Chapter 5: Choice under Uncertainty
CHAPTER 5
CHOICE UNDER UNCERTAINTY
REVIEW QUESTIONS
1. What does it mean to say that a person is risk averse? Why are some people likely to be risk averse, while others are risk lovers?
A riskaverse person has a diminishing marginal utility of income and prefers a certain income to a gamble with the same expected income. A risk lover has an increasing marginal utility of income and prefers an uncertain income to a certain income. The economic explanation of whether an individual is risk averse or risk loving depends on the shape of the individual’s utility function for wealth. Also, a person’s risk aversion (or risk loving) depends on the nature of the risk involved and on the person’s income.
2. Why is the variance a better measure of variability than the range?
Range is the difference between the highest possible outcome and the lowest possible outcome. Range does not indicate the probabilities of observing these high or low outcomes. Variance weighs the difference of each outcome from the mean outcome by its probability and, thus, is a more useful measure of variability than the range.
3. What does it mean for consumers to maximize expected utility? Can you think of a case where a person might not maximize expected utility?
The expected utility is the sum of the utilities associated with all possible outcomes, weighted by the probability that each outcome will occur. To maximize expected utility means that the individual chooses the option that yields the highest average utility, where average utility is a probabilityweighted sum of all utilities. This theory requires that the consumer knows the probability of every outcome. At times, consumers either do not know the relevant probabilities or have difficulty in evaluating lowprobability, highpayoff events. In some cases, consumers cannot assign a utility level to these highpayoff events, such as when the payoff is the loss of the consumer’s life.
4. Why do people want to fully insure against uncertain situations when insurance is actuarially fair?
If the cost of insurance is equal to the expected loss, (i.e., if the insurance is actuarially fair), riskaverse individuals will fully insure against monetary loss. The insurance premium assures the individual of having the same income regardless of whether or not a loss occurs. Because the insurance is actuarially fair, this certain income is equal to the expected income if the individual takes the risky option of not purchasing insurance. This guarantee of the same income, whatever the outcome, generates more utility for a riskaverse person than the average utility of a high income when there was no loss and the utility of a low income with a loss (i.e., because of risk aversion, E[U(x)] U(E[x]).
5. Why is an insurance company likely to behave as if it is risk neutral even if its managers are riskaverse individuals? (Hint: How many projects are insured by an entire insurance company? How many does each individual manager deal with?)
Most large companies have opportunities for diversifying risk. Managers acting for the owners of a company choose a portfolio of independent, profitable projects at different levels of risk. Of course, shareholders may diversify their risk by investing in several projects in the same way that the insurance company itself diversifies risk by insuring many people. By operating on a sufficiently large scale, insurance companies can assure themselves that over many outcomes the total premiums paid to the company will be equal to the total amount of money paid out to compensate the losses of the insured. Thus, the insurance company behaves as if it is risk neutral, while the managers, as individuals, might be risk averse.
6. When is it worth paying to obtain more information to reduce uncertainty?
Individuals are willing to pay more for information when the utility of the choice with more information, including the cost of gathering the information, is greater than the expected utility of the choice without the information.
7. How does the diversification of an investor’s portfolio avoid risk?
An investor reduces risk by investing in many unrelated assets. For example, a mutual fund is a portfolio of stocks of independent companies. If the variance of the return on one company’s stock is unrelated to the variance of the return on another company’s stock, a portfolio of both stocks will have a lower variance than either stock held separately. As the number of stocks increases, the variance in the rate of return on the portfolio as a whole decreases. While there is less risk in a portfolio of stocks, risk is not eliminated altogether; there is still some market risk in holding such a portfolio, compared to a lowrisk asset, such as a U.S. government savings bond.
8. Why do some investors put a large portion of their portfolios into risky assets, while others invest largely in riskfree alternatives? (Hint: Do the two investors receive exactly the same return on average? Why?)
In a market for risky assets, where investors are risk averse, investors demand a higher return on investments that have a higher level of risk (a higher variance in returns). Although some individuals are willing to accept a higher level of risk in exchange for a higher rate of return, this does not mean that these individuals are less risk averse. On the contrary, they will not invest in risky assets unless they are compensated for the increased risk.
EXERCISES
1. Consider a lottery with three possible outcomes: $100 will be received with probability .1, $50 with probability .2, and $10 with probability .7.
a. What is the expected value of the lottery?
The expected value, EV, of the lottery is equal to the sum of the returns weighted by their probabilities:
EV = (0.1)($100) + (0.2)($50) + (0.7)($10) = $27.
b. What is the variance of the outcomes of the lottery?
The variance, ^{2}, is the sum of the squared deviations from the mean, $27, weighted by their probabilities:
^{2} = (0.1)(100  27)^{2} + (0.2)(50  27)^{2} + (0.7)(10  27)^{2} = $841.
c. What would a riskneutral person pay to play the lottery?
A riskneutral person would pay the expected value of the lottery: $27.
2. Suppose you have invested in a new computer company whose profitability depends on (1) whether the U.S. Congress passes a tariff that raises the cost of Japanese computers and (2) whether the U.S. economy grows slowly or quickly. What are the four mutually exclusive states of the world that you should be concerned about?
The four mutually exclusive states may be represented as:

Congress passes tariff

Congress does not pass tariff

Slow growth rate

State 1:
Slow growth with tariff

State 2:
Slow growth without tariff

Fast growth rate

State 3:
Fast growth with tariff

State 4:
Fast growth without tariff

3. Richard is deciding whether to buy a state lottery ticket. Each ticket costs $1, and the probability of the following winning payoffs is given as follows:
Probability

Return

0.50

$0.00

0.05

$7.50

a. What is the expected value of Richard’s payoff if he buys a lottery ticket? What is the variance?
The expected value of the lottery is equal to the sum of the returns weighted by their probabilities:
EV = (0.5)(0) + (0.25)($1.00) + (0.2)($2.00) + (0.05)($7.50) = $1.025
The variance is the sum of the squared deviation from the mean, $1.025, weighted by their probabilities:
^{2} = (0.5)(0  1.025)^{2} + (0.25)(1  1.025)^{2} + (0.2)(2  1.025)^{2} + (0.05)(7.5  1.025)^{2}, or
^{2} = $2.812.
b. Richard’s nickname is “Norisk Rick.” He is an extremely riskaverse individual. Would he buy the ticket?
An extremely riskaverse individual will probably not buy the ticket, even though the expected outcome is higher than the price, $1.025 > $1.00. The difference in the expected return is not enough to compensate Rick for the risk. For example, if his wealth is $10 and he buys a $1.00 ticket, he would have $9.00, $10.00, $11.00, and $16.50, respectively, under the four possible outcomes. Let us assume that his utility function is U = W^{0.5}, where W is his wealth. Then his expected utility is:
This is less than 3.162, which is the utility associated with not buying the ticket
(U(10) = 10^{0.5} = 3.162). He would prefer the sure thing, i.e., $10.
c. Suppose Richard was offered insurance against losing any money. If he buys 1,000 lottery tickets, how much would he be willing to pay to insure his gamble?
If Rick buys 1,000 tickets, it is likely that he will have $1,025 minus the $1,000 he paid, or $25. He would not buy any insurance, as the expected return, $1,025, is greater than the cost, $1,000. He has insured himself by buying a large number of tickets.
d. In the long run, given the price of the lottery ticket and the probability/return table, what do you think the state would do about the lottery?
In the long run, the state lottery will be bankrupt! Given the price of the ticket and the probabilities, the lottery is a money loser. The state must either raise the price of a ticket or lower the probability of positive payoffs.
4. Suppose an investor is concerned about a business choice in which there are three prospects, whose probability and returns are given below:
Probability

Return

0.2

$100

0.4

25

What is the expected value of the uncertain investment? What is the variance?
The expected value of the return on this investment is
EV = (0.2)(100) + (0.4)(50) + (0.4)(25) = $30.
The variance is
^{2} = (0.2)(100  30)^{2} + (0.4)(50  30)^{2} + (0.4)(25  30)^{2} = $2,350.
5. You are an insurance agent who has to write a policy for a new client named Sam. His company, Society for Creative Alternatives to Mayonnaise (SCAM), is working on a lowfat, lowcholesterol mayonnaise substitute for the sandwich condiment industry. The sandwich industry will pay top dollar to whoever invents such a mayonnaise substitute first. Sam’s SCAM seems like a very risky proposition to you. You have calculated his possible returns table as follows.
Probability

Return


.999

$1,000,000

(he fails)

.001

$1,000,000,000

(he succeeds and
sells the formula)

a. What is the expected return of his project? What is the variance?
The expected return, ER, of the investment is
ER = (0.999)(1,000,000) + (0.001)(1,000,000,000) = $1,000.
The variance is
^{2} = (0.999)(1,000,000  1,000)^{2} + (0.001)(1,000,000,000  1,000)^{2} , or
^{2} = 1,000,998,999,000,000.
b. What is the most Sam is willing to pay for insurance? Assume Sam is risk neutral.
Because Sam is risk neutral and because the expected outcome is $1,000, Sam is unwilling to buy insurance.
c. Suppose you found out that the Japanese are on the verge of introducing their own mayonnaise substitute next month. Sam does not know this and has just turned down your final offer of $1,000 for the insurance. If Sam tells you his SCAM is only six months away from perfecting his mayonnaise substitute and knowing what you know about the Japanese, would you raise or lower your policy premium on any subsequent proposal to Sam? Based on his information, would Sam accept?
The entry of the Japanese lowers Sam’s probability of a high payoff. For example, assume that the probability of the billion dollar payoff is lowered to 0.0001. Then the expected outcome is:
(0.9999)($1,000,000) + (0.0001)(($1,000,000,000) = $899,900.
Therefore, you should raise the policy premium substantially. But Sam, not knowing about the Japanese entry, will continue to refuse your offers to insure his losses.
6. Suppose that Natasha’s utility function is given by u(I) = y^{0.5}, where I represents annual income in thousands of dollars.
a. Is Natasha risk loving, risk neutral, or risk averse? Explain.
Natasha is risk averse. To show this, assume that she has $10,000 and is offered a gamble of a $1,000 gain with 50 percent probability and a $1,000 loss with 50 percent probability. Her utility of $10,000 is 3.162, (u(I) = 10^{0.5} = 3.162). Her expected utility is:
EU = (0.5)(9^{0.5 }) + (0.5)(11^{0.5 }) = 3.158 < 3.162.
She would avoid the gamble. If she were risk neutral, she would be indifferent between the $10,000 and the gamble; whereas, if she were risk loving, she would prefer the gamble.
You can also see that she is risk averse by plotting the function for a few values (see Figure 5.6.a) and noting that it displays a diminishing marginal utility. (Or, note that the second derivative is negative, again implying diminishing marginal utility.)
Figure 5.6
b. Suppose that Natasha is currently earning an income of $10,000 (I = 10) and can earn that income next year with certainty. She is offered a chance to take a new job that offers a .5 probability of earning $16,000, and a .5 probability of earning $5,000. Should she take the new job?
The utility of her current salary is 10^{0.5}, which is 3.162. The expected utility of the new job is
EU = (0.5)(5^{0.5 }) + (0.5)(16^{0.5 }) = 3.118,
which is less than 3.162. Therefore, she should not take the job.
c. In (b), would Natasha be willing to buy insurance to protect against the variable income associated with the new job? If so, how much would she be willing to pay for that insurance? (Hint: What is the risk premium?)
Assuming that she takes the new job, Natasha would be willing to pay a risk premium equal to the difference between $10,000 and the utility of the gamble so as to ensure that she obtains a level of utility equal to 3.162. We know the utility of the gamble is equal to 3.118. Substituting into her utility function we have, 3.118 = y^{0.5}, and solving for y we find the income associated with the gamble to be $9,722. Thus, Natasha would be willing to pay for insurance equal to the risk premium, $10,000  $9,722 = $278.
7. Draw a utility function over income u(I) that has the property that a man is a risk lover when his income is low but a risk averter when his income is high. Can you explain why such a utility function might reasonably describe a person’s tastes?
Consider an individual who needs a certain level of income, I*, in order to stay alive. An increase in income above I* will have a diminishing marginal utility. Below I*, the individual will be a risk lover and will take unfair gambles in an effort to make large gains in income. Above I*, the individual will purchase insurance against losses.
Figure 5.7
8. A city is considering how much to spend monitoring parking meters. The following information is available to the city manager:
i. Hiring each metermonitor costs $10,000 per year.
ii. With one monitoring person hired, the probability of a driver getting a ticket each time he or she parks illegally is equal to .25.
iii. With two monitors hired, the probability of getting a ticket is .5, with three monitors the probability is .75, and with four the probability is equal to 1.
iv. The current fine for overtime parking with two metering persons hired is $20.
a. Assume first that all drivers are riskneutral. What parking fine would you levy and how many meter monitors would you hire (1, 2, 3, or 4) to achieve the current level of deterrence against illegal parking at the minimum cost?
If drivers are risk neutral, their behavior is only influenced by the expected fine. With two metermonitors, the probability of detection is 0.5 and the fine is $20. So, the expected fine is $10 = (0.5)($20). To maintain this expected fine, the city can hire one metermonitor and increase the fine to $40, or hire three metermonitors and decrease the fine to $13.33, or hire four metermonitors and decrease the fine to $10.
If the only cost to be minimized is the cost of hiring metermonitors, i.e., $10,000 per year, you as the city manager, should minimize the number of metermonitors. Hire only one monitor and increase the fine to $40 to maintain the current level of deterrence.
b. Now assume that drivers are very risk averse. How would your answer to (a) change?
If drivers are risk averse, their utility of a certain outcome is greater than their utility of an expected value equal to the certain outcome. They will avoid the possibility of paying a parking fine more than would riskneutral drivers. Therefore, a fine of less than $40 will maintain the current level of deterrence.
c. (For discussion) What if drivers could insure themselves against the risk of parking fines? Would it make good public policy to allow such insurance to be available?
Drivers engage in many forms of behavior to insure themselves against the risk of parking fines, such as parking blocks away from their destination in a nonmetered spot or taking public transportation. A private insurance firm could offer an insurance policy to pay fines if a ticket is received. Of course, the premium for such insurance would be based on each driver’s probability of receiving a parking ticket and on the opportunity cost of providing service. (Note: full insurance leads to moral hazard problems, to be discussed in Chapter 17.)
Public policy should attempt to maximize the difference between the benefits and costs to all parties. Private insurance may not be optimal, because of the increase in transactions costs. Instead, as the city manager, consider offering another form of insurance, e.g., the selling of parking stickers, and give tickets for inappropriately parked cars.
