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:
\pi'_L(L,K)&=&pY'_L(L,K) - w = 0 \\
\pi'_K(L,K)&=&pY'_K(L,K) - r = 0.