(Jehle and Reny 8.15) Consider once again the moral hazard insurance model. Let the consumers von Neumann- Morgenstern utility of wealth be u(w) = w, let his initial wealth be w0= $100, and suppose that there are but two loss levels, l = 0 and l = $51. As usual, there are two effort levels, e = 0 and e = 1. The consumers disutility of effort is given by the function d(e), where d(0) = 0 and d(1) = 1/3. Finally, suppose that the loss probabilities are given by the following entries, where the rows correspond to effort and the columns to loss levels. So, for example, the probability that a loss of $51 occurs when the consumer exerts high effort is 1/3.

(a) Verify that the probabilities given in the table satisfy the monotone likelihood ratio property.

(b) Find the consumers reservation utility assuming that there is only one insurance company and that the consumers only other option is to self-insure.

(c) What effort level will the consumer exert if no insurance is available?

(d) Show that if information is symmetric, then it is optimal for the insurance company to offer a policy that induces high effort.

(e) Show that the policy in part (d) will not induce high effort if information is asymmetric.

(f) Find the optimal policy when information is asymmetric.

(g) Compare the insurance companys profits in the symmetric and asymmetric information cases. Also, compare the consumers utility in the two cases. Argue that the symmetric information solution Pareto dominates that with asymmetric information.