Game theory and the Nash equilibrium concept can be used to study the interac- tion between criminals and law-enforcement agencies. Gary Becker, a Nobel Prize-winning economist, led the way on this kind of research and showed that economic analysis is extremely useful in this policy arena. According to Becker's theory, "The optimal amount of enforcement is shown to depend on, among other things, the cost of catching and convicting offenders, the nature of punishments—for example, whether they are fines or prison terms—and the response of offenders to changes in enforcement."6 Becker also argued that, with the optimal enforcement system, crime does occur.

Here is a game that illustrates how the government balances the social cost of crime with law-enforcement costs and how criminals balance the value of illegal activity with the probability of arrest. The game has two players: a criminal (C) and the government (G). The government selects a level of law enforcement, which is a number x Ú 0. The criminal selects a level of crime, 

y Ú 0. These choices are made simultaneously and independently. The government's payoff is given by 

uG = −xc4 − y2>x with the interpretation that −y2>x is the negative effect of crime on society (moderated by law enforcement) and c4 is the cost of law enforcement, per unit of enforcement. The number c is a positive constant. The criminal's payoff is given by uC = y1>2>(1 + xy), with the interpretation that y1>2 is the value of criminal activity when the criminal is not caught, whereas 1>(1 + xy) is the probability that the criminal evades capture. Exercise 4 of this chapter asks you to compute the Nash equilibrium of this game.

Consider the game between a criminal and the government described in this chapter.

(a) Write the first-order conditions that define the players' best-response functions and solve them to find the best-response functions. Graph the best-response functions.

(b) Compute the Nash equilibrium of this game.

(c) Explain how the equilibrium levels of crime and enforcement change as c increases.