lesson 2 equilibrium and tâtonnement in walrass eléments ph.d. program in economics university of...

Lesson 2

Equilibrium and tâtonnement in Walras’s Eléments

Ph.D. Program in Economics

University of York

February-March 2008

Franco Donzelli

Topics in the History of Equilibrium Analysis

Lesson 2 - Equilibrium in Walras 2Franco 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 (1896-97)

M. Morishima (1977), (1980)

W.E. Diewert (1977)

F. Donzelli (1986), (1990), (1993), (1997)

Von Witteloostuijn and Maks (1988), (1890)


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.»


The conjecture

1. There exists some ambiguity or inconsistency in Walras’s 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 Walras’s theory of the tâtonnement


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)


Walras’s 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 d’équations, 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.


Walras’s 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 Walras’s answers to the questions raised by his own approach?


Walras’s 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 Walras’s “proofs”;

Walras reacts by replacing everywhere the expression “certain convergence” of the tâtonnement process with the less demanding expression “probable convergence”.


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 (Xi, ui, ωic), where Xi = {xi}

= ℝ2+ is i’s consumption set, ui : ℝ2

+ → ℝ is i’s utility function, and ωi

c ∈ ℝ2+ \ {0} is i’s endowment of consumers’ goods

The price system is p = (p12,1) ∈ ℝ+ x {1} , where commodity 2 is taken as the numéraire

Trader i’s optimization problem is:

maxxi ∈ Xi ui (xi)

s.t.

p12x1i + x2i = p12 ω1ic + ω2i

c


The exchange model (two commodities) 2 By solving i’s optimization problem, one gets:

zi = xi - ωic : ℝ++ → ℝ2 , i’s excess demand function

By aggregating over the traders one gets:

z = x - ωc : 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*) – ω1c = 0

The equilibrium concept in the exchange model is an intrinsically statical concept: consistency, hence executability, of plans of action.

Walras’s rule of price adjustment, governing the tâtonnement process in exchange, stipulates that p12 increases (resp., decreases) iff z1(p12) > 0 (resp., < 0 ). Hence, p12 changes, in the appropriate direction, iff the market for commodity 1 is out of equilibrium.


Disequilibrium in the exchange model 1

Walras’s theory allows the theorist to explain (predict) the traders’ disequilibrium plans of action, rather than disequilibrium actions.

Moreover, disequilibrium behavior, Hicks’s “false trading” (1939), if allowed to take place, would affect the traders’ endowments, hence the data of the economy, even if Walras’s 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).



« 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 l’offre.

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 l’enchère.

3e Hypothèse. Les agents vendeurs ne trouvent plus leur contrepartie. […] Ils vont au rabais. »

(Walras, 1874, p. 71, 5e Leçon, § 43, p. 71)



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 Bertrand’s 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.



« 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 l’offre.

1re Hypothèse. On demande à 60 F une quantité égale à celle qui est offerte à ce même prix. […] L’échange 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, l’échange doit être suspendu. Ils vont à l’enchère.

3e Hypothèse. Les agents vendeurs ne trouvent plus leur contrepartie. […] Suspension de l’échange. Ils vont au rabais. »

(Walras, 1874, p. 71, 5e Leçon, § 43, p. 71)


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, d’un échange général. »


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 (Xi x Yi, ui, ωis), where

Xi x Yi = {(xi, yi)} = ℝL+M+ is i’s generalized consumption set,

ui : ℝL+M+ → ℝ is i’s utility function, and ωs

i ∈ ℝM+ \ {0} is i’s

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 i’s optimization problem is:

maxxi ∈ Xi ui (xi, yi) s.t.

pxi + wyi = wωsi or pxi = w(ωs

i – yi)



By solving i’s optimization problem, one gets:

(xi, yi) : {1} x ℝL-1+M+ → ℝL+M

+ , i’s demand function, where

xi : {1} x ℝL-1+M+ → ℝL

+ is i’s demand function for consumers’ goods, and

yi : {1} x ℝL-1+M+ → ℝM

+ is i’s demand function for services

By aggregating over the consumers-owners one gets:

(x, y) : {1} x ℝL-1+M+ → ℝL+M

+ , the aggregate demand function, where

x : {1} x ℝL-1+M+ → ℝL

+ is the aggr. demand function for consumers’ goods,

y : {1} x ℝL-1+M+ → ℝM

+ is the aggr. demand function for services,

and

ωs is the aggregate endowment of services



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:

π(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 πl > 0, then ∄ a profit-maximizing choice

if πl = 0, then ql* ∈ [0, ∞)

if πl < 0, then ql* = 0



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 →

Walras’s rule of quantity adjustment, governing the tâtonnement process in production, stipulates that the quantity produced of consumers’ good l, ql, increases (resp., decreases) iff πl > 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* = ωs

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.


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: “Walras’s rule of price adjustment” vs. “Walras’s 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: sinceπ* = π(p*, x*) = 0, all equilibrium behavior (both consumers’ and producers’) can be interpreted as optimizing equilibrium behavior.



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”) p ̂0 s.t. x(p ̂0,w0) = q0.

But problem: Who is going to sell to the producers the quantities of inputs they need to supply q0?



Certainly not the consumers-owners, since at prices (p ̂0,w0) they want to sell the input quantities ωs - y(p ̂0,w0), and typically

ωs - y(p ̂0,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 Walras’s approach or take a critical stance towards it, do not realize the impossibility of Walras’s tâtonnement in production, unless special devices are contrived to take care of the problem (see, particularly, Walker (1972), (1987), (1996)).


Tâtonnement and equilibrium in the production model 1

Walras’s 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.



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 d’étudier les effets de leurs variations.» (Walras, 1988, p. 308, 2-3)



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 n’existait pas en matière d'échange. Dans l'échange, il n’y a pas de modification des marchandises. Un prix étant crié, et la demande et l’offre effective correspondant à ce prix n’étant 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 d’équilibre, il faudra non seulement crier d’autres prix, mais fabriquer d’autres 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).



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.



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 Walras’s 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 (Hicks’s 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 Walras’s 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.



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.


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.


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.


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.


Walras’s 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 d’échange, tout en tenant compte de cette circonstance, il n’y a qu’à supposer les entrepreneurs représentant par des bons des quantités successives de produits déterminées d’abord au hasard puis en augmentation ou diminution suivant qu’il y aura excédent du prix de vente sur le prix de revient ou réciproquement, jusqu’à 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 d’abord au hasard puis en hausse ou baisse suivant qu’il y aura excédent de la demande sur l’offre ou réciproquement, jusqu’à égalité de l’une et de l’autre.» (Walras, 1988, p. 309, 4-5)


Walras’s solution in the IV edition of the Eléments 2 The time structure of the analysis is specified 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, 1988, pp. 447, 449, 4-5)



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 .



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”.

lesson 2 equilibrium and tâtonnement in walrass eléments ph.d. program in economics university of...

Documents

equilibrium concept

trader i

stable equilibrium

special equilibrium

ttonnement page

general equilibrium

c page

market equilibrium condition