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.