Why are some countries rich and others poor? Why do some grow faster than others? Modern growth theory has roughly seventy years of formal work on this. The skeleton is the Solow-Swan model (1956). The flesh is what came after — endogenous growth, institutions, human capital, the long arc back to factor accumulation. This module is the formal core.
The Solow-Swan model
Robert Solow and Trevor Swan independently in 1956 set out the workhorse model. Output is produced from capital and labour via a constant-returns-to-scale production function:
Solow-Swan production function
Y = A × K^α × L^(1−α) • Y = aggregate output (real GDP) • A = total factor productivity (the 'technology' or 'efficiency' term) • K = capital stock • L = labour input • α = capital's share of national income (typically 0.3-0.4) In per-worker terms: y = A × k^α • y = output per worker • k = capital per worker Diminishing returns to capital: each new unit of capital per worker raises output by less than the previous one. This is the key mathematical feature that drives the steady-state result.
The steady state
Capital accumulation per worker is governed by the saving-investment identity:
Capital accumulation per worker
Δk = s × y − (n + δ) × k • s = savings/investment rate (share of output saved) • y = output per worker (= A × k^α in Solow) • n = labour-force growth rate • δ = depreciation rate of capital The steady state is where Δk = 0 — capital per worker neither rises nor falls. At steady state: s × y* = (n + δ) × k* Substituting y* = A × (k*)^α and rearranging: k* = (s × A / (n + δ))^(1/(1−α)) y* = A × (s × A / (n + δ))^(α/(1−α)) In the steady state, output per worker depends positively on the savings rate and TFP, negatively on the labour-force growth rate and depreciation.
Convergence
The Solow model has a famous prediction: poor countries with the same fundamentals as rich countries should grow faster, eventually converging. Why? Because diminishing returns to capital mean that the marginal product of capital is HIGHER in poor (low-k) economies than in rich (high-k) ones. Capital flows there; growth is faster; the gap closes.
- Absolute convergence — all countries converge to the same steady state. Empirically rejected — no global convergence across all economies
- Conditional convergence — countries converge to their OWN steady state, which depends on country-specific savings, technology, population growth, institutions. Empirically supported. Mankiw-Romer-Weil (1992) is the classic estimation
- Convergence speed — empirically about 2% per year — meaning half the gap to steady state closes in ~35 years. Slow
Growth accounting and the Solow residual
Taking the production function Y = A × K^α × L^(1−α) and differentiating with respect to time, expressed as growth rates:
Growth accounting identity
g_Y = g_A + α × g_K + (1 − α) × g_L GDP growth = TFP growth + (capital share × capital growth) + (labour share × labour growth) Rearranged to isolate TFP growth (the 'Solow residual'): g_A = g_Y − α × g_K − (1 − α) × g_L The Solow residual is what's left after we account for the contributions of capital accumulation and labour growth. It's the 'measure of our ignorance' — empirically, TFP growth accounts for 50%+ of growth differences across countries and over time. Capital and labour are not enough.
African growth accounting
Decomposing African growth (Easterly and Levine 2001; Berthelemy and Soderling 2001; updated Bosworth-Collins 2003):
- Sub-Saharan Africa 1960-1990: GDP growth ~3.4%/year. TFP growth: roughly ZERO. Growth came almost entirely from factor accumulation, primarily labour-force growth, not productivity. This is the 'African growth tragedy' literature
- Sub-Saharan Africa 1995-2014: GDP growth ~5.0%/year. TFP growth: ~1.0%/year. Improvement primarily from better policy, post-conflict reconstruction, commodity tailwinds
- Sub-Saharan Africa 2015-2024: GDP growth ~3.2%/year. TFP growth: ~0.3%/year. Growth has decelerated; TFP has too. The 'commodities super-cycle' tailwind has receded; structural-reform headwinds returned
The TFP-deficit problem
African economies CAN grow without sustained TFP improvement — by accumulating more capital and labour. But the Solow model says such growth has diminishing returns: each new shilling of capital adds less output than the last, and labour-force growth alone can't sustain rising per-capita income. Sustained per-capita-income growth REQUIRES TFP growth. This is why the 'where does growth come from' debate matters: if African growth is mostly factor accumulation, it will slow. If it's TFP growth from technology adoption, institutional improvement, and structural transformation, it can be sustained. The honest empirical record is that 1995-2014 had a TFP component but it was modest, and post-2014 has been thin.
Endogenous growth — Romer, Lucas, and beyond
Paul Romer (1986, 1990) and Robert Lucas (1988) developed models where TFP growth is endogenous — driven by economic decisions about R&D, education, and learning — rather than exogenous as in Solow.
- Romer 1990: R&D produces non-rival knowledge that raises TFP. Spillovers mean private R&D under-provides relative to social optimum; public support for R&D is welfare-improving
- Lucas 1988: human capital accumulation. Education is itself an investment good; education externalities mean individual choices under-provide the socially optimal amount of education
- Acemoglu-Robinson and Aghion-Howitt: Schumpeterian creative destruction. New ideas displace old ones; the rate of innovation depends on incentives (patents, profits, competition)
What this means for African growth policy
- Factor accumulation matters. Savings rates, physical-capital investment, and labour-force growth and education are necessary conditions. Africa's investment rate (~20% of GDP) is below the level seen in fast-growing economies (East Asia 35%+)
- TFP matters more. Sustainable per-capita-income growth requires productivity improvement. Productivity improvement requires technology adoption, structural reform, and human-capital deepening — not just more savings
- Education quality matters. The Lucas 'human capital' channel works only if education actually produces useful skills. Years-of-schooling has gone up massively in Africa; learning outcomes have not kept pace (Pritchett 2013 'The Rebirth of Education')
- Spillovers and externalities provide a legitimate role for industrial policy. Pure Solow has no role for government in growth; endogenous growth theory has a wide one. This is the foundation of the industrial-policy debate in module 5
Exercise
Suppose Kenya's TFP growth doubled from 0.5%/year to 1.0%/year over a 20-year period, while savings rate and labour-force growth stayed at current levels. Capital share α = 0.35. (1) Using the Solow framework, compute what happens to the steady-state level of per-capita income. (2) Estimate the cumulative impact on per-capita income over 20 years. (3) What structural reforms would plausibly produce this TFP improvement? (4) What are the principal obstacles?