Mr. Fogg is planning an around-the-world trip. The utility from the trip is a function of how much he spends on it (Y) given by

U(Y) = log Y

Mr. Fogg has $10,000 to spend on the trip. If he spends all of it, his utility will be

U (10,000) = log 10,000 = 4

(In this problem, we are using logarithms to the base 10 for ease of computation.)

a. If there is a 25 percent probability that Mr. Fogg will lose $1,000 of his cash on the trip, what is the trip’s expected utility?

b. Suppose that Mr. Fogg can buy insurance against losing the $1,000 (say, by purchasing traveler’s checks) at an actuarially fair premium of $250. Show that his utility is higher if he purchases this insurance than if he faces the chance of losing the $1,000 without insurance.

c. What is the maximum amount that Mr. Fogg would be willing to pay to insure his $1,000?

d. Suppose that people who buy insurance tend to become more careless with their cash than those who do not, and assume that the probability of their losing $1,000 is 30 percent. What will be the actuarially fair insurance premium? Will Mr. Fogg buy insurance in this situation?