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101A_lec13 - Economics 101A(Lecture 13 Dan Acland and...

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Economics 101A (Lecture 13) Dan Acland and Mariana Carrera March 5, 2009
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Outline 1. Expected Utility 2. Insurance 3. Investment in Risky Asset
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1 Expected Utility Nicholson, Ch. 7, pp. 202-209 (Ch. 18, pp. 533— 541, 9th) Consumer at time 0 asks: what is utility in time 1? At t = 1 consumer maximizes max U ( c 1 ) s.t. c 1 i M 1 i + (1 + r ) ( M 0 c 0 ) with i = 1 , 2 , 3 . What is utility at optimum at t = 1 if U 0 > 0? Assume for now M 0 c 0 = 0 Utility U ³ M 1 i ´ This is uncertain, depends on which i is realized!
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How do we evaluate future uncertain utility? Expected utility EU = 3 X i =1 p i U ³ M 1 i ´ In example: EU = 1 / 3 U (20) + 1 / 3 U (25) + 1 / 3 U (30) Compare with U ( EC ) = U (25) . Agents prefer riskless outcome EM to uncertain out- come M if 1 / 3 U (20) + 1 / 3 U (25) + 1 / 3 U (30) < U (25) or 1 / 3 U (20) + 1 / 3 U (30) < 2 / 3 U (25) or 1 / 2 U (20) + 1 / 2 U (30) < U (25)
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Depends on sign of U 00 , on concavity/convepwxity Three cases: U 00 ( x ) = 0 for all x. (linearity of U ) U ( x ) = a + bx 1 / 2 U (20) + 1 / 2 U (30) = U (25) U 00 ( x ) < 0 for all x. (concavity of U ) 1 / 2 U (20) + 1 / 2
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