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I am not an economist but a mathematics student. I heard about the Pareto law - the mathematical modelling of the distribution of wealth, how did Pareto come up with this law? technically speaking from the data perspective?

Is it a consequence of pure chance? The De Matthew principal in economics?

Are there any strong criticism to the law?

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The Pareto distribution was derived based on the observation of Vilfredo Pareto that in Italy $80\%$ share of land belonged to $20\%$ of the country’s population, and he showed this held for many other countries as well (see Pareto Manual of political economy - the book is still being reprinted).

Consequently, based on this empirical observation it was argued that number of people $N$ with incomes higher than $x$ can be modeled as:

$$N = A x^{-\alpha}$$

and the Pareto distribution is based on the above.

In fact Pareto distribution is just another case of more general power law like Zipf's law (see Newman 2005). Power laws are quite common in nature for various reasons (see this article) - for example even human language follows Zipf's law (vsauce has nice video about this).

It is definitely not just due to chance wealth distributions can be approximated by pareto distribution as it is unlikely that this would be observed again and again in various countries across time if this would be just due to random chance. There are also economic models that imply that wealth distribution should follow pareto distribution. For example, Wold & Whittle (1957) show that if you try to model the ratio of growth rate of wealth and mortality rate (which is important as wealth is distributed through inheritances) as a system of differential equations you can show that wealth should follow the Pareto distribution. However, this being said Pareto distribution pops out in many economic relationships (it even matters for income distribution which is log-normal with pareto tail), and I don't know of any generally accepted explanation for why this is (aside for the potential solutions offered in the sources mentioned here which still claim that this is not fully resolved puzzle).

As to whether there is any criticism of Pareto distribution in economics I am not aware of any substantial criticism. Pareto distribution is an empirical distribution so even in theoretical modeling it is mainly applied in situations where we can actually empirically verify that such distribution is present.

For example, optimal income and wealth taxation theory would work with any income/wealth distribution you can imagine. Pareto distribution was used for wealth for some time but until recently when it comes to income people used mainly log-normal distribution (e.g. Mirrlees; 1971), but recently people realized based on empirical observations that we should actually use log-normal distribution with Pareto tail for the upper incomes (e.g. Atkinson, A. B., Piketty, T., & Saez, E. (2011)). Hence if anything when it comes to research on income inequality the criticism is that it was not used enough. This being said I suppose if you would dig deep enough in literature you could probably find examples when it was assumed/used even if it should not be. However, this is not really criticism of the Pareto distribution itself but rather its (mis)application in some specific instances.

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