Chen's car is worth $5,000. But she is sometimes careless and leaves the door open and the keys in the ignition. Consequently her car will be stolen with probability 0.4. If it is stolen, she will never get it back. Chen also has $5,000 in the bank and her utility function for wealth is u(w) = ln(w).
(a) If Chen does not buy any insurance for her car, what is the expected monetary value of her wealth?
(b) What's her expected utility of having a car without insurance?
(c) What would be the price of a "Fair" full insurance plan? Would Chen buy full insurance for her car at this price?
(d) Suppose the insurance company knows Chen's preferences. What's the highest price the insurance company can charge Chen? Is Chen better-off at this new price than she is without full insurance?
(e) The insurance company is thinking about offering a partial insurance to Chen. Under this insurance option, the insurance company would charge Chen a premium of $1000, but it will cover only 50% of the value of the car ($2500) in case it is stolen. Will Chen prefer this insurance over the plan offered in part (d)? Will the insurance company make more profits under this contract?