Chimnesia has two equal-sized groups of people: smokers and nonsmokers. Both types of people have utility U = ln(C), where C is the amount of consumption that people have in any period. So long as they are healthy, individuals will consume their entire income of $15,000. If they need medical attention (and have no insurance), they will have to spend $10,000 to get healthy again, leaving them with only $5,000 to consume. Smokers have a 12% chance of requiring major medical attention, while nonsmokers have a 2% chance.

Insurance companies in Chimnesia can sell two types of policies. The "low-deductible" (L-) policy covers all medical costs above $3,000, while the "high-deductible" (H-) policy covers only medical costs above $8,000.

a. What is the actuarially fair premium for each type of policy and for each group?

b. If insurance companies can tell who is a smoker and who is a nonsmoker and charge the actuarially fair premiums for each policy and group, show that both groups will purchase the L-policy.
Suppose that smoking status represents asymmetric information: each individual knows whether or not they are a smoker, but the insurance company does not.

c. Explain why it is impossible, at any price, for both groups to purchase L-policies in this setting. Which groups, if any, do you expect to buy L-policies, and at what price?

d. Show that it is possible for both groups to purchase insurance, with one group buying L-policies and one group buying H-policies.