The New Palgrave Dictionary of Economics

2018 Edition
| Editors: Macmillan Publishers Ltd


  • Edwin Burmeister
Reference work entry


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
$$ \max \limits_{x}f(x)\mathrm{subject}\kern0.17em \mathrm{to}\;g(x)\le \alpha $$
$$ {\displaystyle \begin{array}{l}x=\left({x}_1,\dots, {x}_n\right)\\ {}f(x)=f\left({x}_1,\dots, {x}_n\right)\\ {}{g}^i(x)={g}^i\left({x}_1,\dots, {x}_n\right)\\ {}g(x)=\left[{g}^1(x),\dots, {g}^m(x)\right]\end{array}} $$
$$ \alpha =\left({\alpha}_1,\dots, {\alpha}_m\right). $$
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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© Macmillan Publishers Ltd. 2018

Authors and Affiliations

  • Edwin Burmeister
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  1. 1.