2) Let v denote the expected healthcare costs of a given individual. Suppose that v is distributed uniformly over the interval [0, 10] in the young population, and uniformly over the interval [10, 30] in the old population. Suppose further that because of risk aversion, a person with expected healthcare cost v is willing to pay up to v + 4 to purchase (complete) health insurance. Assume that 1 2 of the total population are young and that 1 2 are old. Each person knows his or her expected healthcare costs, but insurance companies can observe only whether a person is young or old. Assume that the insurance industry is perfectly competitive.

a) Suppose all insurance companies set premium p for the young. Identify the set of young people who purchase health insurance at that premium, and find expected insurance-company profits as a function of p. Do the same for the old, when all companies set premium q for them.

b) Find equilibrium premiums p for the young and q for the old. Show that at these premiums, the proportion of the old people who purchase health insurance is smaller than that of the young. Hint: Note that the profits of the insurance company from young consumers are (p) = p E[v|insured young given p] and the profits from old consumers are (q) = q E[v|insured old given q]. Use the fact that there is perfect competition among insurance companies (so that they earn zero profits from both types of consumers) to derive the equilibrium prices.