Lecture17-2010 - Risk Aversion in the Large and Small:...

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Risk Aversion in the Large and Small: Lecture XVII Charles B. Moss October 4, 2010 I. Basics of Risk Aversion A. Back to Friedman and Savage: 1. An economic agen with a von Neumann-Morgenstern utility function v : R R is weakly risk averse if and only if E[ v ( c )] v (E [ c ]) (1) for every consumption plan c . 2. An economic agent is risk neutral if v ( c )] = v (E [ c ]) (2) 3. Similarly, an economic agent is strictly risk averse if v ( c )] <v (E [ c ]) (3) B. Risk Aversion and Concavity 1. Theorem 4.1 p-87 a) The agent is risk averse if and only if his or her von Neumann-Morgenstern utility function v is concave. b) An agent is risk neutral if and only if his or her von Neumann-Morgenstern utility function v is linear. c) An agent is strictly risk avers if and only if his or her von Neumann-Morgenstern utility function v is strictly concave. 1
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AEB 6182 Agricultural Risk Analysis and Decision Making Professor Charles B. Moss Lecture XVII Fall 2010 ()( )() 12 1 ux ux αα +− () 1 ux x 1 x 2 x 1 ux 2 Figure 1: Graphical Depiction of Risk Aversion C. Proof: 1. To justify this theorem we start with the basic concept of concavity. If the utility function is concave u ( αx 1 +(1 α ) x 2 ) αu ( x 1 )+(1 α ) u ( x 2 )( 4 ) 2. This relationship is depicted graphically in Figure 1 3. Interpreting α as a probability completes the linkage between concavity and risk aversion.
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Lecture17-2010 - Risk Aversion in the Large and Small:...

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