### Which of the drivers will buy the policy

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##### Reference no: EM13183385

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 \$324 per person. In any given year, the probability that each type of driver gets into an accident is: .3 for safe drivers, .5 for inexperienced drivers, and .7 for crazy drivers. An accident leads to repair costs of \$100.

Each individual's utility only depends on her wealth (w): 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 \$100 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: the insurance company earns zero profits.

a) Suppose the insurance company knows which type of driver a person is and offers an actuarially fair policy to each of them. Which of the drivers will buy the policy? Why? [Note: you do not need to calculate the policy offered to each type of driver]

b) Now suppose that the insurance company cannot distinguish the three types of drivers. Therefore, it has to offer 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 offer 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

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