Why economies grow, why they stop, and why the poor can catch up. Capital per worker climbs while investment beats depreciation and stalls where they balance: k̇ = s·f(k) − (n+δ)k, with f(k)=kα. Down the center the parameters go in, the ODE integrates, and a proven steady state comes out. The blue team builds and defends it; the red team tries to break it. These are models, not investment advice.
source Solow, A Contribution to the Theory of Economic Growth, Quarterly Journal of Economics 70 (1956), 65–94 — doi.org/10.2307/1884513. Rendered, not quoted.
One state variable, capital per worker k. It rises by saved output s·f(k) and falls by break-even requirements (n+δ)k — new workers to equip (n) and worn capital to replace (δ). Diminishing returns (α<1) bend investment below the straight break-even line, so they must cross once, at a unique steady state:
k* = (s / (n+δ))1/(1−α), where s·f(k*) = (n+δ)·k*.
For the current settings, the live fixed point:
| quantity | value |
|---|
The engine f(k)=kα is a per-worker Cobb–Douglas production function — the neighbouring sphere. Solow builds directly on it.
Raising the saving rate lifts k* and output once — a level effect. It does not raise the long-run growth rate: at any steady state k̇=0, so per-worker growth is zero. Sustained per-capita growth needs the function itself to shift — technology (A), which Solow leaves exogenous. Each sphere is the next one's premise.
The blue team's live check: integrate the ODE from below and above k* and confirm both land on the analytic fixed point, that the residual s·f(k*)−(n+δ)k* is ~0, and that the golden-rule rate equals α. If red flips a sign, this badge is where it shows.
Four numbers describe the economy, plus a starting capital stock:
| symbol | meaning |
|---|---|
| s | saving rate — the share of output invested |
| α | capital share / elasticity in f(k)=kα |
| n | labour-force growth rate |
| δ | depreciation rate of capital |
| k₀ | initial capital per worker |
These feed the panel below. The steady state depends only on s, α, n, δ — not on where you start (k₀): convergence is the whole point.
Curve = investment s·f(k); line = break-even (n+δ)k; they cross at k*. The dot slides from k₀ toward k*.
Move any slider — k* is re-solved from the closed form and the path is re-integrated on the spot, never looked up. Forecasts of any real economy from this would be AMBER: it is a model, not advice.
What the machine produces, proven: a unique, stable steady state k* where investment equals break-even; a Runge–Kutta path from any k₀ that lands on it to 1e−6; the level-not-growth reading of saving; and the golden-rule rate s=α that maximises steady-state consumption. The current numbers are above; the totals are the output.
The blue team's witness (left) re-integrates these live; the red team (right) tries to make capital diverge.
Its scaffolding is idealization (AMBER): perfect competition, one representative agent, one homogeneous good, a smooth Cobb–Douglas with constant returns and diminishing marginal product, a closed economy, all other things equal. Endogenous-growth theory (Romer 1986, Lucas 1988) makes technology a choice and breaks the model open. Economics is not physics; a "law" here is a modelling assumption, and this is a model, not investment advice.
"Save more and you grow faster forever." Cut. Higher s is a level effect: it lifts k* once, then long-run per-worker growth returns to zero. The engine verifies k̇=0 at every steady state.
"Solow explains what causes growth." Corrected. The driver, technical progress, is the exogenous Solow residual — measured, not explained.
"Poor countries always catch up." Corrected. Only conditional convergence: each economy approaches its own k*, set by its own s, n, δ. Absolute convergence across all countries fails in the data.
The red team's move: flip the sign of depreciation so worn capital is added instead of removed — k̇ = s·f(k) + (n+δ)k. Now capital only ever grows: no steady state, no convergence. The blue team's witness (window 7) is watching.
Flip the sign and the ODE diverges — the witness re-integrates, cannot reach k*, and turns red. Nothing is faked; the attack is real and it is caught.