Y = Kα(AL)1−α — watch capital per worker converge to steady state as you adjust savings, depreciation and technology growth
The Solow growth model (Robert Solow & Trevor Swan, 1956) is the foundation of modern macroeconomics. Effective capital per worker k̃ = K/(AL) evolves by: dk̃/dt = s·f(k̃) − (δ+g+n)·k̃, where f(k̃) = k̃α is the Cobb-Douglas production function in intensive form.
The steady state k* is where investment s·f(k̃) exactly covers capital widening (δ+g+n)·k̃. Below k* the economy grows; above k* it contracts — guaranteeing convergence. The Golden Rule savings rate maximises steady-state consumption: sgold = α (capital share).
The left panel shows time-series of y, k, and c = (1−s)y per effective worker. The right phase diagram shows the sf(k̃) (actual investment) curve and the (δ+g+n)k̃ (break-even) line; their intersection is k*.
This interactive model traces the Solow-Swan theory of long-run growth. It numerically integrates the law of motion for effective capital per worker, dk̃/dt = s·k̃^α − (δ+g+n)·k̃, advancing time in half-year steps and plotting output, capital and consumption per worker until they settle at the steady state k*. The right-hand phase diagram overlays the actual-investment curve sf(k̃) against the break-even line (δ+g+n)k̃, so you can see exactly why capital converges.
Capital accumulation under a Cobb-Douglas production function f(k̃)=k̃^α. The time-series panel draws output y, capital k and consumption c=(1−s)y per effective worker, while the phase diagram shows where investment sf(k̃) crosses break-even (δ+g+n)k̃ — their intersection is the steady state k*.
Drag five sliders: savings rate s (0.05–0.70), depreciation δ (0.01–0.20), technology growth g (0–0.10), population growth n (0–0.05) and capital share α (0.20–0.60). Use Pause/Play to halt the animation, Reset to restart from low capital, and Phase Diagram to toggle the right panel. Live stats report k*, y* and the golden-rule savings rate.
The golden-rule savings rate that maximises steady-state consumption equals the capital share α. Because real economies typically save less than α, most are "dynamically efficient" — saving more would raise long-run consumption, a result Robert Solow's 1956 paper helped formalise.
It is a neoclassical model of long-run economic growth published by Robert Solow and Trevor Swan in 1956. Output is produced from capital and effective labour via a Cobb-Douglas function, Y=K^α(AL)^(1−α). The model explains how an economy converges to a stable steady state and why sustained per-capita growth ultimately requires technological progress rather than saving alone.
Each frame it integrates dk̃/dt = s·k̃^α − (δ+g+n)·k̃ using a forward Euler step of dt=0.5 years. The steady state is found analytically as k* = [s/(δ+g+n)]^(1/(1−α)), shown as a dashed line on the time-series chart and as the crossing point in the phase diagram. The simulation starts from low capital and animates the approach to k*.
Savings rate s sets the fraction of output invested; depreciation δ is the rate at which capital wears out; technology growth g and population growth n together with δ form the break-even rate (δ+g+n) that capital must replace; and capital share α is the exponent in the production function. Raising s lifts k* and y*, while raising δ, g, n or lowering α reduces k*.
The equations are the standard textbook Solow-Swan equations and the steady-state and golden-rule formulae are exact. Simplifications are deliberate: it uses a single Cobb-Douglas sector, treats parameters as constant, employs a coarse Euler integrator for speed, and works in per-effective-worker units. It captures the qualitative dynamics and comparative statics faithfully rather than fitting any specific country's data.
A higher savings rate raises the steady-state level of capital and output per worker, giving a temporary burst of faster growth during the transition. Once the new k* is reached, growth in output per worker returns to the rate of technological progress g. This is the model's central lesson: capital accumulation has diminishing returns, so only technology drives sustained long-run growth in living standards.