Basic math tools:Variables, absolute change, and percent change

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September 5, 2011

Topics

Variables

Absolute change

Relative or percent change

Variables

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

Absolute change

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

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 thisboy’s height.Let t = 12/15/08 and xt = 118. Then, t − 1 = 12/15/07 andxt−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 thecalculations mechanically:

∆xt = 118 − 100 = 18

The annual change in the boy’s height is 18 centimeters. Or: theboy’s height increased in 18 centimeters from 12/15/07 to samedate in 2008.

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, Johnworked for an economic consulting firm as a junior analyst and hisannual income during 2007 amounted to $50,000.Let xt be John’s annual income in 2007. The following calculatesthe change in John’s annual income from the beginning of 2007 tothe beginning of 2008 (in thousands, dollar signs omitted):

xt = 50 − 10 = 40

Relative or percent change

Mary, who, on 1/1/07, was a firm’s lawyer on Wall Street, whereher 2006 income was $400,000. On 1/1/08, Mary still worked atsame firm and her 2007 annual income was $440,000. Let yt beMary’s 2007 income. Then, Mary’s annual income from thebeginning of 2007 to the beginning of 2008 (again, in thousandsand omitting the dollar signs):

yt = 440 − 400 = 40

Relative or percent change

The 2007 change in annual income for both John and Mary wasthe same, $40,000. However, it would not feel right to say thatboth John and Mary had a similar experience. John started from amuch lower income in 2006 and his $40,000 increase in incomerepresents a dramatic turnaround in his life. Mary was alreadymaking a hefty income in 2006 and an additional income of$40,000 does not alter her lifestyle significantly. If we only look atthe change in income without putting things in the context of theirinitial or 2006 incomes, it would seem as if the same thinghappened to both of them. How do we capture the significantlydifferent experience that John and Mary had between 2006 and2007?

Relative or percent change

We express their changes in annual income as percentages of theirinitial (2006) income. The result is the relative or percentchange in their annual incomes. When, as in this case, thepercentage change is over time (rather than cross-sectional), it isalso called the growth rate. (A cross-sectional percent changeis, e.g., the difference in John’s and Mary’s income levels for agiven (the same) year expressed as a percentage of any one ofthem.) A hat on top of a variable symbol (x) will denote thepercentage change in the variable from one point in time toanother. Thus, for John:

xt =xtxt−1

− 1 =xt − xt−1

xt−1

xt =40

10= 4 = 400%

This reads as, “John’s annual income grew by 400 per cent” or“John’s annual income quadrupled between 2006 and 2007.”

Relative or percent change

For Mary:

yt =ytyt−1

− 1 =yt − yt−1

yt−1

xt =40

400= .1 = 10%

This result reads as, “Mary’s annual income grew by 10%” or“Mary’s annual income increased by one tenth over the year.”

Relative or percent change

Note that we could have put in perspective the change in John orMary’s annual income by dividing it over by the final (2007) annualincome level or by some average between the annual income levelsin 2006 and 2007, rather than by dividing it by the initial or 2006annual income level. As long as we are consistent in thedenominator we use, the interpretation of the results should bestraightforward. Although there are some exceptions (e.g. thecalculation of mid point elasticities), most often, when economistsrefer to percentage changes or growth rates, they refer to changesin the variable of interest divided by the level of the variable at theinitial point.