Rubinstein2005-page124

Rubinstein2005-page124 - x ) r 1 ( x ) for all x , where r...

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October 21, 2005 12:18 master Sheet number 122 Page number 106 106 Lecture Nine If (3) then (2). By deFnition, Eu i ( p ) = u i ( CE i ( p )) . Thus, CE i ( p ) = u 1 i ( Eu i ( p )) . If ϕ = u 1 u 1 2 is concave, then by the Jensen Inequality: u 1 ( CE 2 ( p )) = u 1 ( u 1 2 ( Eu 2 ( p )) = ϕ( X k p ( x k ) u 2 ( x k )) ( X k p ( x k u 2 ( x k )) = X k p ( x k ) u 1 ( x k ) = E ( u 1 ( p )) = u 1 ( CE 1 ( p )). Thus, CE 2 ( p ) CE 1 ( p ) . If (1), then (3). Consider three numbers u 2 ( x )< u 2 ( y )< u 2 ( z ) in the range of u 2 and let λ ( 0, 1 ) satisfy u 2 ( y ) = λ u 2 ( x ) + ( 1 λ) u 2 ( z ) . Let us see that u 1 ( y ) λ u 1 ( x ) + ( 1 λ) u 1 ( z ) . If u 1 ( y )<λ u 1 ( x ) + ( 1 λ) u 1 ( z ) , then for some w > y close enough to y , we have both w 1 λ x ( 1 λ) z and w  2 λ x ( 1 λ) z , which contradicts (1). Thus, y % 1 λ x ( 1 λ) z and u 1 ( y ) λ u 1 ( x ) + ( 1 λ) u 1 ( z ) , from which it follows that ϕ( u 2 ( y )) λϕ( u 2 ( x )) + ( 1 λ)ϕ( u 2 ( z )) . Thus, ϕ is concave. The CoefFcient of Absolute Risk Aversion The following is another deFnition of the relation “more risk averse” applied to the case in which vNM utility functions are differentiable: 4. Let u 1 and u 2 be differentiable vNM utility functions repre- senting % 1 and % 2 , respectively. The preference relation % 1 is more risk averse than % 2 if r 2 (
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Unformatted text preview: x ) r 1 ( x ) for all x , where r i ( x ) = u 00 i ( x )/ u i ( x ). The number r ( x ) = u 00 ( x )/ u ( x ) is called the coefFcient of absolute risk aversion of u at x . We will see that a higher coefFcient of absolute risk aversion means a more risk-averse decision maker. To see that (3) and (4) are equivalent, note the following chain of equivalences: DeFnition (3) (that is, u 1 u 1 2 is concave) is satisFed iff the function d / dt [ u 1 ( u 1 2 ( t )) ] is nonincreasing in t iff u 1 ( u 1 2 ( t ))/ u 2 ( u 1 2 ( t )) is nonincreasing in t iff (since ( 1 ) ( t ) = 1 / ( 1 ( t )) ) u 1 ( x )/ u 2 ( x ) is nonincreasing in x (since u 1 2 ( t ) is increasing in t ) iff...
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