Growth Rates Add When Quantities Multiply

A country's GDP is population times output per person. If population grows 1% a year and output per person 2%, GDP grows about 3%. Take logs before differentiating and products become sums: percentage growth rates simply add.

How it works

Logarithmic differentiation

In one sentence

For $y = f(x)g(x) > 0$ , $\ln y = \ln f + \ln g$ , so $\frac{y'}{y} = \frac{f'}{f} + \frac{g'}{g}$ ; in general, differentiate $\ln y$ and multiply through by $y$ .

General formula

{\htmlData{sym=0}{G}} = {\htmlData{sym=1}{N}}\cdot {\htmlData{sym=2}{q}} \;\Rightarrow\; \ln {\htmlData{sym=0}{G}} = \ln {\htmlData{sym=1}{N}} + \ln {\htmlData{sym=2}{q}} \;\Rightarrow\; {\htmlData{sym=3}{\frac{G'}{G}}} = \frac{{\htmlData{sym=1}{N}}'}{\htmlData{sym=1}{{N}}} + \frac{{\htmlData{sym=2}{q}}'}{\htmlData{sym=2}{{q}}}

$\frac{y'}{y}$ is a percentage rate of change. Logs turn multiplying into adding, so the percentage rates of factors add.

What the symbols mean

$G$
GDP
Total output of the economy.
$N$
Population
Number of people.
$q$
Output per person
Productivity.
$\frac{G'}{G}$
GDP growth rate
Fraction GDP grows per year.

Take the natural log of both sides: the product becomes a sum.
Differentiate: $\frac{d}{dt}\ln G = \frac{G'}{G}$ , the relative (percentage) growth rate.
So the growth rate of a product is the sum of the growth rates, and of a quotient the difference.
The same trick differentiates messy products and powers like $x^x$ that the basic rules can't handle directly.

Learn the math

Logarithmic DifferentiationCalc I · Lesson 26, Derivatives