Economics Question

Problem 1: Health Insurance. Every individual has $5,000 to spend. There are 100 healthy type and 100 sick type individuals. Both types needs a minor treatment with probability Pm = 0.2. Any minor treatment costs $300. Healthy people need surgery treat- ment with probability p, = 0.1, while sick people need surgery treatment with probability ps = 0.15. Any surgery treatment costs $2,000. All individuals have the following expected utility function: EU = (1 - Pm - Ps) VCn + Pm VC'm + Ps VC's where C, represent consumption if no treatment is needed, Cmrepresent consumption if a minor treatment is needed, and Cs represent consumption if surgery treatment is needed. (Note: Individuals require at most one treatment.) 1. What is the actuarially fair price of full insurance that covers both treatment types for a healthy person? For a sick person? (The insurer has to pay for minor treatments with probability pm and surgery treatments with probability p,). 2. Suppose the insurance company cannot distinguish between healthy and sick people, so the insurance company decides to charge one price to everyone for full insurance. a) At what price x does the insurance company make zero expected profits if everyone buys? b) If the insurance company charges this price to everyone for full insurance, who will buy full insurance? Who will not? 3. Instead, suppose that the insurance company sells two packages: 1. Full insurance at premium pi 2. Partial insurance at premium py insurance company pays $300 for both treatments a) If the insurance company is attempting to create a separating equilibrium, what prices p, and p2 will the insurance company choose? b) At these prices, will a healthy person buy none, partial, or full insurance? Will a sick person buy none, partial, or full insurance? c) Is this a successful separating equilibrium? Briefly explain