Lesson 2 Equilibrium and t

1
Lesson 2Equilibrium and tâtonnement in Walrass
Eléments
Franco Donzelli Topics in the History of
Equilibrium Analysis

• Ph.D. Program in Economics
• University of York
• February-March 2008

2
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 (1896-97)
• M. Morishima (1977), (1980)
• W.E. Diewert (1977)
• F. Donzelli (1986), (1990), (1993), (1997)
• Von Witteloostuijn and Maks (1988), (1890)

3
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
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
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. 1874-1877
• II ed. 1889
• III ed. 1896
• IV ed. 1900
• 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
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
Walrass two-step procedure 1

• To prove that the two solutions are identical
  Walras suggests the following two-step procedure
• the tâtonnement process does not change the
  data (parameters, functions, relations) of the
  economy
• 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
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
The exchange model (two commodities) 1

• Two commodities, indexed by l 1, 2
• I competitive traders (consumers), indexed by i
  1, , I, with 2 lt I lt 8
• Each trader i is characterized by (Xi, ui, ?ic),
  where Xi xi R2 is is consumption set, ui
  R2 ? R is is utility function, and ?ic ? R2
  \ 0 is is endowment of consumers goods
• The price system is p (p12,1) ? R x 1 ,
  where commodity 2 is taken as the numéraire
• Trader is optimization problem is
• maxxi ? Xi ui (xi)
• s.t.
• p12x1i x2i p12 ?1ic ?2ic

10
The exchange model (two commodities) 2

• By solving is optimization problem, one gets
• zi xi - ?ic R ? R2 , is excess demand
  function
• By aggregating over the traders one gets
• z x - ?c R ? R2 , the aggregate excess
  demand function
• The market equilibrium condition for commodity 1,
  which implicitly defines that for commodity 2 as
  well, is
• z1(p12) x1(p12) ?1c 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
  p12 increases (resp., decreases) iff z1(p12) gt 0
  (resp., lt 0 ). Hence, p12 changes, in the
  appropriate direction, iff the market for
  commodity 1 is out of equilibrium.

11
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).

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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.
• 1re 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.
• 2e Hypothèse. Les agents acheteurs ne trouvent
  plus leur contrepartie. Ils vont à lenchère.
• 3e Hypothèse. Les agents vendeurs ne trouvent
  plus leur contrepartie. Ils vont au rabais. 
• (Walras, 1874, p. 71, 5e Leçon, 43, p. 71)

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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
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.
• 1re 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.
• 2e Hypothèse. Les agents acheteurs ne trouvent
  plus leur contrepartie. Théoriquement,
  léchange doit être suspendu. Ils vont à
  lenchère.
• 3e Hypothèse. Les agents vendeurs ne trouvent
  plus leur contrepartie. Suspension de
  léchange. Ils vont au rabais. 
• (Walras, 1874, p. 71, 5e Leçon, 43, p. 71)

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Tâtonnement and equilibrium in the exchange model

• As a consequence of the no-trade-out-of-equilibriu
  m 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. 

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The production model 1

• I competitive consumers-owners, indexed by i 1,
  , I, with 2 lt I lt 8
• L consumers goods, indexed by l 1, , L
• M services, indexed by m 1, , M
• Each consumer-owner i is characterized by (Xi x
  Yi, ui, ?is), where
• Xi x Yi (xi, yi) RLM is is generalized
  consumption set,
• ui RLM ? R is is utility function, and ?si
  ? RM \ 0 is is endowment of services
• The price system is (p,w) (1, p2, , pL, w1, ,
  wM) ? 1 x RL-1 x RM, 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
• maxxi ? Xi ui (xi, yi) s.t.
• pxi wyi w?si or pxi w(?si yi)

17
The production model 2

• By solving is optimization problem, one gets
• (xi, yi) 1 x RL-1M ? RLM , is demand
  function, where
• xi 1 x RL-1M ? RL is is demand function
  for consumers goods, and
• yi 1 x RL-1M ? RM is is demand function
  for services
• By aggregating over the consumers-owners one
  gets
• (x, y) 1 x RL-1M ? RLM , the aggregate
  demand function, where
• x 1 x RL-1M ? RL is the aggr. demand
  function for consumers goods,
• y 1 x RL-1M ? RM is the aggr. demand
  function for services,
• and
• ?s is the aggregate endowment of services

18
The production model 3

• Walras assumes a single-output technology with
  fixed coefficients.
• A aml a1, , al, , aL
• is the M x L matrix of the technical
  coefficients
• Given (p, w), the unit profits are
• p(p, w) p wA
• For each consumers good l 1, , L, the
  profit-maximizing netput vector (ql, alql) is
  not a well-defined function of (p, w), for
• if pl gt 0, then ? a profit-maximizing choice
• if pl 0, then ql ? 0, 8)
• if pl lt 0, then ql 0

19
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,
  ql, increases (resp., decreases) iff pl gt 0
  (resp., lt 0).
• An equilibrium of production is defined as a
  price system (p, w) and an aggregate demand
  vector (x, y) s.t.
• p wA
• Ax y ?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
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
• p p(p, x) 0, all equilibrium behavior
  (both consumers and producers) can be
  interpreted as optimizing equilibrium behavior.

21
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 w0 ? RM and q0 ? RL
  be the randomly announced vectors of input prices
  and quantities to be produced, respectively.
• To supply q0 the producers need to employ the
  quantities Aq0 ? RM of services.
• q0 is sold at market-clearing selling prices
  (prix de vente) p0 s.t. x(p0,w0) q0.
• But problem Who is going to sell to the
  producers the quantities of inputs they need to
  supply q0?

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Disequilibrium in the production model 3

• Certainly not the consumers-owners, since at
  prices (p0,w0) they want to sell the input
  quantities ?s - y(p0,w0), and typically
• ?s - y(p0,w0) ? Aq0 .
• 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
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
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
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
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
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
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
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
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
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
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
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
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
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.