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Consider a price taking firm with fixed costs $fc \geq 0$: \begin{align*} \Pi &= \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 $(P_c, w)$ & allocations $(n^D, c, n^S)$ s.t. all optimize & markets clear:
1 (Labor market) $n^D = n^S$
2 (Goods market) $c = F(n)$

Rewrite the household's constraint:
\begin{align*} c &=\frac{w}{P_c} n^S + \frac{\Pi}{P_c} \\ &=\frac{w}{P_c} n^S + F(n^D) - \frac{w}{P_c} n^D - \frac{fc}{P_c} \tag{plug-in $\Pi$} \\ &=\frac{w}{P_c} \left(n^S - n^D \right) + F(n^D) - \frac{fc}{P_c} \tag{rearrange} \\ &= 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{ } \&\text{ } n^D(w)= A \frac{P_c}{w} \text{ } \&\text{ } Y= A \log\left( A \frac{P_c}{w} \right) \text{ } \&\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{ } \&\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:

  1. is Walras law not supposed to hold here w/ $fc>0$ ?
  2. 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 $n^D = n^S$

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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 "true GE model" 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 &= \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 $(P_c,w,r)$ & allocations $(n^D,k^D,c,n^S,k^S)$ all optimize & 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 &= w\times n^S + r\times k^S + \Pi \\ &= w\times n^S + r\times k^S + (P_c F(n^D) - w\times n^D - r\times k^D) \\ &= P_c F(n^D) \tag{k, n clear} \\ c &= 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/ & rent out the factor that is fixed.

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