Unit 9 Uneven development on a global scale

9.2 Measuring economic growth: Ratio scales and growth rates

To explore and compare how economies grow, we first need to understand how economic growth is measured.

The growth rate (of GDP per capita, or any other quantity) means the percentage change. Section 3.5 introduced the formula to calculate percentage changes:

\[\text{growth rate of GDP per capita} = \frac{\text{change in GDP per capita}}{\text{original level of GDP per capita}} \times 100\]

hockey stick growth

For most of history, living standards changed little from year to year. In the last two centuries, many countries have shifted to a pattern in which incomes (as measured by GDP) tend to increase year on year, at an average rate of 2% or more. This is known as hockey stick growth because the shape of the graph of GDP over time resembles an ice hockey stick: almost flat for a long period, then a sudden kink upwards. We see hockey stick growth in other variables too: for example, atmospheric carbon dioxide. See also: GDP.

Let’s apply this calculation to South Korea, China, and India. Figure 9.3 shows how these countries have grown. The top panel shows that they were all very poor countries in 1960, but in subsequent decades, they all experienced the characteristic hockey stick growth first observed for Great Britain in the Industrial Revolution.

China’s GDP per capita was $1,057 in 1960 and $9,658.42 in 2010, so its growth rate over that period is:

\[\begin{align*} \text{growth rate of GDP per capita} &= \frac{\text{change in GDP per capita}}{\text{original level of GDP per capita}} \\ &= \frac{9,658.42 - 1,057}{1,057} \times 100 \\ &= 814 \% \end{align*}\]

Applying the same calculation to India (GDP per capita of $1,200 in 1960 to $4,525.75 in 2010) and South Korea ($1,547.69 in 1960 to $31,537.77 in 2010), we obtain growth rates of 277% and 1,938%, respectively.

ratio scale

Graphs are usually plotted using linear scales: the points marked on the axis are a fixed distance apart, and as we move along the axis from one point to the next, the corresponding variable increases by a constant amount. If instead we use a ratio scale, moving from one point to the next represents a constant proportional increase. For example, if a ratio scale with a factor of 2 starts with a value of 50 at the origin, the next points shown along the axis would be at values 100, 200, 400, 800… Ratio scales are useful for variables like aggregate output that tend to change in a proportional way. If output grows at a constant rate (e.g. 3% per annum) and is plotted against time using a ratio scale on the vertical axis, the graph will be a straight line.

These calculations give the growth rate over an entire period, but each country was not growing at the same rate every year. This is shown in the bottom panel of Figure 9.3. To make it easier to compare growth rates over time, we use a ratio scale: the values on the vertical axis increase by a constant factor rather than a constant amount. The top panel of Figure 9.3 plots GDP per capita using a linear scale: the vertical axis values increase by a constant amount ($5,000). The bottom panel plots the same GDP per capita data but using a ratio scale: the vertical axis values increase by a factor of 2 ($500, $1,000, $2,000, and so on).

On a ratio scale, equal vertical distances represent equal percentage changes, or equal ratios. So, with a ratio scale, a straight line means a constant growth rate.

In Figure 9.3, the ratio scale with a factor of 2 helps us visualize how many times GDP per capita has doubled. If GDP per capita doubles every decade, the line would be straight, sloping upwards. If, instead of doubling, the level quadrupled every decade (the growth rate was twice as high), the line would still be straight, but it would be twice as steep. So, with a ratio scale, a steeper line means a faster growth rate.

Comparing the two panels in Figure 9.3 shows how the ratio scale is useful in enabling us to read off directly the rate of growth from the slope of the line. With a linear scale, the upward-curving lines (top panel) make it difficult to visually judge the growth rates. In South Korea, there were four decades of rapid growth (before a slowing down from the year 2000). In China, the acceleration of growth from the beginning of the reforms at the end of the 1970s is evident, growth paused towards the end of the 1990s and resumed at its previous rapid pace coinciding with China’s entry to the World Trade Organization (WTO) in 2001. Growth has slowed more recently. India’s reform period began in 1991, about a decade after China’s, and the growth rate increased. Since the 2000s, its growth rate has been faster. The initial difference in incomes between the three countries is also hardly distinguishable in the upper chart, but using the ratio scale in the lower chart reveals that incomes in 1960 in South Korea at around $1,500 per year were 30% higher than in India and 45% higher than in China.

Compound annual growth rates

Sometimes we want a single number to summarize the average growth rate over several years. Averaging the annual growth rates can be misleading, so instead, we use the compound annual growth rate (CAGR). To calculate the CAGR, we do not take averages, but instead use the principle of compounding. We usually calculate compound annual growth rates over long periods, such as decades.

The CAGR is the constant annual percentage growth rate that would be required for a variable to grow from its starting value to its ending value over the specified period:

\[\text{CAGR (in %)} = \left[\left(\frac{\text{ending value}}{\text{starting value}}\right)^{\frac{1}{\text{number of years}}} - 1\right] \times 100\]

Using the GDP per capita values for China in 1960 and 2010 as an example:

\[\text{CAGR (in %)} = \left[\left(\frac{\text{9,658.42}}{\text{1,057}}\right)^{\frac{1}{\text{50}}} - 1\right] \times 100 = 4.52\%\]

CAGR gives us a smoothed average growth rate, which is useful for comparing overall performance across different periods or countries. It is conceptually similar to finding the average slope of the line connecting the start and end points on a graph using a ratio scale.

The CAGR also helps us recognize if a country’s economy has grown faster or slower than usual. For example, if China’s real GDP per capita grew by 6% this year, but on average it grew by 4.52% between 1960 and 2010, then we can say that the economy’s growth performance this year is above the long-term average.

Compounding is important: failing to account for it would give vastly different growth rate figures. Using China as an example, if we took the average of the growth rate over the same period (calculated as 814% above), we would get:

\[\text{growth rate of GDP per capita} = \frac{1}{50}(814) = 16.28\%\]

In this example, failing to account for compounding gives an answer that is almost four times as large as the actual annual growth rate.

The rule of 70 for growth rates

Another way to interpret growth rates is using the rule of 70: if the economy is growing at a constant rate (a straight line on a ratio scale), the number of years it will take for real GDP to double is approximately 70 divided by the annual growth rate in per cent (for example, with a constant growth rate of 2% per year, the level will double in 35 years, that is, in a generation).

The rule of 70 works well for relatively low, stable growth rates. However, it can be misleading when:

• Growth rates are high: The rule becomes less accurate at higher growth rates. It will tend to overestimate the true doubling time calculated from the CAGR formula or direct calculation.
• Growth rates are volatile or changing: The rule assumes a constant growth rate. If the growth rate fluctuates wildly or consistently declines or increases over the period, applying the rule using just the initial or average rate can give a very poor estimate of the actual time it took (or will take) to double.
• It is used over very short periods: The rule of 70 was intended as a rule of thumb for long periods such as decades, rather than a few years.

For example, the rule of 70 would not be accurate for the countries in Figure 9.3 over the period shown, as they were all growing rapidly and the growth rates were changing. Using China’s compound annual growth rate, we would estimate that it takes 70/4.52 = 15.49 years for its GDP per capita in 1960 to double, when in fact it took 22 years ($2,128 in 1982) and then 13 years to double again ($3,999.73 in 1995).