CHAPTER 12: Uncertainty
MULTIPLE CHOICE
1. In Problem 9, Billy 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 dollars. If he is injured, his income will be only
10,000 dollars. 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.
4,510 dollars.
b.
between 24 and 25 million dollars.
c.
100,000 dollars.
d.
9,020 dollars.
e.
18,040 dollars.
2. In Problem 9, Billy has a von Neumann-Morgenstern utility function U(c) = c1/2. If Billy is not injured
this season, he will receive an income of 4 million dollars. If he is injured, his income will be only
10,000 dollars. 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.
3,620 dollars.
b.
1,810 dollars.
c.
between 3 and 4 million dollars.
d.
100,000 dollars.
e.
7,240 dollars.
3. In Problem 9, Billy has a von Neumann-Morgenstern utility function U(c) = c1/2. If Billy is not injured
this season, he will receive an income of 4 million dollars. If he is injured, his income will be only
10,000 dollars. 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 dollars.
b.
1,810 dollars.
c.
3,620 dollars.
d.
between 3 and 4 million dollars.
e.
7,240 dollars.
4. In Problem 9, Billy 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 dollars. If he is injured, his income will be only
10,000 dollars. 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 dollars.
b.
7,220 dollars.
c.
100,000 dollars.
d.
between 15 and 16 million dollars.
e.
14,440 dollars.
5. In Problem 9, Billy has a von Neumann-Morgenstern utility function U(c) = c1/2. If Billy is not injured
this season, he will receive an income of 4 million dollars. If he is injured, his income will be only
10,000 dollars. 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.
3,620 dollars.
b.
1,810 dollars.
c.
100,000 dollars.
d.
between 3 and 4 million dollars.
e.
7,240 dollars.
6. (See Problem 2.) 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 =
1/13. 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 $3x/15
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 1/9 of what it would be if there is 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 is a flood or not.
d.
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 is no flood.
e.
enough insurance so that if there is a flood, after he collects his insurance, his wealth will
be 1/7 of what it would be if there is no flood.
7. (See Problem 2.) 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 =
1/14. 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 $4x/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.
enough insurance so that if there is a flood, after he collects his insurance, his wealth will
be the same whether there is a flood or not.
b.
enough insurance so that if there is a flood, after he collects his insurance, his wealth will
be 1/16 of what it would be if there is no flood.
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 1/5 of what it would be if there is no flood.
e.
enough insurance so that if there is a flood, after he collects his insurance, his wealth will
be 1/9 of what it would be if there is no flood.
8. (See Problem 2.) 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 =
1/20. 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 $4x/23
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 1/5 of what it would be if there is no flood.
b.
enough insurance so that if there is a flood, after he collects his insurance, his wealth will
be 1/16 of what it would be if there is no flood.
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 the same whether there is a flood or not.
e.
enough insurance so that if there is a flood, after he collects his insurance, his wealth will
be 1/9 of what it would be if there is no flood.
9. (See Problem 2.) 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 =
1/11. The value of Willy’s factory is $800,000 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 $4/4x
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 the same whether there is a flood or not.
b.
no insurance since the cost per dollar of insurance exceeds the probability of a flood.
c.
enough insurance so that if there is a flood, after he collects his insurance, his wealth will
be 1/16 of what it would be if there is no flood.
d.
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 is no flood.
e.
enough insurance so that if there is a flood, after he collects his insurance, his wealth will
be 1/9 of what it would be if there is no flood.
10. (See Problem 2.) 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 =
1/14. The value of Willy’s factory is $400,000 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 the same whether there is a flood or not.
b.
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 is no flood.
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 1/25 of what it would be if there is 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 is no flood.
11. Sally Kink is an expected utility maximizer with utility function pu(c1) + (1 − p)u(c2), where for any x
6,000, u(x) = 2x, and for x greater than or equal to 6,000, u(x) = 12,000 + x.
a.
Sally will be risk averse if her income is less than $6,000 but risk loving if her income is
more than $6,000.
b.
Sally will be risk neutral if her income is less than $6,000 and risk averse if her income is
more than $6,000.
c.
For bets that involve no chance of her wealth exceeding $6,000, 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
$12,000.
e.
None of the above.
12. Sally Kink is an expected utility maximizer with utility function pu(c1) + (1 − p)u(c2), where for any x
6,000, u(x) = 2x, and for x greater than or equal to 6,000, u(x) = 12,000 + x.
a.
Sally will be risk neutral if her income is less than $6,000 and risk averse if her income is
more than $6,000.
b.
Sally will be risk averse if her income is less than $6,000 but risk loving if her income is
more than $6,000.
c.
For bets that involve no chance of her wealth exceeding $6,000, 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
$12,000.
e.
None of the above.
13. Sally Kink is an expected utility maximizer with utility function pu(c1) + (1 − p)u(c2), where for any x
8,000, u(x) = 2x, and for x greater than or equal to 8,000, u(x) = 16,000 + x.
a.
Sally will never take a bet if there is a chance that it leaves her with wealth less than
$16,000.
b.
Sally will be risk neutral if her income is less than $8,000 and risk averse if her income is
more than $8,000.
c.
Sally will be risk averse if her income is less than $8,000 but risk loving if her income is
more than $8,000.
d.
For bets that involve no chance of her wealth exceeding $8,000, Sally will take any bet
that has a positive expected net payoff.
e.
None of the above.
14. Sally Kink is an expected utility maximizer with utility function pu(c1) + (1 − p)u(c2), where for any x
7,000, u(x) = 2x, and for x greater than or equal to 7,000, u(x) = 14,000 + x.
a.
Sally will be risk averse if her income is less than $7,000 but risk loving if her income is
more than $7,000.
b.
Sally will be risk neutral if her income is less than $7,000 and risk averse if her income is
more than $7,000.
c.
Sally will never take a bet if there is a chance that it leaves her with wealth less than
$14,000.
d.
For bets that involve no chance of her wealth exceeding $7,000, Sally will take any bet
that has a positive expected net payoff.
e.
None of the above.
15. Sally Kink is an expected utility maximizer with utility function pu(c1) + (1 − p)u(c2), where for any x
1,000, u(x) = 2x, and for x greater than or equal to 1,000, u(x) = 2,000 + x.
a.
Sally will be risk averse if her income is less than $1,000 but risk loving if her income is
more than $1,000.
b.
Sally will never take a bet if there is a chance that it leaves her with wealth less than
$2,000.
c.
Sally will be risk neutral if her income is less than $1,000 and risk averse if her income is
more than $1,000.
d.
For bets that involve no chance of her wealth exceeding $1,000, Sally will take any bet
that has a positive expected net payoff.
e.
None of the above.
16. (See Problem 11.) Jonas’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. Jonas is offered a choice between
getting a sure payment of $Z or a lottery in which he receives $3,600 with probability .10 or $6,400
with probability .90. Jonas will choose the sure payment if
a.
Z 6,084 and the lottery if Z 6,084.
b.
Z 4,842 and the lottery if Z 4,842.
c.
Z 6,400 and the lottery if Z 6,400.
d.
Z 6,242 and the lottery if Z 6,242.
e.
Z 6,120 and the lottery if Z 6,120.
17. (See Problem 11.) Albert’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. Albert is offered a choice between
getting a sure payment of $Z or a lottery in which he receives $400 with probability .30 or $2,500 with
probability .70. Albert will choose the sure payment if
a.
Z 2,090.50 and the lottery if Z 2,090.50.
b.
Z 1,040.50 and the lottery if Z 1,040.50.
c.
Z 2,500 and the lottery if Z 2,500.
d.
Z 1,681 and the lottery if Z 1,681.
e.
Z 1,870 and the lottery if Z 1,870.
18. (See Problem 11.) Pete’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. Pete is offered a choice between
getting a sure payment of $Z or a lottery in which he receives $1,600 with probability .80 or $14,400
with probability .20. Pete will choose the sure payment if
a.
Z 3,136 and the lottery if Z 3,136.
b.
Z 8,768 and the lottery if Z 8,768.
c.
Z 14,400 and the lottery if Z 14,400.
d.
Z 2,368 and the lottery if Z 2,368.
e.
Z 4,160 and the lottery if Z 4,160.
19. (See Problem 11.) Lawrence’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. Lawrence is offered a
choice between getting a sure payment of $Z or a lottery in which he receives $400 with probability
.30 or $2,500 with probability .70. Lawrence will choose the sure payment if
a.
Z 1,040.50 and the lottery if Z 1,040.50.
b.
Z 2,500 and the lottery if Z 2,500.
c.
Z 2,090.50 and the lottery if Z 2,090.50.
d.
Z 1,681 and the lottery if Z 1,681.
e.
Z 1,870 and the lottery if Z 1,870.
20. (See Problem 11.) Wilfred’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. Wilfred is offered a choice
between getting a sure payment of $Z or a lottery in which he receives $2,500 with probability .40 or
$6,400 with probability .60. Wilfred will choose the sure payment if
a.
Z 4,624 and the lottery if Z 4,624.
b.
Z 3,562 and the lottery if Z 3,562.
c.
Z 5,512 and the lottery if Z 5,512.
d.
Z 6,400 and the lottery if Z 6,400.
e.
Z 4,840 and the lottery if Z 4,840.
21. 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 each 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.
22. Clancy has $1,800. He plans to bet on a boxing match between Sullivan and Flanagan. He finds that he
can buy coupons for $1 each 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 $9 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 450 Sullivan tickets and 50 Flanagan tickets.
c.
Buy exactly as many Flanagan tickets as Sullivan tickets.
d.
Buy 900 Sullivan tickets and 100 Flanagan tickets.
e.
Buy 450 Sullivan tickets and 100 Flanagan tickets.
23. Clancy has $4,800. He plans to bet on a boxing match between Sullivan and Flanagan. He finds that he
can buy coupons for $4 each 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 $6 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 300 Sullivan tickets and 200 Flanagan tickets.
c.
Buy exactly as many Flanagan tickets as Sullivan tickets.
d.
Buy 600 Sullivan tickets and 400 Flanagan tickets.
e.
Buy 300 Sullivan tickets and 400 Flanagan tickets.
24. Clancy has $4,200. He plans to bet on a boxing match between Sullivan and Flanagan. He finds that he
can buy coupons for $7 each 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 $3 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 150 Sullivan tickets and 350 Flanagan tickets.
c.
Buy 300 Sullivan tickets and 700 Flanagan tickets.
d.
Buy exactly as many Flanagan tickets as Sullivan tickets.
e.
Buy 150 Sullivan tickets and 700 Flanagan tickets.