# More or less equal?

Yesterday’s publication of the latest set of statistics on income inequality by the Office of National Statistics was hailed by many as showing that inequality was continuing to diminish (albeit slowly) as top incomes shrank while lower incomes grew.

See for example Chris Giles in the Financial Times

These seemingly mistaken perceptions result from low income growth since the financial crisis (and it is of course quite incontrovertible that real wages have seen an unprecedented lost decade of growth)

But is this the whole story? One element that does not seem to be getting the attention it deserves is that we are not looking at how incomes are distributed between a fixed number of households. The number of households is growing, with growth driven primarily by migration. The latest statistics show the number of non-retired households increasing by 448,000 between 2014/15 and 2015/16, not an inconsiderable amount as this represents growth of 2% in the number of such households. The number of households are divided by income into fifths or tenths of the whole to give quintiles and deciles of the income distribution. Growth in the number of households means that each quintile of the income distribution increased by about 90,000 households compared with the previous year. But unless the new households have the same income distribution as existing households they will make a compositional difference to the quintile (or decile or percentile) distribution.

Simply for the purposes of illustrating the point, imagine that existing households saw no change in their income between 2014/15 and 2015/16 and that all of the new households had incomes falling within the income range of the second quintile of households in 2014/15. All quintiles must expand by the same amount as they are simply equal fifths of whatever the household population happens to be. When the 2015/16 household population is divided up into quintiles, each of which must contain 90,000 more households than in 2014/15, the income of the bottom quintile would increase because the 90,000 lowest earners in the previous second quintile would be displaced into the bottom quintile, increasing the average income there. At the other end of the scale, 90,000 more households must be allocated to the top quintile but as all of the new households have nowhere near that income, the top quintile can only increase in size by expanding to include households previously in the 4th quintile, thus reducing average top quintile income.

Thus if the increase in the number of households is composed mainly of lower income households, this will reduce the income of the highest quintile (or decile) because requiring it to expand has the effect of diluting it. This will affect the 80/20 Gini curve (or any similar measure) for the same reason. Thus, paradoxically, inequality -as defined- can be reduced by increasing the number of households with low incomes.

So, does a reduction in the average income in the top quintile mean that incomes have fallen for anyone in the top quintile? Does an increase in the average income in the bottom quintile mean that any of the lowest earners are better off? Does a reduction in the Gini coefficient mean that inequality is decreasing?

Where the size of the household population is increasing, the answer is in each case that …… it ain’t necessarily so.

Now of course this is entirely theoretical. Could it possibly be the case that in reality the growth in the number of households over the year (or indeed longer periods) has been disproportionately in lower-earning households? I’ve written before about the ‘mist of migration’ obscuring what’s really happening in all sorts of areas, particularly in the labour market. This isn’t to make any value judgement about migration, but as migration is a key driver of growth in the number of households, and as new migrants tend to earn rather less than average (again compositionally contributing to lowering earnings overall even if they do not cause lower earnings for anyone else), it seems at least possible that some of this mist might be rising over the official statistics on income inequality, and if so, obscuring that what one might describe as underlying income inequality (i.e. among life-long residents) might in fact be increasing more or decreasing less than the data suggest….

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