<p>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 <a href="http://people.hss.caltech.edu/~kcb/Notes/EulerHomogeneity.pdf" rel="nofollow">here</a>.) 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 $\Pi(K^*,L^*) = 0$.</p>

<p>By <a href="http://people.hss.caltech.edu/~kcb/Notes/EulerHomogeneity.pdf" rel="nofollow">Euler's Homogeneous Function Theorem</a> we have $\forall K,L$:</p>

<p>\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}</p>

<p>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}</p>

<p>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}</p>

CC BY-SA 3.0
2015-10-21T13:15:26.147Z
Sorry, added the linear homogeneinty of $F$. Actually I'd like to show the reverse case. Is it possible to get the profit maximization conditions $R = F_K$ and $w = F_L$ _from_ cost minimization, as Acemoglu claims? Here you show that the cost minimization conditions can be obtained from profit maximization.
clueless
CC BY-SA 3.0
2015-10-21T14:04:31.597Z
@clueless Yes, and I think my answer covers this, but I will edit to clarify.
Giskard
CC BY-SA 3.0
2015-10-21T11:42:48.113Z
2015-10-21T15:36:39.793Z
2015-10-21T15:36:39.793Z
Giskard
<p>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 <a href="http://press.princeton.edu/chapters/s2_8764.pdf" rel="nofollow noreferrer">Acemoglu (2009)</a> tells us that the FOCs can also derived by cost minimization. </p>

<p><a href="https://i.sstatic.net/IdkiB.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/IdkiB.png" alt="enter image description here"></a> </p>

<p>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}</p>

<p>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}</p>

<ul>
<li>I don't see, how we can conjecture $R = F_K$ and $w = F_L$ from those conditions?</li>
</ul>

<optimization><profit-maximization><firm>
Duality of cost minimization and profit maximization
clueless
# Duality of cost minimization and profit maximization

(Asked by: username_0 on 2015-10-21)

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)](http://press.princeton.edu/chapters/s2_8764.pdf) tells us that the FOCs can also derived by cost minimization.

[![enter image description here](https://i.sstatic.net/IdkiB.png)](https://i.sstatic.net/IdkiB.png)

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?

**Tags:** optimization, profit-maximization, firm

---

## 1 answers

### Answer by username_1 on 2015-10-21. Score: 3 | Accepted answer

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](http://people.hss.caltech.edu/~kcb/Notes/EulerHomogeneity.pdf).) 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](http://people.hss.caltech.edu/~kcb/Notes/EulerHomogeneity.pdf) 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}

> **Comments:**
> **username_0**: Sorry, added the linear homogeneinty of $F$. Actually I'd like to show the reverse case. Is it possible to get the profit maximization conditions $R = F_K$ and $w = F_L$ _from_ cost minimization, as Acemoglu claims? Here you show that the cost minimization conditions can be obtained from profit maximization.  
> **username_1**: @clueless Yes, and I think my answer covers this, but I will edit to clarify.  

---

