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.

Handy Book of Economic Growth

NOTE: The Growth Economics Blog has moved sites. Click here to find this post at the new site.

I thought it would be nice to post some overview articles of significant research in economic growth.

  1. Culture, Entrepreneurship, and Growth. Doepke and Zilibotti
  2. Trust, Growth, and Well-Being, Algann and Cahuc
  3. Long-term Barriers to Economic Development, Spolaore and Wacziarg
  4. Family Ties, Alesina and Giuliano
  5. The Industrial Revolution, Clark
  6. Twentieth-Century Growth, Crafts and O’Rourke
  7. Historical Development, Nunn
  8. Institutions and Economic Growth in Historical Perspective, Ogilvie and Carus
  9. What Do We Learn from Schumpeterian Growth Theory? Aghion, Akcigit, and Howitt
  10. Technology Diffusion: Measurement, Causes, and Consequences, Comin and Mestieri
  11. Health and Economic Growth, Weil
  12. Regional Growth and Regional Decline, Breinlich, Ottaviano, and Temple
  13. The Growth of Cities, Duranton and Puga
  14. Growth and Structural Transformation, Herrendorf, Rogerson, and Valentinyi
  15. The Chinese Growth Miracle, Yang Yao
  16. Growth From Globalization? A View from the Very Long Run, Meissner

If I were an enterprising publisher, I would go find some editors. Maybe Philippe Aghion and Steven Durlauf, just to throw some names off the top of my head. I’d have them put these together into a nice volume. Oh, wait

Quick update: I posted this list under the “Papers” page on this site if you want a more permanent place to find them.

More updates: Thanks to Pseudoerasmus for the links on the Yao and Meissner papers.

Last Links of 2014

NOTE: The Growth Economics Blog has moved sites. Click here to find this post at the new site.

So now that I’ve yanked myself out of the nanaimo bar-induced coma of last week to return to the living, I’ve got a stack of links that have been piling up. So here is a last post of 2014 for you to scan through on your phone while you watch Pit Bull’s NYE countdown. (What, you’re *not* watching Pit Bull? And is it Pit Bull or Pitbull?)

Two books I read recently that were quite good:

  1. The First European Revolution by R.I. Moore. This was a fun read. Moore traces the origin of a unique “European” culture to the time around 900-1000 AD. This was after the Carolingian Empire had broken up, and the remaining areas were scrambling in some sense to re-order themselves. Key events are the clear separation of clergy from nobles (enforced through celibacy of clergy) and strict primogeniture (which eliminates contested lordship of warrior clans). Need to process this alongside Mitterauer’s Why Europe? which is also about creation of unique European family structure in place of clans.
  2. Adam Tooze’s The Deluge, about America’s plunge into world leadership during and after WWI. What stood out for me was the ridiculous self-righteousness of Woodrow Wilson. I had this sense that he had been too idealistic to solve the real-world problems at Versailles. But this book makes it even more clear that what we call “idealism” was really a entitled feeling of superiority of Protestant white dudes over the unwashed masses.

I also read a lot of garbage, but these two books make me sound smart, so there you go.

Assorted links of interest:

  1. Stephen Gordon piles on a willfully stupid article regarding Canada and the “resource curse”. Key quote, and this is one to pin up on your wall: “If God provides you with an abundance of something that the rest of the world values highly and is willing to pay through the nose to obtain, then this is a blessing, not a curse. If the ‘resource curse’ has any meaning, it has to do with politics, not economics.” There is never a time when you *don’t* want to have more of a valuable stock of a natural resource.
  2. More on the “resource curse”. Countries with resources are richer, but grow more slowly. So to the extent that you think being rich but growing slowly is a curse, there you have it. Recent paper by Alexander James says that the slow growth we see in resource exporters is due to slow growth in their resource earnings. Drops in commodity prices make overall growth look bad (think of Russia today). What James shows is that over the last few decades, the growth rate of the non-resource sectors of these resource-exporters grow just as fast as everyone else. So “resource curse” or no?
  3. Speaking of willfully stupid. Scientific American, of all places, published this econo-crank piece on the digital economy and secular stagnation. The lede is that Twitch sold for 970 million, but employs only 170 people. Presumably this means that economic growth might end. Uh, what? The market value per employee of tech companies has little to do with the rate of economic growth. If anything, the higher this is, the *greater* will be growth, as the incentives to start tech companies are so huge. We can have reasonable discussions about the relative value of different types of labor and how they might fare as innovation occurs, but that is an entirely different discussion from the “end of growth”.
  4. Send LateX code in Gmail. You heard me. Go check it out.
  5. A very nice comment by Scott Sumner on Noah Smith‘s comment regarding taxes and labor effort. Scott’s good point is that people often try to have it both ways in arguing for European-style social safety nets. When you say that GDP per person is only 70% of the US level, they say that this is because they work fewer hours, but have just as high utility. But if you talk about high taxes in Europe, they claim that Europeans work just as much as US workers. Which is it?

Growth Class Slides

NOTE: The Growth Economics Blog has moved sites. Click here to find this post at the new site.

I’ve posted up slides for use with Introduction to Economic Growth on the Class Materials page.

I am a bit of a slide minimalist, so they are not done with Beamer, but they were created using TEX. They contain all the important equations and figures from the book, but there is not a lot of verbiage. I tend to leave that off the slides to force myself to talk through things more slowly.

Both the PDFs and the original TEX are posted (along with the necessary figure files). So you can use them as-is or feel free to edit them as necessary. Happy to hear back comments on them if you do use them in an undergrad class.

The Simplest Growth Model

NOTE: The Growth Economics Blog has moved sites. Click here to find this post at the new site.

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.