Solution to problem set3.pdf

Solution to problem set3.pdf - Amherst College Economics 58...

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Amherst College Economics 58 Fall 2010 Solutions to PS # 3 1 . CES Utility a. 1 ( ) x y U/ x MRS = = x y = p /p U/ y δ for utility maximization. Hence, 1 ( 1) ( ) ( ) where 1 (1 ) x y x y x/y = p p p p δ σ σ δ = = . b. If δ = 0 , so y x x y x y p p p x p y = = . c. Part a shows 1 ( ) x y x y p x p y p p σ = Hence, for 1 σ < the relative share of income devoted to good x is positively correlated with its relative price. This is a sign of low substitutability. For 1 σ > the relative share of income devoted to good x is negatively correlated with its relative price – a sign of high substitutability. d. The algebra is a bit tricky here, but worth doing once. Let’s solve for indirect utility σ σ = = y x y x p p y x or p p y x Substituting into the budget constraint yields σ σ σ σ + = + = 1 1 y x y y y x x p p Ip y or y p p p y p I Similarly, σ σ σ + = 1 1 y x x p p Ip x Hence, δ σ σ σ δ δ σ σ σ δ δ δ δ + + + = + = 1 1 1 1 y x
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