The maths of inequality is hard. Intuitions that usually serve us well fail. Results we wouldn't expect are true. Here I use real UK data on the economies of Manchester and London to show one of the most important and most unexpected results.
First, some definitions. The smallest area for which the ONS reports mean household income is the MSOA. Each MSOA contains about 8000 people. Greater Manchester has 345. London has 983.
Below is a histogram of the mean household income (after housing costs have been paid) in Manchester (1) and London (2). The four poorest MSOAs in Manchester have an average disposable income below £250 per week. The two wealthiest MSOAs in London have an average disposable income above £950 per week.
In these graphs we see quite clearly that London is much wealthier than Manchester; the mean household income is over £500. In Manchester it is below £400.
For each wage distribution I have calculated the Gini index*. The higher the Gini index, the greater the inequality. London is significantly more unequal than Manchester. Manchester and London together (3) is more unequal still, with a Gini index of 15.2%.
Let's imagine (4) a Manchester that is 35% richer, making its mean income equal to London's. But this Manchester is much more unequal; the poorest households are no better off than they were before and the richest households have nearly doubled their income. Our intuition would tell us that the combination of London as it is (3) and a much more unequal Manchester (4) would also be more unequal... right?
Wrong. By growing quickly, even if that leads to higher inequality, growth in Manchester would contribute to a great equalising of income across the two cities(5). Extend this metaphor carefully across the whole UK and it is a challenge to our intuitions about inequality.
It's not the result we expect but it's what the maths tells us is true. We see something similar in Germany.
All my workings are in this Excel spreadsheet if you want to check them. And please remember that this is a simplified example just to show that inequality isn't always what it seems. The principle holds for more complicated systems but the details that I've left out here always matter.
ps 1. in advance of the complaints that I am confusing wealth with wages by using the terms 'poor' and 'rich'. I know. It is a deliberate choice. I find it results in fewer complaints and less misunderstanding than the other options.
* ps 2. These Gini indexes are calculated for a set of large-area averages and are not comparable with Gini indexes calculate for a whole population as usually reported for a country.