The ambiguous effect of full automation on wages

By trammell @ 2025-02-07T02:53 (+13)

This is one of two posts I’m putting up today on how little economic theory alone has to say about the effects full automation would have on familiar economic variables.
The other is “The ambiguous effect of full automation on GDP growth”.

Introduction

Let full automation mean a state of affairs in which AI and robotics can single-handedly produce everything we like to consume, and which can produce any unit of output using inputs (e.g. electricity) that currently cost less than what humans currently earn just for the labor contribution to producing the same unit. Many people think that, though partial automation has greatly increased wages historically, full automation would inevitably lower wages. Some people have the opposite intuition: that even full automation would inevitably raise wages, or at least not lower them.

Almost any economic model agrees that full automation would (1) greatly increase output and (2) greatly decrease the labor share, meaning the fraction of output paid out as wages. But at least if we’re just trying to reason from econ theory first principles, wages—the product of output and the labor share—could rise or fall. Which effect prevails ((1) or (2)) depends on details which, in my experience, people with strong opinions on the question don’t always notice. So the goal of this post is to clarify this at a high level.

Even if you think the ultimate answer is obvious one way or the other, hopefully this summary is helpful for seeing what model assumptions are implied by that answer, and for pinpointing where others might disagree.

Throughout, will denote output, $K$ will denote capital (meaning equipment used in production, including computers and robots, not money), and $L$ will denote labor, and we will assume that the wage rate equals the marginal product of labor $d Y / d L$ , as it would in a fully competitive setting. Nothing qualitative will change if we instead assume that employers (or employees) have some market power, as long as the wage stays within some fraction (or multiple) of $d Y / d L$ .

Homogeneous factors

Suppose for now that production is a function of capital and labor, each of which is homogeneous. Each of these inputs (called “factors”, even though they’re not generally factors in the mathematical sense) has a “factor-augmenting technology” term. That is,

Y = F (A K, B L),

where $A$ and $B$ are capital- and labor-augmenting technology respectively, in the sense that (say) doubling $B$ means we now produce as much as if we had twice as much labor as before. $A K$ and $B L$ will be called “effective capital” and “effective labor” respectively.

Currently, because we have not yet achieved full automation, $F (\cdot)$ is such that the factors are gross complements: fixing $B L$ , output would rise only toward an upper bound if $A K$ grew indefinitely. Factories eventually produce very little with hardly anyone around to operate each one. So even if all of output were invested as capital each year and not consumed, GDP would not be able to grow beyond the upper bound imposed by the fixed $B L$ . Long-run output growth requires growth in $B$ and/or $L$ , and wage growth roughly tracks growth in $B$ .

This section considers what happens when full automation is achieved by a change to the production function such that it can still be written as a function of effective capital and effective labor—we still have $Y = F (A K, B L)$ , for some $F (\cdot)$ —but $F (\cdot)$ is now consistent with full automation.

We will also assume that $K$ grows large and that $B$ stays fixed. The first assumption lets us set aside the case in which full automation is technologically feasible but the robots are never built. The second lets us set aside the case in which full automation in isolation lowers wages but this is offset by a big advance in labor-augmenting technology (or in isolation raises wages but this is offset by a big fall in labor-augmenting technology).

Constant returns to scale

When modeling production at this level of abstraction, $F (\cdot)$ is typically assumed to be constant returns to scale (“CRS”) in $K$ and $L$ jointly, on the thought that, fixing technology, if you doubled the world with all its labor and all its capital, you would double output. In fact $F (\cdot)$ might actually be somewhat decreasing returns to scale since we’re not accounting for natural resources, like land, or somewhat increasing returns to scale due to the fact that a doubled world would allow for more specialization,^[1] but these scenarios are left to the next section.

If automation progresses, with capital able to do ever more of what a human can do, the substitutability between the factors will rise—albeit perhaps not monotonically or smoothly—to the point that they are eventually gross substitutes: capital will be able to produce everything that we consume and every form of capital (or a good enough substitute for the things we currently consume and use as capital). In this case, even fixing $B L$ , output grows in effective capital indefinitely. Note that this is a weaker condition than the condition that capital can perfectly substitute for labor: capital and labor might still be somewhat complementary. But in conjunction with high $A$ (so that even a little capital—electricity etc.—is enough to replace a worker), this state of affairs is equivalent to full automation.

A simple example of a production function in which $A K$ and $B L$ are gross but not perfect substitutes would be

Y = (\sqrt{A K} + \sqrt{B L})^{2} .

Observe that the function is CRS; that labor is not necessary, since output is linear in $K$ (it equals $A K$ ) if no one does any work at all and the marginal product of capital, $d Y / d K$ , is always above $A$ if $B L > 0$ ; and nevertheless that there is some complementarity, with wages

d Y / d L = (\sqrt{A K} + \sqrt{B L}) \sqrt{B / L}

increasing in $A K$ . So given high $A$ and $K$ , wages are high.^[2] This is true even though, again, the labor share is low.

If the factors are perfect substitutes, production now takes the form

Y = A K + B L .

High values of A or K now no longer imply high wages. But they don’t drive wages to zero, either; wages just equal B. If one believes that B will in fact stop growing given full automation even though it has grown historically, the implication is simply that wages stagnate. As for where wages stagnate in this case, first principles offer no guidance. Korinek and Suh (2024) do a calculation that they argue shows that wages in this scenario would stagnate at a low level, but the calculation is misguided.^[3]

Increasing or decreasing returns to scale

If the factors are perfect substitutes and there are increasing returns to scale (IRS), e.g. if

Y = (A K + B L)^{2},

then we again get the conclusion that high values of $A$ and $K$ imply high wages, for the same reason that in this case a larger population implies higher wages.

If the factors are perfect substitutes and there are decreasing returns to scale (DRS), e.g. if

Y = \sqrt{A K + B L},

then high values of A and K imply low wages, as the world gets crowded with robots and there is little for each human (or each robot) to add.

Summary

Given only “neutral” factor-augmenting technology, to reliably get the result that the increase in substitutability between capital and labor lowers wages, we need

decreasing returns to scale and
substitutability great enough that the decreasing returns to scale outweighs the fact that effective capital is now plentiful and maybe complementing labor a little bit. In the extreme, as shown above, decreasing returns to scale + perfect substitutability lowers wages.

I think this is the closest story to the intuition that people often have when they think perfectly human-replacing AIs/robots will lower wages. But the decreasing returns premise is missing from the logic of the intuition as it’s often first expressed: “why will anyone hire me at a decent wage if there’s a robot that can do the same work for the cost of a kilowatt-hour of electricity?” In the CRS version of this scenario, a kwh of electricity can now produce a lot of valuable goods and services, so the power plant owners will be paid a lot for each one, just as you were paid a lot for your own efforts all along. But this is no skin off your back.

The intuition expressed by the quote is perhaps driven by the fact that the value produced by a given job-type exhibits decreasing returns to scale: if we double some industry in isolation, fixing all others, the cost of its inputs will typically rise and the cost of its outputs will typically fall. So if cheap human-substitutable robots are introduced to a particular line of work, the marginal value of a human-sized unit of work—the wage—falls. Given CRS or IRS, as discussed, this logic doesn’t extend to the economy as a whole. But it does if even aggregate production is DRS: if the existence of many robots in the world lowers the productivity of each person (or robot), e.g. because each then has fewer natural resources to work with.

In sum, letting PS stand for perfect substitution and GS stand for gross but not perfect substitution:

	GS	PS
DRS	?	–
CRS	+	?
IRS	+	+

Heterogeneous factors

Heterogeneous labor

Suppose capital is interchangeable, but labor comes in multiple varieties, which people can’t move between—e.g. people have different innate skills, or we’re just thinking on a time horizon shorter than the time it takes to retrain. Then full automation can have different impacts on wages for each labor type.

To analyze the impact of full automation on wages for a given type of labor, we can just analyze production as in the “homogeneous labor” case, with the type of labor we’re focusing on taking the place of labor and the existence of the fixed quantities of the other types of labor being baked into the production function.

For instance, suppose there are two labor types, denoted $L_{1}$ and $L_{2}$ , and the full-automation production function we get is

Y = (\sqrt{A K} + \sqrt{B_{1} L_{1}})^{2} + B_{2} L_{2} .

This is CRS and GS in labor as a whole, but this does not mean that the introduction of full automation cannot lower anyone’s wages. It may lower almost everyone’s.

To see what happens to $L_{1}$ ’s wages $d Y / d L_{1}$ in this case, let $B L$ replace $B_{1} L_{1}$ and think of $B_{2} L_{2}$ as a constant baked into the production function. We’re now in the old CRS + GS case; $d Y / d L_{1}$ rises.

But to see what happens to $L_{2}$ ’s wages $d Y / d L_{2}$ in this case, let $B L$ replace $B_{2} L_{2}$ and think of $B_{1} L_{1}$ as a constant. We’re now roughly in the old CRS + PS case^[4]; the effect on $d Y / d L_{2}$ is ambiguous. (It just depends on $B_{2}$ , which is unaffected by $A$ and $K$ .)

Heterogeneous capital

Above we noted the mistake of concluding “no one will hire me at a decent wage if there’s a robot that can do the same work for the cost of a kilowatt-hour of electricity”, without realizing that (at least given homogeneous factors) this conclusion relies on the forgotten assumption of decreasing returns to scale. In my experience, this mistake is disproportionately made by people without economics backgrounds.

There is a parallel mistake I disproportionately see made by people with economics backgrounds: concluding that given GS + CRS (or IRS), growth in capital or capital-augmenting technology must raise wages, or at least cannot lower them. It seems clear enough where this intuition comes from: “since the factors are at least a little complementary (GS), increases in $A$ or $K$ raise wages, and there’s no ‘crowding’ from DRS to offset this”. But this conclusion in turn relies on the forgotten assumption of homogenous capital. It is not true in general.

When modeling heterogeneous capital, we won’t assume that one type of capital can’t be turned into another, as we did with labor. We’ll assume CRS throughout this section, to isolate the effect of heterogeneous capital from the effect of aggregate returns to scale, but introducing DRS or IRS as well has the usual wage-decreasing or -increasing effects.

Suppose

that capital can be used in different ways, some of them complementing labor and some not;
that each of these uses of capital comes with its own technology term; and
that the technology augmenting the more substitutable applications of capital grows more than that augmenting the more complementary applications of capital.

For instance, suppose the production function is

Y = A (1 - z) K + (z K)^{α} (B L)^{1 - α} .

(Note that this production function is CRS + GS.) The interpretation is, there are two sectors, i.e. two ways of producing things: with capital only (the “ $A K$ ” bit) and with both capital and labor (the other bit). With a given amount of capital, firms allocate it between the two sectors—they set $z \in (0, 1]$ —so that the marginal product of capital is equal in each sector:

d Y / d ((1 - z) K) = d Y / d (z K)

⟺ A = α (z K)^{α - 1} (B L)^{1 - α} .

(Or they set $z = 1$ if $A$ and $K$ are low enough that, even putting all the capital into the second sector, its marginal product—the right-hand side of the equation above at $z = 1$ —is still less than $A$ . But we’re assuming large $A$ and $K$ .) Then if $A$ rises to infinity, to keep the marginal products equal, $z K$ falls to zero regardless of how much $K$ grows. Capital is reallocated from the second sector to the first. If capital can be very productive on its own, only the most necessary tools for complementing human work remain worth using capital for. Since wages are

d Y / d L = (z K)^{α} B^{1 - α} (1 - α) L^{- α},

wages also fall toward zero.

Introducing a technology term on the $z K$ is roughly equivalent to allowing for the possibility of growth in $B$ , so let us now allow for this. Growth in $B$ can partially offset the decline in wages, both directly via the equation just above, and indirectly via the equation before it, by mitigating the extent to which increases in $A$ induce a fall in $z$ . If $B$ and $A$ grow at the same rate, it follows from the equation before last (“ $A = . . .$ ”) that $z K$ will be constant. In combination with growth in $B$ , this implies that in this case—in which we reach full automation with two kinds of capital, and the technology augmenting the kinds of capital that complement labor grows at the same rate as the technology augmenting the robot-only production environment—wages rise. There is an intermediate growth rate of $B$ , greater than 0 but less than the growth rate of $A$ , such that wages are stagnant.

In sum, given full automation (with advanced capital-augmenting technology) and CRS, and given that there are different uses capital which complement labor to different extents, whether wages rise or fall depends on how much more quickly the technology term on the substitutionary uses of capital rises than that on the more labor-complementary uses of capital.

^{^}
Since a relatively small share of output is currently paid for the use of natural resources, this force for decreasing returns to scale appears weak. Since opening borders to trade between similar countries doesn’t increase output in either country very much, the specialization increasing returns to scale also appears weak. (Recall that we’re holding technology fixed here.) So in practice I agree with the standard view that CRS is a reasonable approximation as long as we’re not too far from current margins.
^{^}
In a dynamic setting, wages grow superexponentially, if A grows without bound and the savings rate doesn’t plummet. See Nordhaus (2021).
^{^}
Their aim is to ask where wages would stagnate in the world where, in some sense, as much as possible stays the same except that (i) capital starts being able to fully substitute for labor and (ii) B is fixed. The way they do this is to
i) write down a constant elasticity of substitution production function in capital and labor,
ii) set the function’s two free parameters (called the “substitution parameter” and the “share parameter”) so that the function reasonably approximates production today, and then
iii) take a limit as the substitution parameter goes to 1 (perfect substitution) and the share parameter is fixed.
But as I’ve confirmed in discussion with Anton, there is no particular justification for taking the limit in this way. Even under the restriction of CES production, you could just as well parametrize a CES production function by (a) the substitution parameter and, say, (b) the share parameter - the substitution parameter, and take the limit as (a) goes to 1 holding (b) fixed (which would imply a rising rather than fixed share parameter). In short, the counterfactual is quite deeply undefined; you could get an arbitrarily different answer to the question of where wages would end up if “everything stayed the same except capital started being able to fully substitute for labor”.
^{^}
Strictly speaking, the extra “constant × $(A K)^{½}$ ” term we get from expanding the squared expression makes this a case of DRS + GS, which is also ambiguous.