Friday, September 11, 2009

The Real Secret of the Super Rich (Part 1)

The super rich are different from the rest of us – way different. According to the numbers, they are of a different kind, a different species even. Back in the old days this was common “knowledge.” The upper classes declared themselves a different lot, and mere commoners bowed before them, addressing them as “lord” “lady” and similar distinguishing honorifics. In this more enlightened age, we know better. We have examples of men working their way from rags to riches, or (think Bill Gates) from upper middle class to owning more than many sovereign nations. Then again, maybe Bill Gates is a mutant; he rather looks like one…

Perhaps the super rich are not mutant superbeings. Maybe the super rich simply exploit a secret. At meetings of Skull and Bones, the Council on Foreign Relations, and the Bilderbergers, the Insiders chant special words of Power, invoking dark forces which allow them dominate mere mortals, and dwell on this earth in imperial luxury at the cost of their immortal souls – and their ability to correctly read CIA reports.

Maybe not. The real secret of the super rich is a bit more mundane, but it is important. While we cannot all fully use of this secret – someone needs to take out the garbage – we can all learn much from it. We can use it to better use our own abilities and become better off. This country can and should have a much larger upper class – more rich people, but a much smaller gap between rich and super rich. Some of you reading this blog may well join this broader upper class by applying the secret I am going to reveal.

While I started off silly, I was quite serious about the super rich being different. The super rich are not merely the upper end of the bell curve in money making ability. The numbers say otherwise.

Back in 1994 Charles Murray and Richard Herrnstein created a firestorm with their book The Bell Curve[1]. They had the temerity to suggest that intelligence accounts heavily for class differences, and that intelligence is hereditary to a significant degree. The politically correct were aghast; how dare anyone suggest that class differences were natural? How dare they suggest that genetics affects class!

The politically correct were wrong to vilify the authors, especially Charles Murray. Murray is no right wing extremist. He advocates for a rather large citizen’s dividend, an unconditional government check for all. (See In Our Hands: A Plan to Replace the Welfare State.[2]) Charles Murray is more progressive than the late Ted Kennedy. The idea of a bell curve is progressive. Wealth and income following a bell curve (a normal distribution) would be a socialist utopia compared to what we have today!

The super rich are outside the bell curve. That’s why my intro riffed on mutants and dark forces. To see this, let’s plot a crude bell curve to fit the rest of us. I don’t need to be precise to make my argument clear. To start with, let’s review what a bell curve is, and why statisticians look for it. I’m going to hand wave through a bit of math. For those not mathematically inclined, you can skip to the charts. For those who don’t like hand-waving, you can go to Wikipedia for more, or better yet, get a statistics book. (Maybe I’ll write a future post with an intuitive derivation.) The central limit theorem theorem states that when some variable (such as household income) is the sum of many independent random variables, said variable approximately follows a Gaussian distribution:



where

and


Now, all we need are a few data points to set the average and standard deviation, and we will have a probability distribution for any income range – assuming that income does follow a normal distribution. Wikipedia has a nice page on household income in the United States. I’m going to yank some numbers off this page and fit a curve. Once again, the numbers do not have to be exact to make my point. According to this page, median household income is about $50,000, with 116 million households in the U.S. Let’s use $50,000 for our average income. (This is incorrect in reality, but would be true if income followed a normal distribution.) Scrolling down, I see that the bottom quintile (the bottom 20%) makes less than $18,500.

Pulling out my handy dandy CRC Standard Mathematical Tables, I see that the integral of the tail end of the bell curve is .2 (i.e., 20%) when x is .84 standard deviations below the norm. So we can now solve for the standard deviation.





We can view this below graphically. The red area represents those in the bottom quintile.





Now let’s look at the quite well off, those making $150,000 or more. These families make $100,000 more than the median, or 2.66 standard deviations. I colored them blue in the graph below. According to the CRC tables, we should expect .36% of the population 418,000 families at this level or higher.

Now, let’s look at those making serious money, ONE MILLION DOLLARS per year. That’s 25.33 standard deviations from the norm. The CRC tables don’t go that high. Neither does Excel. Breaking out some more heavy duty math software[3], I come up with 7e-142! That 7 times 10 to the negative 142 power! Multiply that by a 116 million and you get 8.6 times 10 to the negative 134 families making a million dollars in one year. The probability of even a single million dollar a year income is infinitesimal! The probability of elves and fairies is higher. Hollywood is indeed a fairyland!

Let’s drop down to $250,000. According to the normal distribution, there should only be 5 to 6 families making more than $250,000 per year! Go up to $300,000 and the number drops to a fraction: .0015 families. Where did all the high-power lawyers go? What about the specialized doctors?

OK, so I may be off in some of these calculations. Perhaps the standard deviation should be higher. Let’s say I’m off by a factor of two. That means we still only have 5 families making over half a million dollars. We still have no 7 figure incomes. CEOs and celebrities are still as improbable as an invasion of the flying saucer men next Wednesday.

The super rich defy the bell curve. They are way outside the norm. The tabloid readers worship and socialists seethe with envy. I, a scientist and liberator, shall debunk both The Secret and socialism. The real secret of the super rich is…nonlinearity.

I will explain further in my next post. Stay tuned.


Notes
[1] Full disclosure: I have not read The Bell Curve. I have only read the backwash.

[2] I haven’t read this book either, but I did listen to a talk by Murray on this subject.

[3] MATLAB or any MATLAB clone (such as Octave) does the trick. Here is my code:




mu = 50000;
sigma = 37500;
income = 300000;
sdev = (income - mu) / sigma;
x = sdev / sqrt(2);
y = .5 * erfc(x);
nRich = 116000000 * y;

6 comments:

Michael Bindner said...

One of the real secrets to being rich is to higher people. Because the labor market is demonstrably not free (because employers are freer to set terms, conditions and salary than employees) and cannot be (because product markets are not free - so workers have less alternatives), employers, particularly the rich, gain economic rent every time someone does work for them - rather than having the wage fully compensate workers for their contribution to the work product.

aumentare il seno said...

Can you explain why there should be a much larger upper class – more rich people, but a much smaller gap between rich and super rich? The rich people are that at expense of the rest. So why not a larger middle-class?

Carl M. said...

@aumentare:

I mean more people making in the low millions, but less in the hundred millions. More decamillionaires, fewer multi-billionaires. I'll explain more in future posts.

Clive at Five said...

I've always looked at income distribution as sort of a blackbody curve, not a bell curve. To me, it makes sense, intuitively, that there's a pooling of incomes near the bottom. Unfortunately, there isn't a real precedent for using a blackbody curve in Statistics, so it's difficult to fit the curve to census data, though one bored evening after reading "Welfare that Works" on Holistic Politics, I managed to do so with help of my Mathematica software.

Part of what makes me think a bell curve is an incorrect model is purely based on most corporate models: for every manager, there are probably ten to twenty peons; but there are fewer managers under Sr. Managers, and even fewer Sr. Mgrs. under V.P.s... The big hurdle is that first jump to a managerial position.

One may argue then, that the corporate model is flawed, corrupt, or unfair. I do feel, however - again, intuitively - that there it takes far fewer people to lead a company than it does to run one.

The other mentionable component, of course, is the pay gap between these "peons" and "lords," which, IMHO, given the added responsibility, is within merit in all but the most extreme cases... such as those found in some of the largest corporations, where the CEOs and Sr. VPs are so far removed from the workers that they're essentially just a figure-head. These are shining examples of where good Capitalism can go bad.

There are certainly counter examples within the corporate structure, and some abnormal instances outside Corporate America (Hollywood, professional sports) where we literally throw money at our celebrities in a sort of peon-lord worship we used to have. The relationship is not as dead as you think!

A long-winded response - sorry for that. Love your articles... most of them. This country needs more common-sense citizen-centric leadership and your ideas will go a long way in establishing that.

Craig said...

I keep checking back for part 2, when is it coming?

Carl M. said...

@Craig: Sorry about the delay. Have had the flu for the past couple of weeks, and a backlog of other tasks to catch up on. One of these days...