Lecture 4 - Economics 101A (Lecture 4) Stefano DellaVigna...

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Economics 101A (Lecture 4) Stefano DellaVigna September 8, 2009

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Outline 1. Constrained Maximization II 2. Envelope Theorem II 3. Preferences 4. Properties of Preferences
1 Constrained Maximization Idea: Use implicit function theorem. Heuristic solution of system max x,y f ( x, y ) s.t. h ( x, y )=0 Assume: continuity and di f erentiability of h h 0 y 6 =0 (or h 0 x 6 =0) Implicit function Theorem: Express y as a function of x x as function of y ) !

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Write system as max x f ( x, g ( x )) f.o.c.: f 0 x ( x, g ( x )) + f 0 y ( x, g ( x )) ∂g ( x ) ∂x =0 What is ( x ) ? Substitute in and get: f 0 x ( x, g ( x )) + f 0 y ( x, g ( x )) ( h 0 x /h 0 y )=0 or f 0 x ( x, g ( x )) f 0 y ( x, g ( x )) = h 0 x ( x, g ( x )) h 0 y ( x, g ( x ))
Lagrange Multiplier Theorem, necessary condi- tion. Consider a problem of the type max x 1 ,...,x n f ( x 1 ,x 2 , ..., x n ; p ) s.t. h 1 ( x 1 2 ,...,x n ; p )=0 h 2 ( x 1 2 n ; p ... h m ( x 1 2 n ; p with n>m. Let x = x ( p ) be a local solution to this problem. Assume: f and h di f erentiable at x the following Jacobian matrix at x has maximal rank J = ∂h 1 ∂x 1 ( x ) ...

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This note was uploaded on 09/02/2010 for the course ECON 101a taught by Professor Staff during the Fall '08 term at University of California, Berkeley.

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Lecture 4 - Economics 101A (Lecture 4) Stefano DellaVigna...

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