# Rule of 70

Rule of 70 is a short-cut method of an economy’s growth accounting which tells us that if an economy’s annual growth rate is g, its output/GDP will double in 70/g years.

For example, if an economy grows by 2.3% constantly, rule of 70 tells us that its total production will double in 70/2.3 years i.e. in 30.43 years.

The formula for rule of 70 can be written as follows:

$$ \text{t}=\frac{\text{70}}{\text{g}} $$

Where t is the time it takes the economy to double and g is the constant percentage growth expected in future.

## Derivation

If an economy grows at a constant rate annual rate g, the value of its gross domestic product (GDP) after t years is given by the following formula:

$$ \ {\rm \text{GDP}} _ \text{t}={\rm \text{GDP}} _ \text{0}\times{(\text{1}+\text{g})}^\text{t} $$

Now, let’s divide both sides by GDP0:

$$ \frac{{\text{GDP}} _ \text{t}}{{\text{GDP}} _ \text{0}}={(\text{1}+\text{g})}^\text{t} $$

If an economy doubles over a period, the rate of GDPt to GDP0 would be 2. Substitute this in the equation:

$$ \text{2}={(\text{1}+\text{g})}^\text{t} $$

Let’s take natural log of both sides of the equation:

$$ \ln{\text{2}}=\ln{{(\text{1}+\text{g})}^\text{t}} $$

Natural log of 2 roughly equals 0.70

$$ \text{0.7}=\text{t}\times\ln{(\text{1}+\text{g})} $$

Since ln (1+g) is equal to g

$$ \text{0.7}=\text{t}\times \text{g} $$

We need to multiply and divide by 100 so as to bring the expression in percentages.

$$ \text{t}=\frac{\text{0.7}}{\text{g}}\times\frac{\text{100}}{\text{100}}=\frac{\text{70}}{\text{g}} $$

## Example

US GDP in 2017 was $18.09 trillion (2016: $17.66 trillion) representing annual growth rate of 2.43%. Using rule of 70, we estimate that if the US economy continues to grow at 2.43%, it will double in 28.80 years.

$$ \text{t}=\frac{\text{70}}{\text{2.43}}=\text{28.80} $$

Now, let’s find out how accurate rule of 70 is by finding the project value of US GDP in 28.80 year using the formula for compound annual growth rate:

$$ {\rm \text{GDP}} _ \text{t}={\rm \text{GDP}} _ \text{0}\times{(\text{1}+\text{g})}^\text{t}\\=\text{\$18.09 trillion}\times{(\text{1}+\text{2.43%})}^{\text{28.80}}=\text{\$36.12 trillion} $$

Since the double of $18.09 trillion is $36.18 trillion, our estimate of $36.12 trillion is very close.

Rule of 70 can be applied regardless of the absolute value of gross domestic project. It means that the rule can be applied to any economy.

by Obaidullah Jan, ACA, CFA and last modified on

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