There are a continuum types of people in the society, whose health status can be represented by x, which lies in between 0 and 1. x follows uniform distribution U[0, 1]. A larger value of x indicates that the persons health condition is better. The probability of getting sick and receiving medical treatment is 0.05 0.03x. The expenditure also depends on x, equal to 300 200x. For example, if x = 1, the probability of getting sick and receiving medical treatment is 0.05 0.03 = 0.02 and the expenditure equals to 300 200 = 100. If x = 0.5, the probability of getting sick and receiving medical treatment is 0.05 0.03 0.5 = 0.035 and the expenditure equals to 300 200 0.5 = 200. All people have an initial wealth of 500.

1. Now we consider one single person with x = 0.4. (15)

(a) What is the maximum amount she is willing to pay for a health insurance policy that covers the medical treatment in full, if she is risk-neutral? (5)

(b) How will your answer to question 1 change if she is risk averse with utility function u(w) = w? (5)

(c) How will your answer to question (a) change if she is risk averse and the policy requires a coinsurance of 20% with a deductible of 50 (i.e., the buyer should pay 50 deductible plus 20% of the total medical expenditure in excess of the deductible)? (5)

2. Now we consider a health insurance policy that covers the total expenditure in full. The premium level is 7. (20)

(a) If all the people in the society are risk neutral. What is the fraction of people that will purchase this policy? Is the insurers profit positive, negative, or zero? [Hint: if X follows U[0, 1], the probability of X x0 when 0 x0 1 is x0. ](10)

(b) If all the people in the society are risk-averse, is the fraction of people that will purchase this policy larger or smaller than your answer to question 2(a)? How will the insurers profit change compared with your answer in question 2(a). (10)

3. Now we consider a health insurance policy that requires a coinsurance of 20% with a deductible of 50. The premium level is 5. (15)

(a) If all the people in the society are risk neutral. Denote x the fraction of people that will purchase this policy, please show an equation that x satisfies. Is the fraction of people that will purchase this policy x larger or smaller than your answer in 2(a)? (10)

(b) Does raising the premium level always improve the insurers profit? Explain your answer. (5)