How to derive the $\alpha$ for the Pareto rule

Question

Suppose we have the CDF for the Pareto Distribution given by:

$$ P(X \leq x) = 1-\left(\frac{x_m}{x}\right)^\alpha \;\;\;\;\;\;\;\;\;\; x \geq x_m$$

What is the intuitive way to find the alpha for which the 80/20 rule holds?

Answer 1

The basic result is: $\alpha=\log_4 5=1.160964...$

The calculation comes from the Lorenz curve; specifically you're asking for the $\alpha$ for which $L(0.8)=0.2$.

$L$ is defined as

$$L(F)=\frac{\int_{x_\mathrm{m}}^{x(F)}xf(x)\,dx}{\int_{x_\mathrm{m}}^\infty xf(x)\,dx} =\frac{\int_0^F x(F')\,dF'}{\int_0^1 x(F')\,dF'}$$

where $x(F)$ is the inverse of the cdf. The denominator is the mean of the distribution.

For the Pareto distribution, the Lorenz curve is $L(F) = 1-(1-F)^{1-\frac{1}{\alpha}}$, from which we obtain the equation we need to solve:

$$0.2 = 1-(1-0.8)^{1-\frac{1}{\alpha}}.\,$$

Hence

$$1-\frac{1}{\alpha} = \log(0.8)/\log(0.2)$$

$$\alpha=\frac{1}{1-\log(0.8)/\log(0.2)}=1.160964\ldots$$

This is mentioned on the Wikipedia page for the Pareto principle, and on the page on the Pareto distribution

The details are given here.