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Lecture3_MathIII

# Lecture3_MathIII - Lecture3 Econ101A:MicroeconomicTheory...

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Lecture 3 Math Preliminaries III Econ 101A: Microeconomic Theory UC Berkeley Spring 2011 Prof. Cristian Santesteban

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1 Implicit function theorem II Multivariate implicit function theorem (Dini) : Consider a set of equations ( f 1 ( p 1 , ..., p n ; x 1 , ..., x s ) = 0; ... ; f s ( p 1 , ..., p n ; x 1 , ..., x s ) = 0 ), and a point ( p 0 ,x 0 ) solution of the equation. Assume: 1. f 1 , ..., f s continuously di ff erentiable in a neigh- bourhood of ( p 0 ,x 0 ) ; 2. The following Jakobian matrix f x evaluated at ( p 0 ,x 0 ) has determinant di ff erent from 0: f x = ∂f 1 ∂x 1 ∂f 1 ∂x s ... ... ... ∂f s ∂x 1 ... ∂f s ∂xs
Then: 1. There is one and only set of functions x = g ( p ) de fi ned in a neighbourhood of p 0 that satisfy f ( p, g ( p )) = 0 and g ( p 0 ) = x 0 ; 2. The partial derivative of x i with respect to p k is ∂g i ∂p k = det μ ( f 1 ,...,f s ) ( x 1 ,...x i 1 ,p k ,x i +1 ...,x s ) det ³ f x ´

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Example 2 (continued): Max h ( x 1 , x 2 ) = p 1 x 2 1 + p 2 x 2 2 2 x 1 5 x 2 f.o.c. x 1 : 2 p 1 x 1 2 = 0 = f 1 ( p,x ) f.o.c. x 2 : 2 p 2 x 2 5 = 0 = f 2 ( p,x ) Comparative statics of x 1 with respect to p 1 ? First compute det ³ f x ´ ∂f 1 ∂x 1 ∂f 1 ∂x 2 ∂f 2 ∂x 1 ∂f 2 ∂x 2 = Ã !
Then compute det μ ( f 1 ,...,f s ) ( x 1 ,...x i 1 ,p k ,x i +1 ...,x s ) ∂f 1 ∂p 1 ∂f 1 ∂x 2 ∂f 2 ∂p 1

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Lecture3_MathIII - Lecture3 Econ101A:MicroeconomicTheory...

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