The optimal fine maximises welfare, W. Remember, however, that any choice of fine will flow through into individual behaviour according to equation (9.3) above. The enforcement authority cannot control illegal activity directly - it can only do so indirectly, through its choice of the fine. What level of f maximises welfare? We assume that revenue is received by the government on behalf of the taxpayer and returned to them in a lump-sum fashion, and that the system of fines is costless to administer at the margin. Thus, at the margin, fines are assumed to be pure wealth transfers from criminals to the rest of society. This assumption regarding punishments can be relaxed, and we do so below.
Using the economic intuition of the Pigouvian approach, we can test the following 'guess'. Recall that the Pigouvian approach sets a tax of t*, where this satisfies:
Thus, economic intuition leads us to the conjecture that if we follow this approach and set the marginal fine f so that pf = H '(x*), the individual will, in accordance with the standard Pigouvian rule, fully internalise the external costs of his actions and choose the efficient rate of illegal activity.
This guess turns out to be correct. To see this, note that with a fine of pf, expected welfare is the sum of the individual's net benefit and the net benefit to the rest of society, both of which include the expected revenue from the fine:
where for any fine, the criminal will choose xc = x(pf), and where, without loss of generality, we have set the costs of administering the system of fines at K = 0. Note that society gains the fine f only when a crime is actually detected (which happens with probability p), but the social harm H occurs whether the crime is detected or not. Plugging xc = x(pf) into (9.4) yields:
Choosing the marginal fine f to maximise welfare yields the first-order condition:
The first term in (9.5) is the marginal change in the individual's benefits when his choice of illegal activity changes. The second term on the left-hand side tells us how responsive illegal activity is to an increase in the expected fine. The third term, p, is the marginal change in the expected fine as f changes. Multiplied together, these terms give the individual's expected marginal loss when the fine increases by a small amount. It is the expected marginal social cost of increasing f.
The first term on the right-hand side of (9.5) is the marginal social harm of illegal activity. The second term is again just a measure of the responsiveness of illegal activity with respect to an increase in the fine. The third term, p, is again the marginal change in the expected fine as f changes. These three terms multiplied together give the expected marginal change in harm when the fine increases by a small amount. In other words, it is the expected marginal social cost of increasing f.
The optimal fine balances out marginal social costs and benefits. Cancelling terms, we get:
This can be simplified by noting from the individual's behaviour that: and so the condition in equation (9.6) becomes:
These two conditions verify that the Pigouvian intuition explored earlier is correct. First, since equation (9.6) has marginal benefits equal to marginal harm, this verifies that it is possible to design a system of fines such that the individual chooses the efficient level of illegal activity. Secondly, equation (9.7) confirms that the efficient solution is to set the expected marginal fine (which in expected terms is equivalent to a
Pigouvian tax here) equal to the marginal external harm evaluated at the optimal rate of illegal activity. Dividing both sides of (9.7) by p yields:
This is a fundamental equation in the economic approach to crime, punishment and illegal activity. It states that for a fixed probability of detection, the welfare-maximising fine f* is equal to the discounted value of marginal harm at the efficient level of x, where the discount rate is the probability of detection p.
There are several points worth noting here:
- 1. This result was obtained under the assumption of risk-neutral behaviour on the part of the individual. Different attitudes towards risk will, in general, lead to a different result.
- 2. The optimal fine is higher, the smaller is the probability of detection and the higher is the marginal harm at the optimum.
- 3. Since p < 1, the efficient marginal fine will exceed the marginal harm created by the illegal activity at the optimum. In a world of imperfect detection, it is simply not good enough to fine individuals an amount that is equal to the marginal external harm. To do so would lead to an inefficiently low expected punishment and in inefficiently high level of illegal activity - indeed, the crime rate would be massively high relative to the efficient level if p is close to zero. Thus, for example, punishment schemes which simply force individuals to return stolen goods or repay amounts of money that they have stolen - without any other punishment - are probably doomed to fail from an efficiency point of view. A similar conclusion holds for 'restorative justice' programmes, which effectively set the 'fine' equal to the marginal social harm that is created. If p < 1 this kind of scheme will tend to encourage inefficiently high rates of illegal activity, since the expected punishment will be far too low from an efficiency point of view.
- 4. Finally, we have fixed the probability of detection at p. But since fines are assumed to be pure transfers from individuals to the rest of society, and since individuals only cares about the expected fine, and since p> 0 means that C(p) > 0, it may be possible to improve upon this solution.
To expand upon the last point, note that for any p, efficiency requires that:
But a risk-neutral individual's choice of x only depends on pf*, not f* by itself. Thus, with the individual assumed to be risk-neutral, it is possible to decrease p and increase f* and equation (9.9) would still hold, with the criminal continuing to choose x*. Since the individual's expected benefit and the external harm remain unchanged, the welfare gain from doing this is:
which is positive if dp< 0 . Thus, if increasing the probability of detection is costly and fines are costless to administer at the margin, then the optimal enforcement policy is to keep increasing f and keep reducing p so that equation (9.9) still holds, and to keep doing so until p~ 0. Once this is achieved, welfare is equal to:
which is the highest level of welfare that can be obtained, given that enforcement is costly. If there are no fixed enforcement costs, then C(0) = 0, and W ~ B(x*) - H(x*) when the efficient fine is chosen.