For any given level of capital (k), the production function, $f\left(k\right)$, determines how much output the economy produces. Meanwhile, the saving rate (s) determines the allocation of output between consumption and investment. Increasing the rate of saving increases the level of investment, and as the capital stock grows, so too does the amount of capital per effective worker.

There are three factors which decrease the capital per effective worker:

1) The depreciation rate ($\mathbf{\delta}$), which accounts for the proportion of the capital stock that wears out each year.

2) The labor force growth rate (n), which reduces k by spreading the existing capital stock more thinly among a larger number of workers.

3) The growth of labor efficiency (g), which is a labor-augmenting form of the technological progress. It causes output to increase as though the labor force had grown by g%. As a result, g is known as the rate of labor-augmenting technical progress.

The term $\left(\mathrm{\δ}\+n\+g\right)k$ defines the break-even investment. This amount of investment is needed to keep the capital per effective worker constant. The term $\mathrm{\δ}k$ is needed to replace the depreciated capital. Furthermore, the term $nk$ is needed to provide capital for new workers, and lastly, the term $gk$ is needed to provide capital created by advances in technology for the new "effective workers".

It is known that investment increases the capital stock; while depreciation, labor force growth, and technological progress reduce it. As a result, the impact of these opposing forces on k can be mathematically expressed as:$\mathrm{Delta;}kequals;i-\left(\mathrm{\delta}plus;nplus;g\right)kequals;sf\left(k\right)-\left(\mathrm{delta;}plus;nplus;g\right)k$.

Steady state represents the equilibrium of the economy in the long term. Equilibrium occurs exactly when the investment equals the break-even investment. As a result, capital stock does not change.

For given values of s, $\mathrm{\δ}$, n, and g, there is only one level of k for which $\mathrm{\Δ}kequals;0$. This is known as the steady state level of capital stock per effective worker, ${k}^{\ast}$. Thus, steady state occurs mathematically when $sf\left({k}^{\ast}\right)equals;\left(\mathrm{delta;}plus;nplus;g\right){k}^{\ast}$.