At today’s pace, closing half of the income gaps between countries would take about two centuries

190 years The half-life of the income gaps across 84 countries, at the speed of convergence measured for 2000–2019

Open with the number. It comes from the appendix notebook (Section E.6): since 2000, poorer countries have grown faster than richer ones, but only fast enough to close about 0.36 percent of the log gap each year. At that speed, half of the gap closes in about 190 years.

Ask the room: the Solow model of Chapter 5 says poor countries should catch up. Do you think they have? How fast? Collect a few guesses: most students expect either “yes, quickly” (China, Korea) or “no, never” (Sub-Saharan Africa). The data say: not at all for most of the period, and very slowly since 2000.

How the deck fits the book: Appendix E is the empirical companion of Chapter 5’s transition dynamics. It uses the Penn World Table and two simple tools, a regression and a variance, to test the model’s catch-up prediction.

Three questions for today

• I. The Solow prediction: why should poor countries grow faster, and how fast should they catch up?
• II. β-convergence in 84 countries: did poorer countries actually grow faster from 1960 to 2019?
• III. σ-convergence and lessons: have incomes across countries become more equal, and what does that mean for the model?

The three acts mirror the mind map on the title slide.

Act I builds the prediction from the Solow model: a speed of convergence and a half-life, then a regression that tests them. Act II runs that regression on 84 countries from 1960 to 2019, and asks whether the answer depends on the period. Act III looks at a second definition of convergence, the spread of incomes, and explains why the two definitions can disagree.

Two words to define now. β-convergence: poorer countries grow faster (a negative slope of growth on initial income). σ-convergence: the dispersion of incomes across countries shrinks over time.

The Solow prediction

Act I

The further below its steady state, the faster an economy grows

• In the Solow model, capital is very productive when it is scarce: each new machine adds a lot of output.
• So an economy far below its steady state invests much more than it loses to depreciation, and grows fast.
• As it approaches the steady state, diminishing returns slow growth down: the principle of transition dynamics of Chapter 5.
• Two economies with the same steady state: the poorer one grows faster and catches up.

Catch-up is a prediction about economies that share a steady state, not about all countries.

Recall Chapter 5: the growth rate of capital per person is larger the further the capital stock is below its steady-state value. In Chapter 5’s notation, g_k = d̄[(k*/k)^(2/3) − 1]: when k is far below k*, the bracket is large.

Use a picture: a ball rolling down a slope toward a valley floor (the mind map’s sketch). It moves fast on the steep part and slows near the floor.

The takeaway card is the key qualifier of the whole deck: the model predicts catch-up among economies with the same saving rate, population growth, depreciation and technology. Keep it in mind for Act III.

The model pins down how fast the gap closes

\[\frac{d\ln\widehat{k}}{dt} = \bar{s}\,\widehat{k}^{\,\alpha-1} - (\bar{n}+\bar{g}+\bar{d})\]

\[\beta = (1-\alpha)(\bar{n}+\bar{g}+\bar{d}) \qquad\qquad t_{1/2} = \frac{\ln 2}{\beta}\]

Each year, a share β of the remaining log gap to the steady state closes.

The full derivation is in Section E.2 of the appendix. Walk through it in three steps.

1. The law of motion for capital per effective worker, k̂ = K/(AL), with Cobb–Douglas production Y = K^α (AL)^(1−α), population growth n̄, technology growth ḡ, saving s̄ and depreciation d̄.
2. Take logs and draw the tangent at the steady state (a first-order approximation). The slope of the tangent is −(1−α)(n̄+ḡ+d̄). That number is the speed of convergence β.
3. The log gap then decays exponentially: ln ŷ(t) − ln ŷ* = e^(−βt) [ln ŷ(0) − ln ŷ*]. Half of the gap is gone when e^(−βt) = 1/2, so the half-life is ln 2 / β.

Notation: β is the Barro–Sala-i-Martin speed. Appendix F, which follows Mankiw, Romer and Weil, calls the speed λ, because β there is human capital’s share.

How fast does the Solow model say an income gap closes?

With a capital share of one third, population growth of 1%, technology growth of 2% and depreciation of 5% a year, how many years does it take to close half of the gap to the steady state: 5, 13, 35 or 100?

Ask for a show of hands for each of the four answers. Then reveal the chart. The answer is on the interpretation slide after the chart.

The gap to the steady state shrinks exponentially

Model: the Solow model with population growth and technology (Appendix E, Section E.2); the economy starts at three-quarters of its steady-state output per effective worker.

How to read it: years on the horizontal axis; the log gap between steady-state output per effective worker and the economy’s current level on the vertical axis (0.29 at the start: the economy is at three-quarters of its steady state). The solid blue line is the exact simulation of the nonlinear law of motion; the dashed teal line is the exponential approximation. The dotted vertical line marks the half-life from the formula ln 2 / β.

What to do live: start at α = 0.33 (title: β = 5.4% a year, half-life 13 years). Then move the slider to 0.75 and read the title again: 2.0% a year, half-life 35 years. Finally try 0.9.

Point at: the exact path lies a little below the approximation. Away from the steady state, convergence is faster than the linear formula says.

The model’s answer: half of the gap closes in about 13 years

• With \(\alpha = 1/3\): \(\beta = \tfrac23(0.08) \approx\) 5.3% a year, half-life about 13 years.
• The exact economy is even faster: starting at three-quarters of its steady state, it halves its gap in about 10.5 years.
• A capital share of 0.75 gives the 2% a year found in many studies, a half-life of 35 years.

In the Solow model with standard numbers, convergence is fast: a matter of decades, not centuries.

Answer to the prediction: 13 years.

The numbers are from the notebook’s check in Section E.2: β = 0.0533 (5.3 percent a year), half-life from the formula 13.0 years, half-life in the simulation 10.5 years.

Why might α be larger than 1/3? One reading (Mankiw, Romer and Weil 1992): “capital” should include human capital, so the share of output paid to all accumulated factors is closer to 2/3 or more. Appendix F takes up that model.

Keep the 13 years in mind: the data in Act II will give a half-life more than ten times longer.

Diminishing returns and dilution set the speed; the saving rate does not

A rise in …	speed \(\beta\)	why
capital share \(\alpha\)	↓	returns diminish more slowly
population growth \(\bar{n}\)	↑	capital is diluted faster
technology growth \(\bar{g}\)	↑	capital is diluted faster
depreciation \(\bar{d}\)	↑	capital wears out faster
saving rate \(\bar{s}\)	—	moves the steady state only

A faster speed means a shorter half-life, \(\ln 2/\beta\).

The saving rate decides where an economy converges to, not how fast it gets there.

All rows follow from β = (1 − α)(n̄ + ḡ + d̄). The saving rate is not in the formula: it changes the steady state k̂* but not the slope of the law of motion at the steady state, once we evaluate it there.

Ask: a country that doubles its saving rate. Does it catch up faster? No: it moves to a higher steady state, and it grows faster for a while because it is now further below its new steady state, but the share of the gap that closes each year is the same.

One regression turns the prediction into a test

\[g_i \equiv \frac{1}{T}\ln\frac{y_{i,T}}{y_{i,0}} = a + \lambda \ln y_{i,0} + \varepsilon_i\]

\[\lambda = -\frac{1-e^{-\beta T}}{T} \qquad\Longleftrightarrow\qquad \beta = -\frac{\ln(1+\lambda T)}{T}\]

A negative slope λ is β-convergence; the formula turns the slope into a speed per year.

This is the Barro–Sala-i-Martin equation (Section E.2, equations 5 to 7 of the appendix). Average annual growth of country i over T years is regressed on the log of its initial income.

The slope λ depends on the length of the period: over long periods most of the gap has closed, so λ is smaller in absolute value than β. Example from the appendix: with β = 0.02 and T = 59 years, λ = −0.0117. That is why we convert slopes into speeds before comparing periods of different lengths.

The conversion needs 1 + λT > 0. Under divergence (λ > 0) the speed is negative, and there is no half-life.

One line for all countries assumes that all countries share one steady state

If the regression has the same intercept \(a\) for every country, poor countries should catch up with rich ones: absolute convergence.

The intercept holds the steady state, the technology level and its growth rate. If countries differ in these, each converges to its own steady state: conditional convergence.

This slide sets up the interpretation of everything in Acts II and III.

Absolute (unconditional) convergence: one regression line for all countries, i.e. one steady state. That is what we test in this appendix, following the tutorial.

Conditional convergence: add controls for each country’s steady state (investment rate, schooling, population growth, institutions). The Solow model predicts conditional convergence, not absolute convergence. The appendix’s Your-turn task in Section E.10 adds population growth as a first control.

β-convergence in 84 countries

Act II

The test uses one balanced panel: 84 countries observed every year from 1960 to 2019

Rule	Why
Penn World Table 10.0, real GDP per person (rgdpe/pop)	the standard source for comparing real income across countries
no oil producers	their income depends on oil, not on capital accumulation
no country-years under 1 million people	very small economies are special cases
only countries with data in 1960	the same countries in every period

A balanced panel keeps the sample fixed, at the cost of 40 countries whose data start later.

The sample follows the tutorial the appendix is based on (Mendez, Beta and Sigma Convergence Across Countries: A Stata Tutorial). It has 84 countries × 60 years = 5,040 observations.

The 40 countries dropped for lack of 1960 data are mostly in Eastern Europe and the former Soviet Union, and in Africa. Many are poor, so this is a sample-selection caveat: results for a larger, unbalanced sample can differ.

Why logs? Income per person is very skewed (mean about 11,951 dollars, median about 5,197 in this panel). The log of income has a nearly symmetric distribution (skewness 0.12).

Did poor countries grow faster from 1960 to 2019?

Over six decades, did the countries that started poor grow faster than those that started rich?

Ask the room before showing the chart. Name a few countries that were poor in 1960 (South Korea, India, Ethiopia, Mozambique) and a few that were rich (the United States, Switzerland, Australia).

From 1960, the cloud of countries has no slope

Data: Penn World Table 10.0 (Feenstra, Inklaar and Timmer 2015), real GDP per person, 84-country balanced panel; prepared file data/appE/pwt100_convergence.csv.

How to read it: each dot is a country. Horizontal axis: log GDP per person in the start year. Vertical axis: average annual growth from the start year to 2019 (0.02 = 2 percent a year). The orange line is the OLS fit; the title gives its slope λ and p-value.

What to do live: hover over the dots at the top (South Korea, Singapore, Taiwan) and at the bottom left. Then move the slider from 1960 to 2000: the line turns downward.

Data caveat: growth is in log points per year; the regression uses robust (HC1) standard errors.

Over 1960–2019 there was no catch-up

• The slope is \(\lambda =\) +0.00057, with a p-value of 0.66: no relation between initial income and growth.
• Initial income explains almost none of the differences in growth: \(R^2 =\) 0.001.
• Some poor countries grew 5–6% a year (South Korea, Singapore, Taiwan); many others grew slowly or not at all.

Over the whole period, poor countries as a group did not catch up.

Numbers from the notebook’s first regression (Section E.4): coefficient 0.0006 on log initial income, standard error 0.001, p = 0.661, R-squared 0.0013, N = 84.

This is the famous “no absolute convergence” result for a broad sample of countries (Pritchett 1997 called the long-run pattern “Divergence, Big Time”).

Point out the fast growers: the East Asian economies show that catch-up is possible; they are not typical.

Does the answer depend on the period?

A single line over 60 years can hide different eras. What happens to the slope if we split the sample in 2000?

Predict the sign of the slope before 2000 and after 2000: positive, zero or negative?

Let students write down two signs. Some will guess “zero and zero” (the full-period result). The chart answers it.

Before 2000 the line slopes up; after 2000 it slopes down

Data: Penn World Table 10.0 (Feenstra, Inklaar and Timmer 2015), real GDP per person, 84-country balanced panel.

How to read it: two panels, the same axes as before. Left: growth from 1960 to the breakpoint. Right: growth from the breakpoint to 2019. The title gives both slopes and their verdict at 5 percent.

What to do live: start at 2000. Then move the breakpoint slider to 1990 and to 2005, and watch the right-hand verdict.

Hover over the right panel: Ethiopia, Romania, India and Bangladesh were among the fastest growers after 2000.

Two eras: divergence until 2000, convergence after

• 1960–2000: \(\lambda =\) +0.00437 (p = 0.007), richer countries grew faster: divergence.
• 2000–2019: \(\lambda =\) −0.00352 (p = 0.019), poorer countries grew faster: convergence.
• The post-break slope is significant only for breakpoints from 2000 to 2008: the turn is recent.

The flat full-period line averages two opposite eras.

Answer to the prediction: positive before 2000, negative after.

Numbers from Section E.5: slope +0.00437 (SE 0.00158, p = 0.007) for 1960–2000; −0.00352 (SE 0.00147, p = 0.019) for 2000–2019.

Caution about choosing breakpoints after looking at the data: we may find a “break” that is only noise. That is why the next sequences use rolling windows and a heatmap, which do not choose a single year. The appendix’s Your-turn task asks students to try 1990 and 2008.

A recent study documents the same shift in a larger sample: Patel, Sandefur and Subramanian (2021), “The new era of unconditional convergence”.

How fast is the recent catch-up?

Since 2000 poor countries have grown faster. How long would it take, at that pace, to close half of the income gap?

Remind the room of the Solow answer from Act I: about 13 years with α = 1/3, 35 years at the 2 percent benchmark.

The measured speeds are far below the 2 percent benchmark

Data: Penn World Table 10.0 (Feenstra, Inklaar and Timmer 2015). The 2 percent benchmark is the speed of conditional convergence in Barro and Sala-i-Martin (1992).

How to read it: one bar per period; the bar is the speed β in percent a year, converted from the OLS slope with β = −ln(1 + λT)/T. Blue: significant convergence; orange: significant divergence; gray: no significant relation. The dashed line is the 2 percent benchmark.

Point at: the only blue bar, 2000–2019, at 0.36; the orange bar for 1960–2000 at −0.40.

Since 2000: 0.36% a year, a half-life of about 190 years

• 2000–2019: \(\beta =\) 0.36% a year, half-life about 190 years.
• Estimating β directly by nonlinear least squares gives the same speed; its 95% interval runs from about 0.05% to 0.68% a year.
• The Solow model with \(\alpha = 1/3\) predicts more than 5% a year: absolute convergence is more than ten times slower.

Countries do not converge to one common steady state.

Numbers from Sections E.6 and E.7: OLS-route speed 0.00365 (shown as 0.36 percent), half-life 190 years; NLS speed 0.00365, SE 0.00157, p = 0.023, 95% interval [0.00052, 0.00678].

OLS and NLS agree to about nine decimal places, because the formula linking λ and β is an exact one-to-one transformation. The standard errors differ a little: NLS measures the uncertainty of β directly, the OLS route measures it for λ.

Why this matters for policy: at 0.36 percent a year, today’s income gaps would persist for generations. Catch-up is not automatic.

When did convergence start?

If we move the start year forward one year at a time, at what start year does convergence become significant?

Explain the rolling window: keep the end year at 2019, move the start year from 1960 to 2010, re-estimate each time. The shortest window is 9 years.

Convergence appears only for start years from the late 1990s

Data: Penn World Table 10.0 (Feenstra, Inklaar and Timmer 2015), real GDP per person, 84-country balanced panel.

How to read it: start year on the horizontal axis; the estimate for the window from that year to 2019 on the vertical axis; the shaded band is the 95 percent confidence interval. The buttons switch between the slope λ, the speed β by the OLS route and the speed β by NLS.

What to do live: start on the speed (OLS route). Then switch to the slope λ: it is the mirror image. Then switch to NLS: the line is the same, only the band differs slightly.

The β band comes from the λ band with the bounds swapped: a larger slope means slower convergence.

Catch-up is a phenomenon of the last two decades

• For start years up to the late 1980s, the band contains zero: no significant convergence.
• From the late 1990s on, the whole band lies above zero.
• The speed peaks for windows starting in 2008, at about 0.5% a year (half-life about 130 years); later windows are short and noisy.

Without choosing a breakpoint, the data still place the turn to catch-up around 2000.

Numbers from Section E.8’s table: start 2000: speed 0.00365; start 2005: 0.00441 (half-life 157 years); peak at start 2008: 0.00525 (half-life 132 years); start 2010: 0.00309, not significant (p = 0.135).

Discussion: what changed around 2000? Candidate explanations discussed in the literature include the commodity boom, better macroeconomic policies in many poor countries and the growth of China and India. The appendix does not test them; treat them as hypotheses.

What if we look at every possible period at once?

Out of all 1,770 periods between 1960 and 2019, in how many did poorer countries grow faster?

Explain the heatmap before revealing: one cell per start year (vertical) and end year (horizontal), 60 × 59 / 2 = 1,770 regressions. Blue cells are convergence, red cells divergence.

Most periods show divergence; the blue corner starts around 2000

Data: Penn World Table 10.0 (Feenstra, Inklaar and Timmer 2015). Colors capped at ±1% a year.

How to read it: start year on the vertical axis, end year on the horizontal axis; only the triangle above the diagonal exists (the end comes after the start). Blue: convergence (β > 0); red: divergence. Colors are capped at ±1 percent a year so that small values stay visible; hover a cell for its exact value.

What to do live: hover the top-right corner (start 2000 or later), then a cell with start 1970 and end 2019. Switch to the λ button: the same picture with the colors reversed in meaning, blue still meaning convergence.

Cells near the diagonal are one- or two-year periods: their colors jump around because short-run shocks dominate.

Only about one period in six shows convergence

• About 16% of the 1,770 periods have \(\beta > 0\).
• Periods that start before the 1990s are divergent almost everywhere.
• The OLS-route and NLS heatmaps are identical; the λ heatmap is their mirror image.

Seen from every angle, convergence is recent, slow and confined to periods that start around 2000.

Numbers from Section E.12’s printout: 1,770 periods, 0 failed NLS searches, largest gap between the two routes about 1.5e-7, share of periods with convergence 16 percent.

This closes Act II: β-convergence exists only recently, and it is slow.

σ-convergence and lessons

Act III

Have incomes across countries become more equal?

Poor countries have grown faster since 2000. Is the spread of incomes across countries in 2019 smaller or larger than in 1960?

Predict: smaller, about the same, or larger?

σ-convergence is about the spread: the variance of log GDP per person across the 84 countries. Logs make the measure independent of units: if every country became twice as rich, the variance would not change.

Collect guesses; many students will say “smaller”, because of the β-convergence they just saw.

The spread of incomes nearly doubled between 1960 and 2019

Data: Penn World Table 10.0 (Feenstra, Inklaar and Timmer 2015), variance of log real GDP per person across 84 countries.

How to read it: two bars, the variance in 1960 (gray) and in a second year (blue); the whiskers are 95 percent confidence intervals from the chi-squared distribution. The slider changes the second year.

What to do live: move the second year through 1970, 1990, 2006 and back to 2019. Every later year is above 1960.

Incomes diverged: the variance rose by 91%

• Variance of log income: 0.92 in 1960, 1.76 in 2019, an increase of 91%.
• Standard deviation: 0.96 → 1.33 log points.
• The two confidence intervals do not overlap: the increase is not sampling noise.

Over 1960–2019 there was σ-divergence: incomes across countries became more unequal.

Answer to the prediction: larger, much larger.

Numbers from Section E.9: variance 0.924 [0.697, 1.285] in 1960 and 1.764 [1.330, 2.452] in 2019; standard deviation 0.961 and 1.328.

Link to Act II: the divergence of 1960–2000 is what pushed the spread up.

When did the spread stop rising?

The variance went up over the whole period. Did it rise all the way to 2019, or did it turn?

Ask the room to sketch the path from 0.92 in 1960 to 1.76 in 2019 on a piece of paper before the reveal.

The spread peaked in 2006 and has narrowed only slightly since

Data: Penn World Table 10.0 (Feenstra, Inklaar and Timmer 2015), variance of log real GDP per person across 84 countries.

How to read it: the variance in every year from 1960 to 2019, with its 95 percent confidence band. The annotation marks the peak.

Point at: the steady rise to the mid-2000s, the peak, and the small decline after it. The band is wide: the recent narrowing is small relative to the uncertainty.

A slight narrowing, far from undoing the divergence

• The variance roughly doubled from 1960 to its peak of 1.93 in 2006.
• By 2019 it was about 8% lower than at the peak.
• It remains far above its 1960 level of 0.92.

At about 0.4% a year, undoing the divergence of 1960–2000 would take generations.

Numbers from Section E.11: peak 1.927 in 2006; 2019 is about 8 percent lower (1.764).

Comparing 2006 with 2019 in the previous chart: the confidence intervals overlap widely, so the recent narrowing is not yet statistically clear.

Do decades tell the same story?

Looking decade by decade: when did poor countries grow faster, and what happened to the spread?

This sequence combines the two definitions in one chart, one decade at a time.

Divergence in every decade until 2000, convergence after

Data: Penn World Table 10.0 (Feenstra, Inklaar and Timmer 2015), 84-country balanced panel.

How to read it: left panel, the slope λ of growth on initial income in each decade (orange: positive, divergence; blue: negative, convergence). Right panel: the variance of log income at the start of each decade. The last decade is nine years long, because the data end in 2019.

Hover the bars for the exact values.

The spread stopped rising when the slopes turned negative

• 1960s to 1990s: positive, significant slopes every decade, and a rising variance (0.92 → 1.54 at the start of the 1990s).
• 2000s and 2010s: negative slopes (p = 0.055 and 0.135) and a variance that stopped rising.
• One decade of data is a weak test: the recent slopes are not significant on their own.

β and σ moved together: divergence widened the spread; recent catch-up has only stopped it from growing.

Numbers from Section E.10’s decade table: slopes 0.00594, 0.00555, 0.00686, 0.00882 (all significant) for the four decades to 2000; −0.00379 (p = 0.055) and −0.00305 (p = 0.135) for 2000–2010 and 2010–2019. Variance at the start of each decade: 0.924, 1.082, 1.289, 1.538, 1.894, 1.826.

β-convergence is necessary for σ-convergence, but not sufficient

\[\sigma_T^2 = (1+\lambda T)^2\,\sigma_0^2 + T^2\,\mathrm{Var}(\varepsilon)\]

2000–2019	variance
in 2000	1.89
catch-up alone, \((1+\lambda T)^2\sigma_0^2\)	1.65
plus new shocks, \(T^2\mathrm{Var}(\varepsilon)\)	+ 0.11
in 2019	1.76

Catch-up pulls countries together; new country-specific shocks push them apart.

Derivation (Section E.10): write the regression in levels, ln y_T = Ta + (1 + λT) ln y_0 + Tε. OLS residuals are uncorrelated with the regressor, so the variance splits exactly into the two terms.

The first term shrinks the spread when λ < 0. The second term (wars, crises, booms, good or bad policies) spreads countries apart. The variance falls only if catch-up beats the new shocks.

Numbers from the notebook’s check: variance in 2000 1.8937; catch-up term 1.6487; shock term 0.1148; sum 1.7635, equal to the actual 2019 variance 1.7635. The identity holds exactly.

Faster growth of the poor does not by itself make the world more equal

Since poor countries have grown faster since 2000, income differences across countries must be shrinking fast.

Catch-up since 2000 cut the variance by only about 0.24 over 19 years, and new shocks added back about 0.11: the spread in 2019 was still 91% above 1960.

This is the β versus σ distinction in one sentence. The numbers are from the decomposition on the previous slide (1.89 → 1.65 from catch-up, then +0.11 from shocks → 1.76) and from Section E.9 (+91 percent since 1960).

No absolute convergence does not refute the Solow model

The data show no catch-up over 1960–2019, so the Solow model is wrong.

The model predicts that each country converges to its own steady state. Absolute convergence tests the extra assumption that all countries share one steady state, and that is what the data reject.

Countries differ in saving rates, population growth, schooling, technology and institutions, which the intercept of the regression absorbs. With controls for these (conditional convergence), studies such as Barro (1991) and Mankiw, Romer and Weil (1992) find negative slopes and speeds near 2 percent a year.

Appendix F estimates the augmented Solow model of Mankiw, Romer and Weil on this question.

Can transfers to poor regions speed up convergence?

The EU spends a large share of its budget on cohesion policy, transfers meant to help its poorest regions catch up. Does convergence analysis say they work?

What the analysis says

• Convergence is about growth rates, so a policy must raise the growth of poor regions for years, not their income once.
• In the Solow logic, transfers that raise investment move a region’s steady state up.

What it leaves out

• Whether the money is spent well: effects differ across regions.
• The counterfactual: regions that receive funds differ from those that do not.

Evidence, presented as contested. Becker, Egger and von Ehrlich (2010) use the EU’s eligibility threshold (75 percent of average EU GDP per head) as a regression discontinuity and find that the poorest regions’ growth rises with Objective 1 transfers. The same authors (2013) find that the effects depend on the regions’ absorptive capacity: human capital and the quality of government. Other studies find smaller or no effects.

Link to the appendix: the same tools (β- and σ-convergence) are standard for measuring the success of regional policy, for example across EU regions.

Catch-up requires raising each country’s own steady state

A finance ministry in a poor country asks: will we catch up with the rich world by ourselves, or does growth policy matter?

What the model predicts

• Without policy change, a country converges to its own steady state, not to the rich world’s.
• Higher investment, more schooling, lower population growth and better technology raise the steady state.

What it leaves out

• Which policies raise investment and productivity: the model takes them as given.
• Institutions and ideas, which Chapter 6 and the growth literature emphasize.

Conditional-convergence studies (Barro 1991; Mankiw, Romer and Weil 1992) find that, controlling for investment, schooling and population growth, poorer countries do grow faster. That means policies that raise those determinants raise the steady state, and catch-up follows.

Caution: these are cross-country correlations, not causal estimates of any single policy. Which policies work remains debated.

A skeptic: “A negative slope may be only regression to the mean”

A skeptic says: “Countries that were unusually poor in the start year will look like fast growers just by bouncing back. A negative slope proves nothing.”

Reply: right, and that is Galton’s fallacy (Quah 1993). It is why we also check σ: since 2000, the slope turned negative and the spread stopped rising, but it fell only about 8% from its peak.

Galton’s fallacy: in any noisy series, extreme observations tend to be followed by less extreme ones. A negative β can then appear even when the distribution is not narrowing. Quah (1993) made this point about convergence regressions.

Two further caveats to mention: the balanced panel drops 40 countries whose data start after 1960, many of them poor; and the breakpoint in 2000 was chosen after looking at the data (the rolling windows and heatmap help here). Convergence clubs, groups converging to different steady states, are another way to read the data.

What to remember

• The Solow prediction: an economy closes a share \(\beta = (1-\alpha)(\bar{n}+\bar{g}+\bar{d})\) of its gap each year; with \(\alpha = 1/3\), a half-life of about 13 years.
• No absolute convergence over 1960–2019: divergence until 2000, then slow convergence, about 0.36% a year.
• The spread of incomes nearly doubled and has barely fallen since its 2006 peak.
• β is necessary but not sufficient for σ: new shocks offset catch-up.
• Countries converge to their own steady states: growth policy is about raising them.

One point per act, plus the two lessons. If time is short, these five bullets are the lecture.

Walk the three branches in one minute each: the Solow prediction (speed and half-life, the regression), β-convergence in 84 countries (no catch-up overall, a break at 2000, slow speed, every period), σ-convergence and lessons (the spread, the peak, β is not enough, own steady states).

Poor countries catch up only with their own steady states: across 84 countries, incomes diverged until 2000 and have converged only slowly since

Next: Appendix F, the augmented Solow growth model

The thesis of the mind map. Appendix F adds human capital to the Solow model and estimates it with the Mankiw, Romer and Weil data, including their conditional-convergence regressions.

Sources: the live figures

• Speed of convergence. Model: the Solow model with population growth and technology, Appendix E, Section E.2.
• β-convergence, the break at 2000, speeds, rolling windows, heatmap. Data: Penn World Table 10.0 (Feenstra, Inklaar and Timmer 2015), real GDP per person (rgdpe/pop), 84-country balanced panel, 1960–2019. Prepared file: data/appE/pwt100_convergence.csv (data notes).
• σ-convergence, σ over time, decades. Same data: variance of log GDP per person with χ² confidence intervals.
• 2 percent benchmark. Barro and Sala-i-Martin (1992).

One line per group of live figures, in deck order. All charts are the appendix notebook’s own figures.

Sources: studies and further reading

• Tutorial. Mendez, C. Beta and Sigma Convergence Across Countries: A Stata Tutorial.
• Barro, R. J., and X. Sala-i-Martin (1992). Convergence. Journal of Political Economy 100(2).
• Barro, R. J. (1991). Economic Growth in a Cross Section of Countries. Quarterly Journal of Economics 106(2).
• Mankiw, N. G., D. Romer and D. N. Weil (1992). A Contribution to the Empirics of Economic Growth. Quarterly Journal of Economics 107(2).
• Quah, D. (1993). Galton’s Fallacy and Tests of the Convergence Hypothesis. Scandinavian Journal of Economics 95(4).
• Pritchett, L. (1997). Divergence, Big Time. Journal of Economic Perspectives 11(3).
• Patel, D., J. Sandefur and A. Subramanian (2021). The New Era of Unconditional Convergence. Journal of Development Economics 152.
• Becker, S. O., P. H. Egger and M. von Ehrlich (2010). Going NUTS. Journal of Public Economics 94; (2013) Absorptive Capacity. AEJ: Economic Policy 5(4).
• Feenstra, R. C., R. Inklaar and M. P. Timmer (2015). The Next Generation of the Penn World Table. American Economic Review 105(10).
• Book appendix. Data Science and Macroeconomic Models, Appendix E (with the Colab notebook). Textbook: Jones, C. I. (2024), Macroeconomics, 6th ed., Chapter 5.

Every study cited on the slides and in the notes, with links. Students who want more should start with Barro and Sala-i-Martin (1992) and Mankiw, Romer and Weil (1992); Appendix F follows the latter.