Expected Utility: Practice 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?

Solution 1d. The maximum Roger would be willing to pay for this insurance will be the premium for which his expected utility will be 3.988560627 and his expected wealth will be 10^3.988560627 = 9740.037464. Thus, Roger will be willing to pay at most 10000 - 9740.037464 = $259.9625357

1e. Suppose people with insurance behave more recklessly than those who do not, and therefore have a probability of 30% of losing $1000. What is the actuarially fair premium? Will Roger buy such insurance?

Solution 1e. With insurance, Roger's expected wealth becomes prior to the compensation becomes 0.7Y + 0.3(Y - 1000), and the expected loss is 0.3*1000 = $300, so the actuarially fair premium is $300.

To see whether Roger would buy such insurance, we compare the expected utility derived from the insurance: log(Y - 300) = log(9700) = 3.986771734 to the expected utility without insurance: 3.988560627

Since 3.986771734 < 3.988560627, Roger will not buy the insurance.

Problem 2. The Hillsdale city council desires to reduce the number of incidences of illegal parking. They are attempting to decide between two mutually exclusive programs. The first would raise parking fines by 10%. The second would increase enforcement, so that the probability a lawbreaker is caught rises by 10%. True or false: if lawbreakers are risk averse, the 10% increase in fines will have a greater disincentive effect than the 10% increase in enforcement rates.

Solution 2. Assume that the initial fine is F and the initial probability of a lawbreaker getting caught is P.

Plan 1: When fines are increased by 10%, F becomes 1.1F.

Plan 2: When probability of a lawbreaker getting caught increases by 10%,

P becomes 1.1P.

Assume that a lawbreaker has an expected utility function for his wealth of u(W) if he does not have to pay any fines and has no probability of getting caught. This function has a positive decreasing slope, since the lawbreaker is risk-averse.

Original expected utility EU₀ was

equal to P*u(W - F) + (1-P)*u(W)

Under Plan 1, expected utility is EU₁ = P*u(W - 1.1F) + (1-P)*u(W)

Under Plan 2, expected utility is EU₂ = (1.1P)*u(W - F) + (1-1.1P)*u(W)

EU₀ - EU₁ = [P*u(W - F) + (1-P)*u(W)] - [P*u(W - 1.1F) + (1-P)*u(W)] =

EU₀ - EU₁ = P*u(W - F) - P*u(W - 1.1F)

EU₀ - EU₂ = [P*u(W - F) + (1-P)*u(W)] - [(1.1P)*u(W - F) + (1-1.1P)*u(W)] =

-0.1Pu(W - F) + 0.1Pu(W) = 0.1P[u(W) - u(W - F)]

Now we must resolve the question of which is greater,

P*u(W - F) - P*u(W - 1.1F), or 0.1P[u(W) - u(W - F)]?

u(W) can be any function with decreasing positive slope. For convenience, we can assume that u(W) = ln(W). This can be done without loss of generality, since the ordinal ranking of EU₀ - EU₁ and EU₀ - EU₂ will be the same for any function with decreasing positive slope.

Then P*u(W - F) - P*u(W - 1.1F) = Pln(W - F) - Pln(W - 1.1F) = P[ln(W - F) - ln(W - 1.1F)] = P[ln[(W-F)/(W - 1.1F)] = P[ln(1 + F/(W - 1.1F)]

0.1P[u(W) - u(W - F)] = 0.1P[ln(W) - ln(W - F)] = 0.1P[ln[(1 + F/(W-F)]]

We know that W - 1.1F < W-F, so F/(W - 1.1F) > F/(W-F) and

1 + F/(W - 1.1F) > 1 + F/(W-F). Thus,

ln[1 + F/(W - 1.1F)] > ln[(1 + F/(W-F)]

Of course, P > 0.1P, so

P*ln[1 + F/(W - 1.1F)] > 0.1P*ln[(1 + F/(W-F)]

and thus P*u(W - F) - P*u(W - 1.1F) > 0.1P[u(W) - u(W - F)], so

EU₀ - EU₁ > EU₀ - EU₂ and the lawbreaker has a greater decrease in expected utility from Plan 1 than from Plan 2. Hence, when probability of a lawbreaker getting caught increases by 10%, the lawbreaker has a smaller disutility than he gets from a 10% increase in fines. Hence, the original statement is true.

Problem 3. A risk averse man faces two possible states of the world. In state 1, his income is Y₁ and in state 2 his income is Y₂. The states occur with probability p and 1 - p respectively. True or false: His indifference curves between income in state 1 and income in state 2 are convex to origin.

Solution 3. Let u(W) be a function with decreasing positive slope representing the utility of wealth W for the risk-averse man.

Expected utility EU₁ = pu(Y₁) + (1-p)u(Y₂) for the situation where the risk-averse man faces uncertainty. If he were to get a guarantee of the expected value of this bet, then his expected utility would be EU₂ = u[pY₁ +(1-p)(Y₂)]

Since the man is risk-averse, for him EU₂ > EU₁, so

u[pY₁ +(1-p)(Y₂)] > pu(Y₁) + (1-p)u(Y₂)

This condition is the condition for strict concavity of a person's utility function.

However, we can graph this problem to show that this risk-averse man's indifference curves are convex.

Draw a set of axes with Y₁ on the horizontal axis and Y₂ on the vertical axis.

Label Y₂ /(1-p) on the vertical axis and Y₁/p on the horizontal axis.

Draw a line between those two points.

Then is a kind of "budget constraint" with slope -[Y₂ /(1-p)]/[Y₁/p] = -[p/(1-p)][Y₂/Y₁]

This is the line representing all possible lotteries with expected value (not expected utility) of pY₁ +(1-p)(Y₂).

The risk-averse individual will prefer the lottery which gives him pY₁ +(1-p)(Y₂) with certainty to all other points on this "budget constraint."This lottery occurs where Y₁ = Y₂ = pY₁ +(1-p)(Y₂). Draw a 45-degree line through the origin intersecting the "budget constraint." The intersection of those two lines will be the point of tangency of the person's indifference curve I₁ for this "budget constraint." We can call this point Q.

If point Q is preferred to all other points on the "budget constraint," then it follows that the indifference curve I₁ is higher than the "budget constraint" for all points other than Q. This means that I₁ has a convex-to-origin shape.

Thus, risk aversion implies convex indifference curves, and the statement is true.

Problem 4. Stephen has a VNM utility function u(x) = x^1/2, where x is his state-contingent wealth. His initial wealth is $160,000. He is considering buying fire insurance, because he faces a 0.05 chance of a small fire that would do $70,000 damage, and a 0.05 chance of a big fire that would do $120,000 damage. He can't suffer both types of fire, i.e. there's a 0.9 chance no fire occurs.

4a. What lottery does he face without insurance?

Solution 4a.

Stephen faces the following lottery: {160000, 90000, 40000; 0.9, 0.05, 0.05}

Without insurance, Stephen has the following expected wealth:

E(w) = 0.9*160000 + 0.05*90000 + 0.05*40000 = $150,500

Stephen has the following expected utility:

EU = 0.9*√(160000) + 0.05*√(90000) + 0.05*√(40000) =

EU_{no_insurance} = 385

4b. Suppose that fire insurance requires payment of a deductible of $7620 in the event of a fire. What lottery does Stephen face with insurance?

Solution 4b. The lottery Stephen faces with insurance (where P is the premium) is

{160000 - P, 160000 - P - 7620, 160000 - P - 7620; 0.9, 0.05, 0.05} =

{160000 - P, 152380 - P; 0.9, 0.1} That is, Stephen can expect to have 160000 - P in wealth with probability 0.9 and 152380 - P in wealth with probability 0.1.

4c. Suppose that fire insurance requires payment of a deductible of $7620 in the event of a fire. What is the most he will pay for full insurance?

Solution 4c. In the event of any fire, Stephen will have 160000 - 7620 = $152380 compensated to him by the insurance company - not counting the premium P. So his expected utility from this lottery will be

EU = 0.9*√(160000 - P) + 0.1√(152380 - P). This expected utility needs to be at least 385, the EU with no insurance. Thus, the maximum premium P* occurs where

385 = 0.9*√(160000 - P) + 0.1√(152380 - P)

Solving this equation by intersecting graphs on the TI-83 calculator,

we get P = $11004. Indeed, we have

0.9*√(160000 - 11004) + 0.1√(152380 - 11004) = 385

So the most Stephen will pay for full insurance is $11004

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