micro-fall2007-9 - Detour The Theory of Non-Linear...

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Detour: The Theory of Non-Linear Optimization Demand Analysis Choose ( x 1 ; x 2 ) so that u ( x 1 ; x 2 ) is maximized under the constraints p 1 x 1 + p 2 x 2 ° y x 1 ± 0 x 2 ± 0. The solution of the household°s problem determines: ( i ) The demand functions for good 1 and 2: x ² 1 = f 1 ( p 1 ; p 2 ; y ) x ² 2 = f 2 ( p 1 ; p 2 ; y ) ( ii ) The indirect utility function u ² = v ( p 1 ; p 2 ; y ) Ani Guerdjikova Fall 2007 107 / 126
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Detour: The Theory of Non-Linear Optimization Example Cobb-Douglas utility function u ( x 1 ; x 2 ) = x 1 x 2 Solution: demand for good 1: f 1 ( p 1 ; p 2 ; y ) = y 2 p 1 ; demand for good 2: f 2 ( p 1 ; p 2 ; y ) = y 2 p 2 ; indirect utility function: v ( p 1 ; p 2 ; y ) = y 2 4 p 1 p 2 . Ani Guerdjikova Fall 2007 108 / 126
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Detour: The Theory of Non-Linear Optimization 6 - @ @ @ @ @ @ @ @ @ @ @ @ @ @ r x 1 x 2 y p 1 y p 2 x ² 1 x ² 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . u ² ( p 1 ; p 2 ; y ) ° ° ± Household°s problem with a Cobb-Douglas utility function Ani Guerdjikova Fall 2007 109 / 126
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Detour: The Theory of Non-Linear Optimization Example Linear utility function: u ( x 1 ; x 2 ) = 5 x 1 + x 2 .
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