# Robots as Factor-Eliminating Technical Change

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A really common thread running through the comments I’ve gotten on the blog involve the replacement of labor. This is tied into the question of the impact of robots/IT on labor market outcomes, and the stagnation of wages for lots of laborers. An intuition that a lot of people have is that robots are going to “replace” people, and this will mean that wages fall and more and more of output gets paid to the owners of the robots. Just today, I saw this figure (h/t to Brad DeLong) from the Center on Budget and Policy Priorities which shows wages for the 10th and 20th percentile workers in the U.S. being stagnant over the last 40 years.

The possible counter-arguments to this are that even with robots, we’ll just find new uses for human labor, and/or that robots will relieve us of the burden of working. We’ll enjoy high living standards without having to work at it, so why worry?

I’ll admit that my usual reaction is the “but we will just find new kinds of jobs for people” type. Even though capital goods like tractors and combines replaced a lot of human labor in agriculture, we now employ people in other industries, for example. But this assumes that labor is somehow still relevant somewhere in the economy, and maybe that isn’t true. So what does “factor-eliminating” technological change look like? As luck would have it, there’s a paper by Pietro Peretto and John Seater called …. “Factor-eliminating Technical Change“. Peretto and Seater focus on the dynamic implications of the model for endogenous growth, and whether factor-eliminating change can produce sustained growth in output per worker. They find that it can under certain circumstances. But the model they set is also a really useful tool for thinking about what the arrival of robots (or further IT innovations in general) may imply for wages and income distribution.

I’m going to ignore the dynamics that Peretto and Seater work through, and focus only on the firm-level decision they describe.

****If you want to skip technical stuff – jump down to the bottom of the post for the punchline****

Firms have a whole menu of available production functions to choose from. The firm-level functions all have the same structure, ${Y = A X^{\alpha}Z^{1-\alpha}}$, and vary only in their value of ${\alpha \in (0,\overline{\alpha})}$. ${X}$ and ${Z}$ are different factors of production (I’ll be more specific about how to interpret these later on). ${A}$ is a measure of total factor productivity.

The idea of having different production functions to choose from isn’t necessarily new, but the novelty comes when Peretto/Seater allow the firm to use more than one of those production functions at once. A firm that has some amount of ${X}$ and ${Z}$ available will choose to do what? It depends on the amount of ${X}$ versus the amount of ${Z}$ they have. If ${X}$ is really big compared to ${Z}$, then it makes sense to only use the maximum ${\overline{\alpha}}$ technology, so ${Y = A X^{\overline{\alpha}}Z^{1-\overline{\alpha}}}$. This makes some sense. If you have lots of some factor ${X}$, then it only makes sense to use a technology that uses this factor really intensely – ${\overline{\alpha}}$.

On the other hand, if you have a lot of ${Z}$ compared to ${X}$, then what do you do? You do the opposite – kind of. With a lot of ${Z}$, you want to use a technology that uses this factor intensely, meaning the technology with ${\alpha=0}$. But, if you use only that technology, then your ${X}$ sits idle, useless. So you’ll run a ${X}$-intense plant as well, and that requires a little of the ${Z}$ factor to operate. So you’ll use two kinds of plants at once – a ${Z}$ intense one and a ${X}$ intense one. You can see their paper for derivations, but in the end the production function when you have lots of ${Z}$ is

$\displaystyle Y = A \left(Z + \beta X\right) \ \ \ \ \ (1)$

where ${\beta}$ is a slurry of terms involving ${\overline{\alpha}}$. What Peretto and Seater show is that over time, if firms can invest in higher levels of ${\overline{\alpha}}$, then by necessity it will be the case that we have “lots” of ${Z}$ compared to little ${X}$, and we use this production function.

What’s so special about this production function? It’s linear in ${Z}$ and ${X}$, so their marginal products do not decline as you use more of them. More importantly, their marginal products do not rise as you acquire more of the other input. That is, the marginal product of ${Z}$ is exactly ${A}$, no matter how much ${X}$ we have.

What does this possibly have to do with robots, stagnant wages, and the labor market? Let ${Z}$ represent labor inputs, and ${X}$ represent capital inputs. This linear production function means that as we acquire more capital (${X}$), this has no effect on the marginal product of labor (${Z}$). If we have something resembling a competitive market for labor, then this implies that wages will be constant even as we acquire more capital.

That’s a big departure from the typical concept we have of production functions and wages. The typical model is more like Peretto and Seater’s case where ${X}$ is really big, and ${Y = A X^{\overline{\alpha}}Z^{1-\overline{\alpha}}}$, a typical Cobb-Douglas. What’s true here is that as we get more ${X}$, the marginal product of ${Z}$ goes up. In other words, if we acquire more capital, then wages should rise as workers get more productive.

The Peretto/Seater setting says that, at some point, technology will progress to the point that wages stop rising with the capital stock. Wages can still go up with general total factor productivity, ${A}$, sure, but just acquiring new capital will no longer raise wages.

While wages are stagnant, this doesn’t mean that output per worker is stagnant. Labor productivity (${Y/Z}$) in this setting is

$\displaystyle \frac{Y}{Z} = A \left(1 + \beta \frac{X}{Z}\right). \ \ \ \ \ (2)$

If capital per worker (${X/Z}$) is rising, then so is output per worker. But wages will remain constant. This implies that labor’s share of output is falling, as

$\displaystyle \frac{wZ}{Y} = \frac{AZ}{A \left(Z + \beta X\right)} = \frac{Z}{\left(Z + \beta X\right)} = \frac{1}{1 + \beta X/Z}. \ \ \ \ \ (3)$

With the ability to use multiple types of technologies, as capital is acquired labor’s share of output falls.

Okay, this Peretto/Seater model gives us an explanation for stagnant wages and a declining labor share in output. Why did I present this using ${X}$ for capital and ${Z}$ for labor, not their traditional ${K}$ and ${L}$? This is mainly because the definition of what counts as “labor”, and what counts as “capital”, are not fixed. “Capital” might include human as well as physical capital, and so “labor” might mean just unskilled labor. And we definitely see that unskilled labor’s wage is stagnant, while college-educated wages have tended to rise.

***** Jump back in here if you skipped the technical stuff *****

The real point here is that whether technological change is good for labor or not depends on whether labor and capital (i.e. robots) are complements or substitutes. If they are complements (as in traditional conceptions of production functions), then adding robots will raise wages, and won’t necessarily lower labor’s share of output. If they are substistutes then adding robots will not raise wages, and will almost certainly lower labor’s share of output. The factor-eliminating model from Peretto and Seater says that firms will always invest in more capital-intense production functions and that this will inevitably make labor and capital substitutes. We happen to live in the period of time in which this shift to being substitutes is taking place. Or one could argue that it already has taken place, as we see those stagnant wages for unskilled workers, at least, from 1980 onwards.

What we should do about this is a different question. There is no equivalent mechanism or incentive here that would drive firms to make labor and capital complements again. From the firms perspective, having labor and capital as complements limits their flexibility, because they then depend on the other. They’d rather have the marginal product of robots and people independent of one other. So once we reach the robot stage of production, we’re going to stay there, absent a policy that actively prohibits certain types of production. The only way to raise labor’s share of output once we get the robots is through straight redistribution from robot owners to workers.

Note that this doesn’t mean that labor’s real wage is falling. They still have jobs, and their wages can still rise if there is total factor productivity change. But that won’t change the share of output that labor earns. I guess a big question is whether the increases in real wages from total factor productivity growth are sufficient to keep workers from grumbling about the smaller share of output that they earn.

I for one welcome….you know the rest.

## 17 thoughts on “Robots as Factor-Eliminating Technical Change”

1. “If they are substistutes then adding robots will not raise wages”

You repeatedly say stuff like this, “will not raise”, as opposed to “may fall”. The possibilities are not just same or higher. Lower is a possibility, even below the minimum wage, or subsistence, right?

This is the possibility that the authors of The Second Machine Age and others are concerned with.

• In terms of the model I set up, no, wages would not fall. They’d remain constant. Now, that’s just in this model. You could consider the possibility that firms get so productive using the capital/robots that they just quit using labor at all, in which case the possibility is that wages would drop.

• I’ll have to look at the model if I can find the time — Seems funny that wages can only move in one direction. I wonder intuitively why.

If the next generation of robots/computers/machines is far more of a substitute than a complement for the unskilled, sure seems like it could shift their demand curve in, maybe way way in, especially since, with advances in telecom, screen resolution, virtual reality, etc., more and more the unskilled in the US will have to compete with the unskilled in China, India, Africa, etc.

Brynholffson and McCafee certainly take this possibility seriously; from page 179 of “The Second Machine Age”:

“In principle the equilibrium wage could be one dollar an hour for some workers, even as other workers command a wage thousands of times higher…

…And in theory this can affect a large number of people, even a majority of the population, even if the overall economic pie is growing.”

And I’ve just finished a thorough, careful read of Oxford’s Frey and Osborne’s paper “The Future of Employment: How Susceptible are Jobs to Computerisation”. Their methodology looks sound. At its core it’s based on careful studied opinions of the near term computerisability of a wide range of occupations by tech experts in the Oxford engineering department. They conclude:

“Figure IV reveals that both wages and educational attainment exhibit a strong negative relationship with the probability of computerisation. We note that this prediction implies a truncation in the current trend towards labour market polarization, with growing employment in high and low-wage occupations, accompanied by a hollowing-out of middle-income jobs. Rather than reducing the demand for middle-income occupations, which has been the pattern over the past decades, our model predicts that computerisation will mainly substitute for low-skill and low-wage jobs in the near future. By contrast, high-skill and high-wage occupations are the least susceptible to computer capital.” (page 42)

“While nineteenth century manufacturing technologies largely substituted for skilled labour through the simplification of tasks (Braverman, 1974; Hounshell, 1985; James and Skinner, 1985; Goldin and Katz, 1998), the Computer Revolution of the twentieth century caused a hollowing-out of middle-income jobs (Goos, et al., 2009; Autor and Dorn, 2013). Our model
predicts a truncation in the current trend towards labour market polarisation, with computerisation being principally confined to low-skill and low-wage occupations. Our findings thus imply that as technology races ahead, low-skill workers will reallocate to tasks that are non-susceptible to computerisation – i.e., tasks requiring creative and social intelligence. For workers to win the
race, however, they will have to acquire creative and social skills.” (page 45)

• Thats something of the detail running beneath the Peretto/Seater set-up. This kind of innovation that Frey and Osborne are talking about is like increasing the alpha parameter in the Peretto/Seater setting. This makes it more likely that we are in the substitutes situation, and wages get de-coupled from capital accumulation.

• And if I might ask, though, can you say the intuition why wages can only move in one direction. I’ve never heard of a model where the price is only capable of moving one way.

• The wage is pinned down by the level of total factor productivity, A, in the model. So it could fall if productivity falls. But nothing in the model endogenously acts to make the wage fall – but that’s just a simplification in the model.

2. I think that this is THE key insight for the less-distant-than-you-think future: “The only way to raise labor’s share of output once we get the robots is through straight redistribution from robot owners to workers.”

It is also the one which engenders the most discomfort in morally straight capitalists who know—just know—that “handouts” would mean the end of life as we know it. (The marginal utility of wealth clearly declines, but greed does not; show me the completely satiated earner of $2,000/month. Or$10,000…)

There is every reason to believe that labor substitution will be “turtles all the way down” and that the marginal value of an hour’s work will tend to zero, even without quibbling about what constitutes perfect AI, etc. But despite all the observational and theoretical reasons for skepticism, people will, when confronted with the reality, start redistributing. Feel free to set a calendar reminder for 2040 and I’ll owe you a beer if I’m wrong.

• Mike – overall I think you’re right on. Even if people remain employed in some capacity while robots do all the “work”, the implication is that the share of output going to workers will be going down. And then ultimately I think some kind of redistribution will occur. Either we’ll redistribute the robots, so that everyone owns one/many, or we’ll redistribute the earnings accruing to those robots.

3. Dietrich,

Thanks for the nice review of my article with Pietro. I want to amplify your remarks in two ways, the second of which bears on the last part of your discussion and also on the comments submitted so far. I also want to make a third point to clarify the discussion of our model’s implications for the path of wages.

(1) One thing you do not mention in your review is that the main idea behind the paper is that firms themselves create the new kinds of production functions by doing R&D that changes alpha_overbar. Firms don’t just have an array of technologies presented to them and then choose from among them. Firms spend resources on changing the array. That is the central idea of the paper. It also leads to one of the major implications, which is that the kind of linearity in the dynamic system that Jones and Growiec identified as necessary for sustained growth arises endogenously in our model rather than being imposed as a (rather difficult to believe) deus ex machina.

(2) Our model omits factor-augmenting technical change, which is the kind considered by virtually all the rest of the literature. We deliberately excluded factor-augmenting technical progress from the model for the sake of analytic convenience. Doing that also can be regarded as good scientific practice because it isolates the effects of factor-eliminating technical change. A complete model, however, would have both kinds of change, and factor-augmenting change can raise wages. It is worth noting, too, that much factor-augmenting change seems to be embodied in physical capital rather than in the workers, which is why so much historical technical progress has been “de-skilling.” That means that even completely unskilled workers can see their wages rise because of factor-augmenting technical progress.

(3) Even in the context of our model, it is wrong to conclude that wages will fall at any time. Wages comprise two components: the return to unskilled labor and the return to skill. Our analysis shows that factor-eliminating change reduces the former but raises the latter. The analysis thus has no prediction for the sign of the change in wages. It does have an implication for the relative wages of “unskilled” and “skilled” workers, but keep in mind that in developed countries there are virtually no workers with no skill at all. So it well may be that all workers’ wages will rise, though some more than others, and that is without even considering the additional effects of factor-augmenting change mentioned in the previous point.

• John – thanks for reading!. I fully admit that I stripped out a lot of nuance from your paper for the post. My goal was to think about what kind of technical change would produce something like stagnant wages and/or a falling labor share of income. In a really simplified version of your model, where there is no human capital, that’s what you can get.

(1) I didn’t want to get into the real interesting dynamic implications of the paper for a blog post, which is why I glossed over the endogenous choice of alpha. Somewhere towards the end I nodded in that direction by saying that your paper implies that firms will inevitably ensure that alpha rises enough that we get perfect substitution – but I skipped all the math behind that. From the perspective of explaining endogenous growth, I’m with you that this is the big feature of the model. For talking about labor’s share/wages, it was less crucial to touch on that.

(2) Right on. Wages could certainly rise if TFP goes up or if there were labor-augmenting technical change. I probably should have added a last paragraph about that to be clear that wages are not stagnating by necessity.

(3) This was just my over-simplification for the post, as I mentioned above (and noting your second comment). However, one could also think of technical change that eliminates human capital from the production function, as long as we’re talking about factor-eliminating changes. Then wages would be ambiguous, I think. You would have a falling share of output going to human capital, but perhaps rising stocks of human capital, so overall wages might go up or down.

• Dietrich,

I understood what you were trying to do and why you simplified the presentation the way you did. I just wanted to alert readers of your blog to aspects of the paper that you did not mention.

An interesting bit of intellectual history sort of related to that point is that Pietro and I did not start out trying to show any particular result. We did not have a set of facts we were trying to explain. Instead, we were motivated entirely by pure intellectual curiosity. The usual treatment of tech change assumes that it takes place through TFP. There are other parameters in the production function, though, so why can’t tech change affect them, too? After all, a production function is just a mathematical representation of technology, and there seems to be no reason to assume that only one parameter of that representation can be changed by tech progress. An extremely simple production function with a second parameter is the Cobb-Douglas, so we asked ourselves what would happen if it were possible to change that second parameter (the exponent) by doing appropriate R&D. We had some idea what the results would look like from some early work I did, but when we did the math we found all sorts of things we had not expected or even suspected. One of them was the thing you emphasize a lot, the bit about firms using more than one technology. We didn’t even have that in our first draft of the paper but figured it out later when we were trying to make sense of some results we had in the first version of the model. One of those results was that R&D apparently could make alpha go up or down. That turned out to be wrong because we had overlooked the possibility of using more than one technology at a time. Using two technologies convexifies the overall production technology and changes the results. In particular, it eliminates the possibility of tech change that reduces alpha. We give the economic intuition for that conclusion in the paper. Once you see it, it’s “obvious.” It sure wasn’t obvious when we started! In any case, the unexpected result was that the exponent on reproducible factors always rises in our model. That rules out the possibility you mention in your reply, that tech change may eliminate human capital, which is a reproducible factor. Perhaps some other model can give that result, but it’s not obvious what that model would be.

A second interesting bit of intellectual history is that we had a hard time getting the paper published. One objection we heard often was that we did not have a motivating fact that we were trying to explain. We regard that view as unimaginative at the very least. We dislike the stifling atmosphere it creates. We hold out our paper as an example of why that attitude is a bad idea. Pure intellectual curiosity with no motivating facts at all led to interesting results that may have important practical application. That seems pretty good.

We hope the paper will encourage others to give free rein to their inquisitiveness and do things just because the question interests them. We hope it will encourage editors and referees to be more open-minded about what constitutes useful research.

• John – that’s a little disconcerting that people didn’t appreciate the paper as an intellectual exploration. As you said, there is nothing insisting that innovation occurs only through some vague TFP term. Changes in alpha seem equally valid.

I’ve wondered about the opposite type of question – factor-adding technical change. I usually think of this in terms of resources. I discover how to use oil, and now my production function includes oil as an input. That’s like saying the alpha goes from zero to some positive number for oil. I guess it’s something like a limit version of your paper with Pietro? I’ll have to mull on that some more.

• Dietrich,

In a way, that’s what happens in my model with Pietro. The world starts not knowing how to use capital (alpha is zero), learns how to make alpha go from zero to positive, and at some point learns enough to make using capital economically justifiable. At that point, K is built, and the most advanced technology goes from linear in L to Cobb-Douglas in L and K. In our model, alpha goes asymptotically to 1, which means 1-alpha goes asymptotically to zero. The latter is the factor-elimination, but it would have been just as valid to emphasize the former, the factor-addition. I believe we never thought about saying it that way. It seems to me that a model in which alpha does not go asymptotically to 1 would be essentially the variety-expansion model.

• John – right, you are doing “factor-adding” change, just adding capital to the mix. I guess what you’d want is to think about some kind of threshold effect. It pays to keep alpha=0 for some time because the fixed cost of adopting alpha>1 is large (but after you have, increasing alpha just requires investment like you have in the paper). Then you’ve got a distinct point at which we decide to “jump” to using the new input.

4. Actually, I misstated the third point. In our model, the wage to unskilled labor is constant and the return to human capital grows. (It is unskilled labor’s *share* that falls.) Wages reflect both the return to unskilled labor and human capital, so the model in fact predicts that wages will rise.