# Optimal enforcement with imprisonment terms

What is the appropriate role of enforcement activities when fines are not available and only costly imprisonment terms can be used as a deterrence device? Do we get a similar result to the one we obtained in equation (9.9) (namely, that the marginal expected imprisonment term *t* should be increased by as much as possible)? To answer this question, suppose that *t* is fixed, but that *p* can be varied. Welfare is:

At the optimum, the change in welfare with respect to *p* is:

*dB*

Once again, noting that — = *plt*, equation (9.22) can be written as:

^{dx}c

The first term on the left-hand side is the marginal change in the crime rate when the expected marginal punishment rises, and the second term is the change in the expected punishment as *p* rises. Together, the product of these two terms equals the change in the crime rate when *p* rises. The term in square brackets is the marginal social cost of illegal activity, and when the illegal activity falls these costs are not incurred. The term is the sum of two effects: the reduction in social harm *h* and the reduction in expected punishment costs, *pta.* Hence, the left-hand side is the marginal social benefit of increasing enforcement activities.

The right-hand side is the marginal social cost of increasing *p* and is the sum of two terms. The first term is the marginal resource cost of increasing enforcement activity. The second term involves an inframarginal resource cost: it is the product of *x _{c}t*, which is simply the

*total*time spent in prison when the crime rate is

*x*and (l + a), which is the per unit social costs of imprisonment. Thus,

_{c},*x*is the total social cost of imprisonment for those who continue to engage in illegal activity. When

_{c}t(l + a)*p*rises, some individuals will be undeterred and will continue to commit crime. The term

*x*(

_{c}t*l + a*) is the change in the social cost of punishing them when

*p*is increased.

Using our definition of elasticity, we can express equation (9.23) as:

But we also know from our analysis of the optimal marginal imprisonment length that for any p, we have:

These two equations together imply that:

which in turn implies that *p* =* 0. Thus, as we had with a fine, when *p* can be changed and *t* can be increased without limit, it is once again optimal to set the marginal punishment at a very high level, and *p* at a very low level, as long as the expected punishment is adjusted to satisfy:

The reason for this result is straightforward: the criminal's behaviour depends solely on the expected punishment *tp.* The resource costs of imprisonment also only depend on *tp.* On the other hand, the resource costs of enforcement depend only on p. Thus, for any expected marginal punishment, it is always possible to hold *tp* fixed (and therefore hold the crime rate and the social costs of crime fixed) and increase welfare by increasing *t* and lowering *p*, thereby reducing enforcement costs.