# 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.

# [insert policy here] Won’t Boost Growth Rates

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

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.

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

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.

# Latitude and Income per Capita in Comparative Development

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New paper out by Holger Strulik and Carl-Johan Dalgaard (who I predict is at this moment taking a smoke break). The paper looks at the reversal of the latitude/income relationship over history, and propose a physiological reason for it.

For starters, if you are familiar at all with basic development statistics, then you probably know that latitude and income per capita are positively correlated. The farther away from the equator you get (higher latitude) the richer you get. This works going north or south. South Africa is richer than Nigeria, for example, and Chile is richer than Ecuador. Dalgaard and Strulik have a nice graph showing this relationship holds not only for all countries, but also within Europe.

The first really interesting fact in the paper is that this gradient reverses if you look at pre-Industrial Revolution data. For 1500 CE, there is a negative relationship, and countries that are closer to the equator are richer. Again, this also holds within Europe. I had some vague concept that it probably reversed in the whole sample, but the within Europe evidence is really fascinating.

Around 1500, the Mediterranean countries in Europe were better off compared to their Northern neighbors. An aside: Were the Greeks and Italians of 1500 tsk-tsk-ing their profligate Teutonic cousins for their lazy attitudes and lack of robust economic institutions? Discuss.

Anyway, the latitude/income reversal, and the fact that it holds up within Europe, are both by themselves the kinds of stylized facts that you should cram into your head when thinking about comparative development.

But given that you have crammed that information in there, you probably have several questions. (1) Why are hot places rich in 1500, and cold places poor? (2) What changed to make the cold places rich today?

Dalgaard and Strulik take a swipe at these questions, focusing on the physiology involved in hot and cold places. There thesis rests on “Bergmann’s Rule”, which is a biological regularity noted in 1847. Bergmann’s rule states that average body mass of organisms rises as they get farther from the equator. This holds for people as well as animals. People generally have higher body mass farther from the equator (and no, that’s not just because of Wisconsin. I kid. Sort of.).

Why does Bergmann’s rule hold? Surface area to mass ratios. Big people have lower surface area to mass ratios, so they are more thermally efficient in cold climates. Thus the optimal body type for high, cold, latitudes is large, while for places closer to the equator small body types are optimal to maximize surface area to mass in order to radiate heat.

Now, large bodies have an additional feature. They require a lot of fuel (food), in particular for mothers when pregnant. Big women having big babies means using a lot of food. Thus people in cold latitudes were able to have fewer babies, given a supply of food, than their peers in equatorial regions. So we have bigger populations in equatorial regions and smaller ones in cold latitudes prior to the IR. Big populations mean more innovation in almost any type of growth model you write down, so equatorial regions had more innovation during the pre-IR era, and hence were richer.

But, eventually even the cold latitudes are going to innovate far enough to get the point of inventing technologies that rely on human capital. And the cold climate physiology gives them a natural tendency to favor quality over quantity of kids. Thus families in higher latitudes are going to more easily adopt the human capital using technology. This then starts a feedback effect, where by having a few, high-education kids means they can use the human-capital technology. Which raises income per capita. Which leads to further investment in kids at the expense of family size, and cold latitudes enter the Demographic Transition ahead of equatorial regions.

The reversal is inevitable in their model, given the initial physiological difference between latitudes. The physiological story is also consistent with differences in marriage patterns and child birth patterns between Europe and much of Asia in the pre-IR era.

They use Europe as an example, and how the latitude/income relationship holds today. But it holds in the U.S. as well. Is the income per capita of states in the U.S. consistent with the implied physiological differences between different areas of Europe, Africa, or Latin America due to population composition?

This paper predicts a reversal, but this reversal has to happen “just so” to avoid becoming a-historical. That is, the reversal has to happen just before the equatorial countries (China, India, various iterations of Islamic empires) become sufficiently rich to colonize Europe, snuffing out their development. This leaves Europe to effect the reversal, and go out to colonize the rest of the world. Did Europe get lucky here, or is there some reason that those places don’t become colonizers? Luck might be the answer, as you’ve got plenty of close-run things in European history [the Mongols turning back, Lepanto, Vienna].

The last thing that comes to mind here is that for this physiological difference to have such persistent effects, the family patterns it determines must become either (a) genetically rooted into populations or (b) some deeply ingrained in culture as to be permanent. Fertility behavior is mutable. For it to continue to be a reason for lack of development in equatorial regions you need some strong force keeping people locked into the “bad” preference for lots of kids. What is that force?

# Why Don’t Growth Economists Study Growth Anymore?

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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.

# The Solow Model

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This is another idea for modifying how to teach the Solow model. One thing I’d like to do is go immediately to including productivity – it follows cleanly from the simplest growth model. Second, I think it might be nice to work with the K/Y ratio immediately. In this way, I think you can actually skip using the whole “k-tilde” thing. And, *gasp*, do away with the traditional Solow diagram.

The simplest growth model doesn’t allow for transitional growth, and this due to the fact that it does not allow for capital, a factor of production that can only be slowly accumulated over time. The Solow Model is a standard model of economic growth that includes capital, and will be better able to account for the transitional growth that we see in several countries.

Production in the Solow Model takes place according to the following function

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

${K}$ is the stock of physical capital used in production, and ${A}$ and ${L}$ are defined just as they were in our simple growth model. So the production function here is just a modification of the simple model to include capital. The coefficient ${\alpha}$ is a weight telling us how important capital or ${AL}$ are in determining output.

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

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

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

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

In per capita terms, this is

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

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

So to understand the role of capital in economic growth, we need to understand the capital-output ratio and how it changes over time. We’ll start by looking at the balanced growth path, and then turn to situations where the economy is not on the balanced growth path (BGP).

One fact about the BGP is that the return to capital, ${r}$, is constant. The return to capital is ${r = \alpha Y/K}$, which depends (negatively) on the capital-output ratio (the return to capital is just the marginal product of capital). If ${r}$ is constant on the BGP, then it must be that ${K/Y}$ is constant on the BGP as well. What does this mean? It means that ${K/Y}$ can have a level effect on output per worker, but has no growth effect. To see this more clearly, take logs of output per worker,

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

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

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

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

The capital/output ratio is constant along the BGP, and has no effect on the growth rate on the BGP. But what if the economy is not on the BGP? Then it will be the case that ${K/Y}$ affects the growth rate of output per worker, because the ${K/Y}$ ratio will not be constant. More precisely, the growth rate of capital/output is

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

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

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

where ${\dot{K}}$ is the change in the capital stock. ${s}$ is the savings rate, the fraction of output that the economy sets aside to invest in new capital goods, so that ${sY}$ is the total amount of new investment. ${\delta}$ is the depreciation rate, the fraction of the existing capital stock that breaks or becomes obsolete at any given moment.

To find the growth rate of capital, divide through the above equation by ${K}$ to get

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

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

The growth rate of output is

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

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

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

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

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

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

and growing if the capital/output ratio is larger than the value on the right-hand side. In other words, if the capital stock is relatively small, then it will have a tendency to grow faster than output, raising the ${K/Y}$ ratio. Eventually ${K/Y = s/(\delta+n+g)}$, the steady state value, and the ${K/Y}$ ratio stops changing.

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

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

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

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