n18 - Atsushi Inoue ECG 765: Mathematical Methods for...

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Atsushi Inoue ECG 765: Mathematical Methods for Economics Fall 2009 Lecture Notes 18 Smooth Dependence on Parameters How does the objective function depend on exogenous model parameters? How do optimal solutions depend on these parameters? We will address these questions by the envelope theorem and implicit function theorem. Outline A. An Interpretation of the Lagrange Multiplier B. Envelope Theorem C. Implicit Function Theorem A. An Interpretation of the Lagrange Multiplier Example. Consider max α log( x ) + β log( y ) subject to px + qy = I The Lagrange multiplier λ is the marginal utility of income. B. Envelope Theorem Let V ( θ ) = max x { f ( x,θ ) | g ( x,θ ) = c } . V ( θ ) is the maximum over x of f ( x,θ ) subject to g ( x ) = c . Write the optimum choice as a function of θ , x ( θ ). Envelope Theorem. V ( θ ) = θ L ( x ( θ ) ) . 1
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where L ( x,θ ) = f ( x,θ ) + λ > ( c - g ( x,θ )). Example: Cost-minimization Problem
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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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n18 - Atsushi Inoue ECG 765: Mathematical Methods for...

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