Part 2 / Section 7: Automation and New Tasks Section 7.2

Displacement and productivity gains

It is clear by now that an increase in the capabilities of capital can harm workers by taking away some of their tasks. However, that only happens if capital becomes a better alternative for production. This can ultimately be good for workers if it allows production to expand, or if workers are hard to substitute in their remaining tasks. We have a tradeoff in our hands. The two key questions that shape this tradeoff are:

How much of workers' existing work moves to machines?
How much cheaper does production become?

We find the answers by measuring the direct displacement of workers, \(D\), and the reduction in unit cost, \(\Gamma\), that follow automation.

\begingroup \titleformat{\paragraph}[runin]{\bfseries\itshape\color{cite_blue}}{}{}{}

How much of workers' existing work moves to machines?

Our previous results show that automation reassigns tasks away from workers, but it does not directly tell us how many workers were displaced because some tasks employ more labor than others. We can measure the direct displacement of workers by automation as the fraction of initial employment devoted to tasks that become automated:

\[\begin{equation} D\;\equiv\;\frac{\text{initial employment in newly automated tasks}}{\text{total initial employment}}. \tag{7.5} \end{equation}\]

This is a measure of displacement, it captures the work transferred to machines before output and wages adjust. It is also the fraction of the initial wage bill paid to workers in those tasks, since all workers receive the same wage.

We can be more precise and link displacement to worker productivity by using the firm's optimal labor demand. From (5.15) we get that labor demand as

\[\begin{equation} L\;=\;Y\left(\frac{c}{w}\right)^\eta H_L, \quad\text{with}\quad H_L\;\equiv\;\int_{\mathcal I_L}a_L(i)^{\eta-1}\dd i. \tag{7.6} \end{equation}\]

Here \(c\) is the unit cost of the final good; under perfect competition its output price is \(P=c\) and the bundle of tasks assigned to labor is \(\mathcal I_L \subset [0,1]\). Why does labor productivity enter with the power \(\eta-1\)? Higher productivity reduces the workers needed per unit of a task, but it also makes that task cheaper, encouraging the firm to use more of it. The task-demand condition and \(\ell(i)=y(i)/a_L(i)\) combine these two effects:

\[\begin{equation} \ell(i) \;=\;\frac{Y}{a_L(i)}\left(\frac{c}{w/a_L(i)}\right)^\eta \;=\;Y\left(\frac{c}{w}\right)^\eta a_L(i)^{\eta-1}, \qquad i\in\mathcal I_L. \tag{7.7} \end{equation}\]

At given \(Y\), \(c\), and \(w\), a one-percent increase in task productivity reduces labor per unit of task service by one percent and increases the quantity of that service demanded by approximately \(\eta\) percent. When \(\eta>1\), the quantity response dominates, so more productive labor tasks employ more workers; when \(\eta<1\), the labor saving dominates, so they employ fewer. When \(\eta=1\), the two effects exactly offset. These are comparisons across tasks within the initial allocation, where \(Y\), \(c\), and \(w\) are common to every task.

Thus \(a_L(i)^{\eta-1}\) weights tasks by the employment they actually account for in the model. The common factor \(Y(c/w)^\eta\) cancels when we divide employment in newly automated tasks by total initial employment, giving

\[\begin{equation} D\;\equiv\;\frac{\displaystyle\int_{\Delta\mathcal A}a_L(i)^{\eta-1}\dd i}{H_L^0}, \qquad H_L^1=(1-D)H_L^0,\qquad 0<D<1. \tag{7.8} \end{equation}\]

The fall in \(H_L\) comes from removing tasks from labor's bundle, not from reducing workers' productivity in the tasks they keep. With Cobb--Douglas tasks, \(\eta=1\), equal-length task intervals employ equal amounts of labor. If automation moves the cutoff from \(I_0\) to \(I_1\), displacement is simply the length of the newly automated interval divided by the length of labor's original interval:

\[\begin{equation} D\;=\;\frac{I_1-I_0}{1-I_0}. \tag{7.9} \end{equation}\]

How much does automation reduce production costs?

Displacement tells us how much work machines take over, but not how much the new technology improves production. For that, we ask how much it costs to produce the same final output before and after automation. The reduction in costs reflects the gains in productivity as tasks are reassigned away from labor and toward capital.

We measure the cost/productivity gain of the firm as the log difference between the initial and final unit costs:

\[\begin{equation} \Gamma\;\equiv\;\log\frac{c^0}{c^1}>0 \,, \quad\text{where}\quad c^t\;=\;\left(\int_0^1\min\{c_L(i),c_K^t(i)\}^{1-\eta}\dd i\right)^{\frac{1}{1-\eta}} \,. \tag{7.10} \end{equation}\]

For a small cost reduction, \(100\Gamma\) is approximately the percentage fall in unit cost. This fall in costs is translated into a lower price for the firm's output and hence higher sales, increasing the overall scale of production as we have seen in previous sections. That expansion in the scale of production can increase demand for the tasks that remain with workers. We therefore need \(\Gamma\) alongside \(D\) to compare the scope for output expansion with the work lost to machines.

The calculation is especially simple with Cobb--Douglas tasks. The log unit cost of the final good is the integral of log task costs, so unchanged task costs cancel when we compare before and after automation:

\[\begin{equation} \Gamma\;=\;\int_{\Delta\mathcal A}\log\frac{c_L(i)}{c_K^1(i)}\dd i, \qquad \eta=1. \tag{7.11} \end{equation}\]

Thus productivity gains depend both on how many tasks become automated and on how much cheaper machines make each task.

\[\begin{equation} \Gamma\;=\;\overbrace{(I_1-I_0)}^{\text{length of newly automated interval}} \;\times\;\underbrace{\text{Average log cost saving per task}}_{\log(\text{old task cost}/\text{new task cost})}. \tag{7.12} \end{equation}\]

We need both measures because the same displacement can come with very different productivity gains. A technology can take over a large fraction of workers' tasks while saving little on each one. Acemoglu and Restrepo (2019); Restrepo (2024) call these so-so automation technologies: machines are cheap enough to be adopted, but the resulting cost savings are small, resulting in small gains for the economy and potential losses for workers. \endgroup