By 'growth rates' we mean annual growth rates. The preferred unit is percent, though for calculations you may have to use the actual number (i.e. the percent / 100). Two examples may help:

- Suppose GDP-per-capita in a country is $5000 and grows at 5% per year. What will GDP-per-capita be in 1 year? In 2 years? In 7 years?
- In 1 year, GDP per capita is simply $5000 + 0.05*$5000 = $5000*(1+0.05) = $5250
- So clearly in two years, GDP per capita is $5250*(1+.05) = $5000*(1+.05)*(1+.05) = $5512.50
- By analogy, for any number of years t, GDP per capita is $5000*(1+0.05)^t. For example, in 7 years GDP per capita will be $5000*(1+0.05)^7 = $7035.50

- In lecture 3, professor DeLong estimates that the world's population was 5 million in 8,000 BC and 750 million in 1800. What was the average annual population growth rate in this period?
- To solve this problem, note that the population growth rate x must solve 5*(1+x)^9800 = 750. (Make sure you understand why!)
- We can solve this equation: divide both sides by 5 to obtain (1+x)^9800 = 150. Then take each side to the power of 1/9800. This yields: 1+x = 150^(1/9800). At this point you'll need a calculator to find that 1+x = 150^(1/9800) = 1.0005. Thus x = 0.0005 or 0.05%.

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