# Which of the three contracts gives the insurance company the highest expected proÖts?

The driver of a car can exert e§ort to avoid an accident (e = 1) or not exert any e§ort (e = 0). If e = 1, the probability of an accident is 1=2. If e = 0, the probability of an accident is 1. The driverís wealth without accident is: w = 100. In case of an accident, the repair of the car costs 64. So, if there is an accident, the driver has w = 100 64 = 36 left. The driverís utility of wealth is p w, that is, the driver is risk averse. The cost of e§ort, C(e), are 0 if e§ort is e = 0, and 1 if e§ort is e = 1. The driverís von Neumann Morgenstern utility function is: u(w; e) = p w C(e). (a) Will the driver choose to exert e§ort? Compare the expected utility of the driver when exerting e§ort (e = 1) with the expected utility when exerting no e§ort (e = 0). 1 (b) Now suppose there is a risk neutral insurance company. This insurance company acts like a principal with the driver being the agent. Suppose the insurance company cannot monitor the driverís behavior. The insurance company considers three contracts, labeled A, B and C. Each contract speciÖes the price p and the amount of money the driver gets in case of an accident, d. Given p and d, the Önal wealth of the driver in case of no accident is w0 = 100 p and the Önal wealth in case of an accident is wA = 36 p + d. The contracts are: Contract Price p Payment d A 36 64 B 19 47 C 19 32 For each of the contracts, calculate the Önal wealths, w0 and wA, in the two outcomes, and list them in a table of the following type: Contract w0 wA A B C Which of these contracts o§ers full insurance to the driver? (c) For each of these contracts, determine which of the two e§ort levels, e = 0 or e = 1, would be expected utility maximizing for the driver if he accepted that contract. Assume that the driver, if both e§ort levels yield the same expected utility, chooses e = 1. (d) Which of these contracts are such that the driver would accept the contract rather than staying uninsured? (For each contract, compare the expected utility from being uninsured with the expected utility when having the contract and choosing the optimal e§ort level. Assume that the driver accepts a contract if indi§erent between insuring and not insuring.) (e) Which of the three contracts gives the insurance company the highest expected proÖts? What are the expected proÖts of the insurance company if it o§ers this contract?

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