The Power-Law Distribution

Original article was published on Artificial Intelligence on Medium

Golden Ratio

The Golden Ratio or ‘80–20’ rule exists as a colloquial natural phenomena. It postulates things like: 20% of the worlds population own 80% of the wealth. Let’s assume for a second that wealth is defined by the Power Law which and is characterised by some α. What fraction W of the total wealth is held by the richest fraction P of the population?

Now we can integrate the power-law function above to derive the fraction of the population whose wealth is at least x, given by the cumulative distribution function:

Moreover, the fraction wealth held by those people is given by:

where α>2. If we now solve the first equation and substitute it into the second, we find an expression that does not depend on wealth (x) at at all:

Now this is crazy to me: by making small assumptions about the distributional properties of wealth distribution, we can remove wealth from the equation and still show how wealth is spread. This extreme top-heaviness is sometimes called the “80–20 rule,” meaning that 80% of the wealth is in the hands of the richest 20% of people.

As an example, say we want to know how much wealth the top 20% of richest people own? Let’s set α=2.2, then (α-2)/(α-1) = 0.2/1.2 = 0.167. Then we set P = 20%, so W = 20% to the power of 1/3, = 76%, which is not far off 80%! Funnily enough, this is actually a pretty good fit for society.

Note: that the relationship can skew if we change the value of α, becoming more extreme as α<2, which shows that wealth is held by a single person.

It’s exactly because of this this functional form being so unique in nature and so eloquent, that we can simplify characteristics as ‘80–20’. It’s not an exact science but social science rarely is. However, deriving an α for these social dynamics goes a long way in telling us exactly how these natural phenomena realise and act.