Suppose people have a utility function of , where is consum…
Suppose people have a utility function of , where is consumption. There are two types of people: high-risk individuals have an 8% chance of getting sick, and low-risk individuals have a 2% chance of getting sick. High-risk individuals represent 55% of the population, and low-risk individuals are 45%. Without insurance, people would consume $50,000 if they are not sick and $10,000 if they are sick (because they have to pay for treatment). Finally, suppose an insurance company offers an insurance plan that pays a benefit equal to $40,000 if a person is sick and $0 if the person is not sick. (a) Assume the insurance company can observe who is high-risk and who is low-risk. Calculate the actuarially fair insurance premium for each type of person. (b) For each type of person, calculate expected utility if the person does not have insurance. (c) Suppose the government prohibits the insurance company from charging different amounts to different people. Instead, it must charge a single premium to everyone. Assuming for now that everyone buys insurance, what premium would allow the insurance company to break even? (d) Calculate the expected utility that each type of person would receive from buying insurance at the premium from part (c). Which type of person would want to buy insurance at this premium? (e) Where would the insurance market end up? I.e., who would buy the insurance plan, and at what premium? (f) Now suppose the insurance company decides to offer two plans. Both plans pay a benefit if a person is sick and nothing if the person is not sick. Plan A pays a benefit . It charges a premium equal to the one calculated in part (e). Plan B pays a benefit . It charges a premium of $300. Calculate the expected utility that each type of person would receive from buying Plan A. (g) For each type of person, calculate expected utility from buying Plan B. (h) Which plan will each type of person choose? (i) Given the type of person who buys Plan B, is the $300 premium actuarially fair? What does this say about firm profits?