Productivity growth of 1% or 2% a year sounds small. But growth compounds, like the exponential growth you have seen in algebra: each year's growth builds on the last.
Key idea
Rule of 70: a quantity growing at $g$ percent a year doubles in about $$\dfrac{70}{g} \text{ years.}$$ So 2% growth doubles income in about 35 years; 7% doubles it in about 10. The exact doubling time is $\dfrac{\ln 2}{\ln(1 + g/100)}$, very close for small rates.
Why it works
Growing at rate $r$ (as a decimal) for $t$ years multiplies a quantity by $(1 + r)^t$. Doubling means $(1 + r)^t = 2$, so $t = \dfrac{\ln 2}{\ln(1 + r)}$. For small $r$, $\ln(1 + r) \approx r$, and $\ln 2 \approx 0.693$, so $t \approx \dfrac{0.69}{r} = \dfrac{69}{g}$ with $g$ in percent. Rounding to 70 makes the arithmetic easy. Adding a fixed amount each year instead would badly understate the result: growth is a percent of an ever larger base.
Worked examples
Worked example: Doubling time
Real GDP per person grows 2.5% a year. About how long until it doubles?
- $\dfrac{70}{2.5} = 28$ years.
- Exactly: $\dfrac{\ln 2}{\ln 1.025} \approx 28.1$ years.
Answer: About 28 years
Worked example: Small differences, big gaps
Two countries start equal. A grows 1% a year and B grows 2%. After 70 years, how do they compare?
- A doubles every 70 years: one doubling, a factor of 2.
- B doubles every 35 years: two doublings, a factor of 4.
- B ends up about twice as rich as A, from a 1-point difference in growth.
Answer: B is about twice as rich
Worked example: How many times larger?
An economy grows 7% a year. About how many times larger is it after 30 years?
- Doubling time $\approx \dfrac{70}{7} = 10$ years, so 30 years is 3 doublings.
- $2^3 = 8$ times larger (exactly $1.07^{30} \approx 7.6$).
Answer: About 8 times
Common mistake
Adding growth year by year instead of compounding it, which badly underestimates how small differences in growth rates add up over decades. 10% a year for 20 years is not "200% more" but about 6.7 times the start. Check: use the rule of 70 and count doublings.
Try the compounding calculator on the Calculators & Links page with different growth rates and see how the doubling time changes.