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Lesson 2 Equilibrium and tâtonnement in Walrass Eléments Ph.D. Program in Economics University of York February-March 2008 Franco Donzelli Topics in the.

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1 Lesson 2 Equilibrium and tâtonnement in Walrass Eléments Ph.D. Program in Economics University of York February-March 2008 Franco Donzelli Topics in the History of Equilibrium Analysis

2 Lesson 2 - Equilibrium in Walras2 Franco Donzelli The problem 1 In the literature there exist two alternative interpretations of the equilibrium concept as employed by Walras: The temporary equilibrium interpretation V. Pareto ( ) M. Morishima (1977), (1980) W.E. Diewert (1977) F. Donzelli (1986), (1990), (1993), (1997) Von Witteloostuijn and Maks (1988), (1890)

3 Lesson 2 - Equilibrium in Walras3 Franco Donzelli The problem 2 The stationary equilibrium interpretation K. Wicksell (1893), (1901) J.A. Schumpeter (1911) G. Cassel (1918), (1932) J. Hicks (1934) M. Magill and M. Quinzii (1998, p.132): «Walras conceived his general equilibrium model as a genuine intertemporal model, that is, time and capital were to play an essential role. In the end, analytical difficulties forced him to confine his attention to a special equilibrium, namely a steady state in which all prices remain unchanged and the sole price linking adjoining periods is the rate of interest.»

4 Lesson 2 - Equilibrium in Walras4 Franco Donzelli The conjecture 1. There exists some ambiguity or inconsistency in Walrass original formulation of GET explaining the contradictory alternative interpretations of the equilibrium concept employed in the Eléments. 2. Such ambiguity or inconsistency is related, at least in part, to Walrass theory of the tâtonnement

5 Lesson 2 - Equilibrium in Walras5 Franco Donzelli The evolution of the theory of the tâtonnement The theory of the tâtonnement undergoes frequent and radical revisions over the various editions of the Eléments. Editions of the Eléments: I ed II ed III ed IV ed V ed. (posthumous) 1926 Walras developed a special model of tâtonnement for each of the four nested models put forward in the Eléments: exchange, production, capital formation, circulation and money (since the IV edition)

6 Lesson 2 - Equilibrium in Walras6 Franco Donzelli Walrass twofold concept of solution Walras put forward two distinct, but related, concepts of solution for each of the four nested models: solution théorique or mathématique or scientifique, obtained by solving un système déquations, en nombre rigoureusement égal à celui des inconnues (system of ordinary, static equations); solution pratique or empirique, directly provided by le mécanisme de la concurrence sur le marché, which is however theoretically described by the relevant tâtonnement model (system of functional, dynamic equations). The two solutions must be identical, that is, the two methods of solution must provide the same result.

7 Lesson 2 - Equilibrium in Walras7 Franco Donzelli Walrass two-step procedure 1 To prove that the two solutions are identical Walras suggests the following two-step procedure: 1. the tâtonnement process does not change the data (parameters, functions, relations) of the economy; 2. assuming uniqueness of the theoretical solution, the unique solution is a globally asymptotically stable equilibrium, with respect to the specified dynamical system describing the tâtonnement process. How satisfactory are Walrass answers to the questions raised by his own approach?

8 Lesson 2 - Equilibrium in Walras8 Franco Donzelli Walrass two-step procedure 2 As to the first step, Walras is aware of the need to avoid any change in the data during the tâtonnement, but he is uncertain on how to face this requirement: we shall examine his various attempts in the following. As to the second step, Walras repeatedly tries to prove that the theoretical solution is a stable equilibrium for the dynamical system describing the tâtonnement process, but his treatment remains highly unsatisfactory: Wicksteed (1884) points out the defective character of Walrass proofs; Walras reacts by replacing everywhere the expression certain convergence of the tâtonnement process with the less demanding expression probable convergence.

9 Lesson 2 - Equilibrium in Walras9 Franco Donzelli The exchange model (two commodities) 1 Two commodities, indexed by l = 1, 2 I competitive traders (consumers), indexed by i = 1, …, I, with 2 < I < Each trader i is characterized by (X i, u i, ω i c ), where X i = {x i } = 2 + is is consumption set, u i : 2 + is is utility function, and ω i c 2 + \ {0} is is endowment of consumers goods The price system is p = (p 12,1) + x {1}, where commodity 2 is taken as the numéraire Trader is optimization problem is: max x i X i u i (x i ) s.t. p 12 x 1i + x 2i = p 12 ω 1i c + ω 2i c

10 Lesson 2 - Equilibrium in Walras10 Franco Donzelli The exchange model (two commodities) 2 By solving is optimization problem, one gets: z i = x i - ω i c : ++ 2, is excess demand function By aggregating over the traders one gets: z = x - ω c : ++ 2, the aggregate excess demand function The market equilibrium condition for commodity 1, which implicitly defines that for commodity 2 as well, is: z 1 (p 12 *) = x 1 (p 12 *) – ω 1 c = 0 The equilibrium concept in the exchange model is an intrinsically statical concept: consistency, hence executability, of plans of action. Walrass rule of price adjustment, governing the tâtonnement process in exchange, stipulates that p 12 increases (resp., decreases) iff z 1 (p 12 ) > 0 (resp., < 0 ). Hence, p 12 changes, in the appropriate direction, iff the market for commodity 1 is out of equilibrium.

11 Lesson 2 - Equilibrium in Walras11 Franco Donzelli Disequilibrium in the exchange model 1 Walrass theory allows the theorist to explain (predict) the traders disequilibrium plans of action, rather than disequilibrium actions. Moreover, disequilibrium behavior, Hickss false trading (1939), if allowed to take place, would affect the traders endowments, hence the data of the economy, even if Walrass theory of the tâtonnement would be unable to predict the ensuing changes. Walras is aware of the need to keep the data unchanged during the tâtonnement process, but apparently he does not exactly perceive the connection between disequilibrium behavior and change in the data. The market for the rente française 3% at the Paris Stock Exchange in the I edition of the Eléments (or the corn market in the 1874 mémoire).

12 Lesson 2 - Equilibrium in Walras12 Franco Donzelli Disequilibrium in the exchange model 2 « Prenons, par exemple, […] les opérations sur la rente française 3%, à la bourse de Paris. Le 3% est, comme on dit, à 60 F. […] Nous avons maintenant trois hypothèses à faire suivant que la demande est égale, supérieure ou inférieure à loffre. 1 re Hypothèse. On demande à 60 F une quantité égale à celle qui est offerte à ce même prix. […] Le cours de 60 F se maintient; il y a état stationnaire ou équilibre du marché. 2 e Hypothèse. Les agents acheteurs ne trouvent plus leur contrepartie. […] Ils vont à lenchère. 3 e Hypothèse. Les agents vendeurs ne trouvent plus leur contrepartie. […] Ils vont au rabais. » (Walras, 1874, p. 71, 5 e Leçon, § 43, p. 71)

13 Lesson 2 - Equilibrium in Walras13 Franco Donzelli Disequilibrium in the exchange model 3 The above passage is indeed ambiguous, for it does not rule out disequilibrium trades. Starting from Bertrand (1883), it has been repeatedly interpreted (with some recantation) as allowing such trades: Hicks (1934), (1939); Goodwin (1951), (1953); Patinkin (1956); Newman (1965); Jaffé (1967), (1980), (1981); Walker (1972), (1987), (1996) Walras reacts to Bertrands attack by explicitly introducing the no-trade-out-of-equilibrium assumption That assumption, cursorily hinted at in a paper on the working of the Stock Exchange (1880), is explicitly introduced first in an obscure paper published in 1885 and then in the II edition of the Eléments (1889), by adding a few explanatory words in the above passage.

14 Lesson 2 - Equilibrium in Walras14 Franco Donzelli Disequilibrium in the exchange model 4 « Prenons, par exemple, […] les opérations sur la rente française 3%, à la bourse de Paris. Le 3% est, comme on dit, à 60 F. […] Nous avons maintenant trois hypothèses à faire suivant que la demande est égale, supérieure ou inférieure à loffre. 1 re Hypothèse. On demande à 60 F une quantité égale à celle qui est offerte à ce même prix. […] Léchange a lieu. Le cours de 60 F se maintient; il y a état stationnaire ou équilibre du marché. 2 e Hypothèse. Les agents acheteurs ne trouvent plus leur contrepartie. […] Théoriquement, léchange doit être suspendu. Ils vont à lenchère. 3 e Hypothèse. Les agents vendeurs ne trouvent plus leur contrepartie. […] Suspension de léchange. Ils vont au rabais. » (Walras, 1874, p. 71, 5 e Leçon, § 43, p. 71)

15 Lesson 2 - Equilibrium in Walras15 Franco Donzelli Tâtonnement and equilibrium in the exchange model As a consequence of the no-trade-out-of-equilibrium assumption, since the II edition in the exchange model: no observable disequilibrium behavior no actions, only plans of action; tâtonnement a purely virtual process; distinction between the logical time of the tâtonnement and the real time of the economy; it takes only one instant of real time for the equilibrium to be reached instantaneous equilibrium, as explained by Walras (1885): « Le prix courant théorique est essentiellement un prix unique résultant, à un moment donné, dun échange général. »

16 Lesson 2 - Equilibrium in Walras16 Franco Donzelli The production model 1 I competitive consumers-owners, indexed by i = 1, …, I, with 2 < I < L consumers goods, indexed by l = 1, …, L M services, indexed by m = 1, …, M Each consumer-owner i is characterized by (X i x Y i, u i, ω i s ), where X i x Y i = {(x i, y i )} = L+M + is is generalized consumption set, u i : L+M + is is utility function, and ω s i M + \ {0} is is endowment of services The price system is (p,w) = (1, p 2, …, p L, w 1, …, w M ) {1} x L-1 + x M +, where p is the price vector of the L consumers goods, w is the price vector of the M services, and commodity 1 is taken as the numéraire. Consumer-owner is optimization problem is: max x i X i u i (x i, y i ) s.t. px i + wy i = wω s i or px i = w(ω s i – y i )

17 Lesson 2 - Equilibrium in Walras17 Franco Donzelli The production model 2 By solving is optimization problem, one gets: (x i, y i ) : {1} x L-1+M + L+M +, is demand function, where x i : {1} x L-1+M + L + is is demand function for consumers goods, and y i : {1} x L-1+M + M + is is demand function for services By aggregating over the consumers-owners one gets: (x, y) : {1} x L-1+M + L+M +, the aggregate demand function, where x : {1} x L-1+M + L + is the aggr. demand function for consumers goods, y : {1} x L-1+M + M + is the aggr. demand function for services, and ω s is the aggregate endowment of services

18 Lesson 2 - Equilibrium in Walras18 Franco Donzelli The production model 3 Walras assumes a single-output technology with fixed coefficients. A = [a ml ] = [a 1, …, a l, …, a L ] is the M x L matrix of the technical coefficients Given (p, w), the unit profits are: π(p, w) = p – wA For each consumers good l = 1, …, L, the profit-maximizing netput vector (q l *, a l q l *) is not a well-defined function of (p, w), for: if π l > 0, then a profit-maximizing choice if π l = 0, then q l * [0, ) if π l < 0, then q l * = 0

19 Lesson 2 - Equilibrium in Walras19 Franco Donzelli The production model 4 Chiefly, but not only, due to the above difficulties with the outcome of profit-maximization in production, Walras does not assume the producers to be profit-maximizers Walrass rule of quantity adjustment, governing the tâtonnement process in production, stipulates that the quantity produced of consumers good l, q l, increases (resp., decreases) iff π l > 0 (resp., < 0). An equilibrium of production is defined as a price system (p*, w*) and an aggregate demand vector (x*, y*) s.t. p* = w*A Ax* + y* = ω s The equilibrium of production concept is defined as an intrinsically dynamical equilibrium concept, based on the rule of quantity adjustment: it is a price-quantity state s.t. profits are nil, hence the quantities produced of consumers goods do not change.

20 Lesson 2 - Equilibrium in Walras20 Franco Donzelli Disequilibrium in the production model 1 In the production model of the Eléments there co-exist, side by side, two completely different: rules of agents behavior: the maximizing behavior of consumers vs. the non-maximizing behavior of producers; rules of adjustment: Walrass rule of price adjustment vs. Walrass rule of quantity adjustment tâtonnement processes: the tâtonnement in exchange and the tâtonnement in production equilibrium concepts: the statical equilibrium concept, implicit in the embedded exchange model, and the dynamical equilibrium concept, concerning production proper Such differences are somewhat disguised in equilibrium: since π* = π(p*, x*) = 0, all equilibrium behavior (both consumers and producers) can be interpreted as optimizing equilibrium behavior.

21 Lesson 2 - Equilibrium in Walras21 Franco Donzelli Disequilibrium in the production model 2 But out of equilibrium the differences, and the difficulties, cannot be concealed. At time 0, the starting point of the overall tâtonnement process, let w 0 M + and q 0 L + be the randomly announced vectors of input prices and quantities to be produced, respectively. To supply q 0 the producers need to employ the quantities Aq 0 M + of services. q 0 is sold at market-clearing selling prices (prix de vente) p ̂ 0 s.t. x(p ̂ 0,w 0 ) = q 0. But problem: Who is going to sell to the producers the quantities of inputs they need to supply q 0 ?

22 Lesson 2 - Equilibrium in Walras22 Franco Donzelli Disequilibrium in the production model 3 Certainly not the consumers-owners, since at prices (p ̂ 0,w 0 ) they want to sell the input quantities ω s - y(p ̂ 0,w 0 ), and typically ω s - y(p ̂ 0,w 0 ) Aq 0. If the voluntary exchange assumption holds, so that unintended actions cannot be enforced, the tâtonnement process would stop before starting. Walras perceived the existence of a serious problem, much more clearly than all his epigones, critics, and later commentators. All these people, irrespective of whether they sympathize with Walrass approach or take a critical stance towards it, do not realize the impossibility of Walrass tâtonnement in production, unless special devices are contrived to take care of the problem (see, particularly, Walker (1972), (1987), (1996)).

23 Lesson 2 - Equilibrium in Walras23 Franco Donzelli Tâtonnement and equilibrium in the production model 1 Walrass attempted solutions varied dramatically over the editions of the Eléments. In the mémoire on the production model (1876) and in the I edition of the Eléments (1877) Walras assumed the existence of a foreign market, accommodating all the needs of the domestic producers. This solution is highly unsatisfactory, for two reasons: the foreign market assumption is an artificial contrivance, lacking any inner justification and patently contrasting with the logic of General Equilibrium Theory; the transactions assumed to take place between the domestic producers, on one side, and the foreign market, on the other, are nothing but swindles, for they do not even respect the quid pro quo clause, characteristic of all voluntary exchange.

24 Lesson 2 - Equilibrium in Walras24 Franco Donzelli Tâtonnement and equilibrium in the production model 2 In the II edition of the Eléments (1889), having already explicitly adopted a purely virtual interpretation of the tâtonnement process in exchange, Walras reconsidered the interpretation of the tâtonnement process in production as well. First he underlined the basic methodological similarities which should be preserved in analyzing the two adjustment processes: «Il s'agit d'arriver à l'équilibre de la production de la même façon que nous sommes arrivés à l'équilibre de l'échange, c'est-à-dire en supposant les données du problème invariables pendant tout le temps que dureront nos tâtonnements, sauf à supposer ensuite ces données variables en vue détudier les effets de leurs variations.» (Walras, 1988, p. 308, 2-3)

25 Lesson 2 - Equilibrium in Walras25 Franco Donzelli Tâtonnement and equilibrium in the production model 3 But then he explains why the approach cannot be the same, due to an important difference between the two processes: «Mais le tâtonnement en matière de production rencontre une complication qui nexistait pas en matière d'échange. Dans l'échange, il ny a pas de modification des marchandises. Un prix étant crié, et la demande et loffre effective correspondant à ce prix nétant pas égales, on crie un autre prix auquel correspondent une autre demande et une autre offre effectives. Dans la production il y a transformation des services producteurs en produits. Certain prix des services étant criés, et certain quantités de produits étant fabriquées, si ces prix et ces quantités ne sont pas prix et quantités déquilibre, il faudra non seulement crier dautres prix, mais fabriquer dautres quantités de produits.» (Walras, 1988, p. 308, 2-5) While in the exchange adjustment process no observable disequilibrium behavior is allowed to take place (no actions are carried out, only plans of action are formulated), in the production adjustment process observable disequilibrium does take place (actions are actually carried out).

26 Lesson 2 - Equilibrium in Walras26 Franco Donzelli Tâtonnement and equilibrium in the production model 4 In view of the postulated difference, the new solution put forward by Walras for the tâtonnement process in production must differ from that already adopted for the tâtonnement process in exchange: «Acceptant cette nécéssité, nous devons supposer que, pour chaque reprise du tâtonnement, nos entrepreneurs trouveront, dans le pays, des proprietaires fonciers, travailleurs et capitalistes possédant les mêmes quantités de services et ayant le mêmes besoins des services et des produits.» (Walras, 1988, p. 308, 2-3) Hence, in the II edition of the Eléments, the tâtonnement in production remains, as before, an actual process in real time, giving rise to observable disequilibrium; but here, unlike in the first edition, Walras suppresses the foreign market, making instead the explicit assumption that the data of the economy are stationary in real time.

27 Lesson 2 - Equilibrium in Walras27 Franco Donzelli Tâtonnement and equilibrium in the production model 5 Given the assumption that the data are stationary in the same real time over which the economy evolves (stationarity assumption), the tâtonnement process in production, as modeled in the II edition, supports a stationary equilibrium notion. Since in the production model there exist only non-durable consumers goods and services, the economy, as described by Walrass production model, is a pure-flow economy. Hence, since in a pure-flow economy there are no links between the successive dates of the economy (Hickss isolated-period economy), the agents behavior cannot affect the data: hence, the stationarity assumption is legitimate in this model. The notion of stationary equilibrium intrinsic to Walrass production model in the II edition of the Eléments is inherited by the so-called Walras-Cassel model and, consequently, by the secondary literature on GET.

28 Lesson 2 - Equilibrium in Walras28 Franco Donzelli Tâtonnement and equilibrium in the production model 6 The solution put forward in the II edition, concerning the tâtonnement process in production, leaves a number of problems unsolved: the problem of the swindles becomes even worse; disequilibrium behavior in production is observable, but remains unpredictable; yet it is also unessential, for it cannot affect the data overtime; hence the unpredictability of disequilibrium in not that significant.

29 Lesson 2 - Equilibrium in Walras29 Franco Donzelli The capital formation model But a further problem arises in the II edition, concerning the capital formation model. Since produced capital goods are the outcome of a production process, like consumers goods in the production model, Walras wants to extend the assumptions of the tâtonnement process in production to the production of capital goods as well. But in the capital formation model the analysis is no longer restricted to a pure-flow, isolated-period economy, for capital goods are durable stocks and provide a natural endogenous link between the various dates the assumption of stationarity of the data in the same real time over which the economy evolves become illegitimate in this context the associated stationary equilibrium notion can no longer be legitimately used.

30 Lesson 2 - Equilibrium in Walras30 Franco Donzelli Problems left unsolved by the II (or III) edition of the Eléments 1 Epistemological problems Problems connected with the dualistic character of the theory, i.e., with the co-existence of two alternative interpretations of the tâtonnement, to which two alternative interpretations of the equilibrium concept are associated: the tâtonnement in exchange, which is virtual, in logical time, with unobservable disequilibrium, supporting an instantaneous equilibrium notion; the tâtonnement in production, which is actual, in real time, with observable disequilibrium, supporting a stationary equilibrium notion.

31 Lesson 2 - Equilibrium in Walras31 Franco Donzelli Problems left unsolved by the II (or III) edition of the Eléments 2 Analytical problems Problems connected with the existence of inconsistencies in various parts of the theory, namely: the problem of the swindles in the tâtonnement process in production; the assumption of stationarity of the data in the capital formation model. All problems are connected to the pseudo-realistic interpretation of the tâtonnement in production as a process in real time.

32 Lesson 2 - Equilibrium in Walras32 Franco Donzelli Walrass solution in the IV edition of the Eléments 1 The solution put forward by Walras in the IV edition of the Eléments consists in suppressing any realistic pretence in the analysis of the tâtonnement process in production. This is realized by means of the hypothèse des bons: Pour réaliser un tâtonnement rigoureux en matière de production comme en matière déchange, tout en tenant compte de cette circonstance, il ny a quà supposer les entrepreneurs représentant par des bons des quantités successives de produits déterminées dabord au hasard puis en augmentation ou diminution suivant quil y aura excédent du prix de vente sur le prix de revient ou réciproquement, jusquà égalité de ces deux prix; et les propriétaires fonciers, travailleurs et capitalistes représentant de même par des bons des quantités successives de services à des prix criés dabord au hasard puis en hausse ou baisse suivant quil y aura excédent de la demande sur loffre ou réciproquement, jusquà égalité de lune et de lautre.» (Walras, 1988, p. 309, 4-5)

33 Lesson 2 - Equilibrium in Walras33 Franco Donzelli Walrass solution in the IV edition of the Eléments 2 The time structure of the analysis is specified as follows: « Au moyen de lhypothèse des bons, on peut distinguer nettement, surtout si lon les suppose successives, les trois phases suivantes: 1° La phase des tâtonnements préliminaires en vue de létablissement de léquilibre en principe; 2° La phase statique de létablissement effectif ab ovo de léquilibre 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 léquilibre par des changements dans ces données et de rétablissement continuel de léquilibre 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, 1988, pp. 447, 449, 4-5)

34 Lesson 2 - Equilibrium in Walras34 Franco Donzelli Walrass solution in the IV edition of the Eléments 2 The entire tâtonnement process, not only in exchange, but in production as well, becomes purely virtual, in logical time, with unobservable disequilibrium. The entire tâtonnement process, in fact, takes place in the phase des tâtonnements préliminaires, which logically precedes both the phase statique, where all observable behavior is supposed to take place, and the phase dynamique, where all change in the data is supposed to occur. Moreover, since the operation of newly produced capital goods can only start in the time period following that of their production, capital formation is sterilized as a source of change in the data, without resorting to any stationarity assumption, which would be self- contradictory in this context.

35 Lesson 2 - Equilibrium in Walras35 Franco Donzelli Walrass solution in the IV edition of the Eléments 4 By means of the hypothèse des bons Walras can solve some of the problems left unsolved by his treatment of the tâtonnement process and the associated equilibrium concept in the II edition. The dualistic character of the theory in the II edition is eliminated: actual tâtonnement and stationary equilibrium are suppressed; only virtual tâtonnement and instantaneous (temporary) equilibrium survive. The analytical inconsistencies of the II edition disappear: no swindles; no stationarity assumption in the capital formation model. Of course there remains the epistemological difficulty of justifying the coexistence of two distinct time concepts (real and logical time) as well as the existence of an instantaneous process.


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