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Intermediate Macroeconomics Unit 6: Economic Growth: The Solow Model

The Solow model explains how saving, population growth and technology determine living standards and growth in the long run, and why only technological progress delivers sustained growth.

Covers: Capital accumulation and the steady state, Saving and the Golden Rule, Population growth, Technological progress and balanced growth, Growth accounting, Convergence, Endogenous growth

Study guide

The Solow model studies capital accumulation. With constant returns, output per worker y = f(k) depends on capital per worker k. A fraction s of output is saved and invested, so investment per worker is sf(k). Capital wears out at rate δ, and the labor force grows at rate n, so (δ + n)k of investment is needed just to keep k constant: this is break-even investment. The change in capital per worker is Δk = sf(k) − (δ + n)k. When k is low, investment exceeds break-even investment and k rises; when k is high, it falls. The economy converges to a steady state k* where sf(k*) = (δ + n)k*.

A higher saving rate raises k* and y*, and growth is faster during the transition, but in the new steady state output per worker stops growing again. Saving affects the level of income, not its long-run growth rate. With Cobb-Douglas f(k) = k^α, the steady state is k* = (s/(δ + n))^(1/(1−α)) and y* = (s/(δ + n))^(α/(1−α)). Higher population growth lowers k* and y*, because more saving goes to equipping new workers. These results help explain why countries with high saving and low population growth tend to be richer.

More capital does not always mean more consumption. Steady-state consumption c* = f(k*) − (δ + n)k* is maximized at the Golden Rule capital stock, where MPK = δ + n (or δ + n + g with technological progress). For Cobb-Douglas, the Golden Rule saving rate equals α. If an economy has less capital than the Golden Rule, raising saving lowers consumption at first but raises it in the long run, a tradeoff between current and future generations. If it has too much capital, cutting saving raises consumption at every date.

To explain sustained growth, the model adds labor-augmenting technological progress: effective labor L × E grows at n + g. Measured per effective worker, the steady state works as before with break-even investment (δ + n + g)k. In this balanced-growth steady state, output per worker grows at g and total output grows at n + g, matching long-run data. Growth accounting decomposes output growth as ΔY/Y = αΔK/K + (1 − α)ΔL/L + ΔA/A, where the Solow residual ΔA/A measures total factor productivity growth. The model also predicts conditional convergence: economies grow faster the further below their own steady state they are.

The Solow model treats technological progress as exogenous. Endogenous growth theory tries to explain it. In the simplest AK model, Y = AK with no diminishing returns to capital, so growth equals sA − δ and a higher saving rate raises growth permanently. Two-sector models add a research sector: shifting workers into universities lowers output today but raises the growth rate of knowledge forever. These models suggest that policies on education, research and the institutions that encourage innovation can affect long-run growth, not just the level of income.

Key ideas

• The steady state occurs where sf(k) = (δ + n)k.
• Higher saving raises steady-state income per worker but not its long-run growth rate.
• Higher population growth lowers steady-state capital and income per worker.
• The Golden Rule maximizes steady-state consumption; for Cobb-Douglas its saving rate is α.
• With technological progress, output per worker grows at g and total output at n + g.
• The Solow residual measures total factor productivity growth.
• Endogenous growth models allow saving and research to raise long-run growth.

Formulas and rules

Common mistakes

• Claiming a higher saving rate raises the long-run growth rate in the Solow model.
• Leaving n or g out of break-even investment or the Golden Rule condition.
• Assuming the Golden Rule maximizes output rather than consumption.
• Confusing growth rates per worker (g), per effective worker (0) and in total (n + g).
• Using simple averages instead of income-share weights in growth accounting.

Before the test, make sure you can

• Find the steady-state k*, y* and c* from a production function and parameters.
• Compute Δk for a given k and say whether the economy is above or below its steady state.
• Explain the effects of changes in s, n and δ on the steady state and the transition path.
• Apply the Golden Rule condition and judge whether an economy has too much or too little capital.
• State steady-state growth rates of per-worker and total variables with technological progress.
• Calculate the Solow residual using growth accounting.
• Contrast the Solow model with AK and two-sector endogenous growth models.

Words to know

Steady state

The capital per worker at which investment equals break-even investment, so k is constant.

Break-even investment

(δ + n + g)k, the investment needed to keep capital per effective worker constant.

Saving rate (s)

The fraction of output saved and invested.

Depreciation rate (δ)

The fraction of the capital stock that wears out each year.

Golden Rule level of capital

The steady state that maximizes consumption per worker; MPK = δ + n + g.

Capital deepening

Growth in capital per worker.

Labor-augmenting technological progress

Growth in the efficiency of labor E at rate g.

Effective worker

A unit of labor adjusted for efficiency, L × E.

Balanced growth

A steady state in which K, Y and effective labor all grow at the same rate.

Growth accounting

Splitting output growth into contributions from capital, labor and productivity.

Solow residual

Output growth not explained by input growth; measured TFP growth.

Total factor productivity

The efficiency with which an economy combines all inputs.

Conditional convergence

Countries grow faster the further they are below their own steady state.

Endogenous growth theory

Models that explain technological progress within the model.

AK model

A model with Y = AK and no diminishing returns to capital, so saving raises growth permanently.

Half-life of convergence

The time needed to close half the gap between current income and its steady state.

Sample practice questions

30 more questions like these, from level 1 to 5, are in the practice set.