Walras Law in a production economy with fixed costs Consider a price taking firm with fixed costs $fc \geq 0$:
\begin{align*}
\Pi 
&amp;= 
\max_{n^D} \left\{ P_c F(n^D) - w\times n^D - fc \right\} 
\end{align*}
A representative household owns this firm:
$$\max_{c,n^S} U(c,n^S) \text{ s.t. } P_c c = wn^S + \Pi$$
Equilibrium: prices $(P_c, w)$ &amp; allocations $(n^D, c, n^S)$ s.t. all optimize &amp; markets clear:
1 (Labor market) $n^D = n^S$
2 (Goods market) $c = F(n)$
Rewrite the household's constraint:
\begin{align*}
c
&amp;=\frac{w}{P_c} n^S + \frac{\Pi}{P_c}
\\
&amp;=\frac{w}{P_c} n^S + F(n^D) - \frac{w}{P_c} n^D - \frac{fc}{P_c}
\tag{plug-in $\Pi$}
\\
&amp;=\frac{w}{P_c} \left(n^S - n^D \right) + F(n^D) - \frac{fc}{P_c}
\tag{rearrange}
\\
&amp;= F(n)  - \frac{fc}{P_c} \tag{Labor Market: $n^D = n^S$}
\end{align*}
Observe the household's constraint $c = F(n)  - \frac{fc}{P_c}$
is inconsistent w/ goods clearing $c = F(n)$.
Example:
$F(n)= A \log(n)$
$\Rightarrow w=\frac{P_c A}{n} 
\text{ } \&amp;\text{ } n^D(w)= A \frac{P_c}{w} 
\text{ } \&amp;\text{ } Y= A \log\left( A \frac{P_c}{w}  \right)  
\text{ } \&amp;\text{ } wn = A P_c   
$
$\Pi = P_c A \log\left( A \frac{P_c}{w}  \right) - A P_c  -fc  $.
$u(c,n)=c-  \frac{n^{1+\frac{1}{\varepsilon} }}{1+\frac{1}{\varepsilon}}$  s.t.
$P_c c = wn + \Pi$
$U(n)= \frac{w}{P_c} n + \frac{\Pi}{P_c} 
-  \frac{n^{1+\frac{1}{\varepsilon} }}{1+\frac{1}{\varepsilon}}
\Rightarrow
\frac{w}{P_c} = n^{\frac{1}{\varepsilon} }
$.
$\Rightarrow n^S(w) = \left(\frac{w}{P_c}\right)^\varepsilon 
\text{ } \&amp;\text{ } c=\left(\frac{w}{P_c}\right)^{1+\varepsilon} + \frac{\Pi}{P_c} 
$
1 (Labor market) $n^D = n^S$
$\Rightarrow A \frac{P_c}{w} = \left(\frac{w}{P_c}\right)^\varepsilon 
\Rightarrow A  = \left(\frac{w}{P_c}\right)^{1+\varepsilon}   
\Rightarrow \frac{w}{P_c}  = \left(A \right)^{\frac{1}{1+\varepsilon}}
$
2 (Goods market) $c = F(n)$
$\Rightarrow 
\left(\frac{w}{P_c}\right)^{1+\varepsilon} + \frac{\Pi}{P_c} 
= A \log\left( A \frac{P_c}{w}  \right)  
$
$\Rightarrow 
\left(\frac{w}{P_c}\right)^{1+\varepsilon} 
+ 
A \log\left( A \frac{P_c}{w}  \right) - A - \frac{fc}{P_c} 
= 
A \log\left( A \frac{P_c}{w}  \right)  
$
$\Rightarrow 
A + \frac{fc}{P_c}
= 
\left(\frac{w}{P_c}\right)^{1+\varepsilon} 
\Rightarrow \frac{w}{P_c}  = \left(A + \frac{fc}{P_c} \right)^{\frac{1}{1+\varepsilon}} 
$
Problem:
Labor market clearing gives:
$\frac{w}{P_c}  = \left(A \right)^{\frac{1}{1+\varepsilon}}  $
Goods market clearing gives:
$\frac{w}{P_c}  = \left(A + \frac{fc}{P_c} \right)^{\frac{1}{1+\varepsilon}} $
They are only identical if $fc=0$.
Question:

is Walras law not supposed to hold here w/ $fc&gt;0$ ?
how do you set up a GE economy w/ production and fixed costs?

Attic:
We can rewrite Goods-market clearing:
$\frac{w}{P_c} n^S + \frac{\Pi}{P_c} = F(n^D)$
$
\Leftrightarrow
\frac{w}{P_c} n^S + F(n^D) - \frac{w}{P_c} n^D - \frac{fc}{P_c} = F(n^D)
$
$
\Leftrightarrow
\frac{w}{P_c} (n^S - n^D) = \frac{fc}{P_c}
$
$
\Leftrightarrow
 \frac{fc}{P_c} = 0
$ if $n^D = n^S$
Partial answer:
for simplicity let $P_c =1$.
The budget constraint: $c= wn + \Pi$
Simplify (plug in $\Pi$): $c= F(n)- fc$
Goods clearing: $c = F(n)$
The household's budget constraint is inconsistent w/ goods market clearing.
The firm pays a fixed cost that doesn't go to anyone. In a &quot;true GE model&quot; all payments have to go to someone in the economy.
One solution is to rewrite the goods market clearing condition:
$c= F(n)- fc$
IE: some of the output good is consumed by the household, and some by the firm...
Alternatively, a common approach in economics is to assume one factor (say capital) is fixed in the short run ($k=\bar{k}$) and rented from households. In this case $fc= r\times\bar{k}$:
\begin{align*}
\Pi 
&amp;= 
\max_{n^D, k^D} \left\{ P_c F(n^D) - w\times n^D - r\times k^D \right\} \text{ s.t. } k^D = \bar{k} \tag{short-run}
\end{align*}
The household's problem is then:
\begin{align*}
\max_{c,n^S, k^S} U(c,n^S) \text{ s.t. } P_c c = w\times n^S + r\times k^S + \Pi
\end{align*}
GE: prices $(P_c,w,r)$ &amp; allocations $(n^D,k^D,c,n^S,k^S)$ all optimize &amp; markets clear:
1 (Labor) $n^D=n^S$
2 (Goods) $c=F(n^D)$
3 (Capital) $k^D=k^S$
Now the household's constraint is no longer inconsistent w/ Goods market clearing.
\begin{align*}
P_c c 
&amp;= 
w\times n^S + r\times k^S + \Pi
\\ &amp;= 
w\times n^S + r\times k^S + (P_c F(n^D) - w\times n^D - r\times k^D)
\\ &amp;= 
P_c F(n^D) \tag{k, n clear}
\\ 
c &amp;= F(n^D)
\end{align*}
It appears there is no good way of modeling fixed costs in GE w/o having some household in the economy endowed w/ &amp; rent out the factor that is fixed.