Principles of Macroeconomics / Unit 4 MAC 0304 worked example, 60 to 90 seconds
Compound Growth
The idea
To find a value after several periods of steady growth, multiply the starting value by one plus the growth rate, raised to the number of periods. Write the rate as a decimal, so three percent is zero point zero three and the factor is one point zero three. Raise that factor to the number of years, then multiply by the starting value. The same formula runs backwards: dividing an ending value by the starting value and taking the appropriate root gives the average growth rate. Compounding is why the answer is always larger than the growth rate times the number of years.
In practice
Worked example: income of one hundred growing at three percent for twenty-five years. One point zero three raised to the twenty-fifth power is about two point zero nine, so income reaches about two hundred and nine. Simple multiplication would have predicted seventy-five percent growth and an answer of one hundred and seventy-five. The extra thirty-four is compounding.
Where people slip
Use real growth rates, not nominal ones, or the answer measures inflation as well as output. And the same formula describes compound interest, so a sum growing at a rate and an economy growing at a rate are the same arithmetic.
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