This document discusses key math tools for analyzing variables, including absolute change and percent change. Absolute change is calculated as the difference between values of a variable at different points in time. Percent change captures the relative difference by expressing the change as a percentage of the initial value. This provides important context when comparing changes that have different starting points. The document provides examples comparing the absolute and percent changes in income for two individuals, showing that percent change properly captures that one change was much more significant despite the same absolute increase.

1. Basic math tools:
Variables, absolute change, and percent change
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September 5, 2011

2. Topics
Variables
Absolute change
Relative or percent change

3. Variables
In general, a variable is an aspect of the world that can vary or
have more than one value.
Examples: A name, the size of a shirt, the color of flowers, the age
of a person, the unemployment rate, the price of milk in a market,
the income of a household, the annual gross domestic product of
an economy. In our course, we will consider only variables that can
be measured and assigned numerical values or that – at the very
least – have values that can be ordered numerically.

4. Absolute change
If we have the value of a variable at different points in time, we
can determine the absolute change or, simply, the change in the
value of this variable.

5. Absolute change
Example: On 12/15/07, the height of a boy is 100 centimeters.
On 12/15/08, it is 118 centimeters. Calculate the change in this
boy’s height.
Let t = 12/15/08 and xt = 118. Then, t − 1 = 12/15/07 and
xt−1 = 100. We read it as, “The level of x at t − 1 is 100.”
Algebraically, the change in a variable x is given by:
∆xt = xt − xt−1
We can now substitute the values given in our example and do the
calculations mechanically:
∆xt = 118 − 100 = 18
The annual change in the boy’s height is 18 centimeters. Or: the
boy’s height increased in 18 centimeters from 12/15/07 to same
date in 2008.

6. Relative or percent change
To introduce the concept of relative or percentage change
(percent change for short), consider this example: On 1/1/07,
John was a senior college student working part time at McBurgers,
where his income during 2007 totaled $10,000. On 1/1/08, John
worked for an economic consulting firm as a junior analyst and his
annual income during 2007 amounted to $50,000.
Let xt be John’s annual income in 2007. The following calculates
the change in John’s annual income from the beginning of 2007 to
the beginning of 2008 (in thousands, dollar signs omitted):
xt = 50 − 10 = 40

7. Relative or percent change
Mary, who, on 1/1/07, was a firm’s lawyer on Wall Street, where
her 2006 income was $400,000. On 1/1/08, Mary still worked at
same firm and her 2007 annual income was $440,000. Let yt be
Mary’s 2007 income. Then, Mary’s annual income from the
beginning of 2007 to the beginning of 2008 (again, in thousands
and omitting the dollar signs):
yt = 440 − 400 = 40

8. Relative or percent change
The 2007 change in annual income for both John and Mary was
the same, $40,000. However, it would not feel right to say that
both John and Mary had a similar experience. John started from a
much lower income in 2006 and his $40,000 increase in income
represents a dramatic turnaround in his life. Mary was already
making a hefty income in 2006 and an additional income of
$40,000 does not alter her lifestyle significantly. If we only look at
the change in income without putting things in the context of their
initial or 2006 incomes, it would seem as if the same thing
happened to both of them. How do we capture the significantly
different experience that John and Mary had between 2006 and
2007?

9. Relative or percent change
We express their changes in annual income as percentages of their
initial (2006) income. The result is the relative or percent
change in their annual incomes. When, as in this case, the
percentage change is over time (rather than cross-sectional), it is
also called the growth rate. (A cross-sectional percent change
is, e.g., the difference in John’s and Mary’s income levels for a
given (the same) year expressed as a percentage of any one of
them.) A hat on top of a variable symbol (ˆ) will denote the
x
percentage change in the variable from one point in time to
another. Thus, for John:
xt xt − xt−1
xt =
ˆ −1=
xt−1 xt−1
40
xt =
ˆ = 4 = 400%
10
This reads as, “John’s annual income grew by 400 per cent” or
“John’s annual income quadrupled between 2006 and 2007.”

10. Relative or percent change
For Mary:
yt yt − yt−1
yt =
ˆ −1=
yt−1 yt−1
40
xt =
ˆ = .1 = 10%
400
This result reads as, “Mary’s annual income grew by 10%” or
“Mary’s annual income increased by one tenth over the year.”

11. Relative or percent change
Note that we could have put in perspective the change in John or
Mary’s annual income by dividing it over by the final (2007) annual
income level or by some average between the annual income levels
in 2006 and 2007, rather than by dividing it by the initial or 2006
annual income level. As long as we are consistent in the
denominator we use, the interpretation of the results should be
straightforward. Although there are some exceptions (e.g. the
calculation of mid point elasticities), most often, when economists
refer to percentage changes or growth rates, they refer to changes
in the variable of interest divided by the level of the variable at the
initial point.