• MonashU
• ECC5650MicroTheory
• ProblemSet04

A useful identity: Roy's Identity

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1. For the general consumer's utility maximization problem, prove the following identities for y=1:

   1. Inverse demand function:

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   2. Direct demand function:

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2. Prove Shephard's Lemma (for the consumer). That is, that if

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   is differentiable in

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   at

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   with

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   then,

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3. For the budget minimization problem in

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   with utility given by the CES utility function

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   (

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   ), show that the Hicksian demand functions are given by

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   where

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   and the corresponding expenditure function is given by

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4. Verify your answer above by using the fact that

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   and

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   .