Part 1 / Section 2: The Cobb--Douglas technology Overview

Overview

The Cobb--Douglas production function is one of the most common functional forms for the production function we use in economics. It is

\[\begin{equation} Y\;=\;zK^{\alpha}L^{1-\alpha}, \qquad 0\,<\,\alpha\,<\,1. \tag{2.1} \end{equation}\]

However, it is also a restrictive function because it implies that the elasticity of substitution is always equal to one. To see this, consider the marginal products of capital and labor:

\[\begin{align} F_K&\;=\;\alpha K^{\alpha-1}L^{1-\alpha}\;=\;\alpha \frac{Y}{K} & F_L&\;=\;(1-\alpha)K^{\alpha}L^{-\alpha}\;=\;(1-\alpha)\frac{Y}{L}. \tag{2.2} \end{align}\]

The MRT is therefore

\[\begin{equation} \operatorname{MRT}_{L,K} \;=\; \frac{F_L}{F_K} \;=\; \frac{1-\alpha}{\alpha}\frac{K}{L}. \tag{2.3} \end{equation}\]

At the cost-minimizing input mix, the MRT must equal \(w/r\). So, we have,

\[\begin{equation} \log \frac{K}{L}\;=\;\log \frac{\alpha}{1-\alpha}\;+\;\log\frac{w}{r}. \tag{2.4} \end{equation}\]

The capital-to-labor ratio is proportional to the relative price of labor and capital. A one-percent increase in \(w/r\) leads to a one-percent increase in \(K/L\), regardless of the firm's initial position. From this we see directly that the elasticity of substitution is constant and equal to one:

\[\begin{equation} \sigma \;=\; \frac{\dd\log(K/L)}{\dd\log\operatorname{MRT}_{L,K}} \;=\; 1. \tag{2.5} \end{equation}\]

There is another implication (or restriction) of the Cobb--Douglas technology: The input shares are constant. To see this, we can use the optimality condition to write

\[\begin{equation} rK\;=\;\frac{\alpha}{1-\alpha}wL. \tag{2.6} \end{equation}\]

With this we directly get the input shares:

\[\begin{align} \alpha_K &\;\equiv\; \frac{rK}{rK+wL} \;=\; \frac{\frac{\alpha}{1-\alpha}wL}{\frac{\alpha}{1-\alpha}wL+wL}\;=\;\frac{\alpha}{1-\alpha+\alpha}\;=\;\alpha, \tag{2.7}\\ \alpha_L &\;\equiv\; \frac{wL}{rK+wL} \;=\; \frac{\frac{1-\alpha}{\alpha}rK}{rK+\frac{1-\alpha}{\alpha}rK} \;=\; \frac{1-\alpha}{1-\alpha+\alpha}\;=\;1-\alpha. \tag{2.8} \end{align}\]

The fact that the input shares are constant is intimately connected to the fact that the elasticity of substitution is constant and equal to one. Recall from our discussion of input shares that

\[\begin{equation} \dd\log\frac{rK}{wL} \;=\; (\sigma-1)\dd\log\frac{w}{r}. \tag{2.9} \end{equation}\]

This is the same as (1.13). When \(\sigma\,=\,1\), there is no change in the relative payments to capital and labor as prices change. Similarly, we have that the change in the demand for labor and capital is proportional to the change in relative prices, with the proportionality constant given by the input share of the other input:

\[\begin{equation} \dd\log L \;=\;-\alpha \dd \log \frac{w}{r} \qquad \dd\log K \;=\; \left(1-\alpha\right) \dd \log \frac{w}{r}. \tag{2.10} \end{equation}\]