Growth Effects, Level Effects, and Transitional Growth

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This post is about a metaphor for explaining growth dynamics to people. It might be useful if you are either trying to learn growth theory, or teach growth theory. I think the metaphor works nicely for explaining what we mean when we talk about level and growth differences by putting into a context that students can understand. Comments are welcome, I’d like to know if it is something I should try out with a live class next year.

Imagine that every country is a car, and those cars are traveling along a two-lane highway. The farther you go along the highway, the richer you are. Instead of mile markers you have GDP per capita markers, $1,000 per person, $2,000 per person, etc. etc. Your growth rate is your speed, as it measures how fast you go from one GDP p.c. marker to the next. Doing 70 MPH? You’re growing really fast. Doing 30 MPH? You’re growing slowly. But your speed does not tell me where you are along the highway. The car going 70 MPH could be way behind the car going 30 MPH, or it could be way ahead, or it could be in the process of passing the 30 MPH car. So we need another piece of information, which is your location. In terms of growth theory, I would call your location along the highway your level. A country could be much poorer than another (way behind on the highway), much richer (way ahead), or equally rich (at the same spot on the highway). Now the level, or location along the highway, is constantly changing. So it is more accurate to think of level as “how far behind the leading car are you?”

Using this metaphor, how do we think about explaining differences in observed GDP per capita across time or across countries?

First, a “level difference” is the distance between two cars traveling along the highway at the same speed. If they are both going 55 MPH, then this distance will remain constant over time, even though both of them will continue to drive forever on the highway. Level differences are about your position on the highway relative to other cars or trucks. Level differences in GDP per capita are about one country’s position relative to another, but holding the growth rate constant.

Second, a “growth difference” is a difference in the speed of the two cars. If one is going 70 MPH and the other 55 MPH, then even if the faster car starts out behind (poorer), it will pass the slower car, and then continue to expand its lead along the highway. The faster car will always end up richer, and the gap will grow over time. Growth differences would generate massive divergence in GDP per capita, just as persistent speed differences would generate massive divergence in your location along the highway relative to a slower car.

Finally, “transitional growth” is like a car accelerating temporarily to pass a truck doing 55 MPH in the right lane. Transitional growth changes your level difference with respect to the truck. You were behind, and now you are ahead. The only way to make that happen is 70 MPH temporarily. Your measured growth rate (the speed at which the GDP pc markers fly by) is higher than 55 MPH for a minute or two, but after you pass the truck you go back to 55 MPH (there is another truck in the way). But you do not have a permanent growth difference with the truck you just passed. You fundamentally are both doing 55 MPH. Transitional growth just means you jumped ahead of the truck. Transitional growth and level differences go hand in hand. Transitional growth is how you change level differences, just like temporary acceleration to 70 MPH changes your position with respect to the truck.

When we look at the advanced economies of the world (US, Japan, W. Europe, etc..), they seem have small level differences, and little to no growth differences. They are all driving at 55 MPH, roughly. The US is ahead of Japan, Germany, and France by a few car lengths, but nothing too major. Maybe Singapore is a little ahead of the US. But they all are driving at 55 MPH.

Why doesn’t the US just accelerate, and get faster economic growth? Here we need to imagine that there is a sheriff driving along in the right lane at exactly 55 MPH. Passing the sheriff is a bad idea – he’ll arrest you if you try. The sheriff dictates the long-run growth rate at the frontier of economic growth. Whatever happens, you cannot pass the sheriff. Now, within the growth literature there is some debate on whether the sheriff himself can speed up. Chad Jones’ semi-endogenous growth theory comes to the conclusion that the sheriff could perhaps temporarily accelerate, allowing all the countries stacked up behind him to accelerate temporarily as well. But the sheriff cannot really change the fundamental speed limit of 55 MPH. Others will argue that yes, the proper set of incentives or policies could permanently allow the sheriff to speed up to 56 or 57 MPH or more. Regardless of the exact nature of the sheriff, he represents some kind of limit to how fast you can move along the highway once you are the front.

How about countries like China, which seems to have been driving at 90 MPH for a few decades? We think of this as transitional growth, not a growth difference. In other words, China will eventually slow back down to 55 MPH like all the leading countries. China was able to grow so fast because it started out miles behind the leaders on the highway. Once it accelerated up to 90 MPH, it was able to keep that speed for a long time as it zipped down the left lane past a bunch of countries. But as it approaches the sheriff, its speed will slow down, and we are already seeing a little evidence that this is happening. Where exactly it ends up relative to the US or Europe is not clear. It could end up a mile behind, a few car lengths behind, a few car lengths ahead. But its rapid growth is probably transitional growth, not a fundamental growth difference. If China really did have a faster fundamental growth rate – if it could drive 65 MPH forever – then it would pass the sheriff. We’ve never seen anyone pass the sheriff yet, so I’m inclined to think you can’t do it. But maybe China knows a guy, or has diplomatic plates or something.

When we talk about particularly poor countries – Somalia, for example – then we perhaps are looking at both growth and level differences. In level terms, they are far, far behind the leaders, miles back. And their speed appears to be even slower than the leaders, maybe only 25 MPH. So not only are these countries poor, but they are falling further and further back from the leaders. Their economic growth is not sufficient for them to catch up to the leaders.

When an Op-Ed About Growth Fails

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There’s a column in the NYT today by Daniel Cohen, titled “When the Growth Model Fails“. It is…well, I don’t know what it is. A lament? A rant?

Daniel Cohen is a good economist, so it is a shame that the column reads like the work of a politician who occasionally reads the business section of a newspaper. It is a series of disconnected tropes without any meaningful point. It is thick with “truthiness”, but nothing in the form of actual facts.

Let’s take a look:

And yet, at least in the West, the growth model is now as fleeting as Proust’s Albertine Simonet: Coming and going, with busts following booms and booms following busts, while an ideal world of steady, inclusive, long-lasting growth fades away.
…
But in its desperate search for scapegoats, the West skirts the key question: What would happen if our quest for never-ending economic growth has become a mirage? Would we find a suitable replacement for the system, or sink into despair and violence?

What does it mean that the “growth model” is fleeting? Is Bob Solow fading in and out of existence? I presume that the implication is that economic growth is fleeting, and is coming and going.

Am I supposed to believe that booms and busts are a new feature of Western economies? That is patently untrue, and Cohen knows this. Business cycles did not start happening in the last decade. A few minutes looking at long-run data (like here) will show you that even in France the frequency and severity of booms and busts were both much, much higher before World War II than after. Took me 10 minutes to download the data for France, plot it, and run some quick regressions. 10 minutes.

“..steady, inclusive, long-lasting growth fades away”. You have to unpack this with care. Steady, inclusive, and long-lasting are three separate characteristics, and there is nothing that necessitates that they appear together or in any particular combination when growth occurs. Steady? Again, look at some data. What I see is from 1820 to 1940 steady growth at about 1.2% per year, punctuated by severe recessions and booms. After 1980, I see steady growth at 1.4% per year, higher than the pre-war rate. In between WWII and 1980 I see a country experiencing a level shift to a higher balanced growth path, probably due in part to integration within Europe and technology adoption.

Long-lasting? France has been experiencing steady GDP per capita growth for 190 years. Am I supposed to believe that the downturn you can see at the tail end of the figure in 2007 represents the end of that? That the dip in French GDP per capita in 2007 implies that we either have to “replace the system”, whatever that means, or sink into despair and violence? Get some perspective.

I think what Prof. Cohen means is that the era of rapid transitional growth that France experience from 1950 to 1980 is over. Yes, it is. But did you really think that growth of 3.8% was going to last forever, when there is not a single example – ever – of a country growing at that rate in the long run? Again, perspective.

Inclusive? Now here is where we get some traction. Cohen cites that 80 percent of Americans have not seen real wage growth in 30 years. You can quibble with the exact figure, but he’s right on. The last three decades have not been good for everyone, particularly in the U.S. We do not have a problem with “the growth model”, meaning a problem with economic growth. We have a problem with the “distribution model”. So write an op-ed proposing changes to tax rules, or supporting education, or opposing excessive licensing of occupations.

Moving on:

Will economic growth return, and if it doesn’t, what then? Experts are sharply divided.

No, not really. Cohen cites Robert Gordon as a growth pessimist. Gordon is, but he doesn’t predict that growth is ending. Gordon thinks that the growth rate of GDP per capita will drop from the historical 1.8-2% per year to about 0.9-1.2% per year. This is primarily due to a slowdown in the accumulation of human capital as the population ages and the rates of college and high school completion level off. So even the pessimists don’t believe growth is over, just that it will be slower. Gordon also assumes that total factor productivity growth will be lower than in the past, which is completely unknowable. Gordon gets very “cranky old man” about how useless innovations today are (those kids and their Insta-Snap-gram-Book!).

To decide who is right, one must first recognize that the two camps aren’t focusing on the same things: For the pessimists, it’s the consumer who counts; for the optimists, it’s the machines.

Uh, no. To decide who is right we need data. Like several more years of data to see if in fact growth rates have fallen significantly. I wrote a post about this a while back. We won’t be able to to definitively say if growth has fallen below 2% per year until about 2025. Until then, there will be too much noise in growth rates to extract a signal.

What matters is whether they will substitute for human labor or whether they will complement it, allowing us to be even more productive.

Uh, no. Regardless of whether machines/robots/Skynet are a substitute or complement for human labor, we as an aggregate economy will be more productive. Whether particular individuals find themselves displaced and unable to find work depends on their own set of skills. How we treat those people is a distributional question, not a growth question.

The logical conclusion, then, is that both sides in this debate are right: We’re living an industrial revolution without economic growth. Powerful software is doing the work of humans, but the humans thus replaced are unable to find productive jobs.

Uh, no. See above regarding economic growth. It hasn’t ended just because we had a recession, and a very bad one at that. On the job replacement thing, see here. We experienced similar kinds of disruptions in the past. Can we handle this with more sympathy towards those temporarily displaced by technology? Yes. Absolutely. Again, that is a distributional problem, not a growth problem.

The point is this: If workers are to be productive again, then we must come up with new motivation schemes. No longer able to promise their employees higher earnings over time, companies will now have to adjust, compensate, and make work more inspiring.

Wait, who said workers were unproductive? Did I miss the part where everyone forgot how to do their job? And this seems close to 180 degrees from how companies would respond to an economy that stopped growing. No growth would mean a lack of new firms and/or new types of jobs, so workers wouldn’t have outside options. Firms would have even more power to motivate through fear of losing your job, because there wouldn’t be new jobs out there to escape to.

Cohen suggests that firms will have to focus on giving workers autonomy to keep them happy. He cites the Danish situation as one that produces happy workers. They are treated respectfully and given autonomy, and in return they are very productive. They have a significant safety net in place so that people don’t have to keep bad jobs just to pay the bills. Denmark self-reports as being very happy.

I am all for “the Danish model”. Here’s the thing. It’s a good idea no matter what happens to economic growth. Why should I wait to see if growth slows down to encourage companies to adopt a more positive work environment? If anything, higher growth rates would make it easier to transition to a system like this because economic growth gives people outside options.

The biggest sin of this op-ed is the lack of perspective. It presumes that we are living through not just a shift in long-run growth rates, but a cataclysmic collapse of them. If you want to make that case, then you have to bring some…what’s the word? Evidence.

But bonus points for the Proust quote to give it that affected tinge of world-weary seriousness.

Mean-Reversion in Growth Rates and Convergence

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Brad DeLong posted about the recent paper by Pritchett and Summers (PS) on “Asiaphoria” and mean-reversion in growth rates. PS found several things:

• Growth rates are not persistent. The growth rate over the last 10 years has very little information about the growth rate over the next 10 years. Growth rates “regress to the mean” as PS say.
• Growth in developing countries tends to take place in bursts of growth and bursts of stagnation. This is different from rich countries where growth variation tends to consist of mild variation around a trend rate.
• There is no reason to believe that rapidly growing economies today (China and India) will necessarily continue to grow rapidly.

Brad’s response is to take their evidence as a fundamental challenge to the standard Solow model explanation for why growth rates differ.

Lant Pritchett and Larry Summers are now trying to blow this up: to say that just as the neoclassical aggregate production function is a very bad guide to understanding the business cycle, as the generation-old failure of RBC models tells us, so the neoclassical aggregate production function and the Solow growth model built on top of it is a bad guide to issues of growth and development as well.

This is an overreaction. The mean-reversion and “bursts” that PS find are perfectly consistent with a Solow model including shocks.

Let’s start with the finding that regressing decadal growth rates on prior-decadal growth rates gives you a coefficient of something like 0.2-0.3. PS call this mean-reversion. I think it’s an artifact of convergence. Let’s imagine an economy that is following the Solow model precisely. It is very poor in 1960, and growth from 1960-1970 is about 10% per year. By 1970 it is much better off, and so growth from 1970-1980 slows to 5% per year. By 1980 this has gotten the country to steady state, so from 1980-1990 it grows at 2% per year. From 1990-2000 it is still at steady state, so grows at 2% a year again.

Now regress decadal growth rates (5,2,2) on prior-decade growth rates (10,5,2). What do you get? A line with a slope of about 0.397. Why? Because growth rates slow down as you approach steady state. Play with the numbers a little and you can make the slope 0.3 if you want to. The point is that convergence will generate just such a pattern in growth rates.

What about the unpredictability of growth rates? PS find that the correlation of growth rates across periods is very low. This is more problematic for convergence, on the face of it. If convergence is true, then growth rates across decades should be tightly correlated. In other words, even if the slope of the toy regression I ran above is less than one, the R-squared should be large.

In my toy example, the country systematically converges to 2% growth, and the R-squared of my little regression is 0.86. PS find much smaller R-squares in their work. The conclusion is that growth rates in the next decade are very unpredictable. So does this mean that convergence and the Solow model are wrong? No. The reason is that once you allow for any kind of meaningful shocks to GDP per capita, the short-run growth rates get very noisy, and you lose track of the convergence. It doesn’t mean it isn’t there, it just is hard to see.

Let me give you a clearer demonstration of what I mean. I’m going to build an economy that strictly obeys convergence, with the growth rate related to the difference between actual GDP per capita and trend GDP per capita.

More formally, let

where is the long-run growth rate of potential GDP, is potential GDP in year , is actual GDP in year , and are random shocks to GDP in year . This formula mechanically captures convergence to trend GDP per capita, but with the additional wrinkle of shocks occurring in any given period that push you either further away or closer to trend. is the convergence parameter, which I said in some recent posts was about 0.02, meaning that 2% of the gap between actual and trend GDP per capita closes every period.

I simulated this over 100 periods, with , , and . The country starts well below potential. I then let there be a shock to every period, drawn from a normal with mean 0, variance 0.25. Here are the results of one run of that simulation.

First, look at the 10-year growth rates over time. There is a downward trend if you look at it, but this is masked by a lot of noise in the growth rate. You have what look distinctly like two growth booms, about period 25 and period 50.

Second, look at the correlation of the average growth rate in one “decade” and the average growth rate in the prior “decade”. This is essentially what Pritchett and Summers do. I’ve also included the fitted regression line, so you can see the relationship. There is none. The coefficient on the prior-decade growth rate is 0.05, so pretty severe mean-reversion. The R-squared is something like 0.16. A high growth rate one decade does not indicate high growth the following decade, and the current decadal growth rate provides very little information on growth over the next decade.

But this model has mechanical convergence built into it, just with some extra noise dropped on top to make things interesting. And with sufficient noise, things are really interesting. If you looked at this plot, you’d start talking about growth accelerations and growth slowdowns. What happened in period 25 to boost growth? Did this economy democratize? Was there an opening to trade? And what about the bust around period 40? For a poor country, that is low growth. Was there a coup? We see plenty of “bursts” of growth and “bursts” of stagnation (or low growth) here. It’s a function of the noise I built in, not a failure of convergence.

By the way, take a look at the log of output per worker over time. This shows a bumpy but steady upward trend. The volatility of the growth rate doesn’t look as dramatic here.

If I turned up the variance of the noise term, I’d be able to get even wilder swings in output, and wilder swings in growth rates. In a couple simulations I played with, you get a negative relationship of current growth rates to past growth rates – but in every case there was convergence going on.

Why are growth rates so un-forecastable, as PS find? Because of convergence, the noise doesn’t just cancel out over time. If a country gets a big negative shock today, then the growth rate is going to be low this year. But now the country is particularly far below trend GDP per capita, and so convergence kicks in and makes the growth rate larger than it normally would be. And because convergence works slowly, it will be larger than normal for several periods afterwards. There is a natural tendency for growth rates to be uncorrelated in the presence of shocks, but that is again partly because of convergence, not evidence of its absence. There are lots of reasons that the Solow model could be the wrong way to look at growth. But this isn’t one of them.

I think the issue here is that convergence gets “lost” behind all the noise in the data. Over long periods of time, convergence wins out. [“The arc of history is long, but it bends towards Robert Solow”? Too much?] Growth rates start relatively high and end up asymptoting towards the trend growth rate. But for any small window of time – say 10 years – noise in GDP per capita can swamp the convergence effects. In the growth literature we tend to look at differences of 5 or 10 years to “smooth out” fluctuations. That’s not sufficient if one wants to think about convergence, which operates over much longer time periods.

PS are absolutely right that we cannot simply extrapolate out China and India’s recent growth rates and assume they’ll continue indefinitely. We should, as growth economists, account for the gravitational pull that convergence puts on growth rates as time goes forward. But just like gravity, convergence is a relatively weak force on growth rates. It can be overcome in the short-run by any reasonably-sized shock to GDP per capita.

You don’t think “Oh my God, gravity is broken!” every time you see an airplane overhead. So don’t take abnormal growth rates or uncorrelated growth rates as evidence that convergence isn’t occurring.

[insert policy here] Won’t Boost Growth Rates

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Over at the Cato Institute, they hosted an online forum about reviving economic growth. There are lots of smart people involved. The web page has lots of big pictures of their heads, I guess to indicate that their brains are like, totally huge.

Anyway, each one wrote up some proposed policy reform that would help boost long-run growth prospects. Brad DeLong responded to many of the proposals here before his head exploded reading Doug Holtz-Eakin’s essay.

I’m not going to quibble with any of the minutiae of the proposals. My point is going to be a general one on the possible growth effects of [insert policy here]. Short answer, there won’t be any.

There are two ways to boost GDP growth. Either

• Actively raise current GDP through increased spending by some sector of the economy.
• Raise potential GDP and let transitional growth speed up.

The second one perhaps deserves a little explanation. Transitional growth is an extra boost to growth that occurs when current GDP is below potential GDP. Why does this occur? Bob Solow is why. In an economy with accumulable factors of production (physical capital, human capital, knowledge capital) being below potential GDP means that the return to these factors is relatively high, and hence more investment in those factors is done, boosting GDP growth. The wider is the gap between current and potential GDP, the stronger this transitional growth.

The issue is that [insert policy here] is a policy to raise potential GDP, not current GDP. But the transitional effects this encourages are inherently small. So even if [insert policy here] opens up a big gap between potential and actual GDP, this doesn’t translate into much extra growth. In fact, the effects are likely so small that they would be unnoticeable against the general noise in growth rates year by year.

To give you an idea of how little an effect [insert policy here] will have on growth, let’s play with math. Output in period can be written in terms of output in period this way

This says that output in is equal to times current output. That is “regular” growth. The term with the is the additional boost in growth we get from being below potential. is potential GDP in period , and is the gap in GDP. tells us how much of that gap we make up from period to . If , then we are stuck below potential (secular stagnation). If , then immediately next period our GDP will be at potential again.

Let’s think about this in terms of growth rates, so

The growth rate from to depends on the ratio of potential to actual GDP today, period . If that ratio were equal to one – meaning that we were at potential – then the growth rate just becomes , the trend growth rate. The larger is – meaning the farther we are from potential – the higher is the actual growth rate.

Now we can go back to thinking about the possible growth impact of [insert policy here]. GDP today () is about 16 trillion. Potential GDP today () is probably about 17 trillion. You can get a lower estimate from the CBO, Robert Gordon, or John Fernald, or a higher estimate from older CBO forecasts. I’m going to err on the high side for potential because this will inflate the growth effect of [insert policy here].

We also need to know the value of , the percent of the GDP gap that is closed in a year. We’ve got lots of evidence that this value is about , or 2% of the gap closes every year. This estimate goes back to the original cross-country convergence literature starting with Barro (1991), but consistently across samples (countries, US states, Japanese prefectures, Canadian provinces, etc..) economies converge to potential GDP at about 2% of the gap per year.

You get higher values of if you assume that economies pursue optimal savings plans, like in the Ramsey model, meaning that they save at a higher rate when they are farther below steady state. But if there is an economy that saves according to the predictions of the Ramsey model, it is populated by unicorns.

Back to the calculation. The last thing we need is a value for , trend growth. Let’s call that , or trend growth in GDP is about 2% per year. Again, we can argue about whether that is higher or lower, but that’s not going to be the important factor here.

Okay, so based on the fact that we are currently 1 trillion below trend, the growth rate today should be

or growth should be 2.13%. Growth will be about 0.13 percentage points higher than normal – that’s a little over one-tenth of one percent – because we are below potential. The value of is really irrelevant. All the action is inside the brackets. Because is small, there isn’t much bite from transitional growth, even though we are $1 trillion below trend.

But what about [insert policy here]? That will *raise* potential GDP, and therefore will induce faster transitional growth to the new, higher potential GDP. Okay. Let’s say that [insert policy here] has an astonishingly positive impact on potential GDP. I mean massive. [insert policy here] adds a full $1 trillion to potential GDP, which is now $18 trillion. Now, growth under the [insert policy here] regime is

Uh, wow? Growth will be an additional 0.12 percentage points higher thanks to [insert policy here]. This is not a massive change in growth. And the growth boost will *decline* over time as we get closer to potential.

Fine, but what if [insert policy here] is truly revolutionary, and raises potential GDP by $2 trillion? Then growth will be 0.0238. This could be generously rounded to 0.025, meaning you added a half-point to the growth rate of GDP. But let’s not kid ourselves that [insert policy here] is going to have that big of an effect on growth. $2 trillion implies that [insert policy here] is raising potential GDP by about 12%. That would be an anomaly of historic proportions.

[insert policy here] will not generate any appreciable extra economic growth, even though in the very long-run [insert policy here] may be a net positive for the level of economic activity. The problem is that it takes a very, very, very long time for those positive effects to manifest themselves, and thus [insert policy here] won’t do anything to fundamentally change GDP growth.

What about the exceptions I mentioned? Among the proposals, there are a few that could boost current GDP (and thus growth) directly and immediately by encouraging spending.

• Scott Sumner’s NGDP targeting. The proposal speaks directly to raising current GDP, as opposed to raising potential GDP. I think of this as solving the balance sheet problems of households. Boost nominal spending and nominal incomes rise, while nominal debts like mortgages remain fixed, leading to extra spending.
• Brad DeLong’s raising K-12 teacher salaries. If you could do it *now*, then it would raise incomes for these folks, and boost spending. The second part of the proposal, to tie this to teacher tenure changes, is more of a potential GDP changer. Question, how big of an impact would this really have on spending?
• A number of people mention infrastructure spending. Yes, if we would spend that money *now*, then it would materially boost GDP growth *now*, and as a bonus have long-run benefits for potential GDP.

Ultimately, the issue in the U.S. right now is not with potential GDP. We do not need policies to raise this potential GDP so much as we need policies to get us back to potential. That requires actively boosting immediate spending.

Piketty and Growth Economics

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Reviews of Thomas Piketty’s “Capital in the 21st Century” are second only to cat videos on the internet, it seems. Not having any cats, I am unable to make a video, so you’re stuck with a review of Piketty’s book.

I was particulary struck by the implications of this work for economic growth theory. The first section of the book studies capital/output ratios, one of the core elements of any model of growth that includes capital. Piketty provides a long time series of this ratio, showing that in Europe it tended to hover around 7 during the 1800’s and early 1900’s, then dropped dramatically following World War I, stayed at around 3 until the 1970’s, and now is rising towards 6. In the U.S., it has been less variable, going from around 4.5 in the 1800’s to about 3 in the 1960’s, and now is back up to about 4.5.

The projection that Piketty makes is that the capital/output ratio will tend to be about 6-7 across the world as we go into the future. The main reason is that he expects population growth to decline, and the capital/output ratio is inversely related to population growth. In a standard Solow model with a fixed savings rate , the capital/output ratio is , where is population growth, is depreciation, and is the growth of output per worker. You can see that as goes down, the ratio rises.

By itself, this doesn’t imply much for growth theory, in that the expected ratio in the future is entirely consistent with Piketty’s claim regarding population growth. He might be wrong about population growth, but if does in fact fall, then any growth model would have predicted will rise.

The interesting implication of Piketty’s work is on the returns to capital. In particular, the share of national income that goes to capital. His figures 6.1-6.3 document that this share has changed over time. From a share of about 35% in the 1800’s in both Britain and France, the share dropped to about 20-25% in both countries by the mid-20th century. Most recently, the capital share is starting to rise across many countries, going up about 10 percentage points between 1970 and 2010.

One of the bedrock assumptions made in most growth models is a Cobb-Douglas production function, which implies (under conditions of perfect competition) that capital’s share in output is fixed by a technological parameter, typically called and typically assumed to be . Over time, the share of output going to capital is constant at this value of . Growth economists lean on this assumption because of work done by Nicholas Kaldor, who established as a “stylized fact” that capital’s share in output is constant at about 0.3–0.35. As Piketty points out, though, Kaldor established this fact using a very small time series of data from a particularly unusual time period (roughly the mid-20th century).

The fact that capital’s share of output has changed distinctly over long time frames means that this baseline assumption is called into question. What does it mean? I have two immediate thoughts.

• Perfect competition is not a good assumption. This is probably trivially true; there is no such thing as a perfectly competitive economy. But what Piketty’s data would then indicate is that the degree of imperfection has possibly changed over time, with economic profits (not accounting ones) rising in the late 20th century. We have lots of models of economic growth that allow for imperfect competition (basically, any model that involves deliberate research and development), but we do not talk much about changes in the degree of that competition over time.
• The production function is not Cobb-Douglas. Piketty talks about this in his book. The implication of rising capital shares that coincide with rising capital/output ratios is that the elasticity of substitution between capital and labor is greater than one. For Piketty, this contributes to increasing inequality because capital tends to be owned by only a small fraction of people. For growth economists, this raises interesting possibilities for what drives growth. With a sufficiently large elasticity of substitution between capital and labor, then growth can be driven by capital accumulation alone. To see this, imagine perfect substitutability between capital and labor in production, or , where is labor-specific productivity. Output per worker is . As the capital/labor ratio rises, so does output per worker. This continues without end, because there are no longer decreasing returns to capital per worker. Even if technology is stagnant ( does not change), then output per worker can go up. We tend to dismiss the role of capital per worker in driving growth, but perhaps that is because we are wedded to the Cobb-Douglas production function.

The remainder of Piketty’s book is very interesting, and his own views on the implications of rising inequality have been subject to an intense debate. But from the perspective of growth economics, it is the initial section of the book that carries some really interesting implications.

The Solow Model

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This is another idea for modifying how to teach the Solow model. One thing I’d like to do is go immediately to including productivity – it follows cleanly from the simplest growth model. Second, I think it might be nice to work with the K/Y ratio immediately. In this way, I think you can actually skip using the whole “k-tilde” thing. And, *gasp*, do away with the traditional Solow diagram.

The simplest growth model doesn’t allow for transitional growth, and this due to the fact that it does not allow for capital, a factor of production that can only be slowly accumulated over time. The Solow Model is a standard model of economic growth that includes capital, and will be better able to account for the transitional growth that we see in several countries.

Production in the Solow Model takes place according to the following function

is the stock of physical capital used in production, and and are defined just as they were in our simple growth model. So the production function here is just a modification of the simple model to include capital. The coefficient is a weight telling us how important capital or are in determining output.

To analyze this model, we’re going to rewrite the production function. Divide both sides of the function by , giving us

and then take both sides to the power, which gives us the following expression

In per capita terms, this is

Output per worker thus depends not just on , but also on the capital-output ratio, .

So to understand the role of capital in economic growth, we need to understand the capital-output ratio and how it changes over time. We’ll start by looking at the balanced growth path, and then turn to situations where the economy is not on the balanced growth path (BGP).

One fact about the BGP is that the return to capital, , is constant. The return to capital is , which depends (negatively) on the capital-output ratio (the return to capital is just the marginal product of capital). If is constant on the BGP, then it must be that is constant on the BGP as well. What does this mean? It means that can have a level effect on output per worker, but has no growth effect. To see this more clearly, take logs of output per worker,

and then plug in what we know about how moves over time,

The capital-output ratio affects the intercept of this line — a level effect — alongside . The slope of this line — the growth rate — is still .

The capital/output ratio is constant along the BGP, and has no effect on the growth rate on the BGP. But what if the economy is not on the BGP? Then it will be the case that affects the growth rate of output per worker, because the ratio will not be constant. More precisely, the growth rate of capital/output is

So the ratio will change if capital grows more quickly or more slowly than output. First, capital accumulates as follows

where is the change in the capital stock. is the savings rate, the fraction of output that the economy sets aside to invest in new capital goods, so that is the total amount of new investment. is the depreciation rate, the fraction of the existing capital stock that breaks or becomes obsolete at any given moment.

To find the growth rate of capital, divide through the above equation by to get

You can see that the growth rate of capital depends on the capital/output ratio itself.

The growth rate of output is

Now, with (7), and using what we know about growth in capital and output, we have

where we’ve plugged in that , and .

Re-arranging a bit, the capital output ratio is growing if

and growing if the capital/output ratio is larger than the value on the right-hand side. In other words, if the capital stock is relatively small, then it will have a tendency to grow faster than output, raising the ratio. Eventually , the steady state value, and the ratio stops changing.

What is happening to growth in output per worker? If then the ratio is growing, and so output per worker is growing faster than . So the temporarily fast growth in output per worker in Germany or Japan would be because they found themselves with a ratio below their steady state value. How would this occur? It’s easier to see how this works if we re-write the ratio slightly

From this we can see that the ratio would be particularly low if the capital stock, , were to be reduced. This is what happened in Germany, to a large extent, after World War II. The capital stock was destroyed, so fell sharply. This made fall below the steady state value, which meant that there was growth in the ratio, and so growth in output per worker greater than .

A slightly different situation describes South Korea. There, we can think of there being a level effect on , an advance in productivity. This also makes fall sharply, and again causes growth in and growth in output per worker faster than . But in both this case and in Germany’s, as the ratio grows it approaches the steady state value and growth in output per worker slows down to again.