A problem frequently encountered in microeconomics is the maximization or
minimization of a function under constraints. Typical examples are, in household
theory, the maximization of benefits under a budget constraint or, in business
theory, the minimization of the costs of producing a certain quantity of goods.
Both problems have been treated graphically in the corresponding sections. In
addition, there are often other conditions such as the non-negativity of
goods and quantities. Now, the formal solution of such a problem will
be presented. We will restrict ourselves to the case with two variables
($x$ and
$y$ or
$K$ and
$L$) ,
which also corresponds to the graphical analysis in the reference chapters. The
general form can be found in many textbooks, e.g. in Sydsaeter, Hammond
"Mathematik fü r Wirtschaftswissenschaftler" oder Bauer, Clausen, Kerber,
Meier-Reinhold "Mathematik fü r Wirtschaftswissenschaftler" (for free
download: https://www.uni-trier.de/index.php?id=47411). There you will also
find information on how to interpret the Lagrange-multipliers as shadow
prices.

The section is divided into the following pages:

- The Lagrange formalism: The formalism is introduced and the solution concept is presented.
- The Lagrange formalism for the example of the consumption problem: solution with the presentation of the equivalence to the graphical solution and interpretation.
- The Lagrange formalism for the example of a Cobb-Douglas utility function.
- The Lagrange formalism for the example of another utility function.
- The Lagrange formalism for any function (Please note: the capabilities of the integrated computer algebra system are limited)
- The dual problem: Here the equivalence of the maximization and minimization problem is explained
- Marginal solutions: Here, cases are considered which contain $x=0$ or $y=0$.
- The derived demand

(c) by Christian Bauer

Prof. Dr. Christian Bauer

Chair of monetary economics

Trier University

D-54296 Trier

Tel.: +49 (0)651/201-2743

E-mail: Bauer@uni-trier.de

URL: https://www.cbauer.de