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Franco Donzelli
Topics in the History of Equilibrium Analysis
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
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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, 1874-1877
 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. 511-524)
 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 Equilibrium 3 Franco Donzelli

 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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 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. school.edhole.com

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 (1896-97, Vol. II, pp.
11-13) and in the Manuale di economia politica (1906, pp. 104-105),
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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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 (1896-97, Vol. II, pp. 11-13; 1906, pp. 104-105) spoke of the
“method of successive equilibria”.
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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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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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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 1874-77:
 “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:
 1st edition 1874-77
 2nd edition 1889
 3rd edition 1896
 4th edition 1900
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The evolution of Walras’s thought 2
 In the four mémoires written in the period 1874-77, as well as in the
four editions of the Eléments d’économie politique pure published
over the period 1874-1900, Walras developed four nested
equilibrium models:
 the models of exchange, production, and capital formation have
been there since the four mémoires and the 1st edition of the
Eléments (1874-77);
 the model of circulation and money was only introduced in the 4th
edition (1900).
 The models are nested in the sense that each encompasses the
previous one (if any).
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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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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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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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[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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« 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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 But in the Ist edition of the Eléments (1874-77), 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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[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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 « 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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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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 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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[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. 84-85)
 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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 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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 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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 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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 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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 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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 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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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. 18-20 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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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”). school.edhole.com

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 (1896-97)
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 (1896-97, 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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 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 … (1896-97, pp. 11-12; 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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[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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« 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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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 … (1896-97, 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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 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. 233-234):
“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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 [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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Pareto on the epistemological status of GET 1
 What is then, for Pareto, the epistemological status of GET?
 In the Cours … (1896-97, 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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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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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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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 (1896-97), 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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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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 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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 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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 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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 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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