Question
Asked Dec 11, 2019
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Derive demand functions for x and y for the general class of Cobb-Douglas preferences represented by the utility function u(x, y) = x α y β . As functions of px, py, I, α, and β, calculate price elasticities, cross-price elasticities, income elasticities, and expenditure shares for x and y.

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Expert Answer

Step 1

The utility function of Cobb-Douglas preferences is given as,

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u(x.y) = x*y®

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Step 2

 Derivation the demand functions for x and y

 Maximize the given utility function subject to budget constraint using Lagrange method.

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Мaximize, u(x, у) %3Dх"у? subject to, xp, + yp,=I Lagrange Function L=x*y® + \(I-xp, + yp,)

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Step 3

First order conditions ...

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SL = ax (-) y® - ^p, =0 бх ах в-)ур = 1 (1) Px SL — Вх"у в) - Хр, 30 бу - Ap,=0 Вх "у (4) = 1 (2) Py SL %3D1-хр, + ур, -0 SI (3) 1- хр, + уру

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Microeconomics

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