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
M92
MATH MODULE 9: GROWTH RATES, INTEREST RATES, AND INFLATION: THE ECONOMICS OF TIME
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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
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 Fall '12
 Danvo
 Inflation, Interest Rates, Interest, present discounted value

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