The Brandeis Ratio is a Faulty Measure.

It's Christmas. Can't sleep. Clown will eat me.

Inequality has been on the minds of many economists, and some are proposing "fancy" or "simple" new ways of measuring it. What a lot of them seem to forget are the yardsticks by which a good inequality measure is determined. There are four principles to adhere to, and these are all outlined in chapter 6 of Debraj Ray's "Development Economics" (Princeton, 1998). Unfortunately I don't see these principles adhered to enough these days, or even mentioned really.

A good inequality measure must satisfy these four principles (translated to English):
1) who owns the wealth does not matter, or the only pertinent attribute of inequality is the amount of income, ("the anonymity principle"),
2) how many people participate in the economy does not matter, so that countries, regions, or sections of varying sizes can be compared ("the population principle")- otherwise you could never compare say New York to Oklahoma or China to Lesotho,
3) make the comparison on relative income shares and not absolute levels of income (the "relative income principle"), and most importantly
4) when you make the poor worse off to benefit the not poor, inequality should increase ("the regressive transfers principle" or "the Dalton Principle").

This is a minimum standard. I encourage my students to add on from there as they please. But let's stick to the minimum.

It's very easy to get 1 to 3. The regressive transfer principle is more tricky. As a side, a regressive transfer is simply "taking from some one who is not rich and giving to some one who is not poor". Think of it as a reverse-Robin Hood.

Which brings us to the "Brandeis Ratio" from Ayres and Edlin. Assume an economy of 10 people with a wealth distribution of ($5, $6, $10);(5, 4, 1). That is five people earn $5, four people earn $6, and one guy gets $10. The total wealth of the economy is $21, and the number of people is 10. The Brandeis Ratio is ($10)/($5.5) as $5.5 is the median of the wealth distribution, and we're going to cheat and use the richest 10% as opposed to 1% since there are only ten people. So that's what? The richest in this economy is 1.8 times richer than the median household. Okay, sounds good.

If we allow a regressive transfer of one dollar from a man who makes $5 and give it to a man who makes $6, the new distribution is ($4, $5, $7, $10);(1, 4, 5, 1). The median is unchanged at 5.5. That means the Brandeis Ratio is unchanged despite what many people would see as an increase in inequality in that ten person economy.

The Brandeis Ratio fails the regressive transfer principle, something that the already extensively used Gini Coefficient does not. Actually, the Brandeis Ratio is just a fancy Kuznets Ratio, which is even more simple for calculation and still maintains the lethal failing of regressive transfers. I do not understand why we should adopt a new tax policy based on bad statistics.

I suppose the argument then is that the Brandeis Ratio only deals with the extreme case of the very rich, and I'm nit-picking about regressive transfers. In that case why not just use a Kuznets ratio and go with the share of the top 1% divided by the share of the bottom 40%?

The next might be the usefulness of measuring inequality in "medians". I'm still not sure what the difference is between measuring something wrong in medians and measuring something wrong in income shares?

Addendum:
Ian Ayres provides an account of why Brandeis instead of Gini here. It is in my opinion extremely interesting and well rationalized, but I still think the regressive transfer flaw is a weakness and not a strength.

Another Addendum:
Some questions on Ayres' reasoning.
1) The Gini, as I know it anyway, only looks at an income or wealth distribution. Unemployment and illegal immigration do not enter into it (see principle 1 above). It's a set of numbers, ordered from least to greatest. The entrance or exit of a household within the distribution by unemployment or (il)legal migration isn't really the issue. The numbers are the issue. I'm not sure where he's going with the Gini rationalization on unemployment and illegal immigration. It seems to be a stretch from the more basic Gini explanation.
2) I don't think "simplicity" equates to "transparency". The Brandeis is not "transparent" in that it ignores any transfers below the median. It is however simple enough for some one reading a newspaper article, but so are a lot of things. That does not mean those things are correct. I think a BA is sufficient to grasp a Lorenz Curve or a Gini Coefficient, and even if it wasn't why should we base policy on a statistic that is less correct?
3) I don't see the democracy reasoning either. The Brandeis Ratio is not a direct linkage to the top 1%'s ability to influence policy any more than the Gini is. The 1% disproportionately fund political campaigns. Okay, yes. But these plutocrats are measured in the Gini or Theil index as well. I don't see the advantage of using Brandeis over Gini.
4) The argument is the same for income, wealth, or assets.

If policy is going to be proposed based on statistics, let's make sure these are the best available. Not just the simplest.