1 Introduction

Pareto’s personality had a great influence on the evolution of economic theory in the last century at least in the Italian context. As already observed by Schumpeter (1954) almost immediately and therefore at the beginning of the twentieth century a small but combative circle of followers aimed to defend and spread the new Paretian approach based on the idea of general economic equilibrium (Amoroso 1909, 1910; Sensini 1912; de Pietri Tonelli 1921). Paretian scholars were well recognizable in the Italian context for their contributions to mathematical and statistical aspects. At least in Italy following McLure (2007) a Paretian school or tradition was formed to develop and extend the Pareto approach. This school spanned three generations up to the seventies of the last century. The first generation was formed by those who had direct contact with Pareto even just by correspondence. Among these scholars stand out the names of Guido Sensini Luigi Amoroso Alfonso de Pietri Tonelli Roberto Murray Pasquale Boninsegni and Gino Borgatta. The members were criticized by Pasquale Jannaccone in the article “Il Paretaio” (1912). In the 1930s a new generation of Paretians came forward mostly scholars of the previous one. This second generation included the names of Giulio La Volpe Eraldo Fossati Arrigo Bordin Felice Vinci and others. Between the two wars the Paretians were well placed at universities. Finally a third generation was formed the one that emerged in the post-war period. Among the economists who were clearly inspired by Pareto we can find Emilio Zaccagnini Raffaele D’Addario Valentino Dominedò. and above all Giuseppe Palomba.

Apart from the individual contributions some of which are also quite remarkable the Paretian tradition was characterized mainly by its methodological approach. The followers of Pareto even more so than their master set out to transform economics into an exact science based on the model of ration mechanics. This assimilation of the economic facts to the mechanical facts did have extraordinary success in the microeconomic field less in the macroeconomic one and still less in the dynamic one. This radical transformation of economic discourse required a new methodological orientation which consisted essentially of a broad application of mathematics to economic analysis. Thus the idea of a pure economy arose to be distinguished from the applied or historical-descriptive one.

The purpose of this contribution is to outline the essential features of the Italian Paretian school in the field of pure economics or as we would say today of mathematical economics. The most important results were obtained during the period between the two wars so we will limit our analysis to this time frame focusing the analysis on the less known but equally interesting aspects.

2 Mathematical economics and the birth of microeconomics

The period between the two world wars can be defined as the golden age of mathematical economics in Italy. Italian analytical economists published in new journals such as Econometrica and actively participated in forming the Econometric Society in the early 1930s (Nicola 2000; Brandolini and Gobbi 1990). This favorable position of the Italian tradition in the field of mathematical economics was basically due to two circumstances. The first was represented by the legacy of Pareto and his school while the second was to be found in the fact that among the students of Pareto there was a pure mathematician of great value Luigi Amoroso. We see then that an important voice even if poorly valued was that of Bruno de Finetti who published a remarkable series of articles on mathematical economics in the second half of the 1930s. After World War II the Paretian tradition in this field weakened considerably and effectively disappeared internationally. For this reason we concentrate our analysis on the years between the two world wars with the aim of deepening and widening some previous contributions (Guerraggio 1990, 1998).

In the period between the two world wars while in Italy the debate was dominated by so-called corporatist economics we can identify three major lines of research in the field of pure or mathematical economics. The first can be identified in the birth and affirmation of microeconomics as a privileged form of economic thinking at the expense of other institutional historical or descriptive approaches. The second concerns the transition from static to dynamic a relevant topic in the years between the two wars which experienced the devastating effects of the crisis of 1929. The third concerns the birth of welfare theory that is the attempt to offer a moral evaluation of the economic structure a theme developed mainly by De Finetti a Paretian suis generis. Among Pareto’s followers mathematical economics is mainly represented by the writings of Luigi Amoroso. The other Paretians made important but minor contributions which were always in the direction of a mathematization of economic discourse. So we will limit ourselves to considering essentially his works with some reference to the other protagonists.

The first issue that the scholars of Pareto faced was to bring to completion the methodological revolution of their master i.e. the project to transform economics into an exact science on the model of rational mechanics a pure science as it was at the time said. This transformation required a considerable analytical effort and could only be realized with the birth of a new discipline mathematical economics which would be added to the various approaches existing among the scholars of the discipline quite hostile to analytical formalization. This project was pushed by Pareto himself who repeatedly invited the young Amoroso to consider the field of mathematical economics starting from the appendix of the 1909 Corso. In a letter of June 1911 (Pareto 1973) Pareto openly invited Amoroso to prepare a text on mathematical economics and urged him to write notes from which he himself would revise. Equally important was the influence on the young Amoroso of the other protagonist of the marginalist revolution in Italy Maffeo Pantaleoni. Therefore following his lessons in Rome Amoroso chose economic studies definitively abandoning the natural sciences. Second it influenced the choice of content. It can be said that Amoroso followed Pareto in the method the transformation of economics into an analytical science but followed Pantaleoni in the topics considered the development of Marshall’s theory of partial equilibrium.

The first part of Amoroso’s scientific career was dedicated to the realization of the Paretian project of a full formalization of economic discourse (Amoroso 1909, 1910). The result of this research program was the (1921) Lezioni di economia matematica the first text in Europe on mathematical economics. The intermediate step in Amoroso’s path was the Corso di economia pura of (1913) a text that contains notes for the lectures of economics at the Faculty of Law in Rome. The peculiarity of the text is that the treatment inspired by the text of Pantaleoni Principii di eocnomica pura (1989) is divided into two distinct parts. The traditional discursive treatment is accompanied by its analytical exposition for the own use of students of mathematics. Amoroso developed it in very modern terms that is with graphs and applying the multipliers of Lagrange the part of the theory of the consumer. We also find a complex application of the multiplicative utility function the well-known Cobb–Douglas. The opportunity for Amoroso to prepare a true text of mathematical economics was given by his position as a full professor at the Bari Scuola Superiore di Commercio in 1917.

The 1921 Lezioni is the result of a decade of intense work. The whole economic matter from consumption to production is analyzed in essentially mathematical terms as a set of maximum constrained problems. As a text of mathematical economics the Lezioni has a peculiar characteristic in the sense that it does not contain a treatment of the mathematical concepts used in economics. It is assumed that the student already knows the basics of calculus. So rather than a mathematical text preparatory to economics it is a textbook of economics treated mathematically. The difference is remarkable. The final result of Amoroso is what we call a handbook of microeconomics that could preserve its relevance even today. Leaving aside the first part which concerns the currency the second one contains an analytical treatment of the consumer theory the third one of the equilibrium of the firm and the markets the fourth of the costs and the fifth one is dedicated to the general economic equilibrium. The method used namely constrained maximization unites the various topics. This analytical expedient derived from rational mechanics gave a methodological coherence to the new marginalist approach with a clarity that in Amoroso is completely new and aware.

If the 1921 Lezioni rather than a text of mathematical economics is a text of microeconomics that is economics treated mathematically; this is confirmed in the manuals published by the other students of Pareto. For example, the Bordin Lezioni of (1934) and (1936) are texts of microeconomics of the Paretian tradition that take up and develop the themes of the lessons of Amoroso. In Italy we are slowly witnessing a process of transformation of the economic discourse into a mathematical discourse by the Paretians even in a difficult context in which the hostility to the formalization of economic discourse was still present and scholars have to write, to get some academic recognition, on corporatist economics. With amoroso the pure economics turned into microeconomics.

3 The building of economic dynamics: The mathematical business cycle theory and the dynamization of the general equilibrium

Economic dynamics was a field of research that the first generation of marginalists left unexplored. It became one of the fundamental research topics of the period between the two wars (Shackle 1967). The contribution of the Paretian school on this topic was analyzed in depth in the last two decades (Pomini 2008, 2018; Tusset 2009; Montesano 2015).

In the period between the two world wars all main economists dealt with economic dynamics and not only purely theoretical issues. The crisis of 1929 demonstrated the limits of the capitalist economic system and from the theoretical point of view the insufficiency of economic statics. Therefore it is not surprising that dynamic theory also became one of the main fields of research for the Paretians in which they made innovative contributions in two main directions: the mathematical theory of the economic cycle and the dynamization of the equations of the general economic equilibrium which we now consider.

The theory of the economic cycle was developed mainly by Amoroso starting from the criticism of the economic barometers of Harvard put forward by Irving Fisher (Fisher 1925) the American economist really admired by Amoroso. The Harvard barometers had a great resonance in the early 1920s because the path of the three-time series considered production stock exchange and interest rate seemed to offer a solid basis for future forecasts. However the crisis of 1929 proved their modest usefulness and was subsequently abandoned. In a (1928) contribution Amoroso also intervened in the debate on the side of Irving Fisher who was very critical of the provisional capabilities of these statistical regularities.

Following the footsteps of the American economist Amoroso built his economic cycle theory on two fundamental equations (Amoroso 1932). The first was that of Fisher according to which production depended on price changes and therefore on a first derivative. Amoroso called this relation the equazione dell’officina. To achieve the cyclical effect he added a second equation which he defined as the equazione della borsa the inverse relationship between the level of prices and the forecasts of production levels. The final result was a differential equation of the second order that provided cyclic movement with respect to the exogenous variable time. In this macroeconomic vision of cyclical dynamics Amoroso closely followed Pareto. Not monetary disturbances or other exogenous causes but complex sectoral interdependencies generate the dynamics of the system. Amoroso’s model well known internationally in the circle of mathematical economists even if initially published in Italian was one of the first macroeconomic formalizations of cyclical dynamics in the 1930s.

There is also a comparison with the Keynesian equations of the economic cycle contained in Keynes’ Treatise on Money of (1930). Keynes is an author whom Amoroso held in high regard especially for his monetary theory. Amoroso dedicated the entire course of the lectures in Rome of 1932/33 to an examination of the Treatise. He discussed the main Keynesian thesis that the economic cycle was dependent on the rise of extra-profits and he found it weak on the analytical level. He concluded that introducing this hypothesis made the system mathematically unstable a result not acceptable for mathematician Amoroso.

However after the 1929 crisis the financial dimension of the economy was too relevant to ignore. In the definitive model of (1935) published on Econometrica the model is integrated with a third equation relating to the banking system. The final result is interesting. The solution to the complicated system of equations is split into two parts. The first represents the cyclical dynamics of the economy and the second incorporates a long-term trend. This connection between cycle and trend was a typical feature of the Italian reflection on the economic cycle (Fanno 1947). The Amoroso model was one of the first dynamic macro representations of the economic cycle. The analytical aspect is also interesting. The interaction between the sectors is represented by a second-order differential equation very similar to that of the pendulum. Analytical mechanics once again provides the model for solving an economic problem (Vinci 1934; Bordin 1935).

In the second half of the 1930s the dynamic analysis of Paretians made a qualitative step forward with the application of functional calculus. The transition from statics to dynamics required a new advancement in mathematical tools represented in this case by substituting infinitesimal calculus with functional calculus. The investigations in this field were initiated by two American mathematicians: Griffith Evans (1922, 1924, 1930) and his pupil Charles Ross (1925, 1927, 1930). The contributions of the two American economists were well known and appreciated in Italy (Amoroso 1929a, b). A first application in the Italian context was an article by Amoroso on the dynamics of the firm of (1933). However the road in Italy was opened quite curiously by a young economic historian Giulio la Volpe with his monograph Studi sulla teoria dell'equilibrio economico dinamico generale of (1936). La Volpe tried to obtain a dynamic theory of the general economic equilibrium of the microeconomic type through the extended application of functional calculation. The problem was to determine the maximum of an intertemporal utility function under a dynamic budget constraint. Moreover a further novelty of La Volpe’s model is that the optimal trajectory also depended on the expectations of the economic agents considered an exogenous element. In order to construct a theory of economic dynamics it was no longer a question of researching the analogies with rational mechanics but rather of broadening the analytical horizon to take account of the economic subject’s forecasts. This was also the new approach emerging internationally (Tinbergen 1934) both analytically and statistically. Consistent with these new developments in dynamic analysis La Volpe also sought to develop a dynamic pattern based on expectations anticipating by a few years the concept of temporary equilibrium developed by Hicks in Value and Capital (1939). An important analytical achievement of La Volpe was the identification of the intertemporal condition of transversality that allowed effective determination of the equations of dynamic equilibrium. This equation will then become a fundamental element in the life cycle theory of consumption formulated in the 1960s.

The path of dynamic equilibrium opened by La Volpe was further developed by Amoroso who toward the end of the 1930s developed his model of dynamic general equilibrium (Amoroso 1938, 19,411,942). Amoroso followed his methodological framework of close analogies with national mechanics. In the construction of his model he assumed that both consumption and production were characterized by the presence of a force of inertia that could accelerate or delay economic activities. The existence of this additional constraint just as in the case of the movement of bodies studied by rational mechanics made it possible to determine the equations of the economic movement of the individual firm and the individual consumer in a fairly easy way. His theory of general dynamic equilibrium was then developed in a complete manner in the dense lectures carried out at the Institute of High Mathematics in Rome and contained in the text Meccanica Economica of (1942).

This approach to the dynamics of general economic equilibrium remained an isolated case in the economic literature of the time and did not have the success it deserved. It was revived only in the 1960s and therefore almost twenty years later with the theory of optimal growth. This decline and the subsequent resumption of the idea of dynamic equilibrium can be explained by a set of factors. The first is that the two Italian economists’ models were formally very complex. This contrasted with the fact that the mathematical preparation of economists was quite limited and dynamic equations seemed a mathematical game rather than a useful tool for understanding economic reality. Second this approach to economic dynamics was hastily assimilated into corporatism and economic dirigisme. After the end of the war this perspective was quickly shelved. The third probably the most important one is that since it was not based on the aggregate saving–investment scheme at least in the case of Amoroso it did not lead to a theory of growth that instead became the dynamic reference model of the post-World War II period. Dynamic theory developed as a macroeconomic theory and the general economic equilibrium approach was completely shelved in the context of international research which when it moved in this direction rediscovered Ramsey (1928) instead of the pioneering contributions of Italian economists.

4 Toward welfare economics

De Finetti joined in the economic debate during a period of crisis and rethinking of economic theory. The 1929 crisis raised questions about the capacity of the economic system to self-regulate through the system of prices. On the contrary laissez-faire policies seemed to aggravate the already difficult economic conditions. In his articles published in (1935) and (1936) de Finetti introduced the general perspective of his criticism of the Paretian system with respect to welfare theory which he subsequently developed in mathematical terms in two articles published in 1937. These two articles exemplify how mathematical research in economics is driven by the need to conduct an in-depth analysis of an economic problem. The main target of de Finetti’s criticism was in current terms the first theorem of welfare economics—the idea that a perfectly competitive equilibrium is Pareto optimum. The criterion of Pareto’s optimality is also today the central tool in the field of welfare economics. It states that social welfare is maximized by a given allocation of resources when it is impossible to reassign inputs or outputs to make any individual strictly better off without making at least one other individual worse off. Pareto’s purpose in introducing this idea was to rid economic theory of utilitarian philosophy and thus transform economics into a positive science like rational mechanics (McLure 2007).

In his writings from this period de Finetti tried to demonstrate that this proposition did not constitute a truly mathematical theorem but instead was a sophism (sofisma) that is a reasoning that appeared to be correct but was in fact misleading. It is worth noting that de Finetti used the same term used thirty years earlier by the young mathematician Gaetano Scorza (1902) in his debate with Pareto on the same issue (Gattei and Guerraggio 1991). Scorza considered the identification of the result of free market competition and the optimal allocation for society a sophism which de Finetti considered tragic because it could potentially produce very harmful consequences for society. De Finetti observed the following:

A more serious and hateful sophism is added to the error at the basis of research into the optimum through the adoption of certain means of approach: the optimistic sophism of liberalism the superstition of self-regulating anarchy according to which the most simple and secure way to reach the maximum welfare for all consists in allowing each individual to try to realize the maximum egoistic profit. (de Finetti 1935b, p. 440).

To better understand de Finetti’s position it is necessary to consider that his criticism of Pareto was not aimed at questioning the validity of Pareto’s theory as such but instead to free it from what he considered to be a merely contingent and ideological interpretation.

De Finetti’s criticism of Pareto’s general equilibrium theory led him to build the theory of simultaneous maxima his most important contribution to pure economic theory. The starting point was a problem that remained open in Pareto’s system. In fact from a given initial allocation the Pareto optimum position could not be determined uniquely but there could be more than one optimum position as in the well-known case of the Edgeworth box. In his 1937 article de Finetti dealt with a very general context which he defined as simultaneous maximization. This type of optimization differs from traditional constraint optimization which is the common case considered by economists because the problem is to obtain the maximum values of many functions simultaneously in a particular way. These maximum results are simultaneous in the sense that it is not possible to increase the value of one function without decreasing that of another. The analogy with the case of the Pareto optimum—whose simultaneous maximization constitutes a generalization—is evident. De Finetti showed that to verify that a specific arbitrary vector represents a position of simultaneous maxima for n functions two conditions must be met: the determinant of the Jacobian matrix must be zero and its cofactors must all be of the same sign. These are necessary conditions that become sufficient as soon as some restrictions are added such as concavity.

What are the implications of simultaneous maxima for welfare theory? They are undoubtedly deep because the optimization process is the basis of the economic agent’s behavior. The fundamental consequence for de Finetti was that in this way it would be possible to prove that there are many optimal points which are eventually infinite and therefore the identification between the Pareto optimum and free competition would be purely arbitrary in the sense that anarchic market forces reach only one of the many positions that have this property (de Finetti 1943, p. 38). Hence de Finetti (1937a, b) believed that he had revealed the logical weakness of economic liberalism:

This fundamental theorem under the hypotheses that are necessary to establish it rigorously therefore has an indisputable validity but it is not legitimate to interpret it in a more concrete sense with the consequence that liberalism leads to an optimum; and even if that were true it would be necessary to observe that there is not a single point of optimum but infinites.

We demonstrate that normally in the case of n individuals the points of optimum are \(\infty^{n - 1}\). Suppose set the ophelimities \(\Theta_{1} = a_{1}\) \(\Theta_{2} = a_{2}\) …. \(\Theta_{n - 1} = a_{n - 1}\) of n − 1 individuals; on the variety so defined the \(\Theta_{n} = a_{n}\) will admit a maximum value and therefore at least a point of optimum. Of such points there are at least \(\infty^{n - 1}\); they actually constitute a variety at n − 1 dimensions. (p. 552).

In this way de Finetti demonstrated a result that was well known among Italian economists especially among Paretian economists. Considering the problem from a different point of view economists could state that the social optimum in Pareto’s sense is always a relative optimum depending on the initial distribution of resources (Bordin 1948).

After exploring the theoretical terms of the question it was necessary to develop some criteria for defining the optimal allocation of resources for society. De Finetti made a substantial contribution to this issue in the 1943 essay “La crisi dei principi e l’economia matematica” (“The crisis of principles and mathematical economics”) in which he criticized once again the traditional view of Paretian theory and introduced a possible solution to the problem of the non-uniqueness of the social optimum. To determine the optimal allocation of resources for society de Finetti proposed introducing a collective preference function (funziona di preferibilità collettiva) which was not a new idea in the Italian context. Bordin in an article about Cournot (1939) already suggested introducing a function of collective welfare meant as an expression of the government’s preferences. The same idea was formulated more rigorously by de Finetti (1943) as follows:

The criteria of preference \(\phi_{h}\) that can be set in a certain ethical social system will consist in making the situation of each single individual as preferable as possible and the circumstances directly involving the society as preferable as possible.

These collective needs can give place to more functions of preference \(\phi_{1}\)\(\phi_{2}\)…\(\phi_{n}\) each one concerning for example the preference towards the interests in the army the navy the air forces etc. but can be summarized—as we will assume—in a single function of collective preference \(\phi_{o} = F(\phi_{1} ,\phi_{2} ,...,\phi_{n} )\) that summarizes the preference of a body (government) that coordinates and manages the various collective needs. (p. 43).

In the article de Finetti did not develop this issue further. He just observed that this social preference function could be based on individual preferences or could be expressed by the choices made by political powers. De Finetti openly opted for the second possibility. In his view only the strong intervention of the state in the economy could correct some of the economic evils of his time. He made no mention in the article on the social welfare function advanced by Bergson a few years before.

From the point of view of the evolution of economic ideas it is interesting to note that this approach is not as far from the Paretian perspective as de Finetti assumed. Yet it is not the Pareto economist but the Pareto sociologist who is worth considering. In the last part of his Treatise of Sociology (1916) Pareto discussed the problem of a comparative evaluation of the various states in which a society can find itself. Pareto introduced the fundamental distinction between a maximum utility for society and a maximum utility of society. In the former case society is considered from an atomistic viewpoint—as a set of molecules using a physical metaphor—and the optimal position is reached when an increase in the utility of an individual cannot be obtained without causing a detriment to others. The case of the maximum utility of society is different. In this hypothesis society is treated as a single entity and Pareto’s solution was to entrust governments with the task of establishing the characteristics of a hypothetical function of social utility (Tarascio 1969). Pareto stated the following:

Suppose a community whose conditions are such that the only choice is between having a very rich community with great inequality of revenues among its components or a very poor one with an income basically the same for all. The research of the maximum utility of the community can draw closer to the first state while the research of the maximum utility of the community can draw closer to the second. This is because the effect depends on the coefficients used to make homogenous the heterogeneous utilities of the various social classes. The admirer of the “superman” will assign a coefficient of approximately zero to the utility of the lower classes and get a point of equilibrium very close to a state where larger inequalities prevail. The lover of equality will assign a high coefficient to the utility of the lower classes and get a point of equilibrium very close to the equalitarian conditions. (Pareto 1963 pp. 208–209).

More specifically Pareto had in mind a linear social welfare function in which the weights assigned to each welfare element represent assessments of the government. De Finetti totally agreed with this normative way of approaching the general problem of evaluating the best position for society. In the Treatise of Sociology we can find a germinal idea of the social welfare function (Bergson 1983).

5 Conclusions

After World War II the research program of Paretians in the field of mathematical economics became to use an expression of the Hungarian philosopher of science Imre Laktos a degenerative research program—that is a program that has lost its vitality and ability to offer new perspectives. Several factors contribute to this. The first element was certainly geographical and generational. The young scholars most of whom left the Faculty of Law went to train in the U.K. or in the United States. There was an extraordinary generational break. The Italian economists even those of European stature were mostly matched with the smoky corporatism and even the best part of their theories were completely shelved with the curious effect that some theoretical results already achieved by Italian economists returned to Italy through other channels such as in the case of functional calculus.

However why were Paretians no longer able to achieve innovative results if not in some sporadic cases? Here the fundamental element to call into question is probably the weight of the Paretian legacy. If the fidelity to Pareto in the 1920s and 1930s allowed us to obtain remarkable results also in the formalization of economic reasoning after World War II we witnessed a sharp reversal and the Paretian theory became a factor of obstacle rather than progress. First Paretian economists did not agree to follow what was definitively the axiomatic turning point in microeconomic research. Here the result is paradoxical. Pure mathematicians like Emilio Zaccagnini openly contested the reduction of economics to a mathematical metaphysics of the Arrow–Debreu type (Zaccagnini 1950). Paretians tried to show the realistic aspects of their mathematical formulations but in this way they lost contact with international research. Second the Paretians did not follow the macroeconomic shift except Fossati who tried to develop the points of contact between Pareto and Keynes (Fossati 1955). For the Paretians the economic analysis could not do without the study of optimizing behavior and the Keynesian theory seemed very weak from this point of view. They therefore did not participate in the tumultuous process of formation and development of macroeconomics remaining firmly linked to the vision of general economic equilibrium.

The only Paretian economist who made significant contributions to the field of mathematical analysis after the war was Giuseppe Palomba. As a pupil of Amoroso he tried to develop the ideas of the master beyond Newtonian mechanics. To do this he ventured into a difficult path to highlight the analogies between the economic discourse and the physics of the twentieth century borrowing some analytical categories. He also tried to develop an axiomatization of the theory of general equilibrium which on the one hand was extremely complex and on the other hand not useful for understanding concrete economic phenomena. A prolific and brilliant author he sought his own personal path of research that took him away from the main currents of international research. With Palomba emerged all the limits of an attempt to build economic reasoning into analogies with physics. In some sense thanks to his remarkable analytical mastery he could have become the Italian Samuelson, the scholar who marked the development of economic science in the United States and beyond, and instead remained an accomplished author very original but also not very influential in Italy.