EC 370
Practice Questions: Chapter 12 - Uncertainty
ANSWER KEY
Multiple-Choice Questions
√
1. Billy has a von Neumann-Morgenstern utility function U (c) = c. 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 0.1 and the probability that he will
not be injured is 0.9. His expected utility is
(a) 3,620 utility units
(b) 1,810 utility units
(c) 100,000 utility units
(d) between 3 and 4 million utility units
(e) 7,240 utility units
Answer: B
2. Based on (1), what is Billy’s expected income?
(a) $3620
(b) $1820
(c) $100,000
(d) between 3 and 4 million dollars
(e) $7240
Answer: D
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3. Willy’s only source of wealth is his chocolate factory. He has the utility function given by
√
√
p cf + (1 − p) cnf ,
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
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company $ 44
x whether there is a flood or not, but he gets back $x from the company if there
is a flood. Willy should buy
(a)
(b)
(c)
(d)
(e)
no insurance.
insurance so that c∗n > c∗nf .
insurance so that c∗n = c∗nf
insurance so that c∗n < c∗nf
There is not enough information to determine whether Willy should buy insurance or
not.
Answer: C
4. Sally Kink is an expected utility maximizer with utility function EU = 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) = 7, 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.
Answer: D
√
√
5. Lawrence’s expected utility function is p c1 + (1 − p) c2 , 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
0.30 or $2,500 with probability 0.70. Lawrence will choose the sure payment if
(a)
(b)
(c)
(d)
(e)
Z
Z
Z
Z
Z
> 1, 040.50 and the lottery if Z < 1, 040.50.
> 2, 500 and the lottery if Z < 2, 500.
> 2, 090.50 and the lottery if Z < 2, 090.50.
> 1, 681 and the lottery if Z < 1, 681.
> 1, 870 and the lottery if Z < 1, 870.
Answer: D
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Short-Answer Questions
Q1 – Q2 are taken from the Varian’s Workout (with some modifications).
1. Willy owns a small chocolate factory, located close to a river that occasionally floods in the
spring, with disastrous consequences. Next summer, Willy plans to sell the factory and retire.
The only income he will have is the proceeds of the sale of his factory. If there is no flood, the
factory will be worth $500,000. If there is a flood, then what is left of the factory will be worth
only $50,000. Willy can buy flood insurance at a cost of $0.10 for each $1 worth of coverage.
Willy thinks that the probability that there will be a flood this spring is 1/10. Let cF denote
the contingent commodity dollars if there is a flood and cN F denote dollars if there is no flood.
√
√
Willy’s von Neumann-Morgenstern utility function is EU (cF , cN F ) = 0.1 cF + 0.9 cN F .
(a) If he buys no insurance, then in each contingency, Willy’s consumption will equal the
value of his factory, so Willy’s contingent commodity bundle will be (cF , cN F ) =?
(cF , cN F ) = ($50, 000, $500, 000)
(b) To buy insurance that pays him $x in case of a flood, Willy must pay an insurance
premium of $0.1x. (The insurance premium must be paid whether or not there is a
flood.) If Willy insures for $x, then if there is a flood, he gets $x in insurance benefits.
Suppose that Willy has contracted for insurance that pays him $x in the event of a flood.
Then after paying his insurance premium, he will be able to consume cF =? and cN F =?
cF = 50, 000 − 0.1x + x
cN F = 500, 000 − 0.1x
(c) What is his marginal rate of substitution (MRS)?
πF M U (cF )
πN F M U (cN F )
0.1M U (cF )
=−
0.9M U (cN F )
M RS = −
0.1(0.5c−0.5
)
F
−0.5
0.9(0.5cN F )
√
0.1 cN F
=−
√
0.9 cF
=−
(d) What is the slope of his budget line?
−
γ
0.1
1
=−
=−
1−γ
1 − 0.1
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(e) How much insurance does he buy (x∗ )?
γ
The optimal bundle is at M RS = − 1−γ
. Then we have
−
√
0.1 cN F
1
=− .
√
0.9 cF
9
This in turn gives c∗F = c∗N F . From (b), we can solve for x∗ :
50, 000 − 0.1x∗ + x∗ = 500, 000 − 0.1x∗ ,
which gives x∗ = 450, 000.
2. Clarence Bunsen is an expected utility maximizer. His preferences among contingent commodity bundles are represented by the expected utility function
√
√
EU = π1 c1 + π2 c2 .
Clarence’s friend, Hjalmer Ingqvist, has oered to bet him $1,000 on the outcome of the toss
of a coin. That is, if the coin comes up heads, Clarence must pay Hjalmer $1,000 and if the
coin comes up tails, Hjalmer must pay Clarence $1,000. The coin is a fair coin, so that the
probability of heads and the probability of tails are both 1/2. If he doesn’t accept the bet,
Clarence will have $10,000 with certainty. In the privacy of his car dealership office over at
Bunsen Motors, Clarence is making his decision. (Clarence uses the pocket calculator that
his son, Elmer, gave him last Christmas. You will find that it will be helpful for you to use
a calculator too.) Let Event 1 be “coin comes up heads” and let Event 2 be “coin comes up
tails.”
(a) If Clarence accepts the bet, then he will have 9,000 dollars and in Event 2, he will have
11,000 dollars.
(b) What is his expected utility if he accepts the bet with Hjalmer?
√
√
EUbet = 0.5 9000 + 0.5 11000 = 99.8746.
(c) If Clarence decides not to bet, then in Event 1, he will have 10,000 dollars and in Event
2, he will have 10,000 dollars. And in this case, what is his expected utility?
√
√
EUno−bet = 0.5 10000 + 0.5 10000 = 100.
(d) Does Clarence take the bet?
No, since EUbet < EUno−bet
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