Let's say a report comes out that the growth of the Gross Domestic Product was 4.8% last quarter. Technically, quarterly reports give the annual rate of growth, what would happen if the same growth rate was seen for four consecutive quarters. How do we find the quarter to quarter growth. The simplest answer would be to divide, 4.8%/4 = 1.2%. This is actually a little high.
1.2% quarterly growth means you take what you had and multiply by 101.2%, which is to say the 100% is what you had last quarter and the 1.2% is the amount of growth this quarter. Instead of adding 1.2% every quarter, the correct method would be to raise 101.2% to the fourth power.
1.012 × 1.012 × 1.012 × 1.012 = 1.0488709...
Rounded to the nearest tenth of a percent, this would be 4.9%. It's a small difference, but it is noticeable in the long run. The standard name for this is compound interest.
The correct method to find the quarter to quarter growth rate is to take the fourth root of 1.048, which is 1.011789855... The difference here is small and if we round to the nearest tenth of a percent, we would still get 1.2%. We have to round to one more place to see the rate is closer to 1.18%.
The differences get more noticeable as we use more numbers. Let's look at GDP data from two countries that were hit hard by last decade's financial crisis.
Here are Ireland's quarterly GDP numbers once things start going bad in 2008.
-0.4%, -0.9%, 0.4%, -7.4%, -6.2%, -4.8%, -6.3%, -4.5%, -2.7%, -1.4%, 1.0%, 0.2%, -0.9%, 2.9%, 1%, 2.8%, 2.7%, 0.2%, 0.9%, 0%
Long lists of numbers like this are hard to read and understand and even the chart doesn't tell the whole story. If we add the numbers up and take the average, the Irish economy contracted about 1.17% a year for five years. The correct method is to add 100% to each of these numbers take the fourth root, multiply those together and take the fifth root, since we are looking the five years from the start of 2008 to the beginning of 2013. Using this method, the annual contraction is about 1.22%.
4.8%, 2%, -0.8%, -0.7%, -5.9%, -6.1%, -6.5%, -8.6%, -6.6%, -6.2%, -3.2%, -0.1%, 3.9%, 2.1%, 3.5%, 2.1%, 4.2%, -1.2%, 2.2%, 1.4%
The numbers in Iceland show contraction as well. Done the incorrect way of taking averages, Iceland's economy has contracted at a rate of 0.985%. Doing the geometric mean instead, the annual contraction is 1.07%.
Notice that the correct contraction rates are bigger using geometric means, but a larger contraction actually means a smaller number. This is in line with yesterday's post that geometric means will be less than arithmetic means. In Ireland, it's 98.83% using average and 98.78% using geometric mean. In Iceland, it's 99.015% using average and 98.93% using geometric mean.
Paul Krugman uses Iceland and Ireland as examples to prove austerity doesn't work. After five years, we see only small differences between GDP contraction, so neither can be seen as a miracle or a disaster by this metric.
The major difference is in the unemployment rate. Before things fell apart, the Irish unemployment rate was 4.7%. It has climbed to about 15% and has changed very little. The last data I could find reported in a quick search was 14.2% last November.
Iceland unemployment rate was a much lower 2.4% when the crash hit and climbed to 7.9% at its worst in February 2011. Since then it has fallen and is reported at 5.3% as of last November.
Ireland chose austerity and the people are still paying for it. Iceland forgave a lot but not all of the debt incurred by people who took on mortgages they could not afford. More than that, bank officials and corrupt politicians were aggressively prosecuted in Iceland, several of them now languishing in jail. The Icelandic economy is not all the way back to normal, but it is showing signs of improvement for working people that the Irish method cannot yet show.
Note: when taking averages of unemployment rates, the arithmetic mean is perfectly acceptable. There is no need to factor in any compound interest in these rates.
