Extra explanation: Shadow price

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