In this exercise, you are asked to analyze the health insurance.

There are a continuum types of people in the society, whose health status can be represented by x, which lies in between 0 and 1. x follows uniform distribution U[0,1]. A larger value of x indicates that the persons health condition is better. The probability of getting sick and receiving medical treatment is 0.05 0.03x. The expenditure is 200, a constant if one is sick and does not depend on x.

For example, if x = 1, the probability of getting sick is 0.05 0.03 = 0.02. If x = 0.5, the probability of getting sick is 0.05 0.03 0.5 = 0.035. All people have an initial wealth of 500.

1. Consider a person A with x = 0.5 buying health insurance that covers the medical treatment. Suppose A is risk-neutral. (15)

   1. (a) What is As acturially fair price? (5)

   2. (b) If A miscalculates her probability of getting sick to be 0.032 and she correctly calculates her expenditure if being sick, is she able to purchase the insurance? If so, at what price? Assume the insurers are perfectly competitive and market entry is free. (5)

   3. (c) Repeat your analysis in (b) if A miscalculates her probability of getting sick to be 0.038. (5)

2. Consider a person B with x = 0.5 and utility function u(w) = w buying the same insurance. (15)

   1. (a) What is the maximum price that B will accept for this insurance, if she correctly cal- culates her probability of getting sick and the expenditure? (5)

   2. (b) How will your answer to (a) change if B miscalculates her probability of getting sick to be 0.032 and she correctly calculates her expenditure if being sick? Is she able to purchase the insurance? If so, at what price? (5)

   3. (c) Repeat your analysis in (b) if B miscalculates her probability of getting sick to be 0.038. (5)