Are Criminals Risk Preferrers? Footnotes

1Becker initially defines O as the "activity level" for a crime without defining exactly what that means, but later treats it as the number of crimes. Most of his article, like most subsequent work on the subject, confines itself to a single homogenous crime, with O occurences committed by indistinguishable offenders.

2The observed structure and popularity of lotteries provides evidence that some people exhibit more risk preference for lotteries involving low probabilities of high positive payoffs than for less risky lotteries; that does not imply that for sufficiently low probabilities of sufficiently large negative payoffs everyone becomes a risk preferrer. Friedman (1980) argues that high degrees of risk preference are unlikely to be observed; since risk is cheap to produce it must, in equilibrium, have little value.

3Becker's omission of such costs is pointed out in Friedman (1981) and earlier in Polinsky and Shavell (1979); the latter observe that if the costs of imposing risks on risk averse criminals are taken into account, an infinite punishment with finite probability (or a "largest possible" punishment with correspondingly small probability) may no longer be optimal. While they derive several further results, they do not note the relevance of the omission to Becker's conclusion that criminals must be risk preferrers.

Carr-Hill and Stern (1977) also criticise the conclusions and specifications of Becker's model, but since their principal points are the absence of an interior solution when b=0 (costless fines), the characteristics of the solution when b<0, and the inappropriateness, in a second best world, of assuming that a pure transfer "does not increase or decrease social welfare," their criticism is almost perfectly orthogonal to mine.

4Since most things we think of as crimes are not punished with fines, b>1 and Becker's conclusion still holds. Becker points out in a footnote that for b<0 (fine collected greater than fine paid!) his result is reversed and risk aversion is necessary to avoid the corner solution.

5A related point is made in Block and Lind (1975); the authors argue that crimes and punishments may have no monetary equivalent, since utility as a function of wealth may be bounded. One implications is that it may be impossible (and not merely costly) to impose "very large" (in utility terms) fines. Neither Block and Lind nor Polinsky and Shavell combine costs associated with risk and those associated with collecting fines or imposing alternative punishments into the more general observation that punishment cost increases with amount of punishment, as is done here and, at greater length, in Friedman (l981).

6In this situation one would never actually impose the lottery, since the same result could be achieved at a still lower cost by catching a smaller fraction of the criminals and imposing punishment g on all of them, thus reducing enforcement costs.