Jack's truck is worth $50,000. But he is careless and leaves the key on a table. Consequently his car will be stolen with probability 0.4. If it is stolen, he will never get it back. Jack also earns $50,000 from the company for which he is working, and his Bernoulli utility function for wealth is u(w) = ln(w).

Suppose that he can buy $K worth of insurance for his car at a price of $4K. Let w1 and w2 denote his wealth level when the car is stolen and not stolen, respectively.

(a) What is Jack's utility function considering the uncertainty involved in his car? (b) If Jack buys $K worth of insurance, how much of w1 and w2 does he end up with? (c) What is Jack's optimal level of wealths, w1 and w2? How much does he pay for the insurance he buys at the optimum? How much on average will the insurance company pay back to Jack?