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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 individuals 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 however a person may be contemplating which of several harmful acts to commitwhether 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 sanctions plays a role in addition to the usual one of deterring individuals from committing 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 Bentham 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 trefor 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 enforcement effort To investigate marginal deterrence I consider a simple model with monetary sanctions 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 Science Foundation Kaplow A Mitchell Polinsky and an anonymous grant no SES 8821400 for support 0 1992 Butterworth Heinemann referee for comments and the National 346 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 individuals 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 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 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 assumption 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 out principal contribution the comparison Wilde 1989 Yl hat restrict optimal Reinganum those Wilde In Shavell assumption agent man are two act one act but However is a simplifying and two act model model sanctions emphasize enter s into the the when to the two act Reinganum case model and Wilde of non monetary enforcement determination is no necessary that contrast general one effort to apprehend to apprehend the policeman acts there enforcement will be able assumption in the deterrence IYYI I analyzeand the model the one act of monetary is specific lb carry 1986 and sanctions observed is hrst in of the relationship between optimal these probabilities optimal of appre probabilities and model has the opportunity a crime two act extreme it compares the case 1986 model 01 general on the beat is that to consider to the of marginal in the in the attention sanctions and Corlsideration hension of this note it is simpler the general and specific is appropriate those both committing car will not necessarily enforcement effort importance of which different thieves and apprehend results enforcement whenever in the is noted types

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