### Understanding the Impact of Growth Rate Differences

When analyzing the effects of differences in economic growth rates over time, it is generally the case that seemingly small differences in annual growth rates result in large differences in the size of economies (usually measured by Gross Domestic Product, or GDP) over long time horizons. Therefore, it's helpful to have a rule of thumb that helps us quickly put growth rates into perspective.

One intuitively appealing summary statistic used to understand economic growth is the number of years it will take for the size of an economy to double. Fortunately, economists have a simple approximation for this time period, namely that the number of years it takes for an economy (or any other quantity, for that matter) to double in size is equal to 70 divided by the growth rate, in percent. This is illustrated by the formula above, and economists refer to this concept as the "rule of 70."

Some sources refer to the "rule of 69" or the "rule of 72," but these are just subtle variations on the rule of 70 concept and merely replace the numerical parameter in the formula above. The different parameters simply reflect different degrees of numerical precision and different assumptions regarding the frequency of compounding. (Specifically, 69 is the most precise parameter for continuous compounding but 70 is an easier number to calculate with, and 72 is a more accurate parameter for less frequent compounding and modest growth rates.)

### Using the Rule of 70

For example, if an economy grows at 1 percent per year, it will take 70/1=70 years for the size of that economy to double. If an economy grows at 2 percent per year, it will take 70/2=35 years for the size of that economy to double. If an economy grows at 7 percent per year, it will take 70/7=10 years for the size of that economy to double, and so on.

Looking at the preceding numbers, it is clear how small differences in growth rates can compound over time to result in significant differences. For example, consider two economies, one of which grows at 1 percent per year and the other of which grows at 2 percent per year. The first economy will double in size every 70 years, and the second economy will double in size every 35 years, so, after 70 years, the first economy will have doubled in size once and the second will have doubled in size twice. Therefore, after 70 years, the second economy will be twice as big as the first!

By the same logic, after 140 years, the first economy will have doubled in size twice and the second economy will have doubled in size four times- in other words, the second economy grows to 16 times its original size, whereas the first economy grows to four times its original size. Therefore, after 140 years, the seemingly small extra one percentage point in growth results in an economy that is four times as large.

### Deriving the Rule of 70

The rule of 70 is simply a result of the mathematics of compounding. Mathematically, an amount after t periods that grows at rate r per period is equal to the starting amount times the exponential of the growth rate r times the number of periods t. This is shown by the formula above. (Note that the amount is represented by Y, since Y is generally used to denote real GDP, which is typically used as the measure of the size of an economy.) To find out how long an amount will take to double, simply substitute in twice the starting amount for the ending amount and then solve for the number of periods t. This gives the relationship that the number of periods t is equal to 70 divided by the growth rate r expressed as a percentage (eg. 5 as opposed to 0.05 to represent 5 percent.)

### The Rule fo 70 Even Applies to Negative Growth

The rule of 70 can even be applied to scenarios where negative growth rates are present. In this context, the rule of 70 approximates the amount of time it will take for a quantity to be reduced by half rather than to double. For example, if a country's economy has a growth rate of -2% per year, after 70/2=35 years that economy will be half the size that it is now.

### The Rule of 70 Applies to More Than Just Economic Growth

This rule of 70 applies to more than just sizes of economies- in finance, for example, the rule of 70 can be used to calculate how long it will take for an investment to double. In biology, the rule of 70 can be used to determine how long it will take for the number of bacteria in a sample to double. The wide applicability of the rule of 70 makes it a simple yet powerful tool.