micro-fall-2006-10

# micro-fall-2006-10 - 141 5.2 Indirect Utility Function and...

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141 5.2 Indirect Utility Function and Expenditure Functions Each household’s optimization problem can be writ- tenintwoforms: ( i ) as a utility maximization problem for a given bud- get constraint ,or ( ii ) as an expenditure minimizing problem for a given utility level . Under some assumptions on the utility functions, both problems are equivalent, i.e. the optimal consumption bundle is the same. This equivalence can be used to understand the prop- erties of the demand functions.

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142 The dual problem: The household’s problem now becomes: Choose ( x 1 ; x 2 ) so that e ( x 1 ; x 2 )= p 1 x 1 + p 2 x 2 is minimized under the constraints u ( x 1 ; x 2 ) ¯ u x 1 0 x 2 0 . The solution of the household’s problem determines: ( i ) The Hicks’ demand functions for good 1 and 2 : ˜ x 1 = h 1 ( p 1 ; p 2 u ) ˜ x 2 = h 2 ( p 1 ; p 2 u ) ( ii ) The expenditure function ˜ e = e ( p 1 ; p 2 u ) .
143 Example 28 Cobb-Douglas utility function u ( x

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micro-fall-2006-10 - 141 5.2 Indirect Utility Function and...

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