Consider two types of firms, one produces good 1 and the other produces good 2. Each firm uses capital and labor to produce their good. The production function for firm \(i\) is \(f_{i}(k,l) = A_{i}k^{\alpha_{i}}l_{1-\alpha_{i}}\). The rental rate of capital is \(r\) and the wage rate is \(w\). The market price of good \(i\) is \(p_{i}\) and is assumed to be exogenous. Notice that in order to simplify the problem we have abstracted away the utility maximization part of the general equilibrium example.
Calculate the relative market shares
Setting up the Lagrangian for firm \(i\)’s cost minimization problem yields first order conditions
\[\begin{aligned}
\begin{aligned}
w - \lambda(1-\alpha_{i})\frac{q_{i}}{l_{i}} & = 0 \\
r - \lambda \alpha_{i} \frac{q_{i}}{k_{i}} & = 0 \\
A_{i}k_{i}^{\alpha_{i}}l_{i}^{1-\alpha_{i}} & = q
\end{aligned}
\end{aligned}\]
Firm \(i\) has conditional factor demands given by
\[\begin{aligned}
\begin{aligned}
l_{i}(w,r,q) & = \frac{q_{i}}{A}\left(\frac{r}{w}B_{i}^{-1}\right)^{\alpha} \\
k_{i}(w,r,q) & = \frac{q_{i}}{A_{i}}\left(\frac{w}{r}B_{i} \right)^{1-\alpha}
\end{aligned}
\end{aligned}\]
where \(B_{i} = \frac{\alpha_{i}}{1-\alpha_{i}}\). This implies a cost function of
\[c_{i}(w,r,q_{i}) = \frac{q_{i}}{A_{i}}\left(\frac{r}{\alpha}\right)^{\alpha}\left(\frac{w}{1-\alpha}\right)^{1-\alpha}.\]
If we assume that all firms operate in a competitive framework and so have zero profits, then it must be true that
\[\begin{aligned}
\begin{aligned}
p_{i}q_{i} & = c_{i}(w,r,q_{i}) \\
p_{i}q_{i} & = \frac{q_{i}}{A_{i}}\left(\frac{r}{\alpha}\right)^{\alpha}\left(\frac{w}{1-\alpha}\right)^{1-\alpha}
\end{aligned}
\end{aligned}\]
which implies that
\[q_{i} = p_{i}A_{i}\left(\frac{\alpha}{r}\right)^{\alpha}\left(\frac{1-\alpha}{w}\right)^{1-\alpha}\]
Given these quantities of output, we can now figure out the prices of the inputs used in production. To do this, start by plugging in the equilibrium quantities into each firm’s factor demands to get
\[\begin{aligned}
\begin{aligned}
l_{i}(w,r) =(1-\alpha_{i})\frac{p_{i}}{w} \\
k_{i}(w,r) = \alpha_{i}\frac{p_{i}}{r}
\end{aligned}
\end{aligned}\]
Plugging these factor demands into the market clearing equation gives
\[\begin{aligned}
\begin{aligned}
l_{1} + l_{2} = (1-\alpha_{1})\frac{p_{1}}{w} + (1-\alpha_{2})\frac{p_{2}}{w} = L \\
k_{1} + k_{2} = \alpha_{1}\frac{p_{1}}{r} + \alpha_{2}\frac{p_{2}}{r} = K
\end{aligned}
\end{aligned}\]
which implies the equilibrium factor prices
\[\begin{aligned}
\begin{aligned}
r & = \frac{\alpha_{1}p_{1} + \alpha_{2}p_{2}}{K}\\
w & = \frac{(1-\alpha_{1})p_{1} + (1-\alpha_{2})p_{2}}{L}\\
\end{aligned}
\end{aligned}\]
Assignment: Calculate the equilibrium factor demands for each type of firm.