Cost minimization producer

Introduction: In this model on producer behavior we determine the minimum cost for which a certain output quantity can be produced.

Model:

• $w$ is the price of labor $L$
• $r$ is the price of capital $K$
• $C(L,K)=wL+rK$ is the cost function
• $y$ is the output quantity
• $P(L,K)=y$ is the production function

Consider the following cost minimization problem

$\begin{array}{ll}
\mbox{minimize}&C(L,K)=wL+rK\\
\mbox{subject to}&P(L,K)=y,\\
\mbox{where} &  L \in D_1 \ \mbox{and} \ K \in D_2.
\end{array}
$

An extremum location $(L,K)=(c,d)$, where $c \in D_1$ and $d \in D_2$ that is not a boundary point, satisfies the following system of equations:

$\left\{
\begin{array}{lcl}
MRTS(L,K) &=&{\displaystyle \frac{w}{r}}\\[3mm]
P(L,K)&=&y.
\end{array}
\right.
$