# In the limit as n the expression 1 1 n n approaches

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In the limit, as n , the expression (1 + 1/ n ) n approaches the value e 2.718281828, the base of the natural logarithms. If the growth rate g (expressed in percent/year) is an instantaneous growth rate with continuous compounding, then the discrete-period for- mula of Equation M.9.7, X t = X 0 (1 + g ) t , becomes X t = X 0 e gt . (M.9.9) This expression may look somewhat forbidding, but actually, it can often simplify our calculations in the analysis of growth rates. E XAMPLE M.9.7 In 2010, the Consumer Price Index (CPI) is at 120 in terms of 2006 base year prices (that is, the CPI for 2006 is 100). In 2025, the CPI is at 268 in terms of 2006 base year prices. What is the average annual inﬂation rate between 2010 and 2025? Calculate using the discrete-time and continuous time formulas. Discrete-Time Method: Setting 2010 as period 0, we have t = 15 years, so that P 15 = 268 = P 0 (1 + g P ) 15 = 120(1 + g P ) 15 . Hence, (1 + g P ) = (268/120)1/15 = 1.05502699, and so the average annual rate of inﬂation = 5.50%/year. Continuous-Time Method: Again with 2010 as period 0 and t = 15 years, we have P 15 = 268 = P 0 e g P t = 120 e g P (15) . This equation gives us (268/120) = e g P (15) , and taking M9-6 MATH MODULE 9: GROWTH RATES, INTEREST RATES, AND INFLATION: THE ECONOMICS OF TIME

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the natural logarithm of both sides and rearranging gives us g P = [ ln (268/120)]/15 = .053566, or 5.36%/year. [The difference in the results using the two methods results from the fact that with continuous compounding, a lower average annual growth rate will reach the same final value. (With only a bit of practice, the continuous-time method is faster than the discrete-time method.)] The continuous-time method has another computational advantage over the dis- crete-time method, in relating nominal and real interest rates. If compounding is instan- taneous, then no “discrepancy” arises of the sort that requires the correction in Equation 15.5 (page 489) of the text. Hence, for example, if the nominal rate of interest n is 6%/year and the rate of inﬂation q is 10%/year, then the real rate of interest i is exactly n q = 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 real income is growing at a rate of exactly 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! 2. Exercises 1. 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?
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