Transcript 082 (MS PowerPoint , 142kb)

Franco Donzelli
Topics in the History of Equilibrium Analysis
Lesson 2
Equilibrium and tâtonnement
in Walras’s Eléments
Ph.D. Program in Economics
University of York
February-March 2008
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)
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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.»
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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
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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)

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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.
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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?
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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”.
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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 ωic ∈ 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 + ω2ic
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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.
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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).
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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 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)
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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 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.
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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 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)
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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. »
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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 ωsi ∈ 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(ωsi – yi)
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The production model 2

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
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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:
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
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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 →
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.
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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: since
π* = π(p*, x*) = 0, all equilibrium behavior (both consumers’ and
producers’) can be interpreted as optimizing equilibrium behavior.
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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”) p̂0 s.t.
x(p̂0,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 (p̂0,w0) they
want to sell the input quantities ωs - y(p̂0,w0), and typically
ωs - y(p̂0,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)).
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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.
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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 d’étudier les effets de leurs variations.»
(Walras, 1988, p. 308, 2-3)
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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 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).
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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.
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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
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.
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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.
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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.
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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.
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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.
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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)
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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)
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Walras’s 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 selfcontradictory in this context .
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Lesson 2 - Equilibrium in Walras
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Walras’s 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”.
Franco Donzelli
Lesson 2 - Equilibrium in Walras
35