There is a large market for health insurance. The fraction of healthy people in this market is

p and the fraction of sick people is 1 p, where 0 < p < 1. Each individual is endowed with

40,000. The expected cost of treatment for any person is 30,000. A healthy person needs

treatment with probability q = 0.25. A sick person needs treatment with probability q = 0.75.

The expected utility of a person who needs treatment with probability q is

U (q, x, y) = qx + (1 q)y

Where x is wealth when treatment is needed and y is wealth when no treatment is needed.

There is a single insurance company which offers full insurance that covers the cost of treat-

ments. The insurance company offers to each individual a premium for full insurance, and the

person can either accept or reject the company's offer.

a)Calculate each individual's reservation price for full insurance, that is, the maximum

amount this person is willing to pay to be fully insured.

b)Suppose the insurance company knows whether an individual is sick or healthy and

sells full insurance at the individuals' reservation prices. Calculate the expected profit.

c)Suppose now the insurance company cannot tell apart healthy from sick people.

What is the minimum value of p such that there exists a market equilibrium in which all

individuals are insured?

d)Suppose now that sick individuals can become healthy by exercising. The expected

utility of a sick person who exercises is

U (q, x, y) = 0.25 x + (1 0.75) y c

Where c > 50. Does there exist an equilibrium in which a positive share of sick individuals

improve their health? Explain