Consider a world composed of two types of people: those without preexisting conditions (N) and the ones with preexisting conditions (C). People in group N get sick with probability πN = 10%, while people in group C with probability πC = 15%. Sick individuals incur a cost of $2, 000. There are 50 people in each group. Each person has an income of $10, 000. Health insurance companies provide a plan that covers all costs above $200, that is, there is a $200 deductible. Individuals obtain utility from consumption u(x) = √ x

Now assume that the insurance company does not observe an individual’s type.

a) Suppose that the insurance company asks the individuals about their type and offers an actuarially fair contract accordingly (there still is a $200 deductible). What would type C declare? Is there a market failure? Why?

d) Now suppose that the insurance company offers a single insurance product. Conditional on both types buying this insurance product, find the premium p ∗ at which the company breaks even (in expectation).

c) Is p ∗ such that both types actually buy insurance? Why or why not?

d) What is the only possible equilibrium in this case? Is it efficient?