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2010_lecture_5 - Economics 101Lecture 5 The Basic Model of...

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Economics 101—Lecture 5 The Basic Model of Consumer Choice III George J. Mailath January 27, 2011
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In our last episode 1 Utility maximization subject to a budget constraint typically requires MRS ij = U i U j = p i p j . 2 illustrated this using Cobb-Douglas. 3 Perfect substitutes and perfect complements illustrate that this is not always true. 4 Demand functions: x ( p , I ) solve max U ( x ) subject to i p i x i I . 5 Indirect utility function V ( p , I ) = U ( x ( p , I )) .
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In our last episode 1 Utility maximization subject to a budget constraint typically requires MRS ij = U i U j = p i p j . 2 illustrated this using Cobb-Douglas. 3 Perfect substitutes and perfect complements illustrate that this is not always true. 4 Demand functions: x ( p , I ) solve max U ( x ) subject to i p i x i I . 5 Indirect utility function V ( p , I ) = U ( x ( p , I )) . 6 And now some examples!
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Example: Cobb-Douglas U ( x 1 , x 2 ) = x α 1 x β 2 , for some α , β > 0.
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Example: perfect complements U ( x 1 , x 2 ) = min { α x 1 , β x 2 } , for some α , β > 0.
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Example: Perfect substitutes U ( x 1 , x 2 ) = α x 1 + β x 2 , for some α , β > 0.
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Interpretation of the multiplier Recall that V ( p , I ) = U ( x ( p , I )) = U ( x ( p , I )) + λ ( p , I )( I i p i x i ( p , I ))
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Interpretation of the multiplier Recall that V ( p , I ) = U ( x ( p , I )) = U ( x ( p , I )) + λ ( p , I )( I i p i x i ( p , I )) so that V ( p , I ) I = i U i ( x ( p , I )) x i ( p , I ) I + λ ( p , I ) 1 i p i x i ( p , I ) I + ∂λ ( p , I ) I ( I i
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