How Economies Grow

Countries build capital by saving part of what they produce, while machines and roads wear out. Output grows more slowly than capital, so saving can only outrun wear up to a point. Every economy heads to the same steady state.

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How it works

Bernoulli equations

In one sentence

A Bernoulli equation $y' + p(t)y = q(t)y^n$ ( $n \ne 0, 1$ ) becomes linear under $u = y^{1-n}$ ; here $y = k$ , $p = \delta$ , $q = s$ and $n = \alpha$ .

General formula

\frac{d{\htmlData{sym=0}{k}}}{dt} = {\htmlData{sym=1}{s}}\,{\htmlData{sym=0}{k}}^{\htmlData{sym=2}{{\alpha}}} - {\htmlData{sym=3}{\delta}}\,{\htmlData{sym=0}{k}} \quad\xrightarrow{\;{\htmlData{sym=4}{u}} = {\htmlData{sym=0}{k}}^{1-{\htmlData{sym=2}{\alpha}}}\;}\quad \frac{d{\htmlData{sym=4}{u}}}{dt} + (1-{\htmlData{sym=2}{\alpha}})\,{\htmlData{sym=3}{\delta}}\,{\htmlData{sym=4}{u}} = (1-{\htmlData{sym=2}{\alpha}})\,{\htmlData{sym=1}{s}}

Output grows like $k^{\alpha}$ with $\alpha < 1$ , slower than wear, which grows like $k$ . Small economies out-save their wear and grow; large ones can't keep up, so all meet in the middle.

What the symbols mean

$k$
Capital per worker
Machines, buildings and infrastructure per person.
$s$
Savings rate
The share of output saved and invested.
$\alpha$
Capital share
How much output responds to capital; about one third in real economies.
$\delta$
Depreciation
The share of capital that wears out each year.
$u$
Substituted variable
u = k^(1−α), the variable that makes the equation linear.

Each year workers save a share $s$ of output $k^{\alpha}$ and invest it, while a share $\delta$ of capital wears out.
The $k^{\alpha}$ term makes this a Bernoulli equation: nonlinear, but the substitution $u = k^{1-\alpha}$ makes it linear.
In $u$ it is $u' + (1-\alpha)\delta u = (1-\alpha)s$ , so $u$ approaches $s/\delta$ exponentially.
Back in $k$ : every economy approaches the steady state $k^* = (s/\delta)^{1/(1-\alpha)}$ , where saving just covers wear.

Learn the math

Bernoulli EquationsDiff Eq · Lesson 7, First Order Differential Equations SubstitutionsDiff Eq · Lesson 8, First Order Differential Equations