Expected Utility: Practice Problems and Solutions

Mathematical Economics Problems and Solutions - Section 11

See Mr. Stolyarov's complete list of Mathematical Economics Problems and Solutions.

Note: Here, I will present solve problems typical of those offered in a mathematical economics or advanced microeconomics course. The problems were authored by Dr. Charles N. Steele and are reprinted with his generous permission. The solutions to the problems are my own work and not necessarily the only way to solve the problems.

Problem 1. Roger, a Von Neumann-Morgenstern utility maximizer, is planning a cross country trip. His utility from the trip is a function of how much of his cash Y he spends, and is given by

u(Y) = log Y (this is the base 10 logarithm) Assume Roger has $10000 to spend.

1a. What is his utility if he spends all his cash?

Solution 1a. Assume Roger has cash holdings equal to $Y. If he spends all his cash, Roger's utility will be log(Y) = log(10000) = 4

1b. Suppose there is a 25% probability he will lose $1000 during the trip. What is his expected utility?

Solution 1b. Roger has a 0.25 probability of spending Y - 1000 cash and a 0.75 probability of spending Y cash. Thus, his expected utility is

0.75log(Y) + 0.25log(Y - 1000) = 0.75log(10000) + 0.25log(9000) = 3.988560627

1c. Suppose Roger can buy insurance against this loss. What is the actuarially fair premium? Show that he will have higher utility with such insurance than without it.

Solution 1c. If Roger has a 0.25 probability of losing $1000, then Roger's expected loss is 0.25*1000 + 0.75*0 = $250. The actuarially fair premium should be equal to the expected loss, so the premium should be $250.

With this premium, Roger will have a guaranteed amount of cash to spend, equal to Y - 250, so his utility will be log(Y - 250) = log(10000 - 250) = log(9750) = 3.989004616.

Since 3.989004616 > 3.988560627, Roger's utility will be greater with insurance than without.

1d. What is the maximum he would be willing to pay for this insurance?