1. A decision-maker faces a lottery that gives her a final wealth of 1 dollar with probability 1/4, 3 dollars with probability 1/2, and 8 dollars with probability 1/4. (a) Suppose this decision-maker is an expected utility maximizer with von Neumann-Morgenstern utility u₁(2) = √2+1, where r is her final wealth. Find the risk premium associated with this lottery.
1. A decision-maker faces a lottery that gives her a final wealth of 1 dollar with probability 1/4, 3 dollars with probability 1/2, and 8 dollars with probability 1/4. (a) Suppose this decision-maker is an expected utility maximizer with von Neumann-Morgenstern utility u₁(2) = √2+1, where r is her final wealth. Find the risk premium associated with this lottery.
Chapter7: Uncertainty
Section: Chapter Questions
Problem 7.10P
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![4. A decision-maker faces a lottery that gives her a final wealth of 1 dollar with probability 1/4,
3 dollars with probability 1/2, and 8 dollars with probability 1/4.
(a) Suppose this decision-maker is an expected utility maximizer with von Neumann-Morgenstern
utility u₁(x) = √x+1, where z is her final wealth. Find the risk premium associated
with this lottery.
Solution: The expected value of the lottery is 1/4+3/2+8/4 = 15/4. The certainty
equivalent, CE, solves
√CE+1=
√2 √4 √9 7+√2
+
Hence CE = (35+ 14√2)/16 and the risk premium is (25-14√/2)/16.
(b) Now suppose that a second decision-maker who maximizes expected utility with von
Neumann-Morgenstern utility u₂(x) = √faces the same lottery. Without calculating
this decision-maker's risk premium, determine whether it is higher than, lower than, or
the same as that for the decision-maker in part (a).
Solution: The first decision-maker has Arrow-Pratt measure
u₁(x)
1
u₁(x) 2(x + 1)
r₁(x) =
and the second has Arrow-Pratt measure
r₂(x):
==
u(x) 1
u₂(x) 2x
-
Since r₂(x) > r₁(x) for every x>0, the second decision-maker is more risk averse than
the first and therefore has a higher risk premium.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa1ab2968-d288-4fd8-b87c-74963c459231%2F792a9faa-2cad-4a39-bc03-70ad1571f4d8%2F6rut0e_processed.png&w=3840&q=75)
Transcribed Image Text:4. A decision-maker faces a lottery that gives her a final wealth of 1 dollar with probability 1/4,
3 dollars with probability 1/2, and 8 dollars with probability 1/4.
(a) Suppose this decision-maker is an expected utility maximizer with von Neumann-Morgenstern
utility u₁(x) = √x+1, where z is her final wealth. Find the risk premium associated
with this lottery.
Solution: The expected value of the lottery is 1/4+3/2+8/4 = 15/4. The certainty
equivalent, CE, solves
√CE+1=
√2 √4 √9 7+√2
+
Hence CE = (35+ 14√2)/16 and the risk premium is (25-14√/2)/16.
(b) Now suppose that a second decision-maker who maximizes expected utility with von
Neumann-Morgenstern utility u₂(x) = √faces the same lottery. Without calculating
this decision-maker's risk premium, determine whether it is higher than, lower than, or
the same as that for the decision-maker in part (a).
Solution: The first decision-maker has Arrow-Pratt measure
u₁(x)
1
u₁(x) 2(x + 1)
r₁(x) =
and the second has Arrow-Pratt measure
r₂(x):
==
u(x) 1
u₂(x) 2x
-
Since r₂(x) > r₁(x) for every x>0, the second decision-maker is more risk averse than
the first and therefore has a higher risk premium.
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