Has the Long-run Growth Rate Changed?

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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.

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:
Papell Prodan
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.

Is the U.S. Really Below Potential GDP?

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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:
Post 1990 Trend
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.
Post War Trend

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.
All Data Trend
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.


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I’ve had a few posts in the past few months (here and here) about the consequences of mechanization for the future of work. In short, what will we do when the robots take our jobs?

I wouldn’t call myself a techno-optimist. I don’t think the arrival of robots necessarily makes everything better. But I do not buy the strong techno-pessimism that comes up in many places. Richard Serlin has been a frequent commenter on this blog, and he generally has a gloomy take on where we are going to end up once the robots arrive. I’m not bringing up Richard to pick on him. He writes thoughtful comments on this subject (and lots of others), and it is those comments that pushed me to try and be more clear on why I’m “techno-neutral”.

The economy is more creative than we can imagine. The coming of robots to mechanize away our jobs is the latest in a long, long, long, history of technology replacing workers. And yet here we still are, working away. Timothy Taylor posted this great selection a few weeks ago. This is a quote from Time Magazine:

The rise in unemployment has raised some new alarms around an old scare word: automation. How much has the rapid spread of technological change contributed to the current high of 5,400,000 out of work? … While no one has yet sorted out the jobs lost because of the overall drop in business from those lost through automation and other technological changes, many a labor expert tends to put much of the blame on automation. … Dr. Russell Ackoff, a Case Institute expert on business problems, feels that automation is reaching into so many fields so fast that it has become “the nation’s second most important problem.” (First: peace.)
The number of jobs lost to more efficient machines is only part of the problem. What worries many job experts more is that automation may prevent the economy from creating enough new jobs. … Throughout industry, the trend has been to bigger production with a smaller work force. … Many of the losses in factory jobs have been countered by an increase in the service industries or in office jobs. But automation is beginning to move in and eliminate office jobs too. … In the past, new industries hired far more people than those they put out of business. But this is not true of many of today’s new industries. … Today’s new industries have comparatively few jobs for the unskilled or semiskilled, just the class of workers whose jobs are being eliminated by automation.

That quote is from 1961. This is almost word for word the argument you will get about robots and automation leading to mass unemployment in the future. 50 years ago we were just as worried about this kind of thing, and in those 50 years we do not have massive armies of unemployed workers wandering the streets. The employment/population ratio in 1961 was about 55%, and then it steadily rose until the late 90’s when it topped out at about 64%. Even after the Great Recession, the ratio is still 59% today, higher than it was in 1961.

This didn’t happen without disruption and dislocation. And the robots will cause similar dislocation and disruption. Luddites weren’t wrong about losing their jobs, they were just wrong about the economy losing jobs in aggregate. But I don’t see why next-generation robots are any different than industrial robots, mainframes, PC’s, tractors, mechanical looms, or any other of the ten million innovations made in history that replaced labor. We can handle this with some sympathy and try to smooth things out for those dislocated, or we can do what usually happens and let them hang out to dry. The robots aren’t the problem here, we are.

What exactly are those new jobs that will be created? If I knew, then I wouldn’t be writing this blog post, I’d be out starting a company. The fact that I cannot conceive of an innovation myself is not evidence that innovation has ceased. But I do believe in the law of large numbers, and somewhere among the 300-odd million Americans is someone who *is* thinking of a new kind of company with new kinds of jobs.

Robots change prices as well as wages. An argument for pessimism goes like this. People have subsistence requirements, meaning they have a wage floor below which they cannot survive. Robots will be able to replace humans in production and this will drive the wage below that subsistence requirement. Either no firm will hire workers at the subsistence wage or people who do work will not meet subsistence.

The problem with this argument is that it ignores the impact of robots on the price of that subsistence requirement. Subsistence requirements are in real terms (I need clothes and housing and food), not nominal terms (I need $2000 dollars). The “subsistence wage” is a a real wage, meaning it is the nominal wage divided by the price level of a subsistence basket of goods. Robots lowering marginal costs of production lowers the nominal human wage, but it also lowers the price of goods. It is not necessary or even obvious that real wages have to fall because of robots. History says that despite all of the labor-saving technological change that has gone on over the last few hundred years, real wages have risen as the lower costs outweigh the downward pressure on wages.

Who is going to buy what the robots produce? Call this the “Henry Ford” argument. If you are going to invest in opening up a factory staffed entirely by robots, then who precisely is supposed to buy that output? Ford raised wages at his highly mechanized (for the time) plants so that he had a ready-made market for his cars. The Henry Fords of robot factories are going to need a market for the stuff they build. Rich people are great, but diminishing marginal utility sets in pretty quick. That means robot owners either need to lower prices or raise wages for the people they do hire in order to generate a big enough market. Depending on the fixed costs involved in getting these proverbial robot factories up and running, robot owners may be a strong force for keeping wages high in the economy, just like Henry Ford was back in the day.

The wealthy are wealthy because they own productive assets. A tiny fraction of the value of those assets is due to the utility to the owner of the widgets they kick out. The majority of the value of those assets is due to the fact that you can *sell* that output for money and use that money to buy other widgets. Rockefeller wasn’t wealthy because he had a lot of oil. He was wealthy because he could sell it to other people. No other people, no wealth. Just barrel after barrel of useless black gunk.

The same holds for robot owners. Those robots and robot factories have value because you can sell them or the goods they make in the wider economy. And that means continuing to exchange with the non-wealthy. You cannot be wealthy in a vacuum. Bill Gates on an island with robots and a stack of 16 billion dollar bills is Gilligan with a lot of kindling.

Wealthy robot owners will do what wealthy (fill in capital stock here) owners have done for eons. They’ll trade access to the capital, or the goods it produces, to the non-wealthy in exchange for services, effort, flattery, and new ideas on what to do with that wealth.

Wealth concentration would be a problem with or without robots. The worry here is that because the wealthy will be the only ones able to build the robots and robot factories, they will control completely the production of goods and the demand for labor. That’s not a problem that arises with robots, that is a problem that arises with, well, settled agriculture 10,000 years ago. Wealth concentration makes owners both monopolists (market power selling goods) and monopsonists (market power buying labor), which is a bad combination. It gives them the ability to drive (real) wages down to minimum subsistence levels. This is bad, absolutely. But this was bad when (fill in example of a landed elite) did it in (fill in historical era here). This is bad in “company towns”. This is bad now, today. So if you want to argue against wealth concentration and the pernicious influence it has on wages, get started. Don’t wait for the robots, they’ve got nothing to do with it.

Again, be clear that in arguing against techno-pessimism I am not arguing that robots will generate a techno-utopia with ponies and rainbows. I just do not buy the dystopian view that somehow it’s all going to come crashing down around our ears because of the very particular innovations coming in the near future.

Why Did Consumption TFP Stagnate?

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I’ve been trying to think more about why consumption-sector TFP flatlined from about 1980 forward. What I mentioned in the last post about this was that the fact that TFP was constant does not imply that technology was constant.

I then speculated that technology in the service sector may not have changed much over the last 30 years, partly explaining the lack of consumption productivity growth. By a lack of change, I mean that the service sector has not found a way to produce more services for a given supply of inputs, and/or produced the same amount of service with a decreasing supply of inputs. Take something that is close to a pure service – a back massage. A one-hour back massage in 1980 is almost identical to a one-hour back massage in 2014. You don’t get twice (or any other multiple) of the massage in 2014 that you got in 1980. And even if the therapist was capable of reducing back tension in 30 minutes rather than 60, you bought a 60-minute massage.

We often buy time when we buy services, not things. And it isn’t so much time as it is attention. And it is very hard to innovate such that you can provide the same amount of attention with fewer inputs (i.e. workers). Because for many services you very specifically want the attention of a specific person for a specific amount of time (the massage). You’d complain to the manager if the therapist tried to massage someone else at the same appointment.

So we don’t have to be surprised that even technology in services may not rise much over 30 years. But there were obviously technological changes in the service sector. As several people brought up to me, inventory management and logistics were dramatically changed by IT. This allows a service firm to operate “leaner”, with a smaller stock of inventory.

But this kind of technological progress need not show up as “technological change” in doing productivity accounting. That is, what we call “technology” when we do productivity accounting is not the only kind of technology there is. The “technology” in productivity accounting is only the ability to produce more goods using the same inputs, and/or produce the same goods using fewer inputs. It doesn’t capture things like a change in the shape of the production function itself, say a shift to using fewer intermediate goods as part of production.

Let’s say a firm has a production function of {Y = AK^{\alpha}L^{\beta}M^{\gamma}} where {A} is technology in the productivity sense, {K} is capital, {L} is labor, and {M} is intermediate goods. Productivity accounting could reveal to us a change in {A}. But what if an innovation in inventory management/logistics means that {\gamma} changes?

If innovation changes the shape of the production function, rather than the level, then our TFP calculations could go anywhere. Here’s an example. Let’s say that in 1980 production is {Y_80 = A_{1980}K_{80}^{.3}L_{80}^{.3}M_{80}^{.4}}. Innovation in logistics and inventory management makes the production function in 2014 {Y_14 = A_{2014}K_{14}^{.4}L_{14}^{.4}M_{14}^{.2}}.

Total factor productivity in 1980 is calculated as

\displaystyle  TFP_{80} = \frac{Y_{80}}{K_{80}^{.3}L_{80}^{.3}M_{80}^{.4}} \ \ \ \ \ (1)

and total factor productivity in 2014 is calculated as

\displaystyle  TFP_{14} = \frac{Y_{14}}{K_{14}^{.4}L_{14}^{.4}M_{14}^{.2}}. \ \ \ \ \ (2)

TFP in 2014 relative to 1980 (the growth in TFP) is

\displaystyle  \frac{TFP_{14}}{TFP_{80}} = \frac{Y_{14}}{K_{14}^{.3}L_{14}^{.3}M_{14}^{.4}} \times \frac{K_{80}^{.3}L_{80}^{.3}M_{80}^{.4}}{Y_{80}} \times \frac{M_{14}^{.2}}{K_{14}^{.1}L_{14}^{.1}} \ \ \ \ \ (3)

which is an unholy mess. The first fraction is TFP in 2014 calculated using the 1980 function. The second fraction is the reciprocal of TFP in 1980, calculated normally. So the first two fractions capture the relative TFP in 2014 to 1980, holding constant the 1980 production function. The last fraction represents the adjustment we have to make because the production function changed.

That last term could literally be anything. Less than one, more than one, more than 100, less than 0.0001. If {K} and {L} rose by a lot while {M} didn’t go up much, this will lower TFP in 2014 relative to 1980. It all depends on the actual units used. If I decide to measure {M} in thousands of units rather than hundreds of units, I just made TFP in 2014 go down by a factor of 4 relative to 1980.

Once the production function changes shape, then comparing TFP levels across time becomes nearly impossible. So in that sense TFP could definitely be “getting it wrong” when measuring service-sector productivity. You’ve got an apples to oranges problem. So if we think that IT innovation really changed the nature of the service-sector production function – meaning that {\alpha}, {\beta}, and/or {\gamma} changed, then TFP isn’t necessarily going to be able to pick that up. It could well be that this looks like flat or even shrinking TFP in the data.

If you’d like, this supports David Beckworth‘s notion that consumption TFP “doesn’t pass the smell test”. We’ve got this intuition that the service sector has changed appreciably over the last 30 years, but it doesn’t show up in the TFP measurements. That could be due to this apples to oranges issue, and in fact consumption TFP doesn’t reflect accurately the innovations that occurred.

To an ambitious graduate student: document changes in the revenue shares of intermediates in consumption and/or services over time. Correct TFP calculations for these changes, or at least provide some notion of the size of that fudge factor in the above equation.

The Limited Effect of Reforms on Growth

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I said in my last post that transitional growth is slow, and therefore changing potential GDP – as many of the recent Cato Growth proposals would do – could not add much to the growth rate of GDP in the near term.

There were several questions that came up in the comments, so let me try to be more clear about distinguishing between influences of trend growth and short-run shocks. Output in period {t+1} is

\displaystyle  y_{t+1} = (1+g)y_t + (1+g)\lambda (y^{\ast}_t - y_t) \ \ \ \ \ (1)

where the first term on the right is the normal trend growth rate, and the second term is the additional transitional growth that occurs because the economy is not at potential GDP, {y^{\ast}_t}.

We need to distinguish between changes in potential GDP and changes in current GDP. Let’s take the above equation, plug in {\lambda=0.02}, and then use it to iterate forward from period 0 (today) until some arbitrary period {t}. You get

\displaystyle  y_t = (1+g)^t \left[(1-0.98^t)y^{\ast}_0 + 0.98^t y_0 \right]. \ \ \ \ \ (2)

In period {t}, GDP will have grown by a factor of {(1+g)^t} due to trend growth in GDP. The term in the brackets shows the cumulative effect of having {y_0 \neq y^{\ast}_0} in the initial period. The 0.98 terms are just {1-.02}, and capture the changing role of this transitional growth over time. Note that as {t} goes up, {0.98^t} goes to zero and the effect of initial GDP {y_0} falls to nothing. As {t} gets big, the economy reaches potential GDP.

Now let’s assume that period 0 is 2014. Potential GDP is 17 trillion and actual GDP is 16 trillion, and the trend growth rate is 2%. Let’s consider two alternative policies to enact today that take effect in 2015.

  • Policy A: a short run spending surge sufficient to make GDP in 2015 equal to potential GDP. Policy A immediately eliminates the gap between actual and potential GDP, but has no other long term effect.
  • Policy B: raises potential GDP by 1 trillion dollars, but adds no immediate spending to GDP. The effect on potential GDP is permanent.

For Policy A, GDP in 2015 (period 1) is

\displaystyle  y_1 = (1.02)^1\left[(1-0.98)\times 17 + 0.98 \times 17 \right] = 17.34. \ \ \ \ \ (3)

The growth rate of GDP from 2014 to 2015 is {(17.34 - 16)/16 = 0.084} or about 8.4%. That’s a massive GDP growth rate for a developed economy like the US. But it is a one-time shock to the growth rate. From 2015 to 2016, and from 2016-2017, and every year thereafter, the growth rate will be exactly 2% because the economy is precisely back on trend. Policy A gives a one-year gigantic boost to the growth rate.

What about Policy B? GDP in 2015 here is

\displaystyle  y_1 = (1.02)^1\left[(1-0.98)\times 18 + 0.98 \times 16 \right] = 16.36. \ \ \ \ \ (4)

This is nearly 1 trillion less than Policy A. The growth rate of GDP from 2014 to 2015 is {(16.36 - 16)/16 = 0.023}. As the prior post noted, reforms that raise potential GDP don’t have big effects on growth rates. But while the effect on growth is small, it is persistent. From 2015-2016, the growth rate of GDP will be roughly…0.023. It’s actually minutely smaller than from 2014-2015, but rounding makes them look the same. It will take a few years before the growth rate declines appreciably. Fifty years from now the growth rate will still be almost 0.021. Changing potential GDP, like with Policy B, is like turning an oil tanker with a tug boat. It doesn’t go fast, but it goes on for a long time.

So is Policy B worse than Policy A? It depends entirely on your time preferences. In 2015 GDP under Policy A is nearly 1 trillion dollars higher than with policy B. But 100 years from now, GDP will be nearly 1 trillion dollars higher with Policy B. We can actually figure out how soon it will be before Policy B passes Policy A. Set

\displaystyle  (1.02)^t \left[(1-0.98^t)17 + 0.98^t 17 \right] = (1.02)^t \left[(1-0.98^t)18 + 0.98^t 16 \right] \ \ \ \ \ (5)

and solve for {t}. This turns out to be roughly 34 years from now, in 2048. It takes a long, long, time for changes in potential GDP to really pay off. If you want to increase the level of GDP in the near term, and hence raise near-term growth rates by implication, then you have to, you know, boost GDP. GDP is a measure of current spending, so raising GDP means raising current spending. There isn’t a trick to get around this.

Now, could I be underselling Policy B as a near-term boost to growth rates and GDP? Let’s consider a few possibilities:

  • I’m underestimating the size of {\lambda}. As I mentioned last time, there is lots of empirical evidence that this is pretty small. But okay, let’s make {\lambda = 0.05}, more than double my 0.02 value. Now in 2015 policy B yields GDP of 16.4 trillion and a growth rate of 2.6%. Yes, it helps policy B, but doesn’t get it anywhere close to Policy A. It is still 14 years before GDP under Policy B is larger than under Policy A.
  • I’m underestimating the boost to potential GDP that Policy B can deliver. So let’s ask, given {\lambda = 0.05}, how much would {y^{\ast}_0} have to go up to match the 8.4% growth rate of Policy A? Potential GDP would have to jump to roughly 36 trillion, meaning it has to roughly **double** in size thanks to the policy. I think it is totally fair to say that this is implausible in a country like the US.
  • But China was able to do it. Right, when China opened up, made reforms, etc.., it was able to raise its potential GDP by a large amount. You could probably plausibly argue that it raised potential GDP by a factor of something like 8-10. But the rapid growth in China over the last 30 years is not some victory lap for good state-led policy reforms, it’s a testament to just how screwed up Maoism was as an economic system. [Egad! An institutions explanation!]
  • What if Policy B raised the trend growth rate, {g}? If it changed {g} appreciably, then Policy B would be something really special. Let’s review for a moment a few of the changes that did not change the long-run growth rate in the US: the introduction of electricity, the income tax, the Great Depression, the New Deal, Medicare, higher tax rates, the Cold War, the oil crisis, lower tax rates, de-regulation, the IT revolution, and New Coke. There have been shifts in the level of potential GDP, such as the IT revolution shifting up potential GDP and inducing a period of relatively rapid transitional growth in the 1990’s. But it’s hard – if not impossible if you take Chad Jones‘ semi-endogenous growth idea seriously – to fundamentally alter {g}. It is dictated by changes in the scale of the global economy, not by policy effects within the US.

I’m all for policy reforms that raise potential GDP, and several of those proposed in the Cato forum would probably do that. We might want to undertake several of them at once to counteract the drags on potential GDP that Robert Gordon has outlined.

But we can’t be fooled into thinking that any of them would make a really appreciable difference to economic growth today. You can revolutionize education, or corporate taxation, or urban planning, or immigration all you want, but the gains those changes induce will take decades to manifest themselves.

[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 {t+1} can be written in terms of output in period {t} this way

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

This says that output in {t+1} is equal to {1+g} times current output. That is “regular” growth. The term with the {\lambda} is the additional boost in growth we get from being below potential. {y^{\ast}_t} is potential GDP in period {t}, and {y^{\ast}_t - y_t} is the gap in GDP. {\lambda} tells us how much of that gap we make up from period {t} to {t+1}. If {\lambda = 0}, then we are stuck below potential (secular stagnation). If {\lambda = 1}, then immediately next period our GDP will be at potential again.

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

\displaystyle  Growth = \frac{y_{t+1}-y_t}{y_t} = (1+g)\left[\lambda \frac{y^{\ast}_t}{y_t} + (1-\lambda)\right] - 1. \ \ \ \ \ (2)

The growth rate from {t} to {t+1} depends on the ratio of potential to actual GDP today, period {t}. If that ratio were equal to one – meaning that we were at potential – then the growth rate just becomes {g}, the trend growth rate. The larger is {y^{\ast}_t/y_t} – 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 ({y_t}) is about 16 trillion. Potential GDP today ({y^{\ast}_t}) 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 {\lambda}, the percent of the GDP gap that is closed in a year. We’ve got lots of evidence that this value is about {\lambda = 0.02}, 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 {\lambda} 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 {g}, trend growth. Let’s call that {g = 0.02}, 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

\displaystyle  Growth = (1+.02)\left[.02 \frac{17}{16} + (1-.02)\right] - 1 = .0213 \ \ \ \ \ (3)

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 {g} is really irrelevant. All the action is inside the brackets. Because {\lambda} 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

\displaystyle  Growth = (1+.02)\left[.02 \frac{18}{16} + (1-.02)\right] - 1 = .0225 \ \ \ \ \ (4)

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.