# Inequality from a different Engel

Last semester, I was introduced to Engel’s Law. In my quantitative methods subject, some students asked about deriving the relationship from income earned and food expenditure for a sample of households. While once regarded as a useful and interesting economic theory, it is now also a cause for worry.

Engel’s Law was first described by the 19th-century statistician Ernst Engel. Usually regarded as one of the most consistent observations in economics, it suggests that higher-income households tend to spend a smaller proportion of their income on food than those with lower incomes.

Although its application is usually limited to food expenditure, this relationship can be extended to consumption expenditure as a whole. On the basis of mean expenditure and pre-tax income for each quintile measured in the Bureau of Labor Statistics’ Consumer Expenditure Survey, I calculated the marginal propensity to consume (MPC) of each quintile of households over the past 30 years and found them to be in complete accordance with Engels’ Law (Chart 1). In fact, the MPC of the lowest quintile was far higher than any of the others, being almost always above 200% while the others never exceeded 150%. It was also much more volatile.

Taking the household sector as a whole, C = MPC × Y,  being the aggregate MPC of all households and  being aggregate income. Aggregate MPC could be easily derived as the weighted average of each quintile’s MPC:

MPC =MPC1 × s1 + MPC2 × s2 + MPC3 × s3 + MPC4 × s4 + MPC5 × s5

where , s1 , s2 etc represent  the share of aggregate income received by each quintile.

Now, allow me to digress by presenting yet another statistic driving home the severity of income inequality.

As the media reminds us ever so often these days, an increasingly large proportion of aggregate income is going to already high-income earners. Chart 2 below shows that in the US, this has been occurring mostly at the expense of middle-income households, with the top 20% of households earning more than half the total money income last year:

But how does this tie in with Engel’s Law?

Consider this. As depicted in Chart 2 above, , or the share of income earned by the top 20%, has been growing at the expense of the other quintiles. In other words, income is being transferred from the higher-spending, low-income households to thriftier, high-income ones. Less and less will be spent out of every dollar (or percentage point) of income transferred to the highest quintile. Eventually, this will have a retarding effect on consumption growth and subsequently economic growth, a concern shared by Jared Bernstein.

Since ,  being aggregate income[1], it then follows that if we expect income disparities to widen into the future, continued consumption growth will require sustained growth in incomes, failing which consumption will begin to fall. This would then have a noticeable effect on the US economy in particular, as consumption expenditure makes up two-thirds of GDP.

So has this happened?

As the Chart 3 below shows, it has not. (Yet.) Although aggregate MPC has fallen, the rise in money incomes has so far been able to keep consumption growing. But as observed in Chart 1, most of the top quintile’s gains in income share have come from the middle-income quintiles, not the lowest quintile, and thus the corresponding drag on MPC has been smaller.

However, as long as the income gap widens, aggregate MPC must necessarily continue to drop towards the MPC of the highest quintile (currently around 0.61). If current trends persist, a simplistic extrapolation suggests consumption growth could reverse around 2051 (Chart 4).

Would falling consumption really be a problem? Of course, to the extent that consumption is a function of wealth rather than income, declining aggregate MPC will have less of an effect on it. However, Chart 3 shows that consumption still seems to be closely linked to income.  Moreover, as Casey Mulligan points out, growth theory holds that capital formation, not consumption, sustains economic growth;but that is only true in the long-run. As it is a component of GDP,any reduction in consumption will surely be a drag on short-run economic growth. On the surface, its effects may seem far off and minor.But they remain a niggling reminder: maybe –just maybe – income inequality might have major repercussions after all.

[1] Aggregate income as measured by the Census Bureau refers to money income before tax, received on a regular basis, and differs from personal income measured by the Bureau of Economic Analysis. In this case, aggregate income does not equal to GDP.