Profit maximization producer

Introduction: We analyse a model in which both revenue and cost depend upon the input variables labor and capital.

Model: In this model we have the variables

• $L$ labor
• $K$ capital,

and the functions

• $Y(L,K)$ production quantity
• $R(L,K)=pY(L,K)$ revenue, with $p$ the selling price on the market
• $C(L,K)=wL+RK$ cost, with $w$ the cost of labor and $r$ the cost of capital
• $\pi(L,K)$ profit,

such that $\pi(L,K)=R(L,K)-C(L,K)=pY(L,K)-wL-RK$.

Note: The maximum location of the profit function is a stationary point of that function:
$$\left\{ \begin{array}{lll} \pi'_L(L,K)&=&pY'_L(L,K) - w = 0 \\ \pi'_K(L,K)&=&pY'_K(L,K) - r = 0. \end{array}\right.$$