4. (25 points) Suppose that an agent is strictly risk averse and has an initial level of wealth y. His v-N.M utility function over money is My) = 1n 3;. The agent runs a risk of a loss of D dollars. The probability of loss is 7r. The agent can buy insurance however, and one unit of insurance costs It} and pays 13? if the loss occurs. Suppose that the agent decides to buy or units of insurance. (a) Set up the agent's utility maximization problem. {2 points) (b) Assume or > 0. Find the First Order Condition (F020) of optimization for the problem. (3 points) (c) Suppose that the insurance is such that q = 1r. Find the optimum level of a (10 points) ((1) (10 points) Suppose that the agent is insuring his car and once a week he goes to La Banquise to have poutine. There is no drive through. Moreover, if the agent actually stops the car and turns off the engine, his car must be pushed in order for for him to restart it which requires him some eort e. If he turns off the car, he has a probability of e) of his car being stolen while he is in the restaurant. If he does not spend the effort e, he has a probability of rm) of his car being stolen, where rr(0) > e). i. {3 points} Suppose that he is able to get the amount of insurance as in part (b), above. Show that he will always choose to leave his car running. ii. (7 points) Note that we have set, v(y) = la y. If the agent lets his car run while inside La Banquise, his utility is (1 ir(0))ln (y 0:11) + arm) ln (y m1 D + or) If the agent turns off his car, his utility is (1 - W())ln(y as) + e} 111((9 - m1 D + all