Lecture03-1996

# Lecture03-1996 - Lecture III Specific Arrow-Pratt Orange...

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1 Lecture III: Specific Arrow-Pratt, Orange Tree Example, Stochastic Dominance, Beginning Mean-Variance I. An examination of the Arrow-Pratt Coefficients for particular functions. A. Quadratic Utility Function: To specify the appropriate shape of the utility function, the quadratic function becomes ( ) ( ) ( ) 2 2 2 U w aw bw U w a bw U w b = = ′′ = − Arrow-Pratt absolute risk aversion coefficient: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 1 2 0 2 A A A b b R w R w a bw a bw d R w f x b d b dw dx f x a bw f x = − = = > = − Arrow-Pratt relative risk aversion coefficient ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 0 2 2 R R b b R w w a a bw b w d R w b b b dw a bw a bw = = = + > B. Power Utility Function: ( ) ( ) ( ) 1 1 1 r r r w U w r U w w U w rw − − = = ′′ = − Arrow-Pratt absolute risk aversion coefficient: ( ) ( ) 1 2 0 r A r A rw r R w w w d R w r dw w − − = − = = − < Arrow-Pratt relative risk aversion coefficient: ( ) ( ) 1 0 r R r R rw w R w r w d R w dw − − = − = = Constant relative risk aversion. C. Negative Exponential Utility Function:

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2 ( ) ( ) ( ) ( ) ( ) ( ) 2 exp exp exp U w w U w w U w w = − −ρ = ρ −ρ ′′ = ρ −ρ Arrow-Pratt absolute risk aversion coefficient ( ) ( ) ( ) ( ) 2 exp exp 0 A A w R w w d R w dw −ρ −ρ = − = ρ ρ −ρ = Constant absolute risk aversion. Arrow-Pratt relative risk aversion coefficient ( ) ( ) ( ) ( ) 2 exp exp 0 R R w R w w w w d R w dw −ρ −ρ = − = ρ ρ −ρ = ρ > D. HARA–Hyperbolic Absolute Risk Aversion: ( ) ( ) ( ) ( ) ( ) 1 1 2 2 2 1 , 0 1 1 1 1 1 1 1 1 1 aw U w b b aw a U w b aw a b aw a U w a b aw a b γ γ− γ− γ− γ− − γ = + > γ − γ = − γ + − γ − γ = + − γ ′′ = γ − +
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