# 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.