Suppose that two agents, Danny and Bob, are deciding how fast to drive their cars. Each agent chooses speed x_i and gets utility u_i(x_i) from this choice; we assume that u_i(x_i) > 0. However, the faster they drive, the more likely it is that they are involved in a mutual accident. Let p(x₁,x₂) be the probability of an accident, assumed to be increasing in each argument, and let c_i > 0 be the cost that the accident impose on agent i. Assume that each agents utility is linear in money.

a) Show that each agent has an incentive to drive too fast from the social point of view.

b) If agent i is fined an amount t_i in the case of an accident, how large should t_i be to internalize the externality?

c) If the optimal fines are being used, what are the total costs, including fines, paid by the agents? How does this compare to the total cost of the accident?