Now make the model dynamic. Write everything per effective worker, meaning per unit of A·L, and let lower-case letters denote those quantities: y = Y/(A·L) and k = K/(A·L). With constant returns the production function collapses to y = k^α, and the capital stock evolves as
Δk = s · k^α − (δ + n + g) · kwhere s is the saving rate, n the growth of labour and g the growth of technology. The first term is investment per effective worker.
The second is break-even investment: the amount needed just to keep k constant, covering depreciation at rate δ, equipping the new workers arriving at rate n, and covering the fact that A itself is growing at g so the denominator keeps expanding.
How do you solve for the Solow steady state?
The steady state is where those two terms are equal, so Δk = 0:
k* = (s / (δ + n + g))^(1/(1−α)) y* = (k*)^α c* = (1 − s) · y*The steady state has a property worth stating flatly because it is where the model earns its keep. At k*, output per effective worker is constant. Output per worker therefore grows at exactly g, the rate of technological progress, and total output grows at n + g. Neither depends on the saving rate at all.
Take a synthetic economy with s = 0.28, δ = 5%, n = 1.0%, g = 1.5% and α = 0.35. Break-even runs at 7.5%, and the steady state solves to k* = 7.588377, y* = 2.032601 and c* = 1.463473.
The capital-output ratio there is k*/y* = 3.733333, which is exactly s/(δ+n+g) = 0.28/0.075. That identity is a free check on any steady-state solve you ever do.
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What happens when the saving rate rises?
Raise the saving rate permanently to 0.35 and the economy moves to a new steady state with k* = 10.696488 and y* = 2.292105, a level 12.77% higher, held for ever. The growth rate along the new path is 1.5% per worker, exactly as before. A saving rate is a level lever rather than a growth lever, so a country that raises investment from 28% to 35% of output buys a one-off 12.77% and then resumes the old trend.
How long is the one-off? The economy converges at rate β = (1 − α)(δ + n + g), which here is 4.875% a year, so the half-life is 14.22 years. After 10 years 61.42% of the initial gap remains, after 20 years 37.72%, after 30 years 23.17%.
A transition measured in decades looks exactly like a growth-rate change to anyone watching a ten-year window, which is why so many miracle-growth stories were later reclassified as long transitions.