Duality of cost minimization and profit maximization 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 &amp;= 0 \Leftrightarrow R = F_k\\
\Pi_L &amp;= 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}
&amp;\min_{K,L}\{RK + wL\}\\
\text{s.t.}~~&amp; 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&amp; \Leftrightarrow R + \lambda F_K = 0\\
\mathcal{L}_L = 0&amp; \Leftrightarrow w + \lambda F_L = 0\\
\mathcal{L}_\lambda = 0&amp; \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?
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) &amp;= \Pi_K(K,L) \cdot K + \Pi_L(K,L) \cdot L  \\
\Pi(K,L) &amp;= (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 &amp; = 0\\
w + \lambda F_L &amp; = 0.
\end{align}
If $\lambda &gt;1$ then
\begin{align}
F_K(K^*,L^*) - R &amp; &lt; 0\\
F_L(K^*,L^*) - w &amp; &lt; 0,
\end{align}
if $\lambda =1$ then
\begin{align}
F_K(K^*,L^*) - R &amp; = 0\\
F_L(K^*,L^*) - w &amp; = 0.
\end{align}
and if $\lambda &lt;1$ then
\begin{align}
F_K(K^*,L^*) - R &amp; &gt; 0\\
F_L(K^*,L^*) - w &amp; &gt; 0,
\end{align}

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