# Optimal enforcement when fines cannot be increased without limit

The prescription for deterring crime in (9.8) is simple but far from adequate. It is usually the case that fines *cannot* be increased without limit. If individuals are wealth constrained for example, then increasing the fine above their wealth level will have *no* additional deterrence effect, since there is no additional financial liability for the individual in the event that they are caught. Indeed, this is exactly analogous to the judgement proof problem in accident law we examined in earlier chapters. So setting the fine at 'the highest level possible' usually involves choosing *f* to be some maximal level, say *f _{m}.*

Let us assume, then, that this has been done, and analyse the choice of p. The criminal's choice of *x* still obeys:

The key lesson that emerges is since increasing *p* is costly, it may no longer be optimal to choose *p* in such a way that *pf _{m} = H'(*x*). To see this, note that fines are still a transfer between individuals and society and are costless to administer at the margin, so social welfare can still be written as:

With *f* fixed at *f _{m},* an increase in

*p*certainly has an additional deterrence effect by increasing the 'price' that individuals face for engaging in illegal activity, but this now comes at a social cost. The optimal

*p*balances out these benefits and costs. Since the criminal's choice of crime obeys

*W = B'(x) = pf*we can write x

_{m},_{c}= x

_{c}

*(pf*

_{m}), and so:

The first-order condition for the efficient choice of *p* is:

or:

The first term on the left-hand side of (9.13) is the individual's marginal benefit of illegal activity. The second term on the left-hand side of (9.13) is a measure of the responsiveness of illegal activity with respect to an increase in the probability of detection. The third term is the marginal change in the expected fine when the probability of detection *p* changes. The product of these three terms is the change in the individual's expected benefit when *p* increases. Finally, the fourth term on the left-hand side of (9.13) is the marginal cost of increasing p, and depends on the technology of enforcement. The left-hand side of (9.13) is therefore a measure of the marginal social cost (MSC) of increasing *p*. It is the sum of the decline in the expected welfare of those who engage in illegal behaviour, plus the additional cost due to the increase in *p*.

The first term on the right-hand side of (9.13) is the marginal social harm from illegal activity. The second term on the left-hand side of (9.13) is once again a measure of the responsiveness of illegal activity with respect to an increase in the expected fine. The third term is the marginal change in the expected fine when the probability of detection *p* changes. The right-hand side is therefore a measure of the marginal social benefit (MSB) of increasing *p*. It is the marginal reduction in social harm that comes about when *p* changes. The rule in (9.13) just says that if increasing the fine is no longer possible, resources should be spent on enforcement activity until the marginal social costs are equal to the marginal social benefits.

We can simplify this expression further by utilising the concept of

elasticity. Let *e* = — be the *elasticity of the crime rate with respect*

^{d(}P^{f}m^{) X}c

*to the expected marginal fine.* Rearranging the efficiency condition in equation (9.13), we get:

Since the marginal costs of enforcement are assumed to be positive and since we will no longer have *p ~* 0, the right-hand side is positive. Thus, the left-hand side must also be positive, since the two sides are

equal. The term *e =* — * ^{Pfm}* is a measure of the responsiveness of

^{q} *d(pfm*) Xc ^{p}

the individual's demand curve to changes in the 'price' that is faced, and is negative, since the demand curve for illegal activity slopes downwards. This means that we must have:

or:

But since we still have *B'(x _{c}*) =

*p* f*this means that, at the optimal p*, we must have:

_{m},

*If it is costly to increase p and fines cannot be increased any further, it is no longer efficient to set H'(x _{c}*) = p*

*f*. Since marginal benefits of crime are less than the marginal social harm, it must be the case that the crime rate is higher here than was suggested by equation (9.9). In other words, the standard Pigouvian result breaks down under these conditions, and some degree of

_{m}*underdeterrence*is optimal. The extent of the underdeterrence depends on the marginal costs of enforcement - the higher they are, the more efficient is a higher crime rate.