Part 1 / Section 3: CES Production and Technological Change Section 3.3

Technology and factor rewards

We now turn to one of the key questions brought up by Hicks (1932):

How does technological change affect the distribution of income between labor and capital?

To tackle this we need to see how prices respond to technology that favors one input factor over another. Because our question is about technology we are going to hold the physical supplies \(K\) and \(L\) fixed and let competitive factor prices adjust to equal marginal products. The cost minimization problem implies

\[\begin{equation} \dd\log(w/r) \;=\; \dd\log \operatorname{MRT}_{L,K} \;=\; -\frac{\sigma-1}{\sigma}\dd\log A_K \;+\; \frac{\sigma-1}{\sigma}\dd\log A_L. \tag{3.26} \end{equation}\]

This gives us our answer: If \(\sigma>1\), capital-augmenting technical change raises the rental price of capital relative to the wage. If \(\sigma<1\), the result reverses. Making capital more productive raises the relative reward to labor because labor and effective capital are hard to substitute for each other. When \(\sigma=1\), all technological change is neutral and does not affect relative factor prices.

Substitutability plays a central role (again). Only when factors are easy to substitute does making one factor more productive raise its relative reward. When factors are hard to substitute, making one factor more productive ends up raising the relative reward to the other factor. In this second case the rewards goes to the factor that is now relatively scarce in effective terms, not to the factor that is now more productive.

What about the level of the wage?

Does a decrease in the relative factor price \((w/r)\) mean the wage itself falls when the rental rate rises by more? The answer under the CES technology is unambiguous: No. In fact, any increase in technology (either capital- or labor-augmenting) raises the level of the wage.

This is consistent with the result we just derived because equation (3.26) concerns the relative factor price \(w/r\), not the level of the wage. To get at the effect of technological change in the level of wages we can look at the optimality condition for labor demand

\[\begin{align} \log \frac{w}{p} \;=\; \frac{\sigma-1}{\sigma}\log z \;+\;\frac{\sigma-1}{\sigma}\log A_L \;-\;\frac{1}{\sigma}\log L \;+\;\frac{1}{\sigma}\log Y. \tag{3.27} \end{align}\]

From here we can get the effect of a change in \(A_K\) on \(w\).

\[\begin{align} \frac{\partial \log \frac{w}{p}}{\partial \log A_K} & \;=\; \frac{1}{\sigma} \frac{\frac{\partial Y}{\partial \log A_K}}{Y} \;=\; \frac{1}{\sigma} \frac{\frac{\partial Y}{\partial A_K}}{Y} A_K \tag{3.28} \\ & \;=\; \frac{1}{\sigma} \frac{ z \left( \alpha\left(A_K K\right)^{\frac{\sigma-1}{\sigma}} \;+\;\left(1-\alpha\right)\left(A_L L\right)^{\frac{\sigma-1}{\sigma}} \right)^{\frac{1}{\sigma-1}} \alpha A_K^{-\frac{1}{\sigma}}K^{\frac{\sigma-1}{\sigma}} }{Y} A_K \nonumber \\ & \;=\; \frac{1}{\sigma} \frac{ Y^{\frac{1}{\sigma}} \alpha A_K^{\frac{\sigma-1}{\sigma}}K^{-\frac{1}{\sigma}} }{Y} K \nonumber \\ & \;=\; \frac{1}{\sigma} \frac{rK}{pY} \;=\; \frac{\alpha_k}{\sigma} \;>\;0 \nonumber \end{align}\]

This applies also to an increase in the level of capital:

\[\begin{equation} \frac{\partial\log w}{\partial\log A_K} =\frac{\partial\log w}{\partial\log K} =\frac{\alpha_K}{\sigma}>0. \tag{3.29} \end{equation}\]

Thus, in this aggregate CES model, both capital deepening (a term used for capital accumulation) and capital-augmenting technical change raise the wage for every \(\sigma>0\). The rental rate may rise even more, causing \(w/r\) to fall, but that does not undo the positive effect on the level of \(w\).

Here we reach the limit of the aggregate production function. The results we have do not prove that innovation can never harm workers. Instead, they show that at this level of aggregation the model does not have a margin through which innovation can harm workers. That is why we introduce tasks later on.