n13 - Atsushi Inoue ECG 765: Mathematical Methods for...

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Unformatted text preview: Atsushi Inoue ECG 765: Mathematical Methods for Economics Fall 2009 Lecture Notes 13 Lagranges Method Many problems in economics can be written as constrained optimization problems. First we will consider equality constraint optimization problems. Outline A. A Simple Version of the Lagrange Theorem B. Intuitive Proof of the Lagrange Theorem C. A More General Version of the Lagrange Theorem A. A Simple Version of the Lagrange Theorem Consider max f ( x 1 ,x 2 ,...,x n ) subject to g ( x 1 ,x 2 ,...,x n ) = c. Define the Lagrangian function by L ( x, ) = f ( x 1 ,x 2 ,...,x n ) + [ c- g ( x 1 ,x 2 ,...,x n )] . The partial derivatives of L are ( /x j ) L ( x, ) = f j ( fx 1 ,x 2 ,...,x n )- g j ( x 1 ,x 2 ,...,x n ) , and ( / ) L ( x, ) = c- g ( x 1 ,x 2 ,...,x n ) . Lagranges Theorem Suppose that x * solves max f ( x ) subject to g ( x ) = c 1 Suppose further that x * is not a critical point of g , that is, ( /x j ) g ( x * ) 6 = 0 for at least one j . Then there is a value of....
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n13 - Atsushi Inoue ECG 765: Mathematical Methods for...

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