Optimal punishment schemes with state-dependent preferences.

Neilson, William S.

I. INTRODUCTION

It is well known in the economics of crime literature that both the certainty and severity of punishment can be manipulated to deter criminal behavior. As Polinsky and Shavell [1979] show, when punishment takes the form of a fine, the optimal combination of the probability of punishment and the magnitude of the fine depends on the risk attitudes of offenders and on the costs of catching and convicting offenders. This paper examines how the optimal combination of certainty and severity are affected if offenders' preferences are state-dependent.

State-dependent utility is commonly used in situations in which the state of the world affects how well individuals are able to enjoy the consumption of their wealth. Death of the individual is an obvious example, and Hirschleifer and Riley [1992] also suggest the death of a child, the loss of use of a limb, or the loss of a precious heirloom. There are many reasons to add conviction for a crime to this list.(1) First, if the individual is sent to prison, his consumption opportunities are obviously diminished. Even without a prison sentence, though, utility might best be modeled as being state-dependent. For utility to be state-independent, there must exist some decrease in wealth that has the same effect on the utility function as being convicted of the crime. If the offender is risk averse and has state-independent utility, a loss of wealth reduces the level of utility but increases marginal utility. It is easy to imagine why being convicted of a crime would reduce the level of utility, but it is difficult to imagine why it would increase the individual's marginal utility of wealth. For example, drivers who are caught speeding are angered beyond the level consistent with the expected loss of income. But, this additional anger does not lead to an additional increase in the marginal utility of wealth.

In addition to these considerations, Lott [1992a,b] provides evidence that conviction leads to increased divorce rates and constrained job opportunities. The former reduces the level of utility but is unlikely to increase marginal utility, and should therefore be modeled as a shift in the utility function. Constrained job opportunities reduce both income and job satisfaction, and the reduction in job satisfaction is again best modeled as a shift in the utility function.

As a final argument supporting the use of state-dependent preferences, Neilson and Winter [1997] show that if offenders' preferences are state-dependent, it is possible for them to be both risk averse and more sensitive to changes in the certainty of punishment than to proportional changes in the severity of punishment. Becker [1968] notes this empirical finding,(2) and concludes that if offenders are expected utility maximizers, they cannot be risk averse.(3)

Polinsky and Shavell [1979] demonstrate that when offenders are risk averse and have state-independent utility, increasing the fine increases social loss by increasing the amount of risk borne by offenders. Consequently, in the extreme case in which it is costless to catch and convict offenders, the optimal policy has a relatively low fine but punishes criminals with probability one. This paper shows that if preferences are state-dependent, there is an additional social cost associated with setting a high probability of punishment, because it makes it more likely that criminals are on their lower utility functions. To avoid the additional social cost, the certainty of punishment should be reduced.

II. THE MODEL

The model presented below is identical to that in Polinsky and Shavell [1979], except for the addition of state-dependent preferences. Identical individuals choose whether to commit a crime or not.(4) Gains from crime are random: with probability q the individual gains a, and with probability 1-q he gains b [greater than] a. Even though the gains are random, it is assumed that the individual knows the true value of the gain before any decision must be made. If he chooses to commit a crime, he is caught and punished with probability p, in which case he must pay a fine off. The government finances its law enforcement activities through a per capita tax, and any fines collected are used to reduce this tax. Individuals insure against the external monetary cost associated with being a victim of a crime, but they cannot insure against possible punishment for committing a crime.

Suppose that an individual commits a crime and gains z, where z [element of] {a,b}. His expected utility, conditional on the crime being worth z, is

(1) (1 - p)[U.sub.H](y - t - [Pi] + z)

+ p[U.sub.L](y - t - [Pi] + z - f)

where y is the individual's wealth prior to committing the crime, t is the per capita tax, and [Pi] is the insurance premium. The individual's utility function is state-dependent, with [U.sub.H] denoting the utility function in the "high" state in which the individual is not punished, and [U.sub.L] denoting the utility function in the "low" state in which the individual is punished. Consistent with punishment being the "low" state, [U.sub.L](x) [less than] [U.sub.H](x) for all x. Now suppose the individual does not commit a crime. Since he faces no risk of punishment, his expected utility is [U.sub.H](y - t - [Pi]). Since the value of the gain from crime is known beforehand, the individual chooses to commit a crime of value z if

(2) (1 - p)[U.sub.H](y - t - [Pi] + z)

+ p[U.sub.L](y - t - [Pi] + z - f) [greater than] [U.sub.H](y - t - [Pi]).

It remains to define the values of the tax t and the insurance premium [Pi]. The tax is defined by t = c(p, [Lambda]) - npf, where n is the proportion of individuals committing the crime and [Lambda] is a shift parameter of the cost function c. Note that this formulation implies that it is costly to increase the probability of punishment but not to increase the magnitude of the fine. It is assumed that [c.sub.p] [greater than] 0, [c.sub.[Lambda]] [greater than] 0, and c(p,0) = 0. Insurance is assumed to be fair, so that [Pi] = ne, where e is the external cost imposed by the crime. It is assumed throughout that a [less than] e [less than] b, so that it is efficient for type-b individuals to commit crimes and for type-a individuals to refrain from criminal activity. This assumption avoids the trivial cases where all crime is deterred and where no crime is deterred.

The goal of the policymaker is to maximize social welfare. Let E[U.sub.z](p,f) denote the level of expected utility when the gain from committing a crime is z, the probability of punishment is p, the fine is f, and the individual makes the optimal decision about whether or not to commit. Per capita social welfare is given by

(3) EU(p,f) = qE[U.sub.a](p,f)

+ (1 - q)E[U.sub.b](P,f).

The policymaker sets p and f to maximize EU(p,f).

III. OPTIMAL PUNISHMENT

The optimal punishment scheme is first characterized under the condition that punishment is costless. This allows an immediate comparison of the state-dependent case with the state-independent case. Polinsky and Shavell [1979] show that if individuals are risk averse expected utility maximizers (so that [U.sub.H] = [U.sub.L] = U with U concave), and if the cost of convicting criminals is zero (so that [Lambda] = 0), then the optimal punishment scheme has p = 1. Risk is reduced by raising p and reducing f, so when it is costless to convict criminals, they should be convicted with probability one. The main result of the current paper is that when preferences are state-dependent, so that punishment shifts the utility function, this result no longer holds.

Before stating the main result, some terminology must be specified. Call an individual who gains a from committing a crime a type-a individual, and call an individual who gains b a type-b individual. Also, an individual is more sensitive to changes in the certainty of punishment than to proportional changes in the severity of punishment if the elasticity of expected utility with respect to p is of greater magnitude than the elasticity of expected utility with respect to f.

PROPOSITION. If all individuals are state-dependent expected utility maximizers and are more sensitive to changes in the certainty of punishment than to proportional changes in the severity of punishment, then the optimal probability of punishment is less than one when punishment is costless.

Proof First, following Neilson and Winter [1997], let expected utility be given by EU = (1 - p) [U.sub.H](x) + p[U.sub.L](x - f). The elasticity of expected utility with respect to the probability of punishment is [[Eta].sub.p] = [[U.sub.H](x)-[U.sub.L](x-f)]p/EU, and the elasticity of expected utility with respect to the fine is [[Eta].sub.f] = p[U.sub.L][prime](x-f)f/EU. The individual is more sensitive to changes in the certainty of punishment than to proportional changes in the severity of punishment if [[Eta].sub.p] [greater than] [[Eta].sub.f] that is, if

(4) [U.sub.H](x) - [U.sub.L](x-f)/f [greater than] [U.sub.L][prime] (x - f).

Now suppose that, contrary to the conclusion of the proposition, the optimal punishment probability is p = 1 when [Lambda] = 0. Let the corresponding fine be f, and suppose that it is optimal for a type-z individual to commit a crime under these circumstances. We will show that a marginal reduction in p, accompanied by an appropriate marginal increase in f, makes the type-z individual better off, and makes the other type no worse off.

When p = 1 and [Lambda] = 0, each individual pays a tax of t = -nf and pays an insurance premium of [Pi] = ne. Now suppose that the punishment probability is lowered to p [less than] 1 and the fine is raised to f/p. Both the tax and the insurance premium are unchanged. A type-z individual's expected utility is given by

(5) E[U.sub.z](p,f/p)

= (1 - p)[U.sub.H](y - ne + z + nf)

+ p[U.sub.L](y - ne + z + nf - f/p).

Differentiating this expression with respect to p yields

(6) dE[U.sub.z] / dp = -[U.sub.H](y - ne + z + nf)

+ [U.sub.L](y - ne + z + nf - f / p)

+ (f / p)[U.sub.L][prime] (y - ne + z + nf - f / p).

Letting x [equivalent to] y - ne + z + nf and evaluating this expression at p = 1 yields

(7) dE[U.sub.z](1,f) / dp = -[U.sub.H](x)

+ [U.sub.L](x - f) + f[U.sub.L][prime](x - f).

By equation (4) this derivative is negative, and a type-z agent can be made better off by reducing the probability of punishment to p [less than] 1 and increasing the fine to f.

It remains to show that the other type of individual is made no worse off. First, if the other type of individual does not commit a crime in this setting, that type is left indifferent because neither the insurance premium nor the tax change. If the other type of individual does commit a crime, then the same analysis shows that he is made better off by the reduction in p and increase in f.

The proposition relates the optimal punishment scheme directly to the empirical finding that criminals are more sensitive to changes in the certainty than severity of punishment, and it states that even when it is costless to catch and convict criminals, it is not optimal to do so with probability one.

To see why this occurs, consider the case in which only type-b individuals commit crimes at the optimum. Then n = 1-q and [Pi] = (1-q)e. Further, assume that [Lambda] = 0, so that t = -(1-q)pf. The proposition states that social welfare is not maximized when p = 1. When punishment is certain, a type-b individual's expected utility from committing a crime is E[U.sub.b] (1, f) = [U.sub.L](y - t - [Pi] + b - f). Let [Mathematical Expression Omitted]. Since the individual chooses to commit a crime, it must be the case that [Mathematical Expression Omitted].

Now suppose instead that the probability of punishment is lowered slightly to p [less than] 1, and that the fine is increased to f[prime] = f / p. The amount of fine revenue returned to individuals remains unchanged and equal to f. The payoff from committing the crime is now

(8) E[U.sub.b](p,f[prime]) = (1 - p)[U.sub.H](y - t - [Pi] + b)

+ p[U.sub.L](y - t - [Pi] + b - f[prime]).

Type-b individuals are made better off by the reduction in the certainty of punishment if E[U.sub.b](p,f[prime]) [greater than] E[U.sub.b](1,f) = [U.sub.L](y - t - [Pi] + b - f).

Figure 1 depicts a case in which this inequality holds. When punishment is certain, the individual is on his lower utility function with probability one. By reducing the probability of punishment, the probability of being on the lower utility function is also reduced. With p [less than] 1, expected utility is given by the height of the line AB connecting the two utility functions in Figure 1, and the line lies everywhere above the lower utility function. Consequently, any reduction in p increases expected utility for the type-b individuals, and it is not optimal to punish with probability one. In contrast, in the state-independent case point A would lie on the same utility function as B, and the line AB would lie everywhere below the utility function. Thus, any reduction in p would decrease expected utility for the type-b individuals, and it would be optimal to punish with probability one.

The lessons from the figure extend to the general case. If individuals are risk averse and have state-independent utility, as in Polinsky and Shavell [1979], the optimal punishment scheme punishes offenders with probability one to minimize the social loss caused by offenders facing risky punishment. But, if individuals have state-dependent utility, risk aversion neither implies nor is implied by the condition that offenders are more sensitive to changes in the certainty than the severity of punishment. The proposition above establishes that if individuals have state-dependent preferences and are more sensitive to changes in the certainty than the severity of punishment, the optimal probability of punishment is less than one.

Next turn attention to the question of what happens when capturing and convicting offenders is costly. Let p([Lambda]) and f ([Lambda]) denote the optimal probability and fine when the cost parameter is [Lambda]. Polinsky and Shavell [1979, proof of Proposition 2] provide a continuity argument showing that p([Lambda]) [approaches] p(0) and f([Lambda]) [approaches] f(0) as [Lambda] [approaches] 0. Their argument can be readily adapted to fit the state-dependent setting.(5) This means that in the state-dependent setting, when punishment costs are positive but small, the optimal punishment probability is bounded away from unity. Again this result can be contrasted with corresponding results based on different assumptions about preferences. When individuals are expected value maximizers and punishment costs are small but positive, the optimal punishment probability is equal to the lowest value consistent with deterring crime for the type-a individuals, (Polinsky and Shavell [1979, Proposition 1]). When individuals are risk averse with state-independent preferences, the optimal punishment probability is near one (Polinsky and Shavell [1979, Proposition 2]). When individuals have state-dependent preferences and are more sensitive to changes in the certainty than the severity of punishment, the optimal probability is bounded away from one.

IV. CONCLUSIONS

It has been shown that when conviction of criminals is costless, if criminals have state-dependent preferences and are more sensitive to changes in the certainty than the severity of punishment, it is not optimal to punish them with probability one. When the probability of punishment is increased, social loss increases because offenders are more likely to be moved to their lower utility functions. When the severity of punishment is increased, social loss increases because offenders are risk averse and face riskier distributions. Since criminals are more sensitive to changes in the probability of punishment than they are to changes in its severity, the former effect outweighs the latter, and social loss is minimized with a lower probability of punishment than would hold if preferences were state-independent.

I am grateful to Harold Winter, Thomas Saving, and two referees for helpful comments and to the Private Enterprise Research Center at Texas A&M University for financial support.

1. Viscusi [1986] uses a state-dependent expected utility model to estimate the risk-reward tradeoff facing criminals, but he does not examine the implications for the optimal punishment scheme.

2. Also see the experimental evidence provided by Block and Gerety [1995].

3. The offender's objective function is (1 - p)U(x) +pU(x - f), where p is the probability of punishment, f is the magnitude of the fine, U is the utility function, and x is the individual's wealth before prosecution. Calculate the elasticities of this objective function with respect to p and with respect to f. Becker [1968, footnote 19] shows that the elasticity with respect to p cannot be of greater magnitude than the elasticity with respect to fir u is concave.

4. Friedman [1981] considers the complementary issue of how offenders should optimally be punished when individuals are not identical. His setting is beyond the scope of this paper.

5. Polinsky and Shavell's proof relies only on the continuity of preferences, not on the curvature of utility functions. The continuity assumption is maintained for state-dependent preferences.

REFERENCES

Becker, Gary S. "Crime and Punishment: An Economic Approach." Journal of Political Economy, March/April 1968, 169-217.

Block, Michael K., and Vernon E. Gerety. "Some Experimental Evidence on Differences Between Student and Prisoner Reactions to Monetary Penalties and Risk." Journal of Legal Studies, January 1995, 123-38.

Friedman, David D. "Reflections on Optimal Punishment or: Should the Rich Pay Higher Fines?" in Research in Law and Economics, edited by R. O. Zerbe, Jr. Greenwich, CT: JAI Press, 1981, 185-205.

Hirschleifer, Jack, and John G. Riley. The Analytics of Uncertainty and Information, Cambridge: Cambridge University Press, 1992.

Lott, Jr., John R. "An Attempt at Measuring the Total Monetary Penalty from Drug Convictions: The Importance of an Individual's Reputation." Journal of Legal Studies, January 1992a, 159-87.

"Do We Punish High Income Criminals Too Heavily?" Economic Inquiry, October 1992b, 583-608.

Neilson, William S., and Harold Winter. "On Criminals' Risk Attitudes." Economics Letters, August 1997, 97-102.

Polinsky, A. Mitchell, and Steven Shavell. "The Optimal Tradeoff Between the Probability and Magnitude of Fines." American Economic Review, December 1979, 880-91.

Viscusi, W. Kip. "The Risks and Rewards of Criminal Activity: A Comprehensive Test of Criminal Deterrence." Journal of Labor Economics. July 1986, 317-40.

Neilson Associate Professor, Department of Economics, Texas A&M University, College Station Phone 1-409-845-9952, Fax 1-409-862-8483 Email ecinquiry@econ.tamu.edu