* University of Chicago Law School and University College, Cork, Ireland. The authors would like to thank Richard Epstein for bringing them together and Douglas Baird for useful comments on a previous draft.

1An old English proverb, of unknown origin, first recorded in J. Ray, A Collection of English Proverbs, in the 1678 edition. The proverb was recorded there as "Hanged for a sheep as a lamb, As good be."

2See, for instance, the classic discussion in Gary S. Becker, Crime and Punishment: An Economic Approach, 76 J. Pol. Econ. 169 (March/April 1968).

3 Throughout this paper, we use "crime" to mean "type of offense" and "offense" to mean "act of committing a crime." Thus "sheep stealing" is one crime, "lamb stealing" is another crime, and "murder" is a third crime. A particular act of sheep stealing, or a particular murder, is one offense.

4 We are not the first writers to consider this issue. Steven Shavell, A Note on Marginal Deterrence, 12 IRLE 345 (1992), cites earlier discussions by Cesare Bonesaria Beccaria, Baron Montesquieu, and Jeremy Bentham. In the modern literature, George J. Stigler, The Optimum Enforcement of Laws, 78 J. Pol. Econ. 526 (1970) offers at 527-8 one of the most commonly cited explanations, that "marginal costs are necessary to marginal deterrence." The argument is that if the criminal is to be executed for a small crime he has no reason not to commit a greater crime. This argument has been widely cited, for example by Richard A. Posner, Killing or Wounding to Protect a Property Interest, 14 J. Law & Econ. 201 (1971); Jon D. Harford, Firm Behavior under Imperfectly Enforceable Pollution Standards and Taxes, 5 J. Environ. Econ. & Mgt. 26 (March 1978); Paul H. Rubin, The Economics of Crime, 28 Atlantic Econ. Rev. 38 (July/August 1978); Donald Keenan & Paul Rubin, Criminal Violations and Civil Violations, 11 J. Legal Stud. 365 (1982); Frank Easterbrook, Criminal Procedure as a Market System, 12 J. Legal Stud. 289 (1983); William M. Landes, Optimal Sanctions for Antitrust Violations, 50 U. Chi. L. Rev. 652 (1983); David Pyle, The Economics of Crime and Law Enforcement (1983); Francis T. Lui, A Dynamic Model of Corruption Deterrence, 3 J. Pub. Econ. 215 (Nov. 1986); and B. Curtis Eaton & William D. White,The Distribution of Wealth and the Efficiency of Institutions, 29 Econ. Inquiry 336 (April 1991).

5 "The severity of punishment of itself emboldens men to commit the very wrongs it is supposed to prevent; they are driven to commit additional crimes to avoid the punishment for a single one." (Cesare Beccaria, On Crimes and Punishment (1764), trans. Henry Paolucci, Bobbs Merrill, Indianopolis (1963) at 43)

6 See Steven Shavell, Deterrence and the Punishment of Attempts, 19 J. Legal Stud. 435 (1990), and David Friedman, Impossibility, Subjective Probability, and Punishment for Attempts, 20 J. Legal Stud. 179 (1991). "The importance of preventing a criminal attempt authorizes punishment, but as there may be an interval between the attempt and the execution, reservation of greater punishment for the accomplished crime may lead to repentance." (Beccaria, supra note 5, at 40.)

7 For a more detailed and technical discussion of these assumptions, see David Friedman, Reflections on Optimal Punishment or Should the Rich Pay Higher Fines? Research in Law and Economics (1981). The cost of punishment includes the cost to the criminal. Thus, a costlessly collected fine has a net cost of zero, since what the criminal loses exactly cancels what the court receives. Execution, ignoring the cost to the state of operating the gallows or electric chair, has a cost equal to the amount of the punishment; the criminal loses one life and nobody gets one. Imprisonment has a cost greater than the amount of the punishment; the criminal loses his freedom and the state must pay for the prison. See also Gordon Tullock & Warren Schwartz, The Costs of a Legal System, 4 J. Legal Stud. 75 (1975) and A.M. Polinsky & Steven Shavell, The Optimal Trade-off between the Probability and Magnitude of Fines, 69 Amer. Econ. Rev. 880 (1979) for related discussions.

8 If the punishment were a cash fine imposed on a risk neutral criminal, this would be the set of probability/punishment pairs with a given expected value--probability of paying the fine times amount of fine. More generally, it is the set of pairs with the same certainty equivalent-a definition that takes account of differing sorts of punishment and differing attitudes towards risk.

9 We use "effective punishment" as a generalization of the more familiar idea of "expected punishment," one which allows us, without increasing the complexity of the analysis, to drop the assumptions of risk neutral criminals and punishments defined as dollar amounts. The analytical device, but not the term, was introduced in Friedman, supra note 7.

10 Since higher levels of deterrence also result in fewer offenses, total cost of apprehending and punishing criminals may rise or fall, depending on the relation between the cost curve for deterrence and the supply curve for offenses.

11 An efficient offense would be one for which the benefit to the offender was greater than the cost to the victim--a driver speeding, for instance, in a situation where the value to him of getting where he was going sooner was larger than the cost he imposed, in additional accident risk, on other drivers. If some offenses are efficient, a system that deters all offenses, even if it does so costlessly, may not be optimal.

12 One can avoid this result in several other ways. One is to assume that there are always some efficient offenses which we would prefer not to deter. A second is to assume that criminals differ in how likely they are to be caught and that there are always some so skilled that nothing we can do will make crime unprofitable for them. A third possibility is to drop our assumption that enforcement cost is zero if there are no offenses. Under such a model, we might be able to deter all offenses, but only at the cost of maintaining a standby enforcement system to guarantee that if any offense did occur the offender would be almost certain to be apprehended and convicted. Part V of this essay explores the implications of such a model.

13 Some readers may be disturbed by the idea that criminals who cannot be deterred should be punished as little as possible so as to minimize punishment costs. Yet this is, from an economic standpoint, precisely the argument for the insanity defense. Since we cannot deter insane criminals there is no point in punishing them. Anything done to them is justified as treatment or as a way of making future offenses impossible, not as a punishment.

14 Such an assumption seems appropriate for many sorts of theft and robbery. The more valuable the items stolen the greater, ceteris paribus, the loss to the victim and the benefit to the criminal.

15 A discussion of the assumptions needed for stronger forms of the theorem is included in a longer version of this article available from the authors.

16 In an efficient combination of probability and punishment, the marginal cost of increasing effective punishment must be the same for both inputs (probability and punishment). Killing the victim raises the total cost of any probability of apprehension, but does not affect the cost function for punishment. If, as seems plausible, it also raises the marginal cost of apprehension at any probability, then maintaining the same level of effective punishment implies a lower probability combined with a higher punishment for the robber who kills his victim.

17 Strictly speaking, this is the average effective penalty, since under these assumptions the effective penalty may be different for different offenders committing the same crime.

18 In addition, there are costs and benefits associated with the effect of a change in the effective punishment for one crime on the rate of commission of the other. Our definition of crimes being substitutes requires that such effects exist but permits no lower bound to be set to their size, so in demonstrating the existence of a possibility (in this case the possibility that the more serious offense might have the lower optimal punishment) we can assume that they are too small to change the conclusion of our analysis.

19 Shavell, supra note 4.

20 Our result, and our argument, are similar to those in Louis L. Wilde, Criminal Choice, Nonmonetary Sanctions and Marginal Deterrence: A Normative Analysis, 12 IRLE 345 (1992), and Jennifer Reinganum & Louis Wilde, Nondeterrables and Marginal Deterrence Cannot Explain Nontrivial Sanctions, Cal. Inst. Tech. (1986).

21 This result depends on our tie-breaking rule, which is somewhat artificial, and on the general tidiness of the model compared to the real world. The more realistic point is that, once we include a standby cost, we have an incentive to keep down the effective punishment for the more severe crime, since a higher effective punishment is costly even if it never has to be imposed.

22 Reinganum & Wilde, supra note 20; Wilde, supra note 20; and Shavell, supra note 4. All three investigate the question of whether the optimal punishment rises with the seriousness of the offense. Shavell also touches briefly on the question of how the possibility of committing one offense changes the optimal punishment for the other.

23 Shavell, supra note 4.

24 Wilde, supra note 20.

25 Reinganum & Wilde, supra note 20.

26 "By now it should be clear what is needed to get a theory of nontrivial sanctions ...: a link between probabilities of apprehension for different crimes." Reinganum & Wilde, p. 8, supra note 20.

"For optimal sanctions to be different for the two acts, enforcement effort cannot be specific to the act. Suppose instead that enforcement effort is of a general nature, affecting in the same way the probability of apprehension for committing different harmful acts; therefore, assume the probability of apprehension for committing act 1 equals that for committing act 2." Shavell, p. 3, supra note 4.

27 Both the assumptions discussed here and Shavell's assumption, discussed in Part V above, that the cost of catching a given fraction of offenders is unaffected by the number of offenses.

28 Shavell briefly mentions the applicability of the analysis to the case of multiple acts (our Part II); he does not discuss the case where offenses are substitutes (our Part IV). His comment on the issue of punishment costs is: "In the case where sanctions are non-monetary, I have not succeeded in obtaining an appealing characterization of the difference between the one-act and the two-act models, although one supposes that in some general sense the results should be similar to those discussed here." p. 10, supra note 4. .

[29] To a considerable extent that relation exists explicitly in present law. Even where it is not, it seems likely that police effort in catching thieves increases, on average, with the amount stolen.