Part 1 / Section 1: Labor and Capital as Aggregate Inputs: Substitutability Section 1.6

Input shares and input responses

The results we just derived give us a way to see how the combination of capital and labor responds optimally to changes in market prices. But we also want to know how capital and labor respond individually. The elasticity of substitution and the income shares determine how strong the response is of the levels of each input.

To see this we start from the output constraint that defines the isoquant in (1.7). We take advantage of the fact that \(\log x\,=\,\frac{\dd x}{x}\) to write

\[\begin{align} KF_K\dd\log K\;+\;L F_L\dd\log L\;=\;0 &\longrightarrow\quad rK\dd\log K\;+\;wL\dd\log L\;=\;0 \tag{1.15} \\ &\longrightarrow\quad \frac{rK}{rK+wL}\dd\log K\;+\;\frac{wL}{rK+wL}\dd\log L\;=\;0 \nonumber \\ &\longrightarrow\quad \alpha_K\dd\log K\;+\;\alpha_L\dd\log L\;=\;0\nonumber \end{align}\]

The only issue outstanding is that we need a way to connect \(K\) and \(L\). The elasticity of substitution does that for us. From its definition and the optimality condition of the firm we write

\[\begin{equation} \dd\log K \,-\,\dd\log L\;=\; \sigma\dd\log \operatorname{MRT} \;=\; \sigma\dd\log \frac{w}{r}. \tag{1.16} \end{equation}\]

This equation gives a link between the percentage change in capital and labor (approximated by the change in their logs). The final step is using it to get isolated expressions for the response of each input. For labor we have

\[\begin{align} \alpha_K\dd\log K\;+\;\alpha_L\dd\log L\;=\;0 & \longrightarrow\quad \alpha_K \dd\log K\;+\;\left(1-\alpha_K\right)\dd\log L\;=\;0 \tag{1.17} \\ & \longrightarrow\quad \alpha_K \left(\sigma\dd\log \frac{w}{r}\,+\, \dd\log L \right) \;+\;\left(1-\alpha_K\right)\dd\log L\;=\;0 \nonumber \\ & \longrightarrow\quad \alpha_K \sigma\dd\log \frac{w}{r} \,+\, \dd\log L \;=\;0 \nonumber \\ & \longrightarrow\quad \dd\log L \;=\; - \alpha_K \sigma\dd\log \frac{w}{r} \nonumber \end{align}\]

Similar steps deliver

\[\begin{align} \dd\log K \;=\; \alpha_L\sigma\dd\log \frac{w}{r} \tag{1.18} \end{align}\]

These are local responses at fixed output and technology. For example, if capital's cost share is \(0.4\) and \(\sigma=0.5\), a one-percent rise in the wage with the rental price fixed reduces conditional labor demand by approximately \(0.2\) percent and raises conditional capital demand by approximately \(0.3\) percent. The ratio rises by \(0.5\) percent.

The responses depend on the input share of the other input, or in other words, on the complement input shares. The higher the input share of the input, the less scope the firm has to substitute into it. Similarly, the larger the input share of the other inputs the more substitutability there can be.