{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

This preview shows pages 1–3. Sign up to view the full content.

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 ) , θ ) . 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
where L ( x, θ ) = f ( x, θ ) + λ > ( c - g ( x, θ )). Example: Cost-minimization Problem max f ( x, θ ) = - θ > x subject to g ( x ) = c. Example: Consumer Demand max u ( x ) subject to p > x = I C. Implicit Function Theorem Let f
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 3

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online