The New Palgrave Dictionary of Economics

2018 Edition
| Editors: Macmillan Publishers Ltd

Saddlepoints

  • Edwin Burmeister
Reference work entry
DOI: https://doi.org/10.1057/978-1-349-95189-5_1701

Abstract

The assumption that economic agents act as if they were maximizing some criterion function subject to feasibility constraints is central to much of modern economic theory. A typical static problem is
$$ \max \limits_{x}f(x)\mathrm{subject}\kern0.17em \mathrm{to}\;g(x)\le \alpha $$
where
$$ {\displaystyle \begin{array}{l}x=\left({x}_1,\dots, {x}_n\right)\\ {}f(x)=f\left({x}_1,\dots, {x}_n\right)\\ {}{g}^i(x)={g}^i\left({x}_1,\dots, {x}_n\right)\\ {}g(x)=\left[{g}^1(x),\dots, {g}^m(x)\right]\end{array}} $$
and
$$ \alpha =\left({\alpha}_1,\dots, {\alpha}_m\right). $$
The Lagrangian function for the constrained maximization problem (1) is
$$ L\left(x,\lambda \right)=f(x)+\lambda {\left[\alpha -g(x)\right]}^{\prime } $$
where the prime denotes the transpose operator and where
$$ \lambda =\left({\lambda}_1,\dots, {\lambda}_m\right) $$
is a vector of Lagrangian multipliers.
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Copyright information

© Macmillan Publishers Ltd. 2018

Authors and Affiliations

  • Edwin Burmeister
    • 1
  1. 1.