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# When the interest rate is lower the future earnings

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When the interest rate is lower, the future earnings are discounted by a smaller factor, and hence their present discounted value is higher. 9.3 DISCRETE-TIME AND CONTINUOUS-TIME GROWTH RATES As Equation M.9.5 suggests, if a variable X grows from an initial value X 0 , at a constant growth rate g (in percent/year), compounded annually , its value at time t (in years) is given by the familiar “compound interest” formula, X t = X 0 (1 + g ) t . (M.9.8) Suppose that we set g =100%/year, and t =1 year. Then we have X t = X 1 = X 0 (1 + 1) 1 = 2 X 0 . That is, after one year, the value of X will double. Suppose now instead that we compound semi -annually, at the same annual rate. In this case, we halve the growth rate per subperiod, to 50% per 6 months, but now we compound twice per year. At the end of the year, X 1 = X 0 (1 + 1/2) 2 = 2.25 X 0 . More fre- quent compounding at the same annual rate increases the value of X at the end of the year. If we compound quarterly at the same annual rate, we have X 1 = X 0 (1 + 1/4) 4 = 2.44140625 X 0 . As we shorten the length of the subperiods and correspondingly increase the number of subperiods per year, we are moving towards continuous compounding. 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 Fnal 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-
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