Billionaires and Calculus
Ratio versus Rate of Change
Is Einstein's Work Function Observed in this Problem?
Immediately following the publication of the 2013 Forbes billionaire list, another article appeared, in Slate magazine with a ranking of countries based 'per capita' calculations. We all understand that these are pretty ridiculous rankings since per capita means we are dealing with the ratio y/x, where y is the number of billionaires and x is the population. The ratio will obviously have a high value for countries with a small population. Even so, such per capita lists are routinely prepared. India, it was noted, had 55 billionaires, a respectable number the article said, but not really if you consider that Germany, with only 6% the Indian population, has 58 billionaires.
This got me interested in analyzing the relationship between the number of billionaires and the population of a country. As in many other complex problems of interest to us, the empirical observations suggest a remarkably simple and linear relation, of the type y = hx + c, where x is the population and y the number of billionaires, is observed when we consider the Forbes data for the countries with the lowest populations. In other words, the number of billionaires increases at a fixed rate h = ∆y/dx with increasing population. Here ∆x and ∆y are, respectively, the changes in population and the number of billionaires when we consider the data for different countries. The per capita ratio y/x = m = h + (c/x) is thus affected by both the size of the population x and the nonzero intercept c.
The linear law observed here can be compared to Einstein's photoelectric law. The nonzero intercept c is exactly similar to the work function W introduced into physics by Einstein. The extension of this important idea of a nonzero work function and Planck's ideas, outside physics, to economics and the social sciences is also discussed briefly in the article I have uploaded (with references being made to earlier articles by the present author, for completeness); see
Table 1: Per capita rank and number of billionaires
Rank (per capita ratio) |
Country |
Population, x |
Billionaires, y |
6 |
Cyprus |
1.1 |
3 |
9 |
Kuwait |
2.8 |
5 |
8 |
Singapore |
5.2 |
10 |
10 |
Switzerland |
7.9 |
13 |
7 |
Israel |
7.8 |
17 |
4 |
Hong Kong |
7.1 |
39 |
Source: Slate magazine. This data is analyzed in what follows here. The key figures are included here without discussion which may be found in the article cited.
Figure 1: The linear law y = hx + c illustrated here with data for six of the top 10 in the per capita list. Hong Kong is an outlier but the other five reveal a nice linear relationship between the number of billionaires y and the population x. This increases at a fixed rate of about 1.8 per million.
Figure 2: The linear law y = hx + c illustrated here with data for four of the top 10 in the per capita list. Cyprus and Kuwait fall on the lower line A with slope h = 1.176 and Singapore and Switzerland fall on the roughly parallel line B with a slope h = 1.111. The difference in the intercept c and the also the population x thus affects the ratio y/x = m = h + (c/x).
Five Countries from Forbes list with Population and Number of Billionaires
Country, No. |
1 |
2 |
3 |
4 |
5 |
Billionaires, y |
3 |
5 |
10 |
13 |
17 |
Population, x |
1.1 |
2.8 |
5.2 |
7.9 |
7.8 |
Planck's distribution of energy from the 1900 paper
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 |
The sum of the values in bottom row equals 100. In Planck problem the average energy of N particles is U and the total energy is UN = NU = Pε where P and N are very large integers. Here N = 10 and P = 100 for illustration. It appears that the same ideas can be extended to many other problems outside physics to economics and the social sciences. We are determining the "average" value of some property of interest in a very complex system, such as described by various distributions, simplified above.