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Wealth and Inequality – Pareto, Gini and Contingency

January 1, 2012

The nature of inequality is a topic that is getting a lot of attention these days.  It’s worth taking a careful look at how inequality arises in our economy.  In fact, inequality is built into capitalism and it perpetuates itself, steadily increasing, even in the absence of greed.  Inequality naturally arises whenever future wealth is contingent upon the wealth you already have — almost the definition of capitalism — even if individual decision-making is no better than a throw of the dice.

Perhaps you’ve heard that 70% of the income is made by only 30% of the workers.  More curiously, if you look at the top 30% you will find that of those 30%, 70% of their incomes is made by 30% of the 30%.  In fact, you can keep doing this kind of dissection and you will come close to the actual income distribution in this country,  the form discovered by Pareto in 1906 to describe the distribution of wealth in society.  The Pareto distribution has a “fat tail”, because there are a lot more individuals at the extreme high-end tail than you would expect from just a random distribution with a similar spread.

The Model

I made a little Excel spread sheet experiment to see if I could generate a Pareto distribution with a simple model.  You can download the spreadsheet here.  In this model, wealth is won or lost with the toss of the dice.  Individuals start off with a modest nest egg that they can invest – but their success in the “market”  reflects the reality that the market as whole is a zero-sum game.  Their chance of winning is just the same as their chance of losing.  Statistically, this keeps the total average wealth more or less constant, but it allows for there to be individual winners and losers.

My little model starts out with all individuals with 100,000 units of wealth.  We roll the dice and the individual’s fortune is equally likely to increase or decrease up to a volatility rate, R.  We continue to roll the dice another 50 times, each time using the previous wealth value as the basis for the random percentage increase or decrease.  The Excel spread sheet contains 30,000 individuals to generate enough statistics to see the trends in the tail, and the results are shown in the charts above and below.

I arbitrarily set the volatility parameter, R, to 22% because after 50 cycles, the resulting distribution had a Gini coefficient  of about 0.46, which is  about the same value as for the U.S. income distribution.   (Note that the actual wealth distribution in the U.S. has a Gini coefficient of about 0.78.  The wealth distribution is much more skewed toward the wealthy than is the income distribution.)   The Lorentz curve for the model’s wealth distribution is shown below.  (Max O. Lorentz and Corrado Gini were turn-of-the-century thinkers that established the quantitative mathematics we still use to describe inequality.)

Using the Lorentz curve, we can do a little graphical math and find the Gini coefficient for the distribution.   The green curve in the upper chart shows the Pareto distribution that would have the same Gini coefficient found in the lower chart.  The wealthy “tail” of the distribution shows a clear power law characteristic.  My eyeball power law fit (red curve), does a pretty good job where the tail statistics are bigger than single digits.  The Pareto fit isn’t bad either, considering it was fit to the entire data set via the Gini parameter, rather than just to the wealthy tail.  For an individual sitting in the power law tail, the landscape looks the same regardless how far up the tail you go.  There are always those few that are richer than you, and the many that are poorer. The psychological stratification seen by an individual is pretty much the same all along the wealthy tail.

A few interesting things come out of playing with this model.  First, the volatility parameter strongly affects the rate of inequality growth.  More volatility means the bets are larger as a fraction of wealth, so changes each cycle are larger.  Second, although the growth rate of inequality slows after several cycles, it never stops getting larger.  The nature of the process is to spread the wealth; however, there is a zero lower bound but no upper bound.  This behavior is illustrated in the chart below.  

Discussion

Our model for wealth distribution generates reasonable looking distributions with only two requirements.  First, changes in wealth are proportional to an individual’s existing wealth.  Second, an individual’s success is merely a matter of contingency, the roll of the dice.    In many ways, these simple requirements describe capitalism.  Wealth is generated based upon the work of accumulated capital (wealth) rather than labor.  In practice, wealthy investors are making bets with peers that have similar levels of wealth.  Since neither side is actually creating wealth from raw goods, our model approximation, that on average there are equal numbers of winners and losers, is justified.

There are good arguments that too much inequality is detrimental to social well-being.  It is instructive to realize that the underlying mechanism for inequality growth need not include deliberate actions by wealthy individuals.  Without deliberate mechanisms to redistribute wealth downward, inequality will grow without bounds out of statistical necessity.  Managing inequality means introducing mechanisms that redistribute wealth more equitably which are at least as strong as the statistical effects which naturally concentrate wealth.

This is where greed comes in.  In practice, those that have attained wealth, even if by the throw of the dice, feel entitled to their fortune and resist any attempt at leveling the playing field.  Whether it is luck or skill that is involved in negotiating one’s bets, winners consistently attribute their success to skill, and with that comes a sense of entitlement to their winnings.

One  mechanism that would reduce the problem would be policies that reduce market volatility, since  limiting the size of gains and losses reduces the growth rate of inequality.  Tobin taxes on financial transactions, which penalize speculation, would seem to offer such a benefit.  Inheritance taxes which limit the transfer of wealth over generations are also extremely important to break otherwise ever-growing concentrations of wealth.

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8 Comments leave one →
  1. Jesse_Fan permalink
    January 1, 2012 11:25 pm

    Gary,

    /* Second, an individual’s success is merely a matter of contingency, the roll of the dice. In many ways, these simple requirements describe capitalism. */

    I cannot agree with the second statement. An entrepreneurial venture is not a roll of dice. It is a well-thought out, risk-evaluated undertaking and therefore cannot be equated to a totally random event.

    /* In practice, wealthy investors are making bets with peers that have similar levels of wealth. Since neither side is actually creating wealth from raw goods, our model approximation, that on average there are equal numbers of winners and losers, is justified.*/

    I think your model is justified based on financial firms that are wealthy betting against each other to get even more wealthy. In fact, reality is even worse. Their losses are socialized, as in the tax payer backstops it.

    While I agree that what you say may apply closely to crony-capitalism, I don’t think that real capitalism (with failure penalization) will work out the same way.

  2. January 2, 2012 4:41 pm

    Hi Jesse-Fan,

    You have to realize that any simple model has to make unrealistic assumptions. My point was more that you need not have “a well thought out, risk-evaluated undertaking” to generate the wealth profile we have in this country. The real question is how you apportion success in my model. I require that there be no net increase in wealth globally by requiring overall and even chance on the flip of the coin. This may not be realistic on an individual basis – but it still must be true overall because my model has no real “production” in it, just “reinvestment” of current wealth.
    Often employing global “boundary conditions” helps get insight on macro-economic trends that you will not get by just considering choices of the individual.

  3. auntiegrav permalink
    January 3, 2012 8:43 am

    “Without deliberate mechanisms to redistribute wealth downward, inequality will grow without bounds out of statistical necessity. Managing inequality means introducing mechanisms that redistribute wealth more equitably which are at least as strong as the statistical effects which naturally concentrate wealth.”
    Actually, a more efficient long-term strategy is to have mechanisms to PREVENT wealth accumulation in the first place, which leaves resources more available to a broader population for local use. Wealth accumulation is a form of denial of access. Preventing initial overaccumulation reduces the need for the size of the mechanisms to redistribute resources, and the overhead costs. It isn’t the ‘bust’ in the bubble and bust process that is the problem, it’s the growth part.

  4. January 3, 2012 6:44 pm

    The easiest solution is to only tax assets, not income or sales.

    This was the case before the 1913 income tax amendment.

  5. January 3, 2012 9:49 pm

    auntiegrav, Would be nice, but my point is that there’s no way to PREVENT accumulation under capitalism.

    John A.B., There are lots of “solutions” – the problem is that those with money have the power to make sure you don’t tax THEM.

  6. no name permalink
    January 4, 2012 5:30 pm

    I found this piece quite interesting. By far the most important part was,

    “Without deliberate mechanisms to redistribute wealth downward, inequality will grow without bounds out of statistical necessity. Managing inequality means introducing mechanisms that redistribute wealth more equitably which are at least as strong as the statistical effects which naturally concentrate wealth.”

    Although I strongly disagree with many of your primacies and the zero-sum game, they aren’t the point of the article. Your mathematical-model shows the inherit inequality compounding due to your current circumstances. I think this is a very valid criticism and I would rewrite the article around your point rather than your introductory primacies. It would be far stronger.
    It allows for the debate segregated into what portion of wealth is accumulated from skill, entrepreneurialism and persistence and what portion of wealth is accumulating on its own.

  7. Richard Crowder permalink
    November 25, 2012 9:42 am

    Gary–great stuff! Question: as your model uses 50 cycles, what would be a realistic equivalent in real time, given the 22% parameter? My thought is that if we can get a good sense of the amount of inequality that would be generated per year, we can estimate the (undoubtedly large) effect of luck in actual wealth inequality over time.

    • November 25, 2012 11:47 am

      Hi Richard,
      As tempting as it is to try to make some quantitative estimate about “actual” effects of chance from my simple model, I would take caution. Simple toy models like this are designed to provide insight rather than have predictive value. My message is that this effect is part of the inequality “problem” that is not considered. There is a natural instability in capitalism that leads to ever increasing inequality and if we wish to have with a similar social hierarchy as we go through life, we can’t sit idly by, invest our 401k’s in the stock market, and assume that doing nothing means nothing happens!.

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