9.1 Lagrange

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:

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