The Average Net Worth of the Forbes 2013 Billionaires

The Average Net Worth of the Forbes 2013 Billionaires The accumulation of the first trillion
§1. Summary
The average net worth U of N billionaires is obtained simply by dividing their combined net worth UN by N. Thus, U = UN/N. When all the billionaires are arranged in descending order of decreasing net worth (as in the Forbes list, with ranks from 1 to 1342), their combined net worth UN is shown to increase at first nonlinearly as N increases from N = 1 to N = 10 and then essentially linearly as N increases to 35 at which point the combined net worth crosses the $1 trillion mark. Correspondingly, the average net worth U is shown to decrease continuously as N increases. The present analysis has been continued only up to N = 35. Both the linear law, y = hx + c, and the nonlinear law, y = Axn, where x is the number of billionaires and y their combined net worth, are shown to described the data for appropriate ranges of x, or N, the number of billionaires. Both the linear and the nonlinear law indicate that the average net worth U decreases as x increases. According to the nonlinear law, the average net worth U = y/x = Axn-1, and, according to the linear law, the average net worth U = y/x = h + (c/x). The nonzero intercept c in the linear law means that the average U = y/x is NOT constant and will depend on x, the number of billionaires. This is also the reason why the average billionaires’ worth, when the list is organized by countries, is found to vary, from country to country. Nigeria with only two billionaires get the top spot in this such a list, whereas the US, with the largest number of billionaires in the world gets the 16th spot. Finally, the problem of determining the average net worth of billionaires is shown to be exactly similar, mathematically speaking, to the problem of determining the average energy U of N microscopic particles, as described by
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Planck in his famous December 1900 paper that marks the birth of quantum physics. The expression for U obtained by Planck was later also obtained by Einstein (in 1907, when he addresses the problem of the anomaly in the specific heat of solids). Indeed, the method used by Einstein is shown here to be, mathematically speaking, exactly analogous to the method used to determine averages in the finance and the business world, such as the average price of N stocks in various indices. The same considerations also apply to the billionaire problem. Thus, the extension of the ideas of Planck and Einstein, when they developed quantum physics, can be extended to interpret the behavior of many other complex problems outside physics where again one is interested in determining the average value of U of some property of interest.

Table of Contents
§ No.
1 2 3 4 4.1 4.2 5 6 7 8 9 10

Topic
Summary Introduction What is the average net worth The net worth of the top 10 billionaires A Simple linear model A Simple nonlinear model Evolution of combined net worth to first $1 trillion Average net worth by country Appendix 1: General expression for the average U any complex system (Einstein’s derivation) Appendix 2: The evolution of the combined and the average net worth for the 17 Israeli billionaires Appendix 3: The marginal tax rate m and the tax equation y = mx + c Reference List and Related Articles

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Posted at Forbes.com and on My Facebook Page (Mar 28, 2013, between 9:15 to 9:20 PM)

o

Vj Laxmanan 5 minutes ago

Dear All: Following up my recent essay on the “Median” Billionaire, I have discussed here (see link given below), the meaning of the average value of a data set is discussed here using the Forbes 2013 Billionaires list. Such lists, with thousands, even millions, of data points are being developed daily. I have also shown that the method of determining the average energy of a complex system of particles by Planck (in 1900) and Einstein (in 1907) can be generalized and applied to all such lists of observations being developed in many different fields where empirical observations are being made on a daily, monthly, quarterly, and annual basis. http://www.scribd.com/doc/132910617/The-Average-Net-Worth-of-the-Forbes2013-Billionaires From my Facebook Page (Mar 28, 2013) around 9:15 PM

Vj Laxmanan The meaning of the average value of a data set is discussed here using the Forbes 2013 Billionaires list. Such lists, with thousands, even millions, of data points are being developed daily. I have also shown that the method of determining the average energy of a complex system of particles by Planck (in 1900) and Einstein (in 1907) can be generalized and applied to all such lists of observations being developed in many different fields where empirical observations are being made on a daily, monthly, quarterly, and annual basis. http://www.scribd.com/doc/132910617/The-Average-Net-Worth-of-the-Forbes-2013-Billionaires

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§2. Introduction
In the companion article entitled, “An essay on the Median Forbes Billionaire in 2013”, see Ref. [1], we discussed the statistical concept of a ‘median’ which is widely used, for example, by the US Census Bureau (USCB) to examine changes in the net worth of US households over time; see the USCB 2011 Current Population Report, Ref. [2]. Another recent article by Gottschalck, Vornovytskyy and Smith (hereafter GVS), of the USCB, Ref. [3], uses the concept of the ‘median’ net worth of US households, exclusively, to discuss how the net worth has declined between 2000 and 2011, see also the discussions by Luhy [4] and Salmon [5] of the same ‘median’ household in other popular internet websites. As we have seen in the essay on the ‘median’ Forbes billionaires, we must be very very wary of this whole idea of a ‘median’ anything, especially the ‘median’ US household and the changes in the net worth of this phantom household as the bellwether for the decline in American prosperity. Now, let us examine another widely used concept, the ‘mean’ or the ‘average’ value. If we have N billionaires with a combined net worth UN the “average” net worth U = UN/N. Conversely, the combined net worth UN = NU where N is the number of billionaires and U their average, or ‘mean’, net worth. As we know, telecom tycoon Carlos Slim Helu is the world’s richest man, four years in a row now, with a net worth of $73 billion. He heads the Forbes 2013 billionaire list, with a total of 1426 billionaires. Their combined net worth is $5.425 trillion for an “average” U = UN/N = $5425/1426 = $3.8 billion each, see Refs. [6-9]. We will now discuss how this “average” evolves as we add more and more billionaires to our list.

§3. What is the average net worth?
The “average” or “mean” value is the sum of all the N values in a data set divided by N. As applied to the billionaire problem, it is the sum of the net worths of all the N billionaires (combined net worth) divided by the number
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of billionaires. This idea of an “average” net worth goes back to the first ever list of the 30 wealthiest Americans compiled by B. C. Forbes, the founder of Forbes magazine, in 1918, see Ref. [10]. In that first ‘Rich List”, the combined net worth of the 30 richest Americans was $3.68 B, for an average net worth of $3.68/30 = $0.123B, compared to the average net worth of $3.8 B in 2013, almost one hundred years later, for the 1426 billionaires of the world. Each year’s billionaire list is a “snapshot” of the net worth of the wealthiest individuals, at a given point in time. During the course of a single year, the net worth of the billionaires changes, on a daily, weekly, monthly, or quarterly basis. Some recent reports mention that Carlos Slim is in danger of losing his top spot because of the plummeting values of the stocks he holds. However, he will continue to remain the world’s richest man until the next Forbes billionaires list is published, sometime in the first week of March 2014. There will be no premature “dethroning” of Mr. Helu from his top spot. In other words, we are always, implicitly, considering the “time-averaged” value of the net worth. Like the molecules of a gas, which have a wide range of energies (kinetic energy, associated with their chaotic motion in different directions), or the microscopic particles within a heated body (that vibrate at various frequencies about some mean position and radiate energy at all possible frequencies), whose energy spectrum was studied by Ludwig Boltzmann (in a famous 1877 paper) and Max Planck (in a famous December 1900 paper), we now associate the property called “money”, or “net worth” with each of our billionaires. It is the time-averaged values of various properties, like energy and entropy, or billionaire net worth, which are of interest. Planck emphasizes this point in the very first paragraph of his famous December 1900 paper, which marks the birth of quantum physics. Indeed, Planck’s expression for the “average” energy U of a complex system of N particles can be derived very easily, as discussed in Appendix 1. Equations 1 and 2 given in Appendix 1 are, mathematically speaking, EXACTLY the same as the equations used to determine the “weighted average” value of the price of stocks in various commonly used market indices such as
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the Dow Jones Industrial Average, the S & P 500, the NASDAQ, and so on. If there are N0 stocks with price P0, then N1 stocks with price P1, and N2 stocks with price P2 and so on, the total number of stocks N is given by equation 1 and the total worth of these stocks (the market value) is given by equation 2, and the “weighted average” is then given by equation 3. N = ∑ Nj = N0 + N1 + N2+ N3 + … Nk UN = ∑ NjPj = N0P0 + N1P1 + N2P2 + …… U = UN/N = (N0P0 + N1P1 + N2P2 +.. ) / (N0 + N1 + N2+ …) ……..(1) ……..(2) ……..(3)

Planck actually shows us how to derive the general expression for U for any complex system. As noted in other articles (see ‘related articles’ list), energy in physics is just like money in economics, or vice versa. Thus, the idea of an “average” and how it is to be determined when we encounter a complex system, such as the system of N billionaires, or N households, is a very fundamental one. In a later paper, published in 1907, Einstein shows how to derive exactly at the same expression for the average U derived by Planck in 1900 using a different approach that has immediate applications to many problems, such as the problem of determining the average net worth of N billionaires, or N households, or the ‘weighted average’ price of N stocks. With this brief digression, let us now return to the discussion of now the “average” evolves as we add more and more billionaires to our list. This will be presented using x-y diagrams, with x being the number of billionaires N and y their combined net worth UN.

§ 4. The net worth of the top ten billionaires
The net worth of the top ten billionaires is listed below with the Forbes rank in the top row and their individual net worths in the bottom row. Indeed, this is exactly similar to the table given by Plank in his December 1900 paper, when he begins his discussion. Planck’s table is also included here for
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comparison. An English translation of Planck’s original paper can be found in the book Great Experiment in Physics, edited by Morris H. Shamos (Dover Publications (1959), see pages 301-314). Planck’s table showing distribution of energy between ten oscillators Particle no. 1 2 3 4 5 6 7 8 9 10 Energy units, ε 7 38 11 0 9 2 20 4 4 5 Total energy = 100 units of elementary ε, UN = NU = Pε where N = 10, P = 100. The top 10 Forbes 2013 Billionaires (Individual net worths) Forbes Rank 1 2 3 4 5 6 7 8 9 10 Net worth Uk 73 67 57 53.5 43 34 34 31 30 29 Combined net worth UN = $451.5 B, Average U = $45.2 B, and the Median M = (43+34)/2 = $38.5 B

Combined net worth, y [US dollars, $]

500 450 400 350 300 250 200 150 100 50 0 0 2 4 6 8 10 12

Number of billionaires, x
Figure 1: The nonlinear increase in the combined net worth y = UN (plotted on the vertical axis) as the number of billionaires x = N (plotted on the horizontal
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axis) increases. The combined net worth of two billionaires is less than the net worth of the first billionaire (the world’s richest). The combined net worth of three billionaires is less than what would be obtained with a tripling of net worth and so on. In other words the ratio y/x is not a constant and we must be careful when we try to draw conclusions based on such a simple ratio analysis. The sum of the individual net worths in the second row is analogous to the total energy of the particles in the second row of Planck’s table. Since this is an ordered list, with an even number of billionaires, the ‘median’ equals the average of the net worths of billionaires number 5 and 6. The ‘average’ or the ‘mean’ value is higher than the ‘median’ value. The latter is being mentioned here in passing. The ‘average’ is higher since the top four in the list each have a higher net worth than the bottom four in the list. The rising combined net worth, with increasing N, is illustrated graphically in Figure 1. The combined net worth figures being plotted here are given in the third row of Table 1. We sum the net worths, one by one. Notice the clearly non-linear trend that is evident here. Table 1: The combined worth of the top 10 Forbes 2013 Billionaires Forbes 1 Rank Individual net worth 73 Uk 2 67 3 57 4 53.5 5 43 6 34 7 34 8 31 9 30 10 29

Combined Net worth 73 140 197 250.5 293.5 327.5 361.5 392.5 422.5 451.5 UN The “average” of the numbers in the third row is ym = 290.95. The average of the number in the first row is xm = 5.5. With one billionaire, we have a net worth of $73 B but with two billionaires, their combined net worth, UN = U1 + U2 increases to $140B. It is not doubled to $146B. With three billionaires, it has increased to $197B, which is less than a

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tripling to $219B. In other words, the law describing this process of increasing billionaire worth is not the simple law y = kx or y/x = k = constant. The ratio y/x is, of course the “average” value with x being the number of billionaires and y their combined net worth. This is illustrated in Figure 2.

Combined net worth, y [US dollars, $]

500

400

y = kx = 73x Or y/x = k = 73

300

y = kx = 45.15x Or y/x = k = 45.15

200

100

0 0 2 4 6 8 10 12 14

Number of billionaires, x
Figure 2: The nonlinear increase in the combined net worth means that the ratio y/x is not a constant. Hence, the law relating the x and y, the number of billionaires and their net worth is NOT the simple law y = kx, or y/x = k. The ratio y/x will keep decreasing as the number of billionaires x increases, as indicated by the decreasing slopes of the two “rays” joining the (x, y) pairs for the first and after the tenth billionaire has been added to the list. The two dashed straight lines, or “rays”, join the origin to the first and the last (x, y) pair in our list. The slope of this “ray” y/x = k is continuously decreasing. In fact, since we are dealing with an ordered list with descending values of
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the net worth, this is a mathematical property of this system. The ratio y/x, the “average” value will always decrease with increase x, or number of billionaires. A random listing of billionaire net worths from different countries, or even from a single country like the USA might have produced a different result. This point is discussed in more detail in Appendix 2, using the example of the 17 Israeli billionaires in the Forbes 2013 list. (Israel was chosen since it appears very high the list of billionaires per capita, see Refs. [..,..]. One of the comments, from rocky2345, at the Forbes.com website, has discussed the Israeli situation. This poster states that Israel is the exception in the billionaires per capita list. Israel is not a tax haven and no one moves to that country for tax purposes.)

§ 4. 1 Simple linear relation for combined net worth for the top ten billionaires
One could now take two approaches to deduce the mathematical relation between x and y. From a purely statistical standpoint, one could still fit a simple linear relation to describe the combined net worth data for the top 10 billionaires in the Forbes 2013 list. This is illustrated in Figure 3. Alternatively, a simple nonlinear model can be used. This will be discussed in the following section after discussing the simpler linear model. The mathematical equation of the “best-fit” line through several points that seem to line up approximately along a straight line is y = hx + c where h is the slope of the line and c is the nonzero intercept made by the straight line on the vertical axis (when x = 0, y = c is nonzero, unlike the situation with the “rays” passing through the origin). Such a “best-fit” line always passes through the point (xm , ym), see solid blue dot. The value xm = 5.5, the “average” of the numbers in the first row of Table 1. This means we are dealing with a fictitious billionaire number 5.5 (between our No. 5 and 6). The combined net worth when we come to this billionaire ym = $290.95B, see third row of Table 1.
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Combined net worth, y [US dollars, $]

600

500

y = hx + c = 40.78x + 66.67 2 r = 0.977

400

300

200

(xm , ym)

100

0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Number of billionaires, x
Figure 3: A statistically significant and simple linear relation can be deduced between the x, the number of billionaires, and their combined net worth, y. The linear law is y = hx + c = 40.78x + 66/67 with a nonzero (and positive) intercept c. Hence, the ratio y/x = k = h + (c/x) = 40.78 + (66.67/x) will keep on decreasing as the number of billionaires x increases. The slope h of the “best-fit” line is determined as follows. Notice that some points fall above the line and some fall below the line. If y is the actual value and yb the value on the best-fit line, the “error” made in the prediction, or the vertical deviation (y – yb) is either positive or negative. The sum of all such deviations, or “errors” is, of course, zero. This is what we mean by an “average” and the best-fit line passes through (xm , ym). The square of the “errors” or vertical deviations, (y – yb)2, on the other hand, is always positive. Hence, the sum of these ‘errors’, ∑ (y – yb)2, will always have a finite positive value. In a famous paper, published in 1805, the French mathematician Legendre proposed that the slope h be determined ‘to render the sum of the squares of the errors a minimum’ (click here, or see Ref. [..], remarks made by
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Legendre to introduce this method). Hence, Legendre’s method is called the least squares method and is widely used in statistical analysis to this day. Legendre then derived the mathematical formula for the slope, see Ref.[..] which provides a simple worked example. The nonzero c is the consequence of the requirement of minimizing the sum of the square of the errors, and also forcing the best-fit line to pass through the “average” point (xm , ym). Notice that we are dealing with the “average” of the cumulative, or combined, net worths, not the average of the individual net worths. (Think of the average profits for four individual quarters Q1, Q2, Q3, and Q4, and the average of the profits for 3 months, 6 months, 9 months, and the 12 months.) This also explains why we often observe the ratio y/x behaving erratically. Sometimes the ratio increases. Sometimes it decreases. In such observations, there is an implicit size effect, determined by the variable x. Hence, in general, the ratio y/x = k = h + (c/x). In the billionaire problem, it is seen to keep on decreasing as x increases since the numerical value of the intercept c > 0. This is illustrated in Figure 3. Notice also that the slope h = 40.78 for the best-fit line is less than the slope y/x = 73 of the “ray” joining the (x, y) pair for Carlos Slim back to the origin. If every billionaire could have the net worth of Mr. Helu, all the data points will follow the dashed line in Figure 2. Indeed, it is only after billionaire No. 3 that we see significant deviation from this highest slope. The data for Bill Gates and Amancio Ortega, who hold the No. 2 and No. 3 ranks, are only slightly off. The least squares line, or the “best-fit” line, is thus a statistically averaged slope. The slope of this line h = 40.78 tells us about the rate of increase of the combined net worth as the number of billionaires x increases. The fixed slope h that we observe here can be compared to the marginal tax rate. The US tax code is a series of straight line segments with nonzero intercepts. The tax owed y is given by the equation y = mx + c where x is the taxable income. The constants m and c can be determined from the tax tables (click here, see Pub 17 tax rate schedule on page 264) and apply for a limited range of incomes. This is discussed further in Appendix 3 so as not to digress too far from the main discussion here.
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In other words, we must be careful in drawing conclusions based on this simplistic use of the y/x ratio (unfortunately widely prevalent), when we analyze many such empirical observations. Instead of the net worth of billionaires, we could be discussing how profits or revenues of a single company (over time, several quarters, years etc.), or different companies in the same sector of the economy (at a single point in time, as we are now doing with billionaires) increases as the number of products (revenue streams) increase, or the number of companies in the sector of the economy increase.

§ 3. 2 Simple nonlinear model for rising combined net worth for the top ten billionaires
The nonlinearity evident in Figure 1 is a very systematic. This is also evident after we superimpose the “best-fit” line and “approximate” this behavior with a simple linear model. The deviations above and below the best-fit line are not due to any kind of a statistical ‘noise’. Thus, we can model the observed nonlinearity, mathematically, using the simplest of all nonlinear laws, the power-law model. In its most general form, this model is described by the equation y = Axn + B where A, B, and n are constants that can be deduced from the (x, y) observations. When n = 1, this reduces to the equation of a straight line y = Ax + B. For n < 1, we get a rising curve with a decreasing slope, or decreasing values of the derivative dy/dx. For n > 1, the curve is again rising but the slope also increases as x increases. The expression for the slope, or the derivative dy/dx, is readily obtained using elementary rules of calculus. The derivative of the function xn is one of the first rules for differentiation that we learn in our calculus courses. The derivative dy/dx = A(nxn-1) = n(Axn/x) = n(y – B)/x. For the special case of B = 0 (curve passes through origin, like a straight line passing through the origin), the derivative dy/dx = n(y/x). As both x and y increase, the slope dy/dx of the curve decreases (for n < 1) since y does
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increase as fast as x does. The opposite is true when n > 1. The power-law fit to the data is illustrated in Figure 4.

Combined net worth, y [US dollars, $]

$600.00

$500.00

y = Axn + B = 99x0.67 = 99x2/3

$400.00

$300.00

$200.00

$100.00

$0.00 0 2 4 6 8 10 12 14 16

Number of billionaires, x
Figure 4: The power-law curve y = Axn + B, with B = 0 is fitted here to the combined net worth data for the top 10 billionaires in the Forbes 2013 list. This curve was generated using Microsoft Excel program. First the value of n = 0.667, or n ≅ 2/3, was chosen, since the values of n close to this value are often observed in many physical processes (including experiments on a novel material with a unique microstructure studied in my doctoral thesis work at MIT, click here). The numerical value of the constant A could then be fixed by choosing any one (x, y) pair. However, the (x, y) pair, which is close to the “average” point of the data point is seen to yield the best match with most the data points here.

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Combined net worth, y [US dollars, $]

700 600 500 400 300 200 100 0 0 2 4 6 8 10 12 14 16

y = Axn + B = 79.54x0.779 r2 = 0.992

Number of billionaires, x
Figure 5: The power-law curve y = Axn + B, with B = 0 is fitted here to the combined net worth data for the top 10 billionaires in the Forbes 2013 list. The numerical values of the constants A and n were determined by first preparing a log x versus log y plot, since log (y – B) = log A + n log x. The slope of graph will yield the value of n and the intercept will yield the value of A from log A.

Rigorously speaking, the constants A and n can be determined by first preparing a plot of log x versus log y (logarithm of x versus logarithm of y) since log (y – B) = log A + n log x, with the nonzero B being matched to the intercept of the graph. The slope of the log x-log y plot equals the exponent ‘n’ and the intercept equals log A from which A can be deduced. The curve-fitting method, using Microsoft Excel, was quick easy way to fit the data. The numerical values A and n were also determined after preparing a log x-log y plot and this yields A = 79.54 and n = 0.779. The linear regression coefficient r2 = 0.992 for this log-log plot. The resulting “best-fit” power-law curve in illustrated in Figure 5. This seems to follow the data much more closely at
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lower values of x but deviates more significantly at the higher values of x compared to the earlier curve with A = 99 and n ≅ 2/3.

Average net worth, U = y/x [US dollars, $]

100

y/x = h + (c/x) Linear law
80

60

40

dy/dx = n(y/x) Nonlinear law

20

0 0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

Number of billionaires, x

Figure 6: Falling values of the “average” net worth U = UN/N as more and more billionaires are included in the group. Both the linear law (upper curve) and the nonlinear power law (lower curve) explain the observed decreasing value of the average net worth U. To summarize the discussion so far, we have considered the net worth data for the top 10 billionaires in the Forbes 2013 Billionaires List. As we add more and more billionaires to our list, their combined net worth UN increases as expected. Both the linear and the nonlinear model explain the observed decrease in the average net worth U = y/x = UN/N as the number of billionaires in our list increases. This is illustrated in Figure 6.

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As we see here, the average net worth U decreases as we add more and more billionaires to the list. Also, this decrease is nonlinear and seems to level off for large value of x, or N, the number of billionaires. Indeed, the “asymptotic” value of U is the slope h of the best-fit straight, since U = y/x = h + (c/x) and the second term (c/x) decreases and goes to zero (theoretically) as x becomes very large. Let us therefore consider now how the combined net worth UN and the average U change as we add the next 10 and the next 20 billionaires to our list. The graphs for the combined net worth are illustrated in Figures 7 and 8.

§ 5. Evolution of the combined net worth to the first trillion dollars
The evolution of the combined net worths of the first 20 billionaires is illustrated in Figure 7. Gridlines have been added here to aid the interpretation of the data. Notice that no fractions are allowed, at least on the horizontal axis. The number of billionaires can only increase by one at a time. Once we get past billionaire number 8 (Li Ka-shing of Hong Kong), the nonlinearity evident earlier in Figure 1 disappears and the graph is essentially linear. This means that the combined net worth is increasing at essentially a fixed rate, although the individual net worths seem to follow a decreasing trend (since our list is arranged in descending order of net worths). Although the individual net worth of a billionaire decreases as we go down the ranks (increasing rank numbers), in an unpredictable way, sometimes by as little as $0.2 B (billionaire 11 at $28.2 B and billionaire 12 at $28 B) and sometimes by as much as $2.2B (billionaire 19 at $25.2 B and billionaire 20 at $23B, Karl Albrecht, Germany, of Aldi supermarket chain, and Larry Page, of Google). The fixed rate of growth of their combined net worth, the slope of the graph is h = ∆y/∆x where ∆y = $322B between billionaires 8 and 20 and ∆x = 12 the change in the number of billionaires. Thus, the fixed rate h = $26.83B per billionaire. The extrapolation of the straight line joining x = 8 and x = 20 will obviously not pass through the origin. The positive intercept c = 177.83.

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Combined net worth, y [US dollars, $]

$800.0 $700.0 $600.0 $500.0 $400.0 $300.0 $200.0 $100.0 $0.0 0 4 8 12 16 20 24

y = hx + c = 26.83x + 177.83

Number of billionaires, x
Figure 7: The essentially fixed rate of increase of the combined net worth as the number of billionaires increases from 10 to 20. Once past billionaire number 8, the graph is essentially linear. The equation joining the data for number 8 (Li Ka-shing)and number 20 (Larry Page, of Google) is indicated here. The straight line makes a finite positive intercept c = 177.83. The average net worth U = y/x = h + (c/x) = 26.83 + (177.83/x) keeps on decreasing as the more billionaires are added, although the combined net worth increases at the fixed rate h = $26.83B Exactly the same pattern is revealed when we add more billionaires to the list and the combined net worth climbs to the first $1 trillion, at billionaire No. 35 (Iris Fontbona and family, of Chile,, at $17.4 B) The graph is essentially linear past billionaire number 8. However, a more careful examination reveals a further decrease in the slope h between billionaires 21 and 35. The fixed rate h = $19.36 B per billionaire, obtained by considering the overall changed in the combined net worth from billionaire 21 to billionaire 35. Correspondingly the positive intercept c also decreases. This may be taken as confirming the continuously decreasing slope predicted by the power law model, y = Axn.
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Combined net worth, y [US dollars, $]

1200
1000

800
600 400 200 0 0 5 10 15 20 25 30 35 40

y = hx + c = 19.36x + 330.65 y = hx + c = 26.83x + 177.83

Number of billionaires, x
Figure 8: Growth in the combined net worth of billionaires, from N = 1 to N = 35 at which the combined net worth crosses the $1 trillion mark. Three linear segments, or a smooth power-law curve can be used to describe this data.

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Average net worth, U = y/x [US dollars, $]

80.00 70.00 60.00 50.00 40.00 30.00 20.00 10.00 0.00 0 10 20 30 40 50

Number of billionaires, x
Figure: 9 The falling values of the average net worth of billionaires as more and more billionaires are added to the list. Only N = 1 to N = 35 are considered here. The falling values are also (at least partly) due to the fact that we are considering an ordered list with decreasing individual net worths for the billionaires from N = 1 to N = 35.

Both the linear and nonlinear model are supported, depending on one’s point of view, with the linear model offering a good approximation. However, it should also be remembered that both the nonlinear power law, Axn + B, as well as the linear law y = hx + c imply that the average U = y/x = h + (c/x) or y/x = Axn-1 is continuously decreasing. Hence, the fundamental implications of the nonzero intercept c in the linear law cannot be overlooked.

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§ 6. Average net worth by country
The very systematic pattern of rising values of the combined net worth UN with increasing number of billionaires N is due to the fact we have, thus far, only considering an ordered billionaire list, with descending individual net worths (or increasing rank numbers, from 1 to 1342). Similar results are obtained when we consider the data after the billionaires are sorted by country. The average billionaire net worth by country is available, see article by Edwin Durgy, and also been discussed by the present author, for both the Forbes 2013 billionaire list and the Hurun 2013 Rich list (published by the group based in Shanghai, China). The following results are obtained if we organized the top 35 billionaires (with combined net worth of just over $1 trillion) by countries.

Table 2: Top 35 Billionaires Organized by Country
Country USA Hong Kong Germany Mexico France Spain Sweden India Italy Canada Saudi Arabia Brazil Russia Chile Total Billionaires, N 16 3 3 2 2 1 1 1 1 1 1 1 1 1 35 Combined net worth UN $, billions 502.5 71.3 64.4 91.2 59 57 28 21.5 20.4 20.3 20 17.8 17.6 17.4 1008.4 Average net worth, U = UN/N $, billions 31.4 23.8 21.5 45.6 29.5 57 28 21.5 20.4 20.3 20.0 17.8 17.6 17.4 28.81

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Fourteen are represented in the list of top 35 billionaires. The USA leads this list with 16 billionaires, followed by Hong Kong and Germany with two each and Mexico and France with two each and nine countries with one billionaire each. The “average” for the nine countries determined in this way varies widely. Hence a more complete analysis which consider all the countries and all the billionaires is required. This has already been presented in the articles listed at the end. In summary, the simplest law relating the number of billionaires and their combined net worth is the linear law y = hx + c. The average billionaire net worth U = y/x = h + (c/x) thus decreases as the number of billionaires increases if the nonzero intercept c is positive (c > 0). It could also increase with increasing number of billionaires (see case of Israel discussed in Appendix 2) if the group yields c < 0 (negative intercept) or even a negative slope (h < 0), depending on how the groups are created.

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Appendix 1 Planck and Einstein’s Solution to the problem of determining the average value of any quantity of interest for a complex system
I have used the mathematical symbols U and UN since they are also used to denote the total energy UN and the average energy U of a complex system of N microscopic particles in physics. In a famous paper, presented to the German Physical Society in Berlin, on December 14, 1900, Planck derived the following expression for the average energy U of N particles (he calls them oscillators, or resonators). This is also now considered to mark the birth of what we now call quantum physics. Very briefly, if there are N0 particles of the same kind (with the lowest energy U0 = 0), N1 particles with energy U1 and N2 with energy U2 and so on, their total energy UN can be obtained by a simple summation. UN = N0U0 + N1U1 + N2U2 + ………… NkUk where, Average, N = N0 + N1 + N2 + ………… Nk U = UN/N U = ε [e-ε/kT / (1 - e-ε/kT) ] ……..(1) ……..(2) ……..(3) ……..(4)

Planck’s expression

The expression given by Planck was also derived later by Einstein, quite simply, as follows, in a paper published in 1907 (click here to see Planck’s original approach, discussed recently by the author). Let N0 particles have the energy U0, let N1 particles have the energy U1 and so on. In order to arrive at the general expression for U, we make the assumption that N1 = N0t, N2 = N0t2, and so on. Here t is a parameter that will be defined later. The same considerations apply when we are interested in any general property of interest, in a complex system, with a complicated distribution of

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the N entities in various “microscopic” states. Hence, the total number of entities N in given by. N = (N0 + N0t + N0t2 + … ) = N0 (1 + t + t2 +….) and, N = N0 /(1 – t) ……..(5a) ……..(5b)

Here we have used the well-known series expansion for (1 – x)-1, click here (see image copied and pasted below). Next, we determine the total UN using exactly the same procedure.

Next, we determine the total UN using exactly the same procedure. UN = N0 x 0 + N1ε + N2 (2ε) + N3 (3ε) + …… = N0 t (ε) + N0t2 (2ε) + N0t3 (3ε) + ….. = ε (N0t) (1 + 2t + 3t2 + …… ) = ε (N0t) /(1 – t)2 ……..(6a) ……..(6b) ……..(6c) ……..(6d)

To determine the total UN , we have again used another well-known series expansion. Both these are given by Longair, as well as by Feynman in his famous Lectures on Physics. The expression for the average U is therefore, U = UN/N = ε [ t /(1 – t)] ……..(7) The image from the above link (from Longair’s book, Theoretical concepts in physics, Cambridge University Press) is given below.

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http://books.google.com/books?id=bA9Lp2GH6OEC&pg=PA352&lpg=PA352 &dq=longair+planck+quantum+average+energy+series+expansion&source=b l&ots=0Gy4_UaDB3&sig=k3I8eDotZODLEWLoWkPoEBUXtA&hl=en&sa=X&ei=jJFUUfapMPS0wHRuIHoBw&ved=0CFQQ6AEwBjgK#v=onepage&q=longair%20planck% 20quantum%20average%20energy%20series%20expansion&f=false

Note that the unknown N0 cancels out in the final expression which is given entirely in terms of the two parameters, an elementary ε and the parameter t. The parameter “t” tells us about the how the various microscopic entities are distributed in the system. The elementary ε is the quantity used for counting purposes. The property U takes on integral values of this elementary ε. In physics, the parameter t = e-ε/kT is associated with another quantity called the temperature T of the system and the Boltzmann constant k. Thus, the numerical values of “t” vary from 0 to 1 as the ratio ε/kT increases. Also, in physics, the elementary ε = hf where h is another constant, called the Planck constant and “f” is the frequency. Recall also that although “h” is a universal constant in physics, the frequency “f” is a continuously divisible quantity. Once the frequency f is fixed, ε takes
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on the integral values of ε, 2ε, 3ε, and so forth. When the frequency increases by ∆f the elementary ε always increases by ε = h∆f. Likewise, ε = ∆y = h∆x, when these concepts outside physics.

Appendix 2: The combined and the average net worth of the Israeli billionaires in the Forbes 2013 list
There are a total of 17 billionaires in Israel, according to the Forbes 2013 Billionaires list, see Refs. [8,9, 11]. Israel ranks number 43 in the list by average billionaire net worth by country, Ref. [12], with an average U of $2.721 billion and appears in the top 10 list of countries ranked by billionaires per capita, see Refs. [13-16]. The Israeli 2013 Billionaires (with Forbes ranks) Forbes 182 198 286 308 308 316 670 670 704 831 Rank Net worth 6.5 6 4.4 4.2 4.2 4.1 2.2 2.2 2.1 1.8 Uk Forbes 922 974 1107 1332 1332 1342 1342 Rank Net worth 1.65 1.5 1.3 1.05 1.05 1 1 Uk Combined net worth UN = $46.25 B, Average U = $2.72 B, Median M =$2.1B Notice that two Israelis each share the ranks 308, 670, 1332 and 1342. Since there are 17 billionaires, the median billionaire net worth is the 9th value in the list. The list can be arranged in either ascending or descending order without changing the median value. Performing the analysis described in the main text, the combined net worth UN increases as the number of billionaires N increases in a highly nonlinear manner. This is illustrated in Figure 10.

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The list could also be rearranged in many different ways leading to the following results for the combined net worth UN and the number of billionaires N. Counting only the 10 billionaires in the top row UN = 37.7 while counting only the 7 billionaires in the bottom row UN = 8.55. Mixing up the lists yields the other values listed in the table that follows and clearly leads to widely different values for the same billionaire count N. For example, the combined net worth for five billionaires could be greater than the combined net worth for seven billionaires, or even 9 or 10 billionaires, depending on who is being counted.

Combined net worth, y [US dollars, $]

70.00 60.00 50.00

y= Axn = 8.65x0.667
40.00 30.00 20.00 10.00 0.00 0 2 4 6 8 10 12 14 16 18 20

y= Axn = 7.88x0.665 Best–fit A and n r2 = 0.974

Number of billionaires, x
Figure 10: Growth in the combined net worth UN for the 17 Israeli billionaires, as the number of billionaires increases from N = 1 to N = 17, when arranged in descending order of net worths and corresponding Forbes ranks.

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The Israeli 2013 Billionaires list rearranged N UN N UN 5 5 5 7 7 7 9 10 6.55 14.8 19.1 8.55 13.55 31.6 17.9 37.7 10 10 11 12 12 12 15 17 14.65 23.05 29.65 29.75 37.95 40.85 40.05 46.25

Combined net worth, y [US dollars, $]

60

50

40

30

20

10

0 0 5 10 15 20 25

Number of billionaires, x
Figure 11: Grouping the 17 Israeli billionaires in many different ways yields this scatter plot of the combined net worth UN versus the number of billionaires N. Notice all the remarkable alignment of eight (x, y) pairs along the straight line y = 3.308x – 9.992 = 3.308x (x – 3.02), with six of these eight points lying practically on the straight line.

A graphical representation of the data after this rearrangement reveals a lot of scatter, as illustrated in Figure 11. However, if we join the two extreme points (5, 6.55) and (17, 46.25), we observe a nice linear trend with several (x, y)
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pairs falling very close to this line. The slope h = ∆y/∆x = $39.7/12 = $3.308 where ∆y and ∆x are the changes in the combined net worth and the number of billionaires, respectively. Indeed, the four (x, y) pairs that lie practically on this straight between the two extreme points were added later by carefully choosing the N billionaires to yield a total that agrees with the predictions on the line, y = hx + c = 3.308x – 9.992 = 3.308(x – 3.02). The slope h is positive but the intercept c is negative giving rise to the positive intercept x0 = -c/h on the horizontal axis. Hence, as the number of billionaires increases, the average net worth, U = y/x = 3.308 – (9.992/x) actually increases (rather than decrease when h > 0 and c > 0) as we move up this line. Also, a careful examination of this plot shows that there are two (x, y) pairs, (5, 14.8) and (12, 37.95) that lie above this trend line on what appears to be a parallel. The slope h = ∆y/∆x = $14.95/7 = 3.307 is virtually identical to h = 3.308 = $39.7/12 deduced earlier. In other words, it appears that when the number of billionaires increases by ∆x their combined net worth always increases by the same fixed amount ∆y = h∆x. The validity of this law has been demonstrated nicely here using the Israeli billionaires data. Indeed, similar analyses could be performed with data from several individual countries, or groups of countries, as discussed in several articles listed here. The discussion of how the combined net worth and the average net worth evolves, when we consider the data much more systematically, serves to confirm the validity of the law y = hx + c. The implications of the nonzero intercept c have also been discussed. When c is positive (c > 0), the average net worth decreases. When c is negative (c < 0), the average net worth decreases. This constant “c” is exactly similar to the ‘work function’ introduced by Einstein to explain some of the observations on the photoelectric effect that puzzled physicists of the late 19th and early 20th century (see Ref. [21], under related articles, also click here).

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The higher the value of x0 = -c/h, the lower the combined net worth of the billionaires in that group. This is related to the barriers experienced to increase net worth, akin to the difficulty of producing an electron from within a metal. Some of the energy of the photon that strikes the metal must be given up to produce the electron that is observed outside the metal. Likewise, the appearance of a billionaire and the increase in their net worth also depends on the “complex” environment in which these billionaires produce their wealth (in this case the industries they are associated with). This changes the “work function”, or the nonzero intercept c, making it positive or negative as we see here.

Appendix 3: The marginal tax rate and the tax equation
If x is the taxable income and y the tax owed, the ratio y/x is the fraction of income paid as taxes, Or, after being multiplied by 100, this represents the percentage of income paid as taxes. The ratio y/x is NOT the same as the marginal tax rate, m. This can be understood as follows.

Table 3: The US 2012 Tax Rate Schedule Converted into the Tax Equation
Taxable Of income (x) But not Tax owed (y) amount Tax equation is over over Is over $0 $8700 10% $0 y = 0.10x $8700 $35,350 $870 + 15% $8700 y = 0.15x -435 35,350 85,650 $4867.5 + 25% $35,350 y = 0.25x – 3,970 85,650 178,650 $17,442.50+28% $85,650 y = 0.28x – 6,539.5 178,650 388,350 $43,482.5+33% $178,650 y = 0.33x – 15, 472 388,350 No limit $112,683.5+35% $388,350 y = 0.38x – 23,239 For the 15% bracket, y = 0.15x – 435 = 0.15(x – 2900) = 870 + 0.15(x – 8700) For the highest 35% bracket, y = 0.38x – 23,239 = 112,683.5 + (x – 388,350)
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$2,000

(15,000, 1815) y/x = 0.121

$1,500

y = mx + c = m(x – x0) y/x = m + (c/x) = 0.15 = 0.15 – (435/x) y = mx (c = 0, x0 = 0) y/x = m = 0.10

Tax owed, y

$1,000

(10,000, 1065) y/x = 0.107

$500

y/x = 0.10 = constant
$0 $0 $4,000 $8,000 $12,000 $16,000

Taxable income, x
Figure 12: The US tax code is a series of straight line segments, with increasing slopes (and increasing intercepts x0 = -c/h, see also Figure 13) as shown here for the lowest two brackets. The ratio y/x, the percent of income paid as taxes, keeps increasing as taxable income x increases. This is true for ALL tax brackets, except the lowest (with c = 0 and x0 = -c/m = 0). The maximum value the ratio y/x = m and is achieved only when x approaches the limit of the tax bracket. This is the significance of the slope m. The maximum value of the ratio y/x is always less than m (when c < 0). If we use the term “tax percent” for the ratio y/x, as opposed to “tax rate” for the slope m, it is clear that no one is the second tax bracket pays 15% of the total income as taxes. Only the marginal tax rate is 15%, which means that if income increases by ∆x within this bracket, tax owed will go up by ∆y = m∆x = 0.15∆x. The ratio y/x < m = ∆y/∆x since c < 0, because of the way the tax code is designed.

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The “tax tables”, used conveniently by the majority of taxpayers with (taxable) incomes of less than $100,000, are actually prepared using the simple linear equation y = mx + c. The slope m is the marginal tax rate and keeps increasing as (taxable) income levels increase. Each slope (the tax rate) applies for a range of (taxable) income levels, x. For a single taxpayer, the tax rates, for tax year 2012, are given in Table 3.

$2,000

The 15% tax bracket y = 0.15x – 435 = 0.15 (x – 2900)
$1,500

Tax owed, y

$1,000

The 10% tax bracket y = 0.10x

$500

The 15% tax bracket x0 = -c/m = $2900
$0
$0 $4,000 $8,000 $12,000 $16,000

Taxable income, x
Figure 13: The increase in the marginal tax rate, between two income brackets, leads to the increased slope, as illustrated here. Extrapolating backwards (see dashed line), the tax equation cuts the income axis (x – axis) at a finite positive value of x0 = - c/m. The tax equations given here, with h > 0 and c < 0, are equivalent to tax computation formula recommended by the Internal Revenue Service (IRS); see the last row of the table.

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Because of the negative intercept c, the ratio y/x = m + (c/x) is always increasing. In other words, EVERYONE, who has an income, always pays a higher percent (the ratio y/x) in taxes as the taxable income increases. For the 15% bracket, y/x = 0.15 – (435/x) and the percent paid as taxes (ratio y/x) increases to the maximum of m = 0.15 or 15% as x increases within this taxable range of $8700 to $35,350. Of course, total taxable income is derived from many sources, not necessarily wages. Incomes other than wages (such as investment income, or capital gains, social security income) are taxed very differently. Note that although ALL taxable income of a person in the 15% tax bracket is being taxed, it is as if all the income above the cut-off x0 = $2900 is being taxed at the fixed rate of 15%. The ratio y/x = 0.15 – (435/x) keeps increasing as taxable income increases. Hence, a higher percent of income is being paid as taxes when income levels increase. This is true for ALL tax payers, at all the higher marginal tax rates except those in the lowest bracket. This is also the reason why as income levels rise to what we call the really “rich levels”, everyone begins to feels that they are paying too much in taxes. Mathematically speaking, the negative value of the intercept c is the reason for this widely held perspective of an increasing “tax burden” with increasing income. This is how the “progressive” tax code has been designed. There is nothing we can do to get around it, except reduce the values of the marginal tax rates, the constant m, which is where are ALL the political and social debates have been centered, especially after President Reagan came to office in 1981 (click here for the historical tax rates from 1913-2013, one hundred years of income taxation in the United States). When Reagan took office, in January 1981, the top tax rate was 70% (click here, see image copied and pasted below) and by the time he left office in 1988, it was down to 28%. During World War I, the top tax rate was 77%

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(President Woodrow Wilson) and during World War II (President Franklin D Roosevelt), it was 94%. And now it is up to 35% (for 2012 and will go up to 39.6% for 2013 tax year) and all of the political posturing and turmoils of the past few years, especially since President Obama took office, have been for this difference of 28% and 35% … no laughing matter when income levels are far in excess of $100,000 or $200,000 or $1,000,000. The main point here, however, is NOT the discussion of tax rates but the meaning of the slope m and the ratio y/x = m + (c/x). The ratio y/x can either increase or decrease as the variable x increases because of the nonzero intercept c, as in the tax problem.

Courtesy: http://www.businessinsider.com/history-of-tax-rates?op=1
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The same, however, applies to many other problems that we encounter in the business world and economics, especially the profits-revenues problem the most fundamental of all. When revenues x increase, profits y increase following the law y = hx + c where the ratio y/x is the profit margin. (This linear law can actually be shown to be a consequence of the classical breakeven model for profitability of a company.) Since y/x = h + (c/x), at least three situations can be envisioned. a) If the intercept c is negative (c < 0), as in the tax problem, both profits and profit margins keep increasing with increasing revenues and the company is rewarded in the market place (as it happened with Microsoft, Apple, Google, and many other successful companies). b) When the intercept c is positive (c > 0), the profits increase with increasing revenues, but profit margins decrease. Then the company starts getting into trouble in the market place. c) Finally, we also see the case of negative h and positive c (h < 0, c > 0). Now, revenues increase but profits and profit margins decrease. Such companies are headed towards bankruptcy. This is what happened with the “old” General Motors before it was forced to file for bankruptcy in June 1 2009 (click here). Indeed, a composite x-y graph of profits versus revenues, over a period time, reveals all these three characteristics – which is manifested in the maximum point on the profits-revenues graph. Several companies (notably Ford Motor company, Yahoo Inc!, Southwest Airlines), are now operating past their maximum point, in the falling part of the profit-revenues curve, with erratic variations in profits and revenues. This has been discussed in detail for several companies, in several articles. For one article which provides important references, please click here).

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Reference List
1. The Median Net Worth of the 2013 Forbes Billionaires: Why the median is a meaningless stat, Published March 26, 2013 http://www.scribd.com/doc/132433860/An-Essay-on-the-MedianForbes-2013-Billionaire-Why-the-Median-is-a-meaningless-stat All readers who have read this “essay” and find themselves in agreement with the conclusions must contemplate taking some steps to stop the US Census Bureau from putting out this highly questionable stat. Why? The flawed conclusions are affecting millions of lives and our perception of the US economy and the world economy. Of course, it is equally important that leading economists and statisticians from the academic world, and the more research minded from the business and financial world, examine this essay and its findings. The discussion of the ‘median’ billionaire calls attention to some of the most fundamental flaws in the use of such ‘median’ stats. One need not be academician with the high credentials of a doctoral degree to understand how seriously flawed this idea of a median really is. 2. Income, Poverty, and Health Insurance Coverage in the United States: 2011, Current Population Reports, P60-243, Published by US Census Bureau, http://www.census.gov/prod/2012pubs/p60-243.pdf Table A1, which starts on page 31 has the required data on number of households N, the mean household income, and the median household income for the years 2000-2011, with detailed breakdown by races gender, and age (click here to get table). 3. Household Wealth in the US: 2000 to 2011, by Alfred Gottschalck, Marina Vornovytskyy and Adam Smith, Posted March 21, 2013 chttp://www.census.gov/people/wealth/files/Wealth%20Highlights%20 2011.pdf The median net worth values for 2000, 2005 and 2011 are also given here. The dollar amounts are in constant 2011 dollars with inflationadjustment factors. Between 2000 and 2011, aggregate net worth of $28.9 trillion to $40.2 trillion. 4. Chart of the day: Median net worth, 1962-2010, Reuters, Posted June 12, 2012, by Felix Salmon (click here), http://blogs.reuters.com/felixsalmon/2012/06/12/chart-of-the-day-median-net-worth-1962-2010/
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5.

6.

7.

8.

9.

10.

11.

12.

The article starts with, “The big news from the Fed this week is, in the words of the NYT headline, that Family Net Worth Drops to Level of Early ’90s. But if you look at the actual report, there isn’t any data in there on family net worth before 2001.” So, after reviewing the data from 1962 to 2010, the author concludes “I think it’s fair to say that the median US household is no richer now than it was 30 years ago.” Where do Americans stash their Wealth? by Tami Luhby, March 22, 2013, http://economy.money.cnn.com/2013/03/22/americanhouseholdwealth/?source=cnn_bin The World’s Billionaires, The Richest People on the Planet 2013, by Luisa Kroll and Kerry A Dolan, March 4, 2013, http://www.forbes.com/billionaires/ Inside the world’s billionaires: Facts and Figures, by Luisa Kroll, http://www.forbes.com/sites/luisakroll/2013/03/04/inside-the-2013billionaires-list-facts-and-figures/ Mapping the wealth of the world’s billionaires, by Luisa Kroll, March 9, 2013, http://www.forbes.com/sites/luisakroll/2013/03/09/mapping-thewealth-of-the-worlds-billionaires/ Mapping the wealth of the world’s richest, by Ricardo Geromel, March 22,, 2013 http://www.forbes.com/sites/ricardogeromel/2013/03/22/forbesbillionaires-map/ The First Rich List, by Arik Hesseldahl and Claudia DeMairo, September 27, 2002, http://www.forbes.com/2002/09/27/0927richestphotos.html see also http://www.forbes.com/2002/09/27/0927richest.html The combined fortune of the 30 richest was $3,680,000,000 ($3.68 B) and the average net worth was $0.123 B ($122,666,666). The full list of the 30 richest men in 1918 may be found on page 15 of 17 of this article. Seventeen Israelis make Forbes billionaire’s list, The Times of Israel, March 4, 2013, Times of Israel Staff, http://www.timesofisrael.com/seventeen-israelis-make-forbesbillionaires-list/ Average Billionaire Net Worth by Country: Full List, by Edwin Durgy, March 13, 2013,
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http://www.forbes.com/sites/edwindurgy/2013/03/13/averagebillionaire-net-worth-by-country-full-list/ 13. Which Country Has the Most Billionaires Per Capita? By Emma Roller, March 4, 2013, Slate magazine, click here, or see http://www.slate.com/blogs/moneybox/2013/03/04/forbes_billionaires_ list_countries_with_the_most_billionaires_per_capita.html 14. 10 Countries with the most billionaires, by Harry Bradford, The Huffington Post, First posted April 12, 2011, updated June 12, 2011, http://www.huffingtonpost.com/2011/04/12/10-countries-with-thebillionaires_n_847693.html#s262776&title=4_India 15.Countries with the most billionaires in the 2013 Forbes Billionaire List, Published March 9, 2013, http://www.scribd.com/doc/129406030/Countrieswith-Most-Billionaires-in-the-Forbes-2013-Billionaires-List 16.Per Capita Trends for Ultra High Net Worth Residents in US Cities, Published March 11, 2013, http://www.scribd.com/doc/129722844/Per-CapitaTrends-for-the-Ultra-High-Net-Worth-UHNW-Residents-in-US-Cities

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Related articles on the Billionaire Problem
1. The Median Net Worth of the 2013 Forbes Billionaires: Why the median is a meaningless stat, Published March 26, 2013 http://www.scribd.com/doc/132433860/An-Essay-on-the-MedianForbes-2013-Billionaire-Why-the-Median-is-a-meaningless-stat 2. The Average Net Worth of the Forbes 2013 Billionaires: Accumulation of the first trillion, Published March 27, 2013 3. Average Net Worth of Billionaires and Ordinary US households, Published March 24, 2013, http://www.scribd.com/doc/132189331/Average-Net-Worth-ofBillionaires-and-Ordinary-US-Households 4. Empirical Evidence for Entropy Outside Physics: Wealth Distribution among billionaires in the 2013 Forbes and Hurun Rich Lists, http://www.scribd.com/doc/132059874/The-Empirical-Evidence-forEntropy-Outside-Physics-Wealth-Distribution-among-the-billionaires-of2013-Forbes-and-Hurun-Rich-Lists Published March 24, 2013. 5. Wealth Distribution between the 2013 Forbes and Hurun Rich Lists, http://www.scribd.com/doc/131830628/Wealth-Distribution-Betweenthe-2013-Forbes-and-the-Hurun-Rich-Lists, Published Mar 22, 2013. 6. Average Net Worth by Country - Part 1: Preliminary Analysis of the 2013 Forbes Billionaire Data, Published March 18, 2013. http://www.scribd.com/doc/131208984/The-Average-Net-Worth-byCountry-Part-I-Analysis-of-the-2013-Forbes-Billionaires-Data 7. Average Net Worth by Country - Part 2: Analysis of the 2013 World’s Billionaire Data, to be Published March xx, 2013. 8. China has fewer billionaires than most other large countries, Published March 21, 2013, http://www.scribd.com/doc/131583456/China-Has-Fewer-BillionairesAccording-to-the-Billionaire-Net-Worth-Analysis2013 9. Average Net Worth of Billionaires by Country: 2013 Hurun Rich List,
http://www.scribd.com/doc/131764299/Average-Net-Worth-of-Billionairesby-Country-The-2013-Hurun-Rich-List Published March 22, 2013.
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10. The Average Net Worth by Country, Part 1: Introducing the x-y Diagram, Preliminary Analysis of the Forbes 2013 World’s Billionaires Data, Published March 19, 2013, http://www.scribd.com/doc/131208984/The-Average-Net-Worth-byCountry-Part-I-Analysis-of-the-2013-Forbes-Billionaires-Data 11. Average Net Worth by Country - Part 2: Analysis of the 2013 World’s Billionaire Data, to be Published March xx, 2013. 12. The Recent Growth in the Average Worth of US Billionaires (’09-‘13) A Curious Limits to Growth Revealed Again, Published March 18, 2013 http://www.scribd.com/doc/131005255/The-Recent-Growth-in-theAverage-Worth-of-US-Billionaires-%E2%80%9909-%E2%80%9813-ACurious-Limits-to-Growth-Revealed-Again 13.Quantitative Comparison of the Billionaires in the Forbes Hall of Fame, http://www.scribd.com/doc/130826078/Quantitative-Comparison-of-theBillionaires-in-the-Forbes-Hall-of-Fame Published March 17, 2013. 14.Growth in the Combined Net Worth of Billionaires from 2000-2013: An Interesting Limits to Growth is Revealed, Published March 15, 2013, http://www.scribd.com/doc/130517722/Growth-in-the-Combined-Net-Worthof-Billionaires-From-2000-2013-An-Interesting-Limit-to-Growth-is-Revealed 15.Growth of Billionaires Worldwide and in the USA: Analysis of the number of billionaires data since 1982, Published March 14, 2013, http://www.scribd.com/doc/130377891/The-Growth-of-BillionairesWorldwide-and-in-USA-Analysis-of-the-data-since-1982 16.Countries with the most billionaires in the 2013 Forbes Billionaire List, Published March 9, 2013, http://www.scribd.com/doc/129406030/Countrieswith-Most-Billionaires-in-the-Forbes-2013-Billionaires-List 17.Average worth of the Top 10 Billionaires in a Country, Published Mar 11, 2013, http://www.scribd.com/doc/129863162/Average-Worth-of-the-Top-10Billionaires-in-a-Country 18.Per Capita Trends for Ultra High Net Worth Residents in US Cities, Published March 11, 2013, http://www.scribd.com/doc/129722844/Per-CapitaTrends-for-the-Ultra-High-Net-Worth-UHNW-Residents-in-US-Cities 19.Europe and Asia-Pacific billionaires in the Forbes 2013 list: What’s the difference? Published March 10, 2013
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http://www.scribd.com/doc/129634796/Europe-and-Asia-Pacific-Billionairesin-the-2013-Forbes-List-What-s-the-difference 20. The Rate of Creation of Billionaires: Analysis of the Forbes 2013 Billionaire List, Published March 6, 2013, http://www.scribd.com/doc/128944910/The-Rate-of-Creation-ofBillionaires-Analysis-of-the-2013-Forbes-Billionaire-s-List 21. Billionaires and Calculus: Ratio versus Rate of Change Is Einstein’s Work Function Observed In this Problem? Published March 5, 2013. http://www.scribd.com/doc/128610494/The-Forbes-Billionaires-andCalculus-Is-Einstein-s-Work-Function-Observed-Here 22. Billionaires and the Population Law: Analysis of the 2013 Forbes Billionaire’s List, to be published shortly.

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About the author V. Laxmanan, Sc. D.
The author obtained his Bachelor’s degree (B. E.) in Mechanical Engineering from the University of Poona and his Master’s degree (M. E.), also in Mechanical Engineering, from the Indian Institute of Science, Bangalore, followed by a Master’s (S. M.) and Doctoral (Sc. D.) degrees in Materials Engineering from the Massachusetts Institute of Technology, Cambridge, MA, USA. He then spent his entire professional career at leading US research institutions (MIT, Allied Chemical Corporate R & D, now part of Honeywell, NASA, Case Western Reserve University (CWRU), and General Motors Research and Development Center in Warren, MI). He holds four patents in materials processing, has co-authored two books and published several scientific papers in leading peer-reviewed international journals. His expertise includes developing simple mathematical models to explain the behavior of complex systems. While at NASA and CWRU, he was responsible for developing material processing experiments to be performed aboard the space shuttle and developed a simple mathematical model to explain the growth Christmas-tree, or snowflake, like structures (called dendrites) widely observed in many types of liquid-to-solid phase transformations (e.g., freezing of all commercial metals and alloys, freezing of water, and, yes, production of snowflakes!). This led to a simple model to explain the growth of dendritic structures in both the groundbased experiments and in the space shuttle experiments. More recently, he has been interested in the analysis of the large volumes of data from financial and economic systems and has developed what may be called the Quantum Business Model (QBM). This extends (to financial and economic systems) the mathematical arguments used by Max Planck to develop quantum physics using the analogy Energy = Money, i.e., energy in physics is like money in economics. Einstein applied Planck’s ideas to describe the photoelectric effect (by treating light as being composed of particles called photons, each with the fixed quantum of energy conceived by Planck). The mathematical law deduced by Planck, referred to here as the generalized power-exponential law, might
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actually have many applications far beyond blackbody radiation studies where it was first conceived. Einstein’s photoelectric law is a simple linear law and was deduced from Planck’s non-linear law for describing blackbody radiation. It appears that financial and economic systems can be modeled using a similar approach. Finance, business, economics and management sciences now essentially seem to operate like astronomy and physics before the advent of Kepler and Newton. Finally, during my professional career, I also twice had the opportunity and great honor to make presentations to two Nobel laureates: first at NASA to Prof. Robert Schrieffer (1972 Physics Nobel Prize), who was the Chairman of the Schrieffer Committee appointed to review NASA’s space flight experiments (following the loss of the space shuttle Challenger on January 28, 1986) and second at GM Research Labs to Prof. Robert Solow (1987 Nobel Prize in economics), who was Chairman of Corporate Research Review Committee, appointed by GM corporate management.

Cover page of AirTran 2000 Annual Report
Can you see that plane flying above the tall tree tops that make a nearly perfect circle? It requires a great deal of imagination to see and to photograph it.

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