Demonstrate how the government balances the social cost of crime with the law-enforcement costs and explain how the criminals balance the value of illegal activity with the probability of arrest. The game has two players: a criminal and the government. The criminal selects a level of crime y ≥ 0. The government selects a level of law enforcement, which is a number x ≥ 0. These choices are made independently and simultaneously. The government's payoff is given by the uG= - xc4 - y2/x with the interpretation that - y2/x is the negative effect of crime on the society 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 the uC= y0.5/(1+xy), with the interpretation that y0.5 is the value of criminal activity when the criminal is not caught, whereas 1/(1+xy) is the probability that the criminal evades capture.
1. Write down the first order conditions that define the players' best response functions and solve them to identify the best response functions.
2. Calculate the Nash Equilibrium of this game.
3. Discuss how the equilibrium levels of enforcement and crime change as c increases.