Old Money Goes Broke

My last post talked about inequality among sand grains, diamonds, firms, and cities. Specifically, that their sizes are distributed like lognormals, but with thicker power law tails. I noted that firms and cities are distributed quite unequally, with a (Zipf’s law) upper tail power near one.

In this post I’ll focus on wealth. In the 1890s Pareto found that the (upper tail of) wealth and income are distributed as power laws. Recent studies of US and world rich folks

estimate powers of 1.3 to 1.5, similar to Pareto’s original findings. This distributes individual wealth more equally than firm and city sizes.

Consider a simple differential equation model:

w‘ = s*w + c*(1-w)

Here the time rate of change w’ of an individual’s wealth w is given by a zero-mean randomly-fluctuating proportional growth s, and a redistribution c. This equation gives a steady state distribution proportional to:

exp((1-a)/w)*w^(-1-a)

This approaches a power law for large wealth, with power a = 1 + c/s. This model illustrates two key points:

1) While a (Zipf’s law) power of one implies no local net change, as with cities and firms, a power above one implies net local change. In particular, the wealth of individual rich (w>1) folk tends to fall on average, while the wealth of individual poor (w<1) folk tends to rise on average. The numbers of the slowly-getting-poorer rich are only held steady by a large influx of recently poorer folks. On average, old money goes broke, while the poorest bounce back.

2) Risk-averse folks (i.e., most everyone) dislike fluctuations s, and would prefer to eliminate them. But when people are forced to suffer larger fluctuations s, the distribution of wealth will spread out, creating more very rich people. Thus policy changes that result in there being more very rich people do not necessarily favor rich people. Policies that induce larger fluctuations s create more very rich but hurt each one of them. In fact, very rich folks are often especially risk averse, investing primarily in bonds. While the US has more very rich folks than other nations, and more than in prior decades, this might be because of policies forcing the rich to suffer more challenges to their positions, and to hold larger stakes in their enterprises.

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  • Vaniver

    This focuses on the very rich; you might find fascinating (if you haven’t already seen it) work that looks at the other 98% of the population. Victor Yakovenko at UMD did most of the work I’m familiar with:

    http://www.physics.umd.edu/~yakovenk/econophysics/

    1.5 probably has the best pictures, but they make the most sense in chronological order.

    Essentially, the lower ~98% of the population has income that follows a Boltzmann distribution, i.e. they’re in thermal equilibrium. The upper 2% follows a power law whose scale tracks the S&P 500.

  • Vacslav

    Or, consider a caricature society where everyone initially has 10000 units of wealth.

    On every time step each member of the society gives a fraction of their wealth to a random person. After a few iterations (hundreds or thousands, depending on the fraction of wealth to be given away on every step) there are extremely wealthy individuals and very poor individuals. Some of the very poor individuals, though, will become extremely rich, and almost all rich will become poor – in the next epoch spanning several hundred or thousands of steps.

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  • genauer

    @Robin,
    what is wrong with your equation is the simple fact,
    that most folks at the top (exception Dell, and I dont know these folks below Ballmer) (Jobs and the Aldi brothers in Germany come to mind as well) came from often poor to just affluent background at best, certainly not the upper 2 %, and made it in just one generation to the very top.
    “Outliers”, only possible in certain technology change periods, with typical growth rates of 20 % and often much more, used to a low consumption life style, “hungry”, high-beta (>4), “putting all your eggs into one basket and watch it” (Andy Grove). This was also similar in Germany after 1850. And from the top it goes downwards. Even many countries seem to have this one/two generation (50 – 70 year) run from like nowhere to the top. “First modern society” Dutch 1585 (Fall of Antwerp) – 1651 (Navigation Acts)in the 1600s, etc., Japan 1854 black ships arrival until 1905 sinking the Russian fleet. Germany 1864 -1914, China 1975 – 2025

    It is not some “continuous” rise, but more of some one giant leap, like a vulcano. Rise with 30 % (per anno) in one generation, and trickle down with 5 % in the next 3 generations.

    @vaniver
    interesting link, thanks !

    • http://www.mccaughan.org.uk/g/ g

      genauer, let’s take a look at the people at the top of that list, in order. Gates: rich family. Buffett: rich family. Allen (I’m assuming this is Paul Allen, though it’s news to me if he’s really #3 in the world): prosperous but not rich family. Waltons: very rich family (though one generation back is a different story). Ellison: middle-class family. Dell: rich family. Ballmer: rich family. Anthony (I think this is Barbara Cox Anthony, who is now dead): very rich family. Chambers (I think this is Anne Cox Chambers): very rich family, the same one as Anthony. Kluge (presumably John Kluge): poor family, so far as I can tell.

      So, of the 14 people called out by name at the top of that chart, we have: half of them (five Waltons, two Coxes) inherited their wealth from their super-rich parents; four more (Gates, Buffett, Dell, Ballmer) were from rich families (surely top 2%); then we have one wealthy-ish (Allen), one middle class (Dell), one poor (Kluge).

      I don’t know about you, but I wouldn’t describe that by saying that “most folks at the top came from often poor to just affluent background”. Most folks at the top come from families that are (at least) very comfortably off. It’s easier to take big risks — starting a company, investing a lot in a high-risk high-return venture, etc. — if you have a good safety net.

      In any case, I don’t think Robin’s equation is at all intended to be a perfectly accurate universal model of how people’s wealth evolves. There’s a reason why both Robin and the paper he quoted it from call it a “simple model”.

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  • genauer

    @g
    the plot above is from 2000. That explains a lot

    Otherwise: Good point. We have to say this a little more carefully.
    I use loosely “affluent” for the upper end of the “thermal distribution” lets say upper 10 % minus upper 1 – 2 % and “rich” for the power law, upper 1 %.

    For Allen, I read on wiki: father associate director of library, which I put at the lower end of affluent.
    Jobs was definitely not even affluent (I assume you know his step parent story from his 2005 Stanford speech?).
    Ballmer grew up “in an affluent community”, …., but that means most likely that his parents where rich by our definitions, I agree now.
    Ellison “grew up in 2-bed room apartment” what I would not even call “affluent”, outside of NYC 🙂
    For the Waltons I took the Father Sam, who basically made most of this and started poor.
    Gates father was a lawyer, something I associated here not with “rich”, but a rich one in deed, I checked.
    In this way I arrived at a picture of about 2/3 “not rich”.

    I wanted to stretch that in most cases it is a one generation fast rise to the top, and your point that it is much easier, when the jumping off point is close to the “rich” area, is well taken.

    I started with a 3-4 bedroom senior director father (3 kids, public officer, safe, but no financial assets, I would call that not even “affluent”), sold my first software during school, but it would have been unthinkable to break off studying and founding a Software company (and where to get the seed money?, like minded co founders?).

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