A country that invests twice as much is about three times as rich

×3 What doubling the investment share of GDP goes with, across 98 countries in 1985

The Solow model with a capital share of one third predicts about ×1.4. Today: why the gap, and how human capital closes it.

Open with the number. In MRW’s non-oil sample, the slope of log income on the log investment rate is about 1.6, so twice the investment share goes with about 2^1.6 ≈ 3 times the income. Ask the room: is that big or small? Then reveal what the textbook Solow model predicts: an elasticity of α/(1−α) = 0.5 with α = 1/3, so doubling investment should raise income by only 2^0.5 ≈ 1.41, about 41 percent.

The whole appendix is about that gap: Mankiw, Romer and Weil (1992), “MRW”, showed that adding human capital to the Solow model closes it.

The number comes from the notebook, Fig. 1 (elasticity 1.57 in the non-oil sample).

Three questions for today

• Does the textbook Solow model fit the differences in income across countries?
• What changes when schooling counts as a second kind of capital?
• Do poor countries catch up, and do MRW’s answers survive today’s data?

The three questions are the three acts of the mind map: the textbook model tested; the augmented model; convergence and today’s data.

Students have seen the Solow model in Chapter 5. Here we confront it with data from 98 countries, as MRW did in 1992, and then repeat the test with the Penn World Table 11.0 for 1990–2023.

Everything we estimate today is done in the Appendix F notebook with a few lines of statsmodels, and every chart there has a widget. In class we only use the charts.

The textbook Solow model, tested

Act I

Do countries that invest more have higher incomes?

Across countries, how strongly does income per worker go with the share of GDP that is invested?

If one country invests twice the share of GDP of another, how much richer do you expect it to be? Write down a number.

Collect two or three guesses. Most students say “richer”, few give a number. Push them to a number: twice as rich? 50 percent richer?

Keep the answer for the interpretation slide.

Investment and income go together in every sample

Data: Mankiw, Romer and Weil (1992), from the Penn World Table 4.0 (Summers and Heston 1988). 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 the average investment share of GDP, 1960–1985; the vertical axis is GDP per working-age person in 1985. Both axes are on a ratio scale, so the slope of the dashed line is an elasticity. Colours mark regions.

Live: click the three sample buttons, Non-oil (98 countries), Intermediate (75) and OECD (22). Watch the elasticity in the title. Hover over a few countries at both ends: Burkina Faso and Niger at the bottom left, the European and North American countries at the top right.

Point out that within the OECD the cloud is almost flat.

The slope is about 1.6, three times what the textbook model predicts

• Non-oil sample: an elasticity of 1.57, so twice the investment share goes with about 3 times the income
• Intermediate sample: 1.54; OECD: only 0.53, because rich countries invest similar shares
• The textbook Solow model with \(\alpha = 1/3\) predicts an elasticity of \(\alpha/(1-\alpha) = 0.5\)

The direction fits the model; the size does not.

Answer the prediction: a country that invests twice as much is about three times as rich in the non-oil sample. The model, with capital’s share of one third, says about 1.41 times.

Caveat: this is a correlation of one variable with income. Population growth is left out, and so is everything else that differs across countries. The next slides add population growth, as the model says we should.

The Solow model turns saving and population growth into a regression

\[\dot{k} = \bar{s}_K\,k^{\alpha} - (\bar{n}+\bar{g}+\bar{d})\,k \quad\Longrightarrow\quad y^* = \left(\frac{\bar{s}_K}{\bar{n}+\bar{g}+\bar{d}}\right)^{\frac{\alpha}{1-\alpha}}\]

\(k\) and \(y\): capital and output per effective worker, \(K/AL\) and \(Y/AL\)

\[\ln\frac{Y}{L} = a + \frac{\alpha}{1-\alpha}\ln\bar{s}_K - \frac{\alpha}{1-\alpha}\ln(\bar{n}+\bar{g}+\bar{d}) + \varepsilon\]

Two predictions: equal and opposite coefficients, each about 0.5.

The steps, briefly (the full derivation is in Section F.4 of the appendix): capital per effective worker k̂ = K/(AL) grows with investment s_K ŷ and is diluted by depreciation, population growth and technology growth. At the steady state the two forces balance, which gives k̂* and ŷ* = (k̂*)^α. Taking logs and moving from effective workers to workers gives the regression.

The constant a holds ln A(0) and ḡt; the error ε is the country-specific part of A(0): institutions, geography, climate. The key assumption for OLS is that saving and population growth are unrelated to ε. MRW, like the Solow model, treat them as given from outside. Come back to this in the Devil’s-advocate slide.

MRW set ḡ + d̄ = 0.05 in every country.

MRW’s first test: the signs are right, the sizes far too large

	Non-oil	Intermediate	OECD
\(\ln\bar{s}_K\)	1.42	1.32	0.50
\(\ln(\bar{n}+\bar{g}+\bar{d})\)	−1.99	−2.02	−0.74
Adjusted \(R^2\)	0.59	0.59	0.01
Countries	98	75	22
Test: coefficients sum to 0, p-value	0.36	0.26	0.81

Two variables explain 59 percent of income differences, with coefficients three times too large.

These are MRW’s Table I, reproduced in the notebook from their own data. (MRW print −1.97 for the non-oil coefficient on ln(n+g+d); their rounded data give −1.99.)

Read the table in three steps: signs (right in all samples); fit (59 percent in the two large samples, nothing in the OECD); the restriction (not rejected anywhere, so we can impose it).

The OECD column is a lesson about variation: 22 rich countries differ little in saving and population growth, so the regression has little to work with.

The estimates imply a capital share of 0.60, not one third

0.60 The capital share \(\alpha\) implied by the restricted regression, non-oil sample (intermediate 0.59)

National accounts pay capital about one third of income (Chapter 4). Saving and population growth matter far more than that share allows.

With the restriction imposed, the coefficient c on ln s_K − ln(n+g+d) equals α/(1−α), so α = c/(1+c). The non-oil estimate c = 1.49 gives α = 0.60; the intermediate sample 0.59; the OECD 0.36, but imprecisely.

This is MRW’s puzzle. Ask: what could make saving matter more than the capital share says? Let students guess: externalities, missing variables, endogeneity. MRW’s answer is a missing kind of capital.

A coefficient that is too large is a clue, not a refutation

The coefficients are far too large, so the Solow model is wrong.

The signs and the fit are right. Too-large effects are what we expect if a second kind of capital, correlated with saving, is missing from the regression.

Omitted-variable bias: if countries that invest in machines also invest in schooling, a regression without schooling credits the machines with the effect of the schools. The fix is not to discard the model but to add the missing variable, which is Act II.

Good question to ask: which way would the bias go if schooling were negatively related to saving? (Downward.)

Adding human capital

Act II

Schooling is a second kind of capital, built by giving up consumption

\[Y = K^{\alpha}H^{\beta}(AL)^{1-\alpha-\beta}, \qquad \alpha + \beta < 1\]

\[\dot{k} = \bar{s}_K\,y - (\bar{n}+\bar{g}+\bar{d})\,k \qquad \dot{h} = \bar{s}_H\,y - (\bar{n}+\bar{g}+\bar{d})\,h \qquad \text{(per effective worker)}\]

\[\ln\frac{Y}{L} = a + \frac{\alpha}{1-\alpha-\beta}\ln\bar{s}_K + \frac{\beta}{1-\alpha-\beta}\ln\bar{s}_H - \frac{\alpha+\beta}{1-\alpha-\beta}\ln(\bar{n}+\bar{g}+\bar{d}) + \varepsilon\]

Human capital H is the stock of skills of the workforce. Like machines, it is built by investing a share s_H of output (schooling), and it depreciates and is diluted by population growth.

α + β < 1 means diminishing returns to all capital together, so there is still a steady state.

The derivation (Section F.6): at the steady state ĥ/k̂ = s_H/s_K; substitute into the condition for k̂ and solve. The exponents of the steady-state income equation are α/(1−α−β), β/(1−α−β) and −(α+β)/(1−α−β). The three coefficients add up to zero: the restriction we test.

We measure s_H with MRW’s SCHOOL: the share of the working-age population in secondary school. Ask the room what it misses: primary and university education, quality, learning on the job, health.

With human capital, saving and population growth matter more

Elasticity of steady-state income to	Textbook	Augmented, \(\alpha=\beta=1/3\)
the investment rate \(\bar{s}_K\)	0.5	1
the investment rate in schooling \(\bar{s}_H\)	—	1
\(\bar{n}+\bar{g}+\bar{d}\)	−0.5	−2

More saving raises income, which raises schooling, which raises income again.

The mechanism: with a fixed share of output going to schooling, a richer economy also invests more in skills. So a rise in s_K raises income twice: directly through machines, and indirectly through the extra human capital it pays for.

Population growth now dilutes both kinds of capital, so its effect is four times larger.

A second point: if s_H is correlated with s_K and n, the textbook regression credits their coefficients with part of the schooling effect. That is the omitted-variable story of the misconception slide.

How much steeper does the line get when schooling counts?

The textbook line has slope 0.5. What happens to the slope when the human capital share \(\beta\) rises from 0 to 1/3?

Doubling the investment rate from 20 to 40 percent raises steady-state income by 41 percent in the textbook model. Guess the rise in the augmented model.

Ask for guesses: the same? a bit more? Collect them before showing the chart.

Human capital makes income more sensitive to investment

Model: the textbook and augmented Solow models, with \(\bar{s}_H = 0.10\), \(\bar{n} = 0.01\), \(\bar{g} = 0.02\), \(\bar{d} = 0.03\). Adapted from the Solow model of Chapter 5 in Jones (2024), Macroeconomics, 6th ed.

How to read it: both axes are on a ratio scale. The horizontal axis is the investment rate s_K; the vertical axis is steady-state income relative to its value at s_K = 0.20 (the white dot). On a ratio scale each line is straight and its slope is the elasticity.

Live: move the β slider from 0.33 down to 0. At β = 0 the two lines coincide. Then move it to 0.4: the augmented line gets steeper still. The title gives both elasticities.

Doubling investment doubles income, instead of raising it 41 percent

• With \(\alpha = \beta = 0.33\) the elasticity is 0.97 instead of 0.49: from 20 to 40 percent, income about doubles
• The larger \(\beta\), the steeper the line: \(1-\alpha-\beta\), the strength of diminishing returns, shrinks
• \(\bar{s}_H\), \(\bar{n}\), \(\bar{g}\) and \(\bar{d}\) move the level of income, not the slope

Human capital makes saving matter twice as much.

Answer the prediction: income roughly doubles, against 41 percent in the textbook model.

Connect to the data: MRW’s textbook regression gave an elasticity of about 1.4. With α = 1/3, the augmented model gets there with β a little above 0.4, which is a check students can do on the slider in the notebook.

MRW’s second test: shares near one third, and 78 percent of income explained

	Non-oil	Intermediate	OECD
\(\ln\bar{s}_K\)	0.70	0.70	0.28
\(\ln(\text{SCHOOL})\)	0.65	0.73	0.77
\(\ln(\bar{n}+\bar{g}+\bar{d})\)	−1.75	−1.50	−1.08
Adjusted \(R^2\)	0.78	0.77	0.24
Implied \(\alpha\), \(\beta\) (restricted)	0.31, 0.27	0.29, 0.30	0.14, 0.37

Counting schooling halves the effect of machines and brings both shares near one third.

MRW’s Table II, reproduced in the notebook. Three things: 1. Schooling enters with a large positive coefficient in every sample. 2. The coefficient on investment falls from 1.42 to 0.70 in the non-oil sample: part of what looked like the effect of machines was the effect of schooling. 3. The fit jumps from 0.59 to 0.78.

The restriction (the three coefficients add to zero) is not rejected: p-values 0.39, 0.88 and 0.97. With it imposed, α = c_K/(1+c_K+c_H) and β = c_H/(1+c_K+c_H).

The OECD estimates are imprecise: 22 similar countries cannot pin down two shares.

How much better does the augmented model predict incomes?

If we predict each country’s income from its saving, schooling and population growth, how close do we get?

Ask: which countries do you expect the models to miss? (Countries with unusual institutions or histories, which sit in A(0).)

Predicted and actual income, textbook and augmented models

Data: Mankiw, Romer and Weil (1992); unrestricted regressions of their Tables I and II. 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 the log income the model predicts, the vertical axis the actual log income in 1985. Dots on the dotted 45-degree line are predicted exactly. The panel titles give the adjusted R².

Live: switch between the samples. Hover over the dots far from the line in the augmented panel and ask what is special about them.

The fit rises from 59 to 78 percent, most among the poorest countries

• Non-oil: adjusted \(R^2\) 0.59 → 0.78; the dots hug the line in the augmented panel
• The gain is largest at the bottom, where schooling is lowest
• The countries still far from the line differ in what the model leaves out: \(A(0)\)

Saving, schooling and population growth explain about four fifths of income differences.

The phrase “almost a complete explanation” is MRW’s spirit, but the remaining fifth matters: that is technology, institutions and history. Appendix D’s development accounting comes back to it with a different method.

After a rise in investment, which economy gains more, and how fast?

An economy on its balanced growth path raises \(\bar{s}_K\) from 0.20 to 0.25 in year 10. What happens to its income in the textbook and augmented models?

Guess the long-run gain in each model, and which one gets there sooner.

Hint for the room: use the elasticities from the table (0.5 and 1) and the fact that a rise from 0.20 to 0.25 is a 25 percent rise, about 22 log points.

The move to a new steady state

Model: the textbook and augmented Solow models from the baseline \(\bar{s}_K = 0.20\), \(\bar{s}_H = 0.10\), \(\alpha = 1/3\). Adapted from Fig. 5.5 in Jones (2024), Macroeconomics, 6th ed.

How to read it: years on the horizontal axis; on the vertical axis, income per worker above the path it would have followed without the change, in log points (100 × the log difference, close to percent for small gaps). Dotted lines mark the long-run gains.

Live: move the slider for the size of the rise. In the notebook the widget also switches the rise to the investment rate in schooling, s_H: then the textbook line stays flat, because the textbook model has no schooling.

Twice the long-run gain, on a longer road

• Long-run gain from \(\bar{s}_K\) 0.20 → 0.25: 11 log points in the textbook model, 22 in the augmented model
• The augmented economy takes much longer: human capital keeps rising as income rises
• A rise in \(\bar{s}_H\) does nothing in the textbook model, by construction

With human capital, policies that raise investment pay more, but later.

Answer the prediction: 11 and 22 log points; the textbook economy is close to its new path after a few decades, the augmented one is still climbing after a century.

This slow transition is the same fact as the slow convergence of Act III: with α + β = 2/3, diminishing returns bite weakly.

Does investing in schools pay off like investing in machines?

A finance ministry can fund more schooling or more public investment. Which raises long-run income more?

What the model predicts

• With \(\alpha = \beta\), the two elasticities are equal
• A rise of 0.05 is 25% of \(\bar{s}_K = 0.20\) but 50% of \(\bar{s}_H = 0.10\), so the same spending share does more in schooling
• Either gain arrives slowly, over decades

What it leaves out

• School quality: years in school are not skills learned
• Costs and lags: a pupil enters the labour force years later
• Whether more schooling causes growth, or growth pays for more schooling

The model gives a clean answer only because it assumes the same share and depreciation for both kinds of capital. The policy debate is about the right-hand column.

Evidence: Hanushek and Woessmann (2012) find that test scores (cognitive skills) are much more strongly related to growth than years of schooling; the causal interpretation is debated. Bring this up as a contested question, not a settled fact.

Ask the MPP students: which column would a ministry of finance care most about, and how would they measure school quality?

Convergence and today’s data

Act III

Human capital halves the speed of convergence

\[\lambda = (\bar{n}+\bar{g}+\bar{d})(1-\alpha-\beta)\]

\[\ln y(t) - \ln y^* = e^{-\lambda t}\left[\ln y(0) - \ln y^*\right], \qquad t_{1/2} = \frac{\ln 2}{\lambda}\]

	Textbook (\(\beta = 0\))	Augmented (\(\alpha=\beta=1/3\))
Speed \(\lambda\), with \(\bar{n}+\bar{g}+\bar{d} = 0.06\)	4% a year	2% a year
Half-life	about 17 years	about 35 years

The speed of convergence is the share of the remaining log gap to the steady state that closes each year. The derivation (Section F.10) linearizes the two laws of motion around the steady state: the 2×2 system has two speeds, D and D(1−α−β) with D = n+g+d. The fast one closes gaps in the ratio of the two capitals; the slow one governs income.

Note on notation: Appendix E writes this speed as β, following Barro and Sala-i-Martin. Here β is human capital’s share, as in MRW, so the speed is λ.

How long does it take to close half the gap?

Two economies start 20 percent below their steady states: one without human capital, one with \(\beta = 1/3\). When has each closed half of its gap?

Guess the two half-lives in years.

Students who remember the previous slide will say 17 and 35. Ask them whether the formula should hold exactly in a simulation, and why not.

The gap to the steady state closes at 4 percent a year without schooling, 2 percent with it

Model: the textbook and augmented Solow models with \(\bar{g} + \bar{d} = 0.05\), starting at 80 percent of the steady state. Adapted from the transition dynamics of Chapter 5 in Jones (2024), Macroeconomics, 6th ed.

How to read it: the vertical axis is the log gap between steady-state income and current income; it starts at 0.22 (80 percent of the steady state) and falls to zero. The dotted vertical lines mark the half-lives of the formula.

Live: move the β slider. At 0 the two lines coincide; at 0.4 the augmented economy is very slow.

Half-lives of 17 and 34 years, as the formula says near the steady state

• With \(\alpha = \beta = 0.33\): 17 years without human capital, 34 with it; the simulated gaps halve close to the dotted lines
• The formula is a local approximation: from a quarter of the steady state the simulated half-lives are 6 and 27 years
• Faster population growth speeds up convergence: it strengthens the pull back to the steady state

The data say about 2 percent a year: the augmented model’s number.

Answer the prediction: 17 and 34 years (the readout uses 0.33 rather than exactly 1/3, so 34 rather than 35).

The second bullet is a teaching point about approximations: far below the steady state the returns to capital are very high, and the economy grows faster than the straight line says. The speed λ describes the last part of the journey. The notebook’s check cell prints all four numbers.

Did poor countries grow faster from 1960 to 1985?

If the Solow model is right, should the poorest countries of 1960 have grown fastest?

Draw the cloud you expect: growth 1960–1985 against income in 1960. Sloping up, down, or flat?

Many students say “down”: diminishing returns. Ask the follow-up: does the model say that all countries share one steady state?

Growth against initial income, without and with controls

Data: Mankiw, Romer and Weil (1992). Chart inspired by Figure I of Mankiw, Romer and Weil (1992) and Fig. 5.9 in Jones (2024), Macroeconomics, 6th ed.

How to read it: panel A is log growth 1960–1985 against log income in 1960, no controls. Panels B and C plot the same relation after removing the part of both variables that the controls explain: saving and population growth (B), and schooling too (C). By the Frisch–Waugh theorem, the slope in each panel is the coefficient on initial income in the multiple regression. Each variable’s mean is added back, so the axes keep their units.

Live: switch between the samples. In the OECD all three panels slope down.

No unconditional convergence, but clear conditional convergence

• Panel A, intermediate sample: slope 0.00: the poorest countries of 1960 did not grow faster
• Panels B and C: −0.23 and −0.37: among countries with the same saving, schooling and population growth, the poorer grew faster
• In the OECD all three panels slope down: similar countries, similar steady states

Countries converge, but each to its own steady state.

Answer the prediction: flat in panel A. Poor countries in 1960 were mostly poor because their steady states were low, so they had no reason to grow fast. Once we hold the steady state constant, the model’s prediction appears.

Appendix E studies β- and σ-convergence on 84 countries up to 2019; this slide is the conditional version.

MRW’s growth regressions imply 1.4 to 2.1 percent a year

Controls	Non-oil	Intermediate	OECD
None: slope on \(\ln y_{1960}\)	0.094	−0.004	−0.341
Saving, population: speed \(\lambda\)	0.61%	1.03%	1.72%
+ schooling: speed \(\lambda\)	1.36%	1.82%	2.03%
Restricted: speed \(\lambda\)	1.41%	1.86%	2.06%

\[\lambda = -\frac{\ln(1+b)}{T}, \qquad T = 25\]

Close to the augmented model’s 2 percent, far from the textbook’s 4.

MRW’s Tables III to VI, reproduced. The coefficient b on log initial income equals −(1−e^{−λT}), which the formula turns into a speed.

Without controls, the non-oil slope is positive: the poorer countries grew a little slower, so the implied “speed” is negative, meaningless.

The restricted regressions also imply α of 0.38–0.48 and β of 0.23, a bit higher α than the levels regressions; MRW read the two sets of results as broadly consistent.

Does a poor country with lower investment rates catch up?

A poor country starts at a tenth of a rich country’s income, with the same technology. What decides whether it catches up?

If it invests half the rich country’s rates, in machines and in schools, does it close the gap? Partly? Not at all?

Students tend to say “it grows faster, so it catches up”. Push them on where it is heading.

Two countries, one technology, different steady states

Model: the augmented Solow model with \(\alpha = \beta = 1/3\), \(\bar{n} = 0.01\), \(\bar{g} = 0.02\), \(\bar{d} = 0.03\) in both countries. Adapted from the transition dynamics of Chapter 5 in Jones (2024), Macroeconomics, 6th ed.

How to read it: income per worker, relative to the rich country in year 0, on a ratio scale. The rich country sits on its balanced growth path, growing at 2 percent. The poor country starts at 0.1. The dotted orange line is the poor country’s own balanced growth path.

Live: move the slider for the poor country’s investment rates (as a share of the rich country’s). At 1.0 the paths converge; at 0.5 the poor country converges to a path a quarter as high; at 0.3 or below it grows slower than the rich country.

Catch-up depends on the poor country’s own steady state

• Same investment rates: growth of 6.9% a year over the first 25 years, against 2.0%, and the gap closes
• Half the rates: its steady state is \(0.5^2\), one quarter of the rich country’s; it grows fast at first, then settles below
• Below about 0.32 of the rich rates, its path lies below where it starts: it grows slower and falls behind

Poverty persists when the steady state is low, not because convergence fails.

Answer the prediction: partly. Its steady state is the share squared times the rich one, because the elasticities to s_K and s_H add up to 2 with α = β = 1/3.

The threshold: with a starting income of 0.1, the poor country grows at the rich rate when its steady state equals its starting income, share² = 0.1, so share ≈ 0.32.

Link to panel A of the previous chart: countries that fell behind between 1960 and 1985 are what this model predicts for low steady states.

Poor countries not catching up does not refute the Solow model

Poor countries did not grow faster than rich ones, so the Solow model’s convergence prediction failed.

The model predicts convergence to each country’s own steady state. Holding saving, schooling and population growth constant, poor countries did grow faster: about 2 percent of the gap closed each year.

This is the most common mistake about convergence. Unconditional convergence needs the same steady state everywhere. In the OECD, where steady states are similar, we do see it.

Has the data itself changed since 1992?

MRW used an early Penn World Table. Do today’s numbers tell the same story for the same countries in the same years?

The appendix rebuilds MRW’s variables from the Penn World Table 11.0 and the World Bank’s WDI for 1960–1985 and 1990–2023. Before re-running the regressions, compare the data.

MRW’s data and PWT 11.0 for the same countries, 1960–1985

Data: Mankiw, Romer and Weil (1992); Penn World Table 11.0 (Feenstra, Inklaar and Timmer 2015).

How to read it: each dot is one of 95 countries. Left: the average investment share in MRW (horizontal) and in PWT 11.0 (vertical). Right: income in 1985, MRW (per working-age person, 1985 dollars) against PWT 11.0 (per person, 2021 dollars). Both on ratio scales.

Hover over the outliers in the left panel.

Incomes agree; investment rates do not

• Income in 1985: a correlation of logs of 0.93: the ranking of countries has hardly changed
• Investment rates: only 0.54. PWT 11.0 measures investment at current PPP prices; MRW used constant prices
• Machines are relatively expensive in poor countries, so by today’s measure many invested a larger share

A new vintage of the same data can change the right-hand side of a regression.

A data lesson for policy students: the same concept, “the investment rate”, measured two ways, gives quite different numbers for poor countries. Results that depend on it can move with the data vintage.

Other differences in our rebuilt data: output and population growth are per person rather than per working-age person (a choice made for this appendix), and the WDI’s secondary enrollment is missing before 1986 for most OECD members, so the 1960–1985 OECD regression with schooling is skipped.

MRW’s estimates, then and now

Data: Penn World Table 11.0 (Feenstra, Inklaar and Timmer 2015); World Bank WDI (SE.SEC.ENRR, SP.POP.1564.TO). Benchmark: MRW (1992), non-oil sample.

How to read it: bars are implied parameters by sample (MRW’s non-oil, intermediate and OECD samples matched by country, plus a full sample of 123 economies in 1990–2023), for two measures of human capital: MRW’s schooling recipe (teal) and PWT’s hc index (orange). Panels: 1960–1985 and 1990–2023. The dotted line is MRW’s own non-oil estimate.

Live: click the buttons for α, β, the speed of convergence and the fit.

Saving still matters too much, β is stable, and the hc index misleads

• Textbook model, 1990–2023: implied \(\alpha\) about 0.7 outside the OECD, even above MRW’s 0.6
• MRW’s schooling measure: \(\beta\) 0.25–0.27 in 1990–2023, next to MRW’s 0.28; the fit 72–84%
• Convergence: \(\lambda\) between 1.2 and 2.2% a year in every window and sample outside the OECD
• PWT’s hc index implies \(\beta\) of 1.2–1.9, impossible in the model: its log varies little across countries

The results survive thirty years of data; the measurement of human capital decides the shares.

Within the OECD in 1990–2023, nothing fits (adjusted R² near zero or negative): rich countries have become too similar in saving, schooling and population growth.

The hc index puzzle: its log has a standard deviation of about 0.3 across countries, against 1.2 for log income, so the regression needs a coefficient of about 2.5 on it. That is the measurement problem that Klenow and Rodríguez-Clare (1997) raise in the opposite direction: MRW’s SCHOOL varies much more across countries than years of schooling do.

To lift a poor country’s path, change its steady state

A development bank asks: will poor countries catch up on their own if we just wait?

What the model predicts

• Only with similar steady states: convergence is conditional
• Raising \(\bar{s}_K\) or \(\bar{s}_H\), or lowering \(\bar{n}\), raises the path
• About 2% of the remaining gap closes each year: half-lives of decades

What it leaves out

• \(A(0)\): technology, institutions, geography, the unexplained fifth
• Why saving and schooling differ in the first place
• Ideas and long-run growth (Chapter 6)

The model’s policy message: waiting is not enough when the steady state is low; the levers are investment, schooling and population growth, and their effects take decades.

The right column is where the debate is. Appendix D’s development accounting finds that TFP carries most of today’s income gaps; Klenow and Rodríguez-Clare (1997) make the same point against MRW. Present the size of the TFP share as contested.

Saving and schooling may be high because countries are rich

A skeptic says: “Rich countries save and school more because they are rich, and good institutions raise both income and saving. The regressions measure correlation, not the model’s causal effect.”

Reply: A fair point, and the main caveat. MRW assume saving and population growth are unrelated to \(A(0)\). Panel-data studies that let each country have its own \(A(0)\) (Islam 1995) find faster convergence and a smaller role for capital. The model’s predictions, signs and orders of magnitude still hold.

Use this to build ethos: the key identifying assumption is strong, and we said so on the regression slide.

Two directions of bias: reverse causality (income raises saving and schooling) and omitted variables (institutions raise both). Both push the coefficients up. Bernanke and Gürkaynak (2001) re-run MRW on later data: the levels results hold, but long-run growth rates are correlated with saving, which the exogenous-growth Solow model cannot explain.

What to remember from this appendix

• The textbook Solow model gets the signs right, but the effects of saving and population growth imply a capital share of about 0.6
• Counting schooling as capital doubles the effect of saving; MRW find \(\alpha\) and \(\beta\) near one third and explain 78% of income differences
• Convergence is conditional: about 2% of the gap to each country’s own steady state closes each year, half the textbook speed
• With PWT 11.0 for 1990–2023 the results survive, but the measure of human capital decides the shares

One takeaway per act, plus the data lesson. If time is short, the second bullet is the one to keep.

Walk the three branches in one minute each: the textbook model tested (signs right, sizes wrong); adding human capital (elasticity doubles, shares near one third, slower transitions); convergence and today’s data (2 percent a year, conditional, results survive, measurement matters).

Saving, schooling and population growth explain most of why countries are rich, and poor countries converge only to their own steady states

Next: Chapter 6, Growth and Ideas

The thesis of the appendix. Chapter 6 asks what the Solow model takes as given: where technology and its growth come from.

Sources

• Fig. 1, 3 and 6. Data: Mankiw, Romer and Weil (1992), Appendix, from the Penn World Table 4.0 (Summers and Heston 1988); prepared file data/appF/mrw1992.csv. Charts inspired by Fig. 5.3 and Fig. 5.9 in Jones (2024) and Figure I of MRW (1992).
• Fig. 2, 4, 5 and 7. Model: the textbook and augmented Solow models of Appendix F. Adapted from the Solow model and Fig. 5.5 of Chapter 5 in Jones (2024).
• Fig. 8 and 9. Data: Penn World Table 11.0 (Feenstra, Inklaar and Timmer 2015); World Bank, World Development Indicators, SE.SEC.ENRR and SP.POP.1564.TO; prepared file data/appF/pwt_mrw.csv.
• Studies. Mankiw, Romer and Weil (1992), QJE 107(2); Islam (1995), QJE 110(4); Klenow and Rodríguez-Clare (1997), NBER Macroeconomics Annual 12; Bernanke and Gürkaynak (2001), NBER Macroeconomics Annual 16; Hanushek and Woessmann (2012), Journal of Economic Growth 17(4).
• Textbook. Jones, C. I. (2024). Macroeconomics, 6th ed. W. W. Norton.
• Book appendix. Data Science and Macroeconomic Models, Appendix F