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Lesson 1 On the Notion of Neoclassical Equilibrium: Walras, Pareto and the Stationary Equilibrium Approach Ph.D. Program in Economics University of York February-March 2008 Franco Donzelli Topics in the History of Equilibrium Analysis

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Lesson 1 - Neoclassical Equilibrium2 Franco Donzelli Neoclassical equilibrium 1 The “neoclassical” or “marginalistic revolution” is conventionally traced back to three works published at the beginning of the 1870s: W. S. Jevons, The Theory of Political Economy, 1871 C. Menger, Grundsätze der Volkswirtschaftslehre, 1871 L. Walras, Eléments d’économie politique pure, As a matter of fact, these works are much less homogeneous than once one used to believe. (see Jaffé (1976), “Menger, Jevons and Walras De-homogeneized”, Economic Inquiry, 14, pp ) Yet one can try to single out some common traits: the assumption that each agent (initially, only consumers-traders) chooses her/his optimal plan of action by maximizing her/his objective function (initially, only utility functions) subject to constraints is common to all the three founders; the assumption that the optimally chosen plans must be compatible, giving rise to an equilibrium state, is common to Jevons and Walras (though not shared by Menger).

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Lesson 1 - Neoclassical Equilibrium3 Franco Donzelli Neoclassical equilibrium 2 Starting from this (almost) common basis, one can tentatively put forward a definition of an abstract concept of neoclassical equilibrium: Given an economy, characterized by a specified set of data, a neoclassical equilibrium is a state of the economy in which all the agents participating in the economy optimally choose their plans of action, given the constraints to which they are subjected, and all the optimally chosen plans are compatible, both with each other and with the external constraints to which the agents are subjected, so that they can all be carried out. With reference to a competitive market economy the definition can be further specialized: the agents are consumers and producers; commodity prices are the signals that the agents need to take into account in solving their optimization programs (since prices enter either the constraints to which they are subjected or the objective functions they try to maximize).

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Lesson 1 - Neoclassical Equilibrium4 Franco Donzelli Neoclassical equilibrium 3 One can tentatively define an abstract concept of neoclassical competitive equilibrium (not shared by Jevons, however, so that it is actually a concept of Walrasian equilibrium): Given a competitive economy, characterized by a specified set of data, a neoclassical competitive equilibrium is an array of prices, one for each commodity existing in the economy, and of plans of action, one for each consumer and/or producer participating in the economy, such that all the plans of action are both optimally chosen, given the prices and the constraints to which the agents are subjected, and compatible, both with each other and with the external constraints to which the agents are subjected, so that they can all be carried out. Yet, since (at least) the last decade of the Nineteenth century, two alternative notions of neoclassical competitive equilibrium have been used by different groups of neoclassical economists, coexisting thereafter side by side during the first four decades of the twentieth century (and beyond): the instantaneous competitive equilibrium notion; the stationary competitive equilibrium notion.

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Lesson 1 - Neoclassical Equilibrium5 Franco Donzelli Instantaneous equilibrium 1 The instantaneous equilibrium notion is the equilibrium notion eventually adopted by the economists of the Lausanne School, that is, by the last Walras, Pareto, and their (very few) immediate followers. Since the late 1920s, such notion started to spread out of the restricted circle of the Lausanne School economists, being rediscovered by Hayek (1928), (1933), (1937); Lindahl (1929), (1930); Hicks (1933), (1939). After the II World War: Arrow-Debreu (1954), Debreu (1959) Walras, in the fourth edition of the Eléments (1900, Lesson 35), and Pareto, both in the Cours d’économie politique ( , Vol. II, pp ) and in the Manuale di economia politica (1906, pp ), make it clear that the equilibrium concept they intend to adopt is an instantaneous equilibrium concept of the temporary type. According to such interpretation, any particular equilibrium of the model, corresponding to an admissible specification of the data, should be viewed as describing the equilibrium state instantaneously reached by the economy at that particular instant (resp., unit period) of its history at which the specified data are supposed to prevail.

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Lesson 1 - Neoclassical Equilibrium6 Franco Donzelli Instantaneous equilibrium 2 Under this interpretation, there is no reason why the data should be unchanging over time; on the contrary, one should in general expect the data, hence the equilibria, to change as time elapses, for both exogenous and endogenous reasons. Hence, under this interpretation, the evolution of the economy over time can (must) be described by means of a chronologically ordered continuum (resp., sequence) of instantaneous (temporary) equilibrium states, differing in general from one another. To refer to this sort of analysis, Walras used the expressions “variable or moving equilibrium” or “dynamic point of view”, while Pareto ( , Vol. II, pp ; 1906, pp ) spoke of the “method of successive equilibria”.

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Lesson 1 - Neoclassical Equilibrium7 Franco Donzelli Stationary equilibrium 1 Since the last decade of the nineteenth century, however, the leading economists belonging to all kinds of neoclassical schools, with the only exception of the Lausanne school, almost invariably started to suggest that their equilibrium models be interpreted as stationary equilibrium models. The “neoclassical stationary equilibrium approach” includes such theorists as: Marshall (1890), Wicksell (1898), (1901), (1906), J.B. Clark (1899), (1907), Schumpeter (1911), (1939), Cassel (1918) and (1925), Knight (1921), Frisch (1935), Pigou (1935). According to such interpretation, any particular equilibrium of the model, corresponding to an admissible specification of the data, should be viewed as a stationary state of the economy under theoretical investigation.

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Lesson 1 - Neoclassical Equilibrium8 Franco Donzelli Stationary equilibrium 2 As such, a stationary equilibrium should not be taken to refer to any specific instant (resp., period) in the history of the economy, but rather to the whole continuum (resp., sequence) of instants (resp., periods) over which the data are supposed to be unchanging. But then this interpretation of the equilibrium concept, unlike the instantaneous one, forces the theorist to assume the stationarity of the data characterizing the economy over time. The adoption, diffusion and eventual supremacy of the stationary viewpoint over a period of about forty years are due to a number of factors. The most important one has to do with the problem of the “empirical justification” of neoclassical equilibrium theory: the equilibrium apparatus ought to be “empirically justified” by showing that an adjustment mechanism is at work in the economy which is capable of driving it towards an equilibrium position.

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Lesson 1 - Neoclassical Equilibrium9 Franco Donzelli The evolution of Walras’s thought 1 The first to tackle this problem was Walras himself, the founder of the general equilibrium approach, to whom both the interpretations of the equilibrium notion – the “instantaneous” and the “stationary” one – can ultimately be traced. General Equilibrium Theory (GET) was founded by Walras in : “Principe d’une théorie mathématique de l’échange” (1874) “Equations de l’échange” (1876) “Equations de la production” (1876) “Equations del la capitalisation” (1877) GET was further developed by Walras over the various editions of the Eléments d’économie politique pure: 1 st edition 2 nd edition 1889 3 rd edition 1896 4 th edition 1900

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Lesson 1 - Neoclassical Equilibrium10 Franco Donzelli The evolution of Walras’s thought 2 In the four mémoires written in the period , as well as in the four editions of the Eléments d’économie politique pure published over the period , Walras developed four nested equilibrium models: the models of exchange, production, and capital formation have been there since the four mémoires and the 1 st edition of the Eléments ( ); the model of circulation and money was only introduced in the 4 th edition (1900). The models are nested in the sense that each encompasses the previous one (if any).

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Lesson 1 - Neoclassical Equilibrium11 Franco Donzelli Walras’s two solution concepts For each of the four models Walras defines two solution concepts: the “theoretical” or “mathematical” or even “scientific” solution, on the one hand, and the “empirical” or “practical” solution, on the other. The “theoretical” solution is the solution of a system of ordinary algebraic equations describing the “static” equilibrium conditions appropriate to the model concerned: e.g., in the exchange model, the “static” equations are the market- clearing equations. The “empirical” solution is the (stationary) solution of a system of functional equations describing the “dynamic” adjustment process appropriate to the model concerned, called by Walras the tâtonnement process: e.g., in the exchange model, the equations are (incompletely formalized) difference equations, embodying the well-known Walrasian price-adjustment rule, later also called the “auctioneer’s rule” or the “law of supply and demand”.

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Lesson 1 - Neoclassical Equilibrium12 Franco Donzelli Walras’s “empirical” solution According to Walras, the system of functional equations describing the tâtonnement process mimics (with some idealization, forced by analytical reasons) the adjustment process concretely realized by the competitive market mechanism: this explains why this solution is called “empirical”. Walras never fully formalized the dynamical system and consequently never solved it analytically: yet, this deficiency was immaterial for Walras, since – according to him – the solution is directly “found” by the market. The stationary state to which the dynamic adjustment process “certainly” (or at least “probably”) converges is assumed to coincide with the “theoretical” solution of the static system of equations. As a consequence of this assumed coincidence, the “theoretical” solution acquires an “empirical” content as well, though only indirectly.

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Lesson 1 - Neoclassical Equilibrium13 Franco Donzelli Walras’s “theoretical” solution 1 The static system of equations, unlike the dynamic one, was fully formalized by Walras (even if Walras’s formalization does not meet contemporary standards of rigor). Can the static system be analytically solved? Two partially conflicting answers were provided by Walras. In Walras’s first theoretical essay (“Principe…”, 1874), the answer is apparently positive: “A priori, this problem is evidently solvable, at least in principle, by means of the mathematical method, as it is actually solvable on the market, by means of the empirical method of the rise and fall in prices. So far, we have supposed the buyers and sellers to be personally present on the market concerned. Yet the presence of these traders is not necessary: should they pass their orders to some agents, the market will be made by the latter. […]

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Lesson 1 - Neoclassical Equilibrium14 Franco Donzelli Walras’s “theoretical” solution 2 [continued] “But, from a theoretical point of view, is the presence of agents more necessary than that of the traders themselves? By no means. These agents are the sheer executors of orders written on block-notes: if, instead of participating in an auction, they hand over their block- notes to a calculator, this calculator will determine the equilibrium price not certainly so quickly, but without fail more rigorously than what could be done by means of the mechanism of the rise and fall in prices. We are such calculator: our demand curves represent the orders of the traders. Should one give us all the necessary time, we must be able to mathematically determine our equilibrium prices.” (Walras,1993, p. 37; my translation, italics added)

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Lesson 1 - Neoclassical Equilibrium15 Franco Donzelli Walras’s “theoretical” solution 3 « A priori, ce problème est évidemment soluble, du moins en principe, par le procédé mathématique, comme il est soluble, en fait, sur le marché, par le procédé empirique de la hausse et de la baisse. Sur notre marché, nous avons supposé les acheteurs et les vendeurs en présence les uns des autres; mais la présence des ces échangeurs n’est pas nécessaire: qu’ils donnent leurs ordres à des agents, le marché ce tiendra entre ces derniers. […] Mais, théoriquement, la présence des agents est-elle plus nécessaire que celle des échangeurs eux-mêmes? Pas le moins du monde. Ces agents sont les exécuteurs purs et simples d’ordres inscrits sur des carnets: qu’au lieu de faire la criée, ils donnent ces carnets à un calculateur, et ce calculateur déterminera le prix d'équilibre non pas certes aussi rapidement, mais à coup sûr plus rigoureusement que cela ne pourrait se faire par le mécanisme de la hausse et de la baisse. Nous sommes ce calculateur ; nos courbes de demande représentent les ordres des échangeurs ; on nous donne tout le temps nécessaire ; nous devons pouvoir déterminer mathématiquement nos prix d’équilibre.» (Walras, 1874, p. 37; italics added)

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Lesson 1 - Neoclassical Equilibrium16 Franco Donzelli Walras’s “theoretical” solution 4 But in the I st edition of the Eléments ( ), published just a few months later, Walras is much more cautious: “We are now in a position to see clearly what the mechanism of market competition is. It is the practical solution, reached through a rise or fall in prices, of the same problem of exchange to which we have just given a theoretical and mathematical solution; but it must be understood that we do not have the slightest idea of substituting one solution for the other. The rapidity and reliability of the practical solution leave no room for improvement. It is a matter of daily experience that even in big markets where there are neither brokers nor auctioneers, the current equilibrium price is determined within a few minutes, and considerable quantities of merchandise are exchanged at that price within half or three quarters of an hour. In fact the theoretical solution would be absolutely impracticable in almost every case.”

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Lesson 1 - Neoclassical Equilibrium17 Franco Donzelli Walras’s “theoretical” solution 5 [continued] “On the other hand, it is not a valid objection against our method to speak of the difficulty of deriving [empirical] curves or equations of exchange. Whether there is any advantage to be found in constructing all or part of either the demand or the offer curve of a given commodity in certain cases, and whether it is possible or impossible to do so, are questions on which we reserve judgement entirely. For the moment, we are examining the problem of exchange in general, and the [abstract] conception, pure and simple, of curves of exchange is sufficient and at the same time indispensable.” (Walras, 1954, p. 106; Jaffé’s translation) Walras is ambiguous about: the very possibility of numerically or analytically specifying the “data” (parameters, functions, relations) of the static system; the solvability of the static system, which however appears to be “absolutely impracticable in almost every case”

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Lesson 1 - Neoclassical Equilibrium18 Franco Donzelli Walras’s “theoretical” solution 6 « On voit clairement à présent ce qu'est le mécanisme de la concurrence sur le marché; c'est la solution pratique, et par hausse et baisse des prix, du problème de l'échange dont nous avons fourni la solution théorique et mathématique. On doit comprendre d'ailleurs que notre intention n'est aucunement de substituer une solution à l'autre. La solution pratique est d'une rapidité et d'une sûreté qui ne laissent rien à désirer. On peut voir, sur de grands marchés fonctionnant même sans courtiers ni crieurs, le prix courant d'équilibre se déterminer en quelques minutes, et des quantités considérables de marchandise s'échanger à ce prix en deux ou trois quarts d'heure. Au contraire, la solution théorique serait, dans presque tous les cas, absolument impraticable. » (Walras, 1988, p. 93; italics added)

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Lesson 1 - Neoclassical Equilibrium19 Franco Donzelli Walras’s view of the equilibration process 1 Once again, the solvability of the static system is immaterial for Walras, as long as the “theoretical” solution coincides with the “empirical” solution “found” by the market. This requirement severely constrains the nature of the tâtonnement process: for such process must not change the “data” of the economy. Yet this requirement conflicts with Walras’s original (pre-analytic) vision of the equilibration process, which is very “realistic”. The tâtonnement process (in both exchange and production) is initially viewed by Walras as an observable disequilibrium process in “real” time, leading to a stationary equilibrium state. In the first edition of the Eléments Walras apparently does not rule out the occurrence of observable disequilibrium trades and certainly allows for the occurrence of observable disequilibrium production.

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Lesson 1 - Neoclassical Equilibrium20 Franco Donzelli Walras’s view of the equilibration process 2 Right at the beginning of Lesson 5 of the Eléments, one finds a long illustrative passage, where the functioning of the market for “3 per cent French Rentes” is described in detail (an almost identical illustration, referring however to the “corn market”, had already been introduced in Walras’s first mémoire, “Principe…”): “Let us take, for example, trading in 3 per cent French Rentes on the Paris Stock Exchange and confine our attention to these operations alone. The three per cent, as they are called, are quoted at 60 francs. [...] We shall apply the term effective offer to any offer made, in this way, of a definite amount of a commodity at a definite price. [...] We shall apply the term effective demand to any such demand for a definite amount of a commodity at a definite price. We have now to make three suppositions according as the demand is equal to, greater than, or less than the offer.

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Lesson 1 - Neoclassical Equilibrium21 Franco Donzelli Walras’s view of the equilibration process 3 [continued] “First supposition. The quantity demanded at 60 francs is equal to the quantity offered at this same price. [...] The rate of 60 francs is maintained. The market is in a stationary state or equilibrium. Second supposition. The brokers with orders to buy can no longer find brokers with orders to sell. [...] Brokers [...] make bids at 60 francs 05 centimes. They raise the market price. Third supposition. Brokers with orders to sell can no longer find brokers with orders to buy. [...] Brokers [...] make offers at 59 francs 95 centimes. They lower the price.” (Walras, 1954, pp ) As can be seen, Walras is apparently willing to allow the traders (“brokers”) not only to change the price, contrary to the standard competitive assumption of price-taking behavior, but actually to trade out of equilibrium.

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Lesson 1 - Neoclassical Equilibrium22 Franco Donzelli Walras’s view of the equilibration process 4 But: no theory of observable disequilibrium behavior is available; observable disequilibrium behavior would generally entail some changes in the “data”; a stationary equilibrium model cannot be easily applied to a changing (“progressive”) economy. In 1883 the mathematician Bertrand criticizes Walras’s assumptions concerning the trading process, maintaining that observable disequilibrium transactions would make the equilibrium indeterminate. As a matter of fact, there is more than that: not only would the equilibrium be indeterminate, but the whole equilibration process would become theoretically intractable, for Walras can provide no theory of observable disequilibrium trade. Yet Bertrand does not raise this further point, nor is he bothered by the limitations of the stationary equilibrium assumption.

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Lesson 1 - Neoclassical Equilibrium23 Franco Donzelli Walras’s view of the equilibration process 5 Walras reacts to Bertrand’s critique first in a footnote in an obscure paper published in 1885, and then in the second edition of the Eléments (1889). In both these writings Walras makes an explicit “no-trade-out-of- equilibrium-assumption”, turning the tâtonnement process in the “exchange model” into a purely virtual process in “logical time”, which takes just a single instant of “real” time to carry its effects through. Specifically, in the second edition of the Eléments (1889), Walras changes the securities example, by adding the expressions: "Exchange takes place" in the case of market equilibrium; "Theoretically, trading should come to a halt" and "Trading stops" in the case of excess demand and excess supply, respectively. In view of this, the equilibrium of the exchange model should be conceived as instantaneously arrived at – what Walras explicitly recognizes in his 1885 paper, when he says that the overall equilibrium exchanges will take place “à un instant donné”.

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Lesson 1 - Neoclassical Equilibrium24 Franco Donzelli Walras’s view of the equilibration process 6 Yet Walras does not change the expression “stationary state or equilibrium”, appearing in the first edition of the Eléments. Moreover he sticks to the original assumption that observable disequilibrium production can occur in the economy. In order to confront a number of obvious analytical difficulties raised by the latter assumption, in the “production model” he assumes the data of the economy (quantities of available services) to remain unchanged over time: thereby the “production model” becomes a stationary equilibrium model of a pure-flow economy. Hence in the second edition of the Eléments there coexist two alternative interpretations of the tâtonnement process: a virtual one in “logical time”, in the “exchange model”, and an observable one in “real time”, in the “production model”.

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Lesson 1 - Neoclassical Equilibrium25 Franco Donzelli Walras’s view of the equilibration process 7 The interpretation of the equilibrium concept is ambiguous: sometimes it appears to be instantaneous (as it happens in Walras 1885), while on other occasions it appears to be stationary (occasionally in Walras 1889). Walras is unable to solve the dilemma in the third edition of the Eléments (1896). Only in the fourth edition of the Eléments (1900), by making the so-called “hypothèse des bons”, Walras eventually generalizes to all models and activities the interpretation of the tâtonnement already adopted with reference to the exchange model only. According to this interpretation, the tâtonnement becomes a purely virtual process, taking place in a sort of “logical time”, to be distinguished from the “real time” over which the economy evolves, and capable of bringing about its full effects without consuming any amount of “real time”. At the same time, Walras drops any surviving reference to a stationary equilibrium concept, adopting instead an instantaneous equilibrium notion (of the temporary type) as his only tool of analysis.

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Lesson 1 - Neoclassical Equilibrium26 Franco Donzelli Walras’s view of the equilibration process 8 In the fourth edition of the Eléments (1900) Walras writes as follows: « Au moyen de l'hypothèse des bons, on peut distinguer nettement, surtout si l'on les suppose successives, les trois phases suivantes: 1° La phase des tâtonnements préliminaires en vue de l'établissement de l'équilibre en principe; 2° La phase statique de l'établissement effectif ab ovo de l'équilibre relatif à la livraison des services producteurs et des produits pendant la période de temps considérée, aux conditions convenues, sans changements dans les données du problème; 3° Une phase dynamique de trouble continuel de l'équilibre par des changements dans ces données et de rétablissement continuel de l'équilibre ainsi troublé. En conséquence de ces définitions, il doit être bien entendu que les capitaux neufs, fixes ou circulants, qui seront livrés pendant la seconde phase [...], ne fonctionneront que dans la troisième phase, constituant ainsi un premier changement dans les données du problème. » (Walras, [1900] 1988, pp. 447, 449, 4-5)

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Lesson 1 - Neoclassical Equilibrium27 Franco Donzelli Walras’s view of the equilibration process 9 Hence Walras’s revised assumptions concerning the tâtonnement process: exchange model: “no-trade-out-of-equilibrium assumption” (1885, 1889); models of production, capital formation, circulation and money: “hypothèse des bons” (1900). Walras’s final conception: the tâtonnement process is eventually viewed as a virtual, unobservable disequilibrium process in “logical” time, leading to an instantaneous equilibrium state (of the “temporary” type) The paradoxical outcome of Walras’s theoretical revisions is the following: the explanatory power and empirical content of Walras’s final version of his general equilibrium theory crucially rest on the convergence of the tâtonnement process; but the process itself can only be described by means of a highly unrealistic, vaguely defined model, whose very stability, never formally discussed by Walras, is nothing more than a tenet of faith.

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Lesson 1 - Neoclassical Equilibrium28 Franco Donzelli Walras on the epistemological status of GET 1 The change in Walras’s perspective is essentially due to his attempt of overcoming the theoretical difficulties related to the assumption of an observable disequilibrium approach in “real time” and the associated stationary equilibrium notion. Yet, even if the reasons for Walras’s change of mind are essentially theoretical, he can also allege some motives of an empirical character. The purely virtual interpretation of the tâtonnement is an empirically acceptable approximation of the real-world equilibration process since, according to Walras, the latter process is very effective and extraordinarily quick (see quotations pp above). By the same token, Walras is led to justify the theoretical use of an instantaneous equilibrium notion (of the temporary type) as an empirically acceptable approximation.

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Lesson 1 - Neoclassical Equilibrium29 Franco Donzelli Walras on the epistemological status of GET 2 What is then the relation between theory and reality according to Walras? Walras’s answer is given in Lesson 35 “The continuous market” (Walras, 1954, 380): “Finally, in order to come still more closely to reality, we must drop the hypothesis of an annual market period and adopt in its place the hypothesis of a continuous market. Thus we pass from the static to the dynamic state. […] Such is the continuous market, which is perpetually tending towards equilibrium without ever attaining it, because the market has no other way of approaching equilibrium except by groping, and, before the goal is reached, it has to renew its efforts and start over again, all the basic data of the problem […] having changed in the meantime.” Yet the description of such continuous disequilibrium process by means of a sequence of temporary equilibria is justified by Walras as a reasonable approximation, also in view of the assumed rapidity of the equilibration process (“the current equilibrium price is determined within a few minutes”).

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Lesson 1 - Neoclassical Equilibrium30 Franco Donzelli Pareto on statics and dynamics 1 Vilfredo Pareto, Walras’s successor on the economics chair of the University of Lausanne, took over the general equilibrium approach from Walras and further developed it chiefly in the following writings: Cours d’économie politique ( ) Manuale di economia politica (1906) Manuel d’économie politique (1909) Did Pareto tackle the analytical and epistemological questions left open by Walras? Was he able to solve them? To what extent? In the Cours ( , Vol. II, 23-24) Pareto sided with Walras in the latter’s controversy with Edgeworth over the theory of the tâtonnement, but his defense of Walras’s theoretical construct was purely formal. Even if Pareto’s knowledge of dynamical systems theory and rational mechanics was much better than Walras’s, he never tried to formalize the latter’s theory of the tâtonnement and he eventually dropped it in the Manuale.

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Lesson 1 - Neoclassical Equilibrium31 Franco Donzelli Pareto on statics and dynamics 2 The reason for this is that Pareto, unlike Walras, was aware, at least to a certain extent, that the relationship between statics, or equilibrium theory, and dynamics is not so simple in economics as it is in rational mechanics. While in mechanics the so-called d’Alembert principle allows one to switch from the static equations defining the equilibrium position of a material system to the dynamic equations defining its motion, no such easy passage is possible in economics. As Pareto explained in the Cours … ( , pp ; my translation): “Society may then be regarded as being in a state of equilibrium, and a stable one. To tell the truth, it is a question here of dynamic, rather than static, equilibrium, as the entire society is driven by a general movement that slowly changes it. Such movement is generally designated by the term evolution.”

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Lesson 1 - Neoclassical Equilibrium32 Franco Donzelli Pareto on statics and dynamics 3 [continued] “In mechanics the d’Alembert principle allows us to study the dynamic state of a system completely. In political economy, for the time being, we can only catch a glimpse of a similar principle. In the social science, even this vague notion is wanting. In both these latter sciences, we are compelled to substitute the analysis of a sequence of static equilibria for the analysis of dynamic equilibrium. To give a somewhat rough but expressive idea of the matter, suppose that a man in a sleigh slides down a slope at the same time as another man steps down the same slope, stopping at every step. The two men start simultaneously from the top, constantly proceed together, and arrive simultaneously at the foot of the hill. Hence, by and large, their motion is approximately the same. But the motion of the man in the sleigh is a continuous one: its study involves a problem of dynamics. The motion of the man stepping down on foot represents a sequence of successive positions of equilibrium: he proceeds from one to the next discontinuously. It is precisely such a sequence of equilibrium positions that we can study in political economy.”

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Lesson 1 - Neoclassical Equilibrium33 Franco Donzelli Pareto on statics and dynamics 4 « Pour donner une idée assez grossière, mais expressive de la chose, supposons qu’un homme qui est dans un traîneau, glisse sur une pente. Un autre homme descend à pied la même pente, en s’arrêtant à chaque pas. Les deux hommes partent en même temps du sommet, voyagent constamment en compagnie, et arrivent en même temps en bas de la pente. En gros, leur mouvement est donc à peu près le même. Mais le mouvement de l’homme qui est dans le traîneau, est un mouvement continu, son étude constitue un problème de dynamique. Le mouvement de l’homme descendant à pied, représente une suite de positions successives d’équilibre. Il passe de l’une à l’autre, d’une manière discontinue. C’est précisément une suite semblable de positions d’équilibre que nous pouvons étudier en économie politique. » (Pareto, 1987, p. 10, par. 587).

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Lesson 1 - Neoclassical Equilibrium34 Franco Donzelli Pareto on the meaning of equilibrium 1 Pareto ended up with adopting a methodological position which is similar to that embraced by the last Walras (that of Lesson 35 “The continuous market”), the only difference being that, for Pareto, only statics (i.e., equilibrium theory) matters, while the theory of the tâtonnement is immaterial. Equilibrium theory, and specifically the “method of successive equilibria”, provides a first approximation to an ever changing economy. As Pareto wrote in the Cours … ( , Vol. I, p. 58; my translation): “In reality, the equilibrium is never reached since, in so far as one tries to approach it, it continuously changes as the technical and economic conditions of production change. The real state is then a state of persistent oscillations around a central equilibrium point, which itself drifts away.”

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Lesson 1 - Neoclassical Equilibrium35 Franco Donzelli Pareto on the meaning of equilibrium 2 As to the insolvability of the static system of equations, Pareto was apparently much sharper than Walras. In the Manuale … (1906) Pareto wrote (1927, pp ): “It may be mentioned here that this determination has by no means the purpose to arrive at a numerical calculation of prices. Let us make the most favourable assumption for such a calculation, let us assume that we have triumphed over all the difficulties of finding the data of the problem and that we know the ophélimités of all the different commodities for each individual, and all the conditions of production of all commodities, etc. This is already an absurd hypothesis to make. Yet it is not sufficient to make the solution of the problem possible. We have seen that in the case of 100 persons and 700 commodities there will be 70,699 conditions (actually a great number of circumstances that we have so far neglected will further increase that number); we will therefore have to solve a system of 70,699 equations.

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Lesson 1 - Neoclassical Equilibrium36 Franco Donzelli Pareto on the meaning of equilibrium 3 [continued] This exceeds practically the power of algebraic analysis, and this is even more true if one contemplates the fabulous number of equations which one obtains for a population of forty million people and several thousand commodities. In this case the rôles would be changed: it would not be mathematics which would assist political economy, but political economy would assist mathematics. In other words, if one really could know all these equations, the only means to solve them which is available to human powers is to observe the practical solution given by the market.” In the above passage Pareto underlines two sorts of practical difficulties: difficulties related to the collection of a huge number of individual data; computational difficulties due to the size of the equational system.

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Lesson 1 - Neoclassical Equilibrium37 Franco Donzelli Pareto on the epistemological status of GET 1 What is then, for Pareto, the epistemological status of GET? In the Cours … ( , Vol. II, 8-9, my translation) Pareto distinguished three sorts of scientific explanations and corresponding types of theories: “Let us consider, in general, certain phenomena A, B, C. Our knowledge concerning their mutual dependence may pass through three successive stages. α) We may only know that such dependence exists […]. β) We may have an idea of the links existing among A, B, C, […]. In other words, we may know the directions of the changes arising in B, C, D … as a consequence of a certain change in A. γ) Finally, we can not only know the directions of such changes, but also exactly calculate their magnitudes. When we reach such point, our knowledge is complete and perfect. As regards the motions of [bodies belonging to] the solar system, astronomy has reached stage (γ) […]. In general, if the numerical data are enough, mechanics allows us to reach the degree of knowledge (γ) as far as a material system is concerned. If numerical data are lacking, it leads us at least to degree (β).

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Lesson 1 - Neoclassical Equilibrium38 Franco Donzelli Pareto on the epistemological status of GET 2 [continued ] Before the new theories were discovered, political economy had stopped at degree (α) […] The new theories have raised political economy to degree (β). The complete system of equilibrium equations not only helps to emphasize the mutual dependence of economic phenomena, but is also of use to enlighten us about the directions of the changes in certain elements when we make other elements change. Still more: such equations show us the way that will allow us to climb to level (γ), as soon as statistics is able to provide the data necessary to this end.” Then, after all, Pareto did not rule out the possibility that, in the more or less distant future, General Equilibrium Theory can be used to obtain some sort of “complete and perfect knowledge” of the economic phenomena (though not deterministic, but only statistical in nature). Yet, neither in the Manuale … (1906), nor in the Manuel … (1909), such sort of statistical approach to GET was ever concretely employed or even exemplified.

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Lesson 1 - Neoclassical Equilibrium39 Franco Donzelli Summary of Walras’s and Pareto’s stances on GET 1 The epistemological status of GET is highly ambiguous from beginning to end. For Walras GET is an empirical theory, but its empirical content is made to depend on a theoretical construct, the tâtonnement, whose empirical content is itself controversial. Starting from a highly realistic interpretation of the tâtonnement, Walras progressively realizes the theoretical difficulties connected with such interpretation, slowly turning the tâtonnement into a less and less realistic construct. In the end, by means of the no-trade-out-of-equilibrium-assumption (1885, 1889) in the exchange model and the “hypothèse des bons” in the models of production, capital formation, and circulation and money (1900), the tâtonnement becomes a purely virtual, durationless process in “logical” time, instantaneously leading to an instantaneous equilibrium at every instant of “real” time.

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Lesson 1 - Neoclassical Equilibrium40 Franco Donzelli Summary of Walras’s and Pareto’s stances on GET 2 Pareto drops the theory of the tâtonnement altogether, focusing on the static notion of equilibrium exclusively. Pareto takes over Walras’s notion of an instantaneous (mobile) equilibrium, explicitly maintaining that the study of successive positions of static equilibrium over time is the only method that economics, unlike mechanics, can make use of. For Pareto it is “absurd” to deem that the system of equations defining a static equilibrium state can be numerically solved, chiefly owing to practical difficulties of data collection and computation. Yet such system, though analytically unsolvable, still retains a sort of intermediate explanatory power, as a first approximation to a highly complex world. For Pareto ( ), if sufficient statistical data were collected, the use of GET to produce quantitative predictions of a statistical kind could not be ruled out in the future. Yet, in (1906) and (1909) Pareto did not produce any significant instance of such a use of GET.

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Lesson 1 - Neoclassical Equilibrium41 Franco Donzelli The stationary equilibrium approach 1 Walras’s and Pareto’s ultimate empirical justification for using an instantaneous equilibrium concept rested on their confidence in the speed and effectiveness of the real-world equilibration processes. Yet this confidence was not shared by most neoclassical economists of the first generations: on the contrary, they were convinced that the adjustment processes at work in real-world markets take a lot of time to carry their effects through. However a time-consuming process can hopefully approach a well- defined equilibrium position only if the data of the economy remain unchanged over time. This means that the use of the equilibrium apparatus can apparently be “empirically justified” only if the economy under investigation is assumed to be characterized by an unchanging set of data.

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Lesson 1 - Neoclassical Equilibrium42 Franco Donzelli The stationary equilibrium approach 2 In conclusion, the chief reasons behind the swift diffusion and almost general adoption of the stationary interpretation of the equilibrium concept around the turn of the Nineteenth century and in the following years were: the concern with the empirical relevance of equilibrium analysis; the belief (or the suspicion) that any real-world adjustment process cannot but be slow and time-consuming. These, on the other hand, are also the main reasons why a stationary interpretation of the equilibrium concept had survived so long in Walras’s own writings, up until the fourth edition of the Eléments (1900), occasionally sneaking in Pareto’s writings as well. Another reason, at least as far as Walras and Pareto are concerned, was their desire to imitate the physical sciences, and rational mechanics specifically: for in those sciences there is no distinction between “real” and “logical” time and any equilibrium necessarily is a stationary state of some sort (in “real” time).

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Lesson 1 - Neoclassical Equilibrium43 Franco Donzelli The stationary equilibrium approach 3 Yet, the stationary equilibrium approach, if consistently employed, has a very limited scope. The assumption of stationarity of the data, which it is obliged to make, forces it to rule out all sorts of activities which might cause an endogenous change in the data. In a general equilibrium model, i.e., a model analyzing a number of interconnected markets, this means to rule out all intertemporal activities, such as savings, investment, credit, dealings in durables, dealings in money or financial assets, etc. The only kind of economy which can be consistently examined within this approach is a pure-flow economy, i.e., an economy where the only kinds of commodities produced, consumed and traded are services (by definition non-durables) and non-durable consumers’ goods.

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Lesson 1 - Neoclassical Equilibrium44 Franco Donzelli The stationary equilibrium approach 4 In such an economy the agents have no way to connect with one another, by means of their economic activities, the time periods over which the economy evolves: for this economy is an “isolated-period” economy (Hicks). This is the essence, e.g., of the so-called “Walras-Cassel model”, the model of a pure flow-economy that Cassel ((1918), (1925)) derived from reinterpreting Walras’s original “production model”. Another example is Wicksell’s “model of a-capitalistic production” ((1901), (1906)) or Schumpeter’s “circular flow model” ((1911), (1939)). A different way to preserve the assumption of stationarity of the data is Marshall’s adoption of the coeteris paribus clause, with its related restriction of the analysis to the partial equilibrium framework of an isolated market (some special supplementary assumptions on the agents’ characteristics, such as the “constant marginal utility of money” assumption, need also to be made).

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Lesson 1 - Neoclassical Equilibrium45 Franco Donzelli The stationary equilibrium approach 5 When the strict boundaries of the approach are not respected, as when “capital” in some sense is incorporated into the stationary equilibrium models, inconsistencies necessarily ensue. This happens, in particular, with Wicksell’s “model of capitalistic production” ((1901), (1906)), J.B. Clark’s “model of a static economy” ((1899), (1907)), Schumpeter’s “model of capitalistic development” ((1911), (1939)), Cassel’s “model of a uniformly progressing economy” ((1918), (1925)). The controversies on the theory of capital, raging in the inter-War period (1920s and 1930s), and again, much later, in the 1960s and 1970s, arise precisely from these attempts to preserve a stationary equilibrium interpretation of the “neoclassical” equilibrium concept in models where such interpretation is logically inconsistent: for, due to the existence of “capital”, the time periods over which the economy evolves are not isolated from one another and the economy is not a pure-flow economy.

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