ECON5002 · Macroeconomic Theory
Economic Growth: The Solow-Swan Model
The Solow-Swan model (Smith's lectures use the Australian label Swan-Solow, after Trevor Swan) explains long-run living standards from one mechanism: capital per worker rises only while saving per worker s·f(k) beats the break-even investment (n + d)k needed to replace depreciation and equip a growing workforce. Because capital has diminishing returns, the two must eventually meet at a steady state where per-capita growth stops. The model's headline result for ECON5002 is that a higher saving rate raises the level of output per worker but not its long-run growth rate — only exogenous technical progress delivers sustained per-capita growth, which is why the IMF "save more to grow faster" prescription is undercut by the model's own logic. It is final-exam material (Topics 4-8), where a numerical steady-state problem and a level-versus-growth essay recur every year.
What this chapter covers
- 011. Supply-led growth — output sits at full employment on the trend path; AD adjusts, factor prices flex (contrast Smith's demand-led super-multiplier alternative)
- 022. Intensive production function — y = A·f(k) in per-worker terms, concave with diminishing returns to capital per worker (f′ > 0, f″ < 0)
- 033. Break-even investment (n + d)k — the investment per worker needed just to hold k constant against depreciation d and population growth n
- 044. Fundamental Swan-Solow equation — Δk = s·A·f(k) − (n + d)k; the steady state is the saving / break-even cross where Δk = 0
- 055. Steady-state growth — k and y per worker are constant, but aggregates Y, K, N all grow at n; per-capita growth is zero without technology
- 066. Saving-rate result — a higher s raises the LEVEL of k* and y* (plus a transitional spurt) but NOT the long-run per-capita growth rate
- 077. Golden rule — the saving rate that maximises consumption per worker, where MPK = n + d (over-save vs under-save diagnosis)
- 088. Technology, convergence & the AK contrast — exogenous progress (the Solow residual) is the only lasting growth source; AK's constant returns make g = sA − d, plus the convergence puzzles and IMF-savings critique
Steady state, consumption and the golden rule
- +2Steady-state condition: saving per worker equals break-even investment, s·f(k*) = (n + d)·k*. Substitute: 0.45·k*^0.5 = (0.04 + 0.05)·k* = 0.09·k*.
- +2Divide both sides by k*^0.5: 0.45 = 0.09·k*^0.5, so k*^0.5 = 5 and k* = 25.
- +2Output per worker y* = √25 = 5; consumption per worker c* = (1 − s)·y* = 0.55 × 5 = 2.75.
- +2Golden rule: set MPK = n + d. With y = k^0.5, MPK = 0.5·k^(−0.5) = 0.09, so k^0.5 = 0.5/0.09 ≈ 5.56 and k_gold ≈ 30.9.
- +1Since k* = 25 < k_gold ≈ 30.9, the economy holds less capital than the golden rule, so it under-saves; a higher saving rate would raise steady-state consumption per worker.
Key terms
- Capital per worker (k)
- The capital stock divided by the labour force, k = K/N; the intensive variable whose accumulation drives output per worker y = A·f(k). Raising it requires net investment above break-even.
- Break-even investment (n + d)k
- The investment per worker needed just to keep k constant — enough to replace the fraction d that depreciates each period and to equip the new workers added by population growth n.
- Steady state
- The capital per worker k* where saving per worker equals break-even investment, s·A·f(k*) = (n + d)k*, so Δk = 0. Per-worker quantities are constant; aggregate output, capital and labour all grow at the population rate n.
- Diminishing returns to capital
- The concavity of f(k) (f″ < 0): each extra unit of capital per worker adds less output than the last. This is why the saving curve must eventually fall below the break-even line, ending per-capita growth — the core mechanism of the whole model.
- Golden rule
- The saving rate that maximises steady-state consumption per worker, characterised by MPK = n + d (the slope of f(k) equals the slope of the break-even line). Above it the economy is dynamically inefficient: it could consume more, forever, by saving less.
- Solow residual (multifactor productivity)
- The part of measured output growth not explained by growth in capital and labour, a = (g_y − n) − b(g_k − n) with b capital's income share. In Swan-Solow it represents exogenous technical progress — the only source of sustained per-capita growth.
- AK / endogenous growth
- A growth model with constant (not diminishing) returns to capital, y = Ak, giving a permanent growth rate g = sA − d. Unlike Swan-Solow, a higher saving rate raises long-run growth forever, because the saving curve never falls below break-even.
- IMF-savings critique
- The point that the policy advice "raise the saving rate to grow faster" is undercut by Swan-Solow's own logic: a higher s changes only the level of output per worker (plus a transitional spurt), not the steady-state growth rate.
Economic Growth: The Solow-Swan Model FAQ
Why doesn't a higher saving rate raise long-run growth in Swan-Solow?
A higher saving rate lifts the saving curve, so capital deepens to a higher steady state and there is a temporary growth spurt during the transition. But diminishing returns to capital shrink the gap between saving and break-even investment until they are equal again, at which point per-capita growth returns to zero. So a higher saving rate is a level effect, not a permanent growth effect — the basis of the IMF-savings critique.
What is the difference between the steady state and the golden rule?
The steady state is where actual investment equals break-even investment, s·f(k) = (n + d)k, so capital per worker stops changing. The golden rule is the particular saving rate that maximises consumption per worker, where the marginal product of capital equals n + d. An economy can be in a steady state that is below, at, or above the golden-rule capital stock — comparing k* with k_gold tells you whether it under-saves or over-saves.
What happens when population growth rises?
Faster population growth n steepens the break-even line (d + n)k, because more investment is needed to equip the extra workers. The saving curve now meets it at a lower capital per worker, so steady-state k* and y* fall — living standards per head drop — even though aggregate output grows faster at the higher n. This is the population paradox of the model.
What is the only source of sustained growth in output per person?
Labour-augmenting technical progress. With Y = F(K, AN), output per person grows at the rate of technological progress in the long run. But in Swan-Solow technology is exogenous — measured as the Solow residual, not explained — which is exactly what motivated endogenous growth theory (the AK and human-capital models).
How is this topic examined in ECON5002?
It is final-exam material (Topics 4-8), tested two ways: a numerical short-answer that asks you to solve s·f(k) = (n + d)k for k*, y* and c* and then find the golden-rule k via MPK = n + d, and a choose-2-of-4 essay on why a higher saving rate is only a level effect (with the AK contrast and the IMF critique). Every essay expects a correct, labelled saving / break-even diagram drawn from memory.
Exam move
Treat Topic 7 as one diagram plus one numerical drill. First make the saving / break-even figure automatic: a concave production curve f(k), a concave saving curve s·f(k) below it, a straight break-even line (n + d)k from the origin, and the steady state where saving crosses break-even — be able to draw and fully label it from memory, because every growth essay demands it. Then drill the numerical core until the setup is reflexive: write s·f(k) = (n + d)k, solve for k*, read off y* and c* = (1 − s)y*, then find the golden rule from MPK = n + d and compare with k* to diagnose over- or under-saving. Finally, rehearse the level-versus-growth discriminator in one clean line — a higher saving rate raises the level of output per worker, not its long-run growth rate, because of diminishing returns, whereas the AK model's constant returns make g = sA − d so saving raises growth forever — and pair it with the convergence puzzles and the IMF-savings critique, since this is where the essay marks sit.
