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Unformatted text preview: Utility Functions, Risk Aversion Coecients and Transformations: Lecture XVIII Charles B. Moss October 6, 2010 I. An examination of the Arrow-Pratt Coecients for particular func- tions. A. Quadratic Utility Function: To specify the appropriate shape of the utility function, the quadratic function becomes U ( w ) = aw bw 2 U ( w ) = a 2 bw U 00 ( w ) = 2 b (1) Arrow-Pratt absolute risk aversion coecient R A ( w ) = 2 b a 2 bw R A ( w ) = 2 b a 2 bw dR A ( w ) dw = 2 b 2 b ( a 2 bw ) 2 > d dx 1 f ( x ) = f ( x ) ( f ( w )) 2 (2) Arrow-Pratt relative risk aversion coecient R R ( w ) = 2 b a 2 bw w = 2 b a w 2 b dR R ( w ) dw = 2 b 2 b ( a 2 bw ) 2 + 2 b a 2 bw > (3) 1 AEB 6182 Agricultural Risk Analysis and Decision Making Professor Charles B. Moss Lecture XVIII Fall 2010 B. Power Utility Function U ( w ) = w 1 r 1 r U ( w ) = w r U 00 ( w ) = rw r 1 (4) Arrow-Pratt absolute risk aversion coecient R A ( w ) = rw r 1 w r = r w dR A ( w ) dw = r w 2 < (5) Arrow-Pratt relative risk aversion coecient R R ( w ) = rw r 1 w w r = r dR R ( w ) dw = 0 (6) Constant relative risk aversion....
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