Optimal deterrence with legal defense expenditure.

Gravelle, Hugh ; Garoupa, Nuno

I. INTRODUCTION

In this article we consider a justice system which makes both type I errors (sometimes convicting the innocent) and type II errors (sometimes acquitting the guilty) and in which legal defense expenditure on behalf of the accused can reduce their probability of conviction. Such expenditure has a private value, but its social value is less clear. We address two sets of issues. First, do individuals accused of crimes take socially optimal decisions about their expenditure on legal defense and, if not, what are the implications for public policy in terms of the regulation of defense expenditure or its provision by the state? Second, what are the implications of legal defense expenditure for optimal deterrence policy?

In much of the literature on optimal deterrence welfare is the unweighted sum of the expected utilities of honest and dishonest individuals (Becker, 1968; Polinsky and Shavell, 1979). With such an efficiency-oriented welfare function, the only socially relevant aspects of legal defense are its effect on deterrence and its cost.

Because defense expenditures reduce the probability of punishment of dishonest individuals, they reduce the expected sanction on dishonest individuals and thus reduce deterrence. However, when there are type I errors, honest individuals face the risk of wrongful arrest and conviction. (1) Defense expenditure by them increases the rewards for honesty. What matters for deterrence are the effects of defense expenditure on the difference between the expected utility of honest and dishonest individuals. We show that permitting defense expenditures may increase deterrence as compared with a system in which no such expenditure is permitted.

We also show that the level of defense expenditure chosen by accuseds is not efficient. The private value of legal defense expenditure is the reduction in the probability of conviction. Both honest and dishonest individuals choose their expenditure, when accused, to maximize their expected utility. The marginal effect of defense expenditure on the expected utility of honest and dishonest individuals at the privately optimal levels is zero. Consequently marginal changes in defense expenditure at the private optimum have no effect on deterrence because they do not change the difference between the expected utilities of honest and dishonest individuals. But defense expenditures do have a positive marginal social cost, so that the marginal private and social values of defense expenditure are different and there is scope for regulation to improve on the market equilibrium.

There is a commonly held view that type I and type II errors have a direct effect on social welfare, rather than an indirect one through their effect on deterrence. The view is reflected in Blackstone's remark that it is "better that ten guilty persons escape, than that one innocent suffer" (Volokh, 1997, 174). (2) With a justice-oriented welfare function reflecting concern with type I and II error, as well as with individuals' expected utilities, increases in defense expenditure by the honest reduce type I errors and thereby, ceteris paribus, increase welfare. Increases in defense expenditure by the dishonest increase type II errors and thereby directly reduce welfare. With a justice oriented welfare function honest individuals spend too little on defense if the social cost of a type I error is greater than the fine imposed if the individual is convicted. The marginal social value of defense expenditure by the dishonest is always negative at their privately optimal level.

The canonical result in efficient deterrence policy is the Becker (1968) demonstration that, because fines have a positive deterrent effect but a zero marginal social cost, the optimal policy is to increase them to a level that leaves convicted accused with zero wealth. There are few examples of deterrence regimes that use maximal fines, and the Becker result has generated a large literature that suggest reasons why the efficient fine should not bankrupt convicted accused (Garoupa, 1997; Polinsky and Shavell, 2000).

The effective fine is limited by the wealth, net of defense expenditure, of those convicted. We show that if the nominal fine is large enough, accused individuals will choose a high level of defense expenditure such that they will be bankrupted and unable to pay all the nominal fine if found guilty but which leaves them with positive wealth if acquitted. With a lower nominal fine they spend less on defense and are not bankrupt even if convicted. There is critical level of the fine such that they are indifferent between the high level of defense expenditure that bankrupts them if found guilty and a lower level that does not.

Suppose that the critical fine is smaller for the dishonest than the honest. Then it can never be efficient to set the fine above the level that is critical for the dishonest. Raising the fine above this level would have no effect on the dishonest, because they cannot pay it in full. Their defense expenditure will, however, jump upward. The honest do pay the fine in full if convicted. They are made worse off and their defense expenditure is increased. Hence raising the fine above the critical level for the dishonest simultaneously reduces deterrence and increases socially costly defense expenditure. The optimal fine cannot exceed this critical level and both dishonest and honest types are left with positive wealth if convicted. Note that it is the fact that defense expenditures are chosen by the accused that drives the conclusion that the optimal fine leave those convicted with positive wealth, not the presence of type I errors.

Alternatively, if less plausibly, suppose the critical fine for the honest is smaller than for the dishonest, so that they are induced to choose bankruptcy before the honest. The fine again should not be greater than the critical level that induces the dishonest to choose bankruptcy. At higher levels there is no effect on deterrence because neither type of individual pays the fine in full, but the expenditure of the dishonest jumps upward. Hence the fine should not exceed a level that leaves the dishonest with positive wealth but that induces the honest to prefer bankruptcy if convicted. We show that if defense expenditures are sufficiently responsive to the fine it will be optimal to set the fine below the critical level for the dishonest. Thus even in this case where the honest are induced to choose bankruptcy before the dishonest, the efficient deterrence policy can leave both the honest and the dishonest with positive wealth if convicted. The result does not require that there be a direct social harm fro m type I errors.

With a justice-oriented welfare function, in which the amount of type I and II error has a direct impact on welfare, assessing the marginal social value of the fine is more complicated. Greater deterrence reduces the amount of type II error because there are fewer dishonest individuals, but, because there are more honest individuals, there is more type I error. A higher fine induces more defense expenditure, which reduces the amount of type I error and increases type II error. Overall a concern for type I and II errors can increase or reduce the optimal fine.

We outline the model and consider the implications of defense expenditure for deterrence in the next section. In the following section, we consider the choice of the fine in the presence of type I and II error and defense expenditure and show that it is never efficient to have a fine that bankrupts all those convicted. At the end of the third section, we consider the justice-oriented welfare function and its implications for optimal fine. Next, policies to regulate the level of legal defense expenditure are discussed. We show that the market equilibrium level of legal defense expenditure is not optimal under either welfare function. We further examine an upper limit on individuals' legal defense expenditures and a tax-financed public defender system in which all accused receive the same amount of defense.

In the remainder of this introduction we relate our analysis to the previous literature. Type I and II errors are considered by Ehrlich (1982) and Miceli (1991), who show that such errors can lead to a less than maximal sanction if there is direct social concern with the wrongful punishment of the honest and the mistaken acquittal of the dishonest. In Png (1986) individuals can decide to take part in an activity that may produce more social benefit than harm if they exercise sufficient care. Because those who do not exercise sufficient care may escape punishment, the fine should exceed the harm caused to ensure that expected punishment equals the harm. Such a fine reduces participation in the activity below its efficient level because some of those who do take sufficient care are wrongly punished. Thus there should also be a subsidy for participation in the activity. In none of these papers are individuals able to reduce their probability of punishment by legal defense expenditure.

The relationship between punishment and legal defense is considered in Lott (1987), where it is assumed there are only type II errors, that punishment is fixed, and that potential criminals have different opportunity costs of imprisonment. Lott (1987) argues that if punishment does not vary by opportunity cost, those with high opportunity costs will be overdeterred. Allowing individuals to choose their defense expenditure leads to those with greater opportunity costs spending more on defense and having lower probability of punishment, thus reducing the extent of overdeterrence. But restrictions on defense expenditure can be welfare-increasing in these circumstances because they also reduce the rewards to dishonesty of individuals with low opportunity costs of imprisonment who are underdeterred (Garoupa and Gravelle, 2000).

Kaplow and Shavell (1990) show that legal advice to those contemplating potentially harmful acts may either raise or lower welfare, though they do not consider the costs of providing the advice. In Malik (1990) there are only type II errors, and individuals who decide to commit a crime can engage in a costly "avoidance" activity that reduces their probability of punishment. Consequently, it is argued that the fine may be less than the wealth of the accused because its beneficial deterrent effect is partially offset by its inducement of costly avoidance activity. Any policy that reduced avoidance activity would therefore be socially beneficial. Our model allows for type I errors and thus avoidance activity (legal defense expenditure) by honest individuals. In these circumstances the issue of whether avoidance activities that raise the expected utilities of both honest and dishonest individuals are socially beneficial is nontrivial. Further, by correctly specifying the bankruptcy constraint on policy, we show that the efficient fine always leaves the convicted dishonest individuals with positive wealth.

II. THE MODEL

The basic model is standard (Polinsky and Shavell, 1979, 2000). All individuals are risk neutral and endowed with an income of y. (3) Each considers the possibility of committing a harmful act with a benefit of b and a social cost of h. Individuals are identical except for their benefit b [member of] [0, [infinity]) which has a distribution function G(b) over individuals. Because h > 0 is finite and fixed, some crimes have a positive net social benefit. This is not essential, though it is of interest to show that an optimal deterrence policy will in general not seek to deter all crimes with a negative net social value. One example might be firms that can choose to abide by pollution control standards or violate them and risk a fine. The social harm from additional pollution may exceed or be less than the costs incurred by the firm in reducing its emissions.

Choice of Defense Expenditure by Accused

Every individual, whether honest or dishonest, has a probability of being arrested and tried for the offense. The arrest probability of the dishonest [q.sub.1] exceeds that of the honest [q.sub.0] (see Table 1):

(i) 1 > [q.sub.1] > [q.sub.0] > 0.

The probability of conviction [p.sub.i] for an accused individual of type i is a decreasing convex function [p.sub.i]([c.sub.i]) of legal defense costs [c.sub.i]: [p'.sub.i] < 0, [p".sub.i] > 0 and satisfies [p.sub.i](0) < 1, [p.sub.i]([infinity]) > 0. There are both type I errors, because the honest have a positive probability of conviction: [q.sub.0][p.sub.0]([infinity]) > 0, and type II errors, because the dishonest have a positive probability of escaping punishment: [q.sub.1][1 - [p.sub.1](0)] > 0.

The nominal fine for the offense is f. If the fine is less than y - [c.sub.0] it is paid in full by a wrongfully convicted honest individual, who is left with y - [c.sub.0] - f. If the fine exceeds y - [c.sub.0] he is left with nothing. The honest accused chooses [c.sub.0] to maximize [u.sub.0]([c.sub.0]) = [1 - [p.sub.0]([c.sub.0])](y - [c.sub.0]) + [p.sub.0]([c.sub.0]) max{y - [c.sub.0] - f, 0}.

The dishonest accused's problem is very similar. We assume that the benefit b from the crime is not recoverable by a fine. (4) If the nominal fine is less than (y - [c.sub.1]) it is paid in full and the convicted dishonest individual is left with y + b - [c.sub.1] - f. If the fine exceeds y - [c.sub.1] he is left with b. The dishonest accused will choose [c.sub.1] to maximize

[u.sub.1]([c.sub.1]) + b = [1 - [p.sub.1]([c.sub.1])](y - [c.sub.1])

+ [p.sub.1]([c.sub.1]) max{y - [c.sub.1] - f, 0} + b,

= [1 - [p.sub.1]([c.sub.1])](y + b - [c.sub.1])

+ [p.sub.1]([c.sub.1]) max{y + b - [c.sub.1] - f, b}.

We assume that [p'.sub.i](0) > -[infinity] so that there is a small enough fine [f.sup.0.sub.i] such that the accused only chooses to spend on defense if f > [f.sup.0.sub.i]. There are two possible solutions with positive defense expenditure: [c.sup.*.sub.i] and [c.sup.**.sub.i], i=0,1. The first holds when the accused chooses a level of expenditure that leaves positive wealth if convicted: [c.sub.i](f) = [c.sup.*.sub.i](f) = argmax{y - [c.sub.i] - [p.sub.i][f.sub.i]}. The second holds when the fine is so large that the individual chooses a level of expenditure that leads to bankruptcy if convicted: [c.sub.i](f) = [c.sup.**.sub.i]=argmax{(y - [c.sub.i])(1 - [p.sub.i])}. The two solutions are characterized by

(1) -[[p'.sub.i]([c.sup.*.sub.i])f + 1] = 0 and

(2) -[p'.sub.i]([c.sup.**.sub.i])[y - [c.sup.**.sub.i]]-[1-[p.sub.i]([c.sup.**.sub.i])] = 0

and are shown in Figure 1.

When the individual is bankrupted if convicted, both the expected marginal cost and the expected marginal gain from defense expenditure are reduced. As we show in the Appendix (in the proof of Proposition 1), the former effect outweighs the latter, so that expenditure when the individual chooses to be bankrupt if convicted is greater than if he does not: [c.sup.**.sub.i] > [c.sup.*.sub.i].

For low enough fines the accused chooses a level of defense expenditure [c.sup.*.sub.i] that leaves positive income even if convicted. His maximized expected utility conditional on being accused is then [v.sup.n.sub.i](f) = y - [c.sup.*.sub.i](f) - [p.sub.i]([c.sup.*.sub.i](f))f and is decreasing with the fine. For a large enough fine the accused will prefer to choose a level of defense expenditure [c.sup.**.sub.i] that bankrupts him if convicted. His maximized expected utility in this case is [v.sup.b.sub.i] = [1 - [p.sub.i]([c.sup.**.sub.i])](y - [c.sup.**.sub.i]), which does not vary with the fine. As we show in the Appendix (in the proof of Proposition 1), there is a level of the fine [F.sub.i] at which the accused is indifferent between [c.sup.*.sub.i]([F.sub.i]) and [c.sup.**.sub.i]. To avoid needless complications we assume that indifferent individuals choose [c.sup.*.sub.i]([F.sub.i]). Defense expenditure is at first increasing in f, jumps upward at [F.sub.i], and is constant thereafter. Expected utility is continuous in the fine, though defense expenditure is not.

Although the accused is indifferent between [c.sup.*.sub.i](F) and [c.sup.**.sub.i] when f = [F.sub.i] and would be bankrupt if convicted if he chose [c.sup.**.sub.i], the accused is not bankrupt if he chooses [c.sup.*.sub.i](F). Let [c.sub.i] be the defense expenditure that just bankrupts him: [c.sub.i] = y - [F.sub.i]. Substituting [F.sub.i] = [p.sub.i] ([c.sub.i])[F.sub.i] + [1 - [p.sub.i]([c.sub.i])][F.sub.i] and rearranging, we see that if he chose this level of expenditure he would have y - [c.sub.i] -[p.sub.i]([c.sub.i])[F.sub.i] = [1 - [p.sub.i]([c.sub.i])] (y - [c.sub.i]). As inspection of Figure 1 shows, this must occur where [c.sub.i] > [c.sup.*.sub.i] ([F.sub.i]).

Hence, if convicted he must have strictly positive wealth at [c.sup.*.sub.i]([F.sub.i]): y - [c.sup.*.sub.i]([F.sub.i]) - [F.sub.i] > 0.

Summarizing, we have the following (proofs of propositions not given in the text are in the Appendix).

PROPOSITION 1. There is a level of the fine [F.sub.i] for each type of individual such that if f [less than or equal to] [F.sub.i] the accused chooses a level of defense expenditure [c.sup.*.sub.i](f) that does not lead to bankruptcy if convicted and if f > [F.sub.i] the accused chooses a level of defense expenditure [c.sup.**.sub.i] that does lead to bankruptcy if convicted. Privately optimal defense expenditure [c.sub.i] (f) is increasing in the fine in the nonbankruptcy case, discontinuous upward in the fine at [F.sub.i], and unaffected by the fine in the bankruptcy case:

[c.sub.i](f) = 0, f [member of] [0, [f.sup.o.sub.i]],

= [c.sup.*.sub.i](f), f [member of] ([f.sup.o.sub.i], [F.sub.i]],

= [c.sup.**.sub.i] > [c.sup.*.sub.i]([F.sub.i]), f [member of] ([F.sub.i], [infinity]).

[c.sub.if] = [c.sup.*.sub.if](f) > 0, f [member of] ([f.sup.o.sub.i], [F.sub.i])

= 0, f [member of] ([F.sub.i], [infinity])

We assume that the legal system is not perverse in the sense that the honest are more likely to be convicted than the dishonest when they spend the same amount on defense. The nonperversity assumption is

(ii) [p.sub.1](c) > [p.sub.0](c).

It is less obvious whether defense expenditure has a larger or smaller marginal effect for the honest or the dishonest. We could argue that a good (high-cost) lawyer will make more difference when the evidence against the accused is stronger (because he is guilty) so that

(iii) [p'.sub.1](c) < [p'.sub.0](c) < 0.

However, defense expenditure could have a greater marginal effect for the innocent: [p'.sub.0](c) < [p'.sub.1](c) < 0. For example, searching for witnesses who will testify that the accused was elsewhere when the crime was committed may be less costly if he is innocent than if he is guilty.

Using the first-order conditions it is straightforward to establish the following.

PROPOSITION 2. For f > [f.sup.o.sub.1] dishonest accuseds spend more on legal defense than honest accused if their defense expenditure has a greater marginal effect (iii holds). For f [member of] ([f.sup.o.sub.i], [F.sub.1]) and for f > [F.sub.0] dishonest accuseds spend less on legal defense than honest accused if their defense expenditure has a smaller marginal effect.

Part (a) of Figure 2 illustrates the case in which defense expenditure is more productive for the dishonest than the honest. In part (b) defense expenditure is more productive for the honest.

If the marginal effect of defense expenditure is larger for honest accused, Proposition 2 and the nonperversity assumption (ii) imply that the honest have a smaller probability of conviction than the dishonest. However, if the marginal effect of defense expenditure is greater for the dishonest, they could spend sufficiently more on defense to offset their disadvantage in having a higher conviction probability at given levels of defense expenditure. Hence there could be a p probability reversal: [p.sub.1]([c.sub.1](f)) < [p.sub.0]([c.sub.0](f)).

If the dishonest accused have a smaller conviction probability than the honest, it is also possible that they have a smaller unconditional probability of being convicted: [q.sub.1][p.sub.1]([c.sub.1](f)) < [q.sub.0][p.sub.0]([c.sub.0](f)). As we will see shortly, such a qp probability reversal implies that when both honest and dishonest prefer a level of defense expenditure that enables them to pay the fine if convicted, increases in the fine will increase the amount of crime. We will restrict attention to the more plausible case in which there is no qp probability reversal and the dishonest have a greater unconditional conviction probability than the honest:

(iv) 1 > [q.sub.1][p.sub.1]([c.sub.1](f)) > [q.sub.0][p.sub.0]([c.sub.0](f)) > 0.

Deterrence policy is considerably affected by whether the honest or the dishonest are the first to choose bankruptcy as the fine increases. Accuseds prefer the optimal nonbankrupting expenditure as long as [v.sup.n.sub.i](f) exceeds [v.sup.b.sub.i]. Increases in the fine reduce [v.sup.n.sub.i] at the rate [p.sub.i]([c.sup.*.sub.i](f)) and have no effect on [v.sup.b.sub.i]. Honest and dishonest have the same expected utility from nonbankruptcy with a zero fine. With no probability reversal [v.sup.n.sub.1] will fall faster than [v.sup.n.sub.0] as f increases. However, this does not imply that the dishonest will be the first to choose bankruptcy as f increases: their expected utility from bankruptcy is smaller than that of honest accuseds. Although [v.sup.n.sub.1] falls faster than [v.sup.n.sub.0], it must fall further before the dishonest prefer bankruptcy.

It there is probability reversal, so that [v.sup.n.sub.1] falls more slowly than [v.sup.n.sub.0] as f increases, the honest will choose bankruptcy before the dishonest: [F.sub.0] < [F.sub.1]. The converse does not hold, so that if there is no probability reversal we cannot conclude that the dishonest will choose bankruptcy before the honest.

The case in which the dishonest choose bankruptcy before the honest seems more plausible and we will often make this assumption:

(v) [F.sub.1] < [F.sub.0].

Honesty versus Dishonesty

If honest, an individual has expected utility [V.sub.0]([c.sub.0]) = (1 - [q.sub.0])y + [q.sub.0][u.sub.0]([c.sub.0]); whereas a dishonest individual has expected utility [V.sub.1]([c.sub.1]) = (1 - [q.sub.1])y + [q.sub.1][u.sub.1]([c.sub.1]) + b. The individual commits a crime if and only if [V.sub.1] > [V.sub.0] or equivalently

(3) b > z([c.sub.0], [c.sub.1], f, y)

= (1 - [q.sub.0])y + [q.sub.0][u.sub.0]([c.sub.0]) - (1 - [q.sub.1])y

- [q.sub.1][u.sub.1]([c.sub.1]).

Because the proportion of individuals choosing honesty is G(z) we can define z as the amount of deterrence produced by the criminal justice system.

When the dishonest choose bankruptcy if convicted at a lower fine than the honest ([F.sub.1] < [F.sub.0]), we have

(4) z = [q.sub.1][[p.sub.1]f + [c.sup.*.sub.1]] - [q.sub.0][[p.sub.0]f + [c.sup.*.sub.0], f [less than or equal to][F.sub.1];

(5) = [q.sub.1][[p.sub.1]y + (1 - [p.sub.1])[c.sup.**.sub.1]] - [q.sup.0][[p.sub.0]f + [c.sup.*.sub.0]],

[F.sub.1] < f [less than or equal to] [F.sub.0]; and

(6) = [q.sub.1][[p.sub.1]y + (1 - [p.sub.1])[c.sup.**.sub.1]]

- [q.sub.0][[p.sub.0]y + (1 - [p.sub.1])[c.sup.**.sub.0]], [F.sub.0] < f.

When [F.sub.0] < [F.sub.1] the deterrence function has the same form as (4) for f [member of](0, [F.sub.0]) and as (6) for f [member of] ([F.sub.1], [infinity]), but

(7) z = [q.sub.1][[p.sub.1]f + [c.sup.*.sub.1]] - [q.sub.0][[p.sub.0]y+(1 - [p.sub.0])[c.sup.**.sub.0]], [F.sub.0] < f [less than or equal to] [F.sub.1].

Examination of the deterrence function establishes the following.

PROPOSITION 3. Deterrence is a continuous function of the fine. If the dishonest have a greater unconditional probability of conviction than the honest (iv holds), then (a) if [F.sub.1] < [F.sub.0], deterrence is increasing in the fine up to [F.sub.1]: dz/df = [q.sub.1][p.sub.1] - [q.sub.0][p.sub.0] > 0, decreasing over the range ([F.sub.1], [F.sub.0]): dz/df = -[q.sub.0][p.sub.0] < 0, and constant thereafter; (b) if [F.sub.0] < [F.sub.1], deterrence is increasing in the fine up to [F.sub.1] and constant thereafter.

Figure 3 illustrates. In part (a), where [F.sub.1] < [F.sub.0], increases in the fine to [F.sub.1] make crooks and honest individuals worse off, even though they respond by increasing their defense expenditure. The crooks suffer a larger fall in expected utility and so more individuals choose to be honest. Over the range [[F.sub.1], [F.sub.0]], crooks are unaffected by increases in the fine because they are bankrupted if found guilty, whereas honest individuals are made worse off. Hence the gain from crime compared to honesty increases. With a fine large enough to make all types of convicted accused prefer bankruptcy if convicted, further increases in the fine have no effect on the expected utility of either crooks or honest individuals.

Figure 3(b) shows the case where [F.sub.0] < [F.sub.1]. The expected utility of the dishonest falls faster than that of the honest up to [F.sub.0], so deterrence increases. Once the honest have chosen to be bankrupt if convicted, increases in the fine only affect the dishonest, so that the fine has a greater marginal deterrence effect and the deterrence function is kinked upward at [F.sub.0]. Once both groups prefer bankruptcy when convicted, further increases in fine have no additional deterrent effect.

Type I and II Errors and Deterrence

It is well known (Miceli, 1991) that the existence of type I and type Illegal errors weakens deterrence because the possibility of wrongful arrest and conviction reduces the expected utility from being honest and the possibility of escaping punishment increases the expected utility from dishonesty. The implications of type I and II errors are less straightforward once legal defense expenditure is allowed for.

The accuseds' ability to spend money to reduce their conviction probability when arrested increases their expected utility compared with no expenditure: [V.sub.i]([c.sub.i](f)) > [V.sub.i](0). What matters for deterrence is the difference between the expected utilities of honest and dishonest individuals. The fact that [V.sub.i]([c.sub.i](f)) > [V.sub.i](0) implies nothing about whether [V.sub.0]([c.sub.0](f)) - [V.sub.1]([c.sub.1](f))is greater or less than [V.sub.0] (0) - [V.sub.1] (0). However if there are no type I errors, so that only the dishonest engage in defense expenditure that increases their expected utility, defense expenditure must reduce deterrence. We have the following.

PROPOSITION 4. In the presence of type I and II errors, permitting defense expenditure by accused individuals may increase or reduce the amount of crime: z([c.sub.0](f), [c.sub.1](f),f) may be less than or greater than z(0, 0, f). If there are no type I errors ([q.sub.0][p.sub.0] = 0) permitting legal defense expenditure reduces deterrence.

III. WELFARE MAXIMIZING DETERRENCE

Efficient Fine

The public sector cost of deterrence activities is the cost of policing less the expected receipts from fines and is covered by lumpsum taxes T imposed elsewhere in the economy. We assume in effect that we are concerned with deterrence in a small area or sector whose potentially dishonest population is small in relation to the total population of the economy. Hence changes in T resulting from changes in this sector have a negligible effect on the tax bills of the population in the sector. The assumption enables us to ignore the potential complications that would result from changes in T leading to changes in the critical levels [F.sub.i] at which accused would prefer bankruptcy if convicted.

To make our results comparable with the deterrence literature we initially assume an efficiency-oriented welfare function that is the unweighted sum of the expected utilities of honest and dishonest individuals and the taxpayers:

(8) W = W([c.sub.0](f), [c.sub.1](f),

z([c.sub.0](f), [c.sub.1](f),f),f)

=[[integral].sup.z.sub.0] [V.sub.0]dG + [[integral].sup.[infinity].sub.z][[V.sub.1] - h]dG - T

= y + [[integral].sup.[infinity].sub.z](b - h)dG - m - G(z)[q.sub.0][c.sub.0]

- [1 - G(z)][q.sub.1][c.sub.1],

where z(*) is defined by (3) and m is the fixed level of policing expenditures. Notice that the social consequences of legal defense, as opposed to its private consequences, are solely determined by its impact on deterrence and by its cost.

We assume that the planner cannot observe whether an individual is honest or not, nor can she observe the level of any individual's defense expenditure. We suppose initially that her only policy instrument is the fine (5) and can establish (see the Appendix) the following.

PROPOSITION 5. Assume that the dishonest have a greater probability of conviction than the honest (iv). The efficient fine always leaves dishonest individuals with positive wealth if convicted: [f.sup.*] [less than or equal to] [F.sub.1]. If [F.sub.1] < [F.sub.0] both honest individuals and dishonest individuals have positive wealth if convicted.

Although, as we noted, deterrence is a kinked function of the fine at [F.sub.1] and [F.sub.0] and that defense expenditure jumps at these points, the derivative of the welfare function with respect to the fine exists and is continuous except at [F.sub.0] and [F.sub.1]. We can therefore obtain some intuition about the result by

examining

(9) dW/df = [W.sub.z][z.sub.f] + [SIGMA][W.sub.ci][c.sub.if]

= [(h - z) + ([q.sub.1][c.sub.1] - [q.sub.0] [c.sub.0])]g(z)[z.sub.f]

-[q.sub.0]G[c.sub.0f]-[q.sub.1](1 - G)[c.sub.1f].

Suppose that the dishonest choose bankruptcy before the honest ([F.sub.1] < [F.sub.0]). Increases in the fine have no effect on defense expenditures for an individual of type i if the fine exceeds [F.sub.i] (see Proposition 1). Hence for f > [F.sub.0] > [F.sub.1] the last two terms in (9) are zero and so is the first term, because deterrence is not affected. Consequently, the marginal value of the fine is zero for f > [F.sub.0] > [F.sub.1].

For [F.sub.0] > f > [F.sub.1] the first term in (9) is negative because deterrence is reduced in this range (Proposition 3). Because [c.sub.0f] is positive and [c.sub.1f] is zero, the marginal value of the fine is negative for this range.

For [F.sub.0] > [F.sub.1] > f increases in the fine do increase deterrence so that the first term in (9) is positive. The second and third terms are negative, so an interior efficient fine is possible in this range.

In the perhaps less plausible case in which the honest are bankrupt before the dishonest ([F.sub.0] < [F.sub.1]) deterrence is increasing with the fine up to f = [F.sub.1] and constant thereafter. Because defense costs increase with f it is again possible that the efficient fine does not maximize deterrence and leaves either the dishonest or both the honest and dishonest with positive wealth if convicted.

These arguments suggest that the welfare-maximizing fine cannot exceed [F.sub.1] and that there are two types of efficient fine depending on the relative marginal importance of the fine's impact on deterrence and in reducing defense expenditure: (6)

* Critical fine: [f.sup.*] = [F.sub.1]. If deterrence has a high marginal value relative to the marginal cost of defense expenditure, then our assumption that all dishonest individuals choose [c.sup.*.sub.1] at [F.sub.1] means deterrence can be increased to its maximal extent without triggering a jump in defense expenditure. Dishonest accuseds are not bankrupted if not convicted. Honest individuals are also left with positive wealth or bankrupt if convicted depending on whether [F.sub.1] < [F.sub.0] or [F.sub.1] > [F.sub.0].

* Interior fine: [f.sup.*] < [F.sub.1]. If increases in the fine lead to an increase in defense expenditure that is large relative to the change in deterrence, there will be a interior solution in which the efficient fine is less than the critical [F.sub.1] and the dishonest are not bankrupted by the fine. If [F.sub.1] < [F.sub.0] the honest will have positive wealth if convicted but may be bankrupt if [F.sub.1] > [F.sub.0].

In models with type I and II errors but no defense expenditures the bankruptcy constraint is just y [less than or equal to] f for both honest and dishonest accuseds. Because there is no qp probability reversal in such models, increases in the fine always increase deterrence and do not increase socially costly defense expenditures. Hence the efficient fine is as large as possible: [f.sup.*] = y. Thus it is defense costs, rather than the presence of type I and type II errors, that potentially overturn the Becker (1968) maximal fine result when there are no direct costs of type I and II errors. Although the fine is a deterrent, it is also a stimulus to socially wasteful expenditure by the accused.

Does punishment fit the crime in the sense that the dishonest face an expected penalty equal to the direct social harm they impose: z = h? In the standard Becker (1968) model the punishment is less than the harm because policing has a positive marginal cost. The marginal value of deterrence is positive at the optimum because deterrence has a positive marginal cost. In our model, where there are defense costs and legal error, the marginal social benefit from deterring an additional criminal will also be positive at the optimum. The marginal social benefit from deterrence is the net social cost of the marginal crime (h - b) = (h - z) plus the expected increase in expenditure of criminals compared to honest individuals ([q.sub.1][c.sub.1] - [q.sub.0][c.sub.0]). When (iii) holds, so that criminals spend more on defense than honest accused, the latter term is positive. Hence a positive marginal social value of deterrence does not necessarily imply that the benefit to the marginal criminal (z) is less than the soci al harm from the crime. Criminals may face expected penalties that exceed the direct harm they inflict.

Justice-Oriented Welfare Function

The unweighted utilitarian or efficiency orientated welfare function (8) is standard but does not reflect the common view that there are additional social costs from a justice system that fails to convict all dishonest accused and convicts some honest individuals.

Suppose that each case of wrongful conviction has a social cost of [k.sub.0](f), where [k'.sub.0](f) > 0 reflects the judgment that the social harm from wrongful conviction is greater the greater the punishment imposed (Ehrlich, 1982; Miceli, 1991). Each dishonest person who is not punished, either by not being tried or by wrongful acquitted, has a social cost of [k.sub.1]. (7) The justice-oriented social welfare function is

(10) W = W - G[q.sub.0][p.sub.0][k.sub.0] - (1 - G)

x (1 - [q.sub.1][p.sub.1])[k.sub.1].

The weights [k.sub.0], [k.sub.1] are the marginal social cost of a single wrongful conviction and a single guilty person escaping punishment. The judgment that it is better that n guilty persons go free than that one innocent person be convicted can be interpreted as fixing the relative weights [k.sub.0] = n[k.sub.1].

The effect of a marginal increase in deterrence on welfare is

[W.sub.z] [W.sub.z] - {[p.sub.0][q.sub.0][k.sub.0] - [(1 - [q.sub.1])

+ [q.sub.1](1 - [p.sub.1])][k.sub.1]}g(z)

=[W.sub.z] - Kg(z).

The term K is the marginal injustice cost of increasing deterrence and could be positive or negative. With increased deterrence more individuals choose to be honest, so that the number of mistaken convictions of the honest increases, thereby increasing the amount of injustice. However, the number of dishonest individuals decreases, so that there are fewer who escape punishment, thereby reducing injustice from type II errors. The greater the relative weight [k.sub.0] on type I errors the more likely is it that increasing deterrence has a positive injustice cost.

The marginal social effect of an increase in the fine is

(11) [W.sub.f] = [W.sub.f] - K g [z.sub.f] - G[q.sub.0][p.sub.0][k'.sub.0]

-G[q.sub.0][p'.sub.0][c.sub.0f][k.sub.0] + (1 - G)[q.sub.1][p'.sub.1][c.sub.1f][k.sub.1].

A concern for injustice may tend to increase or decrease the optimal fine. The second term in (11) is negative if the marginal injustice cost of deterrence K is positive. The third effect tends to reduce the optimal fine (Ehrlich, 1982; Miceli, 1991), because increases in the fine raise the social cost of type I errors. The last two terms reflect the impact of the fine in increasing accuseds' defense expenditures and hence reducing their probability of conviction. The fourth term is the social benefit from reduction in the probability of conviction of the honest is a social benefit and works to increase the optimal fine. The last term is the social cost from increasing the probability of acquittal of the dishonest and works to reduce the optimal fine.

A greater concern for type I errors, reflected in an increase in [k.sub.0], has an ambiguous effect on the optimal fine. Increases in [k.sub.0] increase the marginal injustice cost of deterrence K and thereby tend to reduce the optimal fine because increases in the fine increase the number of type I errors because there are more honest individuals. But increases in [k.sub.0] mean that there is a greater value in defense expenditure of accused innocents and increases in the fine increase that expenditure. Hence [k.sub.0] has offsetting effects on the second and fourth terms in (11).

IV. REGULATING DEFENSE COSTS

In this section we consider various policies that can be adopted to affect the defense expenditure of accused. We start by temporarily assuming that it is possible to identify crooks and honest individuals and to control their defense expenditure. The aim of the assumption is to further investigate the welfare implications of defense expenditure as a background to choice of policy under more realistic circumstances.

Social Value of Legal Defense Expenditure

When the defense expenditures of honest and dishonest individuals are directly controlled the welfare function is written as W ([c.sub.0], [c.sub.1], z([c.sub.0], [c.sub.1], f), f). The marginal social values of defense expenditure are

(12) dW / d[c.sub.0] = [W.sub.z] [z.sub.[c.sub.0]] -- K g [z.sub.[c.sub.0]] + [W.sub.[c.sub.0]] -- G[q.sub.0][p'.sub.0][k.sub.0];

(13) dW / d[c.sub.1] = [W.sub.z][z.sub.[c.sub.1]] -- K g [z.sub.[c.sub.1]] + [W.sub.[c.sub.1]] + (1 - G)[q.sub.1][p'.sub.1][k.sub.1].

At the privately optimal defense expenditures the first and second terms in both (12) and (13) are zero. z is the difference between expected utilities that are maximized by private choice of defense expenditures and so [z.sub.[c.sub.1]] = 0. The third term in both expressions is negative and reflects the direct cost of defense expenditure.

An increase in defense expenditure for honest individuals beyond the private optimum has two offsetting effects on welfare: the cost of defense expenditure and the reduction in the conviction probability of honest individuals. If f [less than or equal to] [F.sub.0], the first-order condition (1) on [c.sup.*.sub.i](f) implies that (12) is (f - [k.sub.0])G[q.sub.0][p'.sub.0]. Hence the private choice of legal expenditure when innocent will be too small from a social point of view if the marginal social cost of convicting an innocent person is greater than the marginal gain from the reduction in taxation implied by the increased fine revenue: [k.sub.0] > f. This seems a plausible value judgment.

The marginal social value of [c.sub.1] at [c.sub.1] = [c.sup.*.sub.1](f) is negative because in addition to its cost, additional defense expenditure increases the chance of wrongful acquittal. But the first best socially optimal value of [c.sub.1] may not be zero. It may be possible to set it at a high level, above [c.sup.*.sub.1](f), such that the dishonest individual has a lower expected utility than if [c.sub.1] was zero. For example, setting [c.sub.1] = y leaves the dishonest individual with zero wealth whether convicted or acquitted and a lower expected utility than if [c.sub.1] = 0. The resulting increase in deterrence may be worth the additional defense costs.

PROPOSITION 6. The choices of defense expenditure by honest and dishonest accused are not first-best socially optimal. If f [less than or equal to] [F.sub.0] and there is a social cost of mistaken convictions, the honest individual's choice of legal defense expenditure is too small if and only if [k.sub.0] > f. The first-best expenditure by dishonest accuseds is either zero or greater than the privately optimal level.

Limit on Private Legal Defense Expenditure

The regulator's information is limited: she observes the results of the trial, but she does not observe b, nor individual defense expenditures, nor whether an individual committed a crime. The regulator's feasible policies, in addition to the fine, include a limit on private defense expenditure by accuseds, and a ban on private defense expenditures coupled with a provision of the same level of defense expenditure to all accused (a public defender system). (8)

When the regulator can fix the maximum amount c that accuseds can spend on their defense, the marginal social value of c is

(14) dW / dc = [W.sub.z][z.sub.c] - K g [z.sub.c] + [SIGMA][W.sub.[c.sub.i]]d[c.sub.i]/dc -[k.sub.0]G[q.sub.0][p'.sub.0]d[c.sub.0]/dc + [k.sub.1](1 - G) x [q.sub.1][p'.sub.1]d[c.sub.1]/dc.

Suppose that (iii) and (v) hold so that the honest spend less on defense and choose bankruptcy at a higher fine than the dishonest. Setting c [greater than or equal to] [c.sup.**.sub.1] does not constrain the choices of either type of accused. A reduction in c in the range c [euro] ([c.sup.*.sub.1](f), [c.sup.**.sub.1]) does not affect the expected utility of the honest or their defense expenditure. Reducing c to [c.sup.*.sub.0](f) reduces the defense expenditure of the dishonest and makes them worse off. Hence deterrence increases and defense expenditure falls. If there is no social cost to type I and II errors, (14) reduces to the first and third terms and welfare would be increased by tightening the constraint on defense expenditure in this range.

With no cost to type I or II errors the regulator can restrict attention to c [less than or equal to] [c.sup.*.sub.0] (f) and honest and dishonest accused will choose [c.sub.0] = [c.sub.1] = c. Deterrence in this range is

(15) z = [q.sub.1][[p.sub.1](c)f + c] - [q.sub.0][[p.sub.0](c)f + c].

We have assumed (iii) that the defense expenditure of the dishonest has a greater marginal product, so deterrence is decreasing in c

(16) [z.sub.c]=[q.sub.1][[p.sub.1](c)f + 1] - [q.sub.0][[p'.sub.0](c)f + 1] < 0

by (i) and the fact that c [less than or equal to] [c.sup.*.sub.0](f) < [c.sup.*.sub.1](f). Hence (14) is negative for c [less than or equal to] [c.sup.*.sub.0](f) and the efficient value of c is zero. Note also that deterrence is increasing in f: [z.sub.f] = [q.sub.1][p.sub.1](c) - [q.sub.0][p.sub.0](c) > 0 by (i) and (ii), and so with no social costs of type I and II errors the optimal fine should be maximal.

If there is only a cost to type II errors the previous arguments hold a fortiori because now increased deterrence is of even more value and the third term in (14) is - K g [z.sub.c] = [k.sub.1] (1 - [q.sub.1][p.sub.1])g[z.sub.c] < 0.

We saw in the previous section that the privately optimal defense expenditure was not first-best efficient. We have now established that the private, unregulated equilibrium expenditure is also not second-best efficient: a complete ban on private defense expenditure yields a higher welfare than the unregulated market equilibrium. Indeed any binding restriction on private legal defense expenditure increases welfare compared with the market equilibrium when there are no costs of type I errors.

When there is a cost to type I errors the optimal value of c may be positive and could even be greater than [c.sup.*.sub.0](f). First, with a cost to type I errors K may be positive, so that there is a cost to increased deterrence because the number of honest individuals, who are subjected to type I errors, is increased. Second, relaxing the constraint on defense expenditures in the range c < [c.sup.*.sub.0](f) reduces the probability of conviction of honest accuseds. In the range c [euro] ([c.sup.*.sub.1](f), [c.sup.**.sub.1]), the second factor does not operate but the first does. Hence it is possible that a concern for type I errors could lead to a constraint on defense expenditure that binds only for the dishonest. The same type of argument we used in the previous section shows that optimal fine may also be less than maximal when there is a cost to type I errors.

PROPOSITION 7. If defense is more productive for the dishonest (iii), they choose bankruptcy before the honest and (v), and there is no cost of type I error ([k.sub.0]=0), then the efficient limit on defense expenditure is zero: [c.sup.*] = 0 and the efficient fine is maximal: [f.sup.*] = y. When there is a cost of type I error ([k.sub.0] > 0), the optimal limit on defense expenditure may be positive and may only constrain the dishonest, and the optimal fine may be less than maximal.

Public Defender System

Under a public defender system all accused are provided with the same amount of legal defense [c.sub.0] = [c.sub.1] = c, which is paid for by taxation rather than by the accused. Accordingly we now constrain policy by y - f [greater than or equal to] 0. Deterrence is

(17) z = [[q.sub.1][p.sub.1](c) - [q.sub.0][p.sub.0](c)]f,

and increases in c reduce deterrence: [z.sub.c] < 0.

The marginal welfare from public expenditure on legal defense is

[W.sub.z][z.sub.c] + [SIGMA] [W.sub.[c.sub.]] - K g [z.sub.c] - [k.sub.0]G[q.sub.0][p'.sub.0] + [k.sub.1] (1 - G)[q.sub.1][p'.sub.1],

which is very similar to (14). Analogous arguments to those in previous sections establish the following.

PROPOSITION 8. If the cost of convicting the innocent is zero, the efficient amount of publicly funded legal defense expenditure is zero, and the efficient fine is maximal. When the cost of type I error is positive ([k.sub.0] > 0) the optimal amount of publicly funded legal defense expenditure may be positive and may be greater than the amount chosen by honest defendants, and the optimal fine may be less than maximal.

V. CONCLUSIONS

When legal defense expenditure reduces the probability of conviction and there is no direct concern with the cost of legal errors, policies are influenced by their implications for deterrence and the level of defense costs. If the only policy variable is the fine, its impact on defense expenditure means that it is never efficient to set it high enough that convicted dishonest accused are bankrupt. If defense expenditures can be directly regulated the efficient regime has no defense expenditure and a fine that is maximal, just bankrupting all accused, both the honest and the dishonest.

Some of these policies would likely be challenged as unconstitutional or grossly unfair to the accused. One way of reflecting such concerns is to suppose that there is a direct social cost of legal errors. Because defense expenditure by the honest reduces the probability that the innocent are wrongly convicted and punished, it can be optimal to permit such expenditure. Direct error costs can also justify the provision of defense from public funds. Furthermore, optimal fines could be less than maximal when there is concern about the conviction of innocent individuals.

APPENDIX

Proof of Proposition 1

We assume that the productivity of defense expenditure ensures that there is an [f.sup.o.sub.i] such that when f > [f.sup.o.sub.i], the optimal level of defense expenditure is positive and characterized by the first-order conditions (1) and (2), depending on whether the accused is bankrupted by the optimal defense expenditure or not.

a. Denote the maximized value of y - [c.sub.1], - [p.sub.i]([c.sub.i])f by [v.sup.n.sub.i](f) and of [1 - [p.sub.i]([c.sub.i])](y - [c.sub.i]) by [v.sup.b.sub.i]. [v.sup.n.sub.i](f) is monotonically decreasing in f and satisfies [v.sup.n.sub.i](0) > [v.sup.b.sub.i] > [v.sup.n.sub.i]([infinity]). Hence there exists a unique [F.sub.i] satisfying [v.sup.n.sub.i]([F.sub.i]) = y - [c.sup.*.sub.i] ([F.sub.i]) - [p.sub.i]([c.sup.*.sub.i]([F.sub.i]))[F.sub.i] = (1 - p([c.sup.**.sub.i]))(y - [c.sup.**.sub.i]) = [v.sup.b.sub.i].

b. Because (1) characterizes the choice of defense expenditure if and only if the individual is not bankrupt at [c.sup.*.sub.i] we have -1 + [p.sub.i]([c.sup.**.sub.i]) - [p'.sub.i]([c.sup.**.sub.i])[y - [c.sup.**.sub.i]] = 0 = -1 - [p'.sub.i]([c.sup.*.sub.i])f [less than or equal to] -1 - [p'.sub.i]([c.sup.*.sub.i])[y - [c.sup.*.sub.i]], which implies 0 < - [p'.sub.i]([c.sup.**.sub.i])[y - [c.sup.**.sub.i] < - [p'.sub.i]([c.sup.*.sub.i])[y - [c.sup.*.sub.i]]. Suppose that [c.sup.**.sub.i] [less than or equal to] [c.sup.*.sub.i]. Then -[p'.sub.i]([c.sup.**.sub.i]). [greater than or equal to] -[p'.sub.i]([c.sup.*]) from the convexity of [p.sub.i](c) and y - [c.sup.**.sub.i] [greater than or equal to] y - [c.sup.*.sub.i]. Because y - [c.sup.**.sub.i] [greater than or equal to] 0 we have a contradiction and have established that [c.sup.*.sub.i](f) < [c.sup.**.sub.i] for all f [member of] (0, [F.sub.i]].

c. From the implicit function theorem, we know that the sign of [c.sup.*.sub.if](f) and [c.sup.**.sub.if](f) are given by [[partial].sup.2][u.sub.i]/[partial][c.sub.i][partial]f. The comparative static results are immediate from (1) and (2).

d. From the definition of [F.sub.i] and the fact that [1 - [p.sub.i]([c.sub.i])](y-[c.sub.i]) is maximised at [c.sup.**.sub.i], y-[c.sup.*.sub.i]-[p.sub.i]([c.sup.*.sub.i])[F.sub.i] = [1- [p.sub.i]([c.sup.**.sub.i])](y-[c.sup.**.sub.i]) > [1-[p.sub.i]([c.sup.*.sub.i])](y-[c.sup.*.sub.i]), and subtracting the right-hand expression from the left-hand expression we get [p.sub.i]([c.sup.*.sub.i]([F.sub.i]))[y-[c.sup.*.sub.i]([F.sub.i])-[F .sub.i]] > 0 and so y-[c.sup.*.sub.i]([F.sub.i]) > [F.sub.i]. The accused is not bankrupt if convicted when he chooses [c.sup.*.sub.i]([F.sub.i]).

e. Similarly, [1-[p.sub.i]([c.sup.**.sub.i])](y-[c.sup.**.sub.i]) = y - [c.sup.*.sub.i]-[p.sub.i]([c.sup.*.sub.i])[F.sub.i] > y - [p.sub.i]([c.sup.**.sub.i])[F.sub.i] - [c.sup.**.sub.i] implies [p.sub.i]([c.sup.*.sub.i])[F.sub.i] > [p.sub.i]([c.sup.**.sub.i]) [F.sub.i] > [p.sub.i]([c.sup.*.sub.i])(y-[c.sup.**.sub.i]) and so [F.sub.i] > y - [c.sup.**.sub.i]. The dishonest accused is bankrupt when convicted at f [greater than or equal to][F.sub.i] when he chooses [c.sup.**.sub.i].

Proof of Proposition 3

Deterrence is continuous because up to a positive constant, it is equal to the difference between the value of two maximized functions [V.sub.0]([c.sub.0]), [V.sub.1]([c.sub.1]), each of which are continuous in the fine. Using the envelope theorem, the expression [z.sub.f] is immediate.

Proof of Proposition 5

The welfare function W(f) = Z(f)) = W([c.sub.0](f), [c.sub.1](f), Z(f))=W([c.sub.0](f),[c.sub.1](f),z([c.sub.0](f),[c.sub.1](f),f)) has a continuous total derivative with respect to f everywhere except at [F.sub.1] and [F.sub.0]. The sign of dW/df is discussed in the text.

a. Consider the case in which [F.sub.1] < [F.sub.0]. Then for [member of] > 0, W([F.sub.0]) - W([F.sub.0] + [epsilon]) = [1 - G(Z([F.sub.0]))][[c.sup.**.sub.0] - [c.sup.*.sub.0]([F.sub.0])] > 0, because Z([F.sub.0] = [epsilon]) = Z([F.sub.0]), [c.sub.1]([F.sub.0] + [epsilon]) = [c.sub.1]([F.sub.0]), and [c.sub.0]([F.sub.0] + [epsilon]) = [c.sup.**.sub.0] > [c.sub.0]([F.sub.0]) = [c.sup.*.sub.0]([F.sub.0]). For f [member of] ([F.sub.1],[F.sub.0]), W (f) is decreasing in f because deterrence is decreasing, honest individuals' defense costs are increasing, and dishonest individuals' defense costs are constant. Because W([F.sub.1]) - [lim.sub.z[right arrow]0] W([F.sub.1] + [epsilon]) = G(Z([F.sub.1])) [[c.sup.**.sub.1] - [c.sup.*.sub.1] ([F.sub.1])] > 0, the optimal fine cannot exceed [F.sub.1].

b. When [F.sub.1] > [F.sub.0] and [epsilon] > 0, W([F.sub.1]) - W ([F.sub.1] + [epsilon]) = G(Z([F.sub.1]))[[c.sup.**.sub.1]-[c.sup.*.sub.1]([F.sub.1])] > 0, because Z([F.sub.1] = [epsilon]) = Z([F.sub.1]), [c.sub.1]([F.sub.1] + [epsilon]) = [c.sup.**.sub.1] > [c.sub.1] ([F.sub.1]) = [c.sup.*.sub.1]([F.sub.1]), and [c.sub.0]([F.sub.0] + [epsilon]) = [c.sup.**.sub.0]. Again, the optimal fine cannot exceed [F.sub.1].

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

TABLE 1

Notation

Variable Definition

h social harm from a crime
b private benefit from a crime
G(b) distribution function of benefits
 from crime
y endowed income
[q.sub.0] probability that honest individuals
 is arrested and charged
[q.sub.1] probability that dishonest individuals
 is arrested and charged
[p.sub.0]([c.sub.0]) probability of conviction of honest
 accused
[p.sub.1]([c.sub.1]) probability of conviction of honest
 accused
[c.sub.i] legal defense expenditure of type
 i = 0, 1 accused
f fine
[F.sub.i] fine that just induces type i accused
 to choose [c.sub.i], which bankrupts
 him if convicted
m policing costs
t tax rate on defense expenditure
[theta] compensation for acquitted accused
[k.sub.0] social cost per convicted honest person
[k.sub.1] social cost per acquitted dishonest
 person
[lambda] social value of relaxing bankruptcy
 constraint on effective fine
T lump-sum tax

(1.) Throughout the article we use the terms honest and dishonest as synonyms for innocent and guilty, respectively.

(2.) It is just about possible to interpret the statement within the standard efficiency framework as an empirical judgment about the marginal rate of substitution between type I and II errors. Such an interpretation does not seem compatible with the terms of the debate in the voluminous literature discussing the number of guilty persons whose wrongful acquittal is equivalent to a single wrongful conviction (Volokh, 1997).

(3.) The assumption that individuals have the same income and that all count equally in our welfare function means that we are not concerned with distributional issues, such as the optimal means of providing subsidies for legal defense expenditure to the poor (Dnes and Rickman, 1998).

(4.) The offense may yield purely nonmonetary benefits. Alternatively, it may yield monetary benefits that have been spent or distributed to shareholders by the time the accused is convicted. A similar assumption is made in Malik (1990) and Polinsky and Shavell (2000).

(5.) Increasing policing expenditures m will increase the arrest rates [q.sub.i]. Because type I and II errors and defense expenditures have no new implications for efficient policing expenditure, we simplify by assuming that m is fixed. The standard maximal fine result is driven by the fact that the fine is a costless instrument that can be substituted for costly policing expenditure. However, our demonstration that the optimal fine does not bankrupt dishonest accused does not depend on the fixity of m. If m was a policy variable it would be efficiently chosen when [(h - z) + ([q.sub.1] [c.sub.1] - [q.sub.0][c.sub.0])]g(z)[z.sub.m] - 1 = 0, implying dW/df = -dm/[df\.sub.z] - [q.sub.0] G[c.sub.0f] - [q.sub.1] (1 - G)[c.sub.1f], so that dW/df need not be positive even though increases in the fine permit reductions in costly policing.

(6.) If any of the dishonest accused choose [c.sup.**.sub.1] rather than [c.sup.*.sub.1]([F.sub.1]) when f = [F.sub.1] the welfare problem may have no solution. Welfare would be always be greater at some fine slightly less than [F.sub.1] than at [F.sub.1] because of the upward jump in defense expenditure by accuseds who choose [c.sup.**.sub.1] at [F.sub.1].

(7.) The social harm from failure to punish the dishonest could plausibly depend on the cost of their crime, as in Miceli (1991), but this is not relevant in the current model, where we assume that all crimes impose the same direct social cost h.

(8.) In a longer version of the article we consider the possibility of taxing defense costs and introducing compensation for acquitted individuals and reach analogous results (Gravelle and Garoupa, 2000).

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Nuno Garoupa *

* We are grateful for helpful comments from the editor, two referees, Mitch Polinsky, Jennifer Reinganum, Ricard Torres, and participants in the 14th Annual meeting of the European Law and Economics Association, Barcelona, September 1997.

Gravelle: Professor, Centre for Health Economics, University of York, Heslington, York YO10 5DD, UK. Phone 44-1904-432663, Fax 44-1904-432700, E-mail [email protected]

Garoupa: Associate Professor, Department d'Economia i Empresa, Universitat Pompeu Fabra, Ramon Trias Fargas 25-27, 08005 Barcelona, Spain. Phone 34-93-542 2639, Fax 34-93-542 1746, E-mail [email protected]