# Walras Law in a production economy with fixed costs

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 $$n^D = n^S$$

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.