# Labor's Share, Profits, and the Productivity Slowdown

There's been a slowdown in measured productivity growth, particularly in the last few years, but generally since about 2000. This is something that I've poked around at several times, and if you're reading economics blogs like this, then this shouldn't be a revelation to you.

At the same time, there has been increasing attention given to the fact that labor's share of GDP has been trending downward over the last 30 years or so. Piketty, perhaps, called the most public attention to the idea, but this is something that other people, like Loukas Karabarbounis and Brent Neiman have been working a lot on lately. The flip side of this declining labor share is a less well-documented sense that this is related to greater rents being collected by firms with more market power (Bob Solow on the topic).

What I want to do here is show how these two trends are related in some fundamental sense through how we measure productivity growth. The TL;DR version is that a falling labor share (and rising profit share of GDP) will necessarily lead to a decline in measured productivity growth, even if underlying innovation doesn't change. The reason is that if firms have increasing market power, then they are using inputs less efficiently from an aggregate perspective, and measured productivity growth is about how efficiently we use inputs. So increased market power - captured by the decline in labor share - will put a drag on productivity growth.

Lots of math follows. None of it is too daunting, but it did end up pretty dense. When we want to measure productivity, we use a residual, because productivty cannot be directly observed. Call this measured residual productivity term ${R}$. You calculate it as

$$R = \frac{Y}{K^{1-s_L}L^{s_L}}$$

where ${Y}$ is GDP, ${K}$ is the capital stock, and ${L}$ is the labor supply (which you could measure in units of human capital if you wanted). The term ${s_L}$ is labor's measured share in total output.

GDP is assumed to be produced according to a Cobb-Douglas function like

$$Y = A K^{\alpha} L^{1-\alpha}$$

where ${A}$ is true'' productivity, which is what we are trying to get a measure of. The really important thing to note here is that ${K}$ and ${L}$ are raised to powers that depend on ${\alpha}$, not ${s_L}$.

${\alpha}$ and ${1-\alpha}$ are true'' technological coefficients. The measure how GDP responds to stocks of capital and labor. But we don't know them. All we know is ${s_L}$, labor's share in GDP. We don't even know capital's share in GDP, all we know is that ${1-s_L}$ is the left-over amount of GDP paid out as returns to capital and profits.

This wouldn't be an issue if somehow ${s_L = 1-\alpha}$. And under a very precise set of conditions, these two things would be equal. If we had competition in output markets, and competition in factor markets, then ${s_L = 1-\alpha}$. But what are the chances that this describes the real world?

We can make a little headway if we allow for market power. The following relationship is something you can get by simply assuming that firms are cost-minimizers

$$s_L = \frac{1-\alpha}{\mu}$$

where ${\mu}$ is the mark-up of price over its marginal cost. For example, if ${\mu = 2}$, then the price charged is twice the marginal cost of production (which is the cost of hiring labor and capital). Under competition, P=MC, so ${\mu=1}$, and ${s_L = 1-\alpha}$. But again, do we think we really have true competition at work in the economy? Probably not. So ${\mu>1}$ to some extent.

Now that we know a little about ${s_L}$, go back to the residual calculation

$$R = \frac{Y}{K^{1-s_L}L^{s_L}} = \frac{A K^{\alpha} L^{1-\alpha}}{K^{1-s_L}L^{s_L}} = A\left(\frac{K}{N}\right)^{s_L(1-\mu)}.$$

The residual measure of productivity captures not only ${A}$ - true productivity - but also this adjustment for the capital/labor ratio. So ${R}$ is not a clean measure of ${A}$ if ${\mu >1}$.

What is the growth rate of the residual measure of productivity? That is

$$\frac{\dot{R}}{R} = \frac{\dot{A}}{A} - s_L(\mu-1)\frac{\dot{k}}{k}$$

where I used ${\dot{k}/k}$ as the growth rate of the capital/labor ratio, ${K/N}$. Again, if we had perfect competition and ${\mu=1}$, then the growth rate of the measured residual, ${\dot{R}/R}$, would be exactly equal to the growth rate of true'' productivity, ${\dot{A}/A}$. But once ${\mu>1}$, this is no longer the case, and what we can measure (${\dot{R}/R}$) need not equal what we want to measure (${\dot{A}/A}$).

This is a general issue. But it may not be totally deadly, because perhaps at least changes in ${\dot{R}/R}$ could tell us about changes in ${\dot{A}/A}$. For example, let's say that ${s_L}$ and ${\mu}$ are constant over time. And assume that the economy is essentially at steady state, so that ${\dot{k}/k}$ is growing at the same rate as true productivity. Then if the growth rate of true productivity went down, ${\dot{R}/R}$ would fall as well. Working that logic backwards, if the economy is at steady state and ${s_L}$ and ${\mu}$ are constant, then changes in the growth rate of ${R}$ are informative about changes in the growth rate of ${A}$. The slowdown in measured productivity growth we see in the data would tell us that true productivity growth (innovation?) is also slowing down.

But, this isn't true if ${s_L}$ and ${\mu}$ are changing. Are they changing? The labor share ${s_L}$ is certainly falling over the last two to three decades. What about the markup, ${\mu}$? Is that changing?

It's hard to measure that directly, but I think there is a way to infer that it almost certainly has been rising. Remember that relationship of ${s_L = (1-\alpha)/\mu}$? That came from assuming that firms are cost-minimizing (not necessarily profit-maximizing even, just cost-minimizing). That cost-minimization problem also implies that the following has to be true

$$\text{Returns to scale} = \mu (1-s_{\pi}).$$

Returns to scale'' captures the returns to scale of the true production function. What I wrote above has constant returns to scale (${\alpha}$ plus ${1-\alpha}$ add up to 1), and so the returns to scale are equal to 1. We can have a long argument about whether that is correct or not, but it isn't actually crucial for the point I'm making here.

${s_{\pi}}$ is the share of GDP that gets paid out as profits - A/K/A rents. What this relationship says is that if the share of output going to rents rises, then so must the markup. Or think about it the other way. If firms can charge higher markups, they must be earning more in rents/profits. This is just a mechanical relationship, so it doesn't necessarily have to be driven by one or the other.

Let's put this all together. We've had a decline in the labor share of GDP, ${s_L}$, over the last few decades. By necessity, this implies that the share of GDP going to rents or payments to capital have risen. If the share of GDP going to rents, ${s_{\pi}}$, went up at all, then the markup being charged by firms, ${\mu}$, must have risen as well.

Let's throw some numbers at this. Assume that ${\dot{k}/k = 0.015}$ over the last 30 years. Let the true growth rate of innovation be ${\dot{A}/A = 0.02}$ over the entire last 30 years (yes, an assumption). Start out 30 years ago by assuming the labor share is ${s_L = 0.65}$ and that the markup is ${\mu=1.1}$, so firms charge 10% over marginal cost. This means that measured productivity growth is

$$\frac{\dot{R}}{R} = 0.02 - 0.65\times(1.1-1)\times0.015 = 0.018$$

or about 1.8% per year. This is pretty close to what you see in the data for the period from 1948-1973.

Now, let the labor share fall to ${s_L = 0.60}$, and let the markup rise to ${\mu = 1.5}$. This is a pretty big markup, but for the moment I'm just trying to establish a point, so bear with me. We get that measured productivity growth is

$$\frac{\dot{R}}{R} = 0.02 - 0.6\times(1.5-1)\times0.015 = 0.015$$

or only about 1.5% per year. Measured productivity growth has fallen, even though the underlying true productivity growth rate did not change at all.

The point is that lower measured productivity growth - ${\dot{R}/R}$ - does not necessarily mean that actual innovation has slowed down. The decline in labor share is consistent with a rise in markups (and profit's share of output), which will produce a drag on measured productivity growth, ${\dot{R}/R}$. I don't think this story explains all of why measured productivity growth has fallen recently, but it probably plays a part.

Measured productivity growth is about how efficiently we use our inputs, and that is only partially related to the true rate of innovation. Measured productivity growth also depends on market power, because that also dictates how efficiently we use our inputs. If firms are gaining market power - meaning they can charge a higher markup - then this implies that they will use inputs less efficiently from a social perspective. Each individual firm is producing less than the amount they would under competition (with costs = marginal costs), and so we are not getting everything we can out of our inputs. If market power has increased, this exacerbates that issue, and so measured productivity - the efficiency of input use - will fall.

You cannot look at measured productivity growth, ${\dot{R}/R}$, and make any definitive conclusions about what is happening to true innovation or productivity growth. You cannot infer that recent innovations are less useful or productive than those that came before just because ${\dot{R}/R}$ is falling. It may be that the policies and norms transfering some share of GDP from labor to profits/rents are pushing down the growth rate of measured productivity as well.

It's also quite possible that you could actively work to curtail the profit share of GDP - through taxes or regulation or whatever - and yet see measured productivity rise as the markup goes down. Think about the example above, and how measured productivity growth is higher even though the markup (and hence the profit share) is lower.

Or think about the opposite situation, where you propose a policy that actively favors the profit share (lower taxes on businesses or entrepreneurs, weaker labor laws, allowing concentration of industries). It isn't even theoretically true that this will necessarily lead to higher measured productivity growth. In the example above, any policy that tried to use lower labor shares and higher markups would have to raise the underlying growth rate of innovation by 15% - from 2% to 2.3% per year - just to break even. That is a massive change, and I think it is fair to be completely skeptical that any of those policies could raise underlying rates of innovation by that much.

There is not an either/or choice between rapid productivity growth and a higher labor share. Repeat after me: there is not an either/or choice between rapid productivity growth and a higher labor share.

A last point is that we do care explicitly about measured productivity growth if we care at all about GDP. Measured productivity growth tells us how efficiently we use inputs to produce GDP, so anything that makes measured productivity go up - better technology (${A}$) or lower markups - is good for us in terms of producing GDP.