“Matt Yglesias”:http://yglesias.thinkprogress.org/2010/09/new-labour-and-inequality/ and “Brad DeLong”:http://delong.typepad.com/sdj/2010/09/in-which-matthew-yglesias-observes-that-innumeracy-is-an-awful-thing.html have argued that this graph, from “Lane Kenworthy”:http://lanekenworthy.net/2009/06/01/did-blair-and-brown-fail-on-inequality/, shows that we shouldn’t be too critical of Labour’s performance with respect to inequality over their 12 years of government in Britain.

Both Matt and Brad are pushing back against “Chris Bertram’s post at Crooked Timber”:http://crookedtimber.org/2010/09/30/its-about-the-distribution-stupid/, which argued that Labour had done very little about equality. (Although in his remark on my comment on his post, Brad now seems to suggest that his post was a pre-emptive strike against what Chris would go on to write in comments.) There’s a natural rejoinder on behalf of Chris, which has been well made in both Matt and Brad’s comments threads. Namely, if the graph really showed that things had gotten better, equality-wise, the Gini coefficient for the UK would have fallen. But in fact it rose, somewhat significantly, over Labour’s term. Indeed, the “IFS Report”:http://www.ifs.org.uk/publications/4524 that the graph is based on shows quite clearly that it rose markedly towards the end of Labour’s term.

So I got to thinking about how good a measure “Gini coefficients”:http://en.wikipedia.org/wiki/Gini_coefficient are of equality. I think the upshot of what I’ll say below is that Chris’s point is right – if things were really going well, you’d expect Gini coefficients to fall. But it’s messy, particularly because Gini’s are much more sensitive to changes at the top than the bottom.

Because my own thoughts on this are such a mess, I thought I’d hand over to two other characters to handle the back-and-forth.

ACHILLES: Imagine you’ve got a reasonably unequal society.

TORTOISE: You mean 1997 UK, don’t you?

ACHILLES: I do, but any will do.

TORTOISE: I’ll imagine 16th Century Venice then. I like canals.

ACHILLES: Suit yourself. Now imagine that this society gets 30% richer over a period of a few years.

TORTOISE: Yay! More canals!

ACHILLES: And this 30% increase in real income is distributed in proportion to how much income people had at the beginning of the example. In other words, since it was a very unequal society to start with, the gains are distributed very unevenly. So the society has gotten worse with respect to equality.

TORTOISE: Wait a minute. In this example the ratios between incomes of people in the society doesn’t change as a result of the increase. So the Gini coefficient, which is only sensitive to ratios of incomes, doesn’t change. And since the Gini coefficient is the perfect measure of equality, the society hasn’t gotten worse with respect to equality; rather, it hasn’t changed a bit.

ACHILLES: But now the society has all the inequality embedded in the initial position, plus the inequality embedded in the distribution of the new wealth. That’s worse than what they started with.

TORTOISE: Do you have a better measure of inequality than Gini?

ACHILLES: No, but one can be sceptical that a model is universally applicable without having a better model to put in its place.

TORTOISE: Seems like lazy talk to me. In any case, that’s not what actually happened in the UK, 1997-2009. (Which I know is what this is really all about for you.) That there graph slopes downward in the middle section. It isn’t flat, like in you’re example.

ACHILLES: Only if you ignore the tails. I agree though that if you ignore the tails it slopes down. That’s in large part an effect of how the y-axis is drawn. If it was drawn in pounds (or real purchasing power) it would slope up, not down. Eyeballing the graph, it looks like most people got a 1.5% per year real income rise (on average) plus something like a £2 per week per year raise. In other words, it was something like 3/4 due to the gains being distributed in proportion to starting income, and 1/4 due to everyone getting one shiny coin per week extra each year.

TORTOISE: Why are we eyeballing this? Don’t you have the numbers?

ACHILLES: Actually, no I don’t. The IFS study includes a graph, but not the table it is based on.

TORTOISE: I’m thinking I’m going to lose my standing as the lazy one in this dialogue at this rate. In any case, wouldn’t that mean Gini went down. And as we went over earlier, Gini is perfect. So Labour wins, Bertram loses, nyah nyah nyah nyah.

ACHILLES: But Gini didn’t go down, it went up. (And I’m not conceding that it’s the perfect measure.)

TORTOISE: Ah yes, I imagine that had something to do with those tails at either end of the graph. It’s probably something to do with those. You know, when there is a large increase in in-kind payments to the very poor, weird things happen at the lower tail. That’s probably all there is to it.

ACHILLES: That’s probably not all there is to it. Although I am too lazy to find the real numbers and confirm this. But I do have a nice toy model to suggest it.

TORTOISE: Ooh, a toy model. Are you a philosopher or something?

ACHILLES: Something like that. Let’s say we have a society with 100 people, n₁, through n₁₀₀, and let’s say the income of each person n_i is i. Surprisingly, this only leads to a Gini of 0.33, which is lower than most existing societies. It does feel fairly unequal though.

TORTOISE: Only 0.33, not too bad, not too bad.

ACHILLES: Now let’s imagine we reduce the income of the bottom 9 people by 100%, i.e., to 0. This raises the Gini obviously, but not by that much. It only goes up to 0.3409. (At least if Wikipedia is right that “this equation”:http://upload.wikimedia.org/math/b/b/5/bb50601acc135c45a24bb0493f7555b4.png works.)

TORTOISE: A Wikipedia equation. Nice, very nice.

ACHILLES: Yes well, you get what you pay for here. Anyway, let’s restore the income to those 9 people, which is pleasing to do, and now double the income of the top nine earners in the society. Now the Gini rockets up to a more real-world like 0.4149. So Gini is much more sensitive to rises at the top end than falls at the bottom end, at least if this toy model is anything to go by.

TORTOISE: I’ll finally admit, this seems like a weakness in the Gini model. After all, when we care about equality, we should care more about what goes on at the lower end than the upper. But it also suggests to me that the rise in Gini under Labour is an artifact of a model that’s too sensitive to the top, combined with a very pleasing rise in the incomes of my City friends.

ACHILLES: So the model is perfect, except when it grounds complaints about Labour and the City, in which cases it should be set aside.

TORTOISE: You may very well think that. I couldn’t possibly comment.

I come out of that liking Achilles’ position more than Tortoise’s, though I also like the conclusion they draw at the end. In principle a rise in Gini that was caused solely by the very rich getting even richer could be accompanied with an intuitive improvement in equality, if we saw in general things getting better for the poorer across the board. But that’s really not what we saw in Britain. And in a period of economic growth, you’d expect Gini to fall even if we were standing still on equality. When it rises, we have an indication that things are getting considerably worse, and I don’t see anything in the data to defeat that indication.

This was “cross-posted at Crooked Timber”:http://crookedtimber.org/2010/10/01/fun-with-gini-coefficients/. On a related, i.e. CT, note, “this blog post on Ed Miliband”:http://d-squareddigest.blogspot.com/2010/10/let-me-be-governed-by-lazy-men-like.html is one of the best blog posts I’ve read in ages.