…the richest man on earth would have a fortune of no more than $43,000! If you don’t believe me read this post!

Have you ever thought about the distribution of wealth as a function of some quality? Especially rich people pride themselves on extraordinary abilities, so that they somehow “deserve” their wealth. Now “abilities” is somewhat hard to measure so let us take “intelligence” as a proxy. Intelligence is distributed normally with mean = 100 and standard deviation = 15. An interesting question is what is the expected maximum intelligence of all persons alive today?

The first thought could be to simulate this but as it soon turns out this approach doesn’t work because you had to create and analyze a vector with nearly 8 billion elements – R will balk at this.

So we will have to look for an analytical solution – fortunately, asking my colleague Prof. Dr. Google brings up the following mathunderflow question and answers: Expected value for maximum of n normal random variable. According to this the expected maximum of independent is:

where (lowercase phi) and (uppercase phi) stand for the pdf and cdf of the normal distribution respectively. I spare you the mathematical details of the derivation.

Unfortunately there doesn’t seem to be a closed form for the integral so we will solve it numerically by making use of the `integral`

function – first we reproduce the values of the accepted answer:

f <- function(x, n) x * dnorm(x) * pnorm(x)^(n-1) for (i in 2:10) print(i * integrate(f, n = i, -Inf, Inf, abs.tol = 1e-20)$value) # test ## [1] 0.5641896 ## [1] 0.8462844 ## [1] 1.029375 ## [1] 1.162964 ## [1] 1.267206 ## [1] 1.352178 ## [1] 1.4236 ## [1] 1.485013 ## [1] 1.538753

This seems to be good enough. Now, let us calculate the maximum IQ that is to be expected for the current world population:

100 + 15 * world_population * integrate(f, n = world_population, -Inf, Inf, abs.tol = 1e-20)$value # expected max IQ of all people alive ## [1] 196.1035

To compare this with the actual record we find that Marilyn vos Savant seems to be (one of) the most intelligent persons on the planet. When you read the Wikipedia article you will find that there is some debate about what the actual number really is but it seems to be around 200… again, the above approximation seems to be good enough.

And now for the final leg: if the wealth distribution followed the same rules as the IQ distribution we could use the same formula to calculate the expected maximum wealth:

global_wealth <- 1.68e+14 # source: Allianz Global Wealth Report 2018 world_population <- 7.7e+09 per_capita_wealth <- global_wealth / world_population per_capita_wealth_sd <- per_capita_wealth * 0.15 # equivalent to IQ per_capita_wealth ## [1] 21818.18 per_capita_wealth + per_capita_wealth_sd * world_population * integrate(f, n = world_population, -Inf, Inf, abs.tol = 1e-20)$value ## [1] 42786.21

There you go: not more than $43.000! But the richest man at the moment, Jeff Bezos, has a net worth of about 140 billion US$!!! Conversely that would translate to an IQ of

1.4e+11 / per_capita_wealth * 100 ## [1] 641666667

So, compared to an IQ of over 641 million Einstein would be the mental equivalent of an amoeba! If IQ were additive, i.e. if two persons with an IQ of 100 each had an combined IQ of 200, you had to take nearly the whole population of Europe (about 738 million) to match that; even the whole North American continent wouldn’t suffice with its 579 million inhabitants!

But if it is not ability/intelligence that determines the distribution of wealth what else could account for the extreme inequality we perceive in the world?

Stay tuned for more to come…

**Update**

The follow-up post with an answer to the last question is now online:

The Rich didn’t earn their Wealth, they just got Lucky