n15 - g ( x ) formed by taking those rows i for which g i (...

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Atsushi Inoue ECG 765: Mathematical Methods for Economics Fall 2009 Lecture Notes 15 Inequality-Constrained Optimization When constraints involve inequalities, the Kuhn-Tucker theorem provides first-order necessary conditions for constraint local optima. Consider optimization problems with inequality constraints. For example, max x X u ( x ) subject to g ( x ) c In the presence of inequality constraints, we will use the Kuhn-Tucker theo- rem: Kuhn-Tucker Theorem: Suppose x is an n -dimensional vector, c an m -dimensional vector, f a scalar-valued function, g an m -dimensional vector-valued function and m < n . Define L ( x,λ ) = f ( x ) + λ > [ c - g ( x )] , where λ is an m -dimensional column vector of Lagrange multipliers. If x * maximizes f ( x ) subject to g ( x ) c , and the constraint qualification holds, that is, the submatrix of the Jacobian matrix of
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Unformatted text preview: g ( x ) formed by taking those rows i for which g i ( x * ) = c i has the maximum possible rank, then there is a value of such that L x ( x * , ) = 0 . L ( x * , ) , , i L i ( x * , ) = 0 . Example: Quasi-Linear Preferences (Dixit, 1990, pp.31-34) Consider max U ( x,y ) = y + a log( x ) subject to px + qy I x y 1 where a &gt; 0. Example: Technological Unemployment (Dixit, 1990, pp.3437) max ln x + ln y subject to 2 x + y 300 x + 2 y 450 x y where , &gt; 0 and + = 1. Suggested Exercises. Read Section 18.318.7 and 19.3 of Simon and Blume (1994) and do exercises 18.12, 18.18 and 19.14. 2...
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This note was uploaded on 10/22/2009 for the course ECG 765 taught by Professor Fackler during the Fall '07 term at N.C. State.

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n15 - g ( x ) formed by taking those rows i for which g i (...

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