Calculator

Solow Steady-State Calculator

Compute the Solow-Swan steady state — capital, output, consumption, and investment per effective worker — from the savings rate, capital share, depreciation, population growth, and technology growth.

Results

Interactive output

Adjust parameters and the frontend will debounce one model run per change before updating the charts and tables in place.

Run the model to populate summary metrics, charts, and tables.

Section 1

How to use this calculator

This calculator solves the Solow–Swan steady state in closed form. Give it five numbers and it returns four: steady-state capital per effective worker $k^*$ , output per effective worker $y^*$ , consumption per effective worker $c^*$ , and investment per effective worker $i^*$ . All four quantities are expressed in the same per-effective-worker units, so they move together as you change the parameters.

What you enter

The five inputs are: the savings rate $s$ (the fraction of output saved and invested each period — a value around $0.20$ is typical for rich economies); the capital share $\alpha$ (the elasticity of output with respect to capital in the Cobb-Douglas production function — empirical estimates cluster near $\tfrac{1}{3}$ ); the depreciation rate $\delta$ (the rate at which capital wears out — a value near $0.05$ per year is standard); the population growth rate $n$ ; and the technology growth rate $g$ . The last three enter the model only through their sum $n + g + \delta$ , which is the break-even investment rate — the fraction of capital that must be replaced each period just to keep capital per effective worker constant.

Section 2

The steady-state formula

Capital per effective worker stops changing when saving exactly covers break-even investment — that is, when the investment the economy generates is just enough to offset the three drains on $k$ : physical depreciation at rate $\delta$ , the dilution from a growing workforce at rate $n$ , and the dilution from rising labour-augmenting technology at rate $g$ . Setting the rate of change to zero and solving gives a clean closed form.

Step 1
$\dot{k} = s f(k) - (n + g + \delta)\, k$
Capital accumulation per effective worker: saving minus break-even investment.
Step 2
$s k^{\alpha} = (n + g + \delta)\, k$
At the steady state $\dot{k} = 0$ with $f(k) = k^{\alpha}$ . The saving curve meets the break-even line.
Step 3
$k^{*} = \left(\dfrac{s}{n + g + \delta}\right)^{\frac{1}{1-\alpha}}$
Divide both sides by $k^{\alpha}$ , then raise to the power $1/(1-\alpha)$ to solve for $k^{*}$ .

y^{*} = (k^{*})^{\alpha} = \left(\dfrac{s}{n + g + \delta}\right)^{\frac{\alpha}{1-\alpha}}

Steady-state output per effective worker, obtained by substituting

k^{*}

into

f(k) = k^{\alpha}

. The exponent

\alpha/(1-\alpha)

is typically around

\tfrac{1}{2}

— a 4-to-1 difference in saving rates produces only a 2-to-1 difference in

y^*

The remaining two steady-state quantities follow immediately from $y^*$ . Steady-state investment per effective worker is $i^{*} = s\, y^{*}$ : the economy saves and invests a fixed fraction of its output. Steady-state consumption per effective worker is $c^{*} = (1 - s)\, y^{*}$ : the fraction of output not saved is consumed.

Section 3

Calculate the steady state

Adjust any of the five controls and watch the steady-state values and the Solow diagram update.

No scalar found for key: steady_state_k

No scalar found for key: steady_state_y

No scalar found for key: steady_state_c

No scalar found for key: steady_state_i

The Solow diagram

The steady state $k^*$ sits where the saving curve $s f(k)$ crosses the break-even line $(n + g + \delta) k$ . To the left of $k^*$ saving exceeds break-even investment and capital rises; to the right break-even investment exceeds saving and capital falls.

Solow Steady-State Calculator

What you enter

The Solow diagram

Saving vs break-even investment