problem set3 solution

# problem set3 solution - Problem Set 3 Solutions 1 Prove...

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Problem Set 3 Solutions 1. Prove that if expected utility function ([ ]) represents preferences over lotteries , then expected utility function ([ ]) also represents the same preferences if for all , where and are real numbers. Given , where and , we have ([ ]) ( ) ( ) ( ) ([ ]) ( ) Since , it must be the case that for any two lotteries , ( ) ( ) ( ) ( ) Since ([ ]) is an expected utility representation of , it follows that ( ) ( ) ( ) ( ) Therefore, ( ) is also an expected utility representation of . This exercise shows that any affine transformation (with a positive slope) of an expected utility function still represents the same preference over lotteries. 2. Jill is contemplating her preferences c oncerning tonight’s activities. She strictly prefers: - going out with 50% chance and working with 50% chance, over - watching TV for sure, over - going out with 40% chance, watching TV with 20% chance and working with 40% chance. a) Show that Jill’s preferences violate the independence axiom. [Hint: Start by decomposing the third lottery into a combination of the first two. That is, if the first lottery is denoted A, the second B, and the third C, find [ ] such that ( ) . Then show that ( ) violates the independence axiom.] Let { } denote the activities “going out” “watching TV” and “working” respectively. The options faced by Jill are three lotteries over the three activities: [ ] [ ] [ ] We know that .

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