# Describing the Decline of Capital per Worker

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The last post I did on the composition of productivity growth documented that recently we appear to be using productivity to reduce our capital/worker, as opposed to increasing the growth of output per worker. The BLS measure of ${K/L}$ is actually shrinking in 2011-2013. That is an anomaly in the post-war era, and seems worth digging into further. Here is some more detail on what is driving the negative growth in ${K/L}$.

• Let me start with a correction. I said in the last post that the BLS was including residential capital in their calculation of ${K}$, but that imputed income from owner-occupied housing was not included in ${Y}$, and that seemed strange. The BLS includes tenant-occupied residential capital in their calculation of ${K}$, and tenant-occupied rents as part of their measure of ${Y}$. They exclude owner-occupied residential housing from ${K}$, and imputed owner-occupied rents from ${Y}$. In short, it seems kosher.
• The decline in ${K/L}$ in the last few years is a function of ${L}$ growing faster than ${K}$, but ${K}$ is still growing. The figure below shows the separate growth rates of both ${K}$ and ${L}$ over the last 20 years.

The growth in ${L}$ is relatively large compared to ${K}$. Why is ${L}$ growth so large? This is a composite measure created by the BLS that measures hours worked, and is weighted by worker type (education, etc..). So it is quite possible to have very strong growth in ${L}$ because hours worked of those employed are higher, even though the absolute number of workers is not growing rapidly. Regardless, ${K/L}$ is falling because growth in ${K}$ is relatively slow. But it is not negative.

• Is the slow growth in ${K}$ caused by any particular type of capital? The BLS has separate measures of equipment, structures (think warehouses), intellectual property (think software), land, rental housing, and inventories. We can look and see which, if any, of these are particularly responsible for the slow growth in ${K}$. What I’ve plotted here is the weighted growth rate of each category of capital. The weighting is their share in total capital income, which is how the BLS weights them to add up total capital growth. This makes the different colors comparable in how they influenced the growth of ${K}$ in a given year.

Looking over the last 4-5 years, there was clearly shrinking inventories (grayish/green) and land (red) during the recession. Since then, there has been negative growth in rental housing capital (yellow) over the last 4 years, but this is a really small effect on aggregate ${K}$ growth.

The rest of the categories are growing. But if you compare them to pre-2007 rates, they are all growing slowly. Equipment grew at about 1.8% per year, for example, in 2011-2013, but at 2.6% per year prior to 2007. Structures grew at 0.6% per year 2011-2013, but 1.5% prior to 2007. IP grew at 2.9% 2011-2013, and 5.2% prior to 2007. Rental housing shrinks at 0.6% 2011-2013, and grew at 1.1% prior to 2007. Inventories and land growth rates are roughly similar in the pre-Great Recession and post-Great Recession periods.

The overall decline in ${K/L}$ is thus not driven by any one single category of capital. Even the reduction in rental housing stock is not really that meaningful in absolute size, and it never was that big of a contributor to ${K}$ growth to begin with. This is a broad-based decline in capital growth rates.

What that indicates about the source of this change, I don’t know. I have to think harder on that. It certainly seems to indicate a secular change in investment behavior, though, rather than reallocation away from some category and into another. So explanations that build on a common drop in savings/investment rates are likely to be successful here.

• Because I love you all, I extracted the BLS aggregate labor input data from a PDF, to see what was going on. The figure shows that the BLS labor input measure (the blue bars) contracts sharply in 2008/09, and then has grown at a relatively normal rate of about 2.5% per year since then. This is driven almost entirely by changes in the growth rate of hours (red bars). The growth rate of labor “composition” (green bars) is basically consistently positive over this whole period, but at a low rate of growth. Composition is capturing the quality of labor; think education levels.

In 2011-2013 you can see that the labor input is growing at really robust rates compared to the historical series. This is the strong ${L}$ growth that, combined with the slow growth in ${K}$, is part of the slow growth in ${K/L}$. Why does it appear that labor input is growing so robustly in the BLS data? This is private business sector data only, excluding the government, which is a huge employer and has not been expanding employment much. So the private business sector labor input has been growing robustly, even though the labor input at the national level may not be growing as fast.

• The labor data and capital data seem to indicate that this is some kind of broad slow-down in investment in capital goods, and not some temporary adjustment by one type of capital. This drop occurs exactly when the Great Recession ends, so it seems that the changing financial conditions since then (ZIRP? Credit tightness?) may be responsible, as opposed to something like demographics. If it was demographics, why did all of the sudden after the GR did people decide to stop investing? Did all the Boomers get old all at once?

Whatever the cause, let me just remind everyone that there is no a priori reason that the decline in ${K/L}$ is a bad thing. A perfectly reasonable response to higher productivity is to reduce the use of inputs. But it an an anomaly, and it seems unlikely that everyone decided all at once that they’d like to shed inputs rather than increase output. Whether it has a detrimental long-run effect on growth is not something I can say given the data I’ve got.

# The Changing Composition of Productivity Growth

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After the post I did recently on profit shares and productivity calculations, I’ve been picking around the BLS, OECD, and Penn World Tables methodologies for calculating productivity growth. This has generated several interesting facts. Interesting in the sense that they raise even more questions than I had starting out. Today’s post is just one of those things that I dug up,and will throw out there in case anyone has some ideas about how to explain this.

The BLS provides a measure of private business-sector multi-factor productivity (MFP). “Multi-factor productivity” is what I’d call “total factor productivity” (TFP), but to be consistent I’ll refer to it as MFP here. Because the BLS is working only with information on private business sector output and inputs, they can do more detailed work building up capital stocks and labor inputs. The cost is that they exclude the government (all labor, no capital) and housing services (all capital, no labor) from their measures of output. So the BLS productivity number isn’t going to line up exactly with a measure of productivity based on aggrete GDP, capital, and labor stocks.

That caveat aside – and that caveat may be part of the explanation for the interesting fact I outline below – here’s the BLS series on MFP growth over the long and short run. I apologize for the x-axis in these – it was hard to get Stata to format these into a readable font size.

The basic story is familiar. MFP growth was relatively high from 1960. MFP growth has been relatively slow in the last 8 years, and I’d guess that when we have final numbers for 2014 and 2015 they’ll look similar. A decade of relatively low productivity growth.

Why?

What I’m going to work through here is not a causal explanation, but simply an accounting one. But it may be a useful accounting exercise just because it highlights that the composition of productivity growth in the last decade, and in particular for the last few years, has been different than in the past.

To get started, we need to establish how exactly you measure MFP growth. We can write growth in multi-factor productivity (MFP) as the difference between growth in output per worker and growth in inputs per worker,

$\displaystyle g_{MFP} = g_{Y/L} - \alpha g_{K/L}. \ \ \ \ \ (1)$

By itself, this is pretty straightforward. If output per worker has a high growth rate, but inputs per worker has a low growth rate, then it must be that MFP is growing quickly. We are getting more output per worker even though we aren’t adding lots of inputs per worker.

There are two things about this calculation that are going to be important for understanding MFP growth in the last decade:

• ${\alpha}$ is the weight on inputs (${K}$) in the production function, which we are assuming is Cobb-Douglas. If ${\alpha}$ were zero, then inputs like ${K}$ don’t matter at all, and all growth in output per worker is, by definition, driven by growth in MFP. The size of ${\alpha}$ influences how much the growth rate of MFP depends on the growth rate of inputs per worker.
• If ${g_{K/L}<0}$, then notice this adds to MFP growth. If we have some growth in output per worker, ${g_{Y/L}}$, but we used fewer inputs to get it, then by implication it must be that MFP was growing very quickly.

Look at what happened to ${g_{K/L}}$ over the last few years. The figure below is the growth rate of ${K/L}$ year-to-year, from 1996 until now. You can see that we have this atypical shrinking of capital per worker in this period.

If you extend the series out to 1961, you get a similar message. It is pretty atypical for ${K/L}$ to shrink, and unprecendented in the post-war era for it to shrink 4 years in a row.

Flip over to look at ${\alpha}$. Here, I have to dip in and remind everyone that this weight is not something that we can observe. We can infer it from capital’s share of costs. One reason working with the BLS data is nice is that they specifically report capital’s share of costs, not just capital’s share of output (that’s a different question for a different post). Take a look at what happens to ${\alpha}$ in the last few years.

There is a clear, completely out of the ordinary, surge in capital’s share of costs in the last few years. Thus the ${\alpha}$ that goes into the calculation of MFP growth is rising. This means that whatever is happening to input per worker is getting amplified in it’s effect on MFP growth.

Combine those two facts: ${g_{K/L}<0}$ and ${\alpha}$ rising. What do you get? You get a distinct positive effect on MFP growth. The composition of MFP growth is different than it used to be.

What we’ve got going on in the last few years is that MFP growth reflects our economy using fewer inputs to produce the same output, rather than producing more output using our existing inputs. You can see the difference in these figures. I’ve plotted ${g_{Y/L}}$ (blue bars) and ${\alpha g_{K/L}}$ (red bars) for each year. The difference between these bars is MFP growth.

Up until about 2005, we generally had high input per worker growth. MFP growth allowed us to use inputs more efficiently, and we took advantage of that by using our increasing inputs to increase output by a lot. In the last few years, though,we have taken advantage of MFP growth by shedding inputs while increasing output only a little. Those red bars below zero from 2009-2013 all imply positive MFP growth.

There is nothing inherently right or wrong about this change. But it is different. A good question is whether this is something that represents a temporary change, or whether we’ve entered on a long-run path towards lower and lower input use while output per worker only grows slowly.

From a pure welfare perspective, there is nothing to say that lowering input use makes us worse off. We have to provide fewer inputs, which is nice. But an economy that is shedding inputs rather than expanding output sure seems like a different animal. What does it imply for asset prices, for example, if we are actively letting capital stocks run down?

It is one of those asset stocks that may play a role in explaining what is going on here, by the way. Remember that caveat I made above. The BLS excludes housing services from it’s measure of output, and by housing services I mean the implicit flow of rents that home-owners receive. However, the BLS does, according to their documentation, include residential capital as part of their measure of ${K}$. The decline in housing investment since 2006/07 is going to actively drag down ${K}$, perhaps so much that it explains most of the ${g_{K/L}<0}$ – I can’t find the detailed breakdown from the BLS to be sure.

If the decline in housing stock is responsible for ${g_{K/L}<0}$, then it is implicitly responsible for a large part of the measured MFP growth that we have enjoyed in the last few years. Will that continue? It’s hard to think of a decline in the housing stock as a permanent state of affairs, so it may be a temporary deviation.

As an aside, this could imply that measured MFP growth in the early 2000’s may have been lower because of the run-up of housing capital in that period.

Regardless, it seems odd that the BLS uses residential capital in their calculation of ${K}$, while excluding housing services from their calculation of ${Y}$. I’d love to see some kind of alternative series of MFP growth where residential capital is excluded from ${K}$.

But if the change in the nature of MFP growth is true regardless of how we treat residential capital, then there is something very odd going on. Remember, I didn’t make any causal claim, solely an accounting claim. The composition of MFP growth has changed demonstrably, and now reflects declining input use per worker. Could it represent a change in the kind of technological change that we pursue or are exposed to? Are we now inventing things to eliminate the need for inputs, where before we used to invent things that made inputs more valuable? If yes, that represents a real change from several decades (and probably even longer) of productivity growth.

# Tyler, Noah, and Bob walk into a Chinese bar…

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I know that in internet-time I’m light-years behind this discussion, but Tyler Cowen recently put up a post questioning whether Chinese growth could be explained by Solow catch-up growth, and Noah Smith had a reply that said, “Yes, it could”. I just wanted to drop in on that to generally agree with Noah, and to indulge in some quibbles.

Tyler says that

It seems obvious to many people that Chinese growth is Solow-like catch-up growth, as the country was applying already-introduced technologies to its development.

and Noah rightly says that this isn’t what Solow-like catch-up growth is about.

Solow catch-up growth (convergence) is just about capital investment. That’s the convergence mechanism. And that mechanism says that if you are well below your potential, you’ll grow really fast as you accumulate capital rapidly. So the Solow story for China is that there was a profound shift(s) starting in the late 1970’s, early 1980’s that created a much higher potential level of output. That generates really rapid growth.

Does 10% growth make sense as being due to convergence? We can use my handy convergence-growth calculating equation from earlier posts to figure this out. In this case, Tyler was talking about aggregate GDP growth, so in what follows, ${y}$ represents GDP.

$\displaystyle Growth = \frac{y_{t+1}-y_t}{y_t} = (1+g)\left[\lambda \frac{y^{\ast}_t}{y_t} + (1-\lambda)\right] - 1. \ \ \ \ \ (1)$

The term ${\lambda}$ is the convergence parameter, which dictates how fast a country closes the gap between actual GDP (${y_t}$) and potential GDP (${y^{\ast}}$). The rate ${g}$ is the steady state growth rate of aggregate output.

${g}$ might be something like 3-4% for China, the combination of about 2% growth in output per capita, along with something like 1-2% population growth. The convergence term ${\lambda}$ is around 0.02. We know that Chinese growth was around 10% per year for a while (not any longer). So what does ${y^{\ast}}$ have to be relative to existing output to generate 10% growth? Turns out that you need to have

$\displaystyle \frac{y^{\ast}_t}{y_t} = 4.4 \ \ \ \ \ (2)$

to get there. That is, starting in 1980-ish, you need Chinese potential GDP to be 4.4 times as high as actual GDP. If that happened, then growth would be 10%, at least for a while.

Is that reasonable? I don’t know for sure. It’s really a statement about how inefficient the Maoist system was, rather than a statment about how high potential GDP could be. GDP per capita in China was only about $220 (US 2005 dollars) in 1980. That’s really, really, poor. A 4.4 fold increase only implies that potential GDP per capita was$880 (US 2005 dollars) in 1980. We’re not talking about a change in potential that is ludicrous. There is a good reason to think that standard Solow-convergence effects could explain Chinese growth.

But not entirely. One issue with this Solow-convergence explanation is that growth should not have stayed at 10% for very long after the reforms. That is, the Solow model says that you close part of the gap between actual and potential GDP every year, so the growth rate should slow down until it hits ${g}$. That happens pretty fast.

After 10 years of convergence – about 1990 – China’s growth rate should have been about 6.7%, and it was lower in the early 90’s than in the 1980s. But after 20 years – about 2000 – China’s growth rate should have been down to 5.3%. Yet Chinese GDP growth has been somewhere between 8-10% since 2000, depending on how you want to average growth rates, and what data source you believe.

So why didn’t Chinese growth slow down as fast as the Solow model would predict? That requires us to think of potential GDP, ${y^{\ast}}$, taking even further jumps up over time. Somewhere in the time frame of 1995-2000, another jump in potential GDP took place in China, which then allowed growth to remain high at 8-10% until now. And now, we see growth in China starting to slow down, as we’d expect in the Solow convergence story.

I think I would take Tyler’s post as being about the source of that additional “jump” in potential GDP that kept growth up around 8-10%. It may be that China had some kind of special ability to absorb foreign technology (perhaps just it’s size?). But then again, in the late 1990’s, China actively negotiated for WTO accession, which took place in 2001. Hong Kong also reverted to Chinese control in 1997. Both could have created big boosts to potential GDP.

We do not necessarily need to think of some kind of special Chinese ability to absorb or adapt technology to explain it’s fast growth. Solow convergence effects get us most of the way there. Whatever happened in the 1990’s may reflect some unique Chinese ability to absorb technology, but I’d be wary of going down that route until I exhausted the ability of open trade and Hong Kong to explain the jump in potential.

Okay, last quibble. In Noah’s post, he said that we’d expect Chinese capital per worker to level off as they get close to potential GDP. No, it wouldn’t! The growth of capital per worker will slow down, yes, but will settle down to a rate about equal to the growth rate of output per worker. The growth rate of capital per worker won’t reach zero, if the Solow model is at all right about what is happening.