# Significant Changes in GDP Growth

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A relatively quick post to highlight two other posts that recently came out regarding GDP growth. First, David Papell and Ruxandra Prodan have a guest post up at Econbrowser regarding the long-run effects of the Great Recession. They use the CBO projections of GDP into the future (similar to what I did here) and look at whether there was a statistically significant break in the level of GDP at the Great Recession. Short answer, yes. Their testing finds that the break was 2008:Q2, not a surprising date to end up with.

It is important to remember that David and Ruxandra are testing for a break in the level of GDP, and not GDP per capita. It is entirely possible to have a structural break in GDP while not having a structural break in GDP per capita. The next thing to remember is that they cannot reject that the growth rate of GDP is the same after 2008:Q2 as it was before. What I mean is easier to see in their figure than it is to explain:

Before and after the break, the growth rate is identical. It is just the level that has changed.

The second post is from Juan Antolin-Diaz, Thomas Drechsel, and Ivan Petrella. They use only existing data (not CBO projections) and find that there is statistical evidence of a change in the growth rate of U.S. GDP. They see a slowdown in growth starting in the mid-2000’s, consistent with John Fernald’s suggestions regarding productivity growth. It takes until 2015 to see this break statistically because you need several years of data to confirm that the growth slowdown was not a temporary phenomenon.

Note the subtle but very, very, very important difference between the two posts. Papell/Prodan find a significant shift in the level of GDP, while Antolin-Diaz, Drechsel, and Petrella (ADP) find a significant shift in the growth rate of GDP. The former sucks, but the latter is far more troubling. If the growth rate is truly lower, then we will get farther and farther away from the pre-GR trend, and the ratio of actual GDP to pre-GR trend GDP will go to zero. If it is just a level shift, then the ratio of actual GDP to pre-GR trend GDP will go to one as both become arbitrarily large.

I find the Papell/Prodan result more convincing. Keep in mind that David is my department chair and if I knocked on my office wall right now I could interrupt the phone call he is on. Ruxandra’s office is all of 20 feet from mine. I see these people every day. But regardless of the fact that I know them personally, I think they are right.

ADP are getting a false result showing slow growth because of the level shift that David and Ruxandra identify. If ADP do not allow for the level shift, then over any window of time that includes 2008:Q2 the growth rate will be calculated to be low. But that is just a statistical artifact of this one-time drop in GDP. It doesn’t mean that the long-run growth rate is in fact different. Put it this way: if they re-run their tests 25 years from now, they’ll find no statistical evidence of a growth change.

Of course, if the CBO is wrong about the path of GDP from 2015-2025, then Papell/Prodan could be wrong and ADP could be right. But given the current CBO projections, there is strong evidence of a negative level shift to GDP, but no change in the long-run growth rate.

# 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.

# Potential “Potential Output” Levels

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John Fernald has a new working paper out at the San Fran Fed on “Productivity and Potential Output Before, During, and After the Great Recession”. The main take-away from the paper is that productivity growth started to slow down even before 2008, particularly in industries that produce IT products or are significant users of IT products. Because of this, even in the absence of the Great Recession, we would have seen slower trend growth in GDP.

What Fernald’s results imply is that the economy is not as far from its potential GDP as we might think. And the idea that we’re way below potential GDP is something lingering underneath a lot of the discussion about economic policy (tapering, stimulus, etc…). Matt Yglesias just had a post noting that while the U.S. is well below it’s pre-2007 trend for GDP, Europe is even farther below it’s trend. Regardless of the conclusion you want to draw from that regarding policy, the assumption is that the pre-2007 trend is where GDP “should” be.

Back to Fernald’s paper. He finds that productivity growth was already declining prior to 2007, and therefore where GDP “should” be is a lot lower than the naive pre-2007 trend line would indicate. This is easier to see in a picture.

The purple dashed line is from the CBO’s 2007 projection, and that is essentially just an extrapolation of the trend in GDP from about 1990-2007. Compared to that measure of potential GDP, we are doing very poorly, with actual GDP (the black line) falling nearly $2 trillion short of potential. Fernald’s alternative calculations that take into account the slowdown in productivity growth that started in the mid-2000’s suggest a much lower estimate of potential GDP. His estimate (the red line) is a prediction of what GDP would have been without the financial crisis, essentially. It falls well below the CBO 2007 estimate. It suggests that the economy today is only perhaps$400 billion short of potential GDP.

His numbers make a big difference in how you think about policy, if only at the quantitative level. If you’re for some kind of further monetary expansion or a new fiscal stimulus, then the size of that boost should be calibrated to a $400 billion shortfall, not a$2 trillion one.

Why does Fernald come up with lower numbers for potential output than the naive forecast in 2007? Without going into the nitty-gritty, he looks at productivity growth (think output per hour) and finds that around 2003Q4, it stops growing as quickly as it did from 1995-2003. What Fernald chalks this up to is the exhaustion of the IT productivity boost. At the time, people thought that the IT revolution might have permanently raised labor productivity growth rates It appears to rather have had a “level effect” – we had a boost in the level of labor productivity, but now it will continue to grow at the normal rate. Again, this is easier to see in pictures, courtesy of Fernald’s paper.

You can see that the 1995-2003 period is exceptional in having high labor productivity growth, and that since 2003 we’ve had growth in labor productivity at about the same rate as 1973-95. Anyone who uses the 1995-2003 period to extrapolate labor productivity growth (like the CBO was implicitly doing in 2007) would overestimate potential output.

This isn’t to say that the CBO or anyone else was being lazy or duplicitous. In 2007, if you looked at the data on labor productivity, there would not be enough evidence to suggest that growth in labor productivity had fallen. The data from 1995-2007 would not be enough to tell you if we had experienced a “level effect” from IT that led to a temporary boost to growth rates, or a “growth effect” from IT that had permanently raised growth rates. You can only tell the difference now because we see the slowdown in productivity growth, so in retrospect it must have been a “level effect”.

Regardless, Fernald’s paper suggests that the scope of the Great Recession is less “Great” than previous estimates would lead you to believe. And given that the trend growth rate in labor productivity is driven primarily by technological innovation, then boosting that growth rate means hoping that someone will invent a new technology that has a transformative power similar to IT.

# 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

$\displaystyle Y = K^{\alpha}(AL)^{1-\alpha}. \ \ \ \ \ (1)$

${K}$ is the stock of physical capital used in production, and ${A}$ and ${L}$ 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 ${\alpha}$ is a weight telling us how important capital or ${AL}$ are in determining output.

To analyze this model, we’re going to rewrite the production function. Divide both sides of the function by ${Y^{\alpha}}$, giving us

$\displaystyle Y^{1-\alpha} = \left(\frac{K}{Y}\right)^{\alpha} (AL)^{1-\alpha} \ \ \ \ \ (2)$

and then take both sides to the ${1/(1-\alpha)}$ power, which gives us the following expression

$\displaystyle Y = \left(\frac{K}{Y}\right)^{\alpha/(1-\alpha)} AL. \ \ \ \ \ (3)$

In per capita terms, this is

$\displaystyle y = \left(\frac{K}{Y}\right)^{\alpha/(1-\alpha)} A. \ \ \ \ \ (4)$

Output per worker thus depends not just on ${A}$, but also on the capital-output ratio, ${K/Y}$.

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, ${r}$, is constant. The return to capital is ${r = \alpha Y/K}$, which depends (negatively) on the capital-output ratio (the return to capital is just the marginal product of capital). If ${r}$ is constant on the BGP, then it must be that ${K/Y}$ is constant on the BGP as well. What does this mean? It means that ${K/Y}$ 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,

$\displaystyle \ln y(t) = \frac{\alpha}{1-\alpha} \ln\left(\frac{K}{Y}\right) + \ln A(t) \ \ \ \ \ (5)$

and then plug in what we know about how ${A(t)}$ moves over time,

$\displaystyle \ln y(t) = \frac{\alpha}{1-\alpha} \ln\left(\frac{K}{Y}\right) + \ln A(0) + gt. \ \ \ \ \ (6)$

The capital-output ratio affects the intercept of this line — a level effect — alongside ${A(0)}$. The slope of this line — the growth rate — is still ${g}$.

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 ${K/Y}$ affects the growth rate of output per worker, because the ${K/Y}$ ratio will not be constant. More precisely, the growth rate of capital/output is

$\displaystyle \frac{\dot{K/Y}}{K/Y} = \frac{\dot{K}}{K} - \frac{\dot{Y}}{Y}. \ \ \ \ \ (7)$

So the ${K/Y}$ ratio will change if capital grows more quickly or more slowly than output. First, capital accumulates as follows

$\displaystyle \dot{K} = s Y - \delta K \ \ \ \ \ (8)$

where ${\dot{K}}$ is the change in the capital stock. ${s}$ is the savings rate, the fraction of output that the economy sets aside to invest in new capital goods, so that ${sY}$ is the total amount of new investment. ${\delta}$ 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 ${K}$ to get

$\displaystyle \frac{\dot{K}}{K} = s\frac{Y}{K} - \delta. \ \ \ \ \ (9)$

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

The growth rate of output is

$\displaystyle \frac{\dot{Y}}{Y} = \alpha \frac{\dot{K}}{K} + (1-\alpha)\frac{\dot{A}}{A} + (1-\alpha)\frac{\dot{L}}{L}. \ \ \ \ \ (10)$

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

$\displaystyle \frac{\dot{K/Y}}{K/Y} = (1-\alpha)\left(s\frac{Y}{K} - \delta - g - n \right) \ \ \ \ \ (11)$

where we’ve plugged in that ${\dot{A}/A = g}$, and ${\dot{L}/L = n}$.

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

$\displaystyle \frac{K}{Y} < \frac{s}{\delta + n + g}, \ \ \ \ \ (12)$

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 ${K/Y}$ ratio. Eventually ${K/Y = s/(\delta+n+g)}$, the steady state value, and the ${K/Y}$ ratio stops changing.

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

$\displaystyle \frac{K}{Y} = \frac{K}{K^{\alpha}(AL)^{1-\alpha}} = \left(\frac{K}{AL}\right)^{1-\alpha}. \ \ \ \ \ (13)$

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

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

# The Simplest Growth Model

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This is an idea for a new way of introducing growth theory. Given that productivity growth is the source of long-run growth, it seems to make sense to start with that, rather than with the Solow model.

Let’s write down a very simple model of economic growth. Let total output ${Y}$ be determined by

$\displaystyle Y = A L \ \ \ \ \ (1)$

where ${A}$ is a measure of labor productivity, and ${L}$ is the number of workers. If we divide through by ${L}$, then we get a measure of output per worker. To keep notation clean, let ${y = Y/L}$ be output per worker, so that now we have

$\displaystyle y = A \ \ \ \ \ (2)$

as our model of economic growth. Basically, output per worker is simply equal to labor productivity ${A}$.

From this we know that the time path of output per worker is simply the same as the time path of labor productivity, ${A}$. So what determines the time path of labor productivity? We’ll assume that it is growing at a constant rate, meaning that it goes up by the same percent every period of time,

$\displaystyle A(t) = A(0) e^{g t}. \ \ \ \ \ (3)$

Here, we’ve written ${A(t)}$ to be clear that we mean labor productivity at any given time ${t}$. ${A(0)}$ is labor productivity in the initial moment of time. The exponential term says that labor productivity grows at the rate ${g}$ over time.

The exponential term implies, perhaps not surprisingly, exponential growth. You get exponential growth when something goes up by the same percent every period of time. If ${g = 0.02}$, then we have 2% growth. At time zero, labor productivity is just ${A(0)}$. When ${t=2}$, then ${A(2) = A(0)e^{.02(2)} = 1.041 A(0)}$, or labor productivity is a little more than 4% higher than at time zero. When ${t=10}$, ${A(10)=A(0)e{.02(10)}=1.221}$, or labor productivity is more than 22% higher than at time zero.

It may not seem obvious, but output per worker in the U.S. and most other developed nations displays exponential growth. Our model matches that, as

$\displaystyle y(t) = A(0) e^{g t}. \ \ \ \ \ (4)$

These countries also tend to have a similar growth rate of about 1.8%, or ${g=0.018}$. Seeing this in a figure, though, is difficult. Graphing ${y}$ over time for the U.S. gives you a curve that quickly accelerates upwards and is almost off the page. Graphs like this will also make it difficult to compare countries to one another.

For that reason, among others, we like to work with the natural log of output per worker, ${\ln{y(t)}}$. Taking natural logs of ${y(t)}$ gives us

$\displaystyle \ln{y(t)} = \ln{A(0)} + g t. \ \ \ \ \ (5)$

This is an equation that says the natural log of output per worker is a linear function of time, ${t}$. If we graph ${\ln{y(t)}}$ against ${t}$, we get a straight line, similar to the trend line we drew in the figure for U.S. output per worker.

We can calculate the growth rate of output per worker by taking the derivative of (5) with respect to time. This results in the following

$\displaystyle \frac{\dot{y}}{y} = g. \ \ \ \ \ (6)$

The value of ${A(0)}$ is fixed, so the derivative of it with respect to time is just zero. The notation ${\dot{y}/y}$ is a shorthand way of writing the growth rate. ${\dot{y}}$ is the absolute change in output per worker at any given moment, and by dividing by ${y}$ we get that change relative to the level of output per worker. This means that ${\dot{y}/y}$ is essentially the percent change in output per worker at any given moment.

That’s it for the simple growth model. Output per worker depends on labor productivity ${A(t)}$, and labor productivity grows at a constant rate ${g}$, which means output per worker grows at that same rate. Despite the mechanical simplicity, this model helps us be clear when we are talking about the growth experiences of different countries. It allows us to distinguish between two forces determining output per worker.

• Level effects: These refer to ${A(0)}$, the intercept of the line in (5)
• Growth effects: These refer to ${g}$, the slope of the line in (5)

Looking at the data over the long run, the general impression we get that the growth rate ${g}$ is similar across countries, and they differ mainly because of level effects. That is, ${A(0)_{Japan}}$ appears to be lower than ${A(0)_{US}}$, but the growth rate ${g}$ is very similar. Theories of economic growth should be consistent with these facts. Things like investment rates, schooling, and social infrastructure are important determinants of level effects, ${A(0)}$, but they have no effect on the growth rate, ${g}$. Under plausible assumptions, theories of endogenous innovation will suggest that the growth rate, ${g}$, is identical across countries.

There are some facts, though, that this simple growth model cannot account for. Namely, there are notable cases where output per worker grows more quickly or more slowly than ${g}$. China, for example, over the last 30 years has grown much faster than the U.S. or Japan. South Korea had a similar growth miracle, starting in about 1960 and lasting until the 2000’s. Germany, from World War II until about 1980, grew at a very accelerated pace compared to the U.S. in the same period. How do we reconcile these facts with the assertion above that ${g}$ is the same for all countries?

The key is noting that these growth accelerations were temporary. Germany grew very quickly, but after 1980 its growth rate fell back to a value nearly identical to the U.S. South Korea’s growth rate has diminshed as well in the 2000’s. What appears to be happening is that once output per worker approaches a frontier level, generally defined by the U.S., growth slows down. While China continues to grow quickly, it has not approached the U.S. level of output per worker.

Looking at these countries, what appears to be happening is that there is a level effect, or their ${A(0)}$ has shifted up. However, it seems to take them a long time to move from their old level to the new, higher level. We call the temporary growth spurt that occurs when a country moves between levels transitional growth. Output per worker grows faster than ${g}$ temporarily – although this could last a few decades – but then growth returns to the rate ${g}$.

Our simple model doesn’t offer a way of understanding this transitional growth. The first major extension we’ll make to this simple model is to add physical capital, which has to be slowly accumulated over time. Because of this slow accumulation, the economy will take an extended time to fully respond to a level effect.