Make sure you understand the power of exponential growth (and of rapid exponenti

Question

Make sure you understand the power of exponential growth (and of rapid exponential inflation) by doing the following exercises: How quickly will a country growing at 10% a year double its income? Quadruple its income? What about a country growing at 5% per year? Suppose a country's per capita income is currently growing at 5% per year. Then it shaves an additional percentage point off its population growth rate for the next twenty years, but overall income continues to grow at the same rate. How much richer would the country be at the end of twenty years (per capita)? Suppose that Brazil experiences inflation at 30% per month. How much is this per year? Do your calculations first without compounding (the answer then is obviously 360%). Now do it properly by compounding the inflation rate.

Explanation / Answer

a. Let the no. of years for doubling the income be n

then, (1.10)n = 2

Taking log on both sides,

n = log 2/log 1.10 = 7.27 years

Time to quadruple would be double this = 7.27 * 2 = 14.54 years

if growth rate is 5%
Then doubling time: (1.05)n = 2
n = log 2/log 1.05 = 14.20 years
Quadrupling time = 2*14.20 = 28.40 years

b. Let the population growth rate be 2% (the peak was around 2.2% and is now closer to 1%, so 2% is a reasonable estimate)

per capita growth at 5% economic growth and 2% pop growth = economic growth/population growth = (1.05/1.02)^20 = 1.7856

per capita growth at 5% economic growth and 2% pop growth = (1.05/1.01)^20 = 2.1745

Increase = (2.1745 - 1.7856) /1.7856 = 21.78%

c. without compounding, annual rate = 12 *30% = 360%

with compounding, annual rate = (1.30)^12 - 1 = 22.2981 or 2229.81%

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