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Beautiful fractals and ugly inequality

UPDATE 4pm: is there any point to this post? see end of text

UPDATE II: 4:30pm Critic cuts me some slack. see end of text

UPDATE III 11am, 9/10/10: Paul Krugman says he had the idea first (see end of text)

In our ceaseless search for trendy themes, let’s consider today the beauty of fractals. The picture below shows one fascinating kind of fractal called a “Koch snowflake.” Fractals have the same amount of “jaggedness” or “unevenness” at every scale.
Income inequality behaves like a fractal: income is very uneven at large scales and at small scales. Here’s a mapping exercise that illustrates this, with a tastefully chosen color scheme that is consistent across all maps (rich is red or brown-red, poor is pale yellow, in between is orange).
We are going to go from global to the US to the New York City metro area to the neighborhood of NYU in Manhattan. At each scale, there is a remarkably high level of inequality across space.
The rich coastal cities in the US and the poor rural South. Rich lower and midtown Manhattan and poor South Bronx. Rich West Village and Soho and poor Lower East Side.
Inequality is one of the hardest policy problems, so more later. A simpler insight of economics is that the most obvious answer to inequality is exactly wrong — complete redistribution (i.e. a 100 percent tax on everyone above average to go to everyone below average) would destroy incentives for wealth creation and make everyone worse off.

UPDATE: a commentator on Facebook asked me what the implications are. Reminds of the World Bank research managers, who if you told them it was raining, they would ask “but what are the policy implications?”  Finally a chance for revenge through my favorite Mark Twain quote (from preface to Huckleberry Finn):

Persons attempting to find a motive in this narrative will be prosecuted; persons attempting to find a moral in it will be banished; persons attempting to find a plot in it will be shot. By Order of the Author.

Please be content every now and then to just contemplate how the world is, which is kinda necessary before you immediately try to fix it.

UPDATE II: my Facebook critic and also @viewfromthecave grant permission for me to meditate on this for a while, and just try to convince you IT’S INTERESTING; note that fractal genius Mandelbrot thought cotton prices were interesting.  (If you do want my Comprehensive Solution, see here.)

UPDATE III: Paul Krugman noted this post and self-deprecatingly notes he had the idea first of the fractal nature of inequality. No problem, Professor Krugman, you can have it. As a long-time fan of your theoretical research, could I request that you take some time out from NYT to come up with a nice theory of why inequality behaves fractally?

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  1. Sina wrote:

    this is really good

    Posted September 9, 2010 at 12:43 am | Permalink
  2. Andrew wrote:

    Can you please share the data sources?

    Posted September 9, 2010 at 1:22 am | Permalink
  3. Love the beauty-ugly counterpoint.

    You might be interested in this 2003 Cornell paper offering a theory on fractal poverty traps:

    From the abstract: “This paper offers an informal theory of fractal poverty traps that lead to chronic poverty at multiple scales of socio-spatial aggregation. Poverty
    traps result from nonlinear processes at individual, household, community, national and international scales that cause the coexistence of high and low equilibrium levels of productivity and income and high and low rates of
    economic growth. Multiple equilibria result from key threshold effects that exist at all scales due to market failures and nonmarket coordination problems. Key implications of fractal poverty traps include (i) the
    importance of recognizing meso-level phenomena in addition to conventional micro- and macro-level issues, (ii) inter-connections across social-spatial scales that foster or ameliorate chronic poverty, (iii) the importance of identifying and overcoming thresholds at which accumulation and productivity dynamics bifurcate, and (iv) the significant potential role of transitory donor and government interventions and safety nets to ignite sustainable growth among the poor.”

    Posted September 9, 2010 at 6:02 am | Permalink
  4. G wrote:

    The ‘American Dream’ dogma is that every self-willed pioneer can become anything he wants, but simple observation of reality demonstrates amply that where you live largely determines how your economic life will turn out. So in order to shore up the dogma it becomes necessary to imagine that the poor perversely ‘choose’ to be poor; that there is a Darwinian selection-process based on character and this ineffable-but-essential quality, ‘willpower’…

    Posted September 10, 2010 at 7:58 am | Permalink
  5. Andy wrote:

    Income inequality is only a “problem” to the extent that the have-nots are going without something relatively essential. Within a given block of the West Village, say, there’s probably a lot of inequality as well. But if one person is making $150,000 and one is making $250,000 is that really a problem?

    We should be focusing on how to help bring up the people at the bottom end. Certainly the behavior of the people at the top can contribute at times to the plight of those at the bottom, but the existence of a gap (per se) is not an issue.

    Posted September 10, 2010 at 8:58 am | Permalink
  6. Andy wrote:

    I should add that income inequality, per se, isn’t “ugly” either. I have a hard time thinking of a single human trait (whether it’s attractiveness, intelligence, earning potential, etc.) that tends to be allocated evenly across people. What makes something beautiful or ugly is what the people given more do with their blessing.

    BTW, I assume they’re just slipping into the bottom of the category, but who knew the Kazakhs and Libyans were doing so well?

    Posted September 10, 2010 at 9:03 am | Permalink
  7. mike wrote:

    Its true that complete redistribution makes everyone worse off, but that’s not the main reason why its wrong. Its wrong because its theft, and because a basic principle of freedom is that people should be able to enjoy the fruits of their labor.

    Posted September 10, 2010 at 9:31 am | Permalink
  8. francis48 wrote:

    The difference between researchers and the rest of us is that only the former get paid for “just contemplate how the world is”. I’m not saying it’s necessarily bad but it’s certainly quite remote from the problems of the mother of seven fetching water in Ethiopia …. (were we contemplating her as well when looking at all these fractals?).

    Posted September 10, 2010 at 9:57 am | Permalink
  9. Graydon wrote:

    Mike —

    No one is working by themselves.

    Someone raised them from infancy; they have to buy tools from somebody; somebody educated them; etc. If you doubt this, consider what it would take — what you, personally, would have to do — to get an ax or a shovel, those very basic tools, if you started by yourself in a wilderness with no other people. (and that gives you raised and educated for free. They’re not.)

    So the whole “fruits of labour” thing is specious. People co-operate to produce prosperity; you can’t get prosperity (just like you can’t have money) as an individual.

    That makes “what does it cost to maintain the mechanisms of co-operation? who should pay what part of that cost? could we arrange things so the co-operation was more effective?” a legitimate list of questions.

    (With answers of lots (but would you want to go back to the neolithic?), in proportion to the benefit received (of course the rich pay more; they have more, so have benefited more), hell yes (we keep wasting geniuses; we keep looking at relative outcomes rather than the absolute capability of the system, too), respectively.)

    It’s a choice between “Am I (is my kid) better off in a world where I have greater wealth than my neighbors, or in a world where the total amount of wealth is much greater?”

    Put like that, it ought to be a no-brainer — do you want half of a hundred or five percent of ten thousand? — but, well, the monkey-brain has troubles with this.

    Posted September 10, 2010 at 11:20 am | Permalink
  10. William Easterly wrote:

    Francis48: I sympathize a lot with what you are saying. But at the same time, what you are saying might seem to imply we should stop research of all kinds and limit solutions to what is already known. To use a medical analogy, does the existence of sick and dying people mean that all resources should be reallocated away from medical research and towards treatment?
    thanks for your comment, Bill

    Posted September 10, 2010 at 12:14 pm | Permalink
  11. Alan Beattie wrote:

    1. What Andy said. Absolutes matter more that relatives when dealing with extreme poverty.
    2. The graphic sort of implies, through the use of the same colours in each snapshot, that inequality is similar on a local, national and global basis. But the scales are hugely different. Our average West Village resident has an income 7 times the average Lower East Side resident; the per capita GDP at the bottom of the poorest category of countries is 36 times smaller than the per capita GDP at the bottom of the richest category. If you are prepared to adjust your scale enough for each degree of disaggregation of course you can find inequality and generate a fractal map like this, unless everyone in a given disaggregation earns exactly and precisely the same, down to the last cent.

    Posted September 10, 2010 at 12:21 pm | Permalink
  12. Dee wrote:

    I’m all for contemplating, but it seems to me that in this post you have cherry-picked examples that fit your hypothesis. I’m not sure why grown up professors would fight about who came up with this nonsense.

    Posted September 10, 2010 at 12:30 pm | Permalink
  13. JPB3 wrote:

    The explanation I think is pretty simple: Rich people don’t like to be around poor people, and have the resources to keep them away.

    Posted September 10, 2010 at 1:20 pm | Permalink
  14. misterb wrote:

    My modest proposal for the fractality of many phenomena is that they are caused by feedback loops in complex networks.Our brains model the patterns these complex networks display when an instantaneous slice is taken by looking for similarities and these similarities are apparent whether it’s because we wish them to be or because they are actually there. Who’s to say which is true?

    Posted September 10, 2010 at 1:55 pm | Permalink
  15. Micha wrote:

    Did you read Mandelbrot’s book “The (Mis) Behaviour of Markets”?

    Posted September 10, 2010 at 2:16 pm | Permalink
  16. francis48 wrote:

    Bill, you are right and my comment was deliberately provocative (ie, enticing debate). Research is fundamental, so is contemplation (perhaps even more, to our sanity). My comment was motivated by your general argument that aid is unproductive, to which I agree to a large extent. But whose to say that research (or at least some research, eg, in economics) is that much more productive. In a way, we all tend to accuse the other ones of behaving like aristocrats, whereas … we may all aspire to be a bit aristocratic ourselves, ie, unproductive.

    Posted September 10, 2010 at 2:36 pm | Permalink
  17. This is actually a gloss on the old chestnut of inequality is relative, and then implying moral equivalence between ultra-rich and utter poverty.

    Mathematically, the moral issue is not that there’s a scale, but the endpoints and the slope/exponent/dimension.

    Posted September 10, 2010 at 3:20 pm | Permalink
  18. Fernando DePaolis wrote:

    great use of fractals to show disparities and scaling effects…in a way, it shows that the world is not flat (sorry Mr. Friedman…)

    Posted September 10, 2010 at 3:51 pm | Permalink
  19. William Easterly wrote:

    Alan, you win 1.5 to 0.5.

    Of course I agree with 1.

    On 2, you are half right. Domestic inequality is not as high as international, which is a qualification on the “fractal,” but nor is it a matter of a few cents.

    Best. Bill


    Posted September 10, 2010 at 4:59 pm | Permalink
  20. Steve Verdon wrote:

    Posted this comment over on another website linking to this….

    A fractal implies deterministic non-linear system of equations that look stochastic. Most searches for “chaos” in economic data have been less than fruitful and if you think you can break new ground in this area…good luck. You can see that Easterly is out of his depth with the cotton price comment. When I worked at the BLS the research division I was in had several people who worked on this kind of thing and had access to unbelievably large and detailed data sets on prices and the conclusion was: nice theory, but really no substance here.

    Maybe we’ll find it in income inequality, but here is something else to consider…notice that he mostly lighter colored portions of the map are also areas that have lower population densities. So would we see the same level of detail if we picked Montana and started zooming in? I doubt it. I bet if we picked Montana then started zooming in on various regions we find quite a few that simply went to zero–i.e. the level of jaggedness would disappear to a nice smooth uniform color.

    So…nice theory, but not much substance here.

    Posted September 10, 2010 at 5:37 pm | Permalink
  21. Jason wrote:

    Here’s a stab at the distribution: most of the wealthy countries have coasts, most of their wealth is on the coasts, much of it in cities on those coasts and much of that near the actual ports in those cities on the coasts.

    It may be a holdover from maritime trade being a source of wealth that produced bustling cities with nice places to live that remain desirable simply because they have been desirable for many years.

    There is also a pretty clear equatorial poverty zone that probably has something to do with agriculture and producing enough to export via maritime trade.

    But that’s just me looking at it for a few minutes. So I could be wrong. Kazakhstan Saudi Arabia and Libya seem like an outliers, but they are all oil producers fueling our current wealth.

    So is the Denver area, but that may just be Aspen.

    Posted September 10, 2010 at 7:25 pm | Permalink
  22. ttj wrote:

    This looks more like a question of geography and sociology than one strictly of economics. Wealth agglomerates. The condition of impoverishment excludes opportunity for mobility. Further, one or few wealthy individuals living in a low income area will not raise the median income of that area, just as a few relatively poorer individuals living in a wealthy area won’t lower the median income of that area. Also, wealthy individuals tend to agglomerate in the most geographically/economically attractive areas. If you take all those factors together, it seems relatively apparent why inequality follows fractal patterns. This idea might be harder to justify on an international scale, but if you take into account geographical and cultural considerations, you could probably justify similar reasoning.

    Posted September 10, 2010 at 8:06 pm | Permalink
  23. LVTfan wrote:

    Aw, come on!

    Please take the time to go read Henry George. We cannot carry on a meaningful conversation about poverty or inequality without a shared awareness of his observations on the subject, and his recommendation of how we might end involuntary poverty.

    If your time tonight is short, start with a speech or two — perhaps “The Crime of Poverty” or “Thou Shalt Not Steal.” Or with your choice of essays from “Social Problems” (all are online at

    Or read a modern abridgment of what is still the best-selling book ever on political economy (approximately 6 million copies sold!), “Progress and Poverty,” at

    Fractile patterns make me think of 80/20 “laws” which some people use to suggest that it is simply a “law of nature” (or something) that 20% of us are entitled to 80% of the goodies.

    It ain’t so.

    Posted September 10, 2010 at 9:19 pm | Permalink
  24. The answer to why this is true is obvious. No matter how large or small a division you make there always need to be poor people to do the work and rich people to enjoy the fruits of their labor.

    Posted September 11, 2010 at 1:06 am | Permalink
  25. Jacob wrote:

    I think the appropriate line about this, from when I was living in West Philly, was “wherever you live, it’s exactly one block from the ghetto.”

    Posted September 11, 2010 at 7:48 am | Permalink
  26. G wrote:

    Gross inequality is theft. A loving father would not leave his wife and children hungry just because he has the strength and hunting-skill or whatever. Unless you want to just take an amoral position (in which case, shut up because you have no place in a moral discussion, which is what this is), there’s this ideas that we should love each other, have compassion and work together. We all each have our role in society – professors depend on farmers more than the other way round. If the poverty of the first-world poor ain’t so bad, trade places with one of them, Graydon. Most of them would be happy to oblige you, I’ll bet.

    And there used to be this everyday word ‘decadence’ – it was a good word describing the worthless pursuit of increasing luxury. Now ‘decadence’ has been reappropriated to mean ‘luxury chocolate’. Ever-increasing luxury never makes anyone happy as one continually adjusts one’s expectations, yet we burn the Earth to feed this desire, which is, ludicrously, ‘needed’ for our economic paradigm to move at all. If Kanye West didn’t want to drive a Hummer around, it would mean a panel-beater and a mechanic have no job – this is farcical; we can surely do better with the world we find ourselves interdependent with.

    Posted September 11, 2010 at 12:33 pm | Permalink
  27. G wrote:

    Actually, a better example than Kanye is thousands of people whose allotted role in the economic machine is to risk their lives every day digging up metals that we only value because of their rarity. African diamond-mines, etc.

    This is the ugliness of inequality. Human lives and a beautiful planet ruined so that rich people can show off how rich they are. Ugliness created in the name of ‘beauty’.

    Posted September 11, 2010 at 12:37 pm | Permalink
  28. Alan Beattie wrote:

    “Domestic inequality is not as high as international, which is a qualification on the “fractal,” but nor is it a matter of a few cents.”

    I didn’t say it was. I merely said that you can generate an inequality map for any locality which has a non-zero Gini coefficient, and that it’s misleading to use the same scale for all localities.

    It would be interesting if you could redo these maps with the same percentage scale for each locality, though you I guess would have to have many more gradations of colour shade corresponding to income to make them meaningful.

    Posted September 12, 2010 at 11:47 am | Permalink
  29. Frank wrote:

    I recently published on “fractal inequality” in access to health care:

    I will take up your invitation to explanation at my own blog this week!

    Posted September 13, 2010 at 4:01 pm | Permalink
  30. Mark wrote:

    I’m sorry, but this is weak. We need to see GDP in PPP terms for the US (preferably with housing costs included) to see what inequality looks like.

    The same goes for the Lower East Side. You’ve got plenty of low-income people living in the Vladecks or the East River co-ops who pay virtually no maintenance or rent (and are sitting on $500k+ of real estate.)

    Posted September 13, 2010 at 4:05 pm | Permalink
  31. Andy Berg wrote:


    Hi there.

    I only skimmed this, but surely you (and Paul) remember the 1960 JIE article by Mandelbrot on all this stuff!?

    (“The Pareto-Lévy Law and the Distribution of Income”, International Economic Review, Vol. 1, No. 2 (May, 1960), pp. 79-106)

    He also had an early JPE article and lots of other papers on this sort of thing, it seems.

    The point is, Pareto discovered in 1906 that the income distribution follows a power law (for the higher part of the distribution). And this has profound implications, one of which is that it is scale invariant (like a fractal, but I think a fractal has a lot of other features), and another of which is that if you add up populations that follow a power law, the sum also follows a power law, and the tails are very fat relative to the normal distribution. I guess this sort-of explains your pictures. Though really, not every population has the same exponent in the power law, as income distributions are not all equally even, say across countries.

    Best regards,


    Posted September 17, 2010 at 1:16 pm | Permalink

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  • About Aid Watch

    The Aid Watch blog is a project of New York University's Development Research Institute (DRI). This blog is principally written by William Easterly, author of "The Elusive Quest for Growth: Economists' Adventures and Misadventures in the Tropics" and "The White Man's Burden: Why the West's Efforts to Aid the Rest Have Done So Much Ill and So Little Good," and Professor of Economics at NYU. It is co-written by Laura Freschi and by occasional guest bloggers. Our work is based on the idea that more aid will reach the poor the more people are watching aid.

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