Extra explanation: Shadow price | Wiskunde op Tilburg University

Introduction: A constrained extremum problem is given by

Optimize $z(x,y)$
Subject to $g(x,y)=k$
Where $x \in D_1$ , $y \in D_2$

Remark: The method of Lagrange results in

$\lambda$ , which can be seen as a shadow price. This shadow price gives an indication of the increase of the object function when the restriction is weakened by increasing

$k$ by one unit.

Example: Consider the following constrained extremum problem (see Example (film)).

Maximize $z(x,y)=2xy+3y$
Subject to $4x+y=10$
Where $x,y>0$

The maximum is

$z(\frac{1}{2},8)=32$ with

$\lambda=4$ . This

$\lambda$ is the shadow price and indicates that if we increase the number

$10$ in the restriction a little bit, then the value of the object function

$z$ will increase fourfold. The same extremum problem with

$k=11$ in stead of

$10$ gives approximately a

$z$ -value of

$32+4\cdot 1=36$ . You can check that the exact value is

$36\frac{1}{4}$ .