Walras Law in a production economy with fixed costs
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.