Why is the gap between the rich and the poor getting wider?

From ....   http://www.quora.com/Why-is-the-gap-between-the-rich-and-the-poor-gettin...

There are 200 trillion possible ways to sort thirty (distinguishable) balls into three urns. Of those, only 5.5 trillion ways place exactly ten balls in each urn. The vast majority of distributions will contain some kind of inequality. Don't believe me? Think of three random numbers. What are the odds that an average person would instantly think of three identical ones?

On a macro scale, wealth inequality exists because it is much more likely for a random distribution to partition a fixed sum of resources between its recipients in a heavily unequal manner.

Now some of you out there (and those who answered below) will protest that hey, no random forces gave me my wealth, I had to earn it fair and square, don't you dare wade in with your mathematical mumbo-jumbo and try and explain this away to random forces. To which I will respond: yes, you are right, human agency plays a part in the rich-poor divide, which is why inequality is even worse than random. Hang on to your seats.

By a brute-force maximization of entropy, the most likely distribution is exponential in the amount of wealth:

Pr(x)∝e−kxPr(x)∝e−kx

This means that the chance of you owning twice as much wealth as you do now, is the square of the chance of you owning what you do now (and since probability is less than 1, the square is far less). If the probability of you owning a thousand dollars is 10%, then the probability of you owning two thousand dollars is only 10% of 10%, or 1%. Or below:

Intuitively, this makes sense because if randomness is truly at play then your chances of owning two thousand dollars should be the chances of you owning a thousand dollars, and then another thousand dollars again. This is also known as a Boltzmann distribution, and its derivation is a matter of standard statistical mechanics, for example: Page on ntu.edu.sg (pages 7-11)

Now the exponential distribution is a shitty, shitty distribution as it stands.

On the above graph, a completely even distribution would be a straight line from bottom left to top right: the poorest 10% of the population would have 10% of wealth, the next 10% would have another 10%, and so on. (The measure of how much the actual line curves from this ideal is the Gini coefficient.)

Instead, the poorest 41% of the population is sharing 10% of the wealth; meanwhile the richest 2% also has 10% of the wealth to itself. In other words, on average the poor people fighting for the lowest slices of pie are 20 times poorer than the rich people fighting for the best.

But wait! If you are at all familiar with the data on income inequality, you know that things are actually much worse than this. In 2013, 10% of the pre-tax income in the United States was earned by the top 0.1% of households. (Remember, in the random exponential distribution, this would have been the top 2%.) You think I'm blaming market forces? I'm not. We are 20 times more unequal than randomness predicts. 

And the reason is that on a macro scale, wealth inequality is as bad as it is now because, via multiplicative wealth transfers, rich people live in a world of their own.

When I told you earlier that randomness predicts an exponential wealth distribution I lied - or rather, concealed half the equation, which is that to predict a distribution you need to specify a transfer mechanism that shuttles wealth between people. (The only thing interesting about a distribution of wealth, after all, is how strongly it holds up in the face of billions of daily transactions between people - if it changed drastically every second it wouldn't be worth studying.)

Most people transfer wealth additively: they add to, or subtract from, their wealth in well-defined amounts that don't depend on their existing wealth. Let's say you own about ten thousand dollars worth of assets, with which you hold down a job and buy yourself stuff. Now let's say rich uncle Bob dies and leaves you another ten thousand dollars. How much does your monthly income and expenditure change? Apart from an initial oh-gosh-I'm-rich shock, your income and expenditure will quickly return to their pre-endowment levels. Becoming twice as rich will not allow you to get a promotion that doubles your salary. Nor will the roadside vendors (or, if your preferences differ, fancy French restaurant) charge you double for a Coke (or Dom Perignon).

It turns out that this additive transfer of wealth, which doesn't take into account your existing wealth levels, plugs into a mathematical device called the Fokker-Planck equation and spits out the Boltzmann distribution we encountered earlier - the top-2%-has-10% distribution.

It also turns out that there exists another kind of transfer of wealth called multiplicative transfers of wealth. Imagine that you have ten million dollars worth of assets, which includes a Coke (or Dom Perignon) factory. Now let's say super-rich uncle Bob dies and leaves you another ten million dollars. How much does your monthly income and expenditure change? For most rich people, it doubles: you can buy another factory. Hence multiplicative: rich people gain and lose money as a fraction of their existing wealth. This creates a tendency for wealth to accumulate with the wealthy: the more you own, the more you earn, and thus the more you own, and thus the more you earn ... (This is sometimes cited in the literature as Gibrat's law, or qualitatively as the Matthew effect.)

When you plug multiplicative transfers of wealth into the Fokker-Planck equation, you get a power-law distribution:

Pr(x)∝x−(1+α)Pr(x)∝x−(1+α)

which makes it even less likelier for people to be rich, and therefore much more likely for people to be poor:

(The αα used in the power law above is 1.5; income inequality in the US stood at 1.9 in 1985, stands at 1.3 today, and when the number reaches 1 the model predicts that rich people become infinitely rich. Like, divide-by-zero-this-makes-no-sense rich.)

The income distribution curve looks all kinds of even worse than exponential:

Congratulations, we've now widened the poor to over 47% (tsk tsk) and concentrated the top 20% of wealth in the richest 0.5% of the population. Even the top 0.5% are pissed at this distribution, because the top 10% of wealth is now concentrated in the richest 0.1% of the population - even the super-rich are four times poorer than the super-super rich.

"But," you may protest, "the super-rich worked hard to get super rich! Too bad for poor people!"

Against which I present, evidence (a), trends of income distribution in America over time.

Diagram from [0912.4898] Universal patterns of inequality. Note firstly that the Gini coefficient (top: higher coeff = more curvy distribution line = more unequal distribution) matches the percentage of total income earned by rich people (bottom graph, red curve), showing that income inequality is caused more by rich people getting disproportionately rich than by poorer people getting poorer. Note secondly that the αα curve looks, again, like the inverse of the total income earned by rich people. (Remember, low αα = richer super-rich people.)

Note thirdly: when did the percentage of income earned by rich people spike? During the dotcom bubble, and then during the housing bubble. In other words, rich people make money off market craziness. Do we really want to argue that rich people worked harder and smarter and deserved to earn more money by fudging around with subprime mortgages? (Note even that the average income of poor people increased steadily during all this crazy up and downs: poor people have always worked hard, rightwing politicians notwithstanding.)

Now let's look at (b), income distribution in Australia.
 

Diagrams (this and the next) from [physics/0601176] A study of the personal income distribution in Australia. Note that the Australian population largely follows the exponential distribution (left graph), but when you zoom in on details (right graph) there is a massive spike around the annual income of just under AUD10,000 (or, in scientific terms, 10^4). That happens to be the minimum wage, which is a strong argument for government welfare: the free market will screw people over, unless some rational force that commands popular allegiance does something to help them - and it can. See that wonderful gap at the top left corner of graph (b)? That's about 10% of the population who should be earning under AUD10k a year but isn't. Progress!

Even with all that nasty government welfare, though, the trend of the super-rich still exists:

See how the graphs break away from the exponential curve around the middle of the graphs (a) and inset (c)? That's the super-rich switching the distribution from exponential to power-law, thus becoming way richer than the exponential distribution predicts. This is the second strong argument for government welfare: rich people are already breaking the free market. All the government does is even the odds on the other end of the scale.

So why are the rich getting richer, and the poor getting poorer? You can't even blame the free market. According to a sensible, objective, mathematical analysis of wealth transfers, capitalism makes rich people super-rich. Is that right or wrong? Math can't answer that question. If you really believe that the super rich top 0.1% are really that much better human beings that they deserve to earn a bajillion times more money than the poorest half of humanity, nothing I can say will dissuade you from your version of the truth.

Thankfully, in the blue corner with the rest of us, there's John Oliver. (I rest my case.)
 

Some technical notes:

1. Savvy eyes will realize that the distributions I cite are income distributions, whereas I talk about wealth transfers. Unfortunately, governments require people to list their income annually and their wealth almost never, which is why physical economists are much better at doing theory with income distributions than wealth distributions. However, to a large extent, income is linear in the time derivative of wealth, so that both are good proxies (i.e. you don't become super rich insta quickly without having a huge income).

2. Savvier eyes will note that I made a blatantly silly probability distribution function when introducing the power law; shouldn't there be some much smoother interpolation between them? To which I say: yes there is a smoother way to do it, but no I am not paid enough to do it for you. Anand Banerjee and Victor M Yakovenko are paid enough to, though, and if you check out their paper you will see the same results to a far larger degree of mathematical rigour.