Solow Growth Model Steady State

The Solow growth model (Robert Solow, 1956) is the foundation of modern macroeconomic growth theory. It explains why some countries are richer than others and why growth rates slow as countries get wealthier.

The model. Each worker produces output y = k^alpha where k is capital per worker and alpha is capital’s share of income (typically 0.33 in developed economies). Each period, a fraction s of income is saved and invested. Capital depreciates at rate delta. The labor force grows at rate n and technology improves at rate g.

The steady-state condition. Capital per worker stabilizes when new investment exactly replaces what is lost to depreciation, population growth, and technological dilution:

s x k^alpha = (delta + n + g) x k

Solving for k*:

k* = (s / (delta + n + g))^(1 / (1 - alpha))

Steady-state output per effective worker: y* = (k*)^alpha

Steady-state consumption per effective worker: c* = (1 - s) x y*

Policy implications. A higher savings rate raises k* and y* permanently. But there is a golden-rule savings rate that maximizes consumption per worker. That rate equals alpha (capital’s share). Above this rate, the economy invests too much and consumption falls even as capital grows.

The central result of the Solow model: long-run growth in output per person comes only from technology growth g, not from capital accumulation. Without ongoing technology progress, growth per worker eventually stops.