Lecture2_MathCont - Lecture2 MoreMathPreliminaries

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Lecture 2 More Math Preliminaries Econ 101A: Microeconomic Theory UC Berkeley Spring 2011 Prof. Cristian Santesteban
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2 Multivariate optimization Nicholson, Ch.2, pp. 23-30 (24-32, 9th Ed) Function from R n to R : y = f ( x 1 ,x 2 ,...,x n ) Partial derivative with respect to x i : ∂f ( x 1 , ..., x n ) ∂x i = lim h 0 f ( x 1 , ..., x i + h, . ..x n ) f ( x 1 , ..., x i ,...x n ) h Slope along dimension i Total di f erential: df = ( x ) 1 dx 1 + ( x ) 2 dx 2 + ... + ( x ) n dx n
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One important economic example Example 1: Partial derivatives of y = f ( L, K )= L . 5 K . 5 f 0 L = (marginal productivity of labor) f 0 K = (marginal productivity of capital) f 00 L,K =
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Maximization over an open set (like R ) Necessary condition for maximum x is ∂f ( x ) ∂x i =0 i (3) or in vectorial form f ( x )=0 These are commonly referred to as f rst order condi- tions (f.o.c.) Su cient conditions? De f ne Hessian matrix H : H = f 00 x 1 ,x 1 f 00 x 1 , x 2 ... f 00 x 1 ,x n ... ... ... ... f 00 x n ,x 1 f 00 x n ,x 2 ... f 00 x n ,x n
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Subdeterminant | H | i of Matrix H is de f ned as the determinant of submatrix formed by f rst i rows and f rst i columns of matrix H. Examples. | H | 1 is determinant of f 00 x 1 ,x 1 , that is, f 00 x 1 ,x 1 | H | 2 is determinant of H = Ã f 00 x 1 ,x 1 f 00 x 1 ,x 2 f 00 x 2 ,x 1 f 00 x 2 ,x 2 !
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This note was uploaded on 01/21/2012 for the course ECON 101a taught by Professor Staff during the Spring '08 term at University of California, Berkeley.

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Lecture2_MathCont - Lecture2 MoreMathPreliminaries

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