# Has the Long-run Growth Rate Changed?

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

My actual job bothered to intrude on my life over the last week, so I’ve got a bit of material stored up for the blog. Today, I’m going to hit on a definitional issue that creates lots of problems in talking about growth. I see it all the time in my undergraduate course, and it is my fault for not being clearer.

If I ask you “Has the long-run growth rate of the U.S. declined?”, the answer depends crucially on what I mean by “long-run growth rate”. I think of there as being two distinct definitions.

• The measured growth rate of GDP over a long period of time: The measured long-run growth rate of GDP from 1985 to 2015 is ${(\ln{Y}_{2015} - \ln{Y}_{1985})/30}$. Note that here the measurement does not have to take place using only past data. We could calculate the expected measured growth rate of GDP from 2015 to 2035 as ${(\ln{Y}_{2035} - \ln{Y}_{2015})/20}$. Measured growth rate depends on the actual path (or expected actual path) of GDP.
• The underlying trend growth of potential GDP: This is the sum of the trend growth rate of potential output per worker (we typically call this ${g}$) and the trend growth rate of the number of workers (which we’ll call ${n}$).

The two ways of thinking about long-run growth inform each other. If I want to calculate the measured growth rate of GDP from 2015 to 2035, then I need some way to guess what GDP in 2035 will be, and this probably depends on my estimate of the underlying trend growth rate.

On the other hand, while there are theoretical avenues to deciding on the underlying trend growth rate (through ${g}$, ${n}$, or both), we often look back at the measured growth rate over long periods of time to help us figure trend growth (particularly for ${g}$).

Despite that, telling me that one of the definitions of the long-run growth rate has fallen does not necessarily inform me about the other. Let’s take the work of Robert Gordon as an example. It is about the underlying trend growth rate. Gordon argues that ${n}$ is going to fall in the next few decades as the US economy ages and hence the growth in number of workers will slow. He also argues that ${g}$ will fall due to us running out of useful things to innovate on. (I find the argument regarding ${n}$ strong and the argument regarding ${g}$ completely unpersuasive. But read the paper, your mileage may vary.)

Now, is Gordon right? Data on the measured long-run growth rate of GDP does not tell me. It is entirely possible that relatively slow measured growth from around 2000 to 2015 reflects some kind of extended cyclical downturn but that ${g}$ and ${n}$ remain just where they were in the 1990s. I’ve talked about this before, but statistically speaking it will be decades before we can even hope to fail to reject Gordon’s hypothesis using measured long-run growth rates.

This brings me back to some current research that I posted about recently. Juan Antolin-Diaz, Thomas Drechsel, and Ivan Petrella have a recent paper that finds “a significant decline in long-run output growth in the United States”. [My interpretation of their results was not quite right in that post. The authors e-mailed with me and cleared things up. Let’s see if I can get things straight here.] Their paper is about the measured growth rate of long-run GDP. They don’t do anything as crude as I suggested above, but after controlling for the common factors in other economic data series with GDP (etc.. etc..) they find that the long-run measured growth rate of GDP has declined over time from 2000 to 2014. Around 2011 they find that the long-run measured growth rate is so low that they can reject that this is just a statistical anomaly driven by business cycle effects.

What does this mean? It means that growth has been particularly low so far in the 21st century. So, yes, the “long-run measured growth rate of GDP has declined” in the U.S., according to the available evidence.

The fact that Antolin-Diaz, Drechsel, and Petrella find a lower measured growth rate similar to the CBO’s projected growth rate of GDP over the next decade does not tell us that ${g}$ or ${n}$ (or both) are lower. It tells us that it is possible to reverse engineer the CBO’s assumptions about ${g}$ and ${n}$ using existing data.

But this does not necessarily mean that the underlying trend growth rate of GDP has actually changed. If you want to establish that ${g}$ or ${n}$ changed, then there is no retrospective GDP data that can prove your point. Fundamentally, predictions about ${g}$ and ${n}$ are guesses. Perhaps educated guesses, but guesses.

# 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.

# Is the U.S. Really Below Potential GDP?

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

The CBO just released a new projection of both GDP and the budget out to 2024. In short, the CBO sees the U.S. staying below potential GDP for several years. Menzie Chinn just did a short review of how people use inflation and/or unemployment to try and figure out the difference difference between actual and potential GDP.

From a growth perspective, I wanted to take a look at the projections a little differently. First, I don’t much care about the level of aggregate GDP, I care about the level of GDP per capita. So I took the CBO numbers and combined them with population figures and projections to get actual and projected GDP per capita for the U.S. Note, I’m using the CBO projections for actual GDP, not their potential GDP numbers. I want to look at the expected GDP numbers.

Second, I wanted to consider how this projected GDP per capita compared to long-run trends, rather than using inflation or unemployment to assess whether GDP per capita is “at potential”. I am looking instead whether GDP per capita has deviated from its long-run path. To do this I merged the GDP per capita projections from the CBO with the Maddison dataset on GDP per capita from 1970 to 2008. (The CBO goes back far enough that the two series overlap and I can adjust the actual levels of GDP per capita to match).

I took the trend in GDP per capita from 1990 to 2007, and extrapolated that out from 2008 to 2024. Then I plotted the actual and CBO-projected GDP per capita data against that trend. Here is what you get:

It’s clear here that in 2007 GDP per capita drops below the 1990-2007 trend line. Moreover, the CBO expects that GDP per capita will stay below that trend line out until 2024. It looks like a distinct “level shift” in the parlance of growth economics. GDP per capita is something like 13% below the 1990-2007 trend.

If you look at the post-war trend in GDP per capita from 1947 to 2007, you get something similar. The gap in 2024, 18% below trend, is actually worse than the gap using the post-1990 era.

But if you extend your view back even further, and incorporate the whole period of 1870-2007 to form the trend line, things look different. Now, if you plot the projected GDP per capita against the trend, it looks as if the U.S. is spot on.

GDP per capita is almost exactly where you’d expect it given the historical trend. The CBO expects GDP per capita to be a little low in 2024, about 2% behind the full trend line. Using the 1870-2007 trend, there doesn’t appear to be anything particularly unusual about the projected path of GDP per capita. The U.S. seems to be moving along the same balanced growth path it always has.

What really looks like the anomaly in U.S. data is the extended period from about 1990 to 2010 that we spent above trend. You could think of this as capturing John Fernald’s argument (or see here) that the IT boom of the 1990’s was a one-time level shift up in GDP. We got a big boost from that, but now the economy is settling back to the long-run growth path.

[You should not – NOT – use this as an argument that the financial crash and subsequent recession were necessary, useful, or welfare-improving. It is quite possible for the economy to have managed a graceful slide back to the long-run trend line after 2007 rather than experiencing it all in one dramatic plunge. The long-run trend is like gravity. Yes, it will win in the end, but that does not mean that I have to leap to the ground after cleaning out my gutters. I have a ladder.]

I really thought when I started playing with this data that I’d be writing a post about how the Great Recession had fundamentally shifted GDP per capita below the long-run trend, and that this represented a really fundamental shock given how stable the long-run trend had been until now. But the current path of GDP per capita doesn’t appear to be that surprising in historical perspective.

The big caveat here is that the CBO could be entirely wrong about future GDP per capita growth. If they have been overly optimistic, then we could certainly find ourselves falling below even the very long-run trend. Then again, they could have been pessimistic, and we might find ourselves above trend for all I know. But even with all the uncertainty, the expectation is that the U.S. economy will find itself right where you would have predicted it would be.

# Harry Potter and the Residual of Doom

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

The productivity term in an aggregate production function is tough to get one’s head around. When I write down

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

for aggregate GDP, the term ${A}$ is the measure of (labor-augmenting) productivity. What exactly does ${A}$ mean, though? Sure, mathematically speaking if ${A}$ goes up then ${Y}$ goes up, but what is that supposed to mean? ${Y}$ is real GDP, so what is this thing ${A}$ that can make real GDP rise even if the stocks of capital (${K}$) and labor (${L}$) are held constant?

I think going to Universal Studios last week provided me with a good example. If you take all the employees (about 12,000 people) and capital (building supplies, etc..) at Universal Studios and set up a series of strip malls along I-4 in Orlando, then you’ll generate a little economic activity between people shopping at the Container Store and eating lunch at Applebee’s. But no one is flying to Orlando to go to those strip malls, and no one is paying hundreds of dollars for the right to walk around and *look* at those strip malls. The productivity, ${A}$, is very low in the sense that the capital and labor do not generate a lot of real GDP.

But call that capital “Diagon Alley” and dress the employees up in funny robes, and it is thick with thousands of people like me shelling out hundreds of dollars just for the right to walk around a copy of a movie set based on a book. Hundreds. Each.

This is pure productivity, ${A}$. The fictional character Harry Potter endows that capital and labor in Orlando with the magical ability to generate a much higher level of real GDP. No Harry Potter, no one visits, and real GDP is lower. The productivity is disembodied. It’s really brilliant. Calling this pile of capital “Gringotts” and pretending that the workers are wizard guards at a goblin bank creates real economic value. Economic transactions occur that would otherwise not have.

We get stuck on the idea that productivity, ${A}$, is some sort of technological change. But that is such a poor choice of words, as it connotes computers and labs and test tubes and machines. Productivity is whatever makes factors of production more productive. That is pretty great, because it means that we need not hinge all of our economic hopes on labs or computers. But it also stinks, because it means that you cannot pin down precisely what productivity is. It is necessarily an ambiguous concept.

A few further thoughts:

• It doesn’t matter what is bought/sold, real GDP is real GDP. Spending 40 dollars at Universal to buy an interactive wand at Ollivander’s counts towards GDP just the same as spending 40 dollars on American Carbide router bits (We bought two. Wands, not router bits). There is no such thing as “good” GDP or “bad” GDP. Certain goods (tools!) do not count extra towards GDP because you can fix something with them.
• Yes, you can create economic value out of “nothing”. Someone, somewhere, is writing the next Harry Potter or Star Wars or Lord of the Rings, and it is going to create significant productivity gains as someone else builds the new theme park, or lunch box, or action figure. This new character or story will endow otherwise unproductive capital and labor with the ability to produce GDP at a faster rate than before. ${A}$ will go up just from imagining something cool.
• This kind of productivity growth makes me think that we won’t necessarily end up working only 10 or 12 hours a week any time soon. The Harry Potter park doesn’t work without having lots of people walking around in robes playing the roles. It’s integral to the experience. So we pay to have those people there. Those people, in turn, pay to go see a Stones concert, where it is integral to have certain people working (Keith and Mick among others). We keep trading our time with each other to entertain ourselves. Markets are really efficient ways of allocating all of these entertainers to the right venues, times, etc.. so it wouldn’t surprise me if we all keep doing market work a lot of our time in the future.
• “Long-tail” creative productivity gains like Harry Potter exacerbate inequality, maybe more than robots ever will. You can buy shares in the robot factory, even in a small amount. But you cannot own even a little bit of Harry Potter. You can’t copy it effectively (*cough* Rick Riordan *cough*). So J.K. Rowling gets redonkulously rich because ownership of the productivity idea is highly concentrated.

# Re-basing GDP and Estimating Growth Rates

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

Leandro Prado de la Escosura recently posted a voxeu column about splicing real GDP series after re-basing. Re-basing of real GDP means adopting a new set of reference prices to value output in each year. Think of what Nigeria did last year, when they re-based from 1990 prices to using 2010 prices, and all of the sudden measured real GDP was about twice as big.

de la Escosura’s point is that when we re-base and “retrocast” real GDP numbers to past years, we may obscure evidence of rapid economic growth. You should go read his post, and his associated paper, to understand his point in full. But let’s use the Nigerian 2013 re-basing to get the basic idea. Let’s say that in 1990 Nigeria produced 1000 units of food, and zero motorcycles. In 2010 Nigeria produced 1000 units of food again, but produced 200 motorcycles. So there clearly is real growth in output.

In 1990, the price of food was 1 naira per unit and motorcycles were 500 naira. 1990 real GDP in 1990 prices is 1000(1) + 0(500) = 1000. 2010 real GDP in 1990 prices is 1000(1) + 200(500) = 101,000. This is a dramatic growth rate of real GDP (10,100% actually).

After re-basing, what do we get? In 2010 the price of food was 2 naira per unit, and motorcycles were 100 naira each. So 1990 real GDP in 2010 prices is 1000(2) + 0(100) = 2000. 2010 real GDP in 2010 prices is 1000(2) + 200(100) = 22,000. Still a lot of growth, but only 1100%. The growth rate of real GDP between 1990 and 2010 went from over 10,000% to about 1100%, an order of magnitude drop. Growth looks much slower in Nigeria after re-basing.

Why? Because with dramatic economic growth came dramatic changes in relative prices. Motorcycles dropped severely in price, while food went up slightly. Combined, this makes food look more valuable compared to motorcycles by 2010. So valuing 1990 output in 2010 prices tends to make 1990 look pretty good, because in 1990 they had lots of food relative to motorcycles.

de la Escosura’s argument is that in 1990, for sure, the 1990 prices are the right way to value real GDP. Similarly, in 2010, for sure, the 2010 prices are the right way to value real GDP. So leave those years priced in their own prices. For the nineteen intervening years, 1991-2009 inclusive, compute their real GDP in both 1990 and 2010 prices. Then average those two estimates depending on how far from each year we are.

So for 1991, let real GDP be (1991 GDP at 1990 prices)(18/19) + (1991 GDP at 2010 prices)(1/19). For 1992, let real GDP be (1992 GDP at 1990 prices)(17/19) + (1992 GDP at 2010 prices)(2/19), and so forth. For de la Escosura, this better captures the growth in real GDP over time. For our example, 1990 real GDP in 1990 prices is 1000, and 2010 real GDP in 2010 prices is 22,000, and the growth rate is 2,200%. It essentially splits the difference of the two different benchmarks, preserving some of the rapid growth seen using the 1990 prices.

This isn’t necessarily a new concept. Johnson, Larson, Papageorgiou, and Subramanian discuss this issue in their paper on the Penn World Tables. Their suggestion for a chained PWT price index amounts to a similar suggestion.

The big point is that by re-basing you are necessarily screwing with the implied growth rate of real GDP because you are screwing with the value of real GDP in the first year (1990 in our example). If there has been a lot of economic growth and relative prices have changed, then almost certainly the first year will have a higher measured real GDP when we re-base. With a higher initial level of GDP, the growth rate will necessarily be smaller.

If your worry about computing growth rates, then this is an issue you have to worry about a lot, and something like de la Escosura’s method or the Johnson et al suggestion is what you should do. If you worry about comparing income levels across countries, then this critique is not crucial (although you have other things to worry about).

# Slow Growth in Potential GDP for the U.S.?

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

Robert Gordon released a paper recently where he presents his estimates of potential GDP for the U.S. going forward. I had planned on writing a longer post discussing about why you should take his projections seriously, and maybe some speculation about what would have to happen to reverse his conclusions. Seriously, I had a few paragraphs written, a couple figures cut out and ready to go, and then Jim Hamilton put up precisely the post I wanted to write. So the first thing you should do is go read Jim’s post.

…and if you are still here, let me provide an uber-quick summary of my own before I talk about how you could convince yourself that Gordon is overly-pessimistic (he’s probably not, but if you want to rock yourself to sleep at night, this might help).

So, what does Gordon find? Basically, GDP growth is equal to growth in GDP per hour worked (productivity) times growth in hours worked. Hours worked are unlikely to grow much, given that unemployment has already fallen back to 6%, that average hours per worker have recovered much of their decline, and that the labor force participation rate is unlikely to recover much of its recent decline. So the only way for GDP to grow fast enough to hit our prior level of potential GDP (which is essentially what the CBO projects it will do) is for productivity (GDP per hour) to grow much faster than it has at any time in the last decade.

You can see his implications in the above figure. The red line is Gordon’s projection for potential GDP based on his assumed lower productivity growth rate, which is closer to recent averages. The CBO potential GDP path is driven by what Gordon says are aggressive assumptions about how fast productivity will grow.

In short, we aren’t going to recover back to the pre-Great Recession trend line for potential GDP any time soon. One might quibble a little about Gordon’s assumptions, and perhaps we won’t diverge from the prior trend line (the yellow one) as much as he suggests. But it’s really hard to come up with reasonable evidence that the CBO is making the right assumption regarding productivity growth.

Now, if I want to go to bed at night believing that we might be able to get back on that prior trend line, what should I tell myself? I’m not going to tell myself that we’ll be magically saved by some kind of technology boom. It could happen, I guess, but that’s not something I could rely on, or having any way of reliably predicting.

What I might tell myself is that productivity – because of the way it is backed out of the data – is not simply a measure of technical productivity, it’s a measure of revenue productivity. I talked about this in a prior post, but the difference is that revenue productivity measures firms ability to generate dollars, not their ability to generate widgets. Revenue productivity can thus experience a temporary burst of growth if firms are able to exert some market power, in the same way that revenue productivity can experience a temporary sag if firms lose pricing power during a recession. So if some of the distinct drop in measured productivity growth over the 2008-2014 period was because firms lost pricing power (and not because of a slowdown in innovation/technology growth), then this could be recovered if firms are able to reassert that pricing power.

A few points on this. Why would firms gain (or why did they lose) pricing power? My guess is that it depends on the willingness of consumers to “shop around”, which in turn is based on economic conditions. When things get bad in 2008, people become more sensitive to price changes, and so firms lose pricing power, and hence revenue productivity falls. If consumers were to recover in the sense of becoming less sensitive to price changes, then firms could gain pricing power and that would raise measured productivity. Will that happen? My guess is yes, it will, I just don’t know when. Will that boost to revenue productivity be sufficient to put potential GDP back on the pre-Great Recession path? I don’t know.

The last thing to point out is that revenue productivity is exactly what we want to measure in Gordon’s case, where he ultimately is worried about Debt/GDP ratios. Because the debt is denominated in dollars, what I care about is the economy’s ability to generate dollars, not widgets. There’s an entirely different post to be written about why the Debt/GDP ratio is a stupid way to measure the debt burden, but I’ll leave that alone for now.

In short, if you want to be optimistic about bouncing back to the pre-Great Recession trend for potential GDP, then part of that optimism is that firms regain lost pricing power, and thus experience a boost to their revenue productivity. This can occur in the absence of any change in the underlying pace of real technological change, and isn’t tied to our expectations about the usefulness or arrival of new technologies.

# What does Real GDP Measure?

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

Nearly all cross-country work on growth and development uses, if only for motivation, Penn World Table (PWT) estimates of real GDP for countries. And the PWT generates a single measure of “real GDP” for each country. How do they do this? Before I answer, let me say that much of what I’m going to say is said more thoroughly in Deaton and Heston (2010). So check that out if you’re into cross-national real GDP comparisons.

To start, let’s simplify and think just about two countries, A and B. To compare real GDP in the two countries, we’d want to value the quantities of goods they produce at some common set of prices. So say phones are $50, and haircuts are$10, and then for each country multiple the quantitify of phones times 50 and add the quantity of haircuts times 10. But what are the right prices to use? Why 50 and 10? Why not 60 and 5? You can imagine that we want the prices we use to be somewhat meaningful, and at least related to the observed prices in countries.

So here’s where it gets weird. We could say, whatever, let’s just use the prices from country B. I just need to pick one set of prices, right? But if you measure country A’s GDP using country B’s prices, then country A will look relatively rich compared to country B. This doesn’t mean it makes country A look absolutely richer than country B, just that country A now looks better in comparison. This works if you flip them around. If you measure country B’s GDP using country A’s prices, then country B will look relatively rich compared to country A. This isn’t a mathematical certainty, but reflects what actually happens if you use the prices underlying the PWT.

Let’s take the U.S. and Nigeria as an example. If we measure real GDP in both the US and Nigeria using Nigerian prices, then the US will appear to have an incredibly large lead over Nigeria in GDP per capita. If we measure real GDP in both the US and Nigeria using US prices, then the gap will appear smaller as this will make Nigeria look particularly good.

This doesn’t necessarily have to happen, it’s not some mathematical rule. But the data underlying the Penn World Tables shows that this is the case almost universally. So what is going on? It means that country A has (relatively) high prices for what country B has a lot of, and country A has (relatively) low prices for what country B has little of.

It’s easiest to see this in an example. So let the US produce ${Q^{US}_{phones} = 100}$ and ${Q^{US}_{haircuts} = 10}$. The US produces a lot of phones relative to haircuts. And in the US, ${P^{US}_{phones} = 10}$ and ${P^{US}_{haircuts} = 10}$, or haircuts and phones cost the same. [No, this doesn’t have to be a realistic relative price for this to work]. At US prices, real GDP in the US is

$\displaystyle GDP^{US} = Q^{US}_{phones} P^{US}_{phones} + Q^{US}_{haircuts} P^{US}_{haircuts} = 100 \times 10 + 10 \times 10 = 1100. \ \ \ \ \ (1)$

In Nigeria, we have ${Q^{N}_{phones} = 10}$ and ${Q^{N}_{haircuts} = 100}$, or Nigeria has very few phones, but lots of haircuts. And the prices in Nigeria reflect this, with ${P^{N}_{phones} = 100}$ and ${P^{N}_{haircuts} = 10}$. At Nigeria’s prices, real GDP in Nigeria is

$\displaystyle GDP^{N} = Q^{N}_{phones} P^{N}_{phones} + Q^{N}_{haircuts} P^{N}_{haircuts} = 10 \times 100 + 100 \times 10 = 2000. \ \ \ \ \ (2)$

Now, those two numbers are not comparable because they use different absolute prices to value the goods. To do a fair comparison of output in the two countries, we have to use the same prices.

Let’s value Nigeria’s output using the US prices

$\displaystyle GDP^{N}_{P-US} = Q^{N}_{phones} P^{US}_{phones} + Q^{N}_{haircuts} P^{US}_{haircuts} = 10 \times 10 + 100 \times 10 = 1100. \ \ \ \ \ (3)$

So using US prices, Nigeria looks really good. Their GDP is 1100, exactly equal to the US. They achieve this with lots of haircuts and few phones, so utility could be different in the two places, but their measured real GDP is as high as the US.

But we could equally argue that we should use Nigerian prices to value GDP in both countries. So for the US we get

$\displaystyle GDP^{US}_{P-N} = Q^{US}_{phones} P^{N}_{phones} + Q^{US}_{haircuts} P^{N}_{haircuts} = 100 \times 100 + 10 \times 10 = 10100. \ \ \ \ \ (4)$

The US now has GDP of 10,100, while Nigeria (at its own prices) only has a GDP of 2000. The US is roughly 5 times richer than Nigeria, when valued at Nigerian prices. Why? Because the US produces a lot of what Nigerians find expensive (phones), and little of what they don’t (haircuts).

Which comparison is right? Neither. There is nothing that says we should use the US prices or the Nigerian prices. For real GDP we simply need to pick some set of prices, and use them consistently across all countries. So much of the work in the Penn World Tables is to come up with a common price index. And the nature of this singular set of prices will matter a lot for real GDP comparisons. If the PWT uses prices that look alot like US prices, then this will make Nigeria (and other developing countries) look relatively well off compared to rich countries. But if the PWT used prices that look like Nigerian prices, then this will exaggerate the gap.

In practice, what do they do? They try to construct some kind of weighted average of the price of each good across all countries. The weights are in the PWT are calculated using what is called a Gheary-Khamis method, which essentially weights the prices from different countries by their share of total spending on that good. For phones, the weight for the U.S. is ${100/(100+10) = 0.91}$ because they produce/use 91% of all the phones. For haircuts, the weight for the U.S. is ${10/(100+10) = 0.09}$ because they produce/use about 9% of all haircuts.

Now in my simple example the weights are basically symmetric, because the US has most of the phones, and Nigeria has most of the haircuts. But in the real data, the US has far more phones and more haircuts than Nigeria. So in practice in the PWT, the weights are very large on U.S. prices, and very small on Nigerian prices. When they do these calculations across all countries, the weights on the US, Western Europe, and Japan dominate because they consume most of the stuff out there in the world. So the prices used by the PWT are really similar to a relatively rich Western nation [People have argued that the prices roughly correspond to Italy’s].

Which all means that every country in the PWT is getting valued at rich country prices. As we saw above, this inflates the real GDP of very poor countries, and makes them look “good” compared to rich countries. That is, the gap between the U.S. and Nigeria is much smaller using rich country (e.g. US) prices than Nigerian prices. So the PWT overall makes poor countries look very good. The true gaps in real GDP are likely larger (much larger?) than what the PWT captures.

This is not some kind of deliberate subterfuge by the PWT. “It does what it says on the tin” is a phrase that comes to mind. But that doesn’t mean it has some cosmic truth to it. The PWT isn’t doing anything wrong, but they are running up against the real fundamental problem: there is no set of prices that gives us a true measure of real living standards across countries.

What we’d like is some number that tells us that living standards in Nigeria are one-tenth, or one-twentieth, or one-fifth of those in the U.S. But what do you mean by living standards? No measure of real GDP captures actual welfare. Even if – as we’d assume was the case in a perfectly competitive market – relative prices capture relative marginal utilities, real GDP doesn’t measure welfare.

Multiplying the total quantity times the marginal utility of a good doesn’t tell me anything about the total utility that people enjoy from that good. The marginal utility of a 3rd car in my family is essentially zero, but that doesn’t mean that we get no utility from having 2. So even if there were some “right” set of prices we could use to value real GDP, it still wouldn’t measure welfare.

I think what would be useful for the PWT would be to have the full distribution of real GDP estimates for a country. That is, show me Nigeria’s real GDP valued at the prices found in every single other country in the PWT. I could plot that distribution of real GDP’s in Nigeria against the same distribution of real GDP’s for the U.S. This would at least show me something about the noise in the relative standing in real GDP for these countries. This sounds like something I can make a grad student do.

One last note about these comparisons. Recall that the result that measuring country A’s GDP in country B’s prices makes country A look relatively rich is not a certainty. It holds because there is a specific correlation of prices and quantities in the data. In each country, goods that are produced in large quantities (e.g. haircuts in Nigeria) tend to have low relative prices, and goods produced in small quantities (e.g. phones in Nigeria) tend to have high relative prices. In other words, price and quantity are negatively related. This implies that the main differences between countries are supply differences, not demand differences.

If Nigeria didn’t have a lot of phones because Nigerians didn’t like phones, then phones in Nigeria would be cheap compared to haircuts. And then valuing Nigeria’s output at the U.S. prices, which also has cheap phones compared to haircuts, wouldn’t make Nigeria look so rich. It might make them look poorer, in fact. So the empirical fact that valuing Nigeria’s output at U.S. prices makes Nigeria look relatively rich is evidence that Nigeria and the U.S. have different supply curves for phones and haircuts, not different demand curves [Yes, demand is probably different too. But relative to supply differences, these appear to be small].