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September 29, 2006
Show me the income!
One of the challenges with addressing the question of income inequality is that you can get very different impressions depending on how you look at the data. "The L-Curve" plots the data as income amount by percentile. The singular noticeable feature on this plot is the huge spike at the far right end of the curve. Even if the curve plotted only realized earnings (e.g. wages and capital gains from stock sales, etc.), rather than gains in net worth due to the value of stock holdings, this feature would still be very prominent.
The problem, however, is it's hard to tell how significant that spike really is in the big picture of the economy. If the length of a football field is to represent all 114 million U.S. households, the 50-km-high spike that represents Bill Gates' $50 billion "income" in 1999 is only 0.8 microns thick--approximately 1/50th the thickness of a human hair! Even if Gates can fairly be said to have had a $50 billion income in 1999, that only amounts to 0.6% of all income (in 2004).
The L-Curve really isn't the sort of distribution that scientists would consider useful. When we want to get a visualization of the spread of characteristics in a system, we typically use a histogram. The chart below illustrates where America's income is going. It is constructed so that every slice under the green curve with the same area represents the same amount of money. Likewise, the red curve shows the distribution of America's households. It is plotted on a logarithmic scale, which is useful when examining data over a large range. (More details about the number crunching below the fold.)
I present this curve because I think it is one useful method for visualizing where the actual bulk of income in America is going. I do not claim that this proves whether income is fairly distributed in the U.S. or not. And if anyone finds something wrong with my method, please let me know.

How this chart was constructed:
1) Income data was obtained from the US Census Bureau ("Income, Poverty and Health Insurance in the United States: 2005") and the Internal Revenue Service ("All Returns: Adjusted Gross Income, Exemptions, Deductions, and Tax Items: 2004").
2) All households were grouped into "bins" as defined by the Census (up to $100,000) and IRS (over $100,000) data tables. $1 was used for the lower limit of the "less than $5000" bin, and $100 million was used as the upper limit of the "more than $10 million" bin.
3) Total income in each bin was calculated by multiplying the number of households in the bin by the mean of the upper and lower limits of the bin. The sum of all these bins is the estimated grand total of all U.S. income.
4) The percent of all U.S. income was calculated for each bin.
5) The normalized income fraction for each bin was calculated by dividing the result in step #4 by the difference in the logarithm of the upper and lower limits. (This controls for the fact that not all the bins are the same size, and that the data will be plotted on a logarithmic scale.)
6) The normalized household fraction for each bin was calculated the same way, substituting the % of households in each bin for the % of all income as stated in step #4.
7) The values in steps #5 and #6 were plotted against the mean income level for each bin (the mean of the upper and lower limits).
Posted by Eric Seymour at September 29, 2006 05:05 PM
Rehashing a theme from comments on previous threads, I don't believe that there is such a thing as a "fair" statistical distribution of incomes; fairness is rooted in the means of making incomes, not on how much is earned.
Income distribution graphs can still serve as an indicator of unfair business practices. If the curve is heavily skewed toward deep poverty, odds are that economic freedom in that country sucks rocks. An economically free country will still have pockets of poverty, but not pandemic poverty as in Cuba or sub-Saharan Africa
The L-shaped graph in the earlier post wasn't really an L shape. It was a parabolic shape, with the Bill Gates end making the rest of the spectrum look flatter than it really was.
Posted by: Alan K. Henderson at September 30, 2006 03:38 AM | permalink
"The L-shaped graph in the earlier post wasn't really an L shape. It was a parabolic shape, with the Bill Gates end making the rest of the spectrum look flatter than it really was."
I don't understand this statement, unless it means "the incredible inequalty of of the distribution of income makes the distribution of income look more unequal that it really is."
Posted by: Phil at September 30, 2006 02:53 PM | permalink
Look at the L-curve chart. The curve appears flat up to the 96 yard line, doubles in height for almost all of the remaining yardage, doubles again just shy of the goal line, and Bill Gates' income goes orbital. The horizontal line is really a slope. But because of the resolution of the chart, we can't see the slope. The chart itself only gives us the gap between Gates and median income. We can't compare the gap between, say, the income gap between the 20% and 80% marks.
The chart in this post is far more effective at graphing the income distribution.
Posted by: Alan K. Henderson at October 1, 2006 03:24 AM | permalink
It's a fundamental mistake to plot income by household. The only way to meaningfully compare incomes is by a per-capita measure.
Households and families will do what they have to do to make ends meet which means that in households with lower per-capita incomes, more in the household will be pressed into wage labor. Not only does Daddy go off to work, but so does Mommy, and little Jimmy and Sally.
The net effect is to skew the picture of income, where a household of four workers each making $15,000 is equal to another household, also of four people, but where only one of the four has to work because that one makes $60,000 a year.
Again, a household survey would lead one to believe that the economic circumstances of both are equal, when in fact they are quite obviously, not.
greg
Posted by: Gregory Travis at October 1, 2006 02:15 PM | permalink
From Wikipedia:
To measure the income of a household, the pre-tax earnings of all residents over the age of 15 are combined. The residents of the household do not have to be related to the householder for their earnings to be considered part of the household's income. The use of household income is often seen as the most dependable measure of personal wealth, as people tend to live in households that include other wage earners besides themselves.
Unfortunately, the article doesn't go further into the rationale for measuring household income. It does make sense if one is trying to get a picture of wealth accumulation. Plain and simple, you net more if you have someone to share expenses with.
But getting a clear picture requires a lot more data, starting with distinguishing between different types of households. Singles often swing from dual- to single-incoem household. (There must be some inverse proportionality between apartment rentals and household income.) Working spouses tend to share more of the expenses than working teenage children do. The number of children and nonworking adults vary per household. And spending and investment habits vary.
Per-capita measure is relevant to a different mission: assessing the job market. How many jobs are there are, and what do they pay? Household income doesn't tell us about jobs, and per-capita income doesn't tell us about net income, to which the household data is relevant.
Posted by: Alan K. Henderson at October 2, 2006 01:33 AM | permalink
I think we can ignore the noise (singles that move in with their girlfriends/boyfriends and then move out), non-related persons living in the same household, etc. I doubt very much that, for these purposes, it's statistically relevant.
We're then left with the canonical household/family. I'm sorry, but a household of two individuals where one works, making $40,000 a year, and the other doesn't isn't economically comparable, much less equivalent, to a household where both spouses work, each making $20,000 a year.
And I don't see how the latter situation affords any "cost sharing." In fact, it's a very inefficient arrangement for a two-worker family/household has to outsource (i.e. pay for) domestic services that were historically provided by the non-working spouse.
Examples are day care, housecleaning, home maintenance, meals, etc. Such costs can be, and are frankly, very large.
Consider the two-earner household again, each making $20,000/yr for a household income of $40,000. Now consider that the two-earner household has to have two cars, for each spouse to get to work, has to send their young child to daycare during the day, has to hire a maid/housekeeper to keep the house clean, has to eat mostly takeout food, etc.
You're easily talkng about a thousand dollars a month in "outsourced" services, giving that $40,000 household a net income, due to the economic inefficiencies, of "only" $28,000. Where the couple next door, with only a single wage earner making $40,000 a year, doesn't have to absorb those outsourced costs.
Just another reason why wealth and income can't, and shouldn't, be compared between households.
greg
Posted by: Gregory Travis at October 2, 2006 08:29 AM | permalink
The reason that my chart as well as the "L-curve" both are based on household income is that's the data which is readily available. If anyone knows where to get a comprehensive picture of individual income, please let me know.
Greg is right that this curve can't tell you how many people are working in that family making $40,000 (or $400,000) a year.
But there would also be problems with graphing individual income. Some people only work part-time because someone else in the household is providing most of the income. Even if you only include income from full-time employment, someone making $20,000 a year whose spouse makes $70,000 a year is in a much different situation than a single person making $20,000 a year.
Posted by: Eric Seymour at October 2, 2006 09:12 AM | permalink
Eric,
The BLS has that info (www.bls.gov). It's hard to find (otherwise I'd provide a link) but it's there, I've used it in the past. You can get exactly what you're looking for, namely wages of full-time workers over time.
I'd really like to see your chart with per-capita income, as opposed to household income.
Speaking of which, there's a reason why household income is more readily reported than per-capita and that reason is called politics.
The government, well politicians actually, are much more inclined to report household income than per-capita income for the same reasons that they use the household survey, as opposed to the employer survey, when reporting unemployment.
The picture painted by the per-capita income figures, just like the picture painted by the employer survey of employment, is pretty grim. And has been for decades.
Which explains the bipartisan preference for reporting household figures. Unlike per-capita income, which has been generally declining since the late 1960s, household income has risen. So presidents, etc. can say "over my term, household income has risen 4%!" (or whatever). That's a much nicer message than "over my term, individual income has dropped 1%!"
Of course the reason that household income is increasing, while individual income is decreasing, is that more people in each household are engaged in wage labor. Not only does Daddy work, but so does Mommy and the two kids.
The labor force participation rate (the percentage of those people who can work that are working -- not to be confused with the unemployment rate) has gone from about 50% in the late 1960s to just shy of 70% today.
That's not a good sign.
greg
Posted by: Gregory Travis at October 2, 2006 09:50 AM | permalink
Greg,
You make a fair point about the labor force participation rate increasing, but the point of my graph (and that of the L-curve) was to illustrate income distribution, not income trends over time.
You are more than welcome to construct your own graph using per capita income, using the method I have posted. Personally, I doubt it would look very much different. My guess is that there are even more dual-income families above the median household income than below it, since there are more likely to be two college-educated adults with marketable skills.
Posted by: Eric Seymour at October 2, 2006 12:08 PM | permalink
In fact, it's a very inefficient arrangement for a two-worker family/household has to outsource (i.e. pay for) domestic services that were historically provided by the non-working spouse.
Daycare is an issue, but only up until the kids are school age.
How many dual-income couples hire housekeepers? I've never met any of those couples.
Posted by: Alan K. Henderson at October 2, 2006 01:36 PM | permalink
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