Two Arithmetic Tricks for Working with Percentage Changes

For manipulating many relationships in economics, there is an arithmetic trick that is useful to know: The percentage changes of a product of two variables is approximately the sum of the percentage changes in each of the variables.

To see how this trick works, consider an example. Let P denote the GDP deflator and Y denote real GDP. Nominal GDP is P × Y. The trick states that

Percentage Change in (P × Y)	approx. =	(Percentage Change in P)
		+ (Percentage Change in Y).

For instance, suppose that in one year, real GDP is 100 and the GDP deflator is 2; the next year, real GDP is 103 and the GDP deflator is 2.1. We can calculate that real GDP rose by 3 percent and that the GDP deflator rose by 5 percent. Nominal GDP rose from 200 the first year to 216.3 the second year, an increase of 8.15 percent. Notice that the growth in nominal GDP (8.15 percent) is approximately the sum of the growth in the GDP deflator (5 percent) and the growth in real GDP (3 percent).

A second arithmetic trick follows as a corollary to the first: The percentage change of a ratio is approximately the percentage change in the numerator minus the percentage change in the denominator. Again, consider an example. Let Y denote GDP and L denote the population, so that Y/L is GDP per person. The second trick states

Percentage Change in (Y/L)	approx. =	(Percentage Change in Y)
		– (Percentage Change in L).

For instance, suppose that in the first year, Y is 100,000 and L is 100, so Y/L is 1,000; in the second year, Y is 110,000 and L is 103, so Y/L is 1,068. Notice that the growth in GDP per person (6.8 percent) is approximately the growth in income (10 percent) minus the growth in population (3 percent).

Source: N. Gregory Mankiw, Macroeconomics 3rd edition, Worth Publishers, 1997, p. 26.