(Moral Hazard) (Jehle & Reny 8.15) Consider the moral hazard insurance model. There is one risk...

 

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(Moral Hazard) (Jehle & Reny 8.15) Consider the moral hazard insurance model. There is one risk neutral insurance company and one risk averse customer. Let the consumer's vNM utility of wealth be u(w) = w, let his initial wealth be wo = = $100, and suppose that an accident may occur causing a loss of $51. There are two effort levels, e = 0 and e1. The consumer's disutility of effort is given by the function d(e), where d(0) = 0 and d(1) = 1/3. Finally, suppose that when e = 0, an accident occurs with probability 2/3, and when e = 1, an accident occurs with probability 1/3. An insurance contract C = (a, ) specifies the premium a that a customer has to pay, and the payment that he receives when accident occurs. (a) Verify that the given probabilities of accident satisfy the monotone likelihood ratio property. (b) Find the consumer's reservation utility assuming that there is only one insurance company and that the consumer's 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 com- pany to offer a contract that induces high effort. (In this case, the insurance offers a policy that pays benefits only if a particular effort level was exerted.) (e) Show that the contract in part d will not induce high effort if information is asymmetric. (f) Find the optimal contract when information is asymmetric. (g) Compare the insurance company's profits in the symmetric and asymmetric in- formation cases. Also, compare the consumer's utility in the two cases. Argue that the symmetric information solution Pareto dominates that with asymmetric information. (Moral Hazard) (Jehle & Reny 8.15) Consider the moral hazard insurance model. There is one risk neutral insurance company and one risk averse customer. Let the consumer's vNM utility of wealth be u(w) = w, let his initial wealth be wo = = $100, and suppose that an accident may occur causing a loss of $51. There are two effort levels, e = 0 and e1. The consumer's disutility of effort is given by the function d(e), where d(0) = 0 and d(1) = 1/3. Finally, suppose that when e = 0, an accident occurs with probability 2/3, and when e = 1, an accident occurs with probability 1/3. An insurance contract C = (a, ) specifies the premium a that a customer has to pay, and the payment that he receives when accident occurs. (a) Verify that the given probabilities of accident satisfy the monotone likelihood ratio property. (b) Find the consumer's reservation utility assuming that there is only one insurance company and that the consumer's 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 com- pany to offer a contract that induces high effort. (In this case, the insurance offers a policy that pays benefits only if a particular effort level was exerted.) (e) Show that the contract in part d will not induce high effort if information is asymmetric. (f) Find the optimal contract when information is asymmetric. (g) Compare the insurance company's profits in the symmetric and asymmetric in- formation cases. Also, compare the consumer's utility in the two cases. Argue that the symmetric information solution Pareto dominates that with asymmetric information.