# 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 Î±, as in z = f (x, Î±). 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(Î±), and the maximum value of the objective function may be expressed as f âˆ— (Î±) â‰¡ f (x(Î±), Î±). Assume differentiability. Then in view of the ï¬rst-order condition of optimization, the effect of a small change in Î± on f âˆ— will be exactly the same as its direct effect on f without optimizing adjustment of the decision variable, that is, df âˆ— /dÎ± = âˆ‚f/âˆ‚Î±. Equivalently, the value of the indirect effect given by the expression (âˆ‚f/âˆ‚x)(dx/dÎ±) equals zero and may be disregarded. Analogous ï¬ndings apply when there are multiple decision variables and multiple parameters and in the case of constrained optimization.3 This fundamental principleâ€”the particular simpliï¬cation due to the ï¬rst-order condition(s) of optimizationâ€”ï¬nds 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

, Volume 36 (1) – Mar 1, 2004
28 pages      /lp/duke-university-press/really-pushing-the-envelope-early-use-of-the-envelope-theorem-by-0jGTL1X01Z
Publisher
Duke University Press
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 Î±, as in z = f (x, Î±). 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(Î±), and the maximum value of the objective function may be expressed as f âˆ— (Î±) â‰¡ f (x(Î±), Î±). Assume differentiability. Then in view of the ï¬rst-order condition of optimization, the effect of a small change in Î± on f âˆ— will be exactly the same as its direct effect on f without optimizing adjustment of the decision variable, that is, df âˆ— /dÎ± = âˆ‚f/âˆ‚Î±. Equivalently, the value of the indirect effect given by the expression (âˆ‚f/âˆ‚x)(dx/dÎ±) equals zero and may be disregarded. Analogous ï¬ndings apply when there are multiple decision variables and multiple parameters and in the case of constrained optimization.3 This fundamental principleâ€”the particular simpliï¬cation due to the ï¬rst-order condition(s) of optimizationâ€”ï¬nds a variety of applications. In the

### Journal

History of Political EconomyDuke University Press

Published: Mar 1, 2004