Attached are some questions below. Try and provide solutions for these.

4. (40 Minutes - 20 Points) Consider the following model in which a worker's choice of health play may signal his health to his insurance company. Suppose nature first chooses the health of the worker choosing @ ( {healthy, not healthy). Assume that the probability that the worker is healthy is q. Player 1 observes whether he is healthy and then chooses a, 0. With some probability # 6 (0. 1) she may get sick, in which case she would need to spend L dollars on her health. There is a health-insurance policy that fully covers her health care expenses in case of sickness and costs P to her. (If she buys the policy, she needs to pay P regardless of whether she gets sick.) (a) Find the set of prices P that she is willing to pay for the policy. How does the maximum price P she is willing to pay varies with the parameters M, L. o, and ? (b) Suppose now that there is a test te {-1, +1} that she can take before she makes her decision on buying the insurance policy. If she takes the test and the test t is positive, her posterior probability of getting sick jumps to a* > > and if the test is negative, then her posterior probability of getting sick becomes 0. What is the maximum price c she is willing to pay in order to take the test? (Take P