The Unbroken Window

There are four broad classes of measures used to capture, via a simple summary statistic, what an income distribution “looks like” and how it changes. In a country of 310 million people it is easy to understand why we rely on such summary statistics. But as with all statistics, we ought to be very careful that we understand what they mean, and be very careful about what they cannot tell us. As you will see, sometimes using statistics can actually put meaning where none ought to be put, or it can change the way you look at a population even if that was not the intention.

Imagine we had twenty people in each of two worlds, call them Sunnyville and Cloudyville. Here is what the list of incomes for people in Sunnyville looks like:

{58 92 45 93 36 28 92 4 23 39 78 31 56 62 99 86 42 34 71 51}

And here is what the list of incomes for the people in Cloudyville looks loke:

{59 96 93 48 45 98 51 31 76 99 75 80 51 35 18 95 73 16 41 44}

Can any of you, upon examining the lists, ascertain which income distribution is in some sense, “better?” Even looking only at twenty numbers is would seem a near impossible task to compare them. But the task may appear easier if y0u use some common statistical metrics to summarize all of these numbers into a few easier to digest bits. (By the way, I used Excel’s random number generator to give me numbers between zero and 100). The average of the Sunnyville incomes is $56 while the average of the Cloudy is $61. Whatever you think of averages, given that you know the population of each place is the same, it is certain that Cloudyville is richer. When it comes to measuring income distributions, these 4 approaches are generally taken:

Approach # 1:  Relative inequality approach

• This approach generally focuses on percentage differences in income between people.
• For example, making the following comparison is using a relative inequality approach:
  • Wintercow earns $80,000 per year
  • Summercow earns $800,000
  • Wintercow’s income is 10% of Summercow’s
  • Similarly, if Wintercow’s income increases to $100,000 and Summercow’s income increases to $1,000,000, the our measure of relative inequality is unchanged – Wintercow’s income is still only 10% of Summercow’s.
• The crude reports of things like, “A Fortune 500 CEO makes 450 times more than a minimum wage worker” is a measure of relative inequality.
• The income inequality measures that you are used to looking at are very likely to be relative inequality measures. For example, if you say something like, “the income share of the poorest 20% of the population is the same today as it was in 1980” you are making a statement about relative inequality.
  • Here is how this relative inequality measure would change for this countries:
    • Before change: {1,2,2,5,5} the poorest 20% of the people earn 6.7% of the income
    • After change: {1,2,2,5,10} the poorest 20% of the people earn 5.0% of the income
    • Therefore relative inequality has increased (notice I did not apply a normative criterion to this)
• The Gini coefficient that I reported in the two previous posts is also a measure of relative inequality. I’ll describe how it is calculated, in all likelihood, in the next post in the series.

Approach #2: Absolute Inequality Approach

• This approach is generally not reported on very much despite its intuitive appeal to people who dislike any notion of inequality. It is generally measuring inequality not by looking at percentage differences in income between people but rather by raw income differences between people.
• For example, making the following comparison is using an absolute inequality approach:
  • Wintercow’s income is $80,000 per year
  • Summercow’s income is $800,000 per year
  • Wintercow’s income is $720,000 lower than Summercow’s
  • You’ll notice two things about this measure. First, if Wintercow and Summercow each experience the same percentage increase in income next year, then while the first approach above would show no change in inequality, while this measure would.
    • For example, if their income increases by 25% each, as in the first approach, the relative inequality measure would be unchanged (Summercow still earns 10x more than Wintercow), the absolute measure will have increased (noticed I did not say, “worse”). Their incomes will now differ by $900,000.
  • Interestingly, using an absolute inequality approach may lead you to some counterintuitive interpretations of changes.
    • Wintercow’s income is $8 per year (yes, eight dollars)
    • Summercow’s income is $700,000 per year
    • In this case absolute inequality is “better” than it was in the case just above.
    • In short, looking just at raw income differences doesn’t tell us much about the goodness or badness or desirability or anything much about income distributions.
      • One more thought experiment. What if I told you that the absolute inequality between the richest dude and the median dude in Wintercowistan was $100,000 while the absolute inequality between the richest dude and the median dude in Summercowistan was only $80,000, would this mean that Summercowistan was “more equal” or even a better place to live?
      • What if the median income in Wintercowistan was $1,000,000 with the richest dude making $1.1 million and in Summercowistan the median income was $50,000 with the richest dude making $130,000, would that change your opinion of things? There are obvious parallels to real world data and not just farm country data.

Approach #3:  Absolute income approach

• How many people receive how much income? For example, you might ask what share of workers earns above a particular wage.
• A special case of this approach is the absolute poverty approach in which a poverty line is drawn and a poverty measure is calculated (such as the percentage of residents falling below this line)

• Here is how this absolute income measure would change for this country:
  • Before change: {1,2,2,5,5} 40% of the workers earn a high wage
  • After change: {1,2,6,8,10} 60% of the workers earn a high wage
  • Therefore this measure of “inequality” has decreased. Of course, the interpretation of this measure depends on exactly how I frame the measurement.
• The measure is vastly different than the relative inequality measure and absolute inequality measure above and changes in those do not correlate well with changes in this and vice versa – so again, using the metric to report on “inequality” depends on what we care about.
  • For example, one might want to “refute” the notion that income inequality is increasing by referencing a table like this one (Table H-17 from the Census Historical Income tables). In 1967 only 14.4% of the population of households earned $75,000 or more. By 2009, this number had risen to 31.6%. In other words, nearly one in three households earns over $75,000 per year. Or, the share of households earning below $50,000 was 63.6% in 1967. Today that number is 50.1%.
  • Those numbers seem to be telling a different story than the “we are getting poorer” story, or the rich is getting richer at the poor’s expense story. Again, I emphasize there are problems with using this figure to make this point, although it does seem to cause some problems for the conventional wisdom. We’ll dig into the claims and the actual data stories a few posts from now.

Approach #4:  Relative poverty approach

• This approach defines a group that is considered “poor” and then computes an income metric for the group.
• Table H-1 of the Historical Census tables is one way to examine this sort of a metric. For example, that table shows that the households in the poorest 20% of households had a maximum income of $16,845 in 1967 and the maximum income of the households in the poorest 20% today is $20,453.
• A more common approach would be to compute a mean or median for the group under study. Note, too, that it is common to compute income dispersion statistics within various subpopulations of the distribution. This post is already too long to get into details or analysis of these points, for now I just need to show you the various ways of doing measurement.

Approach #5:  Income mobility analysis

• Can only be done when data are available over time.  And this can only be done correctly when we can actually follow the same people and households and families over time. Even if we are able to track the same people, it is not clear what measure is of most interest.
• For me, I would care about things like, “the probability that a given family’s standard of living can increase by ABC% or $ABC over some particular time frame” with ABC changing to suit how much mobility you care about. There are literally dozens of other ways to think about mobility. The general point about mobility is that if there is a single concept that captures the “on the ground” concerns of people, it is some notion of mobility – whether or not the American Dream is achievable for particular persons.
• Two quick points:
  • Commonly presented measures of “income-mobility” do not capture what I would consider “mobility” or what I think you would either. For example, in the Development Economics literature, we often see reported the measure of income mobility that asks what the total change in the difference in the shares of income held by the poor and non-poor is. That looks and sounds much more like one of the first three measures above than it does a true mobility measure.
    • Consider again the following country:
  • • Before change: {1,2,2,5,5} the poorest 20% of the people earn 6.7% of the income; the richest 20% earn 33% of the income.
    • After change: {1,2,2,5,10} the poorest 20% of the people earn 5.0% of the income; the richest 20% earn 50% of the income
      • Therefore, the income share of the poorest fell by 1.7% and the income share of the richest increased by 16.7%. So the total change in the share of income held by the rich versus the poor was 18.4%. I actually have  a hard time conceptualizing this.
  • For all that folks (including myself) celebrate the notion of mobility, I actually am having a hard time convincing myself it is important because the way we think about mobility is framed in a zero-sum manner – we think of our mobility within the distribution of people. This notion totally abstracts from how well off we are, on our own merits. I think some of the same criticisms levied against relative inequality measures (which we will get into) can be thrown out mobility measures as well. It’s probably the best we can do, but it is best if we are cognizant of it.

This post is already massively long, so we’ll leave it for tomorrow’s post to comment on what these all mean, and what we might look for in a desirable income inequality measure.