Understanding the Pareto Distribution

The Pareto Distribution

The Pareto distribution was named after an Italian sociologist, economist and civil engineer known as Vilfredo Pareto. It is a probability distribution power-law which is applied in describing actuarial, geophysical, scientific and social as well as other observable phenomena. It was originally used in describing wealth distribution in the society. It led to the trend that a bigger size of wealth is owned by a small number of people in the population. The Pareto distribution became known colloquially as the 80-20 rule or the Pareto principle. It is also occasionally known as the Mathew principle. According to this rule, 80 percent of the entire society’s wealth is owned by 20 percent of the whole population (Nirei and Aoki, 2016).

Application of the Pareto Distribution

Originally, Vilfredo Pareto applied this distribution to illustrate or show how wealth was distributed among people in the society as is properly portrayed how a large fraction of wealth in any society was owned by a few individuals. Vilfredo Pareto also applied it to show how income was distributed in societies. This was the beginning of the 80-20 rule or the Pareto principles which shows that 20 percent of the entire population is in control of at least 80 percent of the whole wealth in a society. Pareto’s 80-20 rule is slightly similar to another one of his value α, British income taxes which shows that at least 30% of the entire population held not less than 70% the whole income (Nirei and Aoki, 2016).

The above probability density graph demonstrates how the fraction or probability of the whole population with wealth per individual that is high and steadily decreases as the size of the wealth increases. Pareto distribution is not restricted to income or wealth alone but can be used in different situations (Rootzén, Segers and Wadsworth, 2018).

Pareto Distribution in Relation to Zipf’s Law

Zipf’s Law is also known as the Zeta distribution. It separates values into simple ranks. Both Pareto distribution and Zipf’s Law have negative exponents which are scaled to make their cumulative distribution to be equivalent to 1 (Arshad, Hu and Ashraf, 2018). Additionally, Zipf’s can be produced from Pareto distribution when the income x value are put into N ranks so that everybody in the ranks follow the 1/rank approach. The distribution is then normalised by the definition xm. Where,

The generalised harmonic figure is

This constitutes a probability density function known as Zipf’s function which is derived from Pareto’s distribution.

Where s=a-1 while x represents ranks ranging from 1-N, N being the highest income rank (Arshad, Hu and Ashraf, 2018).

References

Arshad, S., Hu, S. and Ashraf, B.N., 2018. Zipf’s law and city size distribution: A survey of the literature and future research agenda. Physica A: Statistical Mechanics and its Applications, 492, pp.75-92.

Nirei, M. and Aoki, S., 2016. Pareto distribution of income in neoclassical growth models. Review of Economic Dynamics, 20, pp.25-42.

Rootzén, H., Segers, J. and Wadsworth, J.L., 2018. Multivariate generalized Pareto distributions: Parametrizations, representations, and properties. Journal of Multivariate Analysis, 165, pp.117-131.