American Law & Economics Association Annual Meetings

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1 American Law & Economics Association Annual Meetings Year 2004 Paper 48 What You Don t See Can t Hurt You: An Economic Analysis of Philip A. Curry Simon Fraser University Steeve Mongrain Simon Fraser University This working paper site is hosted by The Berkeley Electronic Press (bepress) and may not be commercially reproduced without the publisher s permission. Copyright c 2004 by the authors.

2 What You Don t See Can t Hurt You: An Economic Analysis of Morality Laws Phil Curry Simon Fraser University Steeve Mongrain Simon Fraser University April 30, 2004 Preliminary Abstract A common justification for morality laws, such as the regulation of drug use and prostitution, is that the government is protecting individuals from themselves. Needless to say, this explanation does not sit well with economists. This paper provides an alternate explanation motivated by the observation that the design of these laws often promotes discretion by the people engaging in the activity in question. We propose that morality laws can be best explained by considering the proscribed activities to impose a negative externality that can be reduced through hiding. In such a case, efficiency requires discretion on behalf of the individual who engages in such activities. Since discretion is often difficult to regulate, the activities are instead proscribed thereby giving individuals incentive to hide their actions from others. In addition, since some level of activity is efficient, the optimal sanctions are not maximal. Key Words: Crime; Externality; Laws; Morality; Enforcement JEL: K42; K32; H32 We would like to thank David Andolfatto for many interesting conversations that culminated in this paper. We also thank Scott Skjei for excellent research assistance. Hosted by The Berkeley Electronic Press

3 1 Introduction Morality laws include prohibitions against exchanging sexual favors for money and the use of narcotics. A common justification for such laws is that the government is protecting individuals from themselves. Needless to say, this explanation does not sit well with economists. This paper offers an explanation for morality laws that is consistent with a number of observations about these laws. First, the design of these laws often promote discretion by the people engaging in the activity in question. For example, prostitution is legal in Canada, while solicitation is not. In other words, it is legal to exchange money for sexual favors in Canada, it is just not legal to discuss it in public 1. The second observation is that the majority the activities covered by morality laws are completely banned. For other forms of externalities, like pollution, a positive legal level or quota is usually set. Finally, the typical penalty for the infraction of these laws seems unusually low. For example, in the city of Vancouver, the maximum penalty for excessive noise while cleaning your carpet with a vehicle mounted cleaner on a Sunday before noon is $2000 while the maximum sentence for a first offense of cocaine possession is six months in prison and $1000. more importantly, the expected penalty does not appear to provide much deterrence. The Alcohol and Other Drugs Survey of 1994 found that 23% of Canadians had tried marijuana at least once 2 In the United States, the National Task Force on Prostitution estimates that over 1 million people and 1% of American women have been employed as prostitutes 3. An estimated 2.7 million Americans were chronic cocaine users in 2000, and an additional 3 million Americans were recreational users. In 1990, Americans consumed $69.9 billion (US), or 447 metric tons worth of cocaine 4. In comparison, an estimated $23 billion (US) 1 The Canadian Criminal Code states that every person who in a public place or in any place open to public view (a) stops or attempts to stop any motor vehicle, (b) impedes the free flow of pedestrian or vehicular traffic or ingress to or egress from premises adjacent to that place, or (c) stops or attempts to stop any person or in any manner communicates or attempts to communicate with any person for the purpose of engaging in prostitution or of obtaining the sexual services of a prostitute is guilty of an offence punishable on summary conviction. 2 Taken from Canada s 2002 Report of Senate Special Committee on Illegal Drugs (p.91). 3 Taken from Prostitutes Education Network, 4 Taken from the Office of National Drug Control Policy webpage, 1

4 worth of merchandise is shoplifted on an annual basis 5. This paper proposes that these stylized facts about morality laws can be best explained by considering the proscribed activities to impose a negative externality that can be reduced through hiding behaviors. In such a case, efficiency requires discretion on behalf of the individual who engages in such activities. Since discretion is often difficult to regulate, the activities are instead proscribed thereby giving individuals incentive to hide their actions from others (including the police). It should be noted that it is not always case that regulations about discretion cannot be implemented. For example, many counties in the United States allow alcohol consumption in public, non-licensed areas, as long as the alcohol is hidden. As mentioned above, prostitution is legal in Canada, however solicitation is not. Optimal policy in the presence of negative externalities has been well studied in economics. One of the most commonly used policies is that of a quota and enforcement consisting of monitoring and sanctions. Examples of quotas include speed limits, blood/alcohol levels while driving, emission of pollutants or noise curfews. On the other hand, some activities that impose negative externalities, such as murder or the emission of deadly pollutants, are banned altogether. This is not surprising as it seems clear that the marginal costs to such activities always outweigh the marginal benefits. Included on the list of activities that are outright banned, however, are prostitution and the consumption of drugs. In these cases, it is not so obvious that the marginal costs exceed the marginal benefits at zero consumption. It should also be noted that if the socially efficient level of consumption were zero, then the penalties associated with morality laws seem to provide an unusually low level of deterrence. For example, in 2000, there were an estimated 14 million current users of marijuana. 6 We demonstrate that the optimal policy may be to proscribe such activities altogether even when the marginal benefits outweigh the marginal costs for some positive levels of consumption. We construct a model in which an agent, called the injurer, chooses the level of consumption for a good which causes harm to another agent, called the victim. 5 Taken from 6 Taken from Canada s 2002 Report of Senate Special Committee on Illegal Drugs (p.101). 2 Hosted by The Berkeley Electronic Press

5 This harm may or may not be reducible through hiding behavior on behalf of the injurer. In the absence of legislation, the injurer will choose to consume until the marginal benefit equals the private marginal cost, leading to consumption greater than the socially efficient level. The government is assumed to have three instruments with which it can affect the injurer s consumption. First, it can set a legal limit to consumption. Second, it can choose a penalty, or sanction, for consumption above the legal limit. Finally, it chooses a level of effort in enforcing the quota, which affects the probability that the injurer is caught when consuming a proscribed level. However, this probability can also be affected by the injurer. The injurer is assumed to be able to reduce the probability of detection through costly hiding. The implications of hiding for social welfare depends on whether it merely reduces the probability of being caught or also reduces the harm incurred by the victim. When the externality is conventional (not affected by the injurer s hiding), hiding is a wasteful activity. The only effect in this case is to reduce the probability of detection of illegal consumption. Consequently, it is always welfare reducing. When hiding reduces the harm of the externality, some level of hiding may be socially efficient. We refer to this situation as a damage reducible externality. It should be noted that an injurer will chose to hide her consumption only if part of this consumption is illegal. That is, if an injurer does not consume above a quota, then they have no incentive to engage in hiding. As a result, the government will wish to set the quota below the socially efficient level but choose sanctions and enforcement such that the injurer will choose to consumer a proscribed level and engage in socially beneficial hiding. It should further be noted that this implies that a certain level of crime is socially desirable. Hiding on behalf of criminals has already been examined in the literature to some extent. Malik (1990) shows that when criminals can invest effort in hiding their crimes, maximal sanctions may be sub-optimal even when sanctions are costless. Higher sanctions increase the incentive to hide, which is welfare reducing. This result also arises in our framework. Since this result is already known, we restrict our analysis to an environment in which this effect is not present. This allows us to focus on additional reasons for the non-optimality of maximal sanctions. First, the government may face a tradeoff between enforcement costs and social costs associated 3

6 with excessive consumption. Second, the government may wish to encourage crime in order to promote hiding. When externalities are conventional and the government is unconstrained in its choice of sanction, the social optimum is implementable by setting the quota at the socially efficient level and setting the sanction arbitrarily high so that neither crime nor hiding occurs. However, if the government is not able to set the sanction above a certain level, the efficient level of consumption can only be induced by incurring enforcement costs. This paper demonstrates in such a case, the government will allow consumption above the efficient level and use costly enforcement such that the marginal cost of each is equalized. As mentioned above, hiding is a wasteful activity that only occurs when the injurer consumes above the legal limit. Increasing the quota reduces hiding and so the optimal policy entails a legal level which is too high compared to the efficient level. In other words, laws are too permissive. The optimality of permissive laws has received little attention in the crime literature. Boadway and Sato (2000) show that, for the case of tax evasion, permissive laws can be optimal if courts or tax payers can make mistakes concerning taxable income. This paper provides a second case for permissive laws. The case of damage reducible externalities constitutes the central analysis in the paper. As mentioned above, hiding in this context can be socially desirable because it reduces the damage incurred by the victim. The only benefit to injurers from hiding, however, occurs if they are committing crime. If a injurer chooses a consumption level within the legal limits, there is no incentive for the injurer to hide. Consequently, the optimal policy portfolio encourages illegal consumption, thereby promoting hiding. In order to induce the correct amount of consumption the optimal level of sanctions are not maximal. In addition, the government will choose a legal level (or quota) which is less than the efficient level. In other words, morality laws seem to be overly restrictive. In many cases, the optimal legal level for consumption is zero, which is consistent with our observation on drug use or prostitution laws in most countries. Alcohol is one example of an activity for which the legal limit is often not zero, although it should be noted that periods of prohibition have been experienced. As pointed out by Levitt (2003) and MacCoun and Reuter (2001) whenever prohibition has been introduced, sanctions were relatively small. The average punishment was 4 Hosted by The Berkeley Electronic Press

7 35 days and a $100 fine, again consistent with our findings. The outline of this paper is as follows. In the next section we present the general model. Section 3 considers the case of conventional externalities. Section 4 analyzes the case of damage reducible externalities. The conclusion follows. Note that all the proofs are in the appendix. 2 The Model Consider the following simple economy with two agents and a government. The first agent, who we call the injurer, I, derives utility from the consumption of a certain good or activity. 7 Denote the injurer s consumption by θ, and let (θ) be the benefit derived from consumption. The marginal benefit is positive, B ( ) > 0, but at a decreasing rate, B ( ) < 0. The second agent we call the victim, V. The victim suffers some disutility from the injurer s consumption. Denote by C (θ) the harm incurred by the victim from the injurer s consumption, where C ( ) > 0 and C ( ) > 0. The injurer and victim have initial wealth m I and m V, respectively. It is assumed that transaction costs are high enough so that bargaining cannot take place. The government maximizes the sum of agents utilities. It does so by choosing a level of legal consumption of the good and through the enforcement of said level. Denote by θ the maximum legal consumption of θ. If the injurer decides to consume more than the legal level, she will be fined with some probability. The probability that illegal consumption is detected is given by p (e, h), where e is the enforcement effort chosen by the government, and h is the effort expended on hiding chosen by the injurer. Hiding effort comes with a marginal cost of one. The cost of enforcement effort is given by κ (e) where κ (e) > 0 and κ (e) 0, it is also assumed that κ (0) = 0. The larger the effort by the government, the larger the probability of being detected (p e ( ) > 0), but at a decreasing rate (p ee ( ) < 0). The injurer s hiding behavior decreases the probability of being caught, p h ( ) < 0, also at a decreasing rate, p hh ( ) > 0. We assume that an increase in the government s monitoring efforts 7 Here we use the example of a consumption externality, but using a production externality will lead to the same results. 5

8 reduces the effectiveness of hiding, so that p he ( ) > 0. Finally, we assume that there exists a small chance that the injurer will get caught when consuming a proscribed amount, even if the government does not expend any enforcement effort. That is, we assume p (0, h) > 0, h 8. The effort of the government is financed through a uniform lump sum tax T. An injurer who gets caught consuming a proscribed level, i.e. θ > θ, receives a sanction S ( θ θ ). The sanction is an increasing function of the difference between the actual consumption and the legal level. To begin, we concentrate our efforts on the case where the sanction is linear and given by S = s [ θ θ ]. In addition, we consider two scenarios. First, we consider the case were there is no limit to the perunit sanction s. We then consider the case in which the government cannot raise the sanction above an certain level, s max. This latter case is introduced to investigate the tradeoffs between the probability of detection and the quota in a second best world. It is assumed that these sanctions can be collected without cost. Finally, the utility of the injurer is given by: { U I m I T + B (θ) h = m I T + B(θ) p(e, h)[θ θ]s h if θ θ if θ > θ The utility of the victim is given by: U V = m V T C (θ) Note that the victim does not make any decisions in this simple model. 2.1 The Injurer s Decision We now characterize the behavior of the injurer for a given set of regulations, { θ, e, s }. The injurer s maximization problem is as follows: { B (θ) h max θ,h B(θ) p(e, h)[θ θ]s h if θ θ if θ > θ 8 This assumption is made purely to simplify the analysis. Without this assumption, attention would be restricted to suprema of the government s objective function as opposed to maxima. 6 Hosted by The Berkeley Electronic Press

9 First, note that the injurer s utility is strictly increasing in θ and strictly decreasing in h for all θ < θ. Thus one possible solution to the injurer s maximization problem is θ = θ and h = 0. The injurer may choose θ > θ, however. In this case, the optimal choices for θ and h are characterized by the following first order conditions: B (θ ) = p (e, h ) s (2.1) p h (e, h ) s [ θ θ ] = 1 (2.2) Equation 2.1 states that when the injurer chooses her level of consumption, she equalizes the marginal benefit of consumption with its marginal cost, which is given by the probability of being detected times the marginal sanction. Simultaneously, the injurer chooses the level of hiding to equalize the marginal reduction in the expected sanction, with its marginal cost of one 9. Denote the solution to equations 2.1 and 2.2 by θ and h. Note that these first order conditions characterize the optimal choices of θ and h when B (θ ) p (e, h ) s [ θ θ ] h > B ( θ). 10 It is important for future reference to understand how the injurer s choices of θ and h vary with the different policy tools, both for the solutions characterized by the first order conditions and the for the case where there is only legal consumption. In the latter case, the injurer chooses θ = θ and h = 0. Note that the injurer s preferred level of consumption is invariant to changes in the government s detection effort, e, as well as to changes in the per-unit sanction, s. Further note that consumption varies one-to-one with θ. The effort spent on hiding is independent of all policies. For the solutions characterized by equations 2.1 and 2.2, an increase in θ leads to a decrease in θ and h. The intuition for this result is as follows. First, as the legal level of consumption increases, the marginal benefit from hiding decreases because the total penalty (θ θ) decreases. As the amount of hiding decreases, the (expected) marginal cost of consumption increases. The result is a decrease in both the level 9 The second order will be satisfied as long as B (θ)p hh ( )[θ θ]s+p h ( ) 2 s 2 < 0, which we assume to be the case. 10 It is assumed that in the case that B (θ ) p (e, h ) s [ θ θ ] h = B ( θ), the injurer chooses θ = θ and h = 0 7

10 of hiding and the level of consumption. 11 Note that criminal activity (consumption above the legal limit) and hiding are strategic complements. Second, an increase in detection efforts, e, increases the expected cost of consumption above the legal limit, and so θ decreases. In addition, since it has been assumed that the government s monitoring reduces the effectiveness of hiding, we also have that h decreases. 12 Finally, as the per-unit sanction, s increases, the effect on consumption and hiding is ambiguous. As noted above, as the sanction increases, the expected cost to consumption above the legal limit increases (holding the level of hiding fixed). However, the benefit to hiding also increases (holding the level of consumption fixed). The effect of the increase in the per-unit sanction thus depends on which effect dominates. For the purposes of this paper we will focus on the more standard case in which the amount of illegal activity and the level of hiding are decreasing in the sanction Consumption, Quotas and Hiding We will start by looking at the case where the harm caused to the victim by the injurer s consumption is unaffected by hiding. Pollution is the classic example of such an externality. In this case, the efficient levels of consumption and hiding are found by solving max B (θ) C (θ) h θ,h Note that the efficient level of hiding is zero. When consumption produces an externality independent of whether the victim observes the injurer s level of consumption, 11 The actual derivatives are given by: θ θ = [p h( )s] 2 soc < 0 and h θ = B (θ)p h ( )s soc < 0 respectively, where soc denotes the relevant second order condition as stated in the previous paragraph. 12 θ The derivatives are given by: e = [p h( )p he ( ) p e( )p hh ( )][θ θ]s 2 soc < 0 and h e = B (θ)p he (θ θ)s+p e( )p h ( )s 2 < 0 respectively. soc 13 The derivatives are given by: θ = [p h( ) 2 p( )p hh ( )][θ θ]s soc and h = B (θ)p h (θ θ)+p( )p h ( )s soc. A sufficient condition for both derivatives to be decreasing in s is p h( ) p( ) decreasing in h. Malik (1990) notes that when the amount of illegal activity increases as the sanction increases maximal sanctions are no longer optimal. In this paper, the non-optimality of maximal sanctions arises as a result of damage reducible externalities. This is discussed in section 4. 8 Hosted by The Berkeley Electronic Press

11 hiding is a waste of resources. The efficient level of consumption, denoted by θ o, is characterized by B (θ o ) = C (θ o ). We will now look at how the government can affect the level of consumption as well as the injurer s hiding efforts. 3.1 The Government s Problem The instruments available to the government are the quota on legal consumption, θ, the per-unit sanction on consumption above the legal level, s, and the effort in monitoring, e. The problem faced by the government is the following: max B ( θ ( θ, )) ( ( θ, )) ( θ, ) e, s C θ e, s κ(e) h e, s θ,e,s In this scenario, the socially efficient level of consumption and hiding, θ o zero respectively, are implementable as the government has a sufficient number of instruments. This is demonstrated formally below. Result 1: When consumption by the injurer imposes a conventional externality on the victim, the optimal policy is for the government to set θ = θ o, e = 0 and s arbitrarily large. This will induce the injurer to choose θ = θ o and h = 0, the socially efficient levels of consumption and hiding, respectively. The intuition for this result is straightforward. When the expected penalty to consumption above the proscribed limit is high enough, the injurer will choose to consume the quota and no more. Given that the sanction is costless and p (0, h) > 0, h, the government can set arbitrarily high expected penalties while incurring no costs from monitoring effort. As such, the government can set the quota at any level and costlessly induce the injurer to consume that amount. The optimal choice for the quota is the efficient level of consumption, θ o. and 3.2 Constrained Sanctions It may be the case that the government cannot choose arbitrarily large sanctions. This is, it may be that for political, technological or constitutional reasons there exists a limit to the per unit sanction level that can be imposed on illegal consumption. 9

12 Denote this limit to the per unit sanction by s max. In particular, we consider the case where s max is not sufficient to deter the injurer from consuming above the quota when θ = θ o and e = 0. In this case, the government faces a tradeoff between the costs of monitoring, and the social costs of activity above the efficient level plus any resources expended on hiding. The optimal choice for the government is to incur both enforcement effort costs and to allow consumption above θ o. This can be seen as follows. Since θ h θ θ < 0 and < 0 when θ > θ, the government can reduce both crime and hiding by increasing the quota, which is costless. Thus the optimal policy for the government is to choose θ and e such that the injurer chooses to consume the quota and such that the marginal cost of monitoring is equal to the marginal social cost of consumption above θ o. This is demonstrated formally in the following result. Result 2: Suppose that consumption by the injurer imposes a conventional externality on the victim and the government is constrained in the level of the per unit sanction that it can choose. In this case, the optimal policy for the government is to set the sanction to the maximal level and to choose the enforcement effort and quota such that the agent fully complies. Further, the quota and level of effort will be such that the marginal social cost of consumption is equal to the marginal cost of enforcement. It should be noted that when consumption imposes a conventional externality on the victim, any optimal policy entails full compliance. That is, the quota and expected penalty are chosen so that the injurer will not exceed the quota. When the government is unconstrained in its choice of sanction, then the quota is set at the socially efficient level. When the government is constrained in the sanction it can set, however, then the quota may exceed the efficient level. 4 Damage Reducible Externalities We now consider the case in which hiding effort by the injurer reduces not only the probability with which the injurer is caught, but also reduces the harm incurred by the victim. This may occur in cases where it is the observation of the activity that causes disutility to the victim, or where the concept of responsible consumption 10 Hosted by The Berkeley Electronic Press

13 applies. Specifically, we consider the case where the victim s utility is given by U V = m V T C (θ, h) where C θ > 0 and C θθ > 0 as before, but C h < 0 and C hh > 0 so that hiding on behalf of the injurer benefits the victim, but with decreasing returns. We further assume that C θh < 0. Note that the injurer s preferences do not change and so the effect of the quota, θ, enforcement effort, e, and per-unit sanction, s, remains the same as above. What does change, however, is the socially efficient level of consumption and hiding. 4.1 Efficiency When hiding reduces the harm incurred by the victim, it now becomes socially optimal for some hiding to take place. The efficient levels of consumption and hiding are found by solving The first order conditions are max B (θ) C (θ, h) h θ,h B (θ) C θ (θ, h) = 0 C h (θ, h) 1 = 0 Denote the efficient levels of consumption and hiding by θ DR and h DR, respectively. Noted that h DR > 0 = h o and θ DR > θ o. 4.2 The Government s Problem It should first be noted that if the government can legislate the injurer s hiding behavior, then the optimal policy would be to set the quota at θ DR, require that the injurer choose h DR and enforce each law by choosing arbitrarily high sanctions and zero enforcement effort. This section examines the case in which legislation on hiding behavior is not possible. 11

14 Recall that the injurer s behavior changes in precisely the same way as noted above when analyzing consumption externalities. The government s instruments therefore have the same effect, although the objective function is now changed to accommodate the change in the victim s utility. The government s problem is given by max B ( θ ( θ, )) ( ( θ, ) ( θ, )) ( θ, ) e, s C θ e, s, h e, s h e, s κ(e) θ,e,s The first order conditions are [B (θ ) C θ (θ, h )] θ θ [C h (θ, h ) + 1] h = 0 θ (4.1) [B (θ ) C θ (θ, h )] θ e [C h (θ, h ) + 1] h e κ (e) = 0 (4.2) [B (θ ) C θ (θ, h )] θ [C h (θ, h ) + 1] h = 0 (4.3) It should first be noted that it is no longer optimal for the government to set the quota at the efficient level, θ DR and to completely deter consumption over that level. Recall that the injurer never engages in hiding unless also committing crime (consuming above the quota). Such a policy would therefore lead the injurer to choose θ = θ DR and h = 0. A policy that induces the injurer to consume above the quota would constitute an improvement because of the hiding that accompanies crime. The following result demonstrates this formally. Result 3: When externalities are damage reducible, the government s optimal policy, { θ, e, s}, is such that there is crime in equilibrium (θ > θ). A corollary to this result is that the optimal sanction is not maximal. Since the solution involves the injurer choosing θ > θ, the sanction s has to be set to a finite value even though it is costless. This provides another example of the non-optimality of maximal sanctions to complement the results of Andreoni (1991), Malik (1990) and Shavell (1991). Also of interest is the fact that efficiency requires crime to be committed even though it is costless to deter it. To our knowledge, this paper provides the first example of such a result. We now consider the set of policies that induce crime and characterize the optimal one. An obvious candidate for the solution would be one such that the injurer chooses θ = θ DR > θ and accompanies the illegal activity with a degree of hiding equal to 12 Hosted by The Berkeley Electronic Press

15 the socially optimal level, h = h DR. The following result demonstrates that it can be feasible to implement the social optimum without cost. Result 4: Suppose that consumption by the injurer imposes a damage reducible externality on the victim. In this case, the optimal policy for the government is to set a quota below the socially efficient level, choose zero enforcement, and set the sanction less than maximally. In particular, the quota and sanction will be such that the injurer chooses θ = θ DR and h = h DR. This is implementable as long as where ɛ ph (0, h DR ) = hdr p h (0,h DR ) p(0,h DR ). B (θ DR )ɛ ph (0, h DR ) hdr θ DR, As mentioned in the result, such a policy can be implemented only under certain circumstances. Loosely, it will not be implementable when hiding is effective at reducing harm ( hdr is high), but hiding is not very effective at reducing the probability θ DR of getting caught (ɛ ph (0, h DR ) is low). In this case, it is socially optimal to have a high level of hiding, but the injurer does not choose to invest much because it is not productive. In such a case, since hiding and crime are complements, the injurer would have to commit a large amount of crime (i.e. consume far above the legal limit) to induce the appropriate level of hiding. However, since hdr is high, then it is possible θ DR that the amount of crime necessary to induce h DR is greater than the socially efficient level of consumption. In other words, the government would have to set a negative quota. In this case, the government will be unable to induce the socially efficient levels of consumption and hiding without incurring costs. Since this situation arises when the government cannot get enough hiding when inducing the injurer to consume the efficient level, θ DR, the government must now tradeoff the costs of enforcement with the social costs associated with excessive consumption and too little hiding. optimal policy in this case entails a positive level of enforcement effort and finite sanctions. As alluded to above, the injurer is induced to consume greater than the efficient level of consumption θ > θ DR and hide less than the efficient level h < h DR. Result 5: Suppose that consumption by the injurer imposes a damage reducible ex- The 13

16 ternality on the victim and that B (θ DR )ɛ ph (0, h DR ) hdr θ DR. In this case, the optimal policy for the government is to set the quota equal to zero, choose zero enforcement, and set the per unit sanction less than maximally. injurer then chooses θ > θ DR and h < h DR. We thus have that in the presence of damage reducible externalities, we should expect to see low quotas, possibly even zero, accompanied by low penalties and enforcement. In particular, the expected penalty should be low enough that a significant amount of crime happens. These predictions correspond well with the stylized facts on morality laws. Further, this model predicts that the government may choose to legislate hiding behaviors when possible. This is something that does occur, with examples including Canada s legalization of prostitution but proscription of solicitation and pervasive laws that limit alcohol consumption to licensed areas. The 14 Hosted by The Berkeley Electronic Press

17 5 Conclusion This paper provides an explanation for morality laws that does not require the government to be protecting individuals from themselves. We assume that the proscribed activities impose a negative externality on others that can be reduced by engaging in hiding behaviors. This assumption seems natural given the presence of legislation aimed at the promotion of discretion on behalf of individuals engaging in these activities. The results of this assumption include the optimality of non-maximal sanctions and the presence of restrictive laws accompanied by seemingly low expected penalties. 15

18 6 Appendix Proof to Result 1: The first order conditions for the government s maximization problem are as follows: [B (θ ) C (θ )] θ h = 0 θ θ (6.1) [B (θ ) C (θ )] θ e κ (e) h e = 0 (6.2) [B (θ ) C (θ )] θ h = 0 (6.3) where all partial derivatives of θ and h are as described in the section on the injurer s problem above. First, note that if the injurer chooses θ = θ, then h = 0 and θ θ h = 1 and = 0. θ In this case, the first order condition for the quota, equation 6.1, is satisfied when θ is such that B ( θ) ( θ) C = 0, which occurs at θ = θ o. In addition, when θ = θ, then θ e = 0, h = 0, θ = 0 and h = 0. In this case, the marginal return to e monitoring effort, e, is zero when e = 0 and negative for all positive levels of effort. Finally, the marginal effect of a change in s is zero as long as s is sufficient to deter the injurer from consuming above θ. This must be a global maximum since θ = θ o and h = 0 maximizes B (θ) C (θ) h as noted in the section on efficiency. As a result, the optimal policy for the government is to set the quota at θ o, e to zero and s arbitrarily large 14. Proof to Result 2: We begin by demonstrating that it is optimal for the government to choose s = s max. Recall that the first order conditions for the government s problem are given by equations 6.1, 6.2 and 6.3. To begin with we can see that, for every e and s, the government would never choose θ such that θ > θ, θ o. In such a case, the government would prefer to raise the quota, as this has the effect of reducing the level of consumption and hiding. This can be seen from the partial derivative with respect to θ since [B (θ ) C (θ )] < 0 when θ > θ o. [B (θ ) C (θ )] θ h θ θ > 0 In addition, suppose that θ is chosen such that θ = θ > θ o. In this case, the government would like to lower θ as long as the injurer would still comply with this 14 Or, at least large enough to induce the injurer to consume the quota and no more. 16 Hosted by The Berkeley Electronic Press

19 new lower quota. This can be seen by noting that the partial derivative with respect to θ is negative. [B (θ ) C (θ )] < 0 Now consider the case where θ is such that lowering the quota means that the injurer would no longer comply. Denote this quota by θ. Then we are in a situation where the partial derivative to the right is negative, and the partial derivative is positive at all θ < θ and so θ maximizes welfare given e and s. There are, however, different levels of θ depending on e and s. First, it should be noted that if a given θ arises from multiple pairs (e, s), then the pair with the lowest e is preferred by the government since e is costly and s is not. This implies that the government will be choosing between different θ such that s = s max. Thus for every θ there exists a unique e and the government s maximization problem can therefore be reformulated as follows. max B ( θ ) C (θ ) κ ( )) e ( θ θ Note that there are no hiding costs because the injurer chooses to comply and not hide at θ. The first order condition is B ( θ ) C ( θ ) e θ κ (e) = 0 In other words, the marginal social cost of consumption is set equal to the marginal cost of enforcement. Proof to Result 3: The solution to the government maximization problem is given by the set of policies { θ, e, s} which satisfy equations 4.1, 4.2 and 4.3. Policies may be of two types. The first type of policy induces the injurer to comply with the quota. That is, { θ, e, s} are such that θ = θ, then h = 0. The second type of policy leads the injurer to commit crime. In other words, { θ, e, s} are such that θ > θ. In this case, it will also be that h > 0. Consider policies of the first type. The only solution that satisfies all the first order conditions is θ = θ DR, s arbitrary large and e = 0. However such solution is not a maximum, but a saddle point. Since the Hessian is only negative semi-definite at the point, the SOC are inconclusive. If h is fixed at zero, any change in θ would lead to reduction in the objective function, however an increase in h, starting from h = 0, leads to an increase in the objective function. The optimal policy is therefore of the second type. 17

20 Proof to Result 4: The solution to the government s maximization problem is given by the set of policies { θ, e, s} which satisfy equation 4.1, 4.2 and 4.3. Note that the first order conditions with respect to the quota and sanction, 4.1 and 4.3, are satisfied when θ = θ DR and h = h DR. Moreover, the first order condition with respect to effort is satisfied at this level of consumption and hiding as long as e = 0. This means that the government effectively has two instruments, the sanction and the quota, with which to induce the injurer to choose θ DR and h DR. This is feasible, as can be seen as follows. The injurer chooses θ and h to solve the first order conditions 2.1 and 2.2. Suppose that the injurer chooses θ = θ DR and h = h DR. Then it must be that B (θ DR ) = p(0, h DR )s and p h (0, h DR )[θ DR θ]s = 1. This will occur when s = B (θ DR ), and p(0,h DR ) h θ = DR ɛ ph (0,h DR )B (θ DR ) θdr, where ɛ ph (0, h DR ) = p h(0,h DR ) p(0,h DR ) hdr. It is only possible to find a positive θ that will induce h DR and θ DR if B (θ DR )ɛ ph (0, h DR ) hdr. θ DR Proof to Result 5: We first show that the solution entails s > 0 and e, θ = 0. Evaluating the government s first order conditions at the optimum would then yield [B (θ ) C θ (θ, h )] θ θ [C h (θ, h ) + 1] h θ 0 [B (θ ) C θ (θ, h )] θ e [C h (θ, h ) + 1] h e κ (0) 0 [B (θ ) C θ (θ, h )] θ [C h (θ, h ) + 1] h = 0 Note that condition 6.4 can be rewritten as B (θ ) C θ (θ, h ) C h (θ, h ) + 1 = h θ Note that h θ > 0 and so it must that B (θ ) C θ (θ, h ) and C h (θ, h ) + 1 have the same sign. If both are positive, then there is too little consumption and too much hiding. If both are negative, then there is too much consumption and too little hiding. Condition 6.4 can similarly be rewritten as B (θ ) C θ (θ,h ) C h (θ, h ) Since B (θ ) C θ (θ,h ) C h (θ,h )+1 = h θ C h (θ, h )+1 0. It can be shown that h θ C h (θ,h )+1, we have that h θ < h θ θ θ h θ θ θ h θ θ θ if and only if if and only if by substituting in the expressions 18 Hosted by The Berkeley Electronic Press

21 for the partial derivatives: B (θ)p h ( ) θ + p( )p h ( )s [p h ( ) 2 p( )p hh ( )] θs B (θ)p h ( ) 2 θs + p( )p h ( ) 2 s [p h ( ) 2 p( )p hh ( )] θs < B (θ) p h ( ) s p h ( ) 2 s 2 > B (θ) Recall that [p h ( ) 2 p( )p hh ( )] θs is negative by assumption. Thus we have that B (θ)p h ( ) 2 θs + p( )p h ( ) 2 s < B (θ) [ p h ( ) 2 p( )p hh ( ) ] θs p h ( ) 2 s + B (θ) p h h ( ) θ < 0 This is satisfied from the second order condition for the injurer s decision of θ and h. Thus we have that C h (θ, h ) + 1 < 0 which occurs at levels of hiding lower than h HD. Thus it must also be that B (θ ) C θ (θ, h ) < 0, and so there is too much consumption. h θ Finally, the optimal level of e is zero if and only if B (θ ) C θ (θ,h ) h e θ. Substituting in yields e C h (θ,h )+1 h e θ, or e B (θ)p h ( ) θ + p( )p h ( )s B (θ) p he ( ) θs + p e ( ) p h ( ) s 2 [p h ( ) 2 p( )p hh ( )] θs [p h ( ) p he ( ) p e ( ) p hh ( )] θs 2 p ( ) p he ( ) [ p h ( ) 2 s + B (θ) p hh ( ) θ ] p e ( ) p h ( ) [ p h ( ) 2 s + B (θ) p hh ( ) θ ] p ( ) p he ( ) p e ( ) p h ( ) Since the left hand side is positive and the right hand side is negative, this is satisfied. 19

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