Example: Reporting a crime

As an example of the how spillover effects and public good characteristics can affect welfare very naturally in a law and economics setting, consider the following example. Suppose there is a community of M > 2 individuals. Suppose that a crime has occurred in this community. Each individual can either report the crime, or not. Let у = {0,1} be the strategy space of individual i, where:

Suppose that there is a non-refundable cost of c < 1 that each individual incurs if they report the crime. If sufficiently many people report the crime, then the criminal is caught and each member of the community benefits, irrespective of whether they report it or not. Hence reporting a crime has the characteristics of a public good - it is non-rival (enjoyment of the benefits of reporting by one person does not reduce enjoyment by another) and non-excludable (once a crime is reported, it is not possible to prevent an individual from benefitting).

Normalise the benefit to each individual of the criminal being caught to 1. Let w be the least number of individuals who must report the crime in order for the criminal to be caught and the benefits to each individual to be secured. Let m be the number of individuals who actually end up reporting the crime. The payoff to individual i is then:

Consider a community of two individuals, for example. Suppose that if at least one person reports the crime, then the criminal is convicted and punished. Let the cost of reporting be c < 1. In this situation, it is efficient for only one person to report the crime, and total welfare in the efficient outcome is W* = 2 - c . Do individuals have the incentive to behave efficiently? The payoff matrix for this game is shown in Table 2.2.1.

Recall the concept of a Nash equilibrium of a strategic game. Define the notation:

Take any collection s_ of other players' strategic choices. Player i's best response BRi(s_i ) to s_ is simply the strategy (or collection of strategies) that gives him the highest payoff in that situation. That is, BRt(s-t) solves:

where щ (st, s_t) is the utility of player i when he chooses the strategy st and the other players in the game are choosing s_. Hence:

Table 2.2.1 The payoff matrix for the crime reporting game

Individual 2

Report

Don't Report

Individual 1

Report

(1- c,1-c)

(1- c,1)

Don't Report

(1,1-c)

(0,0)

A (pure strategy) Nash equilibrium is simply a collection of strategies s* = (s*,s*,...,s*) such that every player i's strategy choice s* is a best response to s* . In other words, the combination (s*,s*,...,s*) is a Nash equilibrium if it satisfies:

There are two pure strategy Nash equilibria in the game in Table 2.2.1, in which either individual 1 or individual 2 reports the crime, and the other individual does not. These equilibria are both efficient.

There is a symmetric mixed strategy Nash equilibrium as well. To find this mixed strategy equilibrium, consider individual 1. To be willing to use a mixed strategy, he must be indifferent between reporting the crime and not reporting it. Suppose that individual 1 believes that individual 2 reports the crime with probability p. Then the expected payoff from individual 1 reporting is 1 - c, since he is guaranteed to receive that amount from reporting. On the other hand, if individual 1 decides not to report the crime, he may still benefit if individual 2 reports it. Since this happens with probability p, individual 1's expected benefit from not reporting the crime is simply px 1 = p. Thus, to be indifferent between reporting and not reporting, we must have 1 - c = p*. This is the intersection of the individual reaction curves as shown in Figure 2.2.1 below.

The expected number of individuals who report the crime in this mixed strategy equilibrium is:

which shows that in expected terms, if c > 1 the mixed strategy equilibrium is expected to have an inefficiently low number of people report the crime. On the other hand, if c < , then we can expect there to be an inefficiently high number of individuals reporting the crime.

The expected welfare level is:

For any level of c, this is lower than welfare in the efficient outcome. Therefore, even if the expected number of reporters is 1, welfare is still expected to be lower in this equilibrium than in the efficient outcome, for any positive value of c.

Reaction curves and equilibria in the crime reporting game

Figure 2.2.1 Reaction curves and equilibria in the crime reporting game

Note that the expected number of individuals who report the crime, expected welfare, and the probability that both individuals will report the crime depends negatively on the cost of reporting the crime. In the extreme case where the cost of reporting the crime is zero, all individuals report. In the other extreme case where the cost of reporting is 1, nobody reports the crime - even though it is still efficient for at least one person to do so.

In addition to issues related to the public good characteristics of certain law enforcement activity, there are a number of other difficulties. As Cowen (1992) points out, foremost among these is the issue of conflict between competing private law enforcement agencies. Suppose that individual A contracts with enforcement company A to protect his rights, and that A's promised punishment for murder is the death penalty. Suppose that individual B is contracted to enforcement company B, which promises to protect its customers from the death penalty. Now suppose that B murders A's son - this now raises the possibility of conflict between the obligations that company A and B promised to carry out on behalf of their clients.

On the other hand, there is a great deal of empirical evidence which shows that throughout history private dispute-resolution markets have developed, in which arbitration services are supplied by private firms and parties agree to be bound by their decisions.1

In summary, even if legal rules and enforcement activities are pure public goods, this does not resolve all of the issues that are of interest. It does not mean, for example, that it will be always efficient for individuals to supply no private enforcement activity themselves, and that the government should undertake all forms of enforcement activity.

It also does not mean that government supply will make a difference to overall levels of enforcement or economic efficiency, or that the marginal benefits of any level of government supply will always outweigh the costs. And it certainly does not mean that governments will automatically supply the efficient amount of the pure public good.

 
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