4. [10] Jimmy's car is worth $100,000. But Jimmy is careless and leaves the top down and the keys in the ignition. Consequently, his car will be stolen with probability .5. If it is stolen, he will never get it back. Jimmy has $100,000 in other wealth and his von Neumann- Morgenstern utility function for wealth is u(w)= ln(w). Suppose that Jimmy can buy $K worth of insurance at a price of $.3K.

a) How much insurance will Jimmy buy?

b) Without solving, explain how you think this compares to the amount of insurance that he would buy if the price were actuarially fair.