Method of Lagrange

Introduction: A constrained extremum problem is given by

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

Method: We define $L$:
$$L(x,y,\lambda)=z(x,y)-\lambda(g(x,y)-k),$$

and differentiate it with respect to $x$, $y$ and $\lambda$:

• $L'_x(x,y,\lambda)=z'_x(x,y)-\lambda\cdot g'_x(x,y)$
  
• $L'_y(x,y,\lambda)=z'_y(x,y)-\lambda\cdot g'_y(x,y)$
  
• $L'_x(x,y,\lambda)= -g(x,y)+k$

Subsequently, we put all the partial derivatives equal to zero and solve the system of equations in order to find the extremum locations.

Remark 1: We have to check whether the extremum is a minimum or a maximum.

Remark 2: $\lambda$ can be interpreted as a 'shadow price'.