Wealth and Inequality – Pareto, Gini and Contingency

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grondeau's picture
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Wealth and Inequality – Pareto, Gini and Contingency

 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.  


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 underling 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.


darbikrash's picture
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Pareto superior

That’s very interesting, fascinating article. This would seem to show an inexorable tendency towards unequal wealth distribution irrespective of merit and work ethic, a principle at odds with many current (and some not so current) sensibilities.

It’s worth noting that the central idea to the replacement of Feudalism and Mercantilism with Capitalism was the notion that an egalitarian society would be the eventual result. No longer would the masses be subservient to the landed nobility and intellectual direction of the Church, instead we would have the Enlightenment principles of science and reason to direct us to recovering individual rights and liberties. Capitalism gave forth the premise that with hard work and perhaps some risk capital anyone could elevate themselves to the station of the landed nobility, and several period thinkers drew on this concept to project Enlightenment inspired ecosystems to justify and valorize this pursuit.

The holy grail of this post Enlightenment thinking was to secularize, and extend the operating theory of Capitalism into maximizing individual utility while at the same time insisting on unfettered personal liberties- a difficult balancing act to be sure. The most influential solutions to this challenge were the thinkers that espoused so called “self regulating” systems of government, of which there were many, such as Adam Smith, who proposed that acting in ones’ self interest (maximizing individual utility- or pursuing Pareto superior choices) would act to benefit society as a whole, and this would mean a minimalist government structure with the capitalists’ self governing. In direct contrast, Marx proposed different means to meet a similar outcome, he saw Capitalism imploding in on itself, and ultimately being replaced by a series of alterative political structures that would also evolve into a structural-less (or non existent) government self modulated by the worker class.

Neither ideology has worked, Adam Smith’s vision killed by capitalistic excesses and a return to precisely the same conditions as it was designed to replace, and Communism defeated by inability to recognize that the principle of superabundance is unachievable, as resource scarcity is endemic to the human condition. 

The probability and statistical outcomes of capitalistic endeavor are exacerbated by individual seeking Pareto superior outcomes (Utilitarianism) at the expense of the rest of society. In addition, the notion of differential accumulation is starting to be discussed, which suggests that capitalism contains not only an unavoidable growth imperative, but that successful capitalists seek differential accumulation, as it matters not so much to them that everyone is making money, but they must make more that everyone else to realize differential gain. 

This is because we tend not to recognize that money is in effect social power, and unequal distribution of money is a de facto acknowledgment of unequal social power, again, right back to the days of Feudalism.

In my view, the issue of personal liberties with respect to Pareto superior pursuits of wealthy individual and corporations has reached the point of national crisis. We have circumstances where “investments” or more accurately, wagers made by high net worth individuals and organizations put non participants at significant risks, unbeknownst to them. The principal actors stand to gain if their “investments” or wagers pay off, and if they do not, the magnitude of the transactions is so large (via leverage) that they cannot pay them off and can and do topple entire industries such as the real estate market, and potentially, the $600+ trillion derivatives market. This practice eventually can, has and will (further) wipe out non participants, and in the process deny them of their own private property and personal liberties- even though they have no upside in these gambles, and in most cases were not even aware they had counterparty risk.

Yet we hear no substantive objection from any of the political parties or ideologies with respect to regulating the free market, nor protecting any private property owned by non participants in this financial swindle, rather we hear very disturbing commentary that de-regulation and reduced taxation is the preferred solution. Another common defense is the statement that what we are describing is not really capitalism at all, rather some deviant strain of socialism that has crept into our psyche when we weren’t looking. Well it most assuredly is capitalism, and regressive idealism will not play for long in the sentimental search for a simpler time.

Ultimately, we will come to realize that capitalism comes in three strains or variants, Industrial Capitalism which is the more or less traditional variant that comes to mind for most people, Mercantilist Capitalism, which concentrates on the sale of the merchandise produced by the Industrialists (for example Wal Mart), and the Money Lending Capitalists, which create profit centers around the loaning of money to the other two variants (Goldman, JP Morgan, etc). Each is different in its organizational and operational structure, but they share common traits:

a.)    Tendency to centralize and consolidate, either in monopoly, monsopy, or social power.

b.)    Tendency to avoid creative destruction by creating subversive internecine trade agreements.

c.)    Capture of regulators

d.)    Capture of legislatures.

e.)    Extensive spending in lobbying campaigns.

f.)     Massive public advertising campaigns designed to advance the valorization of their role in society.

g.)    Advancement of a political system, regardless of political affiliation, that will allow the status quo to continue, which is to say, accelerate the differential accumulation.

h.)    Disproportional (low) taxation, while again valorizing their contribution.

Of all of these the Financial Capitalists have the least effective regulation. It is my belief that taxation is but one tool to try and balance the influence of outsized social power, intervention into the types of  transactions must also be examined and regulated. Ponzi type, as well as speculative, finance schemes must be stopped, as many of these are now of the size that non participants are losing, or at risk of losing, their private property, as well as individual liberties due to transactions that are taking place without their  knowledge.

I’ve enjoyed your post very much, and hope you continue adding content to the forum.


Differential Accumulation




xraymike79's picture
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Found a nugget here.

Some intuitively know capitalism is a rigged game without being able to illustrate statistically and mathematically as Grondeau has done. Thanks for a gem of a post and an equally good response by DK. I love the graffiti pic.

 It's no wonder most don't really understand that the very foundation of their economic system is what's the problem, and not some aberration thereof. As Jim Morrison is purported to have said: 'Whoever controls the media, controls the mind'.


Rihter's picture
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love the chart

Steve Keen built a model that shows some good info. Thought I's add to the discussion by sharing it.

Keen crash course part 1

There are 3 parts.

TheSilverJournal.com's picture
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Wealth Redistribution Shrinks the Pie

Getting rid of the wealth gap completely means that there is no longer an incentive for anybody to work since if you go to the beach or teach your grandchildren, you'll still be as wealthy as the next guy who is working. Therefore, we can see that some wealth gap is necessary. I can also make an argument that a larger wealth gap also gives a greater incentive to gain wealth.

The problem area with gaining wealth is if it is done buy leaching off of the government through crony capitalism. The more centrally planned an economy becomes, the more necessary it is to become politically connected if you're trying to become wealthy. Redistributing wealth by taking it from those who have not leached off the government only shrinks the economic pie because it is essentially taking wealth from the productive and giving it to the unproductive.


Jim H's picture
Jim H
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Models are too simplistic...

I think the fact that volatility can work against the average investor is a useful conclusion.. but I don't think you can draw any conclusions about the usefulness of capitalism with a few variables.  

Xray says that Capitalism is rigged... and indeed, ours is these days.. but I don't see how this is a knock on capitalism in general.  If we went back to sound money, had a transparent market in trading derivatives (and everything else), got the money out of politics, enforced laws on the books with regard to financial malfeasance, separated savings from investment banking (again), and either got rid of the FED, or vastly limited it's powers (to something like preventing bank runs)... then would the system still be rigged?  I contend it would be, "unrigged" then.   

Jim H's picture
Jim H
Status: Diamond Member (Offline)
Joined: Jun 8 2009
Posts: 2391
Models are too simplistic...

I think the fact that volatility can work against the average investor is a useful conclusion.. but I don't think you can draw any conclusions about the usefulness of capitalism with a few variables.  

Xray says that Capitalism is rigged... and indeed, ours is these days.. but I don't see how this is a knock on capitalism in general.  If we went back to sound money, had a transparent market in trading derivatives (and everything else), got the money out of politics, enforced laws on the books with regard to financial malfeasance, separated savings from investment banking (again), and either got rid of the FED, or vastly limited it's powers (to something like preventing bank runs)... then would the system still be rigged?  I contend it would be, "unrigged" then.   

grondeau's picture
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Posts: 35
re: Wealth and Inequality

Darbicrash, I like your division of capitalism into your three varieties.  It reminds me that a few years ago, GM was making more profit in its financing operations than it was in building cars.  Certainly the financial sector is almost entirely making its profits with bets between its members and with the other sectors of capitalism you have mentioned.  This evolving nature of capitalism toward more financial activity and less actually providing goods and services only makes clearer the way present-day capitalism is more about gambling that it ever was.  I introduced the concept of contingency to highlight this aspect of capitalism and to make my simplistic model tractable.

 Xray, Thanks for the kudos.

Rihter,  Thanks for the link.  I’ve been interested in dynamic system modeling since the days of Meadows and Forrester’s Limits to Growth studies.  One point of understanding such models is that any quasi-realistic system will be cyclical or have issues with stability that need to be addressed.  In practice, these issues of stability are resolved by negative feedbacks that are supplied by governments.  The desire to get the government out of our business I think is potential very destructive because of the natural dynamics of complex economic systems left unchecked. 

Silver Journal, I don’t have a problem with reasonable wealth distribution.  I agree that the incentive to advance up the ladder is an important motivator for many of us, and can make us more productive.  My point is merely that unregulated capitalism does not lead to equilibrium with regard to a wealth distribution – it gets progressively steeper with time (even without a concerted effort by those at the top to get more).   I point this out because this poses a conundrum for those who are both populist and libertarian in perspective.   

Jim H, All useful models are too simplistic!  If they weren’t simplistic, it would be much harder to tease out the wisdom they suggest.  Your desire to have government work like it is supposed to, with a well regulated financial system and a democracy without influence of vast sums of money, illustrates the real problem when power comes with money, and money tends to accumulate to the few who can then control the government.  The wisdom of my model suggests you should beware of solutions that ask for less regulation and taxation of the wealthy! 

Thanks for all the comments!

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