While self-employed workers have the option to purchase private health insurance, many-especially younger-workers do not, due to adverse selection. Suppose that half the population is healthy and the other half is unhealthy. The cost of getting sick is $1,000 for healthy people and $10,000 for unhealthy people. In a given year, any one person (regardless of health) either becomes sick or does not become sick. The probability that any one person gets sick is 0.4. Each person's utility of wealth function is U(Y) = Y0.5, where Y is the person's wealth. Each worker's initial wealth is $30,000. Although each person knows whether he or she is healthy, the insurance company does not have this information. The insurance company offers complete, actuarially fair insurance. Because the insurance company cannot determine whether a person is healthy or not, it must offer each person the same coverage at the same price. The only costs to the company are the medical expenses of the coverage. Under these conditions, the insurance company covers all the medical expenses of its policyholders, and its expected profit is zero.
a. If everyone purchases insurance, what is the price of the insurance?
b. At the price you determined in part a, do healthy people purchase the optimal amount of insurance?
c. If only unhealthy people purchase insurance, what is the price?
d. At the price you determined in part c, do unhealthy people purchase the optimal amount of insurance?
e. Given that each person has the option to purchase insurance, which type actually purchases insurance? What is the price of the insurance? Discuss the adverse selection problem.