# Efficient imprisonment

In the previous section we analysed a setting in which fines could be increased without limit, and also situations in which that assumption did not hold, where fines could not increase above some maximal level *f _{m}.* In some situations fines may not be available at all, or may be used in conjunction with imprisonment terms. This section therefore examines the economics of imprisonment. The critical difference between our analysis of fines and imprisonment is that we will assume that imprisoning individuals uses up economic resources at the margin, whereas in the case of fines we assumed that there were no marginal administration costs. The analysis in this section therefore applies to any punishment which is costly to administer at the margin.

To simplify the analysis, we will assume throughout this section that the marginal social harm from crime is constant, so that *H* (*x) = hx*.

Let:

- •
*t*be the marginal imprisonment term: that is, the length of imprisonment imposed per unit of illegal activity. Thus, if an individual commits illegal activity at a rate of*x*and is caught, he faces a punishment of*tx.* - •
*a >*0 be the marginal social cost of imprisoning a convicted individual for one unit of time. Thus, if an individual commits illegal activity of*x*, and is caught, the resource costs of imposing this punishment are*atx.* - •
*l >*0 be the subjective disutility or loss to the individual per unit of time of imprisonment. Thus, if an individual commits illegal activity of*x*and is caught, the subjective disutility costs to the individual are*Itx.*

Let us examine the efficient choice of marginal imprisonment term, *t*. To illustrate the nature of the trade-offs involved, we assume that *f* = 0 and that *p* is fixed at p. The individual's expected payoff from engaging in criminal activity of *x* is:

The individual engages in illegal activity up to the point where marginal benefits equal expected marginal costs:

So we can write the individual's 'demand curve' as:

where again the demand curve slopes downwards. For any imprisonment term t, the individual's choice of activity must satisfy this equation. When this level of illegal activity is chosen, the individual's net welfare is:

Unlike fines, imprisonment is not
purely a transfer between individuals and the rest of society. Imprisonment creates social costs of its own. These costs, however, are only incurred if the individual is caught and punished (which happens with probability p). In other words, if the level of illegal activity is *x _{c},* then expected social harm is equal to:

Thus, aggregate expected social welfare is now:

Once again, since *x _{c} = x_{c} (plt*), we can write:

A marginal increase in the marginal imprisonment term *t* now has three separate welfare effects. Two are marginal effects, and one is an *inframarginal* effect. The marginal effects are as follows:

- • A marginal increase in
*t*reduces the level of illegal activity by raising the expected 'price'. This reduces the welfare of individuals who undertake the illegal activity, and is a welfare loss. - • Because it reduces illegal activity, the marginal increase in
*t*reduces the social harm caused by crime. And since illegal activity falls, this saves on imprisonment costs for those illegal activities which no longer occur because the individual is deterred from committing them. This is a welfare gain.

These changes only occur at the margin. The inframarginal effect is as follows: ^{[1]}

The optimal imprisonment term balances out these welfare gains and losses. Differentiating *W* with respect to *t* and setting the result equal to zero, we get:

or:

But we know that for any t, the individual's behaviour satisfies:

Thus, the first term on the left-hand side of (9.17) is zero, and we are left with:

We can simplify equation (9.18) further by again utilising the concept

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

*d(plt ^{[2]}) x_{c}*

with respect to the expected marginal imprisonment term. Then multiplying both sides of equation (9.18) by *t ^{[2]}/x_{c}* gives:

and so the efficient expected marginal imprisonment length satisfies:

Suppose that the elasticity of demand is constant. There are several important conclusions to draw from the results in equations (9.19) and (9.20):

will deter them from committing a desired level of illegal activity, and shorter sentences reduce the costs to society of imprisoning criminals.

- • The expected marginal punishment may be increasing or decreasing in
*a,*the marginal resource costs of imprisonment per unit of time. The sign depends on whether the demand for illegal activity is elastic or inelastic. Why do we get this result? Go back to equation (9.19), which has marginal benefits of changing the expected punishment on the left-hand side, and marginal social costs on the right-hand side. Note that*a*enters both sides. Now suppose that*a*rises. Then welfare must fall. But the welfare loss can be partially mitigated by changing the expected marginal imprisonment term. Equation (9.19) indicates that when*a*rises both the marginal benefits and marginal costs of changing*pt*rise. But if demand for illegal activity is elastic (i.e. if*? <*-1) then the marginal costs (the left-hand side) rise by more than the marginal benefits (the right-hand side). The only way that equality can be maintained between the two sides is if*pt** falls. In these circumstances, the efficient response to an increase in the marginal costs of imprisonment is to reduce the marginal expected imprisonment term (and hence allow the crime rate*x*to rise). On the other hand, if the demand for illegal activity is inelastic (i.e. if*?*> -1), then when*a*rises, the marginal costs of expected imprisonment rise by less than the marginal benefit, and the efficient response is to*increase*the expected marginal imprisonment term, thus*reducing*the crime rate. - • The expected marginal punishment is increasing in ?, the elasticity of the crime rate with respect to the expected marginal punishment. Remember that
*? <*0, so this means that individuals whose behaviour is less (more) responsive to the expected marginal punishment require higher (lower) expected*marginal*punishments in order to deter them in an efficient manner.

The efficient rate of illegal activity when these additional costs of impro- sonment are taken into account may be higher or lower than that obtained when only fines were available and could be increased without limit. To see this, note that equation (9.19) implies that:

But the right-hand side of equation (9.21) is equal to *B'(x _{c})* -

*H'(x*which was equal to zero when fines could be increased without limit. The left-hand side of (9.21) can be positive or negative, depending on the elasticity of the crime rate with respect to the expected punishment. It is positive if

_{c}),*?*< -1 (that is, if demand is elastic). In this case, we must have B'(x

_{c})

*> h*which implies that the crime rate is lower than the rate implied in equation (9.9). On the other hand, if

*? > -*1 (that is, if demand is inelastic), then

*B'(x*which implies that the crime rate is higher than the rate implied in equation (9.9).

_{c}) < h- [1] The increase in t makes prison sentences longer. This reduces theexpected welfare of criminals and has a social cost because prisonsare costly to run. These costs must be incurred with probability p,on those illegal activities which are not deterred (that is, all level ofcriminal activity for which x < xc). These are the inframarginal illegalactivities - all those activities which are not deterred by a smallchange in t. This too is a welfare loss.
- [2] 2 The expected marginal punishment is increasing in h, the marginalsocial harm of crime. Illegal activities that are more harmful at themargin should receive higher expected marginal punishments. • The expected marginal punishment is decreasing in l, the individual'ssubjective marginal disutility costs of imprisonment. Individuals whohave a higher marginal disutility of being punished should receivelower expected marginal punishments. The reason is obvious: themore an individual dislikes prison, the more likely a shorter sentence
- [3] 2 The expected marginal punishment is increasing in h, the marginalsocial harm of crime. Illegal activities that are more harmful at themargin should receive higher expected marginal punishments. • The expected marginal punishment is decreasing in l, the individual'ssubjective marginal disutility costs of imprisonment. Individuals whohave a higher marginal disutility of being punished should receivelower expected marginal punishments. The reason is obvious: themore an individual dislikes prison, the more likely a shorter sentence