<p>Partial answer:
for simplicity let <span class="math-container">$P_c =1$</span>.<br />
The budget constraint: <span class="math-container">$c= wn + \Pi$</span><br />
Simplify (plug in <span class="math-container">$\Pi$</span>): <span class="math-container">$c= F(n)- fc$</span><br />
Goods clearing: <span class="math-container">$c = F(n)$</span></p>
<p>The household's budget constraint is inconsistent w/ goods market clearing.<br />
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.</p>
<p>One solution is to rewrite the goods market clearing condition:
<span class="math-container">$c= F(n)- fc$</span><br />
IE: some of the output good is consumed by the household, and some by the firm...</p>
<p>Alternatively, a common approach in economics is to assume one factor (say capital) is fixed in the short run (<span class="math-container">$k=\bar{k}$</span>) and rented from households. In this case <span class="math-container">$fc= r\times\bar{k}$</span>:
<span class="math-container">\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*}</span><br />
The household's problem is then:
<span class="math-container">\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*}</span><br />
<strong>GE</strong>: prices <span class="math-container">$(P_c,w,r)$</span> &amp; allocations <span class="math-container">$(n^D,k^D,c,n^S,k^S)$</span> all optimize &amp; markets clear:<br />
1 (Labor) <span class="math-container">$n^D=n^S$</span><br />
2 (Goods) <span class="math-container">$c=F(n^D)$</span><br />
3 (Capital) <span class="math-container">$k^D=k^S$</span></p>
<p>Now the household's constraint is no longer inconsistent w/ Goods market clearing.<br />
<span class="math-container">\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*}</span>
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.</p>

CC BY-SA 4.0
2021-01-03T02:44:15.933Z
2021-01-03T18:56:22.090Z
2021-01-03T18:56:22.090Z
Albert Zevelev
<p>Consider a price taking firm with fixed costs <span class="math-container">$fc \geq 0$</span>:
<span class="math-container">\begin{align*}
\Pi 
&amp;= 
\max_{n^D} \left\{ P_c F(n^D) - w\times n^D - fc \right\} 
\end{align*}</span><br />
A representative household owns this firm:
<span class="math-container">$$\max_{c,n^S} U(c,n^S) \text{ s.t. } P_c c = wn^S + \Pi$$</span><br />
<strong>Equilibrium</strong>: prices <span class="math-container">$(P_c, w)$</span> &amp; allocations <span class="math-container">$(n^D, c, n^S)$</span> s.t. all optimize &amp; markets clear:<br />
1 (Labor market) <span class="math-container">$n^D = n^S$</span><br />
2 (Goods market) <span class="math-container">$c = F(n)$</span></p>
<p>Rewrite the household's constraint:<br />
<span class="math-container">\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*}</span>
Observe the household's constraint <span class="math-container">$c = F(n)  - \frac{fc}{P_c}$</span>
is inconsistent w/ goods clearing <span class="math-container">$c = F(n)$</span>.</p>
<p>Example:<br />
<span class="math-container">$F(n)= A \log(n)$</span><br />
<span class="math-container">$\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   
$</span><br />
<span class="math-container">$\Pi = P_c A \log\left( A \frac{P_c}{w}  \right) - A P_c  -fc  $</span>.<br />
<span class="math-container">$u(c,n)=c-  \frac{n^{1+\frac{1}{\varepsilon} }}{1+\frac{1}{\varepsilon}}$</span>  s.t.
<span class="math-container">$P_c c = wn + \Pi$</span><br />
<span class="math-container">$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} }
$</span>.<br />
<span class="math-container">$\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} 
$</span></p>
<p>1 (<strong>Labor market</strong>) <span class="math-container">$n^D = n^S$</span><br />
<span class="math-container">$\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}}
$</span><br />
2 (<strong>Goods market</strong>) <span class="math-container">$c = F(n)$</span><br />
<span class="math-container">$\Rightarrow 
\left(\frac{w}{P_c}\right)^{1+\varepsilon} + \frac{\Pi}{P_c} 
= A \log\left( A \frac{P_c}{w}  \right)  
$</span><br />
<span class="math-container">$\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)  
$</span><br />
<span class="math-container">$\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}} 
$</span></p>
<p>Problem:<br />
Labor market clearing gives:
<span class="math-container">$\frac{w}{P_c}  = \left(A \right)^{\frac{1}{1+\varepsilon}}  $</span><br />
Goods market clearing gives:
<span class="math-container">$\frac{w}{P_c}  = \left(A + \frac{fc}{P_c} \right)^{\frac{1}{1+\varepsilon}} $</span></p>
<p>They are only identical if <span class="math-container">$fc=0$</span>.</p>
<p><strong>Question</strong>:</p>
<ol>
<li>is Walras law not supposed to hold here w/ <span class="math-container">$fc&gt;0$</span> ?</li>
<li>how do you set up a GE economy w/ production and fixed costs?</li>
</ol>
<p>Attic:
We can rewrite Goods-market clearing:<br />
<span class="math-container">$\frac{w}{P_c} n^S + \frac{\Pi}{P_c} = F(n^D)$</span><br />
<span class="math-container">$
\Leftrightarrow
\frac{w}{P_c} n^S + F(n^D) - \frac{w}{P_c} n^D - \frac{fc}{P_c} = F(n^D)
$</span><br />
<span class="math-container">$
\Leftrightarrow
\frac{w}{P_c} (n^S - n^D) = \frac{fc}{P_c}
$</span><br />
<span class="math-container">$
\Leftrightarrow
 \frac{fc}{P_c} = 0
$</span> if <span class="math-container">$n^D = n^S$</span></p>

<labor-economics><production-function><general-equilibrium><producer-theory><walrasian>
Walras Law in a production economy with fixed costs
Albert Zevelev
# Walras Law in a production economy with fixed costs

(Asked by: username_0 on 2021-01-03)

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 $n^D = n^S$

**Tags:** labor-economics, production-function, general-equilibrium, producer-theory, walrasian


---

## 1 answers

### Answer by username_0 on 2021-01-03. Score: 5 | Accepted answer

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.


---

