Saddlepoints

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.