Solving the firm's problem: Lagrangian approach¶
We solved the problem graphically, but we can also tackle the problem directly using constrained optimization. The Lagrangian of the cost minimization problem is
\[\begin{equation}
\mathcal{L}\;=\;rK\,+\,wL\,+\,\lambda\left(\overline{Y}\,-\,zF(K,L)\right). \tag{1.19}
\end{equation}\]
The first-order conditions are
\[\begin{align}
r &=\; \lambda zF_K(K,L),\tag{1.20}\\
w &=\; \lambda zF_L(K,L),\tag{1.21}\\
\overline{Y} &=\; zF(K,L).\tag{1.22}
\end{align}\]
The multiplier \(\lambda\) is marginal cost: by the envelope theorem, \(\lambda\,=\,\frac{\partial C(\overline{Y},r,w)}{\partial\overline{Y}}\).
The solution is characterized by the ratio of the first two conditions, which we obtain by dividing (1.21) by (1.20):
\[\begin{equation}
\frac{F_L(K,L)}{F_K(K,L)}\;=\;\frac{w}{r}.
\tag{1.23}
\end{equation}\]
Under constant returns, the first-order conditions determine factor proportions but not a unique scale.