Stephen Gordon recently posted an excellent analysis of trends in income inequality in Canada and elsewhere. Stephen, like almost all of the other authors cited in his post and the subsequent discussion, measured inequality using the Gini coefficient.
Talk about deja vu all over again. Various limitations of the Gini inequality index have been known for years. Tony Atkinson described some and proposed an alternative to back in 1970; other indices for measuring inequality are the Theil index, and the Hoover index. Greselin and co-authors set out new arguments, and make a convincing case for replacing the Gini. But I don't expect to see the Zenga index in wide use any time soon.
We keep on using the Gini because that's the way people have always done it. But why did people even start using the Gini in the first place?
In statistics, the standard deviation is usually used to measure how spread out or "unequal" a distribution is. If a "unit free" measure is desired - one that does not depend upon whether income is measured in Euros or dollars - the coefficient of variation, that is, the standard deviation divided by the mean, or some similar measure can be used.
So why, when people started measuring income inequality, didn't they just use the the standard deviation or the coefficient of variation? Why did people even begin using the Gini?
Max Lorenz can assume part of the blame.
In 1905, as a 28-year-old PhD student, Lorenz published one of the first English-language investigations into income inequality measurement. (A topic which had nothing to do with his PhD thesis, which was on the Economic Theory of Railroad Rates.)
Lorenz begins by considering and rejecting the numerical methods that were being used by American economists at the time to measure inequality. He also appears to reject the frequency distribution approach to graphing income distributions:
Turning now to the graphic measures, a simple plotting of wealth along one axis and the numbers of the population along another is not satisfactory for the reason that changes in the shape of the curve will not show accurately changes in the relationships of individuals.
Instead, he advocated representing income distribution through what is now known as a Lorenz curve - like this:
The horizontal axis shows the cumulative percentage of income, the vertical the cumulative percentage of wealth. So the diagram shows, for example, that in Prussia in 1901, the poorest 60 percent of the population had 32 percent of the income.
Today Lorenz curves are generally drawn the other way around, like this one (source):
This picture shows how inequality in receipt of health care spending decreases as people get older. At age 65, 80 percent of the population have received almost no medicare spending; almost all spending is on a small proportion of high need patients. Twenty years later, however, there is much less inequality - the cheapest 80 percent receive, over their lifetimes, about half of Medicare spending.
Lorenz did not advocate any numeric measure of inequality. His point was the value of graphic representations. Indeed, there are good reasons to prefer pictures to numerical measures. If Lorenz curves do not cross, as in the Medicare spending diagram above, just about any measure will show inequality decreasing over time, so in some sense it does not matter which measure you use. But what if the Lorenz curves do cross, as in this picture:
Compare, in this example, UK 1913 with Brazil 1980. The bottom 80 percent of the population had a larger income share in 1913 UK than in 1980 Brazil. But the 8th decile did better in 1980 Brazil than in 1913 UK - in 1913 income was more concentrated among the wealthiest of the wealthy. So which distribution is less unequal? Which distribution is, in a sense, better? It depends upon how one weights the interests of different groups in society. With a sufficiently high weight on the upper middle class, the 1980 Brazil distribution could conceivably be preferred to the 1913 UK distribution.
It was many years before people recognized this fact. Instead, they continued searching for the Holy Grail of income inequality, a single numerical measure that perfectly captures the shape of the income distribution.
For example, Warren Persons, in a 1909 Quarterly Journal of Economics article, drawing inspiration from the latest research in biology, concluded, "The coefficient of variability is recommended as the most satisfactory measure of variability."
Around the time that Lorenz and Persons were writing, Italian statististicians and sociologists such as Corrado Gini were independently developing methods of inequality measurement. (Savour for a moment, if you wish, the irony of the most widely used measure of inequality being named after the author of The Scientific Basis of Fascism)
Hugh Dalton, in 1920 brought these two strands of inequality measurement together, introducing English readers to Italian research in a highly influential article. He concluded that both the coefficient of variation and "Professor Gini's mean difference" were satisfactory measures of inequality. But, he argued,
If a single measure is to be used, the relative mean difference [Gini coefficient] is, perhaps, slightly preferable, owing to the graphical convenience of the Lorenz curve.
What does that mean? It turns out that the Gini coefficient is equal to two times the area between the Lorenz curve and the line of complete equality. Just look at the Lorenz curve diagrams above - you can see the Gini coefficient.
Simple. Intuitive. Because people were used to visualizing inequality with Lorenz curves, and not with frequency distributions. But the Gini coefficient is seriously defective as a measure of inequality, for two reasons.
Reason 1. Suppose society is composed of two groups, reds and greens. One group is more privileged than the other. A reasonable question to ask is: how much of the inequality in society is attributable to differences between these two groups, and how much inequality is attributable to differences within these groups.
The Gini coefficient cannot answer this question, because it is not easily decomposed into between group and within group inequality. Other measures can be, and that's one reason to use the Theil or another index in preference to the Gini.
Reason 2. Suppose that there is diminishing marginal utility of income - that is, people get more satisfaction from their 100th dollar than their 100,000th. In this case, the best way to get the maximum possible amount of happiness from a given amount of resources is to distribute them as equally as possible. Just like serving a blueberry pie to a group of friends and family. Sure, some might end up with a bit less because they're dieting or a bit more because they're growing. But basically happiness is maximized by sharing the pie, not letting one or two people to hog it all.
In othe words, the reason we care about inequality is that it reduces the happiness achievable from a given amount of income. How much depends upon the happiness/income relationship. Does the marginal utility of income fall rapidly? Or is the happiness from the 100,000th dollar almost as great as the happiness from the 100th?
The Atkinson index is a measure of inequality can be adjusted to take into account society's attitudes towards inequality - placing either more or less emphasis on the extremes of the distribution, depending upon the "inequality aversion parameter" chosen. Because the Atkinson measure makes explicit the welfare judgements underlying inequality measurement, it is preferable to the Gini.
I could go on. The Zenga measure, for example, is preferable to the Gini because it can be estimated with a smaller margin of error.
So why do we use the Gini? I suspect it's the same as the reason why we use QWERTY keyboards. People are familiar with it, it's a lot of effort to change, and it's good enough for most intents and purposes.