In 1960 South Korea was poorer than the Philippines; then it grew more than twice as fast

6.0% vs 2.5% Average growth of GDP per person a year, 1960–2023. In 1960: South Korea $1,408 and the Philippines $2,183 per person (2021 dollars)

Was South Korea’s growth a matter of saving and building more machines? And could it go on forever?

Open with the two countries. The numbers are from the notebook’s Figs. 8 and 9 (Penn World Table 11.0): growth of GDP per person from 1960 to 2023, and the 1960 levels in 2021 dollars.

Make the gap concrete. Growing 6 percent a year doubles income about every 12 years; growing 2.5 percent a year doubles it about every 28 years (the rule of 70 from Chapter 3). Over six decades the first country multiplies its income about 40 times, the second about 5 times.

Ask the room: what changed in South Korea? Collect answers: more factories, education, exports, government policy, technology. Today’s model takes the first answer seriously, investment in machines and buildings, and asks how far it can go.

Today: three questions

• The Solow model. Where does capital come from, and where does it stop?
• Experiments. What happens when a country saves more, its capital wears out faster, its population grows, or an earthquake strikes?
• The model meets the data. Does it explain capital, catch-up and South Korea, and how much of the rich–poor gap?

These are the three acts and the three branches of the mind map on the title slide. Act I builds the model: the production function of Chapter 4 plus one new equation, for capital. Act II runs experiments on it. Act III takes its predictions to data on more than a hundred countries and ends with four policy questions.

The headline result is a disappointment and a lesson at the same time: saving and investment can raise income for a while, but not forever.

The Solow model

Act I

Capital is a stock: investment fills it and depreciation drains it

What the model assumes	Equation
Production, as in Chapter 4	\(Y_t = \bar{A}K_t^{1/3}L_t^{2/3}\)
Output is consumed or invested	\(C_t + I_t = Y_t\)
Capital next year	\(K_{t+1} = K_t + I_t - \bar{d}K_t\)
A fixed workforce	\(L_t = \bar{L}\)
A fixed share of output is invested	\(I_t = \bar{s}Y_t\)

One new idea relative to Chapter 4: capital is no longer given. It is built from past investment.

Five equations, five unknowns in every year: Y, K, L, C and I. Time runs in years. The parameters are productivity Ā, the depreciation rate d̄ (the share of machines and buildings that wear out each year), the investment rate s̄ and the workforce L̄, plus a starting capital stock K̄₀.

The bathtub picture: the capital stock is the water in the tub, investment is the tap and depreciation is the drain. A stock carries over from year to year; investment and depreciation are flows during a year.

The resource constraint also says that saving equals investment: Y − C = I. There is no government and no trade. The consumption rule C = (1 − s̄)Y is not a separate assumption; it follows from the second and the fifth rows.

The markets for labor and capital are in the background, as in Chapter 4: the wage is the marginal product of labor, and the rental price of capital is the marginal product of capital.

Two equations hold the whole model

\[y_t = \bar{A}k_t^{1/3} \qquad\qquad \Delta k_{t+1} = \underbrace{\bar{s}y_t}_{\text{investment}} - \underbrace{\bar{d}k_t}_{\text{depreciation}}\]

• \(y \equiv Y/L\) and \(k \equiv K/L\): output and capital per person.
• The gap between the two terms is net investment: capital per person grows when it is positive and shrinks when it is negative.

There is no formula for \(k_t\) year by year, but a diagram, a steady-state solution and a simulation tell us everything.

How we get there: divide the production function by L. Because the exponents add to one, L^(2/3)/L = L^(−1/3), so Y/L = Ā (K/L)^(1/3). Then put I = s̄Y into the capital equation and divide by the constant L̄.

Substituting the first equation into the second gives one equation that moves capital per person from one year to the next: Δk = s̄Ā k^(1/3) − d̄k. Everything else in the chapter is a way of reading that one line.

The three tools come next: the Solow diagram (Fig. 1), the steady-state solution, and year-by-year simulations (Act II).

Experiment: an economy with little capital

An economy invests 40% of its output (\(\bar{s} = 0.4\)), and 10% of its capital wears out each year (\(\bar{d} = 0.1\)). It starts with capital per person \(k_0 = 2.7\). Will capital per person rise, fall or stay put? Does it keep going forever?

Collect guesses before showing the figure. Most students expect capital to rise; fewer have a view on whether it stops.

You can help with a quick calculation: at k₀ = 2.7, output is 2.7^(1/3) ≈ 1.39, so investment is 0.4 × 1.39 ≈ 0.56, and depreciation is 0.1 × 2.7 = 0.27. Investment is larger, so capital rises. The question is what happens as k grows.

Capital grows until investment just replaces what wears out

Model figure: the Solow diagram with \(\bar{A} = 1\) and \(\bar{d} = 0.1\). Adapted from Fig. 5.1 in Jones (2024), Macroeconomics, 6th ed.

How to read it: both curves are plotted against capital per person k. The curve is investment s̄y = s̄Āk^(1/3), bent because of diminishing returns. The straight line is depreciation d̄k. At k₀ the green bar is net investment, the vertical gap, and the green arrow on the axis shows the direction capital moves. The slider sets the investment rate s̄ from 0.2 to 0.6. The title reports k* and net investment at k₀.

Do live:

1. At s̄ = 0.4, read the title: k* = 8.00, and net investment at k₀ = 2.7 is +0.287. That answers the prediction: capital rises.
2. Point at the gap shrinking as k moves right: capital keeps rising, but by less each year, and it stops where the curves cross.
3. Move the slider to 0.6: the investment curve rotates up and k* moves to 14.70. Move it to 0.2: k* = 2.83, almost exactly k₀, so net investment is close to zero (+0.008).
4. Ask: what happens if an economy starts to the right of k*? (Depreciation exceeds investment, so capital falls back.)

Wherever it starts, capital per person heads for the same steady state

Start at \(k = 1\)	Capital per person
Year 10	3.883
Year 50	7.709
Year 100	7.991
Year 300	8.000

Investment rate \(\bar{s}\)	\(k^*\)
0.2	2.83
0.4	8.00
0.6	14.70

Why it stops: each extra machine adds less output, and so less investment, than the one before, but the same amount of depreciation.

Left table: the notebook’s Check 1. Starting far below the steady state, it applies Δk = s̄y − d̄k for 300 years, one year at a time. Capital climbs fast at first, then more and more slowly, and settles at 8.000, the number the formula on the next slide gives.

Right table: the slider titles of Fig. 1.

The takeaway is the key sentence of the chapter. Depreciation grows in proportion to k, a straight line. Investment grows with k^(1/3), which flattens. Sooner or later the line catches the curve. That is diminishing returns at work, the same property we met in Chapter 4.

Ask: what would happen if the exponent on capital were 1, so that investment were a straight line too? (If s̄Ā > d̄, capital would grow forever. That is one way to see where long-run growth would have to come from.)

The steady state can be solved exactly

\[\bar{s}\bar{A}(k^*)^{1/3} = \bar{d}k^* \quad\Longrightarrow\quad k^* = \left(\frac{\bar{s}\bar{A}}{\bar{d}}\right)^{3/2}, \qquad y^* = \bar{A}^{3/2}\left(\frac{\bar{s}}{\bar{d}}\right)^{1/2}\]

With \(\bar{s} = 0.4\), \(\bar{d} = 0.1\), \(\bar{A} = 1\): \(k^* =\) 8, \(y^* =\) 2, consumption \(c^* = (1-\bar{s})y^* = 1.2\), capital worth \(k^*/y^* = 4\) years of output, and a real interest rate \(r^* = \tfrac{1}{3}\,\bar{d}/\bar{s} \approx\) 8.3%.

Higher saving and productivity raise the steady state; faster depreciation lowers it.

The algebra: divide both sides by d̄ and by (k)^(1/3), so (k)^(2/3) = s̄Ā/d̄. Raise both sides to the power 3/2. For y, put k into the production function: the exponents multiply, (3/2)(1/3) = 1/2, and Ā × Ā^(1/2) = Ā^(3/2).

The numbers are the notebook’s first key numbers: k* = 8.00, y* = 2.00, c* = 1.20, k/y = 4.00 = s̄/d̄, r* = 8.3%.

The real interest rate: a unit of output saved becomes a unit of capital, which earns its marginal product, (1/3) y/k. In the steady state y/k = d̄/s̄. So a country that invests more ends up with a lower return on its last unit of capital. Keep this in mind for the capital-flows question of Chapter 4.

The capital-output ratio s̄/d̄ does not depend on productivity at all. Act III tests it with data.

Saving more: what happens to consumption?

Consumption is what is left after investment. If a country invests a larger share of its output, does its steady-state consumption per person rise or fall?

Let students argue both sides. “Rise”: more saving means more capital and more output. “Fall”: a larger share of output goes to investment. Both effects are real; the figure shows which one wins, and when.

Consumption is the gap between output and investment

Model figure: the Solow diagram with output, \(\bar{A} = 1\) and \(\bar{d} = 0.1\). Adapted from Fig. 5.2 in Jones (2024), Macroeconomics, 6th ed.

How to read it: the top curve is output y = Āk^(1/3). Below it are the investment curve s̄y and the depreciation line d̄k, as in Fig. 1. At k, the black bar between output and investment is steady-state consumption c. The slider sets s̄ from 0.1 to 0.7; the title reports y, investment and c.

Do live:

1. At s̄ = 0.4: y* = 2.00, investment 0.80, consumption 1.20.
2. Move s̄ up step by step and read c* in the title: 0.90 at 0.1, 1.13 at 0.2, 1.21 at 0.3, 1.20 at 0.4, then 1.12, 0.98 and 0.79.
3. Point out that y* keeps rising all the way, while c* rises and then falls.

Consumption peaks at a middle investment rate

Investment rate \(\bar{s}\)	0.1	0.3	0.4	0.6	0.7
Output \(y^*\)	1.00	1.73	2.00	2.45	2.65
Consumption \(c^*\)	0.90	1.21	1.20	0.98	0.79

\[c^* = (1-\bar{s})\,y^* = (1-\bar{s})\,\bar{s}^{1/2}\left(\frac{\bar{A}^3}{\bar{d}}\right)^{1/2} \quad\text{is largest at}\quad \bar{s} = \tfrac{1}{3}\]

Past some point, more saving means more machines to maintain and less left to consume.

The numbers are the slider titles of Fig. 2.

The calculus, for those who want it: maximize (1 − s) s^(1/2). The derivative is s^(−1/2)[(1 − s)/2 − s] = 0, so s = 1/3. With Cobb-Douglas, the consumption-maximizing investment rate equals capital’s exponent. Economists call it the golden rule.

Why consumption falls at high s̄: the capital stock grows, but so does the amount that must be replaced every year, d̄k*. Beyond s̄ = 1/3, the extra output from more capital is smaller than the extra investment needed to keep it.

Careful with policy conclusions: comparing steady states ignores the transition. Raising s̄ toward 1/3 lowers consumption today in exchange for more later.

More investment does not always mean more consumption

A country that invests more will eventually be able to consume more.

Only up to a point. In the steady state, consumption per person is highest at an investment rate of 1/3. Above it, raising \(\bar{s}\) from 0.4 to 0.6 cuts \(c^*\) from 1.20 to 0.98, even though output rises.

The myth confuses output with consumption. Output per person rises with every increase in s̄, but people consume (1 − s̄)y.

Ask: why might a government still want a high investment rate? (A country far below its steady state, or one that expects to need capital in the future, may accept lower consumption now; and the golden rule ignores how people value consumption today against consumption later.) The point here is narrower: more investment is not free.

Productivity counts three times as much as investment

A 1% rise in … changes	\(k^*\) by	\(y^*\) by	Curve that moves
investment rate \(\bar{s}\)	+1.5%	+0.5%	investment rotates up
productivity \(\bar{A}\)	+1.5%	+1.5%	investment and output rotate up
depreciation rate \(\bar{d}\)	−1.5%	−0.5%	depreciation rotates up
workforce \(\bar{L}\)	0	0	none

Doubling \(\bar{s}\) multiplies \(y^*\) by \(2^{1/2} \approx 1.41\); doubling \(\bar{A}\) multiplies it by \(2^{3/2} \approx 2.83\).

Read each row as an elasticity: take logs of the steady-state formulas, ln y* = (1/2) ln s̄ + (3/2) ln Ā − (1/2) ln d̄. The notebook checks them with 1 percent changes: +0.499%, +1.504% and −0.496%.

Why 3/2 for productivity? Write y* = Ā × (k*)^(1/3). Productivity raises output directly (exponent 1), and the extra output means more investment, more capital and more output again (another 1/2). Investment works only through the second channel, and diminishing returns shrink it to 1/2.

In Chapter 4, with capital given, the exponent on Ā was 1. Once capital responds to productivity, productivity matters even more. That is the main lesson for Act III.

The workforce row: a bigger economy has proportionally more capital and output, but per person nothing changes. That is constant returns to scale.

Capital accumulation cannot be the engine of long-run growth

• In the steady state, \(k^*\), \(y^*\) and \(c^*\) are constant: growth per person is zero.
• Accumulation raises income only during the transition, and diminishing returns end it.
• Chapter 3 showed rich countries growing for well over a century. Something other than machines must keep raising output per person.

The Solow model rules out its own candidate for long-run growth, and that is a result.

This is the headline negative result. It surprises many students, because saving and investing look like the obvious recipe for getting rich.

The intuition in one sentence: every new machine needs the same maintenance, but adds less output than the last one, so eventually all new investment goes to replacing worn-out capital.

Where to look instead: productivity, Ā. If Ā keeps rising, the investment curve keeps rotating up, and capital and output keep growing. Chapter 6 models where rising productivity comes from: new ideas.

Ask: does this mean saving does not matter? (No. It matters for the level of income and for growth during a transition, which is what Act II is about.)

Experiments

Act II

Every experiment follows the same three steps

1. Which curve moves? Only the one that contains the changed parameter.
2. Follow the arrows. Where investment now exceeds depreciation, capital grows; where it falls short, capital shrinks.
3. Check the algebra. The steady-state formulas give the new \(k^{**}\) and \(y^{**}\).

Change	What moves in the diagram
\(\bar{s}\) or \(\bar{A}\)	the investment curve rotates
\(\bar{d}\) or \(\bar{n}\)	the depreciation line rotates
an earthquake	nothing: the economy jumps left along the axis

This is the recipe for every exercise in the chapter. Students who follow it rarely go wrong.

Population growth n̄ appears later in this act; it works like extra depreciation. The earthquake is the only experiment in which no curve moves: it changes where the economy is, not where it is going.

Ask a quick check: which curve moves if the government taxes investment so that s̄ falls? (The investment curve rotates down.)

Experiment: the investment rate rises for good

In 2030 the investment rate rises permanently from 0.4 to 0.6. Sketch output per person from 2015 to 2115 on a ratio scale. When is growth fastest? What is the growth rate in 2100?

Ask students to draw the path on paper before the reveal. Common sketches: a straight line going up forever (growth forever), a step up in 2030 (an immediate jump), or a smooth S-curve. Ask them to label when growth is fastest.

Reminder: on a ratio scale, the slope of the line is the growth rate.

Output rises to a higher level, and growth fades back to zero

Model figure: simulation of \(\Delta k = \bar{s}y - \bar{d}k\), with \(\bar{A} = 1\), \(\bar{d} = 0.1\) and \(\bar{s} = 0.4\) until 2030. Adapted from Fig. 5.5 in Jones (2024), Macroeconomics, 6th ed.

How to read it: panel (a) is the Solow diagram with output; the old investment curve is dashed and the new one solid. Panel (b) is output per person, year by year, from 2015 to 2115. The slider sets the new investment rate s̄′ from 0.3 to 0.8; the buttons switch panel (b) between a ratio and a linear scale. The title reports the old and new steady states and the growth rate in 2031.

Do live:

1. At s̄′ = 0.6: y* = 2.00 → y** = 2.45, and growth in 2031 is +1.6 percent.
2. On the ratio scale, point at the slope: steepest right after 2030, flattening toward zero. By 2100 the line is almost flat.
3. Move s̄′ to 0.8: y** = 2.83 and growth in 2031 of +3.2 percent, but the path still flattens.
4. Move s̄′ to 0.3: output falls toward 1.73, with growth of −0.8 percent in 2031. That is the textbook’s exercise on a tax that discourages investment.

A higher investment rate raises the level of income, not its long-run growth

New investment rate \(\bar{s}'\)	0.3	0.5	0.6	0.8
New steady state \(y^{**}\)	1.73	2.24	2.45	2.83
Growth in 2031	−0.8%	+0.8%	+1.6%	+3.2%
Growth in the long run	0	0	0	0

The new steady state follows from the formula: \(y^{**}/y^* = (\bar{s}'/\bar{s})^{1/2} = 1.5^{1/2} \approx 1.22\).

The numbers are the slider titles of Fig. 5. Capital rises by a factor of 1.5^(3/2) ≈ 1.84 (the notebook’s Fig. 4), output by 1.5^(1/2) ≈ 1.22.

Reveal the predictions: growth is fastest immediately after the change, when the gap between investment and depreciation is widest, and it falls toward zero as capital approaches the new steady state. There is no jump in 2030: capital takes time to build.

Policy reading: a reform that permanently raises investment, through taxes or better financial markets, gives a period of faster growth and a permanently higher income level. Both are valuable; neither is permanent faster growth.

Saving more raises income, but not growth forever

If a country permanently raises its saving rate, it will grow faster permanently.

It grows faster only during the transition. Raising \(\bar{s}\) from 0.4 to 0.6 lifts growth to 1.6% in the first year, and then growth fades back to zero at a higher level of income.

This is the most common misreading of the Solow model, and a common claim in policy debates. The distinction between a level effect and a growth effect is the key idea to take away from Act II.

Ask: if growth eventually returns to zero, is the higher saving rate still worth it? (Output and capital are permanently higher; whether people are better off depends on the consumption sacrificed during the transition, as the golden-rule slide showed.)

Experiment: capital wears out faster

Suppose severe storms make machines and buildings wear out faster: the depreciation rate rises from 0.10 to 0.15 in 2030. Which curve moves? What happens to output per person, and is the change fastest at the start or at the end?

Use the three-step recipe from the start of the act. The depreciation line rotates up; the investment curve does not move. At the old k*, depreciation now exceeds investment.

Faster depreciation lowers output, quickly at first

Model figure: simulation of \(\Delta k = \bar{s}y - \bar{d}k\), with \(\bar{A} = 1\), \(\bar{s} = 0.4\) and \(\bar{d} = 0.1\) until 2030. Adapted from Fig. 5.7 in Jones (2024), Macroeconomics, 6th ed.

How to read it: panel (a) is the Solow diagram, with the old depreciation line dashed and the new one solid. Panel (b) is output per person over time. The slider sets the new depreciation rate d̄′ from 0.06 to 0.20; the buttons switch the scale.

Do live:

1. At d̄′ = 0.15: y* = 2.00 → y** = 1.63, and growth in 2031 is −1.7 percent.
2. Switch between the ratio and the linear scale. On the ratio scale, the steep slope right after 2030 shows the fastest decline.
3. Move d̄′ to 0.06: depreciation falls, and output rises to 2.58, the same experiment in reverse.

In panel (a), capital falls to about 54 percent of its old level, because (0.10/0.15)^(3/2) ≈ 0.54.

The farther from the steady state, the faster an economy moves

\[g_{k} = \frac{\Delta k_{t+1}}{k_t} = \bar{d}\left[\left(\frac{k^*}{k_t}\right)^{2/3} - 1\right]\]

Capital per person	50% of \(k^*\)	95% of \(k^*\)	150% of \(k^*\)
Growth of \(k\) a year	+5.87%	+0.35%	−2.37%

The principle of transition dynamics: growth depends on the percentage gap to the steady state. Below it an economy grows; above it, it shrinks.

Derivation: divide Δk = s̄y − d̄k by k to get the growth rate, s̄Āk^(−2/3) − d̄. The steady-state condition says s̄Ā = d̄(k)^(2/3). Substitute, and the growth rate depends only on k/k.

The numbers are the notebook’s Check 3, which compares the formula with growth computed directly from the model: they agree to two decimals.

Output per person grows about a third as fast as capital, g_y ≈ (1/3) g_k, by the growth rule for powers.

This one formula explains both experiments: right after the change, the economy is farthest from its new steady state, so it moves fastest. It is also the tool Act III takes to cross-country data.

Population growth works like extra depreciation

\[\Delta k_{t+1} = \bar{s}y_t - (\bar{n} + \bar{d})\,k_t \qquad\Longrightarrow\qquad y^* = \bar{A}^{3/2}\left(\frac{\bar{s}}{\bar{n}+\bar{d}}\right)^{1/2}\]

• Investment now has two jobs: replace worn-out capital, and equip each new person with the same capital as everyone else.
• With \(\bar{n} = 0.02\) and \(\bar{d} = 0.08\), \(\bar{n} + \bar{d} = 0.10\): the same steady state as before, \(k^* = 8\) and \(y^* = 2\).

Derivation: the growth rate of a ratio is the difference of the growth rates, g_k = g_K − g_L. Capital grows at s̄Y/K − d̄, the population at n̄. Use Y/K = y/k and multiply by k.

The production function per person does not change. Only the drain on capital per person gets bigger: d̄ is replaced by n̄ + d̄.

The starting values are the notebook’s for Fig. 11, chosen so that the baseline matches Fig. 1.

Population growth: richer or poorer?

If a country’s population grows faster, from 2% to 3% a year, what happens to income per person in the steady state? And to total GDP?

Let students vote. Some think more people means more output and more income. Ask them to separate the two questions: income per person and total income.

Faster population growth lowers income per person

Model figure: the Solow diagram with population growth, \(\bar{A} = 1\), \(\bar{s} = 0.4\) and \(\bar{d} = 0.08\). Adapted from Fig. 5.11 in Jones (2024), Macroeconomics, 6th ed.

How to read it: the solid straight line is (n̄ + d̄)k; the dashed gray line is d̄k alone, the case without population growth. The slider sets n̄ from 0 to 5 percent. The title reports k, y and the growth rate of total GDP.

Do live:

1. Start at n̄ = 0.02: k* = 8.00 and y* = 2.00; total GDP grows at 2.0 percent a year.
2. Move to 0.03: the line rotates up, k* falls to 6.93 and y* to 1.91.
3. Move to 0: the line falls back to d̄k, and y* = 2.24.
4. Read the end of the title as you move: total GDP grows at n̄, whatever its value.

A faster-growing population makes an economy bigger, not richer per person

Population growth \(\bar{n}\)	0%	1%	2%	3%	5%
Income per person \(y^*\)	2.24	2.11	2.00	1.91	1.75
Growth of total GDP	0%	1%	2%	3%	5%

From 2% to 3%: \(y^*\) falls by 4.7%, because \((0.10/0.11)^{1/2} \approx 0.95\). Total GDP \(Y = yL\) grows at \(\bar{n}\) in the steady state.

The numbers are the slider titles of Fig. 11 and the notebook’s Check 4 (y* = 1.907, −4.7 percent).

Two separate messages. Per person: faster population growth dilutes capital, so y* is lower. In total: output, capital and population all grow at n̄, so a growing population gives a growing economy, but still no growth per person.

Constant returns to scale are behind the second message: an economy with 100 times more people and 100 times more machines has the same output per person. Chapter 6 shows how ideas break this, because an idea can be used by everyone at once.

Does slower population growth make countries richer?

In the model, a lower \(\bar{n}\) raises income per person. Should governments try to slow population growth?

What the evidence suggests

• Across countries, faster population growth goes with lower income.
• Falling birth rates raised East Asia’s share of working-age adults.
• That shift may explain part of its fast growth.

Why it is contested

• Income also lowers birth rates: causality runs both ways.
• Simulated effects of lower fertility are positive but modest.
• More people also means more inventors of new ideas.

Left column: Mankiw, Romer and Weil (1992, Quarterly Journal of Economics) estimate the Solow model across countries and find that population growth enters with the predicted negative sign. Bloom and Williamson (1998, World Bank Economic Review) argue that the “demographic dividend”, a rising share of working-age people as birth rates fell, accounts for a sizeable part of East Asia’s growth miracle.

Right column: as countries get richer, families have fewer children, so the correlation does not prove that population growth causes poverty. Ashraf, Weil and Wilde (2013, Population and Development Review) simulate a fall in fertility in a typical high-fertility country and find gains in income per person that are real but modest, and slow to arrive. Kremer (1993, Quarterly Journal of Economics) argues that over the very long run, more people meant faster technological progress, the subject of Chapter 6.

For the policy students: coercive population policies have large human costs. Most of the gains in the evidence come from voluntary falls in fertility, through education and health. Ask: which policy levers would you consider, and how would you measure their effects on income?

Experiment: an earthquake destroys half the capital

An earthquake destroys half of the machines and buildings, but no one is hurt, and the investment rate, depreciation and productivity are all unchanged. By what percentage does output per person fall right away?

Collect numbers. The most common guess is 50 percent (“half the capital, half the output”). Some will remember diminishing returns from Chapter 4 and guess less.

Use the recipe: which curve moves? None. The economy simply jumps left along the k axis.

No curve moves: the economy jumps below its steady state

Model figure: the Solow diagram with output, \(\bar{A} = 1\), \(\bar{s} = 0.4\) and \(\bar{d} = 0.1\). Adapted from Fig. 5.12 in Jones (2024), Macroeconomics, 6th ed.

How to read it: the Solow diagram with output, investment and depreciation. The slider sets the share of capital destroyed, from 10 to 90 percent. The economy starts at k* = 8 and jumps to the left; the title reports the new capital and output relative to y*.

Do live:

1. At a share of 0.5: k falls from 8.0 to 4.0, and output falls to 79 percent of y*. That answers the prediction: a 21 percent fall, not 50.
2. Point at the gap between investment and depreciation at k = 4: it is positive, so capital grows back.
3. Move the share to 0.9: output falls to 46 percent of y*.

Losing half the capital costs about a fifth of output

Share of capital destroyed	10%	30%	50%	70%	90%
Output right after, % of \(y^*\)	97%	89%	79%	67%	46%

\[\frac{y_{\text{after}}}{y^*} = (1 - \text{share})^{1/3}: \qquad (1/2)^{1/3} \approx 0.79\]

Diminishing returns in reverse: the machines lost were the least productive ones. The steady state itself does not change.

The numbers are the slider titles of Fig. 12.

The same property that stops growth in the Solow model also cushions disasters: when capital is scarce, each remaining machine is worth more.

Ask: which share of capital must be destroyed to halve output? ((1 − share)^(1/3) = 1/2, so the share is 1 − 1/8 = 87.5 percent.)

The model’s earthquake hurts no one. Real disasters kill people and destroy human capital and institutions too; the policy slide at the end of the act comes back to this.

After the disaster: how long to recover?

After losing half its capital, the economy grows back toward the same steady state. When is growth fastest? Does a disaster twice as big take twice as long to recover from?

Use the principle of transition dynamics: the farther below the steady state, the faster the growth. Ask students to sketch the path of output on a ratio scale.

Output drops at once, then grows back fastest right after the shock

Model figure: simulation of \(\Delta k = \bar{s}y - \bar{d}k\), with \(\bar{A} = 1\), \(\bar{s} = 0.4\), \(\bar{d} = 0.1\) and capital destroyed in 2030. Adapted from Fig. 5.13 in Jones (2024), Macroeconomics, 6th ed.

How to read it: output per person from 2015 to 2115, with the disaster in 2030. The slider sets the share of capital destroyed; the buttons switch between a ratio and a linear scale. The title reports the fall in 2030 and the year output is back within 1 percent of y*.

Do live:

1. Share 0.5: output falls by 21 percent in 2030 and is back within 1 percent of y* in 2073.
2. On the ratio scale, point at the steepest slope right after the drop.
3. Compare 0.1, 0.5 and 0.9: recovery years 2048, 2073 and 2084.

A disaster five times as large does not take five times as long

Share of capital destroyed	10%	30%	50%	70%	90%
Fall in output in 2030	3%	11%	21%	33%	54%
Back within 1% of \(y^*\)	2048	2065	2073	2079	2084

A bigger shock leaves the economy farther below its steady state, so it starts its recovery with faster growth.

The numbers are the slider titles of Fig. 13. From a 10 percent loss, recovery to within 1 percent takes 18 years; from a 50 percent loss, 43 years; from a 90 percent loss, 54 years. Nine times the destruction adds only three times the recovery time.

This is the principle of transition dynamics in its cleanest form: no parameter changed, so the economy returns to exactly where it was. Only the starting point moved.

Fast rebuilding is not a gain from the disaster

After an earthquake, the economy grows unusually fast for years. So the disaster was, in the end, good for the economy.

The fast growth only makes up lost ground. Output returns to the same steady state it had before, so every year of the recovery the economy produces less than it would have without the disaster.

This is a version of the broken-window fallacy: rebuilding creates measured activity, but the resources used to rebuild could have produced something else.

On the ratio scale of Fig. 13, the recovery path lies below the no-disaster line every year after 2030. Add up the gap and you get the total loss.

Ask: is there any way a disaster could leave a country better off in the long run? (Only if it changes a parameter, for example if rebuilding brings better technology or better institutions. That would be a change in Ā or s̄, not the disaster itself.)

Rebuilding after wars and disasters

Cities and regions destroyed by bombing often recovered within a generation. What does that mean for reconstruction policy?

What the evidence suggests

• Hiroshima and Nagasaki regained their place among Japan’s cities within decades.
• The most heavily bombed provinces of Vietnam caught up with the rest.
• As in the model, lost capital is rebuilt fastest where it is scarcest.

Why it is contested

• Some disasters, such as severe cyclones, leave lasting income losses.
• Disasters that bring political upheaval cause long declines.
• Lives, skills and institutions are not machines.

Left column: Davis and Weinstein (2002, American Economic Review) show that Japanese cities bombed in World War II, including Hiroshima and Nagasaki, returned to their prewar relative size within about two decades. Miguel and Roland (2011, Journal of Development Economics) find that by 2002, the most heavily bombed districts of Vietnam did not differ from the others in poverty, consumption or literacy.

Right column: Hsiang and Jina (2014, NBER Working Paper 20352) study 6,700 cyclones and find that national incomes do not recover to their pre-disaster trend for decades. Cavallo, Galiani, Noy and Pantano (2013, Review of Economics and Statistics) find that even very large natural disasters have no significant long-run effect on growth, unless they are followed by a radical political revolution.

In Solow terms: if a disaster only destroys capital, the economy rebuilds. If it also lowers Ā or s̄, through lost skills, broken institutions or lasting risk, the steady state itself falls.

For the policy students: reconstruction aid can speed up the transition, but the long-run outcome depends on whether the parameters, institutions and productivity, survive. Ask: what would you protect first after a disaster?

The model meets the data

Act III

Testing the model’s prediction on capital

In the steady state, capital relative to output equals \(\bar{s}/\bar{d}\). If depreciation is similar everywhere, which countries should hold the most capital relative to their output?

Students should answer: the countries that invest the most. Ask them to name some candidates before the figure appears. Singapore, China and South Korea usually come up.

Countries that invest more hold more capital per unit of output

Data: Penn World Table 11.0 (csh_i, rgdpo; Feenstra, Inklaar and Timmer 2015); investment rate averaged 1990–2023, capital-output ratio in 2023; capital is our perpetual-inventory estimate. Line: the model’s \(K/Y = s/d\). Chart inspired by Fig. 5.3 in Jones (2024), Macroeconomics, 6th ed.

How to read it: each dot is a country. The horizontal axis is its average investment rate since 1990, the vertical axis its capital-output ratio in 2023. The straight line is the model’s prediction K/Y = s/d. The slider sets the depreciation rate d from 0.04 to 0.12; a lower d makes the line steeper (its slope is 1/d).

Do live:

1. Point at the upward slope of the cloud: the prediction holds.
2. Hover over the far right: Singapore, China and South Korea invest the most and hold the most capital per unit of output. The United States lies in the middle.
3. Move the slider: a depreciation rate of about 7 to 9 percent makes the line run through the middle of the cloud.
4. Hover over countries far from the line, and ask why (measurement, recent changes in investment, natural resources).

Data caveat: capital is not observed directly. The notebook builds it from past investment, with 5 percent depreciation, and that choice affects the vertical axis.

The model explains capital well, and income gaps poorly

\[\frac{y^*_{rich}}{y^*_{poor}} = \left(\frac{\bar{A}_{rich}}{\bar{A}_{poor}}\right)^{3/2} \times \left(\frac{\bar{s}_{rich}}{\bar{s}_{poor}}\right)^{1/2}\]

\[115 \;\approx\; \underset{\text{productivity}}{70} \;\times\; \underset{\text{investment}}{1.6}\]

• Five richest against five poorest in 2023: investment rates differ by a factor of only 2.7, so investment explains a factor of \(2.7^{1/2} \approx 1.6\).
• The Solow model leaves an even larger role for productivity than the production model of Chapter 4.

The numbers are the notebook’s key numbers. The five richest: Luxembourg, Norway, Singapore, Qatar and Ireland. The five poorest: South Sudan, Yemen, Burundi, the D.R. of the Congo and the Central African Republic. Venezuela is dropped, because its GDP data broke down during hyperinflation; always look at your data.

The textbook, with older data, finds 75 = 37.5 × 2. Our gap is bigger (115), but the lesson is the same: investment explains a small factor, productivity a large one. The two factors are rounded; 1.64 × 70 ≈ 115.

Why bigger than in Chapter 4? There, part of the capital gap counted as “capital”. In the Solow model, capital is itself partly caused by productivity, so that part moves to the productivity side of the ledger.

Data caveat: a few tiny economies report investment rates far above 100 percent of GDP (Bermuda in 2000). None is among the ten countries here, but drop such rows before averaging over all countries.

Catch-up within the rich world

Among 24 rich OECD countries, did the ones that were poorest in 1960 grow faster or slower over the next six decades?

Ask students to apply the principle of transition dynamics. If the OECD countries share similar steady states, the poorer ones in 1960 were farther below theirs, so they should have grown faster.

Within the OECD, the countries that started poorer grew faster

Data: Penn World Table 11.0 (rgdpo / pop, 2021 dollars; Feenstra, Inklaar and Timmer 2015); OECD members: OECD, Members and partners. Chart inspired by Fig. 5.8 in Jones (2024), Macroeconomics, 6th ed.

How to read it: each dot is an OECD country. The horizontal axis is GDP per person in the starting year, on a ratio scale; the vertical axis is average growth a year from then to 2023. The line is a regression line. Press play to move the starting year from 1960 to 2000, or drag the slider. The later members start hidden: click “Joined the OECD later” in the legend.

Do live:

1. 1960: Portugal, Türkiye, Greece, Japan, Spain and Ireland were the poorest and grew about 3 percent a year or more (Ireland 4.7 percent). The United States, Canada, New Zealand and Switzerland were the richest and grew about 2 percent or less.
2. Hover over the line: for 1960 it falls about 2.2 percentage points of growth for every tenfold difference in starting income.
3. Press play: the line slopes down for every starting year.
4. Click “Joined the OECD later”. South Korea, the poorest in 1960, grew fastest (6.0 percent a year). Mexico grew only 1.5 percent from a similar start as Spain.

The rich world converged: the gaps of 1960 have narrowed

1960 start	GDP per person, 1960	Growth a year to 2023
Portugal	$5,075	3.2%
Japan	$6,093	3.2%
Ireland	$7,504	4.7%
United States	$21,165	2.0%
Switzerland	$25,405	2.0%

Consistent with similar steady states: countries far below theirs in 1960 grew fastest. But who is in the sample matters: the 24 countries joined the OECD early.

The numbers are the hover values of Fig. 8 (Penn World Table 11.0, 2021 dollars). The regression line for 1960 has a slope of −2.2 and an R² of 0.50.

Sample selection: the textbook’s 24 were OECD members by the 1970s, already rich or catching up fast. Colombia, Chile, Costa Rica and Mexico, which joined later, were about as poor as Portugal or Greece in 1960, but grew only 1.5 to 2.4 percent a year. A sample chosen because its members became rich will tend to show catch-up. Here the slope barely changes when the later members are added, but the cloud is wider.

Data note: with Penn World Table 11.0, the 1960 incomes differ from the textbook’s (which uses version 10.0 and 2019), but the pattern is the same.

Experiment: catch-up across the whole world

Now take every country with data since 1960, about 110 of them. Will the poorer ones have grown faster on average? Sketch the cloud and its regression line.

Ask students to sketch before the reveal. Many expect a downward slope, as in the OECD. Ask what the model would need to be true for a downward slope (similar steady states everywhere) and what a flat line would mean.

Across the world, poorer countries did not grow faster on average

Data: Penn World Table 11.0 (rgdpo / pop, 2021 dollars; Feenstra, Inklaar and Timmer 2015). Chart inspired by Fig. 5.9 in Jones (2024), Macroeconomics, 6th ed.

How to read it: the same chart for every country with data; OECD members in one color, the rest of the world in another. Click an entry in the legend to hide or show a group. Press play to change the starting year; the number of countries rises from 110 in 1960 to 181 from 1990.

Do live:

1. 1960: point at the cloud: poor countries are spread from fast growers to countries whose income fell. The line is nearly flat.

2. Hover over the top: South Korea 6.0 percent, China 5.9, Singapore 5.7, Botswana 5.5. Hover over the bottom: the D.R. of the Congo −1.8 percent, the Central African Republic −0.5.

3. Hide the OECD and ask which countries are the fast growers. They are the subject of the next figure.

4. Press play: the grey line (all countries) tilts down for later starting years, but it always stays flatter than the OECD line of the previous chart.

Most poor countries seem close to a low steady state, not far below a high one

• If poor countries shared rich countries’ steady states, they would be far below them and grow fastest. Most did not.
• The Solow reading: their steady states are low, because investment rates and productivity are low.
• The fast growers, South Korea, China, Singapore and Botswana, are countries whose steady state probably moved up.

Convergence is conditional: countries approach their own steady states, not a common one.

This is the main interpretation of Fig. 9. Economists distinguish unconditional convergence (poor countries catch up with rich ones, period), which the world data reject, from conditional convergence (countries grow faster the farther they are below their own steady states), which the Solow model predicts.

Policy reading: for most poor countries, catching up requires changing the steady state itself, through investment and above all productivity, rather than waiting for transition dynamics to do the work.

Mankiw, Romer and Weil (1992) show that, once differences in saving and population growth are held constant, poorer countries do tend to grow faster, at a slow rate.

What changed in South Korea?

South Korea started poorer than the Philippines in 1960 and grew more than twice as fast. What would the Solow model look for in the data?

Students should answer with the recipe: a parameter that raised Korea’s steady state, so that the economy was far below its new steady state and grew fast during the transition. The investment rate is the parameter we can observe.

South Korea’s investment rate rose steeply; the Philippines’ stayed flatter

Data: Penn World Table 11.0 (csh_i, investment as a share of GDP; Feenstra, Inklaar and Timmer 2015); regions: World Bank country groups. Chart inspired by Fig. 5.10 in Jones (2024), Macroeconomics, 6th ed.

How to read it: investment as a percent of GDP, year by year, for South Korea, the Philippines and the United States. The menu picks a region; clicking a country in the legend adds it. The vertical axis is 0 to 60 percent; drag on the chart to zoom out for the few countries above it. The title reports average investment rates since 1990.

Do live:

1. Follow Korea: from about 13 percent in the 1950s to more than 40 percent in the 1990s. The United States and the Philippines stay much flatter.
2. Read the title: since 1990, South Korea 37 percent, the Philippines 18 percent, the United States 24 percent.
3. Pick East Asia & Pacific and add China and Singapore. Singapore invested about 40 percent around 1960 and peaked at 68 percent in 1982.
4. Pick Sub-Saharan Africa and add Botswana: its rate jumped from below 10 percent in 1965 to about 60 percent in the early 1970s, when its diamond mines were built.

Investment explains only part of Korea’s rise

\[\frac{y_{Korea}}{y_{US}} = \left(\frac{\bar{A}_{Korea}}{\bar{A}_{US}}\right)^{3/2} \times \left(\frac{37.1\%}{23.8\%}\right)^{1/2}\]

\[\underset{\text{actual, 2023}}{0.77} \;\approx\; \underset{\text{productivity}}{0.73^{3/2}} \;\times\; \underset{\text{investment}}{1.25}\]

South Korea invests much more than the United States, yet it is poorer, because its productivity is about 73% of the U.S. level.

The numbers are the notebook’s key numbers: average investment rates since 1990 of 37.1 percent (Korea) and 23.8 percent (United States), an investment factor of 1.25, an actual income ratio of 0.77 in 2023, and an implied productivity ratio of 0.73.

Read it as a Solow story in two parts. Korea’s rising investment rate moved its steady state up and produced fast growth during the transition. But on investment alone Korea would be 25 percent richer than the United States; it is 23 percent poorer. The rest is productivity.

Ask: what might have raised Korea’s productivity since 1960? (Schooling, technology adopted from abroad, exporting firms, better management.) The next slide is the debate about how much was capital and how much was productivity.

The East Asian miracle: capital or productivity?

Was East Asia’s fast growth mostly capital accumulation, which must fade, or rising productivity, which can last?

Mostly accumulation

• More capital, schooling and workers explain most of it.
• Measured productivity growth looks ordinary.
• If so, growth must slow as capital piles up.

Productivity mattered more

• Measured with wages and returns, productivity grew fast.
• Exporters learned and adopted foreign technology.
• Investment may follow productivity, not lead it.

Left column: Young (1995, Quarterly Journal of Economics) carefully measures inputs in Hong Kong, Singapore, South Korea and Taiwan and finds that rising investment, labor force participation and schooling account for most of their growth, with productivity growth close to that of other countries. Krugman (1994, Foreign Affairs) popularized the conclusion: the “miracle” was perspiration rather than inspiration, and, like Soviet growth, it would slow.

Right column: Hsieh (2002, American Economic Review) measures productivity growth from factor prices (wages and returns to capital) instead of quantities and finds much faster productivity growth in Singapore than Young does. The World Bank’s report “The East Asian Miracle” (1993) stresses exports, learning and technology adoption alongside high investment.

The Solow link: the elasticity of y* with respect to Ā is 3/2, so even part of the story being productivity matters a lot. And the Korea calculation on the previous slide leaves productivity at 73 percent of the U.S. level, so there is still a gap to close.

Ask the policy students: if you advised a fast-growing country today, would you push investment higher, or something else?

Can aid that finances investment lift growth?

Aid was long planned to fill a “financing gap”: the investment needed for a growth target. What does the Solow model say?

What the evidence suggests

• Aid did not raise investment one for one.
• Investment did not raise growth for long.
• In the model, investment gives a transition, not lasting growth.

Why it is contested

• Aid may work where policies are good.
• That result is fragile in more data.
• Aid aimed at quick results shows modest gains.

The financing-gap approach (based on the Harrod-Domar model) assumed growth proportional to investment, with no diminishing returns. Easterly (1999, Journal of Development Economics) tests it and finds that aid did not lead to one-for-one increases in investment, and that investment and growth were not linked as the model assumed. Zambia is his famous example: had the model worked, its income would have been many times higher.

Contested: Burnside and Dollar (2000, American Economic Review) find that aid raises growth in countries with good fiscal, monetary and trade policies. Easterly, Levine and Roodman (2004, American Economic Review) show that the result disappears with more years and countries. Clemens, Radelet, Bhavnani and Bazzi (2012, Economic Journal) find modest positive effects of aid on growth once the timing of aid is accounted for. Rajan and Subramanian (2008, Review of Economics and Statistics) find little robust effect.

The Solow lesson for the policy students: financing investment can raise the level of income through the transition, but lasting growth needs rising productivity, and the steady state depends on productivity three times as much as on investment. Ask: what would you fund instead, or as well?

“Why teach a growth model that has no growth?”

A skeptic says: “The Solow model ends with zero growth per person, while rich countries have grown for two centuries. A growth model that cannot explain growth is a failure.”

Reply: it explains a great deal: how much capital countries hold, why investment raises income levels, why growth is fast during a transition, and why disasters are followed by quick recoveries. Its failure is informative too: it shows that sustained growth must come from rising productivity, which is where the next chapter looks.

Take the objection seriously: the skeptic is right that the model has no theory of long-run growth.

The reply has three parts. First, the model’s successes are in the data: the capital-output ratio (Fig. 3), catch-up within the OECD (Fig. 8), the recoveries after bombing (the policy slide), and the transition after Korea’s investment rose (Fig. 10). Second, the model is a building block: its two equations, production and capital accumulation, are inside nearly every modern macroeconomic model, including those of later chapters. Third, the failure is sharp: because of diminishing returns, no saving rate can sustain growth, so the search must turn to productivity. Solow (1956, Quarterly Journal of Economics) made this point in the original paper.

Ask: what would have to be true for capital accumulation alone to sustain growth? (No diminishing returns: an exponent of 1 on capital. Chapter 6 shows that ideas, not machines, have this property.)

What to remember from this chapter

• Capital grows when investment \(\bar{s}y\) exceeds depreciation \(\bar{d}k\), and settles at the steady state \(k^* = (\bar{s}\bar{A}/\bar{d})^{3/2}\), \(y^* = \bar{A}^{3/2}(\bar{s}/\bar{d})^{1/2}\).
• Diminishing returns rule out long-run growth from capital alone; consumption peaks at an investment rate of 1/3.
• Changes in \(\bar{s}\), \(\bar{d}\) or \(\bar{n}\) change the level of income; growth is fastest when the economy is farthest from its steady state.
• The data agree on capital-output ratios and on catch-up within the OECD; across the world, convergence is only conditional.
• Investment explains a small part of income gaps; productivity, with an elasticity of 3/2, explains most of them.

Ask each student to write one sentence answering the opening question: could South Korea’s saving and investment alone have made it rich forever? Then ask what the model still cannot explain: why productivity differs across countries and why it grows. That is Chapter 6.

Walk the three branches of the mind map in about a minute each: the Solow model (cyan), experiments (lavender), the model meets the data (green). Click the image to zoom in.

Saving and investment explain capital and fast catch-up, but not lasting growth; productivity is the key

Next: Chapter 6, Growth and Ideas

The Solow model makes capital endogenous and shows its limits. Chapter 6 makes productivity endogenous: ideas can be used by everyone at once, so they escape diminishing returns, and that is how long-run growth becomes possible.

Sources

• Figs. 1, 2, 5, 7, 11, 12 and 13, the Solow model. Model figures adapted from Figs. 5.1, 5.2, 5.5, 5.7, 5.11, 5.12 and 5.13 in Jones (2024).
• Figs. 3, 8, 9 and 10, capital, growth and investment. Penn World Table 11.0 (rgdpo, pop, csh_i; Feenstra, R., R. Inklaar and M. Timmer 2015, American Economic Review 105(10), doi:10.34894/FABVLR); capital is our perpetual-inventory estimate. OECD members: OECD, Members and partners. Regions: World Bank country groups. Inspired by Figs. 5.3 and 5.8 to 5.10 in Jones (2024).
• Population. Mankiw, N. G., D. Romer and D. Weil (1992), Quarterly Journal of Economics 107(2); Bloom, D. and J. Williamson (1998), World Bank Economic Review 12(3); Ashraf, Q., D. Weil and J. Wilde (2013), Population and Development Review 39(1); Kremer, M. (1993), Quarterly Journal of Economics 108(3).
• Disasters. Davis, D. and D. Weinstein (2002), American Economic Review 92(5); Miguel, E. and G. Roland (2011), Journal of Development Economics 96(1); Hsiang, S. and A. Jina (2014), NBER Working Paper 20352; Cavallo, E., S. Galiani, I. Noy and J. Pantano (2013), Review of Economics and Statistics 95(5).
• East Asia. Young, A. (1995), Quarterly Journal of Economics 110(3); Krugman, P. (1994), Foreign Affairs 73(6); Hsieh, C.-T. (2002), American Economic Review 92(3); World Bank (1993), The East Asian Miracle, Oxford University Press.
• Aid. Easterly, W. (1999), Journal of Development Economics 60(2); Burnside, C. and D. Dollar (2000), American Economic Review 90(4); Easterly, W., R. Levine and D. Roodman (2004), American Economic Review 94(3); Clemens, M., S. Radelet, R. Bhavnani and S. Bazzi (2012), Economic Journal 122(561); Rajan, R. and A. Subramanian (2008), Review of Economics and Statistics 90(4).
• The model. Solow, R. (1956), Quarterly Journal of Economics 70(1).
• Textbook. Jones, C. I. (2024). Macroeconomics, 6th ed. W. W. Norton.
• Book chapter, with the data, code and every figure. Data Science and Macroeconomic Models, Chapter 5.