We will assume that the individual is risk-neutral, and therefore acts as if he is maximising his expected net benefits, solving:
The individual optimises by setting x equal to xc, where xc solves B'(xc ) = pf. That is, the individual engages in illegal activity up until the point where his marginal benefit equals the 'price' of the activity, where the price is the expected marginal fine. This gives us the following 'demand curve' for illegal activity:
and where the second last equality follows from the inverse function theorem, and the last inequality follows from the assumption that B" < 0.
The 'demand curve' for illegal activity is downward sloping as a function of the 'price' Pf. This is the deterrence hypothesis: a higher expected marginal punishment reduces the individual's crime rate. Note that the deterrence hypothesis by itself says nothing about the magnitude of the deterrence effect. That depends on the slope of the demand curve or the elasticity of the demand for crime with respect to the expected marginal punishment. We will examine this elasticity below.
Figure 9.2.2 Individual behaviour in response to a fine