particularly active this season involves betting on the winner of the U.S. presidential election. Thus, for example, if one trader buys one share of "Obama to win" from another trader, then if Obama wins the election the seller pays 1 dollar to the buyer, and if Obama does not win the seller pays nothing; either way, the seller keeps the money paid by the buyer for the purchase of the share. (a) Leila is an expected utility maximizer with von Neumann-Morgenstern utility u(x) = In(x + 1) and initial wealth 4. Leila believes that Obama will win with probability 1/2. Find the highest price p, at which she is willing to buy at least one share. Solution: The price pa solves
particularly active this season involves betting on the winner of the U.S. presidential election. Thus, for example, if one trader buys one share of "Obama to win" from another trader, then if Obama wins the election the seller pays 1 dollar to the buyer, and if Obama does not win the seller pays nothing; either way, the seller keeps the money paid by the buyer for the purchase of the share. (a) Leila is an expected utility maximizer with von Neumann-Morgenstern utility u(x) = In(x + 1) and initial wealth 4. Leila believes that Obama will win with probability 1/2. Find the highest price p, at which she is willing to buy at least one share. Solution: The price pa solves
Chapter7: Uncertainty
Section: Chapter Questions
Problem 7.3P
Related questions
Question
please only do: if you can teach explain steps
How does pb compare to the lowest price ps at which Leila is willing to sell at least one
share? Are they same? Explain.
Solution: Since Leila is risk averse, she will only agree to buy or sell if doing so yields
a positive expected value. Therefore, we must have
how to solve for this: ps > 1/2 > pb (and in particular,
ps > pb).
b) c
how to solve for this:
for Vadim is 4, and therefore p0
b < pb.
![particularly active this season involves betting on the winner of the U.S. presidential election.
Thus, for example, if one trader buys one share of "Obama to win" from another trader,
then if Obama wins the election the seller pays 1 dollar to the buyer, and if Obama does not
win the seller pays nothing; either way, the seller keeps the money paid by the buyer for the
purchase of the share.
(a) Leila is an expected utility maximizer with von Neumann-Morgenstern utility u(x) =
In(x + 1) and initial wealth 4. Leila believes that Obama will win with probability 1/2.
Find the highest price p, at which she is willing to buy at least one share.
Solution: The price po solves
which implies that
and therefore,
This equation has solutions
1
In(5-ps) +:
+In(6-pt) = ln 5,
(5-Pb) (6-Pb) = 5,
P² - 11ps + 5 = 0.
11 ± √101
Pb=
2
Since the price must clearly be less than 1 for Leila to buy, we obtain
Pb=
11 - √101
2
(b) How does p, compare to the lowest price p, at which Leila is willing to sell at least one
share? Are they same? Explain.
Solution: Since Leila is risk averse, she will only agree to buy or sell if doing so yields
a positive expected value. Therefore, we must have p. > 1/2> p (and in particular,
Ps > Pb).
(c) Vadim is more risk averse than Leila. He also believes that Obama will win with prob-
ability 1/2. How does the highest price at which he is willing to buy at least one share
compare to ps (i.e. to the corresponding price for Leila)? Prove your answer.
Solution: Let p denote the highest price at which Vadim is willing to buy, and let L(p)
denote the lottery associated with buying one share at price p. For Leila, the certainty
equivalent of the lottery L(ps) is 4. Since Vadim is more risk averse than Leila, the
certainty equivalent of L(P) for him is less than 4. But the certainty equivalent of L(ps)
for Vadim is 4, and therefore p < P-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcc93dd3a-e660-4464-bc8c-e382a2c34aae%2Ffa83407c-c755-40d7-bab2-ef98ce76fa69%2Fjzeef98_processed.png&w=3840&q=75)
Transcribed Image Text:particularly active this season involves betting on the winner of the U.S. presidential election.
Thus, for example, if one trader buys one share of "Obama to win" from another trader,
then if Obama wins the election the seller pays 1 dollar to the buyer, and if Obama does not
win the seller pays nothing; either way, the seller keeps the money paid by the buyer for the
purchase of the share.
(a) Leila is an expected utility maximizer with von Neumann-Morgenstern utility u(x) =
In(x + 1) and initial wealth 4. Leila believes that Obama will win with probability 1/2.
Find the highest price p, at which she is willing to buy at least one share.
Solution: The price po solves
which implies that
and therefore,
This equation has solutions
1
In(5-ps) +:
+In(6-pt) = ln 5,
(5-Pb) (6-Pb) = 5,
P² - 11ps + 5 = 0.
11 ± √101
Pb=
2
Since the price must clearly be less than 1 for Leila to buy, we obtain
Pb=
11 - √101
2
(b) How does p, compare to the lowest price p, at which Leila is willing to sell at least one
share? Are they same? Explain.
Solution: Since Leila is risk averse, she will only agree to buy or sell if doing so yields
a positive expected value. Therefore, we must have p. > 1/2> p (and in particular,
Ps > Pb).
(c) Vadim is more risk averse than Leila. He also believes that Obama will win with prob-
ability 1/2. How does the highest price at which he is willing to buy at least one share
compare to ps (i.e. to the corresponding price for Leila)? Prove your answer.
Solution: Let p denote the highest price at which Vadim is willing to buy, and let L(p)
denote the lottery associated with buying one share at price p. For Leila, the certainty
equivalent of the lottery L(ps) is 4. Since Vadim is more risk averse than Leila, the
certainty equivalent of L(P) for him is less than 4. But the certainty equivalent of L(ps)
for Vadim is 4, and therefore p < P-
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VIEWStep 3: B) Assess the willingness to buy and sell
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