Convergence: do poor countries catch up?

In Lesson 6 the growth rate of capital per effective worker was k̂̇/k̂ = s k̂^α−1 − (n + g + δ). The further k̂ is below its steady state, the higher this is. A poor economy has a high marginal product of capital, so its saving is spread over less capital and buys more growth. Hence the prediction that poor economies grow faster.

But whether that means all poor economies depends on whether they share a destination. Two forms of the prediction are tested:

The tests use β-convergence: regress each country’s average growth over the period on the logarithm of its initial income. A negative coefficient, −β, means convergence. Because the Solow model says the log gap closes at rate λ, the coefficient over T years is

σ-convergence is a different idea: the dispersion of log incomes, measured by its standard deviation σ_t, falls over time. The two are linked but not equal. Poor countries growing faster (β > 0) tends to reduce σ, but random shocks to each country push σ up. A country that overtakes a former leader can make β-convergence look strong while σ barely moves. This point, sometimes called Galton’s fallacy (the regression-to-the-mean mistake), is the subject of Quah (1993).

Ten invented economies (A to J) start with the same spread of incomes, from 40% to 120% of the common steady-state level. With the slider at “identical”, every one has the same saving rate and population growth, so the points fall on one downward line: absolute convergence. Slide it to the right and the economies get different saving and population growth rates, so they have different destinations. The line fades into a scatter, β shrinks, and starting income becomes a poor guide to growth, even though every single economy is still converging to its own steady state at the same speed. The fitted speed λ comes out above the 5.4% a year of the linear formula, because these economies start far from their steady states, where convergence is faster; near the steady state the formula is accurate.

This is the central lesson of the section: weak β-convergence in the data does not refute the Solow model. If countries differ in their parameters, the model predicts conditional convergence, and only a regression that holds the parameters fixed should show it.

Here is the real data: GDP per head, in 2021 international dollars, for 76 countries in 1990 and 2024, from the World Bank. Each dot is a country; the dashed line is the fitted β-convergence regression. Try the second button, which restricts the sample to the 24 rich members of the OECD in 1990 (a group that is more alike in saving rates, institutions and schooling).

What the data show, and what they do not:

• There is a negative slope in both samples. In the 76-country sample it is −0.44 percentage points of growth per unit of log income, with R² = 0.18, which is an implied speed of about 0.5% a year. Of the 20 poorest countries in 1990, 14 grew faster than the United States (1.6% a year), and the 20 poorest averaged 2.7% growth against 1.3% for the 20 richest.
• But the scatter is wide. Among the poorest 20 growth ranged from 0.2% (Cameroon) to 7.8% (China). The 16 sub-Saharan African countries in the sample grew on average 1.8% a year, about the same as the United States, while the 11 Asian developing countries grew 3.7%. Starting poor did not guarantee catching up.
• Among the rich OECD countries the line is steeper, a slope of −1.04 and a speed of about 1.3% a year (R² = 0.24), which is nearer the classic estimates of 2% a year for similar groups. Ireland is the outlier above the line. Greece, Italy and Japan are among those below.
• σ-convergence is mild. The standard deviation of log income across the 76 countries fell from 1.19 in 1990 to 1.08 in 2024: the world is a little more equal in log terms, with the slow growth of the very poorest offsetting the rise of China and India.
• India grew faster than the United States (4.4% against 1.6% a year), so the ratio of US to Indian income per head fell from 21 in 1990 to 8 in 2024: convergence in progress, but still a long way to go.

Treat this as an illustration, not a test: the 76 countries were chosen for data coverage, the regression leaves out saving, schooling and institutions, and the 1990–2024 period includes the rise of China.

Mankiw, Romer and Weil (1992) asked a sharper question: does the Solow model, with saving and population growth measured, explain the differences in income per head across countries? From the steady state of Lesson 5, the log of income per worker depends on ln s and ln(n + g + δ), with coefficients ±α/(1 − α). They found the right signs, and the right rough sizes, but with α implausibly large (about 0.6, when capital’s share is about a third).

Their fix was to broaden capital to include human capital h, with its own saving (schooling). Production becomes Y = K^αH^β(AL)^1−α−β, the steady state gains a term in schooling, and the estimates fell into line with factor shares: the augmented model explains about 80% of the variation in income per head across countries, with α and β both around 0.3. Conditional on saving, population growth and schooling, countries converge at about the rate this augmented model predicts, a slow few per cent a year at most. Barro and Sala-i-Martin’s estimates of the speed, from countries and from regions, are about 2%.

Why does capital not flow to the poor?

If two countries have the same production function y = Ak^α, the marginal product of capital is α A^1/α y^{−(1−α)/α}, so it is higher where income per head is lower:

Take the United States and India in 2024, a ratio of income per head of about 8. With α = 1/3 the marginal product of capital in India would be 8² = 64 times higher; even with α = 0.4 it is 8^1.5 ≈ 23 times higher. Investors would pile into India. They do not, or not nearly to that degree. Robert Lucas (1990) asked why, and gave the answer that the assumption of identical production functions fails: workers in rich countries have more human capital and the economies more productive technology, so the gap in the marginal product of capital is far smaller than the gap in capital per worker suggests. Other explanations are institutions and political risk (capital may be taxed or seized) and the high price of capital goods in poor countries.

Convergence clubs

Maybe the trouble is the assumption of a single world steady state. If economies have different technologies, institutions or education, they converge to different places, and countries in the same club converge to each other while the clubs themselves do not converge. Quah (1996) looked at the whole distribution of income per head across countries, not a single regression line, and found “twin peaks”: a poor group and a rich group, with few countries in between. Durlauf and Johnson (1995) found that countries split into regimes that follow different growth models.

Later research asks whether the picture has changed. Johnson and Papageorgiou (2020) review the evidence; Patel, Sandefur and Subramanian (2021) find that a pattern of unconditional convergence has appeared in recent decades, and the data above lean the same way.

Questions on convergence and on why poor countries may fail to catch up, from JNU and GATE. The next lesson, on where saving itself comes from, has no past-paper questions of its own. The endogenous growth models that follow in Lesson 10 give another answer to the convergence puzzle.