The question Solow answers
Robert Solow and Trevor Swan each published the model in 1956, and Solow’s growth work later earned him the Nobel prize. It explains output per worker by capital per worker, and asks what happens as an economy saves part of its income and turns that saving into machines, buildings and equipment.
Everything is measured per worker, so the diagram has capital per worker, k, on the horizontal axis. Output per worker follows a production function with diminishing returns: in the explorer y = k^α with a capital share α of 0.33, so each extra unit of capital adds a little less than the last.
Investment, break-even investment and the steady state
Each year the economy invests a fraction s of output, so investment per worker is s·f(k). Some of that is used up just standing still: capital wears out at the depreciation rate δ, and new workers arriving at the rate n need equipping too. Break-even investment is therefore (n + δ)k, a straight line from the origin.
Where s·f(k) is above the line, capital per worker grows; where it is below, it shrinks. The steady state k* is where they cross. Solving s·k^α = (n + δ)k gives k* = (s/(n + δ))^(1/(1 − α)). With a saving rate of 25%, population growth of 1% and depreciation of 5%, k* is about 8.4 and output per worker about 2.0.
Saving more: a level effect, not a growth effect
Raise the saving rate from 25% to 35% and the investment curve moves up. Capital accumulates until the economy reaches a new steady state with k* of about 13.9 and output per worker of about 2.4. During the transition, output per worker grows faster. Once the new steady state is reached, it stops growing again.
That is the model’s most examined result: without technological progress, nothing a country does with its saving rate changes its long-run growth rate of output per worker. Saving changes the level of income, not its growth. Sustained growth in living standards needs technology to keep improving, which the basic model takes as given from outside.
Population growth and depreciation
Faster population growth makes the break-even line steeper, because more investment goes on equipping new workers. Raise n from 1% to 2% in the explorer and k* falls from about 8.4 to about 6.7, with output per worker falling from 2.0 to about 1.9. Total output grows faster, at the rate of population growth, while output per worker settles lower.
Higher depreciation does the same thing to the break-even line for a different reason: more of each year’s investment replaces worn-out capital.
The golden rule
A country could save almost everything and have lots of capital, but then it would consume very little. Steady-state consumption is c* = f(k*) − (n + δ)k*, which is highest where the slope of the production function equals n + δ, the golden-rule level of capital.
With Cobb-Douglas production the golden rule has a neat answer: save a fraction equal to the capital share, s = α. In the explorer that means 33%, giving consumption per worker of about 1.55 against 1.51 at a 25% saving rate. An economy saving more than α is dynamically inefficient: it could consume more today and in every future year by saving less. One saving less than α faces a real trade-off, since getting to the golden rule means consuming less during the transition.
Convergence and what the model leaves out
Because of diminishing returns, an economy far below its steady state grows quickly and one close to it grows slowly. Countries with the same saving rate, population growth and technology should therefore converge on the same income per worker: conditional convergence, which much of the empirical growth literature tests.
The basic model has no technological progress and no explanation of where technology comes from. Courses usually add labour-augmenting technology next, so that output per effective worker settles while output per worker grows at the rate of technical progress, and then move on to endogenous growth models in which the growth rate itself responds to policy.