CES_notes

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

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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 δ δ . This function g : R + → R is just a power function, and is strictly increasing. (When δ < 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 δ ( U ( x 1 ,x 2 )) δ = 2 δ ± ( 1 2 x δ 1 + 1 2 x δ 2 ) 1 /δ ² δ = 2 δ ( 1 2 x δ 1 + 1 2 x δ 2 ) = 1 δ x δ 1 + 1 δ x δ 2 = W ( x 1 ,x 2 ) . 1 I am ignoring zero consumption. 1...
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