Volterra, Vito (1860–1940)
Abstract
A mathematician by vocation, Volterra graduated at the Scuola Normale in Pisa in 1882 and obtained the Chair of Rational Mechanics at the University of Pisa in 1883. Subsequently he held chairs at the Universities of Turin and Rome. He became a Senator in 1905, was President of the Consiglio Nazionale delle Ricerche, of the Academia dei Lincei, Fellow of the Royal Society, etc. In 1931 he refused to take the required oath of loyalty to the Fascist government and was deprived of his Rome chair and forced to resign from all Italian scientific academies.
A mathematician by vocation, Volterra graduated at the Scuola Normale in Pisa in 1882 and obtained the Chair of Rational Mechanics at the University of Pisa in 1883. Subsequently he held chairs at the Universities of Turin and Rome. He became a Senator in 1905, was President of the Consiglio Nazionale delle Ricerche, of the Academia dei Lincei, Fellow of the Royal Society, etc. In 1931 he refused to take the required oath of loyalty to the Fascist government and was deprived of his Rome chair and forced to resign from all Italian scientific academies.
Volterra is renowned for his contributions to pure and applied mathematics. He is recognized as the founder of the general theory of functionals (1887, 1927a, 1929). In biological mathematics (independently of Lotka, who had examined the twospecies case earlier) he introduced the prey–predator equations generalized to n species (1926, 1927b, 1931).
In 1906, Volterra reviewed Pareto’s Manuale. Pareto, in treating the problem of integrating the differential equation of the indifference curve to obtain the ‘ophelimity’ (the utility function) had stressed the case in which the ‘elementary ophelimity’ (the marginal utility) of each good was a function solely of the quantity of that good, giving the impression that this was the case in which the integration could be performed with certainty. Volterra reminded Pareto that in the twovariable case there always exists an integrating factor so that it is always possible to perform the integration; he also pointed out that – as there exists an infinite number of integrating factors – the utility function is, in general, indeterminate. The real integrability problem arises when one has to deal with more than two commodities, and Volterra invited Pareto to go more fully into this problem. This was the beginning of the integrability problem in the theory of consumer demand.
Although (1906) was Volterra’s only contribution to economic theory, his work is of interest to economists for at least other two reasons. One is his functional analysis, now so important in problems involving infinite horizons, numbers of goods, etc. This, however, is like any other important mathematical tool whose availability enabled and continues to enable mathematical economists to solve their problems (for example, fixed point theorems or Liapunov’s second method). The other and more important reason is his study of predator–prey equations, which directly inspired an economic model, Goodwin’s growth cycle (1965): ‘Finally, at some happy moment, I remembered Vito Volterra’s formulation of the struggle for existence, and suddenly all became clear to me’ (Goodwin’s foreword to Vercelli (ed.), 1982, p. 72). This is a twoclass model which can be reduced to a system of two differential equations of the Lotka–Volterra type (the variables are the workers’ share of the product and the employment ratio). The result is a growth cycle; i.e. the economy grows, but with cycles in growth rates. Goodwin’s was the first successful attempt at integrating (not merely superimposing) growth and cycles, and his seminal paper has given rise to many important developments which use predator–prey equations as the basic tool (see, e.g., Izzo 1971; Desai 1973; Vercelli (ed.), 1982; Goodwin et al. (eds) 1984).
See Also
Selected Works

A full bibliography is included in Whittaker’s biography of Vito Volterra (originally published in ‘Obituary Notices of the Royal Society’ 1941) as reproduced and completed in the 1959 reprint of (1929). This biography also contains a detailed evaluation of Volterra’s scientific work. Volterra’s scientific papers have been collected in five volumes by the Accademia Nazionale dei Lincei as V. Volterra, Opere matematiche: memorie e note. Rome: Cremonese for the Accademia nazionale dei Lincei, 1954–62; the fifth volume includes the complete bibliography of Volterra’s works.

1887. Sopra le funzioni che dipendono da altre funzioni. Rendiconti della R. Accademia dei Lincei, series IV, 3(2). Reprinted in Opere, vol. I.

1906. L’economia matematica ed il nuovo manuale del Prof. Pareto. Giornale degli economisti 32: 296–301. Reprinted in Opere, vol. III.

1926. Variazioni e fluttuazioni del numero di individui in specie animali conviventi. Memorie della R. Academia dei Lincei, series VI, 2: 31–113.

1927a. Teoria de los funcionales y de las ecuaciones integrales e integrodiferenciales. Conferencias explicadas en la Facultad de Ciencias de la Universidad, 1925, redactadas por L. Fantappié. Madrid: Imprenta Clasica Española.

1927b. Variazioni e fluttuazioni in specie animali conviventi. Rendiconti del R. Comitato talassografico italiano, Memoria CXXXI. Reprinted in Opere, vol. V.

1929. Theory of functionals and of integral and integrodifferential equations. London: Blackie & Son (revised English translation of 1927a). Reprinted (with a Preface by G.C. Evans and the Biography by E. Whittaker). New York: Dover, 1959.

1931. Leçons sur la théorie mathématique de la lutte pour la vie. Paris: GauthierVillars.
Bibliography
 Desai, M. 1973. Growth cycles and inflation in a model of the class struggle. Journal of Economic Theory 6(6): 527–545.CrossRefGoogle Scholar
 Gandolfo, G. 1971. Mathematical methods and models in economic dynamics, 409–416. Amsterdam: North Holland and 436–442.Google Scholar
 Goodwin, R.M. 1965. A growth cycle. Paper presented at the First World Congress of the Econometric Society. Rome. Published in Socialism, capitalism and economic growth: Essays presented to Maurice Dobb, ed. C.H. Feinstein. Cambridge: Cambridge University Press, 1967. Reprinted in Essays in economic dynamics, ed. R.M. Goodwin. London: Macmillan, 1982.Google Scholar
 Goodwin, R.M., M. Krüger, and A. Vercelli (eds.). 1984. Nonlinear models of fluctuating growth. Berlin: SpringerVerlag.Google Scholar
 Izzo, L. 1971. La moneta in un modello di sviluppo ciclico. In Saggi di analisi e teoria monetariaII, ed. L. Izzo. Milano: F. Angeli.Google Scholar
 Vercelli, A. (ed.) 1982. Nonlinear theory of fluctuating growth. Economic Notes: 69–190.Google Scholar