Basic math tools:Variables, absolute change, and percent change firstname.lastname@example.org September 5, 2011
TopicsVariablesAbsolute changeRelative or percent change
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.
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.
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.
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
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
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?
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.”
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.”
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.