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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) = √2+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+V+V_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 7₁(z) = and the second has Arrow-Pratt measure r₂(x) = (2) 1 u₁(x) 2(x+1) 22 (3 Since r₂(x) > r₁(x) for every x > 0, the second decision-maker is more risk averse than

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) = √2+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+V+V_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 7₁(z) = and the second has Arrow-Pratt measure r₂(x) = (2) 1 u₁(x) 2(x+1) 22 (3 Since r₂(x) > r₁(x) for every x > 0, the second decision-maker is more risk averse than

ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN:9780190931919
Author:NEWNAN
Publisher:NEWNAN
Chapter1: Making Economics Decisions
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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.
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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