# Optimal enforcement when imprisonment terms cannot be increased without limit

Finally, suppose that *t* cannot be increased without limit, so that there is a maximum marginal imprisonment length t, which we denote by *t _{m}. *Then applying equation (9.24) yields:

or

The left-hand side is the difference between the marginal benefits and marginal costs of increasing *pt* when *t* can be increased without limit. Since the right-hand side is positive, this difference is also positive,

implying an expected marginal punishment that is less than that implied by equation (9.26). Hence, the level of illegal activity must be higher. In other words, when *t* cannot be increased without limit and enforcement is costly, we again get the result that some degree of underdeterrence is optimal, relative to the case where *t* can be increased without limit.

# Fault-based criminal liability

For some illegal activities, punishments are not meted out unless the illegal activity exceeds some threshold level. For example, in many jurisdictions it is not illegal to have a low positive blood alcohol level whilst driving, but it does become illegal if that level increases above some minimum threshold. This is an example of a fault-based criminal liability rule. Under such a rule, the government defines some threshold level of activity *x,* and punishes acts which meet or exceed *x* but not those which do not. Such rules very much resemble the negligence rule we examined in Chapters 4 and 5.

The previous analytical framework can easily be extended to a fault-based liability regime. Consider, for example, the case of fines. Let *p* be the probability of detection and let *f* be the fine that is imposed (imprisonment terms are examined below). The individual's expected net benefits are now:

How does the individual respond to such a rule? First, since the individual is not punished for illegal activities for which *x < x*, the individual will commit those acts, irrespective of the level of the expected marginal fine. Secondly, for activities for which *x > x*, if the expected marginal fine is below B'(x), then even though the individual is at risk of being found criminally liable for these activities they will still commit those acts for which B'(x) > *pf,* since marginal benefits exceed marginal expected costs. Hence, the individual's choice of illegal activity is:

where *x =* min{x, *pf*}.

What is the efficient threshold and fine? Recall that the efficient level of illegal activity is x*, where the marginal benefits of illegal activity equal the marginal social costs. Therefore, if the threshold is set at *x = x** and the expected marginal fine is at least as large as *pf=H'(x*)* (with *p* again as small as possible), then the individual will choose to commit *x _{c} =* x*, the efficient level of crime. Moreover, since the individual does not choose

*x >*x*, no fine is actually levied, and no punishment actually occurs.

*Figure 9.2.3* Fault-based criminal liability

This last result is particularly interesting in the case of imprisonment. If fines are not available and if the threshold is set at *x = x** and the expected marginal imprisonment term is at least as large as *pt = Hâ€™(x*) *with *p* again as small as possible), then the individual will choose *x _{c}* =

*x**, the efficient level of illegal activity. Moreover, since the individual does not choose

*x*> x*,

*no punishment actually occurs.*Thus, in the case of imprisonment terms and an optimally chosen criminal liability threshold, no social costs of imprisonment are incurred, and individuals do not bear any costs of going to prison. Assuming that the threshold is chosen appropriately and assuming both fines and imprisonment can be increased without limit, welfare under both fines and imprisonment is equal to

*W* = B(x*) -*H(x*). In this case, imprisonment is equivalent to a system of monetary fines.