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17.21 Percentage Changes and Growth Rates

If some variable *x* (say, the number of gallons of gasoline sold in a week) changes from *x*_{1} to *x*_{2}, then we can simply define the change in that variable as Δ*x* = *x*_{2} − *x*_{1}. But there are problems with this simple definition. The number that we calculate will change depending on the units in which we measure *x*. If we measure in millions of gallons, *x* will be a much smaller number than if we measure in gallons. If we measured *x* in liters rather than gallons (as it is indeed measured in most countries), it would be a bigger number. So the number we calculate depends on the units we choose. To avoid these problems, we look at percentage changes and express the change as a fraction of the individual value. In what follows, we use the notation %Δ*x* to mean the percentage change in *x*, and we define it as follows: %Δ*x* = (*x*_{2} − *x*_{1})/*x*_{1}. A percentage change equal to 0.1 means that gasoline consumption increased by 10 percent. Why? Because 10 percent means 10 “per hundred,” so 10 percent 10/100 = 0.1.

Very often in economics, we are interested in changes that take place over time. Thus we might want to compare gross domestic product (a measure of how much our economy has produced) between 2012 and 2013. Suppose we know that gross domestic product in the United States in 2012 was $14 trillion and that gross domestic product in 2013 was $14.7 trillion. Using the letter *Y* to denote gross domestic product measured in trillions, we write: *Y*_{2012} = 14.0 and *Y*_{2013} = 14.7. If we want to talk about gross domestic product at different points in time without specifying a particular year, we use the notation *Y*_{t}. We express the change in a variable over time in the form of a growth rate, which is just an example of a percentage change. Thus the growth rate of gross domestic product in 2013 is calculated as

%Δ *Y*_{2013} = (*Y*_{2013} − *Y*_{2012})/ *Y*_{20126} = (14.7 − 14)/14 = 0.05.
The growth rate equals 5 percent. In general, we write %Δ *Y*_{t}_{+1} = (*Y*_{t}_{+1} − *Y*_{t})/ *Y*_{t}.

## The Main Uses of This Tool