### Discussion Paper

Pareto tails in socio-economic phenomena: a kinetic description

(Published in Special Issue
Agent-based modelling and complexity economics)

## Abstract

Various phenomena related to socio-economic aspects of our daily life exhibit equilibrium densities characterized by a power law decay. Maybe the most known example of this property is concerned with wealth distribution in a western society. In this case the polynomial decay at infinity is referred to as Pareto tails phenomenon (Pareto, *Cours d’économie politique*, 1964). In this note, the authors discuss a possible source of this behavior by resorting to the powerful approach of statistical mechanics, which enlightens the analogies with the classical kinetic theory of rarefied gases. Among other examples, the distribution of populations in towns and cities is illustrated and discussed.

## Comments and Questions

This paper presents a mathematical investigation of the phenomenon of Pareto tails, which is ubiquitous in socio-economic systems. It consists in the fact that often the equilibrium statistical distribution of a certain physical quantity shows a power law-type decay at infinity. This implies that the distribution is fat-tailed, as opposed ...[more]

... to the classical Gaussian or Maxwellian distributions which instead decay exponentially fast at infinity. As a consequence, very high values of that physical quantity may be rare but their probability is in general not so negligible. The terminology originates from the fact that such a trend was observed, probably for the first time, by the Italian economist Vilfredo Pareto (Paris, 1848 - Céligny 1923) with reference to the wealth distribution in western societies.

By appealing to conceptual tools of collisional kinetic theory, the authors show that it is possible to explain the emergence of Pareto tails starting from simple binary, i.e. one-to-one, interactions among the agents of a system. The case studies considered in the paper refer to: traders that produce a wealth redistribution by exchanging goods, which is a prototype of Pareto's investigations; educated people who shape the distribution of knowledge depending on the school degree that they hold; people who migrate to or from a city, thereby determining the size of the urban population.

The paper provides a limpid evidence of the fact that the kinetic approach applied to socio-economic systems is a valuable methodological tool to shed light on how complex collective trends are generated by simple individual interactions empirically experienceable in personal lives. This is indeed the same spirit as the one which, in the late 1800, motivated the studies of the Austrian physicist Ludwig Boltzmann, who aimed at explaining the complex concepts of thermodynamics starting from the simple mechanics of colliding gas molecules. The legacy of his theory is nowadays successfully permeating fields very distant from the original one, as this paper clearly demonstrates.

"Pareto tails in socio-economic phenomena: A kinetic description" by Gualandi and Toscani is a nice overview of the activity performed within mathematics on the kinetic-theory approach to random exchange models. I suggest to accept this paper after minor revisions.

Here is a list of suggested amendments.

1. Page ...[more]

... 2 (preferential attachment). I suggest that the authors consult the paper by Simkin and Roychowdhury, Re-Inventing Willis: https://arxiv.org/pdf/physics/0601192.pdf

They might also consider the Introduction of the book by Durrett "Random graph dynamics, CUP, 2007":

https://services.math.duke.edu/~rtd/RGD/fch1.pdf

Mentioning Willis and Yule in the context of preferential attachment would be appreciated.

2. Page 3 (probability density). If $w$ is a scalar in (1.1), one should mention the role of exchangeability. This is discussed in a recent book chapter by Duering, Georgiou and Scalas

(see https://arxiv.org/pdf/1609.08978.pdf and the published version: During, Bertram, Georgiou, Nicos and Scalas, Enrico (2017) A stylised model for wealth distribution. In: Akura, Yuji and Kirman, Alan (eds.) Economic Foundations of Social Complexity Science. Springer Singapore, Singapore, pp. 95-117).

3. Page 3:

"1,5 e 3" --> "1.5 and 3"

"reach" --> "rich"

4. Page 5: I would replace "indistinguishable" with "exchangeable".

5. Page 7:

"into {\it capitalistic}" --> "into a {\it capitalistic} regime"

"that equation (3.11)" --> "that solutions of equation (3.11)"

"$\Gamma$-like" --> "Gamma-like"

6. Page 8:

"( gain" --> "(gain"

7. Page 9:

"genial" --> "learned"

I would remove "clearly" after figure 4.1

8. Page 11:

I would mention gravity models before (5.15). What is the parameter $\nu$ in $\nu \to \infty$?

9. Page 12:

"the both" --> "both"

We thank the referee for appreciating the content of the work. We will take care of the suggested changes.