CES_notes

CES_notes - U only take on positive values, i.e., values in...

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Economics 101 Notes on the CES utility function February 2, 2011 George J. Mailath In problem set 2, we saw the Constant Elasticity of Substitution utility function: U ( x 1 ,x 2 ) = ( αx η 1 + (1 α ) x η 2 ) 1 , where α (0 , 1) and η ≥ − 1, but η ± = 0, are two constants. In class on Feb 3 (and in the text), we claimed that the utility function, W ( x 1 ,x 2 ) = x δ 1 δ + x δ 2 δ , δ 1 ± = 0 , is also a CES utility function. In particular, I claim that U , with η = δ and α = 1 2 , and W describe the same preferences. Note first that the restrictions on δ and η are consistent. Imposing η = δ and α = 1 2 , we have U ( x 1 ,x 2 ) = ( 1 2 x δ 1 + 1 2 x δ 2 ) 1 . Note first that
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Unformatted text preview: U only take on positive values, i.e., values in R + . 1 Let g ( y ) = 2 y . This function g : R + R is just a power function, and is strictly increasing. (When < 0, the two negative deltas cancel.) Thus U and g U (i.e., g ( U )) describe the same preferences, and g ( U ( x 1 ,x 2 )) = 2 ( U ( x 1 ,x 2 )) = 2 ( 1 2 x 1 + 1 2 x 2 ) 1 / = 2 ( 1 2 x 1 + 1 2 x 2 ) = 1 x 1 + 1 x 2 = W ( x 1 ,x 2 ) . 1 I am ignoring zero consumption. 1...
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