Question 3 Suppose Smith is considering buying theft insurance for a car. Before he buys insurance, his wealth is 0 if there is a theft, and the value of the car, i.e., 12, otherwise. If Smith is careful where he parks, a theft occurs with probability 0.5. However, if he is careless, the probability rises to 0.75. Smith is risk averse and, other things equal, he has a mild preference to be careless, a preference worth only some small amount (close to zero) > 0 to him. The insurance company is risk neutral, and the market is competitive. The insurance company offers a contract C = (x, y), in which Smith pays premium x and receives compensation y if there is a theft.

(a) [15 marks] Derive the equilibrium contract C = (x, y) when the insurance company can observe whether Smith has been careful or careless. Illustrate using a state-space diagram.

(b) [20 marks] Suppose that the company cannot observe Smiths action (i.e., care is observable only to Smith). Determine graphically the equilibrium contract C = (x, y) under asymmetric information. Explain in a diagram the trade-off between efficient risk-sharing and incentives.