Part 1 / Section 2: The Cobb--Douglas technology Section 2.1

The direction of technical change

So far, technological change has been summarized by \(z\), which we call total factor productivity (TFP). A higher \(z\) raises output from any given combination of capital and labor by the same proportion. It is neutral: it raises both marginal products proportionally without changing the MRT at a given input mix. But do all technological improvements affect capital and labor in the same way?

Better organizational practices can help workers spend less time coordinating and more time producing, making each hour of labor more productive. Advances in information technology can make computers and robots faster or more accurate, increasing the services provided by each unit of capital. We describe these possibilities as factor-augmenting technological change. Let \(A_L\) measure the productivity of labor and \(A_K\) the productivity of capital: \(A_LL\) and \(A_KK\) are the corresponding effective inputs. For example, a 10 percent increase in \(A_L\) lets the same workers provide 10 percent more effective labor, without increasing their hours. These are useful aggregate descriptions of changes in how particular tasks are performed. In Section 5, we will examine those tasks directly and distinguish making an input better at its existing tasks from changing which tasks it performs.

Can Cobb--Douglas capture the distinction between a technology that augments capital and one that augments labor? To isolate the two factor-augmenting terms, initially set the separate neutral multiplier to one and write

\[\begin{align} Y &\;=\; (A_KK)^\alpha(A_LL)^{1-\alpha} \;=\; \underbrace{A_K^\alpha A_L^{1-\alpha}}_{z}\,K^\alpha L^{1-\alpha}. \tag{2.11} \end{align}\]

Thus, defining \(z=A_K^\alpha A_L^{1-\alpha}\) gives exactly our original production function. An increase in either factor-augmenting technology is therefore equivalent to an increase in TFP.

This restriction matters for both input choices and factor rewards. The technology terms cancel from the MRT, so at given factor prices the optimal ratio remains

\[\begin{equation} \frac{K}{L}\;=\;\frac{\alpha}{1-\alpha}\frac{w}{r}. \tag{2.12} \end{equation}\]

Neither \(A_K\) nor \(A_L\) induces a change in the chosen physical capital--labor ratio when \(w/r\) is held fixed. For a fixed output target, either improvement lets the firm reduce both inputs in the same proportion. This does not mean input demand cannot change: output expansion can also affect how much capital and labor the firm uses.

At fixed quantities of capital and labor, either improvement raises output and both marginal products by the same percentage. Under competitive factor pricing, the real rewards are \(r/p=\alpha Y/K\) and \(w/p=(1-\alpha)Y/L\), so both rise proportionally as well. The gains are shared in the existing proportions \(\alpha\) and \(1-\alpha\); they are not necessarily split equally. Even a technology that directly augments capital benefits labor in this sense, and neither factor's income share changes.

Cobb--Douglas therefore rules out technology-driven changes in the input mix at fixed factor prices and in the division of income between capital and labor. We want a model that allows these possibilities when studying technologies that change what workers and machines can do. Section 3 introduces the CES production function, where the elasticity of substitution determines whether a factor-augmenting improvement raises or lowers the relative use and income share of that factor.