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-Morgensternutility u₁(x) = √x+1, where z is her final wealth. Find the risk premium associatedwith this lottery.Solution: The expected value of the lottery is 1/4+3/2+8/4 = 15/4. The certaintyequivalent, 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 vonNeumann-Morgenstern utility u₂(x) = √faces the same lottery. Without calculatingthis decision-maker's risk premium, determine whether it is higher than, lower than, orthe same as that for the decision-maker in part (a).Solution: The first decision-maker has Arrow-Pratt measureu₁(x)1u₁(x) 2(x + 1)r₁(x) =and the second has Arrow-Pratt measurer₂(x):==u(x) 1u₂(x) 2x-Since r₂(x) > r₁(x) for every x>0, the second decision-maker is more risk averse thanthe first and therefore has a higher risk premium.

Answered: 1. A decision-maker faces a lottery…

ENGR.ECONOMIC ANALYSIS

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ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

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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.

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A probability distribution refers to a statistical function that encompasses all of these possibilities. A random variable may have any number of different values and likelihoods within a certain range. The lowest and highest possible values will serve as a boundary for this range, but other factors will determine where the prospective value will appear on the probability distribution.

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