ECON1002 · Introductory Macroeconomics
The Solow-Swan Growth Model
The Solow-Swan model turns the per-capita production function into a dynamic story. Households save a fixed fraction (s = θy), capital wears out and is diluted by population growth (break-even investment (d+n)k), and capital per worker evolves as Δk = θy − (d+n)k. Where saving equals break-even investment, the economy reaches a steady state k* and y*, at which long-run growth of output per worker is zero unless TFP rises.
It is a flagship Section-B diagram and calculation: solve for k* and y*, show how a higher saving rate or higher population growth shifts the steady state, and explain conditional convergence.
What this chapter covers
- 011. Behavioural saving: s = θy (θ = saving rate, 0 < θ < 1)
- 022. Break-even / replacement investment (d + n)k (d = depreciation, n = population growth)
- 033. Capital accumulation Δk = θy − (d + n)k = θ·Akᵃ − (d+n)k
- 044. Steady state: Δk = 0 ⇒ θAkᵃ = (d+n)k; solve for k* then y* = A(k*)ᵃ
- 055. Long-run steady-state growth of output per worker = 0 without TFP growth
- 066. Higher θ ⇒ higher k*, y* (level, not long-run growth); higher n ⇒ lower k*, y*
- 077. TFP (A) overcomes diminishing returns — doubling A raises the whole y curve and k*
- 088. Conditional convergence; endogenous growth (AK model: Y = AK, no diminishing returns)
Solow-Swan steady state and the effect of a higher saving rate
- 1 mark(a) Steady state requires Δk = 0, i.e. saving equals break-even investment: θAk^0.5 = (d+n)k ⇒ 0.3·k^0.5 = 0.10·k.
- 2 marksDivide by k^0.5: 0.3 = 0.10·k^0.5 ⇒ k^0.5 = 3 ⇒ k* = 9. Then y* = A·(k*)^0.5 = 1 × 9^0.5 = 3.
- 1 mark(b) With θ = 0.4: 0.4·k^0.5 = 0.10·k ⇒ k^0.5 = 4 ⇒ k* = 16 (and y* = 4). A higher saving rate raises the steady-state level of capital and income per worker.
- 1 markBut the LONG-RUN growth rate of output per worker is still 0 at the new steady state — a higher saving rate raises the level, not the permanent growth rate (only TFP growth does that).
Key terms
- Behavioural saving rule (s = θy)
- The Solow-Swan assumption that households save a constant fraction θ of output, so saving (and hence investment, in a closed economy) per worker is θy = θAkᵃ. It is the upward-curving saving line below the production function.
- Break-even investment ((d+n)k)
- The investment per worker needed just to keep capital per worker constant: replacing depreciated capital (rate d) and equipping new workers as the population grows (rate n). It is a straight line through the origin with slope (d+n).
- Capital accumulation equation
- Δk = θy − (d+n)k = θAkᵃ − (d+n)k: capital per worker rises when saving exceeds break-even investment and falls when it falls short. It drives the economy toward the steady state.
- Steady state (k*, y*)
- The point where Δk = 0, so saving exactly funds break-even investment: θA(k*)ᵃ = (d+n)k*. Capital and output per worker are then constant, and (without TFP growth) per-worker growth is zero.
- Conditional convergence
- The prediction that economies with the SAME saving rate, population growth, depreciation and technology converge to the same steady state — so poorer such economies grow faster and catch up. It holds across similar countries (e.g. the OECD) but not unconditionally across the whole world.
- Endogenous growth (AK model)
- A model where production is Y = AK with no diminishing returns to broadly-defined capital, so investment can sustain permanent growth. It motivates a role for policy — education, R&D, institutions — in driving long-run growth, unlike Solow-Swan where long-run growth needs exogenous TFP.
The Solow-Swan Growth Model FAQ
How is Solow-Swan examined in ECON1002?
As a flagship: a Section-B diagram and a calculation. You solve for the steady-state k* and y* by setting saving equal to break-even investment, show how a higher saving rate raises the saving curve (higher k*) or higher population growth steepens break-even (lower k*) on the diagram, and explain conditional convergence. The sample and 2025 finals both featured a steady-state computation.
How do you find the steady state in Solow-Swan?
Set capital per worker constant, Δk = 0, which means saving equals break-even investment: θAkᵃ = (d+n)k. For y = Ak^0.5 this becomes θAk^0.5 = (d+n)k; divide both sides by k^0.5 to isolate k and solve, then substitute back into the production function for y*. The general result is k* = (θA/(d+n))^(1/(1−a)).
Does a higher saving rate raise long-run growth?
No — it raises the LEVEL of capital and income per worker, not the long-run growth rate. A higher saving rate lifts the saving curve, so the new steady state has a higher k* and y*, and there is faster growth during the transition. But once the new steady state is reached, per-worker growth is again zero. Sustained long-run growth of output per worker requires rising TFP (A), not just more saving.
Why does the Solow-Swan model predict only conditional convergence?
Because diminishing returns mean an economy below its steady state grows faster and catches up — but only to ITS OWN steady state, which depends on its saving rate, population growth, depreciation and technology. Economies sharing those fundamentals converge to the same level (conditional convergence, seen across the OECD), but economies with different fundamentals converge to different steady states, so there is no unconditional catch-up across the whole world.
Exam move
Over-learn the steady-state solve, because it appears every paper: set Δk = 0, write θAkᵃ = (d+n)k, divide by the kᵃ term, and solve for k* then y* — and keep the generic k* = (θA/(d+n))^(1/(1−a)) as a backstop. The decisive conceptual point is the level-versus-growth distinction: higher θ or higher A raises the LEVEL (and gives transitional growth), but long-run per-worker growth is zero unless A keeps rising. For the diagram, rehearse the three shifts — saving rate up (saving curve up, higher k*), population growth up (break-even steeper, lower k*), and TFP up (production and saving curves up, higher k*) — and narrate each. Finally, be ready to explain conditional convergence (same fundamentals ⇒ same steady state) since it is the model's signature prediction.
