Gini or brickbat?

Thursday, March 16, 2006
The Gini coefficient (Gini index), a measure of income inequality, is one of my least favorite statistics. According to the US Census Bureau, the US's Gini has been rising for the last several decades, reaching 0.46 in 2000, rocketing up from 0.40 in 1990 (partly, or perhaps wholly, due to a change in that way that the data from which it is computed is collected). This is higher than the Gini coefficient in other "developed" countries, meaning that income is more unequally distributed in the US. This is almost always reported as a bad thing.

The fundamental problem with the Gini is that there is no a priori way to determine what the "ideal" number is. Zero is the theoretical minimum, which would mean that everyone (or every household) had exactly the same income, but I suspect that most thinking people would toss that out immediately as not being a reasonable goal. Such a system has been attempted (at least nominally in the form of Communism), and it failed miserably. On the other hand, the coefficient could be as high as 1.0, meaning that all income is earned by a single household, with all other households earning nothing. That, also, is easy to reject as an ideal.

So what is the ideal? Well, obviously, somewhere between 0.0 and 1.0. Not very informative, I suppose, but it seems to me that's part of the problem with the Gini. The Gini coefficient is not meaningful in itself; it is, rather, an artifact of an economic system that either works well or doesn't. If the system is working well (making people richer), I would say that's a pretty good system, regardless of the Gini that falls out. If not, I'd say that's probably not a good system, and again the Gini has nothing to do with it.

Hence, the use of the Gini as a bludgeon against the US economic system is strictly political, and is, I think, based on the implicit, unjustified (and as we'll see, probably unjustifiable) assertion that the increasing Gini means the rich are getting richer and the poor are getting poorer (at the very least it relies on an appeal to the Gini being "too high").

The common response to claims that the increase in the Gini is a bad thing is along the lines of "so what if Bill Gates's income increased faster than mine did, mine increased, too. Why should I be jealous?" That's a reasonable response, and in line with my thinking - if the system is making everyone richer, why should we care that there are more really rich people, or richer really rich people, relative to the bulk of us? But it's anecdotal - your income increased, but it's possible that the bulk of the distribution of incomes fell back towards zero and only the part incorporating the wealthy got wealthier. Maybe you're just one of the lucky rabble who got bounced up a bit while those around you generally moved down.

So let's look at the relative distribution of household incomes in 1989 and 1999, adjusted to reflect changes in purchasing power, and see if that provides any insight into how incomes have changed over time.

The US Census publishes household-level income data with every national census - the data available online reflect incomes in 1989 and 1999. Because the cutoffs that are used to group incomes vary from one census to the next, and because the purchasing power of dollar fell (due to inflation) over that time, direct comparison of the two censuses is difficult. Therefore, I've fit the data for each year with a 2-parameter Weibull distribution [technical note: The fits were performed using maximum likelihood fits to the cumulative observed distribution], allowing me to compare the parameters of the fitted distributions instead of the chunked data.

I suppose some sort of theoretical case might be made for the use of the Weibull, but I chose it because a) it fits the data reasonably well (at least for incomes up to about $150,000), and b) its two parameters, the shape parameter and the scale parameter, get directly at the issue at hand. Changes in the shape parameter reflect changes in the way income is distributed across households - the rich getting richer faster than the poor, for example. Changes in the scale parameter, on the other hand, reflect increases (or decreases) in household income that have been equally shared by all households - that is, if everyone got richer by 10%, the scale parameter should increase by 10%.

There is also a practical advantage to this - the adjustment for inflation becomes much easier. I can fit to the original data, and simply adjust the scale parameter by an amount equivalent to the change in the purchasing power of a dollar over the decade.

I've overlaid the best-fitting 2-parameter Weibulls onto the observed income distributions for 1989:

and 1999:

As you can see, the fits are pretty good. They're not perfect - the discrepancies are statistically significant by any standard, but since we're only interested in whether the general shape of the distribution changed that doesn't bother me. The fits are quite good over the ranges up to $150,000 or so, beyond that the Weibull greatly underestimates the number of households (i.e. there are more people with very high incomes than would be the case if household incomes were truly Weibull distributed). This is not an indictment. It doesn't mean there are "too many" people earning a lot of money. What it does mean is that the Weibull is going to be more useful for comparing the big lumpy part of the distribution across the decade than it will be for comparing the long tails.

Here are the results. The shape of the best fitting distributions is exactly the same. I'm using "exactly" only very slightly inexactly - the shape parameters differ (1.3458 in 1989 versus 1.3464 in 1999), but the difference is so small that the curves appear to lie on top of one another. I'd put up another graph, but just trust me - holding the scale parameter constant, the two shapes are so similar you can only see one line.

On the other hand, the scale parameter increased from 1989 to 1999, with the result that the two distributions (normalized to the same income scale and expressed in 1999 dollars) look like this:

What's different? Well, there are more households reflected in 1999 (105 million versus 92 million), and the scale parameter has increased by 13% - estimated mean household income went from $47,784 to $53,961. The result is that there are actually fewer households in the lowest economic brackets (those with household incomes below $17,500) in 1999 than in 1989, despite the number of households having increased by 15%.

On the other hand, there are many more households earning more than that. Here I've just subtracted the number of people estimated to be in each income bracket in 1989 from those in that bracket in 1999, providing a rough idea of the incomes of households "added" to the group during those years:

The peak occurs between about $40,000 and $100,000, an interesting tidbit if you've been hearing, over and over again, that the middle class is disappearing.

The upshot is this - it looks like the usual defense against the Gini brickbat is dead on. Inequality may have increased (though if it did, that must have occurred out in the right-hand tail of the distribution - the identical shapes found for the "lower" income strata (up to $150,000) imply identical Gini coefficients over that range), but the lumpen masses became quite a bit better off as well.

[Two caveats apply. First, remember that 1999 was at the tail end of an economic boom, and that may have exaggerated the differences between the two times. Second, remember that the fitted curves are estimates, not actual data. I will vouch for there being no obvious contradictions between the above analysis based on the estimates and the original data, but the original data are difficult to compare directly, which was why I fitted curves to them in the first place.]


Fresh Air said...


Two cents to add to your very impressive analysis:

My understanding is when the Gini rises, it's largely due to the fact that the very lowest income people experience virtually no increase in wages, whereas higher income earners (obviously) have comparatively high dollar-amount deltas in their wages year-to-year.

Since the lowest earning households in the U.S. are largely dependent upon federal and state government for assistance, and not, frankly, distinguished by a nascent ability to invent things like operating systems and windshield wiper delays, their income is going to see only fractional improvement unless their status changes dramatically (see Hillbillies, Beverly).

A 5% increase on $15,000 is $750. A 5% increase on $100,000 is $5,000. But if I understand correctly, in the Gini, this phenomenon shows "inequality" as increasing significantly. Well, as Al Gore used to say, "Duh-uh!"

That's capitalism, baby.

terrye said...

I know it is capitalism but I think the huge sums of money made by some high profile people have a lot to do with the perception that the rich are getting richer and the poor are getting poorer.

Athletes, movie stars, entertainers, etc make such ridiculous money that it kind of takes away from the idea that performing a useful function has anything to do with reward.

That simply exacerbates the perception of inequality and unfairness. Add in the trust funders like Paris Hilton and you have a class of wealth that is no more related to work than Marie Antoinette or the Czarina.

Right or wrong, people who have to work for a living resent that.

David Thomson said...

One should not give a damn about “inequality.” The real question is this: is everybody’s boat rising? Are the poor vastly better off today than even a few years ago? The answer, of course, is an adamant yes. Our least wealthy are often obese and not starving to death.

terrye said...


That is because macaroni and cheese is cheap.

David Thomson said...

My family lives in a lower middle-class Houston neighborhood. White Anglos like myself are definitely outnumbered when we visit the local Wal-Mart. Trust me, the majority blacks and hispanics are buying a lot more than mere macaroni and cheese!

Morgan said...

fresh air:

I think the Gini is innocent of increasing for the reason you describe - if everyone's income increases by 10%, it should stay the same. I can't think of any particular reason to think that everyone's income should increase by the same amount, though. As you said, people in the lower brackets tend to be engaged in qualitatively different pursuits than those in the upper brcakets.

There is also the fact that some chunk of the upper brackets represents people who just sold a business, and therefore have a great deal more income (for a year or a few years) than they had previously. I haven't done anything to see what impact that would really have, but liquidation of businesses as the owners retire are expected to amount to something like $4 trillion over the next eight years.


I agree that the reporting of the high earnings of high profile people tends to make others feel shortchanged, and I also agree that the occupations of these people often conflict with some long-founded notions of what makes work valuable.

David and Terrye:

It's true that, in general, boats are rising, but even in a rising tide some boats sink. It's also true that many people "feel" poor despite having enough to eat, owning cars, and living in homes with plumbing and climate control. It's also true that some people don't have all these things but struggle mightily to make ends meet.

Often these are young people with better days ahead. Other times they are victims of poor choices.

And still other times they are victims of bad luck or poor health. That's the impulse behind social welfare programs - no one wants those people to starve.

Knucklehead said...


Very interesting analysis. It seems plausible that a case can be made that the "rich are getting richer" (and/or that there are more, and richer, rich people) but it doesn't follow that this means that the "poor are getting poorer".

One fallacy at work here is that wealth is a zero-sum game. The idea that wealth is like a wedding cake with some uniform sized serving matched to the number of people at the reception. The only way for some people to get a bigger piece, or two pieces, of wedding cake is for other people to get a smaller, or no, piece of the wedding cake.

This clearly is not the case but a lot of people tenaciously hold on to just such a silly notion. The only way some rich guy can have a mansion and luxury autos is by greedily forcing some other guy to live in a roach infested cardboard box and walk, shoeless, in search of a job.

If they see one guy making $1M/yr and 40 guys making $25K/yr they can't fathom how that is "fair" and why it can't be the case that 41 guys each make $48,780/yr. In fairness it isn't really that bad. They think more along the lines of, "There's no good reason that the one guy is worth, or needs, $1M/yr and we should cap him at, say, $200K/yr and distribute the remaining $800K/yr among the remaining 40 guys and then they'd be making $45K/yr rather than their meager $25K/yr.

So there! All we need to do is cap the income of all those greedy rich people and distribute, among the working stiffs, the excess between the cap and what they would be paid if there were no cap, then there'd be no working-poor, right? It's just arithmetic - what's so hard?

A lot of folks look at their own employers and see, as an example, that the highest paid 5 people received, say, $55M and they run the arithmetic and come up with the notion that if $50M of that had been distributed among the 20,000 employees they'd each get another $2500 and the "rich" guys would still be millionaires so where's the rub? OR, better yet, that the $50M could have saved (or added) 1,000 jobs at $50K.

Another thing is the whole notion of "poor". What the heck does that mean? What does it mean to be poor? Does it mean one goes hungry or unclothed or has no roof over one's head? Or does it mean, to some people, that some people have more than they do regardless of the overall adequacy of what they themselves have?

When I was a kid it was pretty easy to see poverty. Perhaps I had an unusual childhood but I saw plenty of people living in what looked like squalor to me. Tin roofs, outhouses. There isn't much of that in the US anymore. Clearly the poor have not gotten poorer in the US in my lifetime. The poor of four decades ago couldn't have imagined the conditions of the poor of today - they'd have been thrilled to have the living conditions of today's poor. And the downright ordinary living conditions of a human lifetime ago were barely a match for today's idea of "poverty".

People, all of us, have lost track of what it really means to have just barely enough to live. That's poverty. What portion of Americans actually live at a real poverty level? We've redefined "poverty" to the nebulous notion of "poor" and then redefined "poor" into something measured by comparison to "rich".

There are people, and there's no shortage of them, who don't have some of what most of us would identify as a "need". They could answer "What is it you need that you don't have? with some legitimate answer like, "I don't have the money to buy both food and the $500/month prescription drugs I need."

But an awful lot of people who would readily describe themselves as "poor", or nearly so, would have no good answer to, "What is it you really need that you don't have?" Instead they'd look out and start talking about something somebody else has that the somebody has no claim to need.

Fresh Air said...


Thanks to government largess, the poor you describe for the most part do not exist anymore. Though it is a fallacy to believe most poor live in urban areas. Because of the vast support networks available in cities, the "dirt-poor" you describe are typically rural.

The fact is that people who get the bulk of their incomes from the government are unlikely to experience a rising tide--or a lowering tide, for that matter. They will just sit there, while inflation and the Dow Jones march on. It's pretty easy to predict an enlarged "gap" in equality as a result.

Your point about defining "poor" is a good one. Politicians don't want to do that because it reduces their abilities to declaim against "injustice," and naturally, to propose further remedies for same.

Rick Ballard said...


My compliments. Another piece worthy of a wider audience.


I agree with FA concerning pols unwillingness to define "poor". Your reading of the situation is spot on concerning using static analysis on "the pie". It's a Marxist technique that's been in use for 150 years and it's just clever enough to fool the dummies in the press.

Knucklehead said...


There's interesting sorts of differences between the notions of poverty between rural and urban people. As a yute I had what I have long since considered to be the good fortune to have seen both. Go back to just the mid-60s and one could see tin-shack level poverty in both urban and rural settings. It was there in worst parts of Atlanta and Baltimore just like it was in the hinterlands of AL or MS.

The urban variety of this is largely, if not completely, gone. That doesn't mean there aren't despicable levels of slums and slumlords still operating in the cities but it isn't nearly the same anymore.

Rural poverty is a somewhat different matter. Barring drought or environmental issues the rural poor aren't generally starving. The one thing they have is food because they can grow and raise it. IIRC correctly (it's been a while) the infamous "poverty study" that claimed that something like 12% (1/8) of US children went to bed hungry took advantage of this little oddity.

As I recall the debunking of the study it was finally revealed that the people who did the study went to an area of rural poverty, surveyed household incomes which were, predictably, very low, and then surveyed how many of those households were taking advantage of welfare like food stamps and then made some claim that people who were poor enough to be eligible for food stamps but didn't take them were, by definition, going hungry.

So they found (pulling numbers out of air to make the point) that, say, 75% of these rural poor people were eligible for food stamps. Not taking advantage of food stamps meant they must be going hungry. That equated to X number of kids "going hungry" which was Y% or the kids "below the poverty line" and viola!, 1/8 or US children "going hungry".

What they knew, and the rural people knew, was that food wasn't in particularly short supply. That was the one thing they had and since they didn't need food they didn't take food stamps. (Or whatever the food purchasing welfare assistance was). If somebody would have offered them a welfare program to purchase building materials they probably would have jumped on it to upgrade their homes but they were decent and proud enough not to take assistance they didn't need - that stuff is for hungry people and they weren't hungry.

Where were we? Oh, yeah.


At least some portion of the "rich getting richer" is exactly this. The economy is more complex and with IRAs and 401Ks and Keoughs and 529s and whatnot an increasing portion of the populace are stockholders and, slowly but surely, become at least a little more sophisticated about "investing". It takes a buck to make a buck as the old saying goes.

Living paycheck to paycheck doesn't necessairly mean one is poor but it does mean one isn't "saving" which, more an more, means one isn't "investing". The more money people have the more they are able to take advantage of investment mechanisms. The rich will always get richer except when you have some sort of cultural collapse or a tax system that leads them to run away and hide their money.

It seems to me we always want more rich people getting ever richer. What we don't want is an increasing number of poor people getting ever poorer (in a real sense, not some arbitrary envy sense). I don't see any evidence, either in the numbers or my own anecdotal observations, that we have an increasing number of increasingly poor people in the country.

But, of course, as you point out, that doesn't do much for politicians looking to get elected by taking money from Peter and giving it to Paul. That want ever more people defining themselves as Pauls and wanting some of what Peter's got.