Rubinstein2005-page125

Rubinstein2005-page125 - 12:18 master Sheet number 123 Page...

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October 21, 2005 12:18 master Sheet number 123 Page number 107 Risk Aversion 107 log [ ( u 1 / u 2 )( x ) ] = logu 1 ( x ) logu 2 ( x ) is nonincreasing in x iff the derivative of logu 1 ( x ) logu 2 ( x ) is nonpositive iff r 2 ( x ) r 1 ( x ) 0 for all x where r i ( x ) = − u i ( x )/ u i ( x ) iff definition (4) is satisfied. For a better understating of the coefficient of absolute risk aver- sion, it is useful to look at the preferences on the restricted domain of lotteries of the type ( x 1 , x 2 ) = px 1 ( 1 p ) x 2 , where the probabil- ity p is fixed. Denote by u a differentiable vNM utility function that represents a risk-averse preference. Let x 2 = ψ( x 1 ) be the function describing the indifference curve through ( t , t ) , the point representing [ t ] . It follows from risk aver- sion that all lotteries with expectation t , that is, all lotteries on the line { ( x 1 , x 2 ) | px 1 + ( 1 p ) x 2 = t } , are not above the indifference curve through ( t , t ) . Thus, ψ ( x 1 ) = − p /( 1 p ) . By definition of u as a vNM utility function representing the pref- erences over the space of lotteries, we have pu ( x 1 ) + ( 1 p ) u (ψ( x 1 )) = u ( t ) . Taking the derivative with respect to x 1 , we obtain pu ( x 1 ) + ( 1 p ) u (ψ( x 1 ))ψ ( x 1 ) = 0. Taking the derivative with respect to
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