= 0.06 – 0.10 = –4%/year. Similarly, if your nominal income grows at an instanta-
neously compounded annual growth rate of 8%/year and all prices are rising at
3%/year, then your
income is growing at a rate of
0.08 – 0.03 = 5%/year. And
so on. Economists use such exponential functions not to make their work harder, but to
make it easier!
Left-handed Louie, your friendly neighbourhood loan shark, has agreed to advance
you $1000. You can choose (a) his regular interest rate of 2%/day on the outstand-
ing balance (calculated daily), or (b) his special interest rate of 10%/week on the out-
standing balance (calculated weekly). If you are unable to pay for 52 weeks (364
days), and Louie simply adds your interest to the outstanding balance, then how
much do you owe him at the end of this time, under each of the plans?
2. In 2010, Unidyne Enterprises’ annual sales hit $100,000,000, and they are projected
to grow at a rate of 25%/year for the foreseeable future. When do you expect
Unidyne’s sales to reach $300,000,000 per year?
3. Between 2010 and 2011, the Consumer Price Index rose from 250 to 275 in Leutonia,
while in Granfaloonia, it rose over the same period from 88 to 97. In which country
the rate of inﬂation higher?
4. Calculate the present discounted value of $100 paid exactly 5 years from today,
when the interest rate
is: (a) 0%/year; (b) 5%/year; (c) 10%/year; (d) 20%/year; (e)
50%/year; (f) 100%/year. Then repeat your calculations if the $100 is paid exactly 10
years from today.
5. Suppose that your nominal income grows at a constant, instantaneously com-
rate of 5%/year, while the Consumer Price Index (CPI) grows at a con-
instantaneously compounded rate of 3%/year. In 2010, your nominal income
$100,000. How fast is your income rising in real terms? In terms of 2010 dollars of
constant purchasing power, what will your real income be in 2015?
6. You are offered an opportunity to receive three payments: $8000 one year from
today, another $8000 two years from today, and a third $8000 three years from today.
If these payments are guaranteed and risk-free, what is the most you should pay for
the opportunity if the rate of interest is 5%/year? if the rate of interest is 15%/year?
MATH MODULE 9: GROWTH RATES, INTEREST RATES, AND IN±LATION: THE ECONOMICS O± TIME