Duality of cost minimization and profit maximization

Question

The firm tries to maximize profits $\Pi$ \begin{align} \max_{K,L}\{\Pi(K,L) = F(K,L) - RK - wL\} \end{align} where $F$ is the linear homogeneous production function, $R$ the rental rate of capital $K$ and $w$ the rental rate (wage) of labor $L$. FOCs are given by \begin{align} \Pi_K &= 0 \Leftrightarrow R = F_k\\ \Pi_L &= 0 \Leftrightarrow w = F_L. \end{align} Footnote 1 of page 33 in Acemoglu (2009) tells us that the FOCs can also derived by cost minimization.

With (2.6) and (2.7) being $w=F_L$ and $R=F_K$ respectively. The firm tries to minize costs \begin{align} &\min_{K,L}\{RK + wL\}\\ \text{s.t.}~~& F(k,L) = Y \end{align} where $Y$ is some output level. Set up Lagrangian \begin{align} \mathcal{L} = RK + wL + \lambda(F(K,L) - Y) \end{align}

FOCs are given by \begin{align} \mathcal{L}_K = 0& \Leftrightarrow R + \lambda F_K = 0\\ \mathcal{L}_L = 0& \Leftrightarrow w + \lambda F_L = 0\\ \mathcal{L}_\lambda = 0& \Leftrightarrow F(K,L) - Y = 0 \end{align}

• I don't see, how we can conjecture $R = F_K$ and $w = F_L$ from those conditions?

Answer 1

If $F(K,L)$ is a homogeneous function of degree one then so is $$ \Pi(K,L) = F(K,L) - R \cdot K - w \cdot L. $$ This follows straight from the definition of homogeneity. (A definition of homogeneous function can be found here.) This means that if a maximal profit exists it is zero. Otherwise you could increase all inputs by say 100%, thereby increasing both revenues and costs and thus profits by 100%. So $\Pi(K^*,L^*) = 0$.

By Euler's Homogeneous Function Theorem we have $\forall K,L$:

\begin{align} \Pi(K,L) &= \Pi_K(K,L) \cdot K + \Pi_L(K,L) \cdot L \\ \Pi(K,L) &= (F_K(K,L) - R) \cdot K + (F_L(K,L) - w) \cdot L. \end{align}

Since $\Pi(K^*,L^*) = 0$, we have $$ -(F_K(K^*,L^*) - R) \cdot K^* = (F_L(K^*,L^*) - w) \cdot L^* $$ We know that $K^*,L^* \geq 0$, so if we can show that the signs of $(F_K(K^*,L^*) - R)$ and $(F_L(K^*,L^*) - w)$ match we will have proven them to be equal to zero. Otherwise one side of the equation would be negative and the other positive. From cost minimization you have \begin{align} R + \lambda F_K & = 0\\ w + \lambda F_L & = 0. \end{align} If $\lambda >1$ then \begin{align} F_K(K^*,L^*) - R & < 0\\ F_L(K^*,L^*) - w & < 0, \end{align} if $\lambda =1$ then \begin{align} F_K(K^*,L^*) - R & = 0\\ F_L(K^*,L^*) - w & = 0. \end{align} and if $\lambda <1$ then \begin{align} F_K(K^*,L^*) - R & > 0\\ F_L(K^*,L^*) - w & > 0, \end{align}

so the signs do indeed match, hence \begin{align} F_K(K^*,L^*) - R & = 0\\ F_L(K^*,L^*) - w & = 0. \end{align}