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ec142f09hw2_sol - Kata Bognar [email protected] Economics...

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Kata Bognar [email protected] Economics 142 Probabilistic Microeconomics UCLA Fall 2009 Homework Assignment 2. - Suggested solutions - 1. (Expected Utility Representation) Assume that there are two possible states of [10] nature, nuclear war or peace. Dr. Strangelove believes that nuclear war is three times as likely as peace. Denote x W the amount of money Dr. Stangelove has in case of war and x P the amount of money he has in case of peace. Dr. Strangelove is very strange, he strictly prefers ( x W , x P ) to ( x 0 W , x 0 P ) if x W > x 0 W or if x W = x 0 W and x P > x 0 P . Is there an expected utility representation of Dr. Strangelove’s preferences? If yes, provide a vNM utility function, if no explain why not. Answer: You may start this problem by guessing the vNM utility function and checking if it works. For example, you may try u ( x ) = x. Dr. Strangelove prefers (2 , 1) to (1 , 5) since x W = 2 > 1 = x 0 W , however EU (2 , 1) = 3 4 × 2 + 1 4 × 1 < 3 4 × 1 + 1 4 × 5 = EU (1 , 5) . Therefore u ( x ) = x is not an appropriate vNM function. Similarly, you may try out u ( x ) = x or u ( x ) = x 2 , etc. and find counter-examples. After a while you will see that for every increasing vNM utility function you can come up with a pair of lotteries for which the expected utility does not represent the actual choice. Basically, if you set x W > x 0 W with the difference being small and x P < x 0 P with the difference being big then although ( x W , x P ) ( x 0 W , x 0 P ) the expected utility of ( x W , x P ) will be smaller than the expected utility of ( x 0 W , x 0 P ) . So now you have a strong suspicion that Dr. Strangelove’s preferences have no expected utility representation. How can you prove this? Remember that the expected utility representation was possible if the preferences are complete, transitive and the Independence and the Continuity Axioms hold. You may check
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