📈 Solow Growth Model

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.