Rubinstein2005-page128

Rubinstein2005-page128 - October 21, 2005 12:18 110 master...

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October 21, 2005 12:18 master Sheet number 126 Page number 110 110 Lecture Nine denoting the indifference through ( t , t ) by x 2 = ψ t ( x 1 ) , we have ψ t ( x 1 ) = ψ 0 ( x 1 t ) + t . Assuming that the function u is differentiable, we derive that ψ ±± t ( t ) = ψ ±± 0 ( 0 ) . We have already seen that ψ ±± t ( t ) =−[ p /( 1 p ) 2 ] [ u ±± i ( t )/ u ± i ( t ) ] and thus there exists a constant α such that u ±± ( t )/ u ± ( t ) = α for all t . This implies that [ logu ± ( t ) ] ± =− α for all t and logu ± ( t ) =− α t + β for some β . It follows that u ± ( t ) = e α t + β .I f α = 0, the function u ( t ) must be linear (implying risk neutrality). If α ²= 0, it must be that u ( t ) = ce α t + d for some c and d . To conclude, if u is a vNM continuous utility function represent- ing preferences that are monotonic and exhibit both risk aversion and invariance to wealth, then u is an affine transformation of either the function t or a function e α t (with α> 0 ) .
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This note was uploaded on 12/29/2011 for the course ECO 443 taught by Professor Aswa during the Fall '10 term at SUNY Stony Brook.

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