Benini, Rodolfo

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Benini, Rodolfo



Rodolfo Benini (1862–1956) was born in Cremona. At the very young age of 27 he was appointed to the chair of history of economics at Bari. His academic life led him from Bari to Perugia in 1896 and then to Pavia, where from 1897 to 1907 he taught political economics and statistics. In 1908 Benini went to Rome and there held the chair of statistics until 1928; from then until 1956 he taught economics at the same university. It is difficult to name any single predecessor or teacher who had a particularly strong influence on Benini, since he worked in a great variety of social science fields and showed in each a great degree of independent thought and creativity. As immediate predecessors of Benini we may certainly list Angelo Messedaglia, Ridolfo Livi, Maffeo Pantaleoni, and especially Vilfredo Pareto. In a broader sense we might well cite the whole system of Italian economic and social thought, which up to this century had developed quite independently and which was ahead of the rest of European thought in many instances. It is the connection of statistical knowledge and economic and social theory in Benini which led him to very constructive results in all the fields he engaged in.

Together with Süssmilch, Quetelet, and Achille Guillard, Benini can be regarded as one of the founders of demography as a separate science (1896; 1901a). In his Principii di demografia he distinguished between a qualitative and a quantitative theory of population. Both his concepts differ from the standard use in the literature. In the “qualitative” theory Benini elaborated the concept of descriptive statistics to include rates of birth and mortality; life expectancies; the fertility of women as a decreasing function of their age; and normal distributions of physiological characteristics of men and women. In the analysis of the cohesion of social groups Benini developed an attraction–repulsion index to measure the association between dichotomous characteristics of husbands and wives (1898; 1901a; 1928a). Classifying a population by any given characteristic (e.g., literate vs. illiterate males—m1, m2)—and females—f1,f2), Benini arranged the relative frequencies pij (i, j = 1, 2) of the possible combinations in 2 x 2 tables such as Table 1. From this classification Benini then derived a first measure of attraction (repulsion) with

γij = pijpi. p. j, i,j=1,2

where y11 = y22 = –y21 = — y12. The index of attraction (when yij > 0) or repulsion (when yij ≤ 0), a later measure he put forth, is given by Benini as

the absolute value of the index ranging from 0 to 1. This statistic was further elaborated on by Benini (1928a) and later by many others in Italy and elsewhere (Goodman & Kruskal 1954–1963).

In the “quantitative” theory Benini tried to discover evolutionary laws of aggregate societies and their latent structures. He denied the general validity of Malthus’ theorem. Uniform predictions cannot be made from empirical evidence, said Benini. He expected, however, that the continuing process of urbanization of societies would lead to an adjustment of the growth rates of social aggregates, not so much by delayed marriages as by advanced education and new methods of birth control. Moreover, because of an instinct of imitation, the lower classes of urban centers would aspire to the behavior and social status of the next higher classes. This would cause a continual process of assimilation—a narrowing and adjustment of the average age of marriages within and between social groups; an adjustment of the proportion of unmarried people; and an adjustment of birth and death rates. Thus eventually, the evolution of societies will lead to a stationary state of social aggregates and a gradual elimination of structural differences.

Benini engaged in statistical research in economics similar to his statistical work in demography. He hoped to reduce economic science by systematization and by empirically verified formulas or laws to the concise system of expressions characteristic of physics and chemistry (1907, p. 1053; 1908, p. 17). In 1894 he tried to estimate the distribution of property among social classes (1894). After Pareto’s work on the distribution of incomes appeared (1896; 1897), Benini extended this work and modified the Pareto distribution in his analysis of the distribution of property. The function pro-

Table 1

posed by Benini for the distribution of property values was

logF(x) =logk – a(logx)2, forxx0,

where F(x) is the proportion of property values greater than or equal to x, that is, the distribution function cumulated to the right. There is a truncation point, x0, forming a lower limit. Another way of expressing Benini’s distribution is to say that it is that of a random variable, X, such that [log (X – x)]2 has a negative exponential distribution. [SeeDistributions, STATISTICAL, article onSPECIAL CONTINUOUS DISTRIBUTIONS.] In contrast, Pareto’s distribution is such that log (X — x0) has a negative exponential distribution. The attraction–repulsion index and the modification of the Pareto distribution are Benini’s most original contributions to statistics.

As early as 1907 Benini established empirical estimates of price elasticities, demand curves, and Engel curves (see 1907; 1908). The demand curve underlying his estimates was of the general hyperbolic form logy = a + b logx. In the case of prices and demand for coffee in Italy, Benini came to estimates of a = 3.63161 and b = –0.384. In an extension of this work (1908) Benini estimated, in addition to other demand curves, income-induced increases in expenditures for housing. The underlying model is similar to that shown above, with b now positive and ranging from 0 to 1 (for Dresden and Breslau b = 0.617). In this Benini preceded A. C. Pigou, Lenoir, Lehfeld, R. Frisch, and H. Schultz. Benini, however, had no means of establishing confidence intervals for his estimates, although the signs of his estimates agreed with what one would expect. [SeeDemand and supply, article Oneconometric studies.]

Benini’s other economic ideas are spread over a long series of articles, and it is hard to consider them all. They can be reduced, however, to some few central results which underlie his writings. Most important among them is Beninf‘s notion that in any exchange transaction there exist a minimum price and a maximum price (p min and p max) at which the transaction still can take place (1928b). According to Benini, the most likely price between two equally endowed parties (equality of bargaining power, in whichever way this is defined) would be the point equidistant from the boundary points p min and p max. Given, however, unequal endowment of the contracting parties, there would result a shift in the price and an exploitation of the weaker partner. On this basis Benini explained how profits arise at the expense of labor income.

Benini believed that protectionism in international trade is justified on three grounds: first, by the existence of the same kind of exploitation of a weaker party described above for the general market place (in this case the parties are foreign enterprises or foreign states); second, by the cumulative effect of that exploitation associated with the power of states in international relations; and third, by the vulnerability of infant industries. For all these reasons the state has to fulfill special functions and is therefore introduced by Benini as an additional factor of production.

In connection with the distribution of property and income Benini observed that in some countries, among them Italy, a doubling of property was associated with a threefold increase in total income (per person or household), at least as long as the income derived from labor constituted a significant part of total income derived from labor and property. From this Benini then derived his fiscal axiom that a proportional taxation of incomes would lead in those countries to a less than proportional taxation of property and that a proportional taxation of property would lead to a progressive taxation of incomes.

His empirical results also led him to extend Galton’s law concerning the progressive elimination of economic and social divergencies (structures). This process occurs as extreme points are continuously eliminated. Their elimination induces an asymptotic approach to stationary states. This process was previously noted in Benini’s quantitative theory of the growth of social aggregates.

In addition to his work in demography, sociology, and economics, Benini developed a unique interest in the works of Dante. He undertook to reveal a “second beauty” in Dante by uncovering the quantitative consistency of the structure of The Divine Comedy, which up to then had generally been neglected (there had been some exceptions—Busnelli, Angelitti, and Moore). (It is apparent that Dante did incorporate quantitative relations and symbols in the structure of The Divine Comedy, as is immediately apparent from the fact that each of the three parts of The Divine Comedy contains 33 canti and The Inferno an additional introductory one, bringing the total to exactly 100. Moreover, each of the three parts ends with the word “stelle,” and so on.) Many enigmas and allegorical elements in The Divine Comedy have yet to be interpreted. To do so it is important to appreciate Dante’s knowledge of astronomy and the way he incorporated this knowledge into The Divine Comedy. Similarly, by quantitative analysis the structure and dimensions of the Inferno and Purgatory may be ascertained.

In addition, this kind of analysis may explain the relationship of the calendar used by Dante and the dates he attributed to events in The Divine Comedy. The exact dates of Dante’s poetic voyage with Vergil and, later, with Beatrice through the regions of the Inferno, Purgatory, and Paradise can thus be established. The main assumption on which Benini based his investigation is that the rigorous structure of the poem will not allow for obvious contradictions or omissions. Explanations may be found by applying medieval concepts to these enigmas. In addition to explanations of the above kind, Benini showed that Dante believed Purgatory to be located on Mount Sinai and established the date of birth of Cacciaguida, an ancestor of Dante, and the date when Cacciaguida’s son Alighiero died. Benini also thought he had discovered that Dante’s poetic technique differed, depending on the seriousness of the moral crimes he was describing. The new perspectives contained in Benini’s contribution to the knowledge of Dante’s poem lead to the very margin where one might discover something in Dante that the poet himself was unaware of. Benini knew of this danger and tried to avoid such pitfalls. It took some time for Benini’s work on Dante to find support, just as Benini’s achievements in combining statistics and social science were not immediately appreciated.

Klaus-Peter Heiss

[For the historical context of Benini’s work, see Population, article on Population theories; and the biographies of Pantaleoniand Pareto. For discussion of the subsequent development of Benini’s ideas, see Statistics, descriptive, article onassociation.]


1892 Sulle dottrine economiche di Antonio Serra: Appunti critici. Giornale degli economisti Series 2 5:222–248.

1894 Distribuzione probabile della ricchezza privata in Italia per classi di popolazione. Riforma sociale 1: 862–869.

1896 Di alcuni punti oscuri della demografia. Giornale degli economisti Series 2 13:97–128, 297–327, 509–534.

1897 Di alcune curve descritte da fenomeni economici aventi relazione colla curva del reddito o con quella del patrimonio. Giornale degli economisti Series 2 14:177–214.

1898 Le combinazioni simpatiche in demografia. Rivista italiana di sociologia 2:152–171.

1899 Gerarchie sociali: Contributo alla teoria qualitativa della popolazione. Rivista italiana di sociologia 3: 17–49.

1901a Principii di demografia. Florence: Barbéra.

1901b Tecnica e logica dei rapporti statistici. Giornale degli economisti Series 2 23:503–516.

1905a I diagrammi a scala logaritmica (a proposito della graduazione per valore delle successioni ereditarie in Italia, Francia e Inghilterra). Giornale degli economisti Series 2 30:222–231.

(1905b) 1923 Principii di statistica metodologica. Turin: Unione Tipografico–Editrice Torinese.

1907 Sull’ uso delle formole empiriche nell’ economia applicata. Giornale degli economisti Series 2 35: 1053–1063.

1908 Una possibile creazione del metodo statistico): “L’economia politica induttiva.” Giornale degli economisti Series 2 36:11–34.

1912 L’azione recente dell’ oro sui prezzi generali delle merci. Società Italiana per il Progresso delle Scienze, Rome, Atti 6:97–123.

1928a Gruppi chiusi e gruppi aperti in alcuni fatti collettivi di combinazioni. International Statistical Institute, Bulletin 23, no. 2:362–383.

1928b Un ritorno ai preliminari dell’ economia politica. Economia New Series 1:411–428.

1952 Dante tra gli splendori de’ suoi enigmi risolti, ed altri saggi. Rome: Edizioni dell’ Ateneo.


Bachi, Roberto 1929 I principali scritti di Rodolfo Benini. Giornale degli economisti Series 4 69:1068–1076.

Bari (city), UniversitÀ, FacoltÀ di Economia e Commercio 1956 Studi in memoria di Rodolfo Benini.Bari: The University.

Goodman, Leo A.; and Kruskal, William H. 1954–1963 Measures of Association for Cross-classifications. Parts 1–3. Journal of the American Statistical Association 49:732–764; 54:123–163; 58:310–364.

Pareto, Vilfredo 1896 La curva delle entrate e le osservazioni del Prof. Edgeworth. Giornale degli economisti Series 2 13:439–448.

Pareto, Vilfredo 1897 Aggiunta allo studio sulla curva delle entrate. Giornale degli economisti Series 2 14: 15–26.

[Rodolfo Benini]. 1929 Giornale degli economisti Series 4 69:837–966. → Contains articles on Benini by Corrado Gini and others.

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