Consider an economy with three types of drivers: safe (s), inexperienced (i), and crazy (c). There is an equal number of each of these drivers, and their wealth when they do not get into an accident is $289 per person. In any given year, the probability that each type of driver gets into an accident is 0.4, 0.5 and 0.6 for type-s, i and c respectively. An accident leads to repair costs of $64, and has no other consequences. Each individuals utility only depends on his/her wealth (w). The utility of each individual is given by: U(w) = √w
and individuals seek to maximize the expectation of this utility function. An insurance company offers full insurance to these drivers: it offers a gross payment of $64 for the damage in case of an accident. The drivers decide whether to buy this full insurance or not; they cannot decide exactly how much insurance to buy. The insurance market is perfectly competitive, i.e., in equilibrium the insurance company earns zero profits.
(a) Suppose the insurance company can tell which type of driver a person is and o§ers an actuarially fair policy to each of them. Which of the drivers will buy the policy? Why? (You do not need to calculate the policy o§ered to each type of driver)
(b) Now suppose that the insurance company cannot distinguish the three types of drivers. There- fore, it has to o§er the policy at the same price to all three types. Assuming that all three types of drivers buy the policy, what would be its price? What is the wealth level of the agents in case of an accident? What is the wealth level of the drivers in case of no accident?
(c) Prove that with the price you found in part (b), safe drivers would not be willing to buy the policy.
(d) Since safe drivers do not buy the policy, the insurance company cannot o§er the policy at the price you found in part (b). Assuming that only inexperienced and crazy types buy the policy, what would be its price?
(e) Prove that with the price you found in part (d), inexperienced types would not be willing to buy the policy.
(f) What you have demonstrated is an example of market unraveling. Explain the intuition for why it happened.
(g) Discuss how the exercise above tells you about the role of individual mandate in the A§ordable Care Act?