Measuring Inequality

Ezra writes about the rising income inequality in the US, in the context of an article saying that rise is largely mythical. The article’s bone of contention is that the rise is attributable entirely to the fact that Americans are getting older and better educated.

Now, that may be true, but it’s legitimate to ask why these explanations are only ever invoked when they suggest the real level of inequality is lower than it seems. The US has one of the youngest populations in the first world, if not the youngest; its unweighted Gini index is .47. Japan has one of the oldest, if not the oldest; its unweighted Gini index is .31.

The comment thread on Ezra’s blog degenerates into a kerfuffle about happiness, largely due to the article’s spurious invocation of inequality in happiness as a legitimate measure. As such, there’s relatively little discussion of scenarios like the one invoked by SamChevre,

Let’s suppose you have a society where 18-25 year olds live in their parents’ households.

Now the society gets slightly richer and builds some cheap rental apartments. The 18-25 year olds move out of their parent’s houses, and live in the cheap rental apartments.

If you measure household inequality, it has greatly increased; however, no one is worse off.

First, some countries weight their Gini indices based on the number of household members. That’s how Norway can boast a Gini index of .3, even though using the same measure the US uses, its actual index is .37.

And second, even without weighting, let’s see what happens when the household splits, under the unrealistic assumption that the cost of living doesn’t increase. Mean household income will go down by a factor of 1 + 1/H, where H is the number of households before the split.

The median can go up or down; it will go up iff both households are above median after the split, and down otherwise. In the most drastic case, that when the household is above median before the split but both households are below after the split, the median will go down by a household and a half. If a below median household splits, it will go down by half a household.

To see what I mean, suppose there are 100 households, whose incomes are 1, 2, 3, …, 100. The mean and median are both 50.5. If household 60 splits into two equal halves, the median will go down to 49, going down by a percentile and a half for the one percentile of households that split. If household 30 splits instead, it will only go down to 50. In both cases, the mean will go down to 50, which is about 1% less than 50.5.

So the median-to-mean ratio will go down in the second case iff, approximately, the ratio of percentile 49.5 to percentile 50 is less than 100/101. In the US, it is lower, by a tiny margin. The US provides data in chunks of $2,500 per year; assuming within each such chunk the distribution is uniform, 100/101 times the median is percentile 49.56.

In other words, splits like that are likely not the reason why inequality is rising in the US. The other suggestions, age and education, don’t hold much water either.

First, high levels of education don’t necessarily correlate with higher inequality. The GI Bill ushered in an era of relatively low inequality in the US; the US Gini index bottomed in 1968, apparently.

Inequality is higher among the educated than among the uneducated, but that just reflects the fact that it’s higher at the top than at the bottom. Americans aren’t becoming richer because of greater levels of education, or else wages wouldn’t be stagnating. Rather, it’s just that the middlebrow jobs of 2007 are likelier to require college education than those of 1967. In other words, as more Americans graduate from college, they’re diluting the upper class composition of that bracket, which should lead to lower inequality among college graduates, rather than higher inequality overall.

And second, the aging of the Baby Boomers is mitigated by the entry of the Millennials into the workforce. In 1999, hardly any Millennial was old enough to work and virtually none lived alone. Today, almost half of all Millennials are and many do.There are more Millennials per age cohort than Xers, so their leaving their parents’ homes and starting to work would exercise downward rather than upward pressure on inequality.

On the other hand, some of the discussion about inequality in the US is grossly exaggerated, courtesy of populists who idealize the 1950s and 60s. Between 1992 and 2005, the US Gini went up by 3.5 points, of which 2 were due to methodological adjustment. The Swedish Gini went up by 2.5 without such an adjustment between 1995 and 2004. In Norway, it went up by 2 between 1998 and 2002.

The real gap isn’t between today’s American and yesterday’s, but between the USA and most other developed countries. At its nadir, the US Gini index was higher than the current one in Germany, Japan, Norway, Sweden, and even Britain.