\bar{s},0)}>\frac{v^{\prime}(\bar{s} )}{p\theta_{H}(\bar{s},0)}. \end{equation}$$(24) The primary difference between (24) and its nontargeted analog in (7) is that it involves the crime rate of the high criminal propensity group as opposed to the crime rate for the whole population. As noted previously, the condition in Proposition 1 (and therefore Corollary 1) is more likely to hold when the crime rate is higher. Thus, using rewards in a socially desirable manner is more likely to be feasible when the government can use different sentences as well as different rewards across the two groups. 3.1.2 Targeting with non-discriminatory sentences When the government can use targeted rewards but is constrained to impose the same sentences across groups, it can no longer leave the policy applicable to the low crime group unchanged while targeting the high crime group. Thus, to target one group with rewards, it will have to reduce the sentences applicable to the entire population. This type of policy will naturally have the effect of reducing the deterrence of the untargeted group. However, this reduction in deterrence can potentially be off-set by an increase in the deterrence of individuals in the targeted group, if the rewards directed towards them more than off-set the impact of reduced sentences. This result is more likely to be observed when the targeted group has a large ratio of marginal offenders to non-convicts (i.e., |$\frac{f_{i}}{1-p\theta_{i}}$|⁠), since this ratio is proportional to the per-dollar deterrence effect of rewards (i.e., |$\frac{f_{i}u^{\prime}(r)}{1-p\theta_{i}}$|⁠). In particular, when the per-dollar deterrence effect of rewards within the targeted group is greater than the per-dollar deterrence effect of imprisonment within the entire population, the targeting scheme described above can be successfully implemented. This is noted via the following corollary, whose proof is relegated to the Appendix since it follows steps very similar to those used in proving Proposition 1. Corollary 2 (i) One can reduce crime, imprisonment sentences, and the tax burden generated by any regime that relies solely on imprisonment (involving a sentence |$\bar{s}>0$|⁠) by using rewards that target group |$i$| if $$\begin{equation} u^{\prime}(0)\frac{f_{i}(b^{r})}{1-p\theta_{i}(\bar{s},0)}>v^{\prime}(\bar {s})\frac{f(b^{r})}{p\theta(\bar{s},0)} \end{equation}$$(25) (ii) Let |$H$| and |$L$| be selected such that |$\frac{f_{H}(b^{r})}{1-p\theta _{H}}>\frac{f_{L}(b^{r})}{1-p\theta_{L}}$|⁠, then when |$i=H$| (25) represents a weaker condition than (7). Proof See Appendix. □ A few points related to corollary 2 are worth highlighting. First, the condition identified for successfully using targeted rewards is independent of the relative size of the group being targeted. This is because, given any population and initial enforcement scheme, the smaller the size of the group being targeted, the smaller are the rewards needed to incentivize the group as a whole, and therefore the smaller are the sentence reductions required to finance those incentives. Second, unlike previously identified targeting mechanisms, it is not necessarily the case that the high-crime group ought to be targeted. Instead, the group with the higher marginal offender to nonconvict ratio ought to be targeted. This group naturally corresponds to the one with the higher crime rate whenever that group is at least as responsive to sanctions as the low crime group. This case is consistent with assumptions made in the theoretical literature (e.g., Bjerk 2007; Mungan 2018) as well as empirical findings regarding some high-crime groups (e.g., Levitt 1998). Finally, targeting groups with higher than average marginal offender to non-convict ratios always expands the conditions under which rewards can be successfully used to reduce crime, sentences, and taxes simultaneously. 3.2 Rewards Targeting Judgment Proof Offenders The optimal enforcement regime where people can be punished through imprisonment as well as monetary fines, but without rewards, is analyzed in Garoupa and Mungan (2019). This framework is discussed in further detail in Section 4, below, and results in a punishment scheme where some offenders are judgment proof and some are not. To incorporate the possibility of using rewards to target judgment proof offenders, I consider an initial punishment scheme where offenders are punished not only through imprisonment sentences (of |$\overline{s}$|⁠), but also monetary fines, whose magnitude is given by |$\underline{m}$|⁠. Following Polinsky and Shavell (1984), I assume that there are two groups with low and high wealth: |$\lambda$| proportion of individuals are judgment proof, i.e., they possess disposable wealth |$w<\underline{m}$|⁠, and the wealth of the remaining population (of proportion |$1-\lambda$|⁠) exceeds |$\underline{m}$|⁠, and, thus, these individuals are capable of paying higher fines. In the initial punishment scheme, there are no rewards, and, thus, judgment proof offenders commit crimes if $$\begin{equation} b>p(v(\overline{s})+w), \end{equation}$$(26) whereas wealthy individuals commit crimes if $$\begin{equation} b>p(v(\overline{s})+\underline{m}) \end{equation}$$(27) The question to be answered is whether the government can reduce crime, sentences and the tax burden of the criminal justice system, jointly, by providing targeted rewards to judgment proof offenders alone. First, I answer this question assuming that the proportion of judgment proof individuals is exogenous, i.e., unaffected by the enforcement scheme employed by the government. Subsequently, I consider the case where the enforcement scheme employed by the government may cause moral hazard problems by distorting the incentives of individuals to engage in productive activities to increase their wealth. 3.2.1 Exogenously determined groups In this subsection, I assume that individuals are either judgment proof or not, and this is unaffected by the enforcement scheme in place. Since, nonjudgment proof offenders are capable of paying additional fines, the government can keep their level of deterrence constant by simultaneously increasing the fine above |$\underline{m}$| and reducing the sentence below |$\overline{s}$|⁠. In particular, one can easily define the monetary fine that achieves this, as follows: $$\begin{equation} \widehat{m}(s)\equiv\underline{m}+(v(\overline{s})-v(s)). \end{equation}$$(28) On the other hand, the rewards that must be used in order to keep the deterrence of judgment proof offenders constant is given by |$\overline{r}(s)$|⁠, whose property is described via (5), and is depicted in Figure 2, above, as an isodeterrence curve. In calculating the tax burden, one thing that needs to be noted is that the collection of fines reduces the tax burden by allowing the criminal justice system to partially self-finance. Bearing this in mind, the tax burden can be calculated as follows $$\begin{equation} T=\lambda\lbrack p\theta_{l}(s-w)+(1-p\theta_{l})r]+(1-\lambda)p\theta _{h}(s-m), \end{equation}$$(29) where |$\theta_{l}$| and |$\theta_{h}$| refer to the crime rates among judgment proof people and wealthy people, respectively.27 Following an approach similar to that in Section 2.1, one can conceive of a hypothetical tax constraint, which corresponds to the rewards |$\widehat{r} ^{T}(s)$| that could be directed towards judgment-proof offenders by using the same tax budget as in regime |$(\overline{s},0,\underline{m})$|⁠, by altering |$s$| while simultaneously adjusting fines according to |$\widehat{m}(s)$|⁠, if deterrence could, somehow, be kept constant. Using (29) reveals that |$\widehat{r}^{T}(s)$| and its derivative are given by $$\begin{equation} \begin{array} &\widehat{r}^{T}(s)\equiv \frac{T(\overline{s},0,\underline{m})} {\lambda(1-p\overline{\theta}_{l})}& - \frac{p\overline{\theta}_{l} }{1-p\overline{\theta}_{l}}(s-w) & - \frac{(1-\lambda)p\overline{\theta }_{h}}{\lambda(1-p\overline{\theta}_{l})}& + \frac{(1-\lambda )p\overline{\theta}_{h}}{\lambda(1-p\overline{\theta}_{l})}\widehat{m}(s)\\ & & & & & & & \\ {\textit{Savings from:}} & \textit{(I) sentence } & \textit{(II) sentence} & \textit{(III) extra fines}\\ & \textit{reductions } & \textit{reductions} & \textit{collected}\\ & \textit{for JP} & \textit{for non-JP} & \textit{from non-JP}\\ & & & & & & \\ \frac{d\widehat{r}^{T}(s)}{ds}= & - \overbrace{\frac{p\overline {\theta}_{l}}{1-p\overline{\theta}_{l}}} & - \overbrace{\frac{1-\lambda }{\lambda}\frac{p\overline{\theta}_{h}}{1-p\overline{\theta}_{l}}} & - \overbrace{\frac{1-\lambda}{\lambda}\frac{p\overline{\theta}_{h} }{1-p\overline{\theta}_{l}}v^{\prime}(s)} \end{array}, \end{equation}$$(30) where |$\overline{\theta}_{i\in\{h,l\}}$| are used to refer to the crime levels in the initial regime to abbreviate the expression. The derivative of |$\widehat{r}^{T}(s)$| is split into components (marked (I), (II), and (III)) to explain how rewards that target the judgment proof population expands the conditions under which they can be used to reduce crime, sentences, and taxes, relative to the condition identified in (9). These different components are illustrated in Figure 2, below, and are subsequently explained in further detail. First, because the deterrence of nonjudgment proof offenders are kept constant by increasing fines, the sole objective of rewards is to keep the deterrence of judgment proof offenders constant. Thus, the relevant crime rate in component (I) is the crime rate among judgment proof offenders, i.e., |$\theta_{l}$|⁠, which is empirically higher than the overall crime rate given by |$\theta=\lambda\theta_{l}+(1-\lambda)\theta_{h}$|⁠, which I assume throughout the analysis. Thus, the tax savings from a sentence reduction for judgment proof offenders (as captured by component (I)) is greater than its analog in (9). Second, the nonjudgment proof population’s sentences are reduced, and, yet, their deterrence is held constant through an increase in fines. Thus, the tax savings obtained from their sentence reductions are not directed towards financing rewards for their own group but are directed towards the financing of rewards that go to the judgment proof population. This generates additional savings which were not previously available, and this is reflected via (II). Finally, the additional fines collected from nonjudgment proof offenders to keep their deterrence constant are also directed towards the financing of rewards for the judgment proof population, and this is reflected through component (III). As previously noted, any point below |$\widehat{r}^{T}(s)$| (if there are any) that leads to the same level of crime for judgment proof offenders would result in tax savings. Thus, the outward expansion of |$\widehat{r}^{T}(s)$| relative to its analog in (6) corresponds to a broadening of the conditions under which rewards can be used to reduce crime, taxes, and sentences. This result is formalized in Proposition 4, below. Proposition 4 (i) When the government can exclusively target judgment proof offenders with rewards, it can reduce crime, imprisonment sentences, and tax burdens generated by a regime which relies on imprisonment (with a sentence of |$\overline{s}>0$|⁠) and fines (of size |$\underline{m}$|⁠) by introducing rewards, if $$\begin{equation} \frac{u^{\prime}(0)}{1-p\theta_{l}(\overline{s},0,\underline{m})} +\frac{1-\lambda}{\lambda}\frac{\theta_{h}(\overline{s},0,\underline{m} )}{\theta_{l}(\overline{s},0,\underline{m})}\frac{u^{\prime}(0)[1+v^{\prime }(\overline{s})]}{1-p\theta_{l}(\overline{s},0,\underline{m})}>\frac {v^{\prime}(\overline{s})}{p\theta_{l}(\overline{s},0,\underline{m})}\text{;} \end{equation}$$(31) and, (ii) this condition holds when the proportion of judgment proof offenders is sufficiently small. Proof See Appendix. □ The condition reported in Proposition 4 is obviously a weaker one compared to that in Proposition 1, due to reasons explained via components (I)–(III) in (30). Specifically, the second term in the LHS of (31) is positive and is not present in (7) (due to the emergence of components (II) and (III)), and the crime rate that appears in the denominator of the remaining terms is |$\theta_{l}$| rather than |$\theta$|⁠, which also makes the condition weaker (due to the emergence of component (I)). The extent to which the condition is broadened depends inversely to the proportion of judgment proof individuals, since more people benefiting from the reward system reduces the per-person impact of these rewards. The implication of this is stated in part (ii) of Proposition 4: when the proportion of judgment proof offenders is sufficiently small, rewards can certainly be used to jointly reduce crime, sentences and the tax burden. It is worth briefly noting that the objective here is to identify the simple dynamics which lead to benefits from targeted rewards. The implementation of the simple reward program presented may, of course, generate moral hazard problems by reducing the gains from not being judgment proof. The next subsection formalizes this possibility by introducing a first period in which potential offenders choose whether or not to work to increase their disposable wealth. 3.2.2 Endogenously determined groups To incorporate possible moral hazard problems, I consider a simple two period extension. In the first period people decide whether or not to engage in efforts to increase their wealth by an amount |$a>0$|⁠, and decide whether to engage in criminal acts in the last period. I call productive acts that increase one’s wealth by |$a$| “working” to abbreviate descriptions. As in Section 3.2.1, I consider two groups (⁠|$l$| and |$h$|⁠) with differing wealth with |$w_{h}>$||$\underline{m}+a$| and |$w_{l}<\underline{m}$|⁠. A person in the latter group who works increases his wealth above the monetary sanction in place, i.e., |$a>\underline{m}-w_{l}$|⁠, in which case he is no longer judgment proof, which is what gives rise to moral hazard problems. Working imposes costs of |$c$| which differs across individuals. This change is easily captured by letting |$f_{i\in\{l,h\}}(b,c)$| denote the joint distribution of |$b$| and |$c$| rather than the distribution of only |$b$| as in the previous sections. As in the previous analysis, the initial punishment scheme consists of the monetary fine |$\underline{m}$| and sentence |$\bar{s}$|⁠. The question to be answered is whether introducing rewards that target judgment proof offenders while slightly adjusting the punishment scheme can lead to an improvement in outcomes. These types of rewards can naturally reduce low wealth individuals’ returns from work, since by working a person can lose his eligibility to receive rewards. To focus on this problem, I restrict attention to schemes which involve monetary fines |$m\in\lbrack\underline{m},w_{l}+a)$|⁠, such that this disincentivizing effect is in fact present. Moral hazard problems can be formalized by ascertaining individuals’ behavior through backward induction. Specifically, in the last period, a judgment-proof person with wealth |$w_{l}$| commits crime if $$\begin{equation} b^{J}(s,r)\equiv p(v(s)+u(r)+w_{l})<b \end{equation}$$(32) since he is entitled to receive rewards of |$u(r)$| if not convicted. On the other hand, an individual who has wealth |$w>m$| commits crime if $$\begin{equation} b^{N}(s,m)\equiv p(v(s)+m)<b. \end{equation}$$(33) Thus, $$\begin{equation} b^{N}(\bar{s},\underline{m})>b^{J}(\bar{s},0), \end{equation}$$(34) and this ranking is preserved for small deviations away from the initial enforcement scheme (i.e., |$u(r)<m-w_{l}$|⁠), which is what the analysis below focuses on. Given these best responses, a person’s expected pay-off from working in the first period can be expressed as: $$\begin{equation} \Pi_{i}^{1}\equiv \begin{array} [c]{lll} w_{i}+a-c & {\rm if} & b\leq b^{N}(s,m)\\ w_{i}+a-c+b-p(m+v(s)) & {\rm if} & b>b^{N}(s,m) \end{array} \ \ \text{ for }i\in\{l,h\}. \end{equation}$$(35) On the other hand, the expected pay-off associated with not working for a person in group |$i\in\{h,l\}$| is $$\begin{align} \Pi_{l}^{0} & \equiv \begin{array} [c]{lll} w_{l}+u(r) & {\rm if} & b\leq b^{J}(s,r)\\ (1-p)(w_{l}+u(r))+b-pv(s) & {\rm if} & b>b^{J}(s,r) \end{array} \ \ \text{; and} \end{align}$$(36) $$\begin{align} \Pi_{h}^{0} & \equiv \begin{array} [c]{lll} w_{h} & {\rm if} & b\leq b^{N}(s,m)\\ w_{h}+b-p(m+v(s)) & {\rm if} & b>b^{N}(s,m) \end{array}. \end{align}$$(37) Thus, a person in group |$h$| chooses to work if |$c<a$|⁠,28 since |$\Pi_{h}^{1}-\Pi_{h}^{0}=a-c$| for all |$b$|⁠. On the other hand, a person in group |$l$| chooses to work if |$\Pi_{l}^{1}>\Pi_{l}^{0}$|⁠, which can be written more explicitly as $$\begin{equation} \begin{array} [c]{lll} c^{L}(r)\equiv a-u(r)>c & {\rm if} & b\leq b^{J}\\ c^{I}(b,s,r)\equiv a-b+p(w_{l}+v(s))-(1-p)u(r)>c & {\rm if} & b\in(b^{J},b^{N} ]\\ c^{H}(r,m)\equiv a-(1-p)u(r)-p(m-w_{l})>c & {\rm if} & b>b^{N} \end{array} \ \ , \end{equation}$$(38) where the superscripts on these thresholds refer to the low, intermediate, and high criminal benefits, in which they are obtained, respectively. Inspecting (32) immediately reveals that rewards for judgment proof offenders increases the critical criminal benefit for these individuals and this contributes to reducing crime. However, this comes at the cost of reducing the work incentives of individuals in group |$l$|⁠, which is apparent from (38). Thus, the introduction of rewards can give rise to a trade-off between deterrence and reduced work incentives. Prior to seeking ways to compare these two effects, it is worth making some observations regarding the possible magnitude of moral hazard problems. First, note that when individuals receive their marginal productivity from work, the work incentive distortions of people who are not interested in crime (i.e., people with |$b<b^{J}$|⁠) have negligible welfare impacts. This is because, as noted via (38), absent rewards, these individuals’ returns from work equal their working costs, which implies that their working decision has no impact on welfare.29 Therefore, any negative welfare consequences that arise due to work incentive distortions caused by the introduction of rewards are limited to those generated by the changed incentives of individuals with greater returns from crime (i.e., with benefits |$b\geq b^{J}$|⁠). This limits the potential negative welfare impacts that might be borne from moral hazard problems. Second, and relatedly, as (38) reveals, moral hazard problems caused by rewards come about by changing the incentives of individuals who are on the margin with respect to their working decisions. Thus, the importance of moral hazard problems are positively related to the size of the population who are on these margins. Quite importantly, work incentives (i.e., |$c$|⁠) and criminal benefits (⁠|$b$|⁠) within the population can be correlated in a way that causes few people to be on the work–incentive margins listed in (38). This would be the case, for instance, when people who are least inclined to work (i.e., people with large |$c$|⁠) also have the largest benefits from crime (⁠|$b$|⁠) and vice versa such that |$b$| and |$c$| are positively correlated within the population. One could then expect few individuals to be marginal with respect to their work efforts, since the work-incentive margins listed in (38) are weakly decreasing in |$b$| (i.e., |$f(b,c)$| is small for |$b\geq b^{J}(\bar{s},0)$| and |$c\leq a$|⁠). These observations may perhaps help in explaining why moral hazard effects in this context have not been interpreted as representing a significant obstacle (Galle 2021). Nevertheless, the second observation above relies on empirical facts which are hard to quantify and describe. Thus, it can be useful to identify more conservative, but easier to interpret, conditions under which moral hazard problems, even if sizeable, might be small compared to deterrence benefits that could be obtained from introducing rewards. Unfortunately, this trade-off, too, hinges on similar assessments of how many people are on work-margins versus criminal activity margins, in addition to the relative costs of moral hazard problems versus benefits from reduced crime. The former comparison involves difficult empirical assessments, and the latter may necessitate making debatable value judgments. These problems can be avoided by noting that the changes in enforcement schemes of interest lead to reductions in the tax burden. Thus, if this reduction in the tax burden can be used to off-set the reduced work incentives problem, one can conclude that the gains from introducing rewards more than off-set their costs. This is because the additional proceeds from introducing rewards can be (although need not be) used to fix incentive distortions that they cause. To identify the conditions under which rewards can indeed be introduced in this manner, I consider an adjusted mechanism which distributes part of the savings to workers with wealth |$w_{l}+a$| in the amount of |$n$| per-worker.30 The remaining components of the mechanism are as in the previous subsection. Using this mechanism, it can be verified that a departure away from an initial scheme (where |$s=\bar{s}$|⁠, |$m=\underline{m}$|⁠, |$n=0,$| and |$r=0$|⁠) to a new scheme with a sentence |$s$| below |$\bar{s}$|⁠, and $$\begin{align} \hat{r}(s) & =u^{-1}(v(\bar{s})-v(s))\\ \hat{m}(s) & =\underline{m}+u(\hat{r}(s))\text{, and}\nonumber\\ \hat{n}(s) & =u(\hat{r}(s))\nonumber \end{align}$$(39) leaves the incentives of all types completely unchanged. This can be verified by adjusting and evaluating the critical benefits and costs listed in (32), (33), and (38) under a scheme that satisfies the conditions in (39). The important question, of course, is how the move from the initial scheme to the new scheme described by (39) affects taxes. If the move increases taxes, then it follows that the moral hazard problems created by the targeting of judgment proof offenders cannot be eliminated by the proposed mechanism without increasing the cost of public funding. However, if the taxes necessary are reduced by such moves, then it follows that any moral hazard problems caused by the introduction of rewards are small enough to be off-set by the tax savings obtained from the introduction of rewards. Which outcome is obtained can be ascertained by writing the taxes necessary to implement this mechanism as follows $$\begin{align} T(s) & = \lambda\big\{ (1-\omega)[(1-p\theta_{l}^{0})\hat{r}(s)+p\theta_{l}^{0}(s-w_{l})]+\omega\lbrack\hat{n}(s) \nonumber\\ & \quad{} +p\theta_{l}^{1}(s-\hat{m}(s))]\big\} +(1-\lambda)p\theta_{h}(s-\hat{m}(s)), \end{align}$$(40) where |$\omega$| is the proportion of individuals in group |$l$| who choose to work, and |$\theta_{l}^{0}$| and |$\theta_{l}^{1}$| are the crime rates among people who work and who do not work in group |$l$|⁠, which can of course differ from each other. Since, all schemes described by (39) hold the work as well as crime incentives of all individuals constant, these values are the same as they are in the initial enforcement scheme. Differentiating |$T$| with respect to sentences reveals that one can introduce rewards and simultaneously eliminate moral hazard problems through payments to workers while still reducing the tax burden, if $$\begin{equation} \begin{array} [c]{ll} \frac{dT(\bar{s})}{ds}= & \lambda\left\{ (1-\omega)[p\theta_{l} ^{0}-(1-p\theta_{l}^{0})\frac{v^{\prime}(\bar{s})}{u^{\prime}(0)} ]+\omega\lbrack p\theta_{l}^{1}-(1-p\theta_{l}^{1})v^{\prime}(\bar {s})]\right\} \\ & +(1-\lambda)p\theta_{h}(1+v^{\prime}(\bar{s}))>0. \end{array} \ \ \ \ \end{equation}$$(41) The next proposition summarizes the implications of this observation by re-expressing (41) to allow a better comparison with the analogous condition obtained when moral hazard problems are not incorporated. Proposition 5 (i) When the government can exclusively target judgment proof offenders with rewards, it can reduce crime, imprisonment sentences, and tax burdens generated by a regime which relies on imprisonment and fines without reducing the work incentives of individuals if $$\begin{align} &\frac{u^{\prime}(0)}{1-p\theta_{l}^{0}} \left[1-\omega+\omega\frac{\theta _{l}^{1}}{\theta_{l}^{0}}\right]\nonumber\\ &\quad{} +\frac{1-\lambda}{\lambda}\frac{\theta_{h} }{\theta_{l}^{0}}\frac{u^{\prime}(0)(1+v^{\prime}(\bar{s}))}{1-p\theta_{l} ^{0}}>\frac{v^{\prime}(\bar{s})}{p\theta_{l}^{0}}\left[ 1-\omega+\omega u^{\prime}(0)\frac{1-p\theta_{l}^{1}}{1-p\theta_{l}^{0}}\right], \end{align}$$(42) and (ii) this condition holds when the proportion of judgment proof offenders is sufficiently small. Proof The discussion preceding the proposition explains how one can obtain the same crime rate and work incentives by employing the scheme described via (39). When (42) holds, employing this scheme through a slight reduction in |$s$| below |$\bar{s}$| leads to tax savings. These savings can be used to slightly adjust the scheme to reduce crime via |$r$| and |$n$|⁠. □ A comparison between (31) and (42) reveals that the two conditions with and without moral hazard problems in the targeting of judgment proof offenders is quite similar. In fact, when the difference in the crime rates among workers and nonworkers within group |$l$| are negligible, and they both equal |$\theta_{l}$|⁠, the condition becomes $$\begin{equation} \frac{u^{\prime}(0)}{1-p\theta_{l}}+\frac{1-\lambda}{\lambda}\frac{\theta_{h} }{\theta_{l}}\frac{u^{\prime}(0)(1+v^{\prime}(\bar{s}))}{1-p\theta_{l}} >\frac{v^{\prime}(\bar{s})}{p\theta_{l}}\left[ 1-\omega+\omega u^{\prime }(0)\right.]. \end{equation}$$(43) This condition differs from (31) only in that the term multiplying |$\frac{v^{\prime}(\bar{s})}{p\theta_{l}}$| is a convex combination of |$1$| and |$u^{\prime}(0)$| rather than simply |$1$|⁠. Thus, the RHS is greater than its analog in (31) only when |$u^{\prime}(0)>1$|⁠, which of course, makes even untargeted uses of rewards more likely to succeed. The rationale behind this result can be ascertained by inspecting the first line of (41). The effect on the tax revenue from introducing rewards according to (39) is very similar across workers and nonworkers in group |$l$|⁠. First, they cause a reduction in the imprisonment financing required within both groups (proportional to their respective crime rates of |$p\theta_{l}^{i\in\{0,1\}}$|⁠). Second, they cause an increase in the rewards receivable by nonworkers (proportional to |$(1-p\theta_{l}^{0} )\frac{v^{\prime}(\bar{s})}{u^{\prime}(0)}$|⁠). Third, they cause an increase in the amount of worker compensation that is made out to workers (proportional to |$v^{\prime}(\bar{s})$|⁠). Fourth, the burden of these worker compensations are partially off-set by the increased fines that workers are able to pay (proportional to |$p\theta_{l}^{1}v^{\prime}(\bar{s})$|⁠). Thus, the impacts on the tax burden caused through workers and nonworkers in group |$l$| are quite similar, and they differ only as explained, above. 4. Discussion The models presented here follow the tradition in the theoretical economics of criminal behavior literature by focusing on the deterrent effects of sanctions. It questions how sanction schemes must be designed to balance deterrence benefits against other costs, such as imprisonment costs and disutilities to offenders. In this regard, it can be interpreted as extending the optimal deterrence literature (see e.g., Polinsky and Shavell (2007) for a review of basic models) which includes articles that incorporate imprisonment costs (Polinsky and Shavell 1984; Shavell 1985; Polinsky 2006). Analyzing the simple framework provided here highlights some of the disadvantages and advantages of rewards compared to imprisonment. The primary disadvantage of rewards is that, unless they can be used in a targeted manner, they must be distributed to a very large population. This property of rewards reduces its deterrent effect, given that the administration of the criminal justice system must rely on a limited budget. On the other hand, one of its comparative advantages stems from the diminished deterrent effect of imprisonment which makes the marginal deterrent effect of a dollar spent on a person via rewards stronger than the marginal deterrent effect of a dollar spent on imprisoning a person for a longer time. A second advantage of rewards is that they operate by increasing people’s utilities through, at worst, wealth transfers, and at best, wealth creation, whereas imprisonment is a tool that operates through wealth destruction. The analysis reveals that the benefits of using rewards are likely to dominate their costs, if imprisonment sentences are very long to begin with. Most of my analysis focuses on a comparison of the benefits and costs of rewards versus imprisonment. However, as Section 3.2 demonstrates, it is possible to extend the analysis to a framework where monetary fines can be used as a third tool. The analysis in that section focuses on an initial regime where some offenders are judgment proof, whereas others can afford to pay greater monetary fines. A natural question to ask is why the government would not increase the fine further, prior to resorting to imprisonment. One answer is that the government must choose a probability of detection that is applicable to all individuals, and therefore fines that optimally deter wealthy individuals, by definition, underdeter judgment proof individuals. Thus, it becomes optimal to impose some degree of imprisonment to reduce the underdeterrence of people who are judgment proof. As noted in Polinsky (2006), this problem can be avoided, if the government can either implement wealth-based sentences, or if it can offer people a choice between two sanction schemes, such that the wealthy choose to pay high fines and a shorter imprisonment term. As noted in Garoupa and Mungan (2019), this solution can become impracticable when it is politically or constitutionally impermissible to have wealth-dependent imprisonment sentences. Under those circumstances, we show, the optimal scheme involves one that is similar to the initial scheme considered in section 3.2, where some individuals are judgment proof and some are not. If, instead, the initial scheme were—contrary to what is observed in reality—one where all individuals are subjected to fines which equal their wealth, the targeted sanctions considered in Section 3.2 could not be used, and one would have to revert to using the solutions discussed in Sections 2 and 3.1. Similarly, most of the analysis focuses on the case where the probability of detection, |$p$|⁠, is fixed. That this assumption is harmless in identifying sufficient conditions is explained in note 23, above, and the accompanying text. However, the incorporation of rewards has an important implication regarding the relative value of the certainty versus the severity of punishment, a topic discussed extensively in the criminology and economics literatures. As explained in this article, the value of rewards is greater when the detected crime rate (i.e. |$p\theta$|⁠) is greater, because this increases the deterrent effect of the budget devoted towards rewards by reducing the number of reward recipients. Thus, unlike the severity of punishment, the certainty of punishment serves an additional function of enhancing the benefits from rewards. Therefore, the presence of rewards is likely to add to the list of reasons as to why the certainty of punishment is more effective than the severity of punishment, a point that is also made in Dari-Mattiacci and Raskolnikov (2020). (For further analysis of this issue, see Polinsky and Shavell (1999) and Mungan and Klick (2014).) It is worth noting that the analysis abstracts from several important issues to focus on the properties of rewards listed above. A few of these issues are discussed here to highlight avenues for future research. Most importantly, the analysis does not incorporate the incapacitative effects of imprisonment, whose analysis may reveal important insights regarding rewards. Previous analyses of incapacitation (Shavell (1987); Miceli (2010); Miceli (2012); Shavell (2015)) reveal the difficulty of incorporating both deterrence and incapacitation in a unified model of criminal decision making. Nevertheless, an intuitive conjecture is that the presence of incapacitative benefits from imprisonment may reduce the attractiveness of substituting some imprisonment time with rewards, since this would remove part of the incapcitative gains. Whether this conjecture is likely to hold is complicated by two important factors. First, a well-known observation in the criminology literature is that the prevalence of offending is related to age in an inverse-U shaped manner.31 This would suggest that average marginal incapacitative benefits are decreasing with the imprisonment sentence and can be quite small given long sentences. Second, and perhaps more importantly, in addition to providing incapacitative benefits, incarceration may give rise to criminogenic effects (Mueller-Smith 2015), i.e., contribute to higher recidivism rates.32 This consideration cuts against the increased incapacitative benefits of imprisonment, and could imply that the marginal benefits from using rewards and shortening imprisonment sentences are, in fact, greater than those presented in this article. Thus, it is not clear, a priori, how the incorporation of incapacitative and criminogenic effects of imprisonment may impact the desirability of rewards compared to punishment, which suggests the need for additional research on this issue. There are also some obvious considerations, which presumably increase the attractiveness of rewards, that were not formally incorporated into the analysis, because this would require making ad hoc assumptions regarding the relative importance of these considerations. Perhaps most importantly, when a person is convicted, there are costs borne by relatives and other close ones of the convict, which were not included in the analysis. The existence of these costs obviously tilt the analysis in favor of rewards. Similarly, the analysis ignores possible benefits or costs to third parties arising from deviations from what the public may deem to be the fair punishment for a crime (these considerations can be incorporated in framework similar to that in Polinsky and Shavell 2000). If, as most of the commentary seems to suggest, current sentences are perceived as being longer than what the public perceives to be the fair punishment, rewards will enjoy an additional benefit over imprisonment. The analysis also does not focus on the possibility of rewards being used in a way to permanently alter recipients’ opportunity costs from committing crime. Programs designed to broaden the skill set of individuals who are at risk of committing crime can have long lasting effects by increasing the legal earning potential of recipients. These can be incorporated into the current model by assuming that |$u^{\prime}$| is a number that is greater than |$1$|⁠, or, in a richer setting with dynamic effects, they can be incorporated by assuming that such programs reduce the recipient’s future |$b$|⁠, i.e., his relative benefits from crime. Another consideration which has predictable implications relates to people’s risk attitudes. As discussed in the literature (e.g., Block and Lind 1975; Polinsky and Shavell 1999; Mungan and Klick 2014; Mungan and Klick 2016), people may exhibit risk-seeking attitudes with respect to prison sentences and yet exhibit risk-averse preferences with respect to monetary outcomes. If true, this would again cut in favor of using rewards more often. This is because rewards increase the monetary and certain benefit to be had by refraining from crime whereas punishment only probabilistically increases the sentence over which people have risk-seeking tendencies. The analysis also does not incorporate the possibility of repeat offenses, and thus abstracts from issues related to recidivism. One question that becomes particularly important is whether a potential reward recipient loses his eligibility to receive rewards upon being convicted. In standard models analyzing the punishment of repeat offenders, the impact of rewards considered herein is similar to the impact of rewards which are not conditional on having a clean criminal record.33 This is because an unconditional reward system of this type affects the incentives of first time as well as repeat offenders in a similar fashion, and thus has effects that are similar to those explained herein. However, if the reward eligibility were made conditional on having a clean record, a familiar trade-off between general and specific deterrence would arise (Mungan, 2017b). Rewards would increase the incentives of people with clean records to comply with the laws (as explained in this article). However, ex-convicts would not be eligible for rewards, and if faced with lower imprisonment sentences, they would be more inclined to commit crime compared to a regime wherein there are longer sentences and no rewards. How this trade-off ought to be addressed is a complicated question, which can hinge, among other things, on the relative sizes of marginal first time and marginal repeat offenders in equilibrium. Nevertheless, a very simple observation is that allowing the reward size to depend on one’s criminal history (just as sentences are dependent on criminal history) can only weakly increase the attractiveness of rewards compared to the case where reward size is independent of criminal history. Thus, extending the current analysis to a setting where repeat offenses are possible may lead to the identification of additional mechanisms in which rewards can be used more effectively than explained here. The ideas presented here also bring to mind prior studies analyzing the impact of inequality and redistribution on crime, and how various redistribution methods compare to punishment mechanisms in achieving deterrence (e.g. Eaton and White (1991); Johnsen (1986); Tabbach (2012)). As explained in these studies, redistribution, like punishment, can achieve deterrence by increasing what a criminal has to lose upon being convicted, and this function is equally served by the rewards considered herein. Additionally, redistribution may also enhance deterrence further by reducing what a criminal has to gain through committing property crimes, because redistribution causes a reduction in the assets of potential victims. Moreover, as noted in Eaton and White (1991), punishment, unlike redistribution, can cause criminals to take costly actions to reduce the negative consequences of convictions (an issue also explored in the avoidance literature following Malik (1990)), which is an additional source of inefficiency. These latter two aspects of redistribution were not considered here, because the types of rewards envisioned in the article do not increase the tax burden placed on individuals in society, but instead are financed through reductions in imprisonment sentences (and, only in Section 3.2, through increases in the fines payable by wealthy offenders). Thus, the type of rewards considered here are not expected to cause additional deterrence effects by reducing the assets of potential victims. Similarly, because rewards are receivable only if a person does not commit crime, they are likely to have similar incentive effects as punishment on the avoidance activity of criminals. Thus, although rewards can be financed through large wealth redistributions, the instant analysis has not focused on this possibility to emphasize how this tool compares to imprisonment in achieving deterrence. 5. Conclusion Rewards and imprisonment can be used to perform similar deterrence functions. Despite this, there exists very little theoretical work considering the possibility of using rewards in the criminal setting. This is perhaps because, as highlighted in the analysis in this article, for rewards to be viable substitutes for imprisonment in achieving deterrence, imprisonment sentences would need to be significantly long. However, given the figures reported in the empirical literature regarding the ineffectiveness of imprisonment in enhancing deterrence, many jurisdictions in the United States may have already passed beyond the sentence threshold above which using rewards is a realistic alternative for achieving deterrence. The primary objective of this article is to highlight this possibility and to provide a theoretical framework that can be extended to incorporate further considerations relevant to the social (un)desirability of rewards. A. Appendix Proof of Proposition 2 (i) When (7) holds, it trivially follows that one can enhance welfare through rewards, e.g., by reducing sentences and increasing rewards above zero to hold deterrence constant. When (7) does not hold, part (ii) shows that when the imprisonment elasticity of crime, |$\varepsilon_{s}\equiv-\frac{\partial\theta}{\partial s}\frac{\bar{s}}{\theta }$|⁠, is smaller than a critical level using rewards is optimal. Thus, using rewards is optimal under broader conditions than (7). (ii) First, note that |$\frac{\partial T(\overline{s},0)}{\partial s} =\frac{\partial\theta}{\partial s}p\bar{s}+p\theta>0$|⁠, if |$\varepsilon_{s}<1$|⁠, which holds when |$\varepsilon_{s}<\frac{\theta ps}{\theta\lbrack ps+h]}$|⁠. Therefore, consider the case where |$\frac{\partial T(\overline{s},0)}{\partial s}>0$|⁠. Next, note that if |$\frac{\partial T(\overline{s},0)}{\partial r}\leq 0$|⁠, welfare can be increased simply by introducing rewards. When this condition does not hold, we have that |$\frac{\partial T(\overline{s} ,0)}{\partial s},\frac{\partial T(\overline{s},0)}{\partial r}>0$|⁠, and rewards are optimal if |$\frac{\frac{\partial W(\overline{s},0)}{\partial s}} {\frac{\partial T(\overline{s},0)}{\partial s}}<\frac{\frac{\partial W(\overline{s},0)}{\partial r}}{\frac{\partial T(\overline{s},0)}{\partial r} }$|⁠. Differentiating the welfare function and the tax burden with respect |$r$| and |$s$| reveals that $$\begin{align} \frac{\frac{\partial W}{\partial s}}{\frac{\partial T}{\partial s}} & =\frac{\frac{\partial\theta}{\partial s}(b^{r}-h)-p\frac{\partial\theta }{\partial s}[u(r)+v(s)]-p\theta v^{\prime}(s)}{\frac{\partial\theta}{\partial s}p(s-r)+p\theta}\text{; and}\\ \frac{\frac{\partial W}{\partial r}}{\frac{\partial T}{\partial r}} & =\frac{\frac{\partial\theta}{\partial r}(b^{r}-h)-p\frac{\partial\theta }{\partial r}[u(r)+v(s)]+(1-p\theta)u^{\prime}(r)}{\frac{\partial\theta }{\partial r}p(s-r)+(1-p\theta)}.\nonumber \end{align}$$(A.1) Evaluating these expressions at |$r=0$| and |$s=\bar{s}$|⁠, rearranging terms, and noting that |$\frac{\partial\theta}{\partial r}=\frac{\partial\theta}{\partial s}\frac{u^{\prime}(r)}{v^{\prime}(s)}$| reveals that |$\frac{\frac{\partial W(\overline{s},0)}{\partial s}}{\frac{\partial T(\overline{s},0)}{\partial s} }<\frac{\frac{\partial W(\overline{s},0)}{\partial r}}{\frac{\partial T(\overline{s},0)}{\partial r}}$| if $$\begin{equation} -\frac{\partial\theta}{\partial s}\left[ h\left[ (1-p\theta)-\frac {u^{\prime}(0)}{v^{\prime}(\overline{s})}p\theta\right] +u^{\prime }(0)p\overline{s}\right] <\left[ u^{\prime}(0)+v^{\prime}(\overline {s})\right] (1-p\theta)p\theta. \end{equation}$$(A.2) Multiplying both sides of this inequality by |$\frac{\overline{s}}{\theta}$|⁠, and noting that |$\varepsilon_{s}=-\frac{\partial\theta}{\partial s}\frac {\bar{s}}{\theta}$| reveals that this inequality is equivalent to $$\begin{equation} \overline{\varepsilon}_{s}\equiv\frac{\left[ 1+\frac{v^{\prime}(\overline {s})}{u^{\prime}(0)}\right] (1-p\theta)}{\frac{h}{p\overline{s}}\left[ \frac{1-p\theta}{u^{\prime}(0)}-\frac{p\theta}{v^{\prime}(\overline{s} )}\right] +1}>\varepsilon_{s} \end{equation}$$(A.3) Thus, |$r>0$| is optimal whenever |$\varepsilon_{s}<\overline{\varepsilon}_{s}$|⁠. When (7) does not hold it follows that |$v^{\prime }(\overline{s})\geq\frac{u^{\prime}(0)p\theta}{1-p\theta}$|⁠, in which case |$\overline{\varepsilon}_{s}>\frac{1}{\frac{h}{u^{\prime}(0)p\overline{s}}+1}$|⁠. This implies that a sufficient condition for the optimality of rewards is $$\begin{equation} \frac{p\overline{s}u^{\prime}(0)}{h+p\overline{s}u^{\prime}(0)}\geq \varepsilon_{s} \end{equation}$$(A.4) which holds when $$\begin{equation} \frac{\text{Total imprisonment costs}}{\text{Total criminal harm plus imprisonment costs}}=\frac{\theta p\overline{s}}{\theta (h+p\overline{s})}\geq\varepsilon_{s} \end{equation}$$(A.5) since |$u^{\prime}(0)\geq1$|⁠. □ Proof of Corollary 2 Part (i): When the government targets group |$i$| the crime rate is $$\begin{equation} \theta(s,r)=\eta_{i}\theta_{i}(s,r)+\eta_{-i}\theta_{-i}(s,0) \end{equation}$$(A.6) and the taxes necessary to implement this enforcement scheme are given by $$\begin{equation} T(s,r)=p\theta(s,r)s+\eta_{i}[1-p\theta_{i}(s,r)]r. \end{equation}$$(A.7) This is because |$p\theta(s,r)$| people are imprisoned, but only non-convicts in group |$i$| receive rewards (who have a measure of |$\eta_{i}[1-p\theta _{i}(s,r)]$|⁠). Letting |$r^{T}(s)$| denote the per-person reward that can be distributed to group |$H$| while keeping the tax burden unchanged at |$T(\bar{s},0)$|⁠, one can calculate the increase in the rewards available to this group as a result of reductions in the sentence as follows: $$\begin{equation} \frac{dr^{T}(s)}{ds}=-\frac{T_{s}(s,r)}{T_{r}(s,r)}=-\frac{p[\frac {\partial\theta(s,r)}{\partial s}s+\theta(s,r)]-\eta_{i}pr\frac{\partial \theta_{i}(s,r)}{\partial s}}{\eta_{i}\left( \frac{\partial\theta_{i} (s,r)}{\partial r}p[s-r]+[1-p\theta_{i}(s,r)]\right) }. \end{equation}$$(A.8) Crime is reduced as a result of reducing |$s$| below |$\bar{s}$| while increasing |$r$| above |$0$| according to (A.8) if $$\begin{equation} -\eta_{i}\frac{\partial\theta_{i}(\bar{s},0)}{\partial r}\frac{dr^{T}(\bar {s})}{ds}<\frac{\partial\theta(\bar{s},0)}{\partial s}. \end{equation}$$(A.9) Plugging (A.8) into the LHS of (A.9) reveals that this condition is equivalent to $$\begin{equation} \frac{f_{i}(b^{r})}{1-p\theta_{i}}u^{\prime}(0)>\frac{f(b^{r})}{p\theta}\text{ }v^{\prime}(\bar{s}) \end{equation}$$(A.10) which is the condition stated in the corollary. Part (ii): A simple manipulation of the condition |$\frac{f_{H}(b^{r} )}{1-p\theta_{H}}>\frac{f_{L}(b^{r})}{1-p\theta_{L}}$| reveals that it is equivalent to |$\frac{f_{H}(b^{r})}{1-p\theta_{H}}>\frac{f(b^{r})}{1-p\theta}$|⁠. Thus, (7) implies (25). □ Proof of Proposition 4 (i) The tax burden generated by all regimes that lead to the same level of deterrence as the initial regime can be expressed as $$\begin{equation} T(s,\overline{r}(s),\widehat{m}(s))=\lambda\lbrack p\theta_{l}(s-w)+(1-p\theta _{l})\overline{r}(s)]+(1-\lambda)p\theta_{h}(s-\widehat{m}(s)). \end{equation}$$(A.11) It follows that $$\begin{equation} \frac{\partial T}{\partial s}+\frac{\partial T}{\partial r}\frac{d\overline {r}}{ds}+\frac{\partial T}{\partial m}\frac{d\widehat{m}}{ds}=\lambda\left[ p\theta_{l}-(1-p\theta_{l})\frac{v^{\prime}(s)}{u^{\prime}(r(s))}\right] +(1-\lambda)p\theta_{h}[1+v^{\prime}(s)]. \end{equation}$$(A.12) This expression, evaluated at |$\overline{s}$|⁠, is positive if: $$\begin{equation} \frac{u^{\prime}(0)}{1-p\theta_{l}}+\frac{1-\lambda}{\lambda}\frac{\theta_{h} }{\theta_{l}}\frac{u^{\prime}(0)[1+v^{\prime}(\overline{s})]}{1-p\theta_{l} }>\frac{v^{\prime}(\overline{s})}{p\theta_{l}} \end{equation}$$(A.13) which is the condition expressed in (31). Due to the same reason noted in the proof of Proposition 1, it follows that when this inequality holds, the government can reduce crime, imprisonment sentences, and tax burdens, by introducing rewards. (ii) Follows immediately from the way |$\lambda$| enters the second term in (31 ). □ Acknowledgement This article replaces the working paper “Positive Sanctions versus Imprisonment” (Mungan 2019). I thank two anonymous referees for their careful review of this article and their useful comments and suggestions. I also thank participants of the 29th Annual Meeting of the American Law and Economics Association, the 36th Annual Meeting of the European Association of Law and Economics, the 2019 Public Choice Society Meetings, and the 2019 Southern Economic Association Annual Meeting for valuable comments and suggestions. Footnotes 1. See, e.g., Lee and McCrary (2017) finding sentence elasticities of crime not exceeding |$0.13\,$| and Chalfin and McCrary (2017) for a review of the existing literature. 2. This number is taken from a report by the Legislative Analyst’s Office which was last updated in January 2019 and last accessed in October 2021 at the following URL: https://lao.ca.gov/policyareas/cj/6_cj_inmatecost. 3. Wittman (1984), explained below, Dari-Mattiacci and De Geest (2009), explained in note 10, below, De Geest and Dari-Mattiacci (2013), and Dari-Mattiacci and Raskolnikov (2019) referred to in note 9, below, consider asymmetries between sticks and carrots in different settings. It is also worth noting that there are similarities between the dynamics I explore here and those that emerge in studies investigating the impact of foreign aid on terrorism. Some studies in this field posit that foreign aid, especially when used in a targeted manner, can be an effective tool (and an alternative to military intervention) in reducing terrorism in the aid-recipient country. See, e.g., Azam and Thelen (2008, 2010) and Bandyopadhyay et al. (2011). 4. See, e.g., Wen et al. (2020), He and Barkowski (2020), Aslim et al. (2020), Vogler (2020), all finding crime reducing effects associated with Medicaid expansions. Relatedly, Bondurant et al. (2018) find that the presence of substance abuse treatment centers have a local crime reducing effect. 5. See, e.g., Cohen (2020) finding a crime reducing effect associated with certain housing assistance programs. 6. Many empirical studies report crime reducing effects of educational programs, see, e.g., Garces et al. (2002) for a prominent example, and others referenced in Welsh (2011). Some of these programs target children based on exogenous categories. Welsh (2011) notes, for instance, that “[p]reschool intellectual enrichment programs are generally targeted on the risk factors of low intelligence and attainment” and that these programs have been found in subsequent evaluations to reduce offending. 7. See, e.g., Mazerolle et al. (1998), where nuisance abatement measures are found to reduce observed drug selling relative to more traditional policing measures. 8. For a lengthier discussion of publicly provided benefits that may have crime reducing effects, see, e.g., Galle (2021) and Welsh (2011). 9. It is important to note that this substitutability between carrots and sticks does not extend to cases wherein actors can decide not to be subject to the regulations through which rewards and sanctions are imposed (Dari-Mattiacci and Raskolnikov (2019)). The current analysis focuses on criminal behavior wherein the agent does not have this option. 10. The most important exception is Demougin and Schwager (2000) mentioned, below. Dari-Mattiacci and De Geest (2009) also consider rewards in noncriminal settings and ask whether sticks and carrots may be used in a way that generate different incentives structures. However, their mechanism is not widely implementable in a criminal setting, because it requires specifying punishment schemes that are a function of the behavior profile of all individuals, which would make it administratively very costly to implement in the criminal setting, and it could violate the Equal Protection Clause, because it requires specifying unequal punishments for different violators in some cases. There is also some empirical literature, focusing on a different kind of carrot, which is not a direct choice variable available to the government, namely employment opportunities. See, e.g., Corman and Mocan (2005) and other references reviewed in Chalfin and McCrary (2017). 11. See, e.g., Garoupa (1997) and Polinsky and Shavell (2007) for an account of some of the primary extensions. 12. Unlike the instant article, Demougin and Schwager (2000) consider redistribution as an alternative to more policing, as opposed to the length of imprisonment. Their approach is also quite different in that it considers a redistribution mechanism as opposed to rewards conferred upon avoiding conviction; assumes that wealth transfers are dissipated if the recipient is subsequently convicted; and the set of potential criminals is observable by the state. This last assumption is particularly problematic in the current context. As will become clear in the next sections, the inability to observe the at-risk population becomes an important obstacle in the way of using rewards in a socially beneficial manner. 13. Here, |$\$X$| is assumed to be the reduction in imprisonment costs at the new equilibrium level of deterrence after both policy changes (including the change described in the second step) have been implemented. 14. See, e.g. Mastrobuoni and Rivers (2019) and Abrams and Rohlfs (2011) on empirical estimates of the disutility from imprisonment. Even with these innovative empirical studies, it is difficult to ascertain the marginal disutility from the last day (or any other unit of time) of prison, which is smaller than the average disutility from one day of prison when the disutility from prison rises less than proportionally as discussed in Polinsky and Shavell (1999). 15. I consider two separate welfare functions for this analysis, because there is a disagreement that dates back to the first articles proposing economic analyses of criminal behavior. Becker includes criminals’ benefits in the social welfare function (Becker 1968) and Stigler (1970) questions whether this approach is defensible. Recently, Curry and Doyle (2016) provided an explanation as to why maximizing a utilitarian social welfare function may be equivalent to minimizing criminal harm in some contexts. 16. For instance, although the crime rate among ex-offenders is higher than the general population, targeting ex-offenders can generate incentives to commit crime in the first place. 17. It is not hard to imagine, for instance, that a system which makes rewards available only to males (based on the fact that the crime rate is much higher among males than females) would lead to concerns that it would contribute to existing gender gaps. 18. Existing educational programs, some of which target children with specific needs, can be viewed as targeted rewards. 19. Some may argue that this type of targeted policy may incentivize a higher rate of father-absent families. However, these effects may be small due to reasons similar to those explained in the targeting of judgment proof offenders explained below. 20. See, e.g., Wittman (1984), and others referenced in Galle (2021). 21. I assume that indifferent individuals refrain from committing crime. 22. One may be curious about whether |$\overline{r}(s)$| is necessarily convex, as depicted in Figure 1, although this is not a necessary property for the derivation of any of the results. A quick look at |$\overline{r}$|’s derivative reveals that it is convex as long as |$v^{\prime\prime},u^{\prime\prime}<0$|⁠. 23. This argument can be made more explicit through the use of symbols. Suppose the function to be maximized is |$W(s,r,p)$|⁠. If |$\widehat{p}$| and |$\widehat{s}$| denote the optimal choices of |$p$| and |$s$| subject to the constraint that |$r=0$|⁠, and if |$r^{\prime}>0$| and |$s^{\prime}$| improve welfare relative to this solution subject to the constraint that |$p=\widehat{p}$|⁠, then it follows that $$ W(s^{\ast},r^{\ast},p^{\ast})\geq W(s^{\prime},r^{\prime},\widehat{p} )>W(\widehat{s},0,\widehat{p}), $$ where |$\ast$| indicate optimal levels when all policy variables are endogenously determined. Thus, it follows that |$r^{\ast}\neq0$|⁠, since otherwise it would follow that |$W(s^{\ast},r^{\ast},p^{\ast})=W(\widehat{s} ,0,\widehat{p})$|⁠, which is a contradiction with the above ranking. 24. When criminals’ utilities are excluded from the welfare function, assuming that criminal harm is uniformly distributed across the population, the welfare function reads $$ W(s,r)=(1-\theta)[u(r)-\theta h]. $$ Thus, the maximization problem becomes $$ \underset{s,r}{\max}(1-\theta)[u(r)-\theta h]\text{ st. }T(s,r)=p\theta s+(1-p\theta)r\leq\overline{T}. $$ An analysis of this problem reveals a bound on the imprisonment elasticity of crime of $$ \widetilde{\varepsilon}_{s}\equiv\frac{1}{\frac{h}{ps}\frac{1-2\theta }{1-\theta}(\frac{1-p\theta}{u^{\prime}(0)}-\frac{p\theta}{v^{\prime}(s)})+1} $$ such that—whenever the denominator of |$\widetilde{\varepsilon}_{s}$| is positive—an elasticity lower than |$\widetilde{\varepsilon}_{s}$| implies that welfare can be enhanced by the use of rewards. This implies the same results reported in Proposition 2. 25. When criminals’ utilities are excluded from the welfare function, assuming the tax burden is uniformly distributed across the population, the welfare function instead reads $$ \widehat{W}(s,r)=(1-\theta)\left[ [u(r)-\theta h]-k[p\theta s+(1-p\theta )r]\right.] $$ Repeating steps that are very similar to those in the proof of Proposition 3 to investigate this function reveals the same results reported in Proposition 3. 26. See, e.g., https://www.nbcphiladelphia.com/news/local/Philadelphia-Prosecutors-Must-Include-Cost-of-Prison-Time-During-Sentencing-476987333.html, last accessed in October 2021. 27. The subscripts |$l$| and |$h$| are used to distinguish these rates from |$\theta_{L}$| and |$\theta_{H}$| used in Section 3.1. 28. I assume that indifferent people choose not to work. 29. This dynamic is very similar to that underlying the optimality of underdeterrence in simple Beckerian law enforcement models, see, e.g., Polinsky and Shavell (2007). 30. Another approach is to calculate the moral hazard losses generated by rewards and compare them to tax savings. This approach generates similar results, but the conditions obtained are less intuitive because they relate to the density of individuals on work margins. 31. See, e.g., Loeber and Farrington (2014) discussing the age-crime curve. 32. Mueller-Smith notes that “the empirical results indicate that incarceration generates net increases in the frequency and severity of recidivism ... A cost–benefit exercise finds that substantial general deterrence effects are necessary to justify incarceration in the marginal population.” 33. There is a broad literature analyzing the punishment of repeat offenders. 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