Tim says that inequality falls in recessions, because top incomes are determined by market forces and so fall whilst bottom incomes are welfare benefits which, barring reforms, are fixed in real terms. The data seem to support his view; the Gini coefficient for post-tax and benefit incomes did fall slightly between 2006-07 and 2009-10.
If we look at the incomes of the poor, C is unequal. There, three people have an income of half or less of median income. In B, no-one has so low an income. In A only two do.
But if we look at the gap between the rich and the median, we get the opposite ranking. Society B is highly unequal, whilst C is egalitarian.
If we take the ratio of top incomes to bottom incomes, C is most unequal and B is most equal. However, if we take the absolute gap in incomes, the opposite is true.
So, which society is the equal one? Gini coefficients cannot adjudicate here. They are all the same, by construction. In society C there are small inequalities at the top, but big ones between average and bottom. In B, there are big inequalities in the top half and small ones near the bottom. These inequalities cancel out.
I don’t think my example is contrived. You can think of society B as one with a “squeezed middle” - individuals ranked 2 to 6 are worse off than in society A - and with “winner take all” features that generate a very wealthy minority, but with a relatively generous welfare state. If you like, it’s a New Labour society. Or if you like, it's a society in which the richest manage to get government bail-outs whilst the middle class suffers.
Society C, however, is the opposite. It‘s a bourgeois society, with a big middle class but a weak welfare state.
Which is preferable? Utilitarians can’t adjudicate, as average incomes are the same. Nor can the egalitarian who looks only at Gini coefficients, as these too are all the same. Rawlsians would prefer society B, as this maximizes the position of the worst-off member. Others, though, might prefer C, as it gives us a six in nine chance of getting an average income or better, whilst society B gives us only a four in nine chance and A gives a five in nine chance. Still others might find B least attractive, if they fear adverse social or political consequences of having a very wealthy elite, or if they have (non-consequential) libertarian objections to the large transfers that fund the welfare state.
My only point here is perhaps a trivial one - that simple measures such as Gini coefficients (or I suppose any other single measure) tell us very little about complex phenomena such as inequality.