Chimnesia has two equally sized groups of people: smokers and nonsmokers. Both types of people have utility U = √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 $16,000. If they need medical attention (and have no insurance), they will have to spend $12,000 to get healthy again, leaving them with only $4,000 to consume. Smokers have a 10% chance of requiring major medical attention, while nonsmokers have a 2% chance. Insurance companies in Chimnesia can sell two types of policy. The low deductible (L-) policy covers all medical costs above $2,000, while the high deductible (H-) policy only covers medical costs above $10,000.
(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 charges the actuarially fair premiums for each policy and group, which policy will two groups purchase? Compute the expected utility of each group with each policy. Now suppose that smoking status represents asymmetric information: each individual knows whether or not they are a smoker, but the insurance company doesn't.
(c) Explain why it is impossible, at any prices, for both groups to purchase L-policies in this setting. Which groups, if any, do you expect to buy L-polices, and at what price?
(d) Mathematically show that it is possible for both groups to purchase insurance, with one group buying L-policies and one group buying H-policies.