The steady state generates the model's most famous prediction. Two economies with the same s, n, g and δ converge on the same k*, and the one that starts further below it grows faster on the way, because its capital is scarcer and its marginal product higher. Poor should therefore catch rich.
That is absolute convergence, and in a broad cross-section of countries it is false. Run growth over three decades against the log of initial output per worker for the illustrative 20-economy panel in worked example 3 and the slope is +0.024 points per log point with a standard error of 0.269, a t statistic of 0.09 and an R-squared of 0.000454. The data will not even sign the relationship, let alone size it.
What is conditional convergence?
Conditional convergence is the same prediction with the qualifier the model always contained: economies converge to their own steady states, not to a common one. Two countries with different saving rates, different population growth and different institutional quality have different k*, so a poor country can be near its own steady state and grow slowly for ever without contradicting anything.
Testing it means either adding the determinants of k* as controls or restricting the sample to countries whose fundamentals are similar. Restrict the panel to the twelve members of the convergence club and the slope becomes −1.570 points per log point with a standard error of 0.129, a t statistic of −12.14 and an R-squared of 0.936. The same panel, the same regression, an opposite verdict.
Club convergence is what the two results look like side by side: groups converge internally and diverge between groups. In the panel, the eight stalled economies have a mean initial log income of 8.99 and mean growth of 1.21%, while the six catch-up economies start higher at 9.93 and grow at 3.12%.
Poor and slow, next to less-poor and fast, and the full-sample slope is the average of two unrelated processes.
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How fast do economies converge?
Converting a slope into a speed is the step that makes the regression useful. With growth measured over T years, the coefficient on initial log income is b = −(1 − e^(−βT))/T, so
β = − ln(1 + b·T) / TThe club's slope of −1.570 points, or −0.01570 in fractional units over T = 30, gives β = 2.12% a year and a half-life of 32.66 years. That figure is close to the roughly 2% a year that Robert Barro and Xavier Sala-i-Martin found across many samples and specifications, and its stability is the most reproducible number in the empirical growth literature.
The awkward part comes next, and it is the reason the following section exists. The Solow model with a conventional capital share predicts a much faster speed than 2%. With α = 1/3 and n + g + δ = 8%, β = (1 − α)(n + g + δ) is 5.33% a year and the half-life 13.00 years, about two and a half times too quick.
Mankiw, Romer and Weil resolved it in 1992 by widening capital to include human capital, which pushes the effective exponent to about 2/3 and drops the predicted speed to 2.67% a year with a half-life of 25.99 years. The model was not wrong about the mechanism. It was wrong about what counts as capital.
What is the middle-income slowdown?
The middle-income slowdown is a conditional-convergence statement wearing different clothes. A country that has exhausted the easy gains, moving workers from farms to factories and raising capital per worker from very low levels, arrives near a steady state defined by its own institutions, human capital and saving rate, and growth then falls to whatever its residual can support. The slowdown is the capital term decelerating exactly as the model says it must, with nothing arriving to replace it, and calling it a cliff misdescribes a process that takes twenty years.