University:
Amherst College
Course:
ECON 301 | Advanced Microeconomics
Academic year:

2023
Views:

191

Pages:

2
Author:

trescandogss

Economics 58 Substitution and Income Effects with the Cobb-Douglas If U ( x1 , x2 ) = x1α x12−α we know that x1 = αI p1 x = α A−1 p1α −1 p12−αU c 1 U = Ap1−α p2− (1−α ) I 1. Own price Slutsky Equation: function directly gives ∂x1 ∂x1c ∂x = − x1 1 . Using the Marshallian demand ∂p1 ∂p1 ∂I ∂x1 −α I = 2 . Calculating the components of the Slutsky Equation ∂p1 p1 gives: ∂x1c = α (α − 1) A−1 p1α − 2 p12−αU = α (α − 1) p1−2 I ∂p1 ∂x1 αI α =− ⋅ = −α 2 p1−2 I p1 p1 ∂I ∂x Summing these two effects gives 1 = Sub + Income Effect= − α p1−2 I just as was ∂p1 derived from direct differentiation of the Marshallian demand function. Note, if α = 0.5 , we have − x1 ∂x1 = −0.5 p1−2 I ∂p1 ∂x1c = −0.25 p1−2 I ∂p1 − x1 ∂x1 = −0.25 p1−2 I ∂I That is, the total price effect in the Marshallian demand function is half income effect and half substitution effect. If, say, α = 0.3 , the substitution effect would be 70 percent of 0.3 ⋅ 0.7 the total effect ( 0.7 = ) and the income effect would be only 30 percent of the 0.3 total. These proportions would be reversed if α = 0.7 . 2. Cross-price Slutsky equation : ∂x1 ∂x1c ∂x = − x2 1 . ∂p2 ∂p2 ∂I ∂x1 = 0 and we wish to know why. To calculate the ∂p2 Slutsky Equation we have to know the Marshallian demand for good 2 which is (1 − α ) I x2 = . So the Slutsky components are: p2 Now direct differentiation gives: ∂x1c = α (1 − α ) A−1 p1α −1 p2−αU = α (1 − α ) p1−1 p2−1 I ∂p2 − x2 ∂x1 = −(1 − α ) Ip2−1 ⋅ α p1−1 = −α (1 − α ) p1−1 p2−1 I ∂I Which shows that the substitution and income effects always precisely cancel out regardless of the value of α . That makes the Cobb-Douglas a very special case.

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