_{legal}are illegal, and more harmful to society as x grows: examples of this are speeding and alcohol driving limits, tax evasion, etc.

What determines the choice of behavior, i.e. the preferred x, for a given individual? If we denote by u(x) the utility of x for the individual, this will choose the x at which u(x) is maximum. In the absence of any law regulating this behavior, u(x) is composed of two kind of components:

- Positive factors that make the individual utility grow as x increases: Pleasure from speeding, money evaded, etc.
- Negative factors detracting from the utility as x increases: Fear of an accident, moral concerns, etc. By definition, law penalties are not included here.

To simplify things, let us assume that positive and negative factors contribute in such a way that u(x) increases up to a single maximum at x_{max} and then decreases from that point on --in other words, u(x) is a unimodal function. When there is no law enforcement, if x_{max} ≤ x_{legal} the individual is a law-abiding citizen, otherwise she is a law offender.

Fig. 1: unimodal utility function.

Consider now the introduction of a law penalty p(x) which is null for x ≤ x_{legal} and non-decreasing for x ≥ x_{legal}. Under this new scenario, the resulting individual utility is

w(x) = u(x) − p(x).

For law-abiding individuals, the penalty has no effect and thus the resulting behavior is unaffected. For law offenders, the utility of their original position decreases; if p(x) is high enough for all x > x_{legal}, the maximum of w(x) will be reached then at x_{legal} and the former offender will opt for law abidance.

There are several approaches to designing an effective penalty function. A zero-tolerance policy is one for which even small offenses on the vicinity of legal behavior are punished severely and without exception. Within our model, a zero-tolerance penalty has the form:

p(x) = p_{0} + r(x), x > x_{legal},

where p_{0} > 0 and r(x) is a positive, non-decreasing function (for x > x_{legal}) which we call the residual penalty. p_{0} is the minimum penalty an offender will get regardless of the degree of the offence. The greater p_{0}, the tougher the policy is. On the other hand, we say that p(x) is a progressive penalty if it is nondecreasing and continuous (zero-tolerance penalties are obviously not continuous at x_{legal}). Please note that the characterizations given here for zero-tolerance and progressive penalties are my own and should not be taken as universally known or accepted. The following two figures show the effect of a zero-tolerance and a progressive penalty on the utility function of a law offender.

Fig. 2: impact on utility of a zero-tolerance penalty.

Fig. 3: impact on utility of a progressive penalty.

In both cases the penalty makes the individual opt for a legal position, but the zero-tolerance policy does it at the cost of stricter than necessary punishment for small offences, as stated by the following

Fact. For any zero-tolerance penalty p(x) succesfully enforcing law on a utility function u(x) there exists an equally effective progressive penalty p'(x) which is less severe than p(x), i.e. p'(x) ≤ p(x) for all x.

(We will formally state and prove this rather simple result in a later entry. Informally speaking, we can transformate p(x) into a progressive p'(x) in such a way that the gap u(x_{legal}) − w(x) tends to 0 as x approaches x_{legal} --for a zero-tolerance penalty this gaps tends to some p_{0} > 0.)

Could you title the figures so we know which are the caracteristics of each of them? It would make it much easier to understand what you say. Tks

ReplyDeleteYep, you're right the unnamed figures can be confusing. I've added some titles, hope it looks clearer now.

ReplyDelete