Opening
HOST
Welcome back. Today we are on the Solow growth model, named after the economist Robert Solow. In 1960, South Korea was poorer than the Philippines. What happened next?
ECONOMIST
From 1960 to 2023, income per person in South Korea grew about 6 percent a year. In the Philippines, it grew about 2.5 percent a year.
HOST
So what did Korea do? I've heard it built a lot of factories.
ECONOMIST
It did. Korea saved a large share of its income and invested it in machines and buildings.
HOST
So here is today's question. Can saving and investing in machines make a country rich forever?
ECONOMIST
The answer is not a simple yes. By the end of this episode, you'll be able to do three things. First, explain why building more tools raises income per person only up to a resting point. Second, predict what happens when a country saves more, when its tools wear out faster, when its population grows faster, or when a disaster strikes. Third, show with the model why productivity, meaning how well a country uses what it has, explains more of the gap between rich and poor countries than investment does.
HOST
And we'll follow the chapter's mind map part by part?
ECONOMIST
Yes. Part one builds the Solow model. Part two runs experiments with it. Part three takes the model to the data. Each part has four leaves, small topics we take in order.
The Solow model
HOST
Part one of the map: The Solow model. The first leaf is capital accumulation. What does that mean in plain words?
ECONOMIST
Capital means tools: machines, buildings, boats. Capital accumulation is how a country builds up its stock of tools, meaning all the tools it has at one moment. Let's visit Islandia, a small imaginary island country. Islanders make one all-purpose product. Each year they can eat it, or they can turn it into tools, like boats, nets and workshops.
HOST
And the tools don't last forever, I assume.
ECONOMIST
No. Every year some boats break and some nets fall apart. New tools add to the stock. That's investment. Worn-out tools take away from it. That's depreciation. The leaf's equation says exactly this: the change in tools equals investment minus depreciation.
HOST
So it's like a bathtub. The water is the stock of tools. The tap is investment, and the drain is depreciation. But what decides how fast each one runs?
ECONOMIST
Two simple rules. Islanders always invest the same share of their output, meaning everything they produce in a year. We call that share the investment rate. Here, saving and investing are the same thing. And the same share of the tools wears out every year. That's the depreciation rate.
HOST
And what decides how much output there is?
ECONOMIST
Tools and productivity. Productivity means how well a country uses its tools and workers. We count everything per person, and output per person is also income per person.
HOST
So more tools always means more output?
ECONOMIST
More, yes, but less and less. Economists call this diminishing returns. Each extra tool adds less output than the one before. The first boat changes a fisher's life. The tenth boat adds very little.
HOST
Okay, let me guess. Islandia adds new tools every single year. My guess is that it keeps getting richer forever.
ECONOMIST
That's the natural guess, but it's wrong. At first, each new tool adds a lot of output, so the stock grows fast. But each new tool adds less output, and so less investment, than the one before. Investment still grows, but more and more slowly. Meanwhile, depreciation is always the same share of the stock, so depreciation grows as the stock grows. More boats means more broken boats. In the end, the new tools only replace the broken ones.
HOST
So that's the second leaf, the steady state.
ECONOMIST
Exactly. The steady state is the resting point where investment exactly equals depreciation. In the bathtub, it's the water level where the flow in equals the flow out. Below it, the stock grows. Above it, the stock shrinks. So wherever Islandia starts, it heads to the same point.
HOST
Where is that point for Islandia?
ECONOMIST
Islandia invests 40 percent of its output, so the investment rate is 0.4. Each year, 10 percent of its tools wear out, so the depreciation rate is 0.1. And productivity is set to 1.
HOST
And with those numbers, where does Islandia settle?
ECONOMIST
At 8 tools per person, with output of 2 units per person. Let's check them step by step. In this model, output per person is productivity times the cube root of tools per person. A cube root is the number that, multiplied by itself three times, gives the original number. With 8 tools you make 2 units, because 2 times 2 times 2 is 8.
HOST
Okay. So at 8 tools, output is 2. Then investment is 40 percent of 2, which is 0.8.
ECONOMIST
Right. And depreciation is 10 percent of 8 tools, which is also 0.8. The new tools exactly replace the broken ones, so the stock stops changing. And what do islanders eat? Output minus investment, so 2 minus 0.8. That means consumption per person is 1.2 units. That's the 60 percent of output they don't invest.
HOST
Next leaf: productivity counts more than investment. Why would that be true in a model that's all about investment?
ECONOMIST
Because productivity works through two channels, and investment through only one. First, the direct effect. With better methods, islanders make more with the same tools. Second, the tools effect. More output means more investment, so more tools, so more output again. Investment works only through this second channel, and diminishing returns shrink it.
HOST
Can we measure that?
ECONOMIST
Yes, with one more term: elasticity. An elasticity tells you by what percent one thing changes when another thing rises by 1 percent. Take output per person at the resting point, which we'll call long-run output. If productivity rises by 1 percent, long-run output rises by three halves of a percent, so one and a half percent. If the investment rate rises by 1 percent of its own size, for example from 40 to 40.4 percent, long-run output rises by only one half of a percent.
HOST
So productivity has three times the effect. But where does the one half come from?
ECONOMIST
In the model, long-run output follows the square root of the investment rate. For a small step of 1 percent, a square root rises by about half a percent. But for a big change, you must use the square root itself. Double the investment rate, and long-run output becomes the square root of 2, about 1.41 times as large. Not 1.5 times.
HOST
Then the last leaf of this part: no long-run growth. At the resting point, nothing grows?
ECONOMIST
Nothing per person. Tools, output and consumption per person all stay constant. Growth happens only on the way to the steady state, and diminishing returns end it. But income per person in the United States has grown about 2 percent a year for more than a century. So building tools cannot be what keeps living standards rising in the long run.
HOST
So in part one we saw that tools wear out, and that every extra tool helps less than the last. That's why saving more leads to a higher resting point, not endless growth. And productivity matters more than investment.
ECONOMIST
Yes. Next branch: we change one number at a time.
Experiments
HOST
Part two of the map: Experiments. What is an experiment in a model like this?
ECONOMIST
We start Islandia at its resting point and change one number permanently. The first leaf is a higher investment rate. Suppose Islandia raises it from 0.4 to 0.6, so from 40 to 60 percent of output. At the old resting point, investment is now larger than depreciation. So tools and output grow until Islandia reaches a new resting point. This path is called the transition.
HOST
And part one told us the investment rate works through a square root. So how much richer does Islandia get?
ECONOMIST
The rate is now one and a half times as large. So long-run output is multiplied by the square root of 1.5, which is about 1.22. It rises from 2 to 2.45.
HOST
Wait. So a country that saves more grows faster, permanently?
ECONOMIST
A lot of students think that, but no. Growth jumps at first, to about 1.6 percent in the first year. Then it gets smaller, year after year, until it is back to zero. The higher investment rate raises the level of income permanently, but not its growth rate. Economists call this a level effect, not a growth effect. Growth is fastest right after the change, when Islandia is farthest from its new resting point. This is the principle of transition dynamics. An economy grows faster the farther it sits below its steady state.
HOST
Next leaf: faster depreciation. When would tools wear out faster?
ECONOMIST
Think of a lasting change in the climate, so every year more buildings and machines are damaged. Suppose the depreciation rate rises from 10 to 15 percent. Now the drain is bigger. At the old resting point, more tools break than are built, so the stock shrinks.
HOST
And the rate is again one and a half times as large, but now the change is to the drain. So output is divided by that same 1.22?
ECONOMIST
Yes, because what matters is the investment rate divided by the depreciation rate. A drain one and a half times bigger cancels a tap one and a half times bigger. So long-run output falls from 2 to about 1.63, and it falls fastest right at the start.
HOST
Then population growth. More people means more workers, so more output. Doesn't that make Islandia richer?
ECONOMIST
Bigger, yes. Richer per person, no. Every newcomer needs tools too. So investment must replace broken tools and also equip each new person. So population growth works just like extra depreciation. For this experiment, we change the starting values a little. Tools now wear out at 8 percent a year, and the population grows at 2 percent a year. Together, that's the same 10 percent drain as before. So the resting point stays at the old 8 tools and 2 units per person.
HOST
And if population growth rises to 3 percent a year?
ECONOMIST
Then 8 percent wear plus 3 percent population growth makes a drain of 11 percent instead of 10. That's 1.1 times as large. So long-run output per person is divided by the square root of 1.1, which makes it about 4.7 percent lower. But total output keeps growing, as fast as the population.
HOST
Last leaf of this part: an earthquake. Half the tools are gone. I'd guess output falls by half too.
ECONOMIST
That's the tempting answer, but output falls by only about 21 percent. On Islandia, think of a huge storm that destroys half the boats and workshops, while nobody is hurt.
HOST
Only 21 percent? Where does that come from?
ECONOMIST
Tools per person fall from 8 to 4. Output is the cube root of tools. The cube root of 4 is about 1.59, compared with 2 before. That's about 79 percent of the old output, so a fall of about 21 percent. Each extra tool added less than the one before, so losing the last half of the tools costs much less than half the output.
HOST
And what happens after the storm?
ECONOMIST
Islandia's investment rate, depreciation rate and productivity have not changed. So the resting point is the same, and the economy grows back to it. Growth is fastest right after the storm. In the model, output is back within 1 percent of its old level after about 43 years.
HOST
So all that fast growth after the disaster is good news in the end?
ECONOMIST
That's a common mistake. The fast growth only replaces what was lost. In every year of the recovery, Islandia produces less than it would have without the storm. Real cases show such recoveries. Hiroshima and Nagasaki were hit by atomic bombs in World War Two. Within a few decades, they regained their place among Japan's cities. The most heavily bombed provinces of Vietnam caught up with the rest of the country. But real disasters also kill people and can destroy skills and organizations, and those losses can last.
HOST
So in part two, every experiment had the same shape. A lasting change, like saving more, moves the resting point, and the economy travels there, fast at first, then slowly. A disaster moves the economy, not the resting point.
ECONOMIST
Next comes the last branch: does the real world behave like this?
The model meets the data
HOST
Part three of the map: The model meets the data. The first leaf is capital across countries. What can we test?
ECONOMIST
A country that invests a bigger share of its output must hold more tools before the yearly wear matches its investment. At the resting point, the stock of tools, counted in years of output, equals the investment rate divided by the depreciation rate. Islandia has 8 tools per person and makes 2 units, so its tools are worth 4 years of output. And 0.4 divided by 0.1 is also 4.
HOST
So a country with better productivity should hold more capital compared with its output.
ECONOMIST
That's a natural guess, but no. Better productivity raises capital and output by the same proportion, so productivity does not change the ratio at all. If depreciation is similar everywhere, only the investment rate matters. The data agree: Singapore and South Korea invest a lot and hold a lot of capital compared with their output.
HOST
Next leaf: who catches up?
ECONOMIST
Here we use the principle of transition dynamics. Suppose a group of countries shares a similar resting point. Then the ones that start poorer are farther below it, so they should grow faster. The group should converge, which means the poorer members close much of the gap. Take 24 countries of the OECD, a club of mostly rich economies. In 1960, Japan, Portugal, Spain and Ireland were relatively poor, and they grew fastest, while the rich United States grew slowly. So the OECD converged.
HOST
Then it should work for the whole world too. Poor countries should grow fastest.
ECONOMIST
It barely works. Across all countries, poor ones did not grow faster on average. The Solow model's answer is that most poor countries already sit close to their own resting point, and that point is low, because their investment and productivity are low. A few countries, like South Korea, China and Botswana, grew very fast. Their resting points probably moved up.
HOST
That brings us to the South Korea leaf. What moved Korea's resting point?
ECONOMIST
Investment, at least in part. GDP is a country's total output. Korea's investment rate climbed steeply after the 1950s. Averaged over the years since 1990, Korea invests about 37 percent of its GDP. The U.S. invests about 24 percent.
HOST
So Korea invests far more than the United States. Then by now it should be richer than the U.S.
ECONOMIST
That sounds logical, but it isn't. In 2023, Korea's income per person was about 77 percent of the U.S. level.
HOST
So what would investment alone predict?
ECONOMIST
37 divided by 24 is a bit more than one and a half. The square root of that is about 1.25. So if both countries were at their resting points, with the same productivity, Korea would be about 25 percent richer than the United States.
HOST
But it's poorer. So something else must be lower in Korea.
ECONOMIST
Exactly. We assume both countries lose tools at the same rate, so the only thing left in the model is productivity. Korea invests more, yet earns less, so its productivity must be lower.
HOST
And the last leaf: richest versus poorest.
ECONOMIST
Compare the five richest countries in 2023, like Luxembourg and Norway, with the five poorest, like South Sudan and Burundi. Income per person in the richest five is about 115 times that in the poorest five. But the richest invest only about 2.7 times as large a share of their income.
HOST
So how much of that gap can investment explain?
ECONOMIST
Take the square root of 2.7, which is a bit more than 1.6. So investment alone would make the richest about 1.6 times richer. Divide 115 by that, and about 70 is left for productivity. The two multiply: a bit more than 1.6, times 70, gives the 115. And the 70 includes the extra tools that higher productivity leads to.
HOST
That's a huge gap. What does that mean for policy, for example for foreign aid?
ECONOMIST
For decades, foreign aid was planned to fill a financing gap, the investment a poor country needed to reach a growth target. The Solow model warns that investment gives a period of fast growth, not lasting growth. One study, by the economist Easterly, found that aid did not raise investment by the full amount of the aid, and investment did not raise growth as planned. Other studies find modest gains, but some of those results were not confirmed with more data. So the evidence is still debated.
HOST
So in part three we saw that the model gets capital right, and it explains catch-up among similar countries. But when we compare rich and poor, productivity does most of the work.
ECONOMIST
Exactly. Let's sum up.
Wrap-up
HOST
Let's go back to our three goals. First, why does building more tools stop raising income per person?
ECONOMIST
Because of diminishing returns. Each extra tool adds less output, but it wears out like the others. So in the end, new tools only replace broken ones.
HOST
Second, the experiments. I'd say saving more raises the level of income, not growth forever. Faster wear and faster population growth lower income per person. And after a disaster, the economy grows back.
ECONOMIST
Yes. And in every case, the economy moves fastest when it is farthest from its resting point.
HOST
And third, rich versus poor.
ECONOMIST
Investment explains a gap of about 1.6 times. Productivity explains a gap of about 70 times. Multiply the two and you get the 115. Why? Investment rates differ by less than three times, and investment works through a square root. So here is the big message. Saving and investing explain how much capital countries have, and why some grow fast for a while. But diminishing returns mean that building capital cannot drive growth in the long run. And differences in productivity, not investment, explain most of the gap between rich and poor countries.
HOST
Here's a question to think about. Suppose Islanders invent a magic kind of boat. Each new boat adds just as much fish as the one before, no matter how many boats they already have. Could building tools now make Islandia rich forever? And why?
ECONOMIST
Pause the episode and think about it for a moment.
HOST
Okay, we're back. What's the answer?
ECONOMIST
Yes, it could. Diminishing returns are what stop growth in the model. If every boat adds the same fish, output grows as fast as the number of boats. So does investment, which is 40 percent of output. Broken boats are also a fixed share of the boats. So if new boats outnumber broken ones at the start, they always will, and output keeps growing. In that world, saving more would raise growth forever. Real machines don't work like that. But ideas might.
HOST
Where can students try all this themselves?
ECONOMIST
Open the chapter's Colab notebook to run every experiment yourself. The Model Apps let you move sliders on the model's charts without any code. And the Data Apps show the country data from part three.
HOST
And next time?
ECONOMIST
In Chapter 6, we look for the engine that the Solow model is missing. We ask: can new ideas keep income growing forever?
HOST
That's it for today. Thanks for listening, and see you next time.