# Optimal Preventive Law Enforcement and Stopping Standards

Optimal Preventive Law Enforcement and Stopping Standards Abstract Preventive law enforcement increases social welfare by hindering the infliction of criminal harm, but produces inconvenience costs to the general public, because it requires interfering with the acts of innocents as well as attempters. The optimal amount of investment in preventive enforcement is greater than that which maximizes deterrence, but, smaller than that which minimizes criminal harm. Thus, ignoring preventive benefits and/or inconvenience costs results in an inefficient investment portfolio over enforcement methods, and in a predictable manner. Stopping standards, which determine the threshold suspicion required to trigger a stop, are tools that can be used to optimally trade-off the costs and benefits associated with preventive enforcement. The optimal stopping standard is weaker than its analogs in the trial context, namely standards of proof, which generally require preponderance of the evidence in civil trials and proof beyond a reasonable doubt in criminal trials. Finally, suspicionless stops can be optimal in a variety of circumstances, and are more likely optimal when enforcers perform poorly in forming suspicions; inconvenience costs are small; the population is unresponsive to deterrence measures; and the attempt rate is high. There is one limit which ought never to be neglected: “No method of prevention should be employed, which is likely to cause a greater mischief than the offence itself.”1 –Jeremy Bentham 1. Introduction Why is it that Terry stops2 can legally be conducted only upon reasonable suspicion, but that vehicles at roadblocks and sobriety checkpoints can legally be stopped without suspicion? Is there a good reason to have reduced standards for detaining people suspected of taking part in terroristic activity? Is there a good reason to increase policing efforts, even when criminal activity is unresponsive to the presence of police officers, as may be the case in many “hot spots,” at infamous derbys, or at crowded outdoor celebrations and events? The objective of this article is to provide an economic framework that can be used in answering these difficult questions. This requires exploring a function of law enforcement, which is virtually ignored in the law and economics literature, namely harm prevention. The most commonly studied objective of enforcement by economists, namely deterrence, aims to reduce the proportion of individuals who engage in criminal acts. On the other hand, some preventive enforcement measures aim to stop ongoing crimes, and thereby reduce the proportion of attempted crimes that result in harm. This objective, namely harm prevention, can therefore be thought of as reducing the proportion of crimes—committed by undeterred individuals—that end up causing harm.3 There are many enforcement methods whose primary goal is harm prevention. For instance, sobriety checkpoints reduce the harm inflicted through driving under the influence offenses by preventing the driver from continuing to drive. In contrast, resources spent on crime scene investigations are generally meant to solve previously committed crimes, and thus, they serve the function of increasing the probability of ex post detection and punishment. Preventive and nonpreventive methods alike can be expected to have deterrence effects, lead to incapacitation of the offender, and lead to wrongful convictions. Thus, these features do not lead to sharp distinctions between preventive and nonpreventive methods. However, because preventive enforcement methods have to rely on information obtained prior to the infliction of harm, it is often quite difficult to ascertain whether the suspect is in fact guilty of carrying out a crime, and enforcers have to, very frequently, target innocent people. For instance, only 6% of the 4.4 million Terry stops conducted by the New York Police Department between 2004 and 2012 resulted in arrests.4 These stops undeniably caused inconveniences for the stopees. Hence, obtaining the benefits of harm prevention comes at the cost of generating sizeable inconvenience costs to the general public. It may be argued that ex post criminal investigations also cause inconvenience costs. Similarly, ex post investigations may also result in some preventive benefits, especially in instances where the target is a professional criminal, who may be in the process of committing another crime. Although these statements are undeniably true, as long as there is some correlation between the emphasis an investment in enforcement places on prevention and the amount of inconvenience costs generated by such investments, one can meaningfully talk about diverting investments from less preventive methods toward more preventive methods. In such instances, a trade-off emerges between preventive benefits and inconvenience costs, which can be addressed by altering the investment portfolio over various enforcement methods. To analyze whether the presence of inconvenience costs have an important impact on how enforcement mechanisms ought to be designed, I construct a simple law enforcement model where different investment portfolios over enforcement mechanisms have different prevention and deterrence consequences. Then I ask whether the size of inconvenience costs affects the optimal investment in preventive versus nonpreventive enforcement methods. The first result is that the optimal investment in preventive enforcement is higher than that which maximizes deterrence, but lower than that which minimizes the amount of criminal harm generated through illegal behavior. Moreover, absent inconvenience costs, the optimal investment in prevention coincides with that which minimizes criminal harm, and as inconvenience costs increase, the optimal investment in prevention moves toward the deterrence maximizing enforcement scheme. Thus, the size of inconvenience costs determines whether the government ought to implement a policy geared primarily toward harm minimization or toward deterrence maximization. Although the model focuses primarily on inconvenience costs to stopees, it can also be used to identify other factors that affect the desirability of using preventive enforcement methods. An important factor is the size of the externalities generated by stops to the general public. For instance, increasing the length of a stop not only increases the amount of time that a stopee is investigated, but also increases the amount of time each person in line at a security check point has to wait to be investigated. Naturally, when these externalities are large, preventive enforcement methods become less attractive. Another factor is the proportion of stops that actually lead to some preventive benefits (i.e., accurate stops), and this depends on the characteristics of the population being regulated: the lower the criminal tendencies of the population, the smaller is this proportion. Similarly, the benefit of using preventive enforcement is increasing in the expected social harm to be prevented. The foregoing simple observations can be used to explain why we use preventive enforcement frequently at airports, hot spots, and high crime areas; as well as why it is used to reduce social harms caused by the mentally ill or hooligans at sports events. All results described above relate to how much preventive enforcement ought to be used, but they do not explain what standard ought to guide whom to stop. To discuss this issue, I define a stopping standard, which refers to the threshold quality that a noisy signal regarding the guilt of an individual must posses to trigger an investigation. Investigating the optimal stopping standard reveals that it is weaker than what is called the “preponderance of the evidence” standard used in civil trials, and much weaker than the “proof beyond a reasonable doubt” standard. This is because a wrongful stop, on its own, causes an inconvenience which is typically smaller than the cost of being falsely found liable. However, the benefit from an accurate investigation is not only deterrence but also harm prevention. Thus, stopping standards are typically weaker than standards of proof used in trials. Moreover, if enforcers perform very poorly in forming suspicions (perhaps because the signal they receive regarding people’s behavior is very noisy), then it is optimal to employ the weakest standard possible, and use suspicionless stops. A similar result holds when inconvenience costs are sufficiently small, illustrating the importance of inconvenience costs in justifying stops based on suspicion. The remainder of this article is structured as follows. Section 2 provides a literature review. Section 3 presents and analyzes a law enforcement model with preventive and nonpreventive law enforcement methods. Section 4 introduces stopping standards and derives optimality conditions. Section 5 contains concluding remarks, and an Appendix at the end contains proofs of propositions. 2. Literature Review An issue in assessing the effectiveness of different law enforcement methods pertains to how one measures enforcement inputs and outputs. Although this issue was noted by Chapman et al. (1975) more than forty years ago, most of the theoretical law enforcement literature has focused on a single-dimensional input for law enforcement (money), and generally a single output (deterrence).5 However, enforcement inputs can be used to serve different functions of law enforcement (e.g., employing crime scene investigators to solve completed crimes versus establishing hot spots to increase prevention).6 As highlighted in this article, one can identify meaningful factors that affect the optimal allocation of inputs toward different enforcement methods. Moreover, one can conceive of different ways to measure outputs. For instance, one can measure the deterrent effect of enforcement, or the harm reducing effect of enforcement.7 The former could be measured by changes in the attempt rate whereas the latter would need to measure changes in the completed crime rate.8 Even then, as the current article highlights, it is important to include negative outputs: just like a factory generates pollution as a by-product of its activity, law enforcement generates inconvenience costs, and these ought to be included in measuring outputs. A failure to do so can result in excessive investments in preventive enforcement. The theoretical literature implicitly proposes ways to measure outputs when they formulate social welfare functions to evaluate the performance of law enforcement mechanisms. However, they heavily focus on deterrence in doing so, with a small number of exceptions.9 In particular, Shavell (1987, 2015) and Miceli (2010, 2012) focus on incapacitation, and highlight a trade-off between the costs of imprisonment and the risk of an individual committing further crimes if released. Friehe and Tabbach (2013) formalize preventive enforcement, and highlight that this type of enforcement aims at eliminating harms from a specific act, and not the future harms that are likely to be caused by the individual who has demonstrated his dangerousness,10 as in the case of incapacitation. This distinction is important, and reveals a fundamental difference between the structure of the errors generated through investigations necessary for prevention versus errors generated through punishment. Prevention requires acting on very noisy signals, because, by definition, harm has not yet been inflicted. Thus, small but frequent errors are almost unavoidable in implementing preventive enforcement methods. On the other hand, the decision to punish a defendant is made either after he has completed his crime, or after he has been identified as an attempter. Moreover, an erroneous decision at this stage amounts to a wrongful conviction, and, thus carries larger costs, which have been studied in the literature (Lando 2009; Kaplow 2011; Mungan 2011; Rizzolli and Saraceno 2013). The latter type of errors are a consequence of punishment, and, therefore, are generated by all enforcement methods that lead to punishment; preventive and nonpreventive alike. Thus, to preserve the focus on what most discretely characterizes preventive enforcement methods, the current article incorporates inconvenience costs and preventive benefits, and does not explicitly model type-I and type-II errors associated with punishment. Although erroneous liability costs have been studied extensively in the optimal standard of proof literature, a similar analysis of standards that can be used to affect inconvenience costs is presented for the first time in this article. Thus, the current article also fills a gap in the literature by extending the analysis of standards of proof to cases where the standard is used to determine whether a person should be merely investigated, instead of whether he ought to be found liable. It demonstrates that the optimal stopping standard is lower than the optimal standard of proof in civil (Demougin and Fluet, 2005, 2006) as well as criminal trials (Miceli 1990). This is because the ratio between inconvenience costs and preventive benefits is typically smaller than the relevant ratios in the liability contexts where the standard of proof becomes applicable. It is also worth comparing inconvenience costs to judgment errors considered in the context of ex ante regulation by Shavell (1984a,b). In both cases, errors are by-products of the imperfection of the monitoring/investigation system employed by law enforcers to prevent crime before it occurs. However, a large proportion of inconvenience costs are incurred by innocent parties, and, therefore, they are present even in cases where judgment errors lead to benefits instead of costs. For instance, ex ante regulation may lead to increased deterrence, if a regulator misjudges the characteristics of a regulatee and thereby subjects him to more stringent standards (as in Shavell 1984a). This type of deterrence enhancing judgment error leads to net benefits when the offenses to be deterred are socially undesirable, yet, similar errors in the investigation context can reduce welfare even if they enhance deterrence. Another important issue to note is that inconvenience costs are a function of the crime rate, which is naturally related to the population’s or region’s characteristics. This aspect is closely related to points made by several criminology scholars regarding the spatial aspects of law enforcement (Braga 2008; Kennedy 2009; Nagin 2013; Nagin et al. 2015; Sherman et al. 1989 and Sherman and Weisburd 1995): ex post enforcement can be used non-discriminatorily to solve crimes that were committed in any region, whereas the locations in which preventive enforcement methods are to be used must be chosen strategically. In line with these observations, the current article suggests that preventive enforcement ought to be used more frequently in regions in which the crime rate is higher. Finally, and most importantly, this article builds on Friehe and Tabbach (2013), which is the first economics article to specifically analyze preventive enforcement methods. The instant article, like Friehe and Tabbach (2013), focuses on the deterrence and prevention functions of different enforcement methods, and subsequently furthers the analysis in several dimensions. First, it incorporates inconvenience costs, which are necessary to meaningfully analyze stopping standards. It also identifies optimal stopping standards, as well as conditions under which suspicionless stops are optimal. Moreover, it compares levels of investments in preventive enforcement which serve different goals (e.g., deterrence maximization, criminal harm mitigation, and social welfare maximization). It thereby illustrates the pivotal role that inconvenience costs play in determining whether the optimal policy is closer to deterrence maximization or criminal harm mitigation. 3. Enforcement Schemes in the Presence of Inconvenience Costs 3.1. Model I consider a continuum of individuals who derive benefit $$b$$ from successfully completing criminal acts that cause social harm $$h$$. These benefits vary from person to person, and, $$g$$ and $$G$$ with support over the interval $$b\in \lbrack\underline{b},\overline{b}]\subset\lbrack0,h]$$, respectively denote the density and cumulative distribution of $$b$$. After a person initiates the criminal act, but prior to completing it, he may be stopped (with probability $$\beta$$) as a result of preventive law enforcement efforts. If stopped, investigations may reveal, with probability $$x\in\lbrack0,\overline{x}]\subset\lbrack0,1)$$, that he is in the process of committing a crime. This probability is proportional to the length (or intensity) of the stop. With the residual probability $$1-x$$, the attempter is released. The stop causes the stopee an inconvenience cost of $$xc$$, where $$c$$ represents the marginal inconvenience cost to the stopee from lengthier stops. If the person is not stopped or is released, he carries out his crime to completion. However, he may also be caught and punished after committing the crime with probability $$p$$. Lengthier stops cause the enforcement agency to reduce the efforts that it devotes toward ex post investigations. This is because the enforcement agency allocates a fixed amount of resources across preventive investigations and ex post investigations. Resources allocated toward preventive investigations are increasing in $$x$$, and, because the resource constraint is binding, the remaining resources allocated towards ex post investigations are decreasing in $$x$$. This is formalized by assuming that $$p = p(x)$$ and that $$p^{\prime} < 0$$. Moreover, positive investments in preventive enforcement are guaranteed by assuming that $$\underset{x\rightarrow\overline{x}}{\lim} p^{\prime}(x)=-\infty$$. If a person is caught either after or in the process of committing a crime he pays a fine of $$f$$.11 Thus, a person’s expected benefit from initiating a crime is:12 $$(1-x\beta)(b-p(x)f)-x\beta(c+f)$$ (1) A person who does not initiate a criminal act may nevertheless be the subject of a preventive stop with probability $$\alpha\leq\beta$$, because law enforcers cannot perfectly distinguish between innocent and guilty individuals.13 The gap between $$\beta$$ and $$\alpha$$ can thus be interpreted as the degree to which the stopping process is able to discriminate between guilty and innocent individuals, or, the randomness of a stop. If stopped, the innocent person suffers the same inconvenience cost as a guilty individual. Thus, an innocent individual’s expected payoff is $$-\alpha xc$$. Therefore, a person initiates the act if: \begin{align} b^{\ast}(x)\equiv x\frac{(\beta-\alpha)c+\beta f}{1-x\beta}+p(x)f<b \end{align} (2) I assume some, but not all, people are deterred, that is, $$b^{\ast}\in (\underline{b},\overline{b})$$ for all $$x$$. A higher degree of deterrence, naturally corresponds to a higher $$b^{\ast}$$, since then condition (2) is satisfied for fewer people. This can be noted by defining the attempt rate, $$\theta$$, as follows: $$\theta(x)\equiv1-G(b^{\ast}(x))$$ (3) It is worth highlighting that while $$1-\theta$$ represents the rate of deterrence, $$x\beta$$ represents the probability (or the degree) of harm prevention, and, hence the proportion of attempted crimes that are prevented through enforcement efforts. Inspecting expression (2) reveals that increasing the resources devoted to preventive enforcement enhances deterrence by increasing the cost of being caught prior to completing one’s attempt, but leads to a counter-effect by reducing the expected costs of being caught subsequent to completing one’s crime. Absent further restrictions, the relationship between these two effects is ambiguous and one cannot conclude that deterrence is maximized by an interior and unique choice of $$x$$. To simplify the analysis, I assume that $$p$$ has a functional form that guarantees that $$b^{\ast}$$ is single peaked in $$x$$, such that it has a unique and interior maximizer. These assumptions imply that increasing the resources devoted to preventive enforcement increases deterrence only up to a unique level of $$x$$, denoted as $$x^{d}$$, defined as follows: $$x^{d}\equiv\underset{x}{\arg\max}b^{\ast}(x)$$ (4) 3.2. Inconvenience Costs The total inconvenience costs generated by preventive enforcement is naturally a function of the length of each search ($$x$$) and also the attempt rate, since attempters are generally stopped more frequently than innocent individuals. Specifically, aggregate inconvenience costs can be expressed as: $$xc(\alpha+\theta(x)(\beta-\alpha))$$ (5) A quick investigation of expression (5) reveals that increasing the investment in preventive enforcement has a direct and positive effect on inconvenience costs, since this leads to an increase in the amount of inconvenience suffered by each stoppee. However, for $$x<x^{d}$$, an increase in preventive enforcement enhances deterrence, and thereby reduces the attempt rate. This leads to a reduction in the number of individuals who are searched, and, thus, for $$x<x^{d}$$, an increase in preventive enforcement can actually reduce inconvenience costs. On the other hand, for $$x>x^{d}$$, the two effects go in the same direction, and, therefore, further investments in preventive enforcement increase aggregate inconvenience costs. The marginal inconvenience cost from lengthier stops, $$c$$, also has a similar feature. An increase in $$c$$ leads to a direct increase in inconvenience costs, but it also enhances deterrence (see expression (2), above). Thus, if the latter (deterrence) effect is stronger than the former direct effect, an increase in $$c$$ can lead to a reduction in aggregate inconvenience costs. It is possible, however, to identify conditions under which these possibilities can be ruled out. These points are noted by the following observation. Observation 1 (i) An increase in $$c$$ or $$x$$, holding all else constant, can lead to a reduction in aggregate inconvenience costs. (ii) But, aggregate inconvenience costs are unambiguously increasing in $$c$$ and $$x$$, if, either (a) stops are sufficiently random (i.e., $$\beta-\alpha$$ is small), or (b) the population is relatively unresponsive to expected sanctions (i.e., $$g(b^{\ast})$$ is small). (iii) Moreover, aggregate inconvenience costs are increasing in $$x$$ for all $$x>x^{d}$$. 3.3. Optimal Preventive Enforcement Inconvenience costs are only one component of a welfare function that consists of the sum of all individuals’ utilities. Specifically, in addition to inconvenience costs, social welfare incorporates the net-criminal harm from crime. Thus, social welfare can be expressed as: $$W=(1-x\beta)\int_{b^{\ast}(x)}^{\overline{b}}(b-h)g(b)db-xc(\alpha +\theta(x)(\beta-\alpha))$$ (6) A quick look at equation (6) reveals that a change in the amount of preventive enforcement affects welfare through three separate channels. First, it affects the attempt rate. This in turn impacts the total criminal harm caused by attempters as well as the number of individuals who suffer inconvenience costs from being stopped. Second, the inconvenience costs suffered by each individual is affected. Third, preventive enforcement reduces the rate at which attempters are able to complete their crimes and inflict social harm. These three effects are expressed, in the order they are described, in separate lines along with their explanations, below. $$\matrix{ {\matrix{ {{{dW} \over {dx}} = b_x^ * g({b^ * }(x))[(1 - x\beta )(h - {b^ * }(x)) + xc(\beta - \alpha )]} \cr } } \hfill & {\} \matrix{ {{\it{marginal\,deterrence\,benefits}}} \hfill \cr } } \hfill \cr {\matrix{ {} \hfill & {} \hfill \cr } - c(\alpha + \theta (x)(\beta - \alpha ))} \hfill & {\} \matrix{ {{\it{marginal\,inconvenience\,costs}}} \hfill \cr } } \hfill \cr {\matrix{ {\matrix{ {} \hfill & {} \hfill \cr } + \beta \int_{{b^ * }(x)}^{\bar b} {(h - b)} g(b)db} \cr } } \hfill & {\} \matrix{ {{\it{marginal\,preventive\,benefits}}} \hfill \cr } } \hfill \cr }$$ (7) An interesting observation is that marginal preventive benefits are inversely related to the amount of deterrence. This reveals three important points. First, in societies with low crime rates, there are small gains from prevention. Second, marginal preventive benefits are quasi-convex in $$x$$: they are minimized at $$x^{d}$$ and maximized when all enforcement resources are devoted to a single type of enforcement (i.e., either preventive or nonpreventive). This leads to the third point, which is that as long as the criminal harms preventable at the maximum level of deterrence are greater than total inconvenience costs, marginal preventive benefits are always greater than marginal inconvenience costs. Specifically, as long as assumption 1, below, holds, increasing the investments in preventive enforcement always leads to greater increases in preventive benefits than increases in inconvenience costs. Since inconvenience costs are considered to be small relative to harms from crime, assumption 1, below, is imposed in the remaining parts of the article to simplify the analysis.14 Assumption 1 $$c\leq\overline{c}\equiv\underset{\max b^{\ast}}{\overset{\overline{b}}{\int}}(h-b)g(b)db$$. Another, less obvious, point revealed by equation (7) is that social welfare may generally have multiple local maxima, because the size of marginal deterrence benefits (or losses) can fluctuate due to changes in the density of individuals who are on the margin, that is, $$g(b^{\ast})$$. In such cases, the local maximum which is generated by a low $$x$$ leads to a low-prevention/high-deterrence enforcement scheme, whereas the maximum generated by a high $$x$$ leads to a high-prevention/low-deterrence enforcement scheme. In such cases, inconvenience costs become particularly important, since increases in $$c$$ cause the low-deterrence/high-prevention scheme to become a less desirable option compared to the high-deterrence/low-prevention scheme. This is because the former strategy leads to more frequent inconvenience costs than the latter. Thus, in some cases, a very small increase in $$c$$ can cause large changes in the optimal investment in preventive enforcement.15 The importance of inconvenience costs in determining optimal preventive enforcement is not limited to cases where there are multiple local maxima. In particular, even when social welfare has a single local maximum, ignoring inconvenience costs (which corresponds to minimizing net criminal harms), or simply attempting to maximize deterrence can lead to inefficiencies. The next proposition highlights these possibilities, and compares the optimal investment in preventive enforcement to investments that serve other objectives commonly studied in legal scholarship. Proposition 1 The optimal investment in preventive enforcement, $$x^{w}$$, is (i) greater than that which maximizes deterrence, and (ii) smaller than that which minimizes net-criminal harm whenever $$c>0$$. Proof. See Appendix. □ Proposition 1 reveals that one ought to invest more in prevention than the amount necessary to maximize deterrence, because there are net preventive gains (i.e., preventive gains minus inconvenience costs) from each marginal dollar invested in prevention. Thus, it is optimal to invest in prevention until doing so generates marginal deterrence losses that equal the marginal net gains from prevention. Hence, focusing solely on deterrence would cause a suboptimal investment in preventive methods. Second, minimizing net criminal harms would result in above optimal investments in preventive enforcement, because this objective simply ignores a form of negative externalities, namely inconvenience costs, associated with investments in prevention. This acts as a reminder that ignoring small but frequent costs to third parties can lead to welfare losses. 3.4. Comparative Statics This subsection identifies conditions under which infinitesimal changes in $$h$$ and $$c$$ cause local maximizers to move unambiguously in one direction or another. To ascertain the effect of $$h$$ on the optimal enforcement scheme, note that each prevented crime brings about benefits that are proportional to the severity of the crime. However, deterrence benefits, too, are proportional to the harm from crime, and, as implied by proposition 1, when investments are chosen optimally, an increase in $$x$$ causes a reduction in deterrence. Thus, without further analysis, it is unclear whether increasing the investment in preventive enforcement would generate preventive benefits that more than offset the deterrence losses it would cause. Making an additional observation resolves this ambiguity: the optimal investment balances preventive benefits against inconvenience costs in addition to losses in deterrence. Thus, an increase in $$h$$ causes a greater increase in marginal preventive benefits compared to marginal deterrence losses, and thereby leads to an increase in the optimal investment in prevention. On the other hand, an increase in $$c$$ has a direct and positive effect on the inconvenience costs from more prevention. However, as noted earlier, expected inconvenience costs also have a deterrent effect. Thus, an increase in $$c$$ can cause a reduction in equilibrium inconvenience costs, if the deterrence effect more than outweighs the increase in the marginal inconvenience effect, as noted in observation 1. This effect is similar to one that is studied frequently in industrial organizations: a price increase can bring about a reduction in profits when it leads to a quantity effect that more than offsets the effect due to an increase in the profit margin. Therefore, a priori, it is impossible to make statements about the impact of $$c$$ on the optimal amount of preventive enforcement. However, when stops are conducted randomly, it follows that $$\beta=\alpha$$, and, thus, $$c$$ has no impact on deterrence, as can be inferred from expression (2), and, conversely, the crime rate has no impact on aggregate inconvenience costs, as equation (6) illustrates. A similar result holds when offenders are relatively unresponsive to expected sanctions, even when $$\beta\neq\alpha$$, since then the deterrent effect of $$c$$ is minimal, but it still affects per-person inconvenience costs. Thus, under either of these conditions, the optimal amount of preventive enforcement is negatively related to $$c$$. These results are reported in proposition 2, below. Proposition 2 Locally welfare maximizing investments in preventive enforcement are (i) increasing in $$h$$, and (ii) decreasing in $$c$$ if either $$\beta=\alpha$$, or if $$g(b^{\ast})$$ and $$|g^{\prime}(b^{\ast})|$$ are sufficiently small. Proof. See Appendix. □ An immediate implication of proposition 2 is that if social welfare is single peaked, the optimal investment in preventive enforcement is increasing in $$h$$ and decreasing in $$c$$ under the conditions identified in proposition 2. Moreover, as marginal inconvenience costs get larger, the social objective converges towards deterrence maximization. In such cases it is relatively harmless to simply target deterrence—an easier to measure output—than the maximization of welfare. This last observation, combined with proposition 1, demonstrates the pivotal role inconvenience costs play in the determination of optimal policies. Proposition 3 Suppose $$W$$ is single peaked in $$x$$ for all $$c\in\lbrack0,\overline{c}]$$ and that $$\beta-\alpha$$ is sufficiently small. Then, the social objective converges (i) toward net harm minimization as marginal inconvenience costs converge toward zero, and (ii) toward deterrence maximization as marginal inconvenience costs converge toward $$\overline{c}$$. (iii) Moreover, as marginal inconvenience costs increase, the optimal investment in prevention moves monotonically from that which minimizes net criminal harms toward that which maximizes deterrence. Proof. See Appendix. □ The findings of propositions 1 and 3, which are illustrated in Figure 1, below, rely on $$\beta-\alpha$$ being small. Therefore, an important question is whether it may be optimal to have stops that are relatively random, rather than those that attempt to discriminate between guilty and innocent individuals. Answering this question requires endogenizing the policy variable which determines $$\beta$$ and $$\alpha$$, namely the stopping standard. This is done in the next section. Figure 1 View largeDownload slide Optimal investment in prevention $$(x^w(c))$$ as a function of inconvenience costs when $$\beta = \alpha$$ and $$W$$ is single-peaked. Figure 1 View largeDownload slide Optimal investment in prevention $$(x^w(c))$$ as a function of inconvenience costs when $$\beta = \alpha$$ and $$W$$ is single-peaked. 4. Optimal Stopping Standards This section introduces stopping standards as a policy tool to adjust $$\beta$$ and $$\alpha$$. An intuitive relationship between stopping standards and these probabilities, which is formalized in Section 4.1., is that they are inversely related to the strength of the standard employed in conducting stops: a stricter requirement for conducting stops reduces the probability with which guilty as well as innocent individuals are stopped. This observation suffices to explain how, absent inconvenience costs, suspicionless standards become optimal, and thereby highlights the importance of inconvenience costs in providing a rationale for requiring enforcers to conduct stops only when there exists a reasonable suspicion that the potential stoppee is guilty. Absent inconvenience costs, there is simply no downside to using very weak standards. This is because, in these cases, deterrence as well as inconvenience costs are unaffected by the frequency of incorrect stops (i.e., $$\alpha$$), since innocent stopees are unharmed by stops. On the other hand, an increase in the probability of accurate stops, that is, $$\beta$$, increases the probability with which guilty people forfeit their criminal benefits, in addition to increasing the likelihood with which they are punished for committing crime. Thus, as expression (2) illustrates, when $$c=0$$, suspicionless stops maximize deterrence, since then $$b^{\ast}$$ is maximized when $$\beta$$ reaches its maximum at $$1$$. Moreover, when there are no inconvenience costs, the only social objective in addition to deterrence, is prevention. This objective is clearly best served by using suspicionless stops, because the probability of prevention, $$x\beta$$, is also maximized by setting $$\beta=1$$. Thus, absent inconvenience costs, both deterrence and welfare are maximized by suspicionless stops. This result highlights the pivotal role of inconvenience costs in providing a rationale for requiring a certain degree of suspicion to conduct stops. In particular, when inconvenience costs are introduced, increases in the probability with which innocent people are stopped are detrimental to both deterrence and social welfare. Thus, making the stopping standard weaker generates costs by increasing $$\alpha$$, which must be traded-off against the preventive and deterrence benefits obtained by increasing the probability of stopping guilty individuals more frequently. Even then, as the analysis reveals, the optimal stopping standard is weaker than standards of proof employed in the trial context, because inconvenience costs are smaller than the preventive gains against which they are balanced. Formalizing this point and highlighting subtler points requires the introduction of stopping standards. 4.1. Stopping Standards To formalize stopping standards, suppose that each individual emits a noisy signal $$s\in R$$ about whether he is in the process of carrying out a crime. Suppose that larger values of $$s$$ are indicators of guilt, because they are more likely to be emitted by guilty individuals than innocent individuals. Thus, an intuitive criterion that enforcers may use is to stop individuals only if they receive a sufficiently high signal from them, that is, they may set a threshold signal $$\underline{s}$$, such that an individual is stopped only if he emits a signal $$s\geq\underline{s}$$. Thus, increasing $$\underline{s}$$ corresponds to choosing a stronger standard, in the sense that enforcers require stronger suspicions to conduct stops. Such increases cause a reduction in both $$\alpha$$ and $$\beta$$ by reducing the likelihood with which a person emits a signal that exceeds the standard, that is, $$\frac{d\alpha}{d\underline{s}},\frac{d\beta}{d\underline{s}}<0$$. An intuitive property frequently assumed about the way signals are produced by innocent and guilty individuals is called the monotone likelihood ratio property (MLRP). In the current context, this property holds whenever the ratio between the likelihoods with which a guilty versus an innocent individual emits a particular signal is increasing in the strength of the signal. When the signal emitting processes satisfy MLRP, it is often more convenient to express $$\beta$$ as a function of $$\alpha$$, instead of expressing both probabilities as functions of $$\underline{s}$$.16 This simplification causes no ambiguities, because both $$\alpha$$ and $$\beta$$ are decreasing in $$\underline{s}$$, and, thus, they are invertible. Moreover, the intuitive properties of MLRP imply the following relationships when $$\beta$$ is expressed as a function of $$\alpha$$ (see, e.g., Demougin and Fluet (2005, pp. 198–9) and Fluet and Mungan (2017)): $$\beta(0)=0,\ \beta(1)=1,\ \beta(\alpha)>\alpha\text{ for all }\alpha \in(0,1)\text{, }\beta^{\prime}>0\text{, and }\beta^{\prime\prime}<0$$ (8) The intuition behind the concavity of $$\beta$$ is that guilty individuals are more likely than innocent individuals to emit large signals, and vice versa. Thus, when the standard is strong (i.e., when $$\beta$$ and $$\alpha$$ are close to $$0$$) a change in the standard affects $$\beta$$ more than it affects $$\alpha$$, and when the standard is weak (i.e., $$\alpha$$ and $$\beta$$ are close to $$1$$) a change in the standard affects $$\beta$$ less than $$\alpha$$. Thus, the change in $$\beta$$ relative to the change in $$\alpha$$ is decreasing as the standard is weakened (i.e. $$\beta^{\prime\prime}<0$$). Given the properties listed in expression (8), it immediately follows that there is a unique $$\alpha^{m}$$ such that $$\alpha^{m}=\underset{\alpha}{\arg\max}[\beta(\alpha)-\alpha]$$ (9)$$\alpha^{m}$$ is noted separately, because the standard that generates this probability of type-I error is often called preponderance of the evidence in the standard of proof literature. This is because using it would require a finding of liability in court, whenever the signal is more likely to be produced by a guilty individual rather than an innocent individual. In what follows, I use this standard as a benchmark, and use the following terms to describe results. Definition 1 A standard is weak if it results in $$\alpha>\alpha^{m}$$, and it is strong if it results in $$\alpha \leq\alpha^{m}$$. A standard leads to suspicionless stops if it results in $$\alpha=1$$. The properties of $$\beta$$ listed in expression (8), $$\alpha^{m}$$, and the terms defined in definition 1 are all depicted in Figure 2, below. Figure 2 View largeDownload slide Properties of $$\alpha$$ and $$\beta$$. Figure 2 View largeDownload slide Properties of $$\alpha$$ and $$\beta$$. The shape of $$\beta(\alpha)$$ around $$\alpha=1$$ also plays an important role, and has an intuitive meaning. In general, $$\frac{d(\beta(\alpha)-\alpha )}{d\alpha}=\beta^{\prime}(\alpha)-1$$ can be thought of as the gain or loss in the accuracy of a search caused by a weakening of the standard, since $$\beta-\alpha$$ represents the gap between the probabilities of stopping the guilty and the innocent. Therefore, $$1-\beta^{\prime}(1)$$ can be interpreted as the accuracy loss associated with suspicionless stops, because $$\beta-\alpha$$ can be increased by exactly this amount by infinitesimally strengthening the standard if one starts with suspicionless stops. The properties of $$\beta$$, listed in expression (8), imply that $$1>\beta^{\prime}(1)>0$$, and for each value $$a\in(0,1)$$ one can easily conceive of a functional form such that $$\beta^{\prime}(1)=a$$.17 Thus, the accuracy losses associated with suspicionless stops can vary greatly depending on the shape of $$\beta$$. This observation is useful in characterizing optimal stopping standards. 4.2. Optimal Standards When there are no inconvenience costs, as previously explained, suspicionless stops are optimal. However, when $$c>0$$, weaker stopping standards reduce the expected costs from committing crime, but, they also increase the expected costs associated with remaining innocent. Therefore, the effect of $$\alpha$$ on deterrence is more complex than in the case where $$c=0$$. However, if one starts with a strong standard, that is, $$\alpha<\alpha^{m}$$, making the standard slightly weaker leads to an increase in the gap between the probabilities of stopping guilty and innocent individuals (i.e., $$\beta-\alpha$$ is increased, as depicted in Figure 2). Moreover, the expected punishment associated with being stopped when guilty is naturally greater than the inconveniences one faces by being stopped while innocent. Thus, for $$\alpha\leq\alpha^{m}$$, an increase in $$\alpha$$ raises the expected costs to committing crime more than it increases the expected costs associated with remaining innocent, and thereby enhances deterrence. When standards are weak, increasing $$\alpha$$ beyond $$\alpha^{m}$$ causes the gap between $$\beta$$ and $$\alpha$$ to get smaller. This is detrimental to deterrence, because it amounts to a reduction in the marginal expected inconvenience cost from committing crime (i.e., a reduction in $$(\beta -\alpha)c$$). Moreover, the reduction in $$\beta-\alpha$$ is a function of the accuracy loss associated with using weaker standards, and these losses are greatest when one uses suspicionless stops due to the concavity of $$\beta$$. Thus, if the accuracy losses associated with suspicionless stops are small, then the reductions in the marginal expected inconvenience costs from committing crime are completely offset by the increased likelihood of being sanctioned upon committing crime. Similarly, if inconvenience costs are small relative to the punishment for committing crime, the reduction in the marginal expected inconvenience costs from committing crime is dwarfed by the increased expected sanction for committing crime. These observations are formalized by the following lemma. Lemma 1 (i) The deterrence maximizing stopping standard is weak. (ii) Deterrence is maximized by suspicionless stops if, and only if, $\frac{\beta^{\prime}(1)}{1-\beta^{\prime}(1)}\geq(1-x^{w})\frac{c}{f}$ Proof. See Appendix. □ Lemma 1 illustrates that, although deterrence is always maximized by weak standards, it may or may not be maximized by suspicionless stops. In particular, when the accuracy losses associated with suspicionless stops are large, or if marginal inconvenience costs are large, then, maximizing deterrence requires the use of weak standards, but not suspicionless stops. Lemma 1 also verifies the previously made claim that deterrence is maximized by suspicionless stops when $$c=0$$. Deterrence is only one of the three factors that affects the optimality of policies. In particular, the stopping standard also impacts the frequency with which harms are prevented, and the aggregate inconvenience costs inflicted on people. A quick look at equation (6) reveals that using weaker standards always increases the probability of harm prevention, which equals $$\beta x^{w}$$. On the other hand, the direct impact of $$\$$weaker standards on inconvenience costs (given by $$x^{w}c(\theta\beta+(1-\theta)\alpha)$$) is also always positive, because weaker standards lead to more people being stopped. However, when the stopping standard is strong, these costs are dominated by preventive benefits as long as marginal inconvenience costs are small compared to criminal harms, as stated in assumption 1. Therefore, welfare, like deterrence, is maximized by weak standards. To characterize the optimal standard, note that increasing $$\alpha$$ beyond $$\alpha^{m}$$ causes an increase in $$\beta$$ that is smaller than the increase in $$\alpha$$. This implies that aggregate inconvenience costs are more responsive to changes in the standard than preventive benefits. In fact, there are minimal preventive gains in cases where the accuracy losses from conducting stops based on weaker evidence is extreme (since $$\beta^{\prime}$$ is close to zero), but the increase in inconvenience costs can still be substantial (since they are proportional to $$\theta\beta+(1-\theta)\alpha$$). Thus, due to reasons similar to those described in explaining deterrence effects, whether or not increasing $$\alpha$$ beyond $$\alpha^{m}$$ leads to positive preventive benefits net of inconvenience costs depends on the accuracy losses associated with using weaker stopping standards as well as the magnitude of inconvenience costs. This is formalized by the following proposition. Proposition 4 (i) Welfare is maximized by weak standards. (ii) Suspicion-based stops can be optimal. (iii) Suspicionless stops are optimal, if either $$c$$ or the accuracy losses from suspicionless stops are sufficiently small. Proof. See Appendix. □ Part (iii) of proposition 4 verifies the previously made claim that suspicionless stops are optimal when there are no inconvenience costs. This result combined with part (ii) of proposition 4 reveals that inconvenience costs are necessary to justify the use of suspicion-based stops. Although proposition 4 focuses on identifying simple conditions under which both deterrence and welfare are maximized by suspicionless stops, additional conditions can be identified. In particular, when the attempt rate is very high, inconvenience costs, like preventive benefits, are almost exclusively affected by the rate at which guilty people are stopped. Therefore, making the stopping standard weaker increases welfare by enlarging the gap between preventive benefits and inconvenience costs. Moreover, this gap is the primary consideration when the population’s criminal behavior is relatively unresponsive to expected sanctions. Thus, suspicionless stops are optimal when the attempt rate is high and the population is unresponsive to sanctions. 4.3. Interactions between $$\alpha$$ and $$x$$ The analysis in this section takes the optimal investment in preventive enforcement, $$x^{w}$$, as given, just as Section 3 takes the stopping standard as given in determining the optimal investment in preventive enforcement. An interesting question is whether these two policy variables complement or substitute each other. Unfortunately, this question cannot be answered without imposing additional restrictions on the shape of the criminal benefit distribution or the nature of the tradeoff between preventive and nonpreventive enforcement methods. This is primarily due to two reasons. First, absent such restrictions, an increase in $$\alpha$$ causes a change in the marginal deterrence effect of $$x$$ whose sign and magnitude are ambiguous, because this value is affected by the curvature of the criminal benefit distribution in the population, as well as the properties of $$\beta$$. Second, even in cases where deterrence and second-order effects are small, making the stopping standard weaker has two conflicting effects on the marginal impact of preventive enforcement: it increases the marginal preventive benefits associated with an increase in $$x$$ while simultaneously leading to an increase in the marginal inconvenience costs caused by it. The comparison of the two effects hinges upon many factors, including the attempt rate, the accuracy gains from strengthening the stopping standard being considered, and the magnitude of the net criminal losses to be prevented relative to the size of inconvenience costs. A larger value associated with any of these three factors makes it more likely for weaker standards to increase the effectiveness of preventive enforcement. These ambiguities make it difficult to ascertain the precise welfare interactions caused by simultaneously setting $$\alpha$$ and $$x$$. This is why the analyses in Sections 3 and 4 focus on identifying features of an optimal enforcement regime, without relying on specific interactions between these two choice variables. Future research focusing on more specific functional forms may shed more light on the interactions between these two policy variables. 5. Conclusion Despite its frequent use, preventive law enforcement has not been analyzed much in the law and economics literature. This article highlights the pivotal role inconvenience costs play in the determination of optimal preventive enforcement policies. Specifically, the optimal policy converges toward net criminal harm minimization as inconvenience costs get smaller, and it is closer to deterrence maximization when inconvenience costs are large. Inconvenience costs also play a crucial role in the determination of optimal stopping standards. Absent inconvenience costs, it is hard to justify a policy of requiring reasonable suspicion to conduct stops, because it is always optimal to use suspicionless stops. However, when inconvenience costs are present, stops based on reasonable suspicion can be justified. Thus, ignoring inconvenience costs can lead to an inefficiently frequent usage of preventive enforcement, and may cause stops to be administered through undesirably weak standards. The effects of some important factors on optimal enforcement schemes are ambiguous in the instant setting. It is unclear, for instance, how the magnitude of punishment costs are likely to affect the analysis, how the frequency of preventive enforcement is likely to affect enforcer misconduct, or how the optimal frequency of preventive enforcement is likely to be affected by various criminal investigation procedures. Therefore, future research is required to gain a better understanding of these considerations. I thank Abraham Wickelgren, two anonymous referees, and the participants of the 26th Annual Meeting of the American Law and Economics Association for valuable comments and suggestions Appendix Footnotes 1. Bentham (1843). 2. See Terry v. Ohio, 392 U.S. 1 (1968). 3. Incapacitation has similar functions to those served by preventive enforcement methods. However, preventive methods aim at precluding harms in the present or very near future, rather than harms inflictable through recidivism. Thus, a difference between incapacitation and harm prevention is that the former often requires the commission of a preceding offense. 4. Floyd v. City of New York, 959 F.Supp.2d 540, 558–9 (2013). 5. Costs associated with imprisonment can be regarded as negative outputs. There are articles that consider such costs, and many others that consider fines which are mere transfers. Polinsky and Shavell (2007) contains a review of both types of models. 6. Some inputs are general (e.g., police cars) in that they can be used to serve many enforcement functions, whereas others may be specific (e.g., specialized forensics units). Although this distinction can be important in making long-term investment decisions, it is not central to the current analysis. The current focus is on the optimal allocation of inputs across preventive versus non-preventive law enforcement methods, regardless of whether they are specific or general. 7. Another issue in measurement is whether criminals’ benefits ought to be accounted for. Curry and Doyle (2016) explains this issue, and contributes to the existing debate. 8. In section 3, $$\theta$$ denotes the attempt rate and $$x\beta$$ (the product of the likelihood of being stopped ($$\beta$$), and the intensity of the search ($$x$$)) denotes the prevention rate. Using this notation, the completed crime rate corresponds to $$(1-x\beta)\theta\,$$. Chapman et al. (1975) discusses issues related to output measurement, and proposes a measurement of the “relative prevented crime rate” (RPCR) which is the difference between the crime rate that would emerge without any enforcement and the completed crime rate that emerges with enforcement. Thus, his definition of RPCR corresponds to $$\theta^{n} -(1-x^{e}\beta^{e})\theta^{e}$$ in the instant article, where $$e$$’s indicate values with enforcement and $$\theta^{n}$$ indicates the natural crime rate. 9. See, e.g., Mungan (2012) explaining the focus on deterrence in the existing literature. 10. Friehe and Tabbach (2013, p. 3) makes this distinction as follows: “Incapacitation targets specific individuals (convicted criminals), seeking to prevent them from committing crimes in the future. In contrast, preventive enforcement is not aimed at specific individuals, but rather at preventing the commission of specific acts that represent imminent threats.” 11. Here, a simplifying assumption that does not affect the analysis is that attempts and results are punished equally severely. 12. This expression reflects the assumption that completion of the crime is necessary to receive the benefits from crime. This is similar to the case considered in Mungan and Klick (2014) where the offender receives no criminal benefits when caught red-handed. 13. Guilty refers to a person who has initiated crime, and innocent to one who has not initiated crime. 14. A violation of assumption 1 leads to interesting, yet, unrealistic results. In particular, if $$c$$ is sufficiently high, it may be optimal to invest less than the deterrence maximizing amount in preventive enforcement. 15. See Mungan (2017) for a more detailed discussion of multiple local maxima. 16. Demougin and Fluet (2005) takes a similar approach to abbreviate expressions. 17. A simple example is $$\beta=\alpha^{a}$$. 18. To see this, note that the derivative of the RHS with respect to $$\alpha$$ is $$-c\beta^{\prime\prime}\frac{1-x^{w}\beta }{(\beta^{\prime})^{2}}>0$$ since $$\beta^{\prime\prime}<0$$. 19. When $$c$$ is not constrained to be smaller than $$\overline{c}$$, it follows that weak standards need not be optimal if $$c$$ is sufficiently large. Assumption 1 rules this possibility out. References Bentham, J. 1843 . “Principles of Penal Law” in Bowring, J. ed., The Works of Jeremy Bentham. Vol. 1. Edinburgh : William Tait , 367 – 580 . Braga, A. 2008 . Crime Prevention Research Review No. 2: Police Enforcement Strategies to Prevent Crime in Hot Spot Areas. Washington, DC : US Department of Justice Office of Community Oriented Policing Services. Curry, P. and Doyle. M. 2016 . “Integrating Market Alternatives into the Economic Theory of Optimal Deterrence,” 54 Economic Inquiry 1873 – 83 . Google Scholar Crossref Search ADS Chapman, J, Hirsch, W. and Sonenblum. S. 1975 . “Crime Prevention, the Police Production Function, and Budgeting” 30 Public Finance 197 – 215 . Demougin, D. and Fluet. C. 2006 . “Preponderance of Evidence.” 50 European Economic Review 963 – 76 . Google Scholar Crossref Search ADS Demougin, D. and Fluet. C. 2005 . “Deterrence versus Judicial Error: A Comparative View of Standards of Proof.” 161 Journal of Institutional and Theoretical Economics 193 – 206 . Google Scholar Crossref Search ADS Fluet, C. and Mungan. M. 2017 . “The Signal-Tuning Function of Liability Regimes” George Mason Law & Economics Research Paper No. 17–37. Friehe, T., and Tabbach. A. 2013 . “Preventive Enforcement,” 35 International Review of Law and Economics 1 – 12 . Google Scholar Crossref Search ADS Kaplow, L. 2011 . “On the Optimal Burden of Proof” 119 Journal of Political Economy 1104 – 40 . Google Scholar Crossref Search ADS Kennedy D. 2009 . “Deterrence and Crime Prevention: Reconsidering the Prospect of Sanction” New York : Routledge. Lando, H. 2009 . “Prevention of Crime and the Optimal Standard of Proof in Criminal Law” 5 Review of Law and Economics 33 – 52 . Google Scholar Crossref Search ADS Miceli, T. 1990 . “Optimal Prosecution of Defendants whose Guilt is Uncertain.” 6 Journal of Law Economics and Organization 189 – 201 . Miceli, T. 2010 . “A Model of Criminal Sanctions that Incorporate both Deterrence and Incapacitation” 107 Economics Letters 205 – 7 . Google Scholar Crossref Search ADS Miceli, T. 2012 . “Deterred or Detained? A Unified Model of Criminal Punishment” 8 Review of Law and Economics 1 – 20 . Google Scholar Crossref Search ADS Mungan, M. 2011 . “A Utilitarian Justification for Heightened Standards of Proof in Criminal Trials” 167 Journal of Institutional and Theoretical Economics 352 – 70 . Google Scholar Crossref Search ADS Mungan, M. 2012 . “The Law and Economics of Fluctuating Criminal Tendencies and Incapacitation” 72 Maryland Law Review 156 – 218 . Mungan, M. 2017 . “Optimal Preventive Law Enforcement and Stopping Standards” FSU College of Law, Public Law Research Paper No. 701. Mungan, M., and Klick. J. 2014 “Forfeiture of Illegal Gains, Attempts, and Implied Risk Preferences” 43 Journal of Legal Studies 137 – 53 . Google Scholar Crossref Search ADS Nagin, D. 2013 . “Deterrence: A Review of the Evidence by a Criminologist for Economists.” 5 Annual Review of Economics 83 – 105 . Google Scholar Crossref Search ADS Nagin, D., Solow, R. and Lum. C. 2015 . “Deterrence, Criminal Opportunities, and Police.” 53 Criminology 74 – 100 . Google Scholar Crossref Search ADS Polinsky, M. and Shavell. S. 2007 . “Public Enforcement of Law” in Polinsky A. M. and Shavell, S. eds. Handjournal of Law and Economics , Amsterdam, Netherlands : Elsevier. 403 – 54 . Rizzolli, M. and Saraceno. M. 2013 . “Better That Ten Guilty Persons Escape: Punishment Costs Explain the Standard of Proof.” 155 Public Choice 395 – 411 . Google Scholar Crossref Search ADS Shavell, S. 1984a . “Liability for Harm versus Regulation of Safety” 13 Journal of Legal Studies 357 – 74 . Google Scholar Crossref Search ADS Shavell, S. 1984b . “A Model of the Optimal Use of Liability and Safety Regulation” 15 Rand Journal of Economics 271 – 80 . Google Scholar Crossref Search ADS Shavell, S. 1987 . “A Model of Optimal Incapacitation” 77 American Economic Review 107 – 10 . Shavell, S. 2015 . “A Simple Model of Optimal Deterrence and Incapacitation.” 42 International Review of Law and Economics 13 – 19 . Google Scholar Crossref Search ADS Sherman, L., Gartin P. and Buerger. M. 1989 . “Hot Spots of Predatory Crime: Routine Activities and the Criminology of Place” 27 Criminology 27 – 55 . Google Scholar Crossref Search ADS Sherman L. and Weisburd. D. 1995 . “General Deterrent Effects of Police Patrol in Crime ‘Hot Spots’: A Randomized Study” 12 Justice Quarterly 625 – 48 . Google Scholar Crossref Search ADS © The Author(s) 2018. Published by Oxford University Press on behalf of the American Law and Economics Association. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png American Law and Economics Review Oxford University Press

# Optimal Preventive Law Enforcement and Stopping Standards

American Law and Economics Review, Volume 20 (2) – Oct 1, 2018
29 pages

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Publisher
Oxford University Press
ISSN
1465-7252
eISSN
1465-7260
D.O.I.
10.1093/aler/ahy003
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See Article on Publisher Site

### Abstract

Abstract Preventive law enforcement increases social welfare by hindering the infliction of criminal harm, but produces inconvenience costs to the general public, because it requires interfering with the acts of innocents as well as attempters. The optimal amount of investment in preventive enforcement is greater than that which maximizes deterrence, but, smaller than that which minimizes criminal harm. Thus, ignoring preventive benefits and/or inconvenience costs results in an inefficient investment portfolio over enforcement methods, and in a predictable manner. Stopping standards, which determine the threshold suspicion required to trigger a stop, are tools that can be used to optimally trade-off the costs and benefits associated with preventive enforcement. The optimal stopping standard is weaker than its analogs in the trial context, namely standards of proof, which generally require preponderance of the evidence in civil trials and proof beyond a reasonable doubt in criminal trials. Finally, suspicionless stops can be optimal in a variety of circumstances, and are more likely optimal when enforcers perform poorly in forming suspicions; inconvenience costs are small; the population is unresponsive to deterrence measures; and the attempt rate is high. There is one limit which ought never to be neglected: “No method of prevention should be employed, which is likely to cause a greater mischief than the offence itself.”1 –Jeremy Bentham 1. Introduction Why is it that Terry stops2 can legally be conducted only upon reasonable suspicion, but that vehicles at roadblocks and sobriety checkpoints can legally be stopped without suspicion? Is there a good reason to have reduced standards for detaining people suspected of taking part in terroristic activity? Is there a good reason to increase policing efforts, even when criminal activity is unresponsive to the presence of police officers, as may be the case in many “hot spots,” at infamous derbys, or at crowded outdoor celebrations and events? The objective of this article is to provide an economic framework that can be used in answering these difficult questions. This requires exploring a function of law enforcement, which is virtually ignored in the law and economics literature, namely harm prevention. The most commonly studied objective of enforcement by economists, namely deterrence, aims to reduce the proportion of individuals who engage in criminal acts. On the other hand, some preventive enforcement measures aim to stop ongoing crimes, and thereby reduce the proportion of attempted crimes that result in harm. This objective, namely harm prevention, can therefore be thought of as reducing the proportion of crimes—committed by undeterred individuals—that end up causing harm.3 There are many enforcement methods whose primary goal is harm prevention. For instance, sobriety checkpoints reduce the harm inflicted through driving under the influence offenses by preventing the driver from continuing to drive. In contrast, resources spent on crime scene investigations are generally meant to solve previously committed crimes, and thus, they serve the function of increasing the probability of ex post detection and punishment. Preventive and nonpreventive methods alike can be expected to have deterrence effects, lead to incapacitation of the offender, and lead to wrongful convictions. Thus, these features do not lead to sharp distinctions between preventive and nonpreventive methods. However, because preventive enforcement methods have to rely on information obtained prior to the infliction of harm, it is often quite difficult to ascertain whether the suspect is in fact guilty of carrying out a crime, and enforcers have to, very frequently, target innocent people. For instance, only 6% of the 4.4 million Terry stops conducted by the New York Police Department between 2004 and 2012 resulted in arrests.4 These stops undeniably caused inconveniences for the stopees. Hence, obtaining the benefits of harm prevention comes at the cost of generating sizeable inconvenience costs to the general public. It may be argued that ex post criminal investigations also cause inconvenience costs. Similarly, ex post investigations may also result in some preventive benefits, especially in instances where the target is a professional criminal, who may be in the process of committing another crime. Although these statements are undeniably true, as long as there is some correlation between the emphasis an investment in enforcement places on prevention and the amount of inconvenience costs generated by such investments, one can meaningfully talk about diverting investments from less preventive methods toward more preventive methods. In such instances, a trade-off emerges between preventive benefits and inconvenience costs, which can be addressed by altering the investment portfolio over various enforcement methods. To analyze whether the presence of inconvenience costs have an important impact on how enforcement mechanisms ought to be designed, I construct a simple law enforcement model where different investment portfolios over enforcement mechanisms have different prevention and deterrence consequences. Then I ask whether the size of inconvenience costs affects the optimal investment in preventive versus nonpreventive enforcement methods. The first result is that the optimal investment in preventive enforcement is higher than that which maximizes deterrence, but lower than that which minimizes the amount of criminal harm generated through illegal behavior. Moreover, absent inconvenience costs, the optimal investment in prevention coincides with that which minimizes criminal harm, and as inconvenience costs increase, the optimal investment in prevention moves toward the deterrence maximizing enforcement scheme. Thus, the size of inconvenience costs determines whether the government ought to implement a policy geared primarily toward harm minimization or toward deterrence maximization. Although the model focuses primarily on inconvenience costs to stopees, it can also be used to identify other factors that affect the desirability of using preventive enforcement methods. An important factor is the size of the externalities generated by stops to the general public. For instance, increasing the length of a stop not only increases the amount of time that a stopee is investigated, but also increases the amount of time each person in line at a security check point has to wait to be investigated. Naturally, when these externalities are large, preventive enforcement methods become less attractive. Another factor is the proportion of stops that actually lead to some preventive benefits (i.e., accurate stops), and this depends on the characteristics of the population being regulated: the lower the criminal tendencies of the population, the smaller is this proportion. Similarly, the benefit of using preventive enforcement is increasing in the expected social harm to be prevented. The foregoing simple observations can be used to explain why we use preventive enforcement frequently at airports, hot spots, and high crime areas; as well as why it is used to reduce social harms caused by the mentally ill or hooligans at sports events. All results described above relate to how much preventive enforcement ought to be used, but they do not explain what standard ought to guide whom to stop. To discuss this issue, I define a stopping standard, which refers to the threshold quality that a noisy signal regarding the guilt of an individual must posses to trigger an investigation. Investigating the optimal stopping standard reveals that it is weaker than what is called the “preponderance of the evidence” standard used in civil trials, and much weaker than the “proof beyond a reasonable doubt” standard. This is because a wrongful stop, on its own, causes an inconvenience which is typically smaller than the cost of being falsely found liable. However, the benefit from an accurate investigation is not only deterrence but also harm prevention. Thus, stopping standards are typically weaker than standards of proof used in trials. Moreover, if enforcers perform very poorly in forming suspicions (perhaps because the signal they receive regarding people’s behavior is very noisy), then it is optimal to employ the weakest standard possible, and use suspicionless stops. A similar result holds when inconvenience costs are sufficiently small, illustrating the importance of inconvenience costs in justifying stops based on suspicion. The remainder of this article is structured as follows. Section 2 provides a literature review. Section 3 presents and analyzes a law enforcement model with preventive and nonpreventive law enforcement methods. Section 4 introduces stopping standards and derives optimality conditions. Section 5 contains concluding remarks, and an Appendix at the end contains proofs of propositions. 2. Literature Review An issue in assessing the effectiveness of different law enforcement methods pertains to how one measures enforcement inputs and outputs. Although this issue was noted by Chapman et al. (1975) more than forty years ago, most of the theoretical law enforcement literature has focused on a single-dimensional input for law enforcement (money), and generally a single output (deterrence).5 However, enforcement inputs can be used to serve different functions of law enforcement (e.g., employing crime scene investigators to solve completed crimes versus establishing hot spots to increase prevention).6 As highlighted in this article, one can identify meaningful factors that affect the optimal allocation of inputs toward different enforcement methods. Moreover, one can conceive of different ways to measure outputs. For instance, one can measure the deterrent effect of enforcement, or the harm reducing effect of enforcement.7 The former could be measured by changes in the attempt rate whereas the latter would need to measure changes in the completed crime rate.8 Even then, as the current article highlights, it is important to include negative outputs: just like a factory generates pollution as a by-product of its activity, law enforcement generates inconvenience costs, and these ought to be included in measuring outputs. A failure to do so can result in excessive investments in preventive enforcement. The theoretical literature implicitly proposes ways to measure outputs when they formulate social welfare functions to evaluate the performance of law enforcement mechanisms. However, they heavily focus on deterrence in doing so, with a small number of exceptions.9 In particular, Shavell (1987, 2015) and Miceli (2010, 2012) focus on incapacitation, and highlight a trade-off between the costs of imprisonment and the risk of an individual committing further crimes if released. Friehe and Tabbach (2013) formalize preventive enforcement, and highlight that this type of enforcement aims at eliminating harms from a specific act, and not the future harms that are likely to be caused by the individual who has demonstrated his dangerousness,10 as in the case of incapacitation. This distinction is important, and reveals a fundamental difference between the structure of the errors generated through investigations necessary for prevention versus errors generated through punishment. Prevention requires acting on very noisy signals, because, by definition, harm has not yet been inflicted. Thus, small but frequent errors are almost unavoidable in implementing preventive enforcement methods. On the other hand, the decision to punish a defendant is made either after he has completed his crime, or after he has been identified as an attempter. Moreover, an erroneous decision at this stage amounts to a wrongful conviction, and, thus carries larger costs, which have been studied in the literature (Lando 2009; Kaplow 2011; Mungan 2011; Rizzolli and Saraceno 2013). The latter type of errors are a consequence of punishment, and, therefore, are generated by all enforcement methods that lead to punishment; preventive and nonpreventive alike. Thus, to preserve the focus on what most discretely characterizes preventive enforcement methods, the current article incorporates inconvenience costs and preventive benefits, and does not explicitly model type-I and type-II errors associated with punishment. Although erroneous liability costs have been studied extensively in the optimal standard of proof literature, a similar analysis of standards that can be used to affect inconvenience costs is presented for the first time in this article. Thus, the current article also fills a gap in the literature by extending the analysis of standards of proof to cases where the standard is used to determine whether a person should be merely investigated, instead of whether he ought to be found liable. It demonstrates that the optimal stopping standard is lower than the optimal standard of proof in civil (Demougin and Fluet, 2005, 2006) as well as criminal trials (Miceli 1990). This is because the ratio between inconvenience costs and preventive benefits is typically smaller than the relevant ratios in the liability contexts where the standard of proof becomes applicable. It is also worth comparing inconvenience costs to judgment errors considered in the context of ex ante regulation by Shavell (1984a,b). In both cases, errors are by-products of the imperfection of the monitoring/investigation system employed by law enforcers to prevent crime before it occurs. However, a large proportion of inconvenience costs are incurred by innocent parties, and, therefore, they are present even in cases where judgment errors lead to benefits instead of costs. For instance, ex ante regulation may lead to increased deterrence, if a regulator misjudges the characteristics of a regulatee and thereby subjects him to more stringent standards (as in Shavell 1984a). This type of deterrence enhancing judgment error leads to net benefits when the offenses to be deterred are socially undesirable, yet, similar errors in the investigation context can reduce welfare even if they enhance deterrence. Another important issue to note is that inconvenience costs are a function of the crime rate, which is naturally related to the population’s or region’s characteristics. This aspect is closely related to points made by several criminology scholars regarding the spatial aspects of law enforcement (Braga 2008; Kennedy 2009; Nagin 2013; Nagin et al. 2015; Sherman et al. 1989 and Sherman and Weisburd 1995): ex post enforcement can be used non-discriminatorily to solve crimes that were committed in any region, whereas the locations in which preventive enforcement methods are to be used must be chosen strategically. In line with these observations, the current article suggests that preventive enforcement ought to be used more frequently in regions in which the crime rate is higher. Finally, and most importantly, this article builds on Friehe and Tabbach (2013), which is the first economics article to specifically analyze preventive enforcement methods. The instant article, like Friehe and Tabbach (2013), focuses on the deterrence and prevention functions of different enforcement methods, and subsequently furthers the analysis in several dimensions. First, it incorporates inconvenience costs, which are necessary to meaningfully analyze stopping standards. It also identifies optimal stopping standards, as well as conditions under which suspicionless stops are optimal. Moreover, it compares levels of investments in preventive enforcement which serve different goals (e.g., deterrence maximization, criminal harm mitigation, and social welfare maximization). It thereby illustrates the pivotal role that inconvenience costs play in determining whether the optimal policy is closer to deterrence maximization or criminal harm mitigation. 3. Enforcement Schemes in the Presence of Inconvenience Costs 3.1. Model I consider a continuum of individuals who derive benefit $$b$$ from successfully completing criminal acts that cause social harm $$h$$. These benefits vary from person to person, and, $$g$$ and $$G$$ with support over the interval $$b\in \lbrack\underline{b},\overline{b}]\subset\lbrack0,h]$$, respectively denote the density and cumulative distribution of $$b$$. After a person initiates the criminal act, but prior to completing it, he may be stopped (with probability $$\beta$$) as a result of preventive law enforcement efforts. If stopped, investigations may reveal, with probability $$x\in\lbrack0,\overline{x}]\subset\lbrack0,1)$$, that he is in the process of committing a crime. This probability is proportional to the length (or intensity) of the stop. With the residual probability $$1-x$$, the attempter is released. The stop causes the stopee an inconvenience cost of $$xc$$, where $$c$$ represents the marginal inconvenience cost to the stopee from lengthier stops. If the person is not stopped or is released, he carries out his crime to completion. However, he may also be caught and punished after committing the crime with probability $$p$$. Lengthier stops cause the enforcement agency to reduce the efforts that it devotes toward ex post investigations. This is because the enforcement agency allocates a fixed amount of resources across preventive investigations and ex post investigations. Resources allocated toward preventive investigations are increasing in $$x$$, and, because the resource constraint is binding, the remaining resources allocated towards ex post investigations are decreasing in $$x$$. This is formalized by assuming that $$p = p(x)$$ and that $$p^{\prime} < 0$$. Moreover, positive investments in preventive enforcement are guaranteed by assuming that $$\underset{x\rightarrow\overline{x}}{\lim} p^{\prime}(x)=-\infty$$. If a person is caught either after or in the process of committing a crime he pays a fine of $$f$$.11 Thus, a person’s expected benefit from initiating a crime is:12 $$(1-x\beta)(b-p(x)f)-x\beta(c+f)$$ (1) A person who does not initiate a criminal act may nevertheless be the subject of a preventive stop with probability $$\alpha\leq\beta$$, because law enforcers cannot perfectly distinguish between innocent and guilty individuals.13 The gap between $$\beta$$ and $$\alpha$$ can thus be interpreted as the degree to which the stopping process is able to discriminate between guilty and innocent individuals, or, the randomness of a stop. If stopped, the innocent person suffers the same inconvenience cost as a guilty individual. Thus, an innocent individual’s expected payoff is $$-\alpha xc$$. Therefore, a person initiates the act if: \begin{align} b^{\ast}(x)\equiv x\frac{(\beta-\alpha)c+\beta f}{1-x\beta}+p(x)f<b \end{align} (2) I assume some, but not all, people are deterred, that is, $$b^{\ast}\in (\underline{b},\overline{b})$$ for all $$x$$. A higher degree of deterrence, naturally corresponds to a higher $$b^{\ast}$$, since then condition (2) is satisfied for fewer people. This can be noted by defining the attempt rate, $$\theta$$, as follows: $$\theta(x)\equiv1-G(b^{\ast}(x))$$ (3) It is worth highlighting that while $$1-\theta$$ represents the rate of deterrence, $$x\beta$$ represents the probability (or the degree) of harm prevention, and, hence the proportion of attempted crimes that are prevented through enforcement efforts. Inspecting expression (2) reveals that increasing the resources devoted to preventive enforcement enhances deterrence by increasing the cost of being caught prior to completing one’s attempt, but leads to a counter-effect by reducing the expected costs of being caught subsequent to completing one’s crime. Absent further restrictions, the relationship between these two effects is ambiguous and one cannot conclude that deterrence is maximized by an interior and unique choice of $$x$$. To simplify the analysis, I assume that $$p$$ has a functional form that guarantees that $$b^{\ast}$$ is single peaked in $$x$$, such that it has a unique and interior maximizer. These assumptions imply that increasing the resources devoted to preventive enforcement increases deterrence only up to a unique level of $$x$$, denoted as $$x^{d}$$, defined as follows: $$x^{d}\equiv\underset{x}{\arg\max}b^{\ast}(x)$$ (4) 3.2. Inconvenience Costs The total inconvenience costs generated by preventive enforcement is naturally a function of the length of each search ($$x$$) and also the attempt rate, since attempters are generally stopped more frequently than innocent individuals. Specifically, aggregate inconvenience costs can be expressed as: $$xc(\alpha+\theta(x)(\beta-\alpha))$$ (5) A quick investigation of expression (5) reveals that increasing the investment in preventive enforcement has a direct and positive effect on inconvenience costs, since this leads to an increase in the amount of inconvenience suffered by each stoppee. However, for $$x<x^{d}$$, an increase in preventive enforcement enhances deterrence, and thereby reduces the attempt rate. This leads to a reduction in the number of individuals who are searched, and, thus, for $$x<x^{d}$$, an increase in preventive enforcement can actually reduce inconvenience costs. On the other hand, for $$x>x^{d}$$, the two effects go in the same direction, and, therefore, further investments in preventive enforcement increase aggregate inconvenience costs. The marginal inconvenience cost from lengthier stops, $$c$$, also has a similar feature. An increase in $$c$$ leads to a direct increase in inconvenience costs, but it also enhances deterrence (see expression (2), above). Thus, if the latter (deterrence) effect is stronger than the former direct effect, an increase in $$c$$ can lead to a reduction in aggregate inconvenience costs. It is possible, however, to identify conditions under which these possibilities can be ruled out. These points are noted by the following observation. Observation 1 (i) An increase in $$c$$ or $$x$$, holding all else constant, can lead to a reduction in aggregate inconvenience costs. (ii) But, aggregate inconvenience costs are unambiguously increasing in $$c$$ and $$x$$, if, either (a) stops are sufficiently random (i.e., $$\beta-\alpha$$ is small), or (b) the population is relatively unresponsive to expected sanctions (i.e., $$g(b^{\ast})$$ is small). (iii) Moreover, aggregate inconvenience costs are increasing in $$x$$ for all $$x>x^{d}$$. 3.3. Optimal Preventive Enforcement Inconvenience costs are only one component of a welfare function that consists of the sum of all individuals’ utilities. Specifically, in addition to inconvenience costs, social welfare incorporates the net-criminal harm from crime. Thus, social welfare can be expressed as: $$W=(1-x\beta)\int_{b^{\ast}(x)}^{\overline{b}}(b-h)g(b)db-xc(\alpha +\theta(x)(\beta-\alpha))$$ (6) A quick look at equation (6) reveals that a change in the amount of preventive enforcement affects welfare through three separate channels. First, it affects the attempt rate. This in turn impacts the total criminal harm caused by attempters as well as the number of individuals who suffer inconvenience costs from being stopped. Second, the inconvenience costs suffered by each individual is affected. Third, preventive enforcement reduces the rate at which attempters are able to complete their crimes and inflict social harm. These three effects are expressed, in the order they are described, in separate lines along with their explanations, below. $$\matrix{ {\matrix{ {{{dW} \over {dx}} = b_x^ * g({b^ * }(x))[(1 - x\beta )(h - {b^ * }(x)) + xc(\beta - \alpha )]} \cr } } \hfill & {\} \matrix{ {{\it{marginal\,deterrence\,benefits}}} \hfill \cr } } \hfill \cr {\matrix{ {} \hfill & {} \hfill \cr } - c(\alpha + \theta (x)(\beta - \alpha ))} \hfill & {\} \matrix{ {{\it{marginal\,inconvenience\,costs}}} \hfill \cr } } \hfill \cr {\matrix{ {\matrix{ {} \hfill & {} \hfill \cr } + \beta \int_{{b^ * }(x)}^{\bar b} {(h - b)} g(b)db} \cr } } \hfill & {\} \matrix{ {{\it{marginal\,preventive\,benefits}}} \hfill \cr } } \hfill \cr }$$ (7) An interesting observation is that marginal preventive benefits are inversely related to the amount of deterrence. This reveals three important points. First, in societies with low crime rates, there are small gains from prevention. Second, marginal preventive benefits are quasi-convex in $$x$$: they are minimized at $$x^{d}$$ and maximized when all enforcement resources are devoted to a single type of enforcement (i.e., either preventive or nonpreventive). This leads to the third point, which is that as long as the criminal harms preventable at the maximum level of deterrence are greater than total inconvenience costs, marginal preventive benefits are always greater than marginal inconvenience costs. Specifically, as long as assumption 1, below, holds, increasing the investments in preventive enforcement always leads to greater increases in preventive benefits than increases in inconvenience costs. Since inconvenience costs are considered to be small relative to harms from crime, assumption 1, below, is imposed in the remaining parts of the article to simplify the analysis.14 Assumption 1 $$c\leq\overline{c}\equiv\underset{\max b^{\ast}}{\overset{\overline{b}}{\int}}(h-b)g(b)db$$. Another, less obvious, point revealed by equation (7) is that social welfare may generally have multiple local maxima, because the size of marginal deterrence benefits (or losses) can fluctuate due to changes in the density of individuals who are on the margin, that is, $$g(b^{\ast})$$. In such cases, the local maximum which is generated by a low $$x$$ leads to a low-prevention/high-deterrence enforcement scheme, whereas the maximum generated by a high $$x$$ leads to a high-prevention/low-deterrence enforcement scheme. In such cases, inconvenience costs become particularly important, since increases in $$c$$ cause the low-deterrence/high-prevention scheme to become a less desirable option compared to the high-deterrence/low-prevention scheme. This is because the former strategy leads to more frequent inconvenience costs than the latter. Thus, in some cases, a very small increase in $$c$$ can cause large changes in the optimal investment in preventive enforcement.15 The importance of inconvenience costs in determining optimal preventive enforcement is not limited to cases where there are multiple local maxima. In particular, even when social welfare has a single local maximum, ignoring inconvenience costs (which corresponds to minimizing net criminal harms), or simply attempting to maximize deterrence can lead to inefficiencies. The next proposition highlights these possibilities, and compares the optimal investment in preventive enforcement to investments that serve other objectives commonly studied in legal scholarship. Proposition 1 The optimal investment in preventive enforcement, $$x^{w}$$, is (i) greater than that which maximizes deterrence, and (ii) smaller than that which minimizes net-criminal harm whenever $$c>0$$. Proof. See Appendix. □ Proposition 1 reveals that one ought to invest more in prevention than the amount necessary to maximize deterrence, because there are net preventive gains (i.e., preventive gains minus inconvenience costs) from each marginal dollar invested in prevention. Thus, it is optimal to invest in prevention until doing so generates marginal deterrence losses that equal the marginal net gains from prevention. Hence, focusing solely on deterrence would cause a suboptimal investment in preventive methods. Second, minimizing net criminal harms would result in above optimal investments in preventive enforcement, because this objective simply ignores a form of negative externalities, namely inconvenience costs, associated with investments in prevention. This acts as a reminder that ignoring small but frequent costs to third parties can lead to welfare losses. 3.4. Comparative Statics This subsection identifies conditions under which infinitesimal changes in $$h$$ and $$c$$ cause local maximizers to move unambiguously in one direction or another. To ascertain the effect of $$h$$ on the optimal enforcement scheme, note that each prevented crime brings about benefits that are proportional to the severity of the crime. However, deterrence benefits, too, are proportional to the harm from crime, and, as implied by proposition 1, when investments are chosen optimally, an increase in $$x$$ causes a reduction in deterrence. Thus, without further analysis, it is unclear whether increasing the investment in preventive enforcement would generate preventive benefits that more than offset the deterrence losses it would cause. Making an additional observation resolves this ambiguity: the optimal investment balances preventive benefits against inconvenience costs in addition to losses in deterrence. Thus, an increase in $$h$$ causes a greater increase in marginal preventive benefits compared to marginal deterrence losses, and thereby leads to an increase in the optimal investment in prevention. On the other hand, an increase in $$c$$ has a direct and positive effect on the inconvenience costs from more prevention. However, as noted earlier, expected inconvenience costs also have a deterrent effect. Thus, an increase in $$c$$ can cause a reduction in equilibrium inconvenience costs, if the deterrence effect more than outweighs the increase in the marginal inconvenience effect, as noted in observation 1. This effect is similar to one that is studied frequently in industrial organizations: a price increase can bring about a reduction in profits when it leads to a quantity effect that more than offsets the effect due to an increase in the profit margin. Therefore, a priori, it is impossible to make statements about the impact of $$c$$ on the optimal amount of preventive enforcement. However, when stops are conducted randomly, it follows that $$\beta=\alpha$$, and, thus, $$c$$ has no impact on deterrence, as can be inferred from expression (2), and, conversely, the crime rate has no impact on aggregate inconvenience costs, as equation (6) illustrates. A similar result holds when offenders are relatively unresponsive to expected sanctions, even when $$\beta\neq\alpha$$, since then the deterrent effect of $$c$$ is minimal, but it still affects per-person inconvenience costs. Thus, under either of these conditions, the optimal amount of preventive enforcement is negatively related to $$c$$. These results are reported in proposition 2, below. Proposition 2 Locally welfare maximizing investments in preventive enforcement are (i) increasing in $$h$$, and (ii) decreasing in $$c$$ if either $$\beta=\alpha$$, or if $$g(b^{\ast})$$ and $$|g^{\prime}(b^{\ast})|$$ are sufficiently small. Proof. See Appendix. □ An immediate implication of proposition 2 is that if social welfare is single peaked, the optimal investment in preventive enforcement is increasing in $$h$$ and decreasing in $$c$$ under the conditions identified in proposition 2. Moreover, as marginal inconvenience costs get larger, the social objective converges towards deterrence maximization. In such cases it is relatively harmless to simply target deterrence—an easier to measure output—than the maximization of welfare. This last observation, combined with proposition 1, demonstrates the pivotal role inconvenience costs play in the determination of optimal policies. Proposition 3 Suppose $$W$$ is single peaked in $$x$$ for all $$c\in\lbrack0,\overline{c}]$$ and that $$\beta-\alpha$$ is sufficiently small. Then, the social objective converges (i) toward net harm minimization as marginal inconvenience costs converge toward zero, and (ii) toward deterrence maximization as marginal inconvenience costs converge toward $$\overline{c}$$. (iii) Moreover, as marginal inconvenience costs increase, the optimal investment in prevention moves monotonically from that which minimizes net criminal harms toward that which maximizes deterrence. Proof. See Appendix. □ The findings of propositions 1 and 3, which are illustrated in Figure 1, below, rely on $$\beta-\alpha$$ being small. Therefore, an important question is whether it may be optimal to have stops that are relatively random, rather than those that attempt to discriminate between guilty and innocent individuals. Answering this question requires endogenizing the policy variable which determines $$\beta$$ and $$\alpha$$, namely the stopping standard. This is done in the next section. Figure 1 View largeDownload slide Optimal investment in prevention $$(x^w(c))$$ as a function of inconvenience costs when $$\beta = \alpha$$ and $$W$$ is single-peaked. Figure 1 View largeDownload slide Optimal investment in prevention $$(x^w(c))$$ as a function of inconvenience costs when $$\beta = \alpha$$ and $$W$$ is single-peaked. 4. Optimal Stopping Standards This section introduces stopping standards as a policy tool to adjust $$\beta$$ and $$\alpha$$. An intuitive relationship between stopping standards and these probabilities, which is formalized in Section 4.1., is that they are inversely related to the strength of the standard employed in conducting stops: a stricter requirement for conducting stops reduces the probability with which guilty as well as innocent individuals are stopped. This observation suffices to explain how, absent inconvenience costs, suspicionless standards become optimal, and thereby highlights the importance of inconvenience costs in providing a rationale for requiring enforcers to conduct stops only when there exists a reasonable suspicion that the potential stoppee is guilty. Absent inconvenience costs, there is simply no downside to using very weak standards. This is because, in these cases, deterrence as well as inconvenience costs are unaffected by the frequency of incorrect stops (i.e., $$\alpha$$), since innocent stopees are unharmed by stops. On the other hand, an increase in the probability of accurate stops, that is, $$\beta$$, increases the probability with which guilty people forfeit their criminal benefits, in addition to increasing the likelihood with which they are punished for committing crime. Thus, as expression (2) illustrates, when $$c=0$$, suspicionless stops maximize deterrence, since then $$b^{\ast}$$ is maximized when $$\beta$$ reaches its maximum at $$1$$. Moreover, when there are no inconvenience costs, the only social objective in addition to deterrence, is prevention. This objective is clearly best served by using suspicionless stops, because the probability of prevention, $$x\beta$$, is also maximized by setting $$\beta=1$$. Thus, absent inconvenience costs, both deterrence and welfare are maximized by suspicionless stops. This result highlights the pivotal role of inconvenience costs in providing a rationale for requiring a certain degree of suspicion to conduct stops. In particular, when inconvenience costs are introduced, increases in the probability with which innocent people are stopped are detrimental to both deterrence and social welfare. Thus, making the stopping standard weaker generates costs by increasing $$\alpha$$, which must be traded-off against the preventive and deterrence benefits obtained by increasing the probability of stopping guilty individuals more frequently. Even then, as the analysis reveals, the optimal stopping standard is weaker than standards of proof employed in the trial context, because inconvenience costs are smaller than the preventive gains against which they are balanced. Formalizing this point and highlighting subtler points requires the introduction of stopping standards. 4.1. Stopping Standards To formalize stopping standards, suppose that each individual emits a noisy signal $$s\in R$$ about whether he is in the process of carrying out a crime. Suppose that larger values of $$s$$ are indicators of guilt, because they are more likely to be emitted by guilty individuals than innocent individuals. Thus, an intuitive criterion that enforcers may use is to stop individuals only if they receive a sufficiently high signal from them, that is, they may set a threshold signal $$\underline{s}$$, such that an individual is stopped only if he emits a signal $$s\geq\underline{s}$$. Thus, increasing $$\underline{s}$$ corresponds to choosing a stronger standard, in the sense that enforcers require stronger suspicions to conduct stops. Such increases cause a reduction in both $$\alpha$$ and $$\beta$$ by reducing the likelihood with which a person emits a signal that exceeds the standard, that is, $$\frac{d\alpha}{d\underline{s}},\frac{d\beta}{d\underline{s}}<0$$. An intuitive property frequently assumed about the way signals are produced by innocent and guilty individuals is called the monotone likelihood ratio property (MLRP). In the current context, this property holds whenever the ratio between the likelihoods with which a guilty versus an innocent individual emits a particular signal is increasing in the strength of the signal. When the signal emitting processes satisfy MLRP, it is often more convenient to express $$\beta$$ as a function of $$\alpha$$, instead of expressing both probabilities as functions of $$\underline{s}$$.16 This simplification causes no ambiguities, because both $$\alpha$$ and $$\beta$$ are decreasing in $$\underline{s}$$, and, thus, they are invertible. Moreover, the intuitive properties of MLRP imply the following relationships when $$\beta$$ is expressed as a function of $$\alpha$$ (see, e.g., Demougin and Fluet (2005, pp. 198–9) and Fluet and Mungan (2017)): $$\beta(0)=0,\ \beta(1)=1,\ \beta(\alpha)>\alpha\text{ for all }\alpha \in(0,1)\text{, }\beta^{\prime}>0\text{, and }\beta^{\prime\prime}<0$$ (8) The intuition behind the concavity of $$\beta$$ is that guilty individuals are more likely than innocent individuals to emit large signals, and vice versa. Thus, when the standard is strong (i.e., when $$\beta$$ and $$\alpha$$ are close to $$0$$) a change in the standard affects $$\beta$$ more than it affects $$\alpha$$, and when the standard is weak (i.e., $$\alpha$$ and $$\beta$$ are close to $$1$$) a change in the standard affects $$\beta$$ less than $$\alpha$$. Thus, the change in $$\beta$$ relative to the change in $$\alpha$$ is decreasing as the standard is weakened (i.e. $$\beta^{\prime\prime}<0$$). Given the properties listed in expression (8), it immediately follows that there is a unique $$\alpha^{m}$$ such that $$\alpha^{m}=\underset{\alpha}{\arg\max}[\beta(\alpha)-\alpha]$$ (9)$$\alpha^{m}$$ is noted separately, because the standard that generates this probability of type-I error is often called preponderance of the evidence in the standard of proof literature. This is because using it would require a finding of liability in court, whenever the signal is more likely to be produced by a guilty individual rather than an innocent individual. In what follows, I use this standard as a benchmark, and use the following terms to describe results. Definition 1 A standard is weak if it results in $$\alpha>\alpha^{m}$$, and it is strong if it results in $$\alpha \leq\alpha^{m}$$. A standard leads to suspicionless stops if it results in $$\alpha=1$$. The properties of $$\beta$$ listed in expression (8), $$\alpha^{m}$$, and the terms defined in definition 1 are all depicted in Figure 2, below. Figure 2 View largeDownload slide Properties of $$\alpha$$ and $$\beta$$. Figure 2 View largeDownload slide Properties of $$\alpha$$ and $$\beta$$. The shape of $$\beta(\alpha)$$ around $$\alpha=1$$ also plays an important role, and has an intuitive meaning. In general, $$\frac{d(\beta(\alpha)-\alpha )}{d\alpha}=\beta^{\prime}(\alpha)-1$$ can be thought of as the gain or loss in the accuracy of a search caused by a weakening of the standard, since $$\beta-\alpha$$ represents the gap between the probabilities of stopping the guilty and the innocent. Therefore, $$1-\beta^{\prime}(1)$$ can be interpreted as the accuracy loss associated with suspicionless stops, because $$\beta-\alpha$$ can be increased by exactly this amount by infinitesimally strengthening the standard if one starts with suspicionless stops. The properties of $$\beta$$, listed in expression (8), imply that $$1>\beta^{\prime}(1)>0$$, and for each value $$a\in(0,1)$$ one can easily conceive of a functional form such that $$\beta^{\prime}(1)=a$$.17 Thus, the accuracy losses associated with suspicionless stops can vary greatly depending on the shape of $$\beta$$. This observation is useful in characterizing optimal stopping standards. 4.2. Optimal Standards When there are no inconvenience costs, as previously explained, suspicionless stops are optimal. However, when $$c>0$$, weaker stopping standards reduce the expected costs from committing crime, but, they also increase the expected costs associated with remaining innocent. Therefore, the effect of $$\alpha$$ on deterrence is more complex than in the case where $$c=0$$. However, if one starts with a strong standard, that is, $$\alpha<\alpha^{m}$$, making the standard slightly weaker leads to an increase in the gap between the probabilities of stopping guilty and innocent individuals (i.e., $$\beta-\alpha$$ is increased, as depicted in Figure 2). Moreover, the expected punishment associated with being stopped when guilty is naturally greater than the inconveniences one faces by being stopped while innocent. Thus, for $$\alpha\leq\alpha^{m}$$, an increase in $$\alpha$$ raises the expected costs to committing crime more than it increases the expected costs associated with remaining innocent, and thereby enhances deterrence. When standards are weak, increasing $$\alpha$$ beyond $$\alpha^{m}$$ causes the gap between $$\beta$$ and $$\alpha$$ to get smaller. This is detrimental to deterrence, because it amounts to a reduction in the marginal expected inconvenience cost from committing crime (i.e., a reduction in $$(\beta -\alpha)c$$). Moreover, the reduction in $$\beta-\alpha$$ is a function of the accuracy loss associated with using weaker standards, and these losses are greatest when one uses suspicionless stops due to the concavity of $$\beta$$. Thus, if the accuracy losses associated with suspicionless stops are small, then the reductions in the marginal expected inconvenience costs from committing crime are completely offset by the increased likelihood of being sanctioned upon committing crime. Similarly, if inconvenience costs are small relative to the punishment for committing crime, the reduction in the marginal expected inconvenience costs from committing crime is dwarfed by the increased expected sanction for committing crime. These observations are formalized by the following lemma. Lemma 1 (i) The deterrence maximizing stopping standard is weak. (ii) Deterrence is maximized by suspicionless stops if, and only if, $\frac{\beta^{\prime}(1)}{1-\beta^{\prime}(1)}\geq(1-x^{w})\frac{c}{f}$ Proof. See Appendix. □ Lemma 1 illustrates that, although deterrence is always maximized by weak standards, it may or may not be maximized by suspicionless stops. In particular, when the accuracy losses associated with suspicionless stops are large, or if marginal inconvenience costs are large, then, maximizing deterrence requires the use of weak standards, but not suspicionless stops. Lemma 1 also verifies the previously made claim that deterrence is maximized by suspicionless stops when $$c=0$$. Deterrence is only one of the three factors that affects the optimality of policies. In particular, the stopping standard also impacts the frequency with which harms are prevented, and the aggregate inconvenience costs inflicted on people. A quick look at equation (6) reveals that using weaker standards always increases the probability of harm prevention, which equals $$\beta x^{w}$$. On the other hand, the direct impact of $$\$$weaker standards on inconvenience costs (given by $$x^{w}c(\theta\beta+(1-\theta)\alpha)$$) is also always positive, because weaker standards lead to more people being stopped. However, when the stopping standard is strong, these costs are dominated by preventive benefits as long as marginal inconvenience costs are small compared to criminal harms, as stated in assumption 1. Therefore, welfare, like deterrence, is maximized by weak standards. To characterize the optimal standard, note that increasing $$\alpha$$ beyond $$\alpha^{m}$$ causes an increase in $$\beta$$ that is smaller than the increase in $$\alpha$$. This implies that aggregate inconvenience costs are more responsive to changes in the standard than preventive benefits. In fact, there are minimal preventive gains in cases where the accuracy losses from conducting stops based on weaker evidence is extreme (since $$\beta^{\prime}$$ is close to zero), but the increase in inconvenience costs can still be substantial (since they are proportional to $$\theta\beta+(1-\theta)\alpha$$). Thus, due to reasons similar to those described in explaining deterrence effects, whether or not increasing $$\alpha$$ beyond $$\alpha^{m}$$ leads to positive preventive benefits net of inconvenience costs depends on the accuracy losses associated with using weaker stopping standards as well as the magnitude of inconvenience costs. This is formalized by the following proposition. Proposition 4 (i) Welfare is maximized by weak standards. (ii) Suspicion-based stops can be optimal. (iii) Suspicionless stops are optimal, if either $$c$$ or the accuracy losses from suspicionless stops are sufficiently small. Proof. See Appendix. □ Part (iii) of proposition 4 verifies the previously made claim that suspicionless stops are optimal when there are no inconvenience costs. This result combined with part (ii) of proposition 4 reveals that inconvenience costs are necessary to justify the use of suspicion-based stops. Although proposition 4 focuses on identifying simple conditions under which both deterrence and welfare are maximized by suspicionless stops, additional conditions can be identified. In particular, when the attempt rate is very high, inconvenience costs, like preventive benefits, are almost exclusively affected by the rate at which guilty people are stopped. Therefore, making the stopping standard weaker increases welfare by enlarging the gap between preventive benefits and inconvenience costs. Moreover, this gap is the primary consideration when the population’s criminal behavior is relatively unresponsive to expected sanctions. Thus, suspicionless stops are optimal when the attempt rate is high and the population is unresponsive to sanctions. 4.3. Interactions between $$\alpha$$ and $$x$$ The analysis in this section takes the optimal investment in preventive enforcement, $$x^{w}$$, as given, just as Section 3 takes the stopping standard as given in determining the optimal investment in preventive enforcement. An interesting question is whether these two policy variables complement or substitute each other. Unfortunately, this question cannot be answered without imposing additional restrictions on the shape of the criminal benefit distribution or the nature of the tradeoff between preventive and nonpreventive enforcement methods. This is primarily due to two reasons. First, absent such restrictions, an increase in $$\alpha$$ causes a change in the marginal deterrence effect of $$x$$ whose sign and magnitude are ambiguous, because this value is affected by the curvature of the criminal benefit distribution in the population, as well as the properties of $$\beta$$. Second, even in cases where deterrence and second-order effects are small, making the stopping standard weaker has two conflicting effects on the marginal impact of preventive enforcement: it increases the marginal preventive benefits associated with an increase in $$x$$ while simultaneously leading to an increase in the marginal inconvenience costs caused by it. The comparison of the two effects hinges upon many factors, including the attempt rate, the accuracy gains from strengthening the stopping standard being considered, and the magnitude of the net criminal losses to be prevented relative to the size of inconvenience costs. A larger value associated with any of these three factors makes it more likely for weaker standards to increase the effectiveness of preventive enforcement. These ambiguities make it difficult to ascertain the precise welfare interactions caused by simultaneously setting $$\alpha$$ and $$x$$. This is why the analyses in Sections 3 and 4 focus on identifying features of an optimal enforcement regime, without relying on specific interactions between these two choice variables. Future research focusing on more specific functional forms may shed more light on the interactions between these two policy variables. 5. Conclusion Despite its frequent use, preventive law enforcement has not been analyzed much in the law and economics literature. This article highlights the pivotal role inconvenience costs play in the determination of optimal preventive enforcement policies. Specifically, the optimal policy converges toward net criminal harm minimization as inconvenience costs get smaller, and it is closer to deterrence maximization when inconvenience costs are large. Inconvenience costs also play a crucial role in the determination of optimal stopping standards. Absent inconvenience costs, it is hard to justify a policy of requiring reasonable suspicion to conduct stops, because it is always optimal to use suspicionless stops. However, when inconvenience costs are present, stops based on reasonable suspicion can be justified. Thus, ignoring inconvenience costs can lead to an inefficiently frequent usage of preventive enforcement, and may cause stops to be administered through undesirably weak standards. The effects of some important factors on optimal enforcement schemes are ambiguous in the instant setting. It is unclear, for instance, how the magnitude of punishment costs are likely to affect the analysis, how the frequency of preventive enforcement is likely to affect enforcer misconduct, or how the optimal frequency of preventive enforcement is likely to be affected by various criminal investigation procedures. Therefore, future research is required to gain a better understanding of these considerations. I thank Abraham Wickelgren, two anonymous referees, and the participants of the 26th Annual Meeting of the American Law and Economics Association for valuable comments and suggestions Appendix Footnotes 1. Bentham (1843). 2. See Terry v. Ohio, 392 U.S. 1 (1968). 3. Incapacitation has similar functions to those served by preventive enforcement methods. However, preventive methods aim at precluding harms in the present or very near future, rather than harms inflictable through recidivism. Thus, a difference between incapacitation and harm prevention is that the former often requires the commission of a preceding offense. 4. Floyd v. City of New York, 959 F.Supp.2d 540, 558–9 (2013). 5. Costs associated with imprisonment can be regarded as negative outputs. There are articles that consider such costs, and many others that consider fines which are mere transfers. Polinsky and Shavell (2007) contains a review of both types of models. 6. Some inputs are general (e.g., police cars) in that they can be used to serve many enforcement functions, whereas others may be specific (e.g., specialized forensics units). Although this distinction can be important in making long-term investment decisions, it is not central to the current analysis. The current focus is on the optimal allocation of inputs across preventive versus non-preventive law enforcement methods, regardless of whether they are specific or general. 7. Another issue in measurement is whether criminals’ benefits ought to be accounted for. Curry and Doyle (2016) explains this issue, and contributes to the existing debate. 8. In section 3, $$\theta$$ denotes the attempt rate and $$x\beta$$ (the product of the likelihood of being stopped ($$\beta$$), and the intensity of the search ($$x$$)) denotes the prevention rate. Using this notation, the completed crime rate corresponds to $$(1-x\beta)\theta\,$$. Chapman et al. (1975) discusses issues related to output measurement, and proposes a measurement of the “relative prevented crime rate” (RPCR) which is the difference between the crime rate that would emerge without any enforcement and the completed crime rate that emerges with enforcement. Thus, his definition of RPCR corresponds to $$\theta^{n} -(1-x^{e}\beta^{e})\theta^{e}$$ in the instant article, where $$e$$’s indicate values with enforcement and $$\theta^{n}$$ indicates the natural crime rate. 9. See, e.g., Mungan (2012) explaining the focus on deterrence in the existing literature. 10. Friehe and Tabbach (2013, p. 3) makes this distinction as follows: “Incapacitation targets specific individuals (convicted criminals), seeking to prevent them from committing crimes in the future. In contrast, preventive enforcement is not aimed at specific individuals, but rather at preventing the commission of specific acts that represent imminent threats.” 11. Here, a simplifying assumption that does not affect the analysis is that attempts and results are punished equally severely. 12. This expression reflects the assumption that completion of the crime is necessary to receive the benefits from crime. This is similar to the case considered in Mungan and Klick (2014) where the offender receives no criminal benefits when caught red-handed. 13. Guilty refers to a person who has initiated crime, and innocent to one who has not initiated crime. 14. A violation of assumption 1 leads to interesting, yet, unrealistic results. In particular, if $$c$$ is sufficiently high, it may be optimal to invest less than the deterrence maximizing amount in preventive enforcement. 15. See Mungan (2017) for a more detailed discussion of multiple local maxima. 16. Demougin and Fluet (2005) takes a similar approach to abbreviate expressions. 17. A simple example is $$\beta=\alpha^{a}$$. 18. To see this, note that the derivative of the RHS with respect to $$\alpha$$ is $$-c\beta^{\prime\prime}\frac{1-x^{w}\beta }{(\beta^{\prime})^{2}}>0$$ since $$\beta^{\prime\prime}<0$$. 19. When $$c$$ is not constrained to be smaller than $$\overline{c}$$, it follows that weak standards need not be optimal if $$c$$ is sufficiently large. Assumption 1 rules this possibility out. References Bentham, J. 1843 . “Principles of Penal Law” in Bowring, J. ed., The Works of Jeremy Bentham. Vol. 1. Edinburgh : William Tait , 367 – 580 . Braga, A. 2008 . Crime Prevention Research Review No. 2: Police Enforcement Strategies to Prevent Crime in Hot Spot Areas. Washington, DC : US Department of Justice Office of Community Oriented Policing Services. Curry, P. and Doyle. M. 2016 . “Integrating Market Alternatives into the Economic Theory of Optimal Deterrence,” 54 Economic Inquiry 1873 – 83 . Google Scholar Crossref Search ADS Chapman, J, Hirsch, W. and Sonenblum. S. 1975 . “Crime Prevention, the Police Production Function, and Budgeting” 30 Public Finance 197 – 215 . Demougin, D. and Fluet. C. 2006 . “Preponderance of Evidence.” 50 European Economic Review 963 – 76 . Google Scholar Crossref Search ADS Demougin, D. and Fluet. C. 2005 . “Deterrence versus Judicial Error: A Comparative View of Standards of Proof.” 161 Journal of Institutional and Theoretical Economics 193 – 206 . Google Scholar Crossref Search ADS Fluet, C. and Mungan. M. 2017 . “The Signal-Tuning Function of Liability Regimes” George Mason Law & Economics Research Paper No. 17–37. Friehe, T., and Tabbach. A. 2013 . “Preventive Enforcement,” 35 International Review of Law and Economics 1 – 12 . Google Scholar Crossref Search ADS Kaplow, L. 2011 . “On the Optimal Burden of Proof” 119 Journal of Political Economy 1104 – 40 . Google Scholar Crossref Search ADS Kennedy D. 2009 . “Deterrence and Crime Prevention: Reconsidering the Prospect of Sanction” New York : Routledge. Lando, H. 2009 . “Prevention of Crime and the Optimal Standard of Proof in Criminal Law” 5 Review of Law and Economics 33 – 52 . Google Scholar Crossref Search ADS Miceli, T. 1990 . “Optimal Prosecution of Defendants whose Guilt is Uncertain.” 6 Journal of Law Economics and Organization 189 – 201 . Miceli, T. 2010 . “A Model of Criminal Sanctions that Incorporate both Deterrence and Incapacitation” 107 Economics Letters 205 – 7 . Google Scholar Crossref Search ADS Miceli, T. 2012 . “Deterred or Detained? A Unified Model of Criminal Punishment” 8 Review of Law and Economics 1 – 20 . Google Scholar Crossref Search ADS Mungan, M. 2011 . “A Utilitarian Justification for Heightened Standards of Proof in Criminal Trials” 167 Journal of Institutional and Theoretical Economics 352 – 70 . Google Scholar Crossref Search ADS Mungan, M. 2012 . “The Law and Economics of Fluctuating Criminal Tendencies and Incapacitation” 72 Maryland Law Review 156 – 218 . Mungan, M. 2017 . “Optimal Preventive Law Enforcement and Stopping Standards” FSU College of Law, Public Law Research Paper No. 701. Mungan, M., and Klick. J. 2014 “Forfeiture of Illegal Gains, Attempts, and Implied Risk Preferences” 43 Journal of Legal Studies 137 – 53 . Google Scholar Crossref Search ADS Nagin, D. 2013 . “Deterrence: A Review of the Evidence by a Criminologist for Economists.” 5 Annual Review of Economics 83 – 105 . 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Shavell, S. 2015 . “A Simple Model of Optimal Deterrence and Incapacitation.” 42 International Review of Law and Economics 13 – 19 . Google Scholar Crossref Search ADS Sherman, L., Gartin P. and Buerger. M. 1989 . “Hot Spots of Predatory Crime: Routine Activities and the Criminology of Place” 27 Criminology 27 – 55 . Google Scholar Crossref Search ADS Sherman L. and Weisburd. D. 1995 . “General Deterrent Effects of Police Patrol in Crime ‘Hot Spots’: A Randomized Study” 12 Justice Quarterly 625 – 48 . Google Scholar Crossref Search ADS © The Author(s) 2018. Published by Oxford University Press on behalf of the American Law and Economics Association. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)

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Published: Oct 1, 2018

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