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Part 2 / Section 7: Automation and New Tasks Section 7.3

Labor demand and the labor share at given factor prices

We can now answer how much labor demand goes down with automation (holding \(w\), \(r\), and output fixed). From (7.6), the before-and-after ratio of labor per unit of output is

\[\begin{equation} \frac{L^1/Y^1}{L^0/Y^0} \;=\;(1-D)\left(\frac{c^1}{c^0}\right)^\eta \quad\longrightarrow\quad \Delta\log(L/Y)\;=\;\log(1-D)-\eta\Gamma\;<\;0. \tag{7.13} \end{equation}\]

There are two forces reducing conditional labor demand:

  1. Labor loses the tasks in \(\Delta\mathcal{A}\), captured by displacement \(D\).
  2. The firm changes the quantities of the remaining tasks as automated services become cheaper, captured by cost-savings \(\Gamma\).

The relevant elasticity here is \(\eta\), which measures substitution between tasks. With output fixed, cheaper automated services lead the firm to use less of the remaining labor services; a larger \(\eta\) makes this quantity response stronger. This is why the cost-saving term in conditional labor demand is \(-\eta\Gamma\). For given \(D\) and \(\Gamma\), a higher \(\eta\) therefore implies a larger decline in labor per unit of output. For a small shock, the expression becomes \(\Delta\log(L/Y)\simeq-D-\eta\Gamma\).

Does the labor share also fall?

We can get a useful expression the labor share \(\alpha_L=\frac{wL}{PY}\) using \(P=c\) and (7.6),

\[\begin{equation} \alpha_L\;=\;\left(\frac{w}{c}\right)^{1-\eta}H_L. \tag{7.14} \end{equation}\]

At unchanged factor prices, automation therefore gives

\[\begin{equation} \Delta\log\alpha_L \;=\;\underbrace{\log(1-D)}_{\text{displacement}} \;+\;\overbrace{(1-\eta)\Gamma}^{\text{change in task cost shares}} \;\approx\; -D+(1-\eta)\Gamma \,. \tag{7.15} \end{equation}\]

Displacement reduces the labor share directly because there is less labor being demanded for production. However, the effect of cost-savings depends on the elasticity of substitution between tasks, \(\eta\). A reduction in costs means the denominator of the labor share (the total expenditure of the firm) goes down. This increases the labor share. But a reduction in costs creates a further substitution away from labor. That is the \(-\eta\Gamma\) term we just found for the change in labor. This gives rise to three (by now) familiar cases:

  1. \(\boldsymbol{\eta\,<\,1}\): task quantities respond weakly to relative costs. The remaining labor tasks are hard to substitute away from, so their wage bill falls proportionally less than total cost through the cost-saving channel. The positive term \((1-\eta)\Gamma\) partly offsets displacement, as in our earlier weak-links discussion.
  2. \(\boldsymbol{\eta\,=\,1}\): the quantity response exactly offsets the effect of lower total cost on the share. Only the loss of labor tasks changes the labor share: \(\alpha_L^1/\alpha_L^0=1-D\).
  3. \(\boldsymbol{\eta\,>\,1}\): task quantities respond strongly to relative costs. Substitution toward automated services reduces the remaining labor wage bill proportionally more than total cost, so the cost-saving channel reinforces displacement.

Compare this with better machines in existing capital tasks

Suppose instead that costs fall only within the original capital bundle and assignment stays fixed. Then \(D=0\), and the labor-share response is \((1-\eta)\Gamma\). It is positive when \(\eta<1\), zero when \(\eta=1\), and negative when \(\eta>1\). This reproduces the distinction between more automated tasks and better machines in the weak-links example in Section 3. A broad capital-productivity improvement or a fall in \(r\) can also move the assignment cutoff, as Section 5.2 showed. In that case we must account for both margins; capital accumulation does not, by definition, hold assignment fixed.