1Gary S. Becker & George J. Stigler, "Law Enforcement, Malfeasance, and Compensation of Enforcers," 3 J. Legal Stud. 1 (1974).

2William M. Landes & Richard A. Posner, "The Private Enforcement of Law," 4 J. Legal Stud. 1 (1975).

3 David D. Friedman, "Reflections on Optimal Punishment, or: Should the Rich Pay Higher Fines," 3 Research in Law and Economics 185 (1981).

4Landes and Posner's claim that treating the supply of offenses as a common pool resource leads to inefficiencies is discussed in a longer version of this article available from the author.

5If the cost of the trial does not depend on the punishment, it may be included with enforcement costs; if it does, it must be considered a punishment cost. I am assuming the former for purposes of expository convenience.

6A very high punishment (execution) might be less costly than a lower punishment (imprisonment), in which case the net cost would decrease as the level of punishment increased. In such a case, however, imprisonment need never be used. There is some probability of execution which the criminal would regard as equivalent to imprisonment. If execution were imposed with that probability, net punishment cost would be less and all other costs would remain the same. Hence "probabilistic execution" dominates certain imprisonment. It follows that as long as we limit ourselves to the least costly among alternative punishments, net cost of punishment is a non-decreasing function of amount of punishment.

7It follows from the argument of footnote 6 that in an efficient system we should never observe imprisonment if execution is less costly. The fact that we do observe imprisonment may be interpreted either to mean that our system is not an efficient one or that execution is more costly (relative to imprisonment) than might at first appear. The question is discussed at some length in Friedman, Supra note 3.

8The exact relationship is derived in Friedman, supra note 3; it may be summarized for risk neutral criminals by the formula "expected punishment equals increase in costs to victims, enforcement costs, and punishment costs produced by a reduction in the expected punishment sufficient to increase the number of offenses by one."

9A proof that private enforcement must under some circumstances lead to underenforcement is provided in A. Mitchell Polinsky, "Private Versus Public Enforcement of Fines," 9 J. Legal Stud. 105 (1980) but does not contradict Landes and Posner's argument since it involves a corner solution at the maximum collectable fine.

10Polinsky, supra note 9 at 120, demonstrates that the appropriate bounty will lead to an efficient outcome in the case of a private competitive industry but not in the case of a private monopoly industry.

11Friedman, supra note 3.

12Since the punishment need not be a fine, f is not a number of dollars but a particular punishment--a fine of a certain amount, imprisonment for a certain length of time, a particular number of blows with a whip.

13Some would argue that benefits to the criminal should not be included in such calculations. I disagree, since I believe that in the economic analysis of law what is or is not a crime should be a conclusion of the theory, not an assumption; robbery should, in this context, be considered bad only because permitting it is inefficient. If benefits to the criminal were eliminated (or discounted) the result would be to change the optimal level of punishment but not the conclusions of sections III and V of the paper; I am not certain of the effect on the conclusions of section IV.

14But see David D. Friedman, "Why There Are No Risk Preferrers," 89 JPE 600 (1981) for why criminals (and others) are unlikely to exhibit much risk preference. Gary S. Becker, "Crime and Punishment: An Economic Approach,"76 JPE 169 (1968) concludes that in the equilibrium of an efficient system criminals must be risk preferrers, but I show in Friedman,Supra note 3, fn 3 that the result depends either on omitting costs and benefits associated with risk preference from the social loss function and assuming Z to be independent of E or on an artificial and implausible assumption about the behaviour of Z. The point is discussed at greater length in David D. Friedman, "Are Criminals Risk Preferrers?--A Belated Comment" (preprint available from the author).

15The result follows in general from the argument given in footnote 6.

16To make the argument general one replaces "expected value" with "certainty equivalent" and notes that Z includes the cost or benefit of imposing a punishment lottery on a criminal who is not risk neutral.

17Friedman, Supra note 3.

18Such arrangements currently exist in several contexts, as discussed in David D. Friedman, "Private Creation and Enforcement of Law: A Historical Case," 8 J. Legal Stud. 399, (l979), which also describes the workings of a real world system of private enforcement in medieval Iceland. That system gave victims the property right in offenses; it imposed actual rather than expected punishments, but also punished the criminal for actions which lowered p. "Murder," for example, was defined as "secret killing" and punished more severely than "open killing."

The possibility of selling offenses for a negative price undercuts the proof of the inefficiency of private enforcement offered by Polinsky, Supra note 9 at 113, which depended on the revenue from fines being insufficient to cover the costs of an optimal level of enforcement if the damage imposed by an offense were sufficiently high. If the damage is high, so is the amount potential victims are willing to pay firms to take charge of offenses against them.

Such negative prices for offenses only make sense, however, for those offenses where the criminal knows the identity of the victim (and hence whether the victim has sold offenses against him in advance) before he commits the crime. Polinsky's argument is correct where that is not the case--reckless driving, for example. In such situations private enforcement leads to an efficient outcome only if the price of offenses implied by the zero-profit condition (for a competitive enforcement industry) is non-negative.

19A completely private system (of courts, enforcement, and legislation) having some of the features of the system analyzed here is proposed in David D. Friedman, The Machinery of Freedom, (1971). Terry L. Anderson & P.J. Hill, "An American Experiment in Anarcho-Capitalism: The Not So Wild, Wild West," 3 Journal of Libertarian Studies 9 (1979) discuss the American west in the nineteenth century as a historical example of such a system.

20This assumes that criminals know which firms accept bribes or fabricate convictions and hence have higher or lower than normal expected punishments, which seems plausible.

21I have assumed away statistical problems, nonhomogeneity of victims, and the like. Within the constraints of the model, where the court system is assumed to know the supply function of offenses, the production functions of the firms, and everything else necessary to calculate the optimal expected punishment, this is reasonable. In a more realistic setting the court would set E, enforce it in terms of permitted punishments given observed values of p, and monitor crime rates in order to spot the firms that ought to be investigated on suspicion of bribery.

22I made some other assumptions designed to simplify the analysis rather than to eliminate obvious sources of inefficiency; I suspect but have not proved that the conclusion would survive the elimination of those assumptions, provided that the optimal public system to which the private system is compared is required to deal with the same complications.

23Landes and Posner, supra note 2, at 14-15 (footnote omitted).

24Id at 15, n. 32.



27Since the location of PR is not affected by either the supply of offenses function O or the damage function D, both of which affect (p*,f* ), the relative position of the two can easily be altered by altering O or D. Since the argument of Landes and Posner does not assume any particular form for O and D (except for the sign of their derivatives), Figures 1b and 2b are just as plausible as Figures 1a and 2a.