# Welfare analysis

Welfare analysis of crime and punishment in a market setting is straightforward. Suppose that the total private benefits from consumption of the good are:

The inverse demand for the good is given by

Suppose also that consumption leads to social external harm of:

Let us find the efficient quantity. Total welfare is given by:

The efficient level of consumption and production is where marginal social benefits equal marginal social costs, or where:

In a free market, however, production and consumption would occur at *Q° > Q*,* where *Q°* satisfies:

The free market outcome here is not efficient.

# Efficient fines

Suppose that in an effort to reduce consumption towards the efficient level, authorities make production and sales of the good illegal. Suppose first that if illegal activity is detected, no goods are confiscated, but a fine is imposed on producers. Suppose that revenue from the fine is returned to consumers in a lump-sum fashion. Let the enforcement costs be C(p), and suppose initially that *p* is fixed. Since no goods are confiscated, marginal revenue for each firm is simply equal to the market price, and so equation (9.28) indicates that the equilibrium market price will be *P* = c + pf.* Since the market price depends on *p* and *f*, it will also be the case that the equilibrium market quantity will also depend on *f*. Welfare is equal to:

The optimal fine can be found by taking the derivative of equation (9.31) and setting it to zero, which gives:

Hence, given a fixed *p*, the efficient fine equates marginal private consumption benefits with marginal social costs. But from the demand curve in equation (9.30), these marginal consumption benefits are equal to the market price, and so from equation (9.32) we get:

or:

As in our earlier analysis, the efficient expected fine is equal to the marginal external harm.

What if *p* can be varied and the per unit fine *f* can be increased without limit? Since increasing *p* is costly whereas increasing *f* is not, it is possible to increase *f* and reduce *p* whilst keeping the expected fine constant, but also improving welfare by reducing enforcement costs. In the limit, if *f* can be increased without limit, welfare is:

This is a similar conclusion to the one we previously derived for fines. If the fine can be increased without limit and firms are risk neutral, the efficient deterrence policy involves setting *f* as high as possible and *p* as low as possible, whilst keeping *pf* = h.*

But what if *f* cannot be increased without limit? Suppose that there is a maximal fine in place, so that *f* = *f _{m}* *. Welfare is now:

and the optimal *p* satisfies:

So that:

where *e ^{D}* is the elasticity of demand for the good. The right-hand side is positive, so the left-hand side must be as well. But since we have assumed that the demand for the good is a downward-sloping function of price, this implies that:

If increasing fines without limit is not possible, some degree of underdeterrence is optimal. This is similar to the result we had in section 9.2.2.