CHAPTER 12: Uncertainty
TRUE/FALSE
1. Of any two gambles, no matter what their expected returns, a risk averter will choose the one with the
smaller variance.
2. An expected utility maximizer’s preferences between two bundles contingent on event 1 happening
must be independent of what he will get if event 2 happens.
3. If someone has strictly convex preferences between all contingent commodity bundles, then he or she
must be risk averse.
4. Wilma is not risk averse. She is offered a chance to pay $10 for a lottery ticket that will give her a
prize of $100 with probability .06, a prize of $50 with probability .1, and no prize with probability .85.
If she understands the odds and makes no mistakes in calculation, she will buy the lottery ticket.
5. If Paul is risk loving and his basketball team has a probability of .5 of winning, then Paul would rather
bet $10 on his team than $100. (When Paul bets X, he wins X if his team wins and loses X if his team
loses.)
6. If the price of insurance goes up, people will become less risk averse.
7. A consumer has a von Neumann-Morgenstern utility function of the form U(cA, cB, pA, pB) = pAv(cA) +
pBv(cB), where pA and pB are the probabilities of events A and B and where cA and cB are consumptions
contingent on events A and B respectively. This consumer must be a risk lover if v is an increasing
function.
MULTIPLE CHOICE
1. Prufrock is risk averse. He is offered a gamble in which with probability 1/4 he will lose $1,000 and
with probability 3/4, he will win $500.
a.
Since he is risk averse, he will certainly not take the gamble.
b.
Since the expected value of the gamble is positive, he will certainly take the gamble.
c.
If Prufrock’s initial wealth is greater than $1,500, he will certainly take the gamble.
d.
If Prufrock’s initial wealth is smaller than $1,500, he will certainly not take the gamble.
e.
Not enough information is given to determine for sure whether he will take the gamble.
2. Timmy Qualm’s uncle gave him a lottery ticket. With probability 1/2 the ticket will be worth $100 and
with probability 1/2 it will be worthless. Let x be Timmy’s wealth if the lottery ticket is a winner, and
y his wealth if it is a loser. Timmy’s preferences over alternative contingent commodity bundles are
represented by the utility function U(x, y) = min2x − y, 2y − x. He has no risks other than the ticket.
a.
Timmy would sell his lottery ticket for $25 but not for less.
b.
Timmy hates risk so much that he’d be willing to throw away the lottery ticket rather than
worry about whether he won.
c.
Timmy satisfies the expected utility hypothesis.
d.
Timmy is misnamed. He is a risk lover.
e.
None of the above.
3. There are two events, 1 and 2. The probability of event 1 is p and the probability of event 2 is 1 − p.
Sally Kink is an expected utility maximizer with a utility function is pu(c1) + (1 − p)u(c2), where for
any number x, u(x) = 2x if x 1,000 and u(x) = 1,000 + x if x is greater than or equal to 1,000.
a.
Sally is a risk lover.
b.
Sally will be a risk averter if she is poor but will be a risk lover if she is rich.
c.
Sally will be a risk lover if she is poor but a risk averter if she is rich.
d.
If there is no chance of her wealth exceeding $1,000, then she will take any bet that has
positive expected net winnings.
e.
None of the above.
4. Socrates owns just one ship. The ship is worth $200 million dollars. If the ship sinks, Socrates loses
$200 million. The probability that it will sink is .02. Socrates’ total wealth including the value of the
ship is $225 million. He is an expected utility maximizer with von Neuman-Morgenstern utility U(W)
equal to the square root of W. What is the maximum amount that Socrates would be willing to pay in
order to be fully insured against the risk of losing his ship?
a.
$4 million
b.
$2 million
c.
$3.84 million
d.
$4.82 million
e.
$5.96 million
5. Buck Columbus is thinking of starting a pinball palace near a large Midwestern university. Buck is an
expected utility maximizer with a von Neuman-Morgenstern utility function, U(W) = 1 − (6,000/W),
where W is his wealth. Buck’s total wealth is $24,000. With probability .2 the palace will be a failure
and he’ll lose $18,000, so that his wealth will be just $6,000. With probability .8 it will succeed and
his wealth will grow to $x. What is the smallest value of x that would be sufficient to make Buck want
to invest in the pinball palace rather than have a wealth of $24,000 with certainty?
a.
$28,500
b.
$150,000
c.
$96,000
d.
$72,000
e.
$30,000
6. Buck Columbus is thinking of starting a pinball palace near a large Midwestern university. Buck is an
expected utility maximizer with a von Neuman-Morgenstern utility function, U(W) = 1 − (3,000/W),
where W is his wealth. Buck’s total wealth is $12,000. With probability .2 the palace will be a failure
and he’ll lose $9,000, so that his wealth will be just $3,000. With probability .8 it will succeed and his
wealth will grow to $x. What is the smallest value of x that would be sufficient to make Buck want to
invest in the pinball palace rather than have a wealth of $12,000 with certainty?
a.
$75,000
b.
$14,250
c.
$36,000
d.
$48,000
e.
$15,000
7. Oskar’s preferences over gambles in which the probability of events 1 and 2 are both 1/2 can be
represented by the von Neuman-Morgenstern utility function .5y.51 + .5y.52, where y1 is his
consumption if event 1 happens and y2 is his consumption if event 2 happens. A gamble that allows
him a consumption of $9 if event 1 happens and $25 if event 2 happens is exactly as good for Oskar as
being sure to have an income of
a.
$12.5.
b.
$9.
c.
$16.
d.
$17.
e.
None of the above.
8. Mabel and Emil were contemplating marriage. They got to talking. Mabel said that she always acted
according to the expected utility hypothesis, where she tried to maximize the expected value of the log
of her income. Emil said that he too was an expected utility maximizer, but he tried to maximize the
expected value of the square of his income. Mabel said, “I fear we must part. Our attitudes toward risk
are too different.” Emil said, “Never fear, my dear, the square of income is a monotonic increasing
function of the log of income, so we really have the same preferences.” Who is right about whether
their preferences toward risk are different?
a.
Mabel is right.
b.
Emil is right.
c.
Emil is right about small risks but wrong about large risks.
d.
Mabel is right about small risks but wrong about large risks.
e.
They are both wrong.
9. Ronald has $18,000. But he is forced to bet it on the flip of a fair coin. If he wins he has $36,000. If he
loses he has nothing. Ronald’s expected utility function is .5x.5 + .5y.5, where x is his wealth if heads
comes up and y is his wealth if tails comes up. Since he must make this bet, he is exactly as well off as
if he had a perfectly safe income of
a.
$16,000.
b.
$15,000.
c.
$12,000.
d.
$11,000.
e.
$9,000.
10. Gary likes to gamble. Donna offers to bet him $70 on the outcome of a boat race. If Gary’s boat wins,
Donna would give him $70. If Gary’s boat does not win, Gary would give her $70. Gary’s utility
function is U(c1, c2, p1, p2) = p1c21 + p2c21, where p1 and p2 are the probabilities of events 1 and 2 and
where c1 and c2 are his consumption if events 1 and 2 occur respectively. Gary’s total wealth is
currently only $80 and he believes that the probability that he will win the race is .3.
a.
Taking the bet would increase his expected utility.
b.
Taking the bet would reduce his expected utility.
c.
Taking the bet would leave his expected utility unchanged.
d.
There is not enough information to determine whether taking the bet would increase or
decrease his expected utility.
e.
The information given in the problem is self-contradictory.
11. Clancy has $1,200. He plans to bet on a boxing match between Sullivan and Flanagan. For $4, he can
buy a coupon that pays $10 if Sullivan wins and nothing otherwise. For $6 he can buy a coupon that
will pay $10 if Flanagan wins and nothing otherwise. Clancy doesn’t agree with these odds. He thinks
that the two fighters each have a probability of 1/2 of winning. If he is an expected utility maximizer
who tries to maximize the expected value of lnW, where lnW is the natural log of his wealth, it would
be rational for him to buy
a.
50 Sullivan coupons and no Flanagan coupons.
b.
100 Sullivan coupons and no Flanagan coupons.
c.
50 Flanagan coupons and no Sullivan coupons.
d.
100 Flanagan coupons and no Sullivan coupons.
e.
100 of each kind of coupon.
12. Diego has $6,400. He plans to bet on a soccer game. Team A is a favorite to win. Assume no ties can
occur. For $.80 one can buy a ticket that will pay $1 if team A wins and nothing if B wins. For $.20
one can buy a ticket that pays $1 if team B wins and nothing if A wins. Diego thinks the two teams are
equally likely to win. He buys tickets so as to maximize the expected value of lnW (the natural log of
his wealth). After he buys his tickets, team A loses a star player and the ticket price moves to $.50 for
either team. Diego buys some new tickets and sells some of his old ones. The game is then played and
team A wins. How much wealth does he end up with?
a.
$5,000
b.
$15,000
c.
$6,400
d.
$8,400
e.
$10,000
13. Joe’s wealth is $100 and he is an expected utility maximizer with a von Neumann-Morgenstern utility
function U(W) = W1/2. Joe is afraid of oversleeping his economics exam. He figures there is only a 1 in
10 chance that he will, but if he does, it will cost him $100 in fees to the university for taking an exam
late. Joe’s neighbor, Mary, never oversleeps. She offers to wake him one hour before the test, but he
must pay her for this service. What is the most that Joe would be willing to pay for this wake-up
service?
a.
$10
b.
$15
c.
$19
d.
$100
e.
$50
14. Portia has waited a long time for her ship to come in, and she has concluded that it will arrive today
with probability 1/4. If it does come, she will receive $16. If it doesn’t come in today, it never will and
she will have zero wealth. She has a von Neumann-Morgenstern utility function equal to the square
root of her total income. What is the minimum price at which she would sell the rights to her ship?
a.
$1
b.
$2
c.
$2.50
d.
$4
e.
None of the above.
15. Harley’s current wealth is $600, but there is a .25 probability that he will lose $100. Harley is risk
neutral. He has an opportunity to buy insurance that would restore his $100 if he lost it.
a.
Harley would be willing to pay a bit more than $25 for this insurance.
b.
Harley would be willing to pay up to $25 for this insurance.
c.
Since Harley is risk neutral, he wouldn’t be willing to pay anything for this insurance.
d.
Since Harley’s utility function is not specified, we can’t tell how much he would be
willing to pay for this insurance.
e.
Harley would not be willing to pay more than $16.66 for this insurance.
16. After graduating, Sallie Handshake’s best job offer will either be with a Big-8 accounting firm for
$160,000 a year or as a State Farm agent in Grand Rapids, Michigan, for $40,000 a year. She can
increase the probability of the former outcome by studying more, but such studying has its costs. If S
represents her amount of studying (where S = 0 is no study and S = 1 is all-out effort), her probability
of getting the job with a Big-8 firm just equals S. Her utility depends on how hard she studies and her
subsequent annual income Y. She tries to maximize the expected value of the von
Neuman-Morgenstern utility function U(S, Y) = Y1/2 − 400S2. If she chooses S to maximize her
expected utility, how much will she study?
a.
S = 0.1.
b.
S = 0.25.
c.
S = 0.5.
d.
S = 0.75.
e.
S = 0.9.
17. Every $1 invested in Safe Sox will yield $2 for sure. Each $1 invested in Wobbly Umbrellas will yield
$8 with probability 1/2 and $0 with probability 1/2. An investor has $10,000 to invest in these two
companies and her von Neumann-Morgenstern utility function is the expected value of the natural
logarithm of the total yield on her investments. If S is the amount of money that she invests in Safe
Sox and $10,000 2 S is the amount that she invests in Wobbly Umbrellas, what should S be to
maximize her expected utility? (Pick the closest answer.)
a.
$1,111
b.
$3,333
c.
$5,000
d.
$6,667
e.
$9,111
18. Billy Pigskin from your workbook has a von Neumann-Morgenstern utility function U(c) = c1/2. If
Billy is not injured this season, he will receive an income of $16 million. If he is injured, his income
will be only $10,000. The probability that he will be injured is .1 and the probability that he will not be
injured is .9. His expected utility is
a.
3,610.
b.
between 15 million and 16 million.
c.
100,000.
d.
7,220.
e.
14,440.
19. Billy Pigskin from your workbook has a von Neumann-Morgenstern utility function U(c) = c1/2. If
Billy is not injured this season, he will receive an income of $25 million. If he is injured, his income
will be only $10,000. The probability that he will be injured is .1 and the probability that he will not be
injured is .9. His expected utility is
a.
100,000.
b.
9,020.
c.
between 24 million and 25 million.
d.
4,510.
e.
18,040.
20. Willy’s only source of wealth is his chocolate factory. He has the utility function pc1/2f +(1 − p)c1/2nf,
where p is the probability of a flood, 1 − p is the probability of no flood, and cf and cnf are his wealth
contingent on a flood and on no flood, respectively. The probability of a flood is p = 1/6. The value of
Willy’s factory is $500,000 if there is no flood and $0 if there is a flood. Willy can buy insurance
where if he buys $x worth of insurance, he must pay the insurance company $2x/17 whether there is a
flood or not but he gets back $x from the company if there is a flood. Willy should buy
a.
no insurance since the cost per dollar of insurance exceeds the probability of a flood.
b.
enough insurance so that if there is a flood, after he collects his insurance, his wealth will
be 1/4 of what it would be if there were no flood.
c.
enough insurance so that if there is a flood, after he collects his insurance, his wealth will
be the same whether there was a flood or not.
d.
enough insurance so that if there is a flood, after he collects his insurance, his wealth will
be 1/3 of what it would be if there were no flood.
e.
enough insurance so that if there is a flood, after he collects his insurance, his wealth will
be 1/5 of what it would be if there were no flood.
21. Willy’s only source of wealth is his chocolate factory. He has the utility function pc1/2f + (1 − p)c1/2nf,
where p is the probability of a flood, 1 − p is the probability of no flood, and cf and cnf are his wealth
contingent on a flood and on no flood, respectively. The probability of a flood is p = 1/4. The value of
Willy’s factory is $300,000 if there is no flood and $0 if there is a flood. Willy can buy insurance
where if he buys $x worth of insurance, he must pay the insurance company $5x/18 whether there is a
flood or not but he gets back $x from the company if there is a flood. Willy should buy
a.
enough insurance so that if there is a flood, after he collects his insurance, his wealth will
be 1/25 of what it would be if there were no flood.
b.
enough insurance so that if there is a flood, after he collects his insurance, his wealth will
be the same whether there was a flood or not.
c.
no insurance since the cost per dollar of insurance exceeds the probability of a flood.
d.
enough insurance so that if there is a flood, after he collects his insurance, his wealth will
be 1/6 of what it would be if there were no flood.
e.
enough insurance so that if there is a flood, after he collects his insurance his wealth will
be 1/11 of what it would be if there were no flood.
22. Sally Kink is an expected utility maximizer with utility function pu(c1) + (1 − p)u(c2), where for any x
$3,000, u(x) = 2x, and for x greater than or equal to $3,000, u(x) = 3,0001 + x.
a.
Sally will be risk averse if her income is less than $3,000 but risk loving if her income is
more than $3,000.
b.
Sally will be risk neutral if her income is less than $3,000 and risk averse if her income is
more than $3,000.
c.
For bets that involve no chance of her wealth exceeding $3,000, Sally will take any bet
that has a positive expected net payoff.
d.
Sally will never take a bet if there is a chance that it leaves her with wealth less than
$6,000.
e.
None of the above are true.
23. Sally Kink is an expected utility maximizer with utility function pu(c1) + (1 − p)u(c2), where for any x
$2,000, u(x) = 2x, and for x greater than or equal to $2,000, u(x) = 2,000 + x.
a.
For bets that involve no chance of her wealth exceeding $2,000, Sally will take any bet
that has a positive expected net payoff.
b.
Sally will be risk averse if her income is less than $2,000 but risk loving if her income is
more than $2,000.
c.
Sally will be risk neutral if her income is less than $2,000 and risk averse if her income is
more than $2,000.
d.
Sally will never take a bet if there is a chance that it leaves her with wealth less than
$4,000.
e.
None of the above are true.
24. Yoram’s expected utility function is pc1/21 + (1 − p)c1/22, where p is the probability that he consumes c1
and 1 − p is the probability that he consumes c2. Wilbur is offered a choice between getting a sure
payment of $Z or a lottery in which he receives $2,500 with probability .30 and $3,600 with
probability .70. Wilbur will choose the sure payment if
a.
Z 3,249 and the lottery if Z 3,249.
b.
Z 2,874.50 and the lottery if Z 2,874.50.
c.
Z 3,600 and the lottery if Z 3,600.
d.
Z 3,424.50 and the lottery if Z 3,424.50.
e.
Z 3,270 and the lottery if Z 3,270.
25. Quincy’s expected utility function is pc1/21 + (1 − p)c1/22, where p is the probability that he consumes c1
and 1 − p is the probability that he consumes c2. Wilbur is offered a choice between getting a sure
payment of $Z or a lottery in which he receives $3,600 with probability .60 and $12,100 with
probability .40. Wilbur will choose the sure payment if
a.
Z 6,400 and the lottery if Z 6,400.
b.
Z 12,100 and the lottery if Z 12,100.
c.
Z 9,250 and the lottery if Z 9,250.
d.
Z 5,000 and the lottery if Z 5,000.
e.
Z 7,000 and the lottery if Z 7,000.
26. Clancy has $1,800. He plans to bet on a boxing match between Sullivan and Flanagan. He finds that he
can buy coupons for $9 that will pay off $10 each if Sullivan wins. He also finds in another store some
coupons that will pay off $10 if Flanagan wins. The Flanagan tickets cost $1 each. Clancy believes that
the two fighters each have a probability of 1/2 of winning. Clancy is a risk averter who tries to
maximize the expected value of the natural log of his wealth. Which of the following strategies would
maximize his expected utility?
a.
Don’t gamble at all.
b.
Buy 100 Sullivan tickets and 900 Flanagan tickets.
c.
Buy exactly as many Flanagan tickets as Sullivan tickets.
d.
Buy 50 Sullivan tickets and 450 Flanagan tickets.
e.
Buy 50 Sullivan tickets and 900 Flanagan tickets.
27. Clancy has $5,000. He plans to bet on a boxing match between Sullivan and Flanagan. He finds that he
can buy coupons for $5 that will pay off $10 each if Sullivan wins. He also finds in another store some
coupons that will pay off $10 if Flanagan wins. The Flanagan tickets cost $5 each. Clancy believes that
the two fighters each have a probability of 1/2 of winning. Clancy is a risk averter who tries to
maximize the expected value of the natural log of his wealth. Which of the following strategies would
maximize his expected utility?
a.
Buy exactly as many Flanagan tickets as Sullivan tickets.
b.
Buy 250 Sullivan tickets and 250 Flanagan tickets.
c.
Don’t gamble at all.
d.
Buy 500 Sullivan tickets and 500 Flanagan tickets.
e.
Buy 250 Sullivan tickets and 500 Flanagan tickets.
28. Tom Cruiser’s car is worth $100,000. But Tom 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.
Tom has $100,000 in other wealth and his von Neumann-Morgenstern utility function for wealth is
u(w) = ln(w). Suppose that Tom can buy $K worth of insurance at a price of $.6K. How much
insurance will Tom buy?
a.
$0
b.
$100,000
c.
More than $0 but less than $50,000
d.
More than $50,000 but less than $100,000
e.
Exactly $50,000
PROBLEM
1. Gaston Gourmand loves good food. Due to an unusual ailment, he has a probability of 1/4 of losing his
sense of smell, which would greatly reduce his enjoyment of food. Gaston finds an insurance company
that will sell him insurance where Gaston gets $3x if he loses his smell and pays $x if he doesn’t. He
can also buy negative insurance, where Gaston pays $3x if he loses his sense of smell and gets $x if he
doesn’t. Gaston says, “Money will be only half as important to me if I lose my sense of smell.” If we
look at his expected utility function, we see what he means. Where c1 is his consumption if he retains
his sense of smell and c2 is his income if he loses his sense of smell, Gaston has the expected utility
function U(c1, c2) = 3/4 c1/21 + 1/8 c1/22. What insurance should he buy?
2. Oliver takes his wealth of $1,000 to a casino. He can bet as much as he likes on the toss of a coin but
the “house” takes a cut. If Oliver bets $x on heads, then if heads comes up, he gets $.8x and, if tails
comes up, he pays $x. Similarly if he bets $x on tails and if tails comes up, he wins $.8x and, if heads
comes up, he pays $x. Draw a graph with dollars contingent on heads and dollars contingent on tails on
the two axes. Show Oliver’s budget constraint. Oliver is an expected utility maximizer with the utility
function U(h, t) = 1/2h2 + 1/2t2, where h is his wealth if heads comes up and t is his wealth if tails
comes up. Draw the highest indifference curve that Oliver can reach with his budget. What bets if any
does he make?
3. Linus Piecewise is an expected utility maximizer. There are two events, H and T, which each have
100. Draw a graph showing the indifference curves for Linus that pass through
a. the point (50, 0)
b. the point (50, 100)
c. the point (100, 100)
d. the point (150, 100)
4. The certainty equivalent of a gamble is defined to be the amount of money which, if you were
promised it with certainty, would be indifferent to the gamble.
a. If an expected utility maximizer has a von Neuman-Morgenstern utility function U(W) = W1/2
(where W is wealth) and if the probability of events 1 and 2 are both 1/2, write a formula for the
certainty equivalent of a gamble that gives you x if event 1 happens and y if event 2 happens.
b. Generalize your formula in part (a) to the case where the probability of event 1 is p and the
probability of event 2 is 1 − p.
c. Generalize the formula in part (a) to the case where U(W) = W a for a 0.