Lecture 3 - Economics 101A (Lecture 3) Stefano DellaVigna...

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Economics 101A (Lecture 3) Stefano DellaVigna September 3, 2009
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Outline 1. Implicit Function Theorem II 2. Envelope Theorem 3. Convexity and concavity 4. Constrained Maximization
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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 n ; x 1 s )=0 ), and a point ( p 0 ,x 0 ) solution of the equation. Assume: 1. f 1 ,...,f s continuously di f 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 f erent from 0: f x = ∂f 1 ∂x 1 1 s ... ... ... s 1 ... s ∂xs
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Then: 1 .The reisoneandon lyseto ffunct ions x = g ( p ) de f 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 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 1 2 2 1 2 2 = Ã!
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Then compute det μ ( f 1 ,...,f s ) ( x 1 ,...x i 1 ,p k ,x i +1 ...,x s ) ∂f 1 ∂p 1 1 ∂x 2 2 1 2 2 = Ã! Finally, 1 1 = Why did you compute det ³ f x ´ already?
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2 Envelope Theorem Ch. 2, pp. 32-36 (33—37, 9th Ed) You now know how x 1 varies if p 1 varies. How does h ( x ( p )) vary as p 1 varies? Di f erentiate h ( x 1 ( p 1 ,p 2 ) ,x 2 ( p 1 2 ) 1 2 ) with respect to p 1 : dh ( x 1 ( p 1 2 ) 2 ( p 1 2 ) 1 2 ) dp 1 = ∂h ( x , p ) ∂x 1 1 ( p ) ∂p 1 + ( x , p ) 2 2 ( p ) 1 + ( x , p ) 1 The f rst two terms are zero.
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Lecture 3 - Economics 101A (Lecture 3) Stefano DellaVigna...

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