# The Limited Effect of Reforms on Growth

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

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.

# Why Don’t Growth Economists Study Growth Anymore?

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

John Seater (NC State) left a really interesting comment on one of my recaps of the NBER Growth session papers.

It appears from the summaries in this blog that none of the other five papers was a growth paper. Now, literally anything in economics can have an effect on growth, so one could say that all five papers had implications for growth. However, it sounds as if none of the five papers summarized here addressed those implications. I am curious about why static analysis dominated a meeting ostensibly dedicated to studying economic growth. My impression from the programs for the Growth meeting in recent years is that most of the papers presented there are not about growth. What is going on? Has the Growth meeting ceased to be a growth meeting?

The short answer, John, is yes. The NBER Growth meeting really has ceased to be about growth, per se. I guess the broader question lurking around is whether this is a good or a bad thing. Let me see if I can take a shot at answering it from both directions.

The positive (or neutral) response is that growth papers aren’t about dynamics any more because the dynamics are determined by changes in steady states. People study the comparative statics of steady states in their models. Transition between those steady states – the dynamics – then just depend on the rate of accumulation of capital stocks (human and physical). Those rates don’t seem to be very different, so the transition rate isn’t the interesting aspect to study. The static difference in steady states is what determines the growth rates.

In terms of trend growth rates (how fast the economy grows in steady state), people probably implicitly have in their heads that those trend rates are similar across countries. Why? Because you look at the long-run paths of output per worker in most countries and they seem parallel, growing at the same rate in steady state. So that seems relatively less important in explaining cross-country differences.

The negative (or skeptical) response is that we’re missing something crucial by ignoring variation in growth rates. We’re assuming that the transitional growth rate is the same no matter what causes the static shift in steady states. Maybe that isn’t right. More importantly, maybe the trend growth rate isn’t identical across countries. While a lot of relatively well-off countries grow at very similar rates in steady state, poor countries don’t. Several of them grow very slowly, so slowly that they are falling behind rich countries.

Differential growth rates mean that we cannot just look at static differences across countries. Those differences are growing over time, so our static stories cannot be enough to explain them. We need explicit theories of why poor countries grow slowly, not just why the are poor to begin with.

Furthermore, even if countries do grow at the same rate in steady state, we’re still really interested in what that rate is. Growth at 2% per year doubles income every 35 years. Growth at 1% doubles it every 70. That’s a big difference in living standards over time. So studying growth rates is important in and of itself, outside of the question of cross-country comparisons.

I’ll freely admit that as a field, growth generally has strayed away from studying “growth”, in the traditional sense. But I don’t have a huge problem with where we are on this – I find the “what makes rich countries rich” question to be somewhat more compelling than the “is growth 1 or 2 percent per year” question. But it’s worth remembering that the latter question on growth rates has huge ramifications for absolute living standards over long periods of time – never underestimate compound growth.

# Potential “Potential Output” Levels

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

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

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

Back to Fernald’s paper. He finds that productivity growth was already declining prior to 2007, and therefore where GDP “should” be is a lot lower than the naive pre-2007 trend line would indicate. This is easier to see in a picture. The purple dashed line is from the CBO’s 2007 projection, and that is essentially just an extrapolation of the trend in GDP from about 1990-2007. Compared to that measure of potential GDP, we are doing very poorly, with actual GDP (the black line) falling nearly $2 trillion short of potential. Fernald’s alternative calculations that take into account the slowdown in productivity growth that started in the mid-2000’s suggest a much lower estimate of potential GDP. His estimate (the red line) is a prediction of what GDP would have been without the financial crisis, essentially. It falls well below the CBO 2007 estimate. It suggests that the economy today is only perhaps$400 billion short of potential GDP.

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

Why does Fernald come up with lower numbers for potential output than the naive forecast in 2007? Without going into the nitty-gritty, he looks at productivity growth (think output per hour) and finds that around 2003Q4, it stops growing as quickly as it did from 1995-2003. What Fernald chalks this up to is the exhaustion of the IT productivity boost. At the time, people thought that the IT revolution might have permanently raised labor productivity growth rates It appears to rather have had a “level effect” – we had a boost in the level of labor productivity, but now it will continue to grow at the normal rate. Again, this is easier to see in pictures, courtesy of Fernald’s paper. You can see that the 1995-2003 period is exceptional in having high labor productivity growth, and that since 2003 we’ve had growth in labor productivity at about the same rate as 1973-95. Anyone who uses the 1995-2003 period to extrapolate labor productivity growth (like the CBO was implicitly doing in 2007) would overestimate potential output.

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

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

# Is Capital Important?

There is kind of a disconnect in teaching economic growth. We spend a lot of time telling students about the Solow model and capital accumulation, but at the same time the general consensus among growth economists is that total factor productivity is more important to understanding levels of output per worker.

Why do we think that capital isn’t terribly important to levels of output per worker? Basically, because the correlation between capital per worker and output per worker is low – or rather, we assume that it is low. Here’s a way of thinking about this in terms of simple regressions. If I was interested in how important capital per worker was in explaining output per worker, I could run this regression for a sample of countries ( ${i}$) $\displaystyle \ln{y}_i = \beta_0 + \beta_1 \ln{k}_i + \epsilon_i \ \ \ \ \ (1)$

where I’ve put output per worker ( ${y_i}$) and capital per worker ( ${k_i}$) in logs. Logs keep countries with very small or very big values of capital per worker from being so influential, and in logs this regression will have an obvious interpretation for the coefficient ${\beta_1}$.

If I run this regression, I’ll get some estimated coefficient ${\hat{\beta}_1}$, which is the elasticity of output per worker with respect to capital per worker. Moreover, I could look at the R-squared of this regression. This R-squared will tell me what fraction of the variance of log output per worker ( ${Var(\ln{y}_i)}$) is explained by variation in log capital per worker ( ${Var(\ln{k}_i)}$). The R-squared is really what I want; it’s the answer the question “How important is capital in explaining differences in output per worker?”. The coefficient by itself doesn’t tell us that answer.

Now, there are some big problems with this regression. Most importantly, it is almost certainly the case that ${\ln{k}_i}$ is correlated with ${\epsilon_i}$, the residual. The residual captures things like technology levels, institutions, human capital, etc.. etc.. and capital per worker tends to be large when these things are “big”, meaning that they have a big positive effect on output per worker.

So that means we cannot trust our estimate ${\hat{\beta}_1}$, and cannot trust our value of R-squared. It’s worth writing out what the “true” R-squared is if we in fact had the right estimate of ${\beta_1}$. I’ll pre-apologize for the fact that this involves a lot of steps, but I’m writing them all out so it is easier to follow. $\displaystyle \begin{array}{rcl} R^2 &=& \frac{{\beta}_1^2 Var(\ln{k}_i)}{Var(\ln{y}_i)} \\ \nonumber &=& \beta_1 \frac{Cov(\ln{k}_i,\ln{y}_i)}{Var(\ln{k}_i)}\frac{Var(\ln{k}_i)}{Var(\ln{y}_i)} \\ \nonumber &=& \beta_1 \frac{Cov(\ln{k}_i,\ln{y}_i)}{Var(\ln{y}_i)} \\ \nonumber &=& \frac{Cov({\beta}_1\ln{k}_i,\ln{y}_i)}{Var(\ln{y}_i)} \\ \nonumber &=& \frac{Cov({\beta}_1\ln{k}_i,{\beta}_1 \ln{k}_i + \epsilon_i}{Var(\ln{y}_i)} \\ \nonumber &=& \frac{ Var({\beta}_1\ln{k}_i) + Cov({\beta}_1 \ln{k}_i,\epsilon_i)}{Var(\ln{y}_i)}. \nonumber \end{array}$

The last line is identical to what Pete Klenow and Andres Rodriguez-Clare (1997, and KRC hereafter) use to evaluate the importance of capital in explaining cross-country output per worker differences. In other words, KRC are just looking for an R-squared. But as they point out, they cannot simply run the regression I proposed above and get the R-squared from that, because almost certainly ${\hat{\beta}_1 \neq \beta_1}$.

Rather than run the regression, KRC suggest that we use some alternative means of estimating ${\beta_1}$. They propose using the share of total output that gets paid to capital. Why? Because under perfect competition and constant returns to scale, that share should be precisely equal to ${\beta_1}$. In data from the U.S., capital’s share of output is usually something between 0.3–0.4, and KRC use ${\hat{\beta}_1 = 0.3}$. The rest of their data ( ${\ln{k}_i}$ and ${\ln{y}_i}$) is exactly the same data that one would use to run the regression. The only thing they are doing differently is plugging in their outside estimate of ${\hat{\beta}_1}$. What KRC find is that their R-squared is about 0.30, or that only 30% of the variation in log output per worker across countries is accounted for by variation in capital per worker across countries. This is a big reason why growth economists don’t think capital is of primary importance in explaining cross-country differences in output per worker.

It’s interesting to consider, though, what could rescue capital as an important explanatory variable. KRC use the idea that capital’s share in output is equal to ${\beta_1}$ under perfect competition and constant returns to scale. But what if there is not perfect competition and/or constant returns to scale? There is a neat little relationship that holds if we assume that firms are cost-minimizers. That is $\displaystyle s_K = \frac{\beta_1}{\mu} \ \ \ \ \ (2)$

where ${s_K}$ is capital’s share in output (which KRC say is about 0.3) and ${\mu \geq 1}$ is the markup over marginal cost for firms. ${\mu = 1}$ only under perfect competition, and if there is imperfect competition or increasing returns to scale then markups are greater than one, meaning that the price charged by firms is greater than their marginal cost. From this we see that capital’s share may understate the value of ${\beta_1}$ if ${\mu>1}$. In particular, if there are increasing returns to scale at the firm level (i.e. fixed costs) but perfect competition (i.e. free entry/exit) then ${s_K}$ still measures the payments to capital accurately, but ${\mu}$ will be greater than one as firms with increasing returns need to charge more than marginal cost in order to cover the fixed costs.

Practically, if ${\beta_1 = 0.55}$, meaning that ${\mu = 1.83}$, or a markup of 83%, then the R-squared for capital goes to one. That is, with ${\beta_1 = 0.55}$, capital perfectly explains the varation in output per worker. Even with ${\beta_1 = 0.45}$, the R-squared is 0.67, meaning capital explains 2/3 of the variation in output per worker. So a relatively slight adjustment in the value of ${\beta_1}$ changes the conclusion regarding capital’s importance for output levels.

The issues with this line of thinking are (1) if there are increasing returns to scale at the firm level, why don’t we see increasing returns to scale at the aggregate/country level? (2) even if capital explains most of the variation in output per worker, there isn’t any data showing that savings rates actually vary across countries meaningfully. The differences in capital are probably the result of different technologies/institutions, and so those are the more fundamental source of variation.
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