Part 1 / Section 3: CES Production and Technological Change Section 3.2

Input shares and the elasticity of substitution

As we hinted at in the discussion of the Cobb--Douglas technology, the elasticity of substitution also determines how the division of income between capital and labor responds to changes in relative prices. When \(\sigma=1\), the shares are constant. When \(\sigma\neq1\), they change.

We have already established that optimality involves equating the relative price \(w/r\) to the marginal rate of transformation, this gave us from (3.7)

\[\begin{align} \frac{w}{r} \;=\;\frac{1-\alpha}{\alpha}\left(\frac{A_L}{A_K}\right)^{\frac{\sigma-1}{\sigma}} \left(\frac{L}{K}\right)^{-\frac{1}{\sigma}} \quad \longleftrightarrow \quad \frac{K}{L} \;=\; \left(\frac{\alpha}{1-\alpha}\right)^{\sigma}\left(\frac{A_L}{A_K}\right)^{1-\sigma} \left(\frac{r}{w}\right)^{-\sigma} \tag{3.20} \end{align}\]

We can use this to get the ratio of input shares. Recall that the input shares are \(\alpha_K\,=\,\frac{rK}{rK+wL}\) and \(\alpha_L\,=\,\frac{wL}{rK+wL}\). So their ratio is,

\[\begin{align} \frac{\alpha_K}{\alpha_L} \;=\; \frac{rK}{wL} \;=\; \left(\frac{r}{w}\right)\left(\frac{K}{L}\right) \tag{3.21} \end{align}\]

Replacing \(K/L\) from the MRT condition gives us the ratio of input shares taking into account how the firm combines capital and labor given prices

\[\begin{align} \frac{\alpha_K}{\alpha_L} \;=\; \left(\frac{r}{w}\right)\left(\left(\frac{\alpha}{1-\alpha}\right)^{\sigma}\left(\frac{A_L}{A_K}\right)^{1-\sigma} \left(\frac{r}{w}\right)^{-\sigma}\right) \;=\; \left(\frac{\alpha}{1-\alpha}\right)^{\sigma}\left(\frac{A_L}{A_K}\frac{r}{w}\right)^{1-\sigma} \tag{3.22} \end{align}\]

At \(\sigma=1\), we recover \(\alpha_K=\alpha\) and \(\alpha_L=1-\alpha\).

These expressions are useful because they tell us about how inequality in income between input factors changes with relative prices and with technology:

\[\begin{equation} \log\left(\frac{\alpha_K}{\alpha_L}\right) \;=\; \sigma\log\left(\frac{\alpha}{1-\alpha}\right) \;+\;(1-\sigma) \left[ \log\left(\frac{r}{w}\right) -\log\left(\frac{A_K}{A_L}\right) \right]. \tag{3.23} \end{equation}\]

Holding technology fixed, the relative-share response is

\[\begin{equation} \dd\log\frac{\alpha_K}{\alpha_L} \;=\;(1-\sigma)\dd\log\frac{r}{w}. \tag{3.24} \end{equation}\]

A fall in \(r/w\) raises capital's share when \(\sigma>1\) and lowers it when \(\sigma<1\). A change in technology has the opposite effect as being more productive is equivalent to being cheaper.

\[\begin{equation} \dd\log\frac{\alpha_K}{\alpha_L} \;=\;(\sigma-1)\dd\log\frac{A_K}{A_L}. \tag{3.25} \end{equation}\]