The Solow Model and Long-Run Growth
The Solow model constitutes the reference framework for understanding long-run economic growth. Mankiw devotes two full chapters to it (ch. 8 and 9, 9th ed.) and considers it one of the most important contributions of 20th-century macroeconomics. It explains why some countries are rich and others poor, and why standards of living differ so much across the world.
Last updated: 9 July 2026
Definition
DefinitionThe Solow model (named after Robert Solow, Nobel Prize 1987) models the long-run growth of GDP per capita as a function of three determinants: the accumulation of physical capital, population growth, and technological progress. The Cobb-Douglas aggregate production function admits two equivalent forms depending on the pedagogical use intended.
Hicks-neutral form — Y = A × Kᵅ × L¹⁻ᵅ, where A is Total Factor Productivity (TFP). This form makes the measurement of technical progress explicit and directly yields the Solow residual decomposition ΔA/A = ΔY/Y − α(ΔK/K) − (1−α)(ΔL/L) used in growth accounting.
Labor-augmenting form — Y = F(K, AL) = Kᵅ × (AL)¹⁻ᵅ, where A is the efficiency of labor (growing at rate g). This is the form that makes the effective-labor dynamics Δk = sf(k) − (δ+n+g)k and the golden rule MPK = δ+n+g used below internally consistent.
The two writings are equivalent up to a reparametrization (A_Hicks = A_labor^(1−α)). The choice between them depends on the question — growth accounting on one hand, dynamic equilibrium on the other — but they describe the same underlying technology.
Notation — Y is total output, K the capital stock, L the quantity of labor (growing at rate n), AL effective labor, and α the share of capital in national income (empirically ≈ 1/3).
In intensive terms per unit of effective labor (labor-augmenting form), the function becomes:
y = kᵅ
where y = Y/(AL) is output per effective worker and k = K/(AL) is capital per effective worker. In the basic version without technological progress (k = K/L), the term g drops out and the dynamic equation simplifies to Δk = sf(k) − (δ+n)k.
It is the indispensable basis for understanding differences in living standards across countries. Mankiw's Chapter 9 ("Economic Growth II: Technology, Empirics, and Policy") develops the technology-augmented version presented here and then extends toward endogenous growth.
Why it matters
Mankiw considers the Solow model the fundamental answer to the question "why are some countries rich and others poor?" The model identifies three policy levers to raise the standard of living: increasing the savings and investment rate, controlling population growth, and fostering technological progress. However, because of diminishing returns to capital, only technological progress can sustain growth in GDP per capita in the long run.
The fundamental dynamic equation of the model is:
Δk = s × f(k) − (δ + n + g) × k
where Δk is the change in capital per unit of effective labor, s × f(k) investment per unit of effective labor, and (δ + n + g) × k the threshold for capital maintenance (offsetting depreciation, dilution by population growth, and technological progress). The steady state is reached when Δk = 0, i.e., s × f(k*) = (δ + n + g) × k*.
With f(k) = kᵅ, one solves explicitly for capital and output per effective worker in the steady state:
s × (k*)ᵅ = (δ + n + g) × k*
(k*)¹⁻ᵅ = s / (δ + n + g)
k* = [s / (δ + n + g)]^(1/(1−α))
y* = (k*)ᵅ = [s / (δ + n + g)]^(α/(1−α))
These closed-form expressions make cross-country comparisons immediately readable. With α ≈ 1/3, doubling the savings rate multiplies k* by 2^(1/(1−α)) ≈ 2.83 (not 2) and y* by 2^(α/(1−α)) ≈ 1.41 — a ~41% gain in the long-run income level, never in the growth rate. Conversely, a country with half the savings rate converges to a steady state about 29% poorer, all else equal. This result directly illustrates the diminishing returns to capital that make any pure-accumulation growth strategy ineffective in the long run.
Key points
A higher savings rate raises the level of GDP per capita in the steady state but not the long-run growth rate. Only technological progress sustains permanent growth
Conditional convergence predicts that poor countries converge to rich countries if their structural parameters are similar. Empirically, East Asian countries confirm this prediction, while many African countries do not converge due to institutional differences
The Solow residual attributes 50 to 70% of the growth of advanced economies to technological progress rather than to factor accumulation, which underscores the central importance of innovation and education
The golden rule (MPK = δ + n + g) indicates that if the economy is above the golden-rule savings rate, reducing saving immediately and durably increases consumption — a counterintuitive result
Concrete example
ExamplesMankiw uses the post-war comparison between Germany and Japan: despite the destruction of their capital stocks, both countries experienced spectacular growth over the following decades, exactly as the Solow model predicts (rapid convergence to the steady state when k is far below k*). China illustrates the same phenomenon since 1978: starting from a very low capital stock per capita, it experienced annual growth of 8-10% thanks to a savings rate of 40-50% combined with massive imports of foreign technology.
Mankiw anecdote
MankiwMankiw recalls that the model comes from two founding articles by Robert Solow: "A Contribution to the Theory of Economic Growth" (Quarterly Journal of Economics, vol. 70, no. 1, February 1956, pp. 65-94), which lays out the formal framework, and "Technical Change and the Aggregate Production Function" (Review of Economics and Statistics, vol. 39, no. 3, August 1957, pp. 312-320), which introduces the growth-accounting decomposition and finds that about 87% of U.S. growth between 1909 and 1949 is attributable to technical progress rather than to factor accumulation — a result that lastingly reshaped growth macroeconomics. The exogenous nature of technological progress in this framework, i.e. the absence of any mechanism explaining the origin and dynamics of A, would later be the main critique addressed to Solow by endogenous-growth theorists, notably Paul Romer ("Endogenous Technological Change", Journal of Political Economy, 1990) and Robert Lucas ("On the Mechanics of Economic Development", Journal of Monetary Economics, 1988). This critique is often summarized by the formula that Solow's growth theory "took growth as given" — a phrasing that comes from the endogenous-growth literature, not from a self-description by Solow. Mankiw presents these models as attempts to "open the black box" of technological progress, while acknowledging that the Solow model, despite its simplicity, fits the empirical data remarkably well.
📊 Modèle de Solow
Market impact
MarketsCountries in convergence phase (heavy capital accumulation, GDP per capita catching up) experience rapid economic growth, but this does not mechanically translate into higher stock market returns — the correlation between GDP growth and equity returns is weak across countries — and volatility is higher. Technological progress is the structural driver of the very long-run rise in equity markets: U.S. indices reflect the continuous innovation of the economy. Emerging markets offer growth premiums linked to Solow-model catch-up.