Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Really Pushing the Envelope: Early Use of the Envelope Theorem by Auspitz and Lieben

Really Pushing the Envelope: Early Use of the Envelope Theorem by Auspitz and Lieben History of Political Economy 36:1 (2004) Why this would be so may be explained using some formal notation while considering the simplest case. The value of an objective z to be maximized is a function of a decision variable x and a single parameter α, as in z = f (x, α). At the solution to the problem, the value of the decision variable generally depends upon the parameter so that it may be written as x(α), and the maximum value of the objective function may be expressed as f ∗ (α) ≡ f (x(α), α). Assume differentiability. Then in view of the first-order condition of optimization, the effect of a small change in α on f ∗ will be exactly the same as its direct effect on f without optimizing adjustment of the decision variable, that is, df ∗ /dα = ∂f/∂α. Equivalently, the value of the indirect effect given by the expression (∂f/∂x)(dx/dα) equals zero and may be disregarded. Analogous findings apply when there are multiple decision variables and multiple parameters and in the case of constrained optimization.3 This fundamental principle—the particular simplification due to the first-order condition(s) of optimization—finds a variety of applications. In the http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png History of Political Economy Duke University Press

Really Pushing the Envelope: Early Use of the Envelope Theorem by Auspitz and Lieben

History of Political Economy , Volume 36 (1) – Mar 1, 2004

Loading next page...
 
/lp/duke-university-press/really-pushing-the-envelope-early-use-of-the-envelope-theorem-by-0jGTL1X01Z
Publisher
Duke University Press
Copyright
Copyright 2004 by Duke University Press
ISSN
0018-2702
eISSN
1527-1919
DOI
10.1215/00182702-36-1-103
Publisher site
See Article on Publisher Site

Abstract

History of Political Economy 36:1 (2004) Why this would be so may be explained using some formal notation while considering the simplest case. The value of an objective z to be maximized is a function of a decision variable x and a single parameter α, as in z = f (x, α). At the solution to the problem, the value of the decision variable generally depends upon the parameter so that it may be written as x(α), and the maximum value of the objective function may be expressed as f ∗ (α) ≡ f (x(α), α). Assume differentiability. Then in view of the first-order condition of optimization, the effect of a small change in α on f ∗ will be exactly the same as its direct effect on f without optimizing adjustment of the decision variable, that is, df ∗ /dα = ∂f/∂α. Equivalently, the value of the indirect effect given by the expression (∂f/∂x)(dx/dα) equals zero and may be disregarded. Analogous findings apply when there are multiple decision variables and multiple parameters and in the case of constrained optimization.3 This fundamental principle—the particular simplification due to the first-order condition(s) of optimization—finds a variety of applications. In the

Journal

History of Political EconomyDuke University Press

Published: Mar 1, 2004

There are no references for this article.