Consider a health insurance market with one insurer and N consumers. Consumers are risk averse and have utility functions u(c) = c

where c is the consumption of the consumer. All consumers have income equal to 1, and, if they become sick, they must spend their entire income on health services. If they dont become sick, they spend their whole income on consumption. In other words, the utility of consumers that become sick is u(0), and the utility of consumers that do not become sick is u(1).

The probability that consumer n = 1, . . . , N becomes sick is n, where n is uniformly distributed over [0, 1]. In this problem, well examine what premiums p the insurer charges for full insurance coverage. That is, if a consumer purchases insurance, then their utility is u(1 p), regardless if they get sick or not, because the insurer pays for the health care in full.

(1) Suppose is common knowledge, and both the consumer and insurer known .

a. Suppose the insurer offers a premium p(). What is the expected utility of a consumer that purchases insurance at this premium? What is the expected utility of a consumer that does not?

b. What is the maximum premium a consumer will pay for insurance? Let this be the function p ^SO().

c. Which consumers purchase health insurance when the insurer offers the premium p ^SO()? Let this function be I ^SO(), where I ^SO(n) = 1 if consumer n purchases insurance and I ^SO(n) = 0 otherwise.

d. What is the total welfare in this market?

(2) Suppose is hidden information and is only known by the consumer.

a. What consumers will purchase insurance at a constant premium p? Let this be the function I (). Note that I () will be a function of both and p .

b. What consumers is the insurer willing to sell insurance to at the constant premium p?

c. What is the expected health of those who purchase insurance at premium p?

d. Using your answers to questions 2(b) and 2(c), solve for p . Note that p should not be a function of .

e. Now, solve for I (), which should not be a function of p .

f. What is the total welfare of this market?

(3) Suppose insurance is unavailable. What is total welfare?

(4) Compare the total welfare values in questions 1(d), 2(f), 3,. Explain their ordering.