International Reuzew of Law and Economics (1992) 12, 345-355 A Note on Marginal Deterrence STEVEN SHAVELL Harvard Law School, Cambridge, MA 02138, USA I. Introduction and Summary The theory of deterrence has been concerned primarily with situations in which in- dividuals consider whether to commit a single harmful act. For instance, a person may be deciding whether to discharge a pollutant into a lake. In some contexts, how- ever, a person may be contemplating which of several harmful acts to commit- whether to discharge a pollutant into a lake or instead to discharge it onto the ground (where it might cause a different level of harm). In such contexts, the threat of sanc- tions plays a role in addition to the usual one of deterring individuals from commit- ting harmful acts: it influences which harmful acts undeterred individuals choose to commit. Notably, undeterred individuals will have a reason to commit less rather than more harmful acts if expected sanctions rise with harm. This tendency is sometimes said to reflect marginal deterrence because an individual will be deterred from committing a more harmful act owing to the difference, or margin, between the expected sanction for it and for a less harmful act. The term “marginal deterrence” seems to be due to Stigler (1970), but the notion has been well known from the time of some of the earliest writing on sanctions. See Beccaria (1770, 32), Montesquieu (1748, Book VI, Ch. 16, 161-62), and Bentham (1789, 171). Ben- tham, for example, states (citing an essentially identical passage of Montesquieu) that an object of punishment is “to induce a man to choose always the least mischievous of two offenses; therefore where two offenses come in competition, the punishment for the greater offense must be sufficient to induce a man to prefer the less.” A point of the present note, however, is that considerations of marginal deterrence are not a r&on d’e^tre for sanctions to rise with harm. Optimal sanctions rise with harm in models with marginal deterrence only if one makes a particular assumption about enforcement effort-that it is of a general nature (in a sense to be defined). But this assumption also implies that optimal sanctions rise with harm in the usual models without marginal deterrence. Still, as will be noted, marginal deterrence does have a more refined implication for optimal sanctions under the assumption of general en- forcement effort. To investigate marginal deterrence, I consider a simple model with monetary sanc- tions in which each person can do nothing or commit one of two harmful acts: either act 1, a low harm act, or act 2, a high harm act. Because individuals choose between I thank Louis Kaplow, A. Mitchell Polinsky, and an anonymous referee for comments and the National Science Foundation (grant no. SES-8821400) for support. 0 1992 Butterworth-Heinemann346 A note on marginal deterrence two harmful acts, the model allows for marginal deterrence. This two-act model was introduced in Reinganum and Wilde (1986) and is studied further in Wilde (1989). I consider also a related one-act model in which some individuals choose whether to commit act 1, others choose whether to commit act 2, but none has the opportunity to choose between the two acts. Hence, there is no possibility for marginal deterrence in the one-act model. Comparison of these models will allow us to determine the influence on optimal enforcement of marginal deterrence, of the opportunity of in- dividuals to choose between committing different harmful acts.i As was indicated, the conclusions depend on the nature of enforcement effort. Suppose first that enforcement effort can be controlled independently for each harmful act, so that the probability of apprehension is specific to each act. Then in both models the optimal sanction for each act is the maximal sanction, the entire wealth of a person. The reason is well known and due essentially to Becker (1968): If the sanction for an act were less than maximal, the sanction could be raised and the probability of apprehension lowered so as to keep the expected sanction for the act constant; deterrence of the act would therefore be maintained, but enforcement resources conserved; hence, social welfare could be improved. Thus, in both models, optimal sanctions are equal, to wealth, for acts 1 and 2.’ Optimal probabilities of apprehension, however, are generally different f-or the acts (higher for act 2 than for act 1 under certain assumptions).” For optimal sanctions to be different for the two acts, enforcement cannot be spe- cific to the act. Suppose, instead, that enforcement effort is of a general nature, af- fecting in the same way the probability of apprehension for committing different harmful acts; therefore, assume that the probability of apprehension for committing act 1 equals that for committing act 2.4 Then the argument of Becker does not apply independently for each act; if the probability of apprehension is lowered for act 1, the probability is simultaneously lowered for act 2. (How, exactly, this alters the Becker argument is best understood from the analysis.) It is shown under this as- sumption that in both the one-act model and the two-act model, the sanction for act 1 is typically lower than that for act 2, which is maximal. (Hence the earlier statement that marginal deterrence is not in itself a reason for sanctions to rise with harm.) The explanation is that in both models, it is best for the expected sanction for act 1 to be ‘The principal contribution of this note is that it compares the one-act model to the two-act model. ‘lb carry out the comparison. it is simpler to