ECON6002 · Macroeconomic Analysis
The Solow-Swan Growth Model
ECON6002 Macroeconomic Analysis at the University of Sydney introduces the Solow-Swan model as the first serious theory of long-run growth, built by Robert Solow at MIT and Trevor Swan at ANU. Working per unit of effective labour, the whole model collapses to one equation — actual investment s·f(k) versus break-even investment (n+g+δ)k — whose crossing pins down the steady state k*. Its central message is that capital accumulation drives catch-up (convergence), but long-run growth in living standards is set entirely by the exogenous technology rate g, so the saving rate moves only the level of income, never the long-run growth rate. This topic sits in Topics 1-6 and is examined by the 30% closed-book In-Semester Test, where the marks live in your derivations, the labelled Solow diagram, and the golden-rule condition.
What this chapter covers
- 011. Set-up — labour-augmenting technology, effective labour AL, and the intensive form y = f(k)
- 022. The fundamental Solow equation — k̇ = s·f(k) − (n+g+δ)k, derived via the quotient rule
- 033. Break-even investment — depreciation δk plus capital dilution (n+g)k just to hold k constant
- 044. The steady state k* — where actual investment equals break-even; k* = [s/(n+g+δ)]^(1/(1−α)) for Cobb-Douglas
- 055. The balanced growth path — k, y, c per effective worker constant; aggregates grow at n+g, per-capita output at g
- 066. Stability and convergence — k̇ > 0 below k* and k̇ < 0 above, so poorer economies grow faster and catch up
- 077. Comparative statics — a rise in s raises k* and y* (a level effect); a rise in n lowers k*; growth still returns to g
- 088. The golden rule — f′(k*) = n+g+δ maximises consumption per effective worker; s > α over-saves, s < α under-saves
Steady state, consumption and the golden rule
- +1First the break-even (effective depreciation) rate: n + g + δ = 0.02 + 0.03 + 0.05 = 0.10. This is the whole coefficient on the break-even line — never drop n or g.
- +2The steady state sets actual investment equal to break-even investment: s·k*^0.5 = (n+g+δ)k*. Solving gives k* = [s/(n+g+δ)]^(1/(1−α)) = [0.4/0.10]^(1/0.5) = 4^2 = 16.
- +1Output per effective worker: y* = k*^0.5 = √16 = 4. Consumption per effective worker: c* = (1 − s)·y* = 0.6 × 4 = 2.4.
- +1For Cobb-Douglas output the golden-rule saving rate always equals capital's share: s_GR = α = 0.5. (It satisfies f′(k*) = n+g+δ.)
- +1Compare: actual s = 0.4 is below s_GR = 0.5, so the economy under-saves — its k* = 16 is below the golden-rule stock, and raising s toward 0.5 would increase long-run consumption per effective worker.
Key terms
- Effective labour (AL)
- Labour scaled by the level of technology, A × L. Measuring capital and output per unit of effective labour is what makes the model solvable: on the balanced growth path these per-effective-worker variables are constant even while the aggregates keep growing.
- Intensive form (y = f(k))
- The production function written per effective worker, y = Y/AL = f(k) with k = K/AL. It follows from constant returns to scale and turns a two-input problem into a single-variable one; the Cobb-Douglas benchmark is y = k^α with 0 < α < 1.
- Break-even investment ((n+g+δ)k)
- The investment per effective worker needed just to hold k constant: δk replaces worn-out capital and (n+g)k offsets capital dilution from a growing, more productive workforce. Actual investment above this level makes k rise; below it, k falls.
- Steady state (k*)
- The resting point where the fundamental Solow equation gives k̇ = 0, so s·f(k*) = (n+g+δ)k*. For Cobb-Douglas, k* = [s/(n+g+δ)]^(1/(1−α)); output and consumption per effective worker then follow as y* = f(k*) and c* = (1−s)y*.
- Balanced growth path (BGP)
- The long-run path along which per-effective-worker variables (k, y, c) are constant, aggregate K, Y and C grow at n+g, and output per person grows at the exogenous technology rate g — independent of the saving rate.
- Capital dilution
- The (n+g)k part of break-even investment: every period there are more effective workers to equip (population grows at n, each made more productive at rate g), so even with zero depreciation some investment is needed just to keep k per effective worker unchanged.
- Golden rule
- The steady-state capital stock (and the saving rate behind it) that maximises consumption per effective worker, characterised by f′(k*) = n+g+δ. Saving above the golden rule is dynamically inefficient over-accumulation; for Cobb-Douglas the golden-rule saving rate equals α.
- Convergence
- The prediction that an economy starting below its k* has a high marginal product of capital and grows faster, catching up toward the steady state, while one above k* grows more slowly — so otherwise-identical poor economies grow faster than rich ones.
The Solow-Swan Growth Model FAQ
Can AI help me with the Solow-Swan growth model?
Yes — ask Sia to walk through any Solow-Swan growth model problem or concept step by step, the way University of Sydney tests it. Sia is an AI tutor that explains each stage of a derivation — collapsing to intensive form, writing k̇ = s·f(k) − (n+g+δ)k, solving for k*, or applying the golden rule — so you can reproduce it yourself in the closed-book test. It is a study aid that builds your understanding, not an answer service.
Does a higher saving rate raise long-run growth in the Solow-Swan model?
No — this is the topic's headline trap. A permanent rise in the saving rate raises the level of capital and output per effective worker (k* and y* both increase), and growth temporarily exceeds g during the transition, but the long-run per-capita growth rate returns to the exogenous technology rate g. Saving changes the level of income, not the long-run growth rate; diminishing returns to capital are why.
Why is break-even investment (n+g+δ)k and not just δk?
Because holding capital per effective worker constant requires offsetting three things, not one. δk replaces depreciated capital, while (n+g)k covers capital dilution — each period there are more workers (growing at n), each more productive (technology growing at g), so the existing capital has to be spread across a larger effective workforce. Dropping n or g is the single most common Solow error and makes every downstream result wrong.
What is the golden rule in the Solow-Swan model?
The golden rule is the steady state that maximises consumption per effective worker, found by setting f′(k*) = n+g+δ — the marginal product of capital equals the rate at which capital must be replaced and diluted. For a Cobb-Douglas economy the golden-rule saving rate is simply α (capital's income share). If the actual saving rate exceeds α the economy over-accumulates (dynamically inefficient); if it is below α, it under-saves and could raise long-run consumption by saving more.
How do I prove the Solow steady state is stable?
Work straight from the fundamental equation k̇ = s·f(k) − (n+g+δ)k. Because f is concave (f′ > 0, f″ < 0), the actual-investment curve starts steeper than the break-even line and ends flatter, so they cross once at k*. Below k*, actual investment exceeds break-even so k̇ > 0 and k rises; above k*, actual investment falls short so k̇ < 0 and k falls. The arrows point to k* from both sides, so the steady state is globally stable — this derivation is a recurring In-Semester Test question.
How is the Solow-Swan model assessed in ECON6002?
It sits in the first half of the unit (Topics 1-6), so it is examined by the 30% In-Semester Test — a supervised, in-person, closed-book paper with no formula sheet and extended-response questions. The final exam covers the later New Keynesian topics (7-10), so the In-Semester Test is your one chance at growth-model marks. Expect to solve for k*, y* and c*, apply the golden rule, do comparative statics on s or n, and prove stability — showing every step, since the marks are in the working and the labelled diagram.
Studying with AI? Sia — free AI economics tutor works through ECON6002 step by step.
Exam move
Treat the Solow-Swan model as one diagram and one equation you can reproduce cold. Master the Solow diagram — output f(k), actual investment s·f(k), the break-even line (n+g+δ)k, and the steady state k* where the last two cross — and practise deriving the fundamental equation k̇ = s·f(k) − (n+g+δ)k via the quotient rule until it is automatic, because the In-Semester Test is closed-book with no formula sheet. Drill the three moves the test reuses: solve for k*, y* and c* from a Cobb-Douglas y = k^α; run comparative statics on a change in s or n and state explicitly that it is a level effect, not a growth effect (long-run per-capita growth is always g); and apply the golden rule f′(k*) = n+g+δ, using the one-line Cobb-Douglas shortcut s_GR = α to classify over- versus under-saving. Rehearse the stability proof (k̇ > 0 below k*, k̇ < 0 above) as a standalone derivation, and never write break-even investment as δk — the (n+g)k dilution term is where easy marks are lost. As you revise, ask Sia to generate fresh Solow-style problems and to check each line of your derivations, so you practise explaining the mechanism rather than just memorising the final formula.
