91 basic calculation of growth and interest rates we

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9.1 BASIC CALCULATION OF GROWTH AND INTEREST RATES We often want to compare the values of a particular variable at two separate times, or to study the changes in the value of a variable as a function of the length of time that has elapsed between two observations of the variable. One of the basic measures we use for such comparisons is the growth rate. The simplest formula for the growth rate gX of a variable X between period t and period ( t + 1) is the following: g X = X t + X 1 t X t = X X t / t t (M.9.1) E XAMPLE M.9.1: The capital value ( V ) of Unidyne Enterprises on January 1, 2010 is $20,000, and on January 1, 2011, it is $21,000. Using formula M.9.1, the growth rate of Unidyne’s capital value over this period is therefore g V = V t + V 1 t V t = = 0.05/year = 5%/year. In this case, since the capital value V is a stock variable (see Module 4), the growth rate g V measures the rate of change in the value of a stock variable over the peri- od of one year, measured as a proportion of its initial value. (21,000 – 20,000)/1 20,000 M9-2 MATH MODULE 9: GROWTH RATES, INTEREST RATES, AND INFLATION: THE ECONOMICS OF TIME
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E XAMPLE M.9.2: In 2010, Unidyne Enterprises’ sales (TR) were $100,000, and in 2011, they were $87,000. Using formula M.9.1, the growth rate of Unidyne’s sales over this period is therefore g TR = ( TR t +1 TR t )/TR t = = –0.13/year = –13%/year. Here, since sales are a flow variable (see Module 4), the growth rate g TR measures the rate of change in the value of a flow variable over the period of one year, mea- sured as a proportion of its initial value. Note that since sales fell in 2011, the growth rate is negative . We use the same basic formula to calculate the rate of inflation (the rate of growth of a price index) and the interest rate (the interest payment per period expressed as a per- centage of the principal sum advanced of a loan. Given any price index, for example, we can express the inflation rate (the growth rate of prices) as follows: g P = P t +1 P t P t = P P t / t t (M.9.2) E XAMPLE M.9.3: Between 2010 and 2014, the Consumer Price Index (CPI), which represents the cost of buying a representative bundle of consumption goods, rose by 10 points per year, as shown in the following table. What is the annual rate of inflation from year to year in this economy? Year 2010 2011 2012 2013 2014 Price Index 90 100 110 120 130 Applying the formula of equation M.9.2, the inflation rate between 2010 and 2011 is 11.11%/year, between 2010 and 2011 it is 10%/year, between 2010 and 2011 it is 9.091%/year, and between 2010 and 2011 it is 8.33%/year. The absolute increase in prices is constant from year to year: a bundle that cost $90 in 2010 would increase in cost by $10 per year in each subsequent year. The inflation rate , however, con- tinually declines , since in each year the base against which the price increase is measured is 10 points higher than in the previous year. Note that if the inflation rate is 10%/year, as it is from 2011 to 2012, then it takes $110 in 2012 to purchase the same bundle of goods that could be purchased for $100 in 2011. Hence each 2012 dollar can purchase only 100/110 = 0.9091 of what a 2011 dollar could purchase. Thus the purchasing power of a dollar has declined between 2011 and 2012. The extent of the decline in purchasing power is also given by our growth rate formula: (0.90909 – 1)/1 = –0.09091 = –9.091%/year. A
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