# 1.4 Economic Growth

So far, we know how to measure the standard of living, how to make meaningful comparisons between countries and within a country over time. But how do we measure changes in the living standard over time? Real GDP per capita growth is the best measure of changes in economic well-being. The simplest way to measure economic growth is by taking the percentage change in real per capita GDP between two consecutive periods. If we denote real GDP per capita in year t as Yt, then the growth in real per capita GDP, g, between any two years, t and t – 1, expressed as a decimal is given by:

 ${g}_{t}=\frac{{Y}_{t}–{Y}_{t–1}}{{Y}_{t–1}}$ (3)

From Equation (3), we can express Yt in terms of Yt-1 and the growth rate:

 ${Y}_{t}=\left(1+{g}_{t}\right){Y}_{t–1}$ (4)

Take the natural logarithm (the logarithm to the base e), ln, of both sides of equation (4):

 $\mathrm{ln}\left({Y}_{t}\right)=\mathrm{ln}\left[\left(1+{g}_{t}\right){Y}_{t–1}\right]$ (5)

A useful Maclaurin expansion-based approximation is that for any small positive number, a,

 $\mathrm{ln}\left(1+a\right)\approx a$ (6)

Three useful properties of logarithms are the product rule, Equation (7), quotient rule, Equation (8), and power rule, Equation (9):

 $\mathrm{ln}\left(ab\right)=\mathrm{ln}\left(a\right)+\mathrm{ln}\left(b\right)$ (7)
 $\mathrm{ln}\left(\frac{a}{b}\right)=\mathrm{ln}\left(a\right)–\mathrm{ln}\left(b\right)$ (8)
 $\mathrm{ln}\left({a}^{b}\right)=b\mathrm{ln}\left(a\right)$ (9)

where b is a positive number. Applying first Equation (7) and then Equation (6) to Equation (5) yields:

 $\mathrm{ln}\left({Y}_{t}\right)=\mathrm{ln}\left(1+{g}_{t}\right)+\mathrm{ln}\left({Y}_{t–1}\right)$ $\mathrm{ln}\left({Y}_{t}\right)\approx {g}_{t}+\mathrm{ln}\left({Y}_{t–1}\right)$ ${g}_{t}\approx \mathrm{ln}\left({Y}_{t}\right)–\mathrm{ln}\left({Y}_{t–1}\right)$ (10)

Equation (10) is a useful approximation for computing growth rates that is commonly used by economists.

Example 6: Computing Real GDP per Capita Growth

Table 3 below shows Canada’s real per capita GDP which we computed in Example 4.  Compute the annual growth rate in real GDP per capita.

We use Equation (3) to compute the annual growth rate in Canada’s GDP in the third column of Table 3. We leave it as an exercise to you to compute the growth rate using the approximation in Equation (10). Notice the large variation in the growth rate of the Canadian economy from one year to another; these are transient cyclical changes. While business cycle fluctuations are a very important macroeconomic concept, our focus here is not on the short run but the long run. It is trends that drive the standard of living over time and differences in long-run growth rates determine differences in standards of living across countries.

 Year Real GDP per capita Real GDP per capita growth 2010 \$51,196 2011 52,449 $=\frac{52,449.46–51,195.79}{51,195.79}$ = 0.0245 or 2.45% 2012 52,636 0.35% 2013 53,738 2.09% 2014 54,707 1.80% 2015 53,606 -2.01% 2016 53,170 -0.81% 2017 54,692 2.86% 2018 55,034 0.63% 2019 54,981 -0.10% 2020 51,524 -6.29%