# Pareto, Vilfredo Federigo Samaso

This article Vilfredo Federigo Samaso Pareto was adapted from an original article by Marc Barbut, which appeared in StatProb: The Encyclopedia Sponsored by Statistics and Probability Societies. The original article ([http://statprob.com/encyclopedia/VilfredoFederigoSamasoPARETO.html StatProb Source], Local Files: pdf | tex) is copyrighted by the author(s), the article has been donated to Encyclopedia of Mathematics, and its further issues are under Creative Commons Attribution Share-Alike License'. All pages from StatProb are contained in the Category StatProb.

Vilfredo Federigo Samaso PARETO

b. 15 July 1848 - d. 19 August 1923

Summary Pareto showed empirically in 1896 that the distribution of income was asymptotically $x^{-\alpha}$, thus discovering the very first case of a stable non-Gaussian probability distribution.

Vilfredo Pareto was born in Paris, where his father, the Marquis Raffaele Pareto (1812-1882), an engineer from an old Genoese family, was in exile at the time for political reasons. His mother, Marie Métenier (1816-1889), was French, so that French was Pareto's mother tongue; it was only after the family's return to the Piedmont in 1852 that he mastered Italian. He had two sisters who left no progeny.

His first wife whom he married in 1889 was a Russian aristocrat, Alexandrina (Dina) Bakounine, who left him in 1900. He remarried to Jeanne Regis, a Frenchwoman with whom he had been living for twenty years, two months before his death. As he had no children, he left his fairly considerable fortune to her, having earlier donated his library to the University of Lausanne in 1908. He is buried at Céligny near Geneva in Switzerland, where he had been residing since 1900.

His life, public, intellectual and professional is fairly neatly divided into three phases.

1. The period from his birth till 1870 was the formative phase. This was basically scientific and technical, although Pareto learned Greek and Latin by himself and became a discriminating connoisseur of these languages. He first attended the Leardi Technical Institute at Casale Monferrato (Piedmont) from 1859-1861, then the Royal Technical Institute of Turin (1861-1864). This was followed by the University of Turin (Torino), where he obtained his degree in the Mathematical and Physical Sciences in 1867, and finally by the School of Applied Engineering of this University. He was awarded his engineering diploma following a thesis on La théorie de l'élasticité des corps solides et l'intégration des e\'quations differentielles qui définissent l'équilibre.

2. From 1870 to 1873, Pareto worked as an engineer on the Italian railways, then in 1873 joined a firm building railway material at San Giovanni Valdarno in Tuscany. He managed this firm from 1880 to 1890, becoming a consultant between 1890 and 1893. It was during this period, when he faced the economic and social problems of managing the firm that he became vitally interested in economics and sociology.

In 1874 he took part in the creation of the Société Adam Smith . During this period he wrote several papers for L'economista from 1876, the Journal des Economistes from 1887 and for the Monde économique and the Giornale degli Economisti. He also presented several communications at conferences.

He was resolutely in favour of a liberal economy, but he was also a pacifist and an anti-colonialist. He stood for election to the Chamber of Deputies but was not successful.

He travelled extensively abroad, residing on occasion in France, Germany, Austria, England, Belgium and Switzerland. In 1891 in Switzerland he met Léon Walras, whom he was to succeed in 1893 in the Chair of Political Economy at the University of Lausanne.

3. From 1893 on, he devoted himself to teaching and research. As an economist he was a follower of A. Cournot (q.v.) and L. Walras in mathematical economics. His main innovation was a definition of equilibrium (Pareto's optimum) which is still in use today. But Pareto valued theory only insofar as it stood the test of reality; it was one of the reasons for his interest in statistics and for his work in this field.

As a sociologist, he was convinced that economic facts were inseparable from the society which engendered them. Thus, from 1897 until his retirement in 1911, he taught sociology as well.

It was during this period that his three main books were published:

Cours d'économie politique, Lausanne, Vol.1 (1896) and Vol.2 (1897).

Manuale di economia politica con una introduzione alla scienza sociale, Milano, 1906 (French translation: Manuel d'Economie politique, Paris, 1909).

Trattato di Sociologia, Firenze, 1916 (French edition: Traité de Sociologie, Lausanne, 1917).

In statistics Pareto worked in three areas:

(a) Interpolation and fitting methods to which he devoted several papers, in particular those in the Journal de la Société Statistique de Paris, November 1897, and in the Zeitschrift für schweizerische Statistik, XXXV, 1899, as well as the Actes du IVème Congrès International de Psychologie (Geneva, 1909). \smallskip

(b) Actuarial studies on mutual insurance systems and the calculation of pensions. He worked on these in 1905 in response to a request from the employees of the Swiss Federal Railways. \smallskip

The main works which he published in the two areas (a) and (b) have been republished in Pareto (1989), where his paper "Sur les fonctions génératrices d'Abel" in pure mathematics, published in Kronecker's "Journal für die reine und angewandte Mathematik , 110 (1892), 290-323, is also to be found. \smallskip

(c) But it was mainly his discovery in 1895 of the Distribution curve for wealth and incomes" that Pareto is renowned as a statistician. He showed empirically that the observed distribution of incomes, like that of wealth was well fitted by theoretical distributions of the type

$$Pr(X > x) = \frac {K}{(x + c)^\alpha}, x \geq x_0, \tag{1}$$

$K$ and $c$ being respectively parameters of scale and location while the exponent $\alpha$, for the series studied by Pareto, takes values between 1 and 2 (see Pareto, 1964, 1965).

Later, this kind of distribution was found to arise in various fields, not only in the social sciences, but also in the natural and physical sciences. Various probabilistic models lead to such distributions: Pareto himself had attempted to develop probabilistic models as early as 1896 (see Pareto, 1964, supplements to Vol.2, pp.416-419). Thirty years later Maurice Fréchet (q.v.) was to perfect Pareto's model.

The most profound reason for the frequency with which Pareto distributions appear was given in 1935 by Paul Lévy (1937), namely that stable distributions, other than the Laplace-Gauss ( Normal) distribution, exhibit the asymptotic behaviour of type (1), with $0 < \alpha < 2$.

Further, when conditions for convergence to the Normal distribution in the Central Limit Theorem do not apply, namely when dealing with independent identically-distributed random summands whose large values have a non-negligible probability (heavy tailed distributions), if these distributions have tail behaviour asymptotically of type (1), then a Central Limit Theorem result still holds, with stable law of exponent $\alpha$ figuring as the limit law, rather than the Normal.

It is thus to Pareto that belongs the immense credit for breaking the almost total monopoly of the Gaussian (Normal) distribution and its domain of attraction in statistics. Unfortunately it only earned him the lack of understanding and hostility of his peers, in particular F.Y. Edgeworth (q.v.).

#### References

 [1] Pareto, V. (1964). Cours d'Économie Politique, Droz, Geneva. [This is a new edition by G.H. Bousquet and G. Busino.] [2] Pareto, V. (1965). Écrits sur la courbe de répartition de la richesse. Droz, Geneva. [Collected with commentary by G. Busino.] [3] Pareto, V. (1989). Statistique et Économie mathématique. Droz, Geneva. [Preface by René Roy.] [4] Lévy, P. (1937). Théorie de l'addition des variables aléatoires. Gauthier-Villars, Paris.

Reprinted with permission from Christopher Charles Heyde and Eugene William Seneta (Editors), Statisticians of the Centuries, Springer-Verlag Inc., New York, USA.

How to Cite This Entry:
Pareto, Vilfredo Federigo Samaso. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pareto,_Vilfredo_Federigo_Samaso&oldid=39243