Transcript Marshall vs. Walras on Equilibrium and

Franco Donzelli
Topics in the History of Equilibrium Analysis
Lesson 3
Marshall vs. Walras on
Equilibrium and Disequilibrium
Ph.D. Program in Economics
University of York
February-March 2008
Introduction 1

The problem:

Do Walras’s and Marshall’s approaches to price theory only differ in
the respective scope of the analysis (general vs. partial analysis)?

Or do they differ in presuppositions, aims, analysis, and results?

The received view as expressed by:

introductory and intermediate textbooks (e.g., Frank, Schotter,
Varian): graphical vs. algebraic development of price theory

advanced textbooks (e.g., Mas-Colell, Whinston, Green):
general analysis as a natural extension of partial analysis
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Lesson 3 - Marshall vs. Walras
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Introduction 2

The working hypothesis:

Walras’s and Marshall’s approaches to price theory differ in essential
respects.

The main differences have to do with:
 the basic assumptions about the functioning of the trading process
 the nature of competition: “perfect competition” vs. “bilateral
bargaining”
 the nature of the disequilibrium process: in either “logical” or “real”
time
 the interpretation of the equilibrium construct: either an
“instantaneous” state or the “limit point of a sequence” in “real” time
 the nature of prices: numeraire normalization vs. money prices
The manifest difference in the scope of the analysis, i.e., general vs.
partial analysis, is the necessary by-product of more fundamental
epistemological and theoretical differences

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Lesson 3 - Marshall vs. Walras
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Introduction 3

The structure of the presentation:
1.
A common ground for the analysis: the pure-exchange, twocommodity economy
2.
Walras’s approach:
1.
2.
3.
4.
3.
Marshall’s approach:
1.
2.
3.
4.
4.
basic assumptions about the trading process
the model of a pure-exchange, two-commodity economy
interpretation and textual evidence
limitations and extensions
basic assumptions about the trading process
the model of an Edgeworth Box economy
the “temporary equilibrium” model
limitations and extensions
Comparison between the two approaches and conclusions
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The pure-exchange, two-commodity economy 1

Walras’s Eléments d’économie politique pure:
I ed.: 1874-1877; II ed.: 1889; III ed.: 1896; IV ed. 1900; V ed.: 1926
Most important changes in II and IV editions
English ed.: 1954

Price theory: 31 Lessons out of 42
Pure-exchange, two-commodity economy: Lessons 5 to 10
A small part of overall price theory, but fundamental (as recognized by
Walras himself)

Marshall’s Principles of Economics
8 editions, from 1890 to 1920
Most important changes in V edition (1907)

Price theory: Book V (out of 6, since II ed.)
Pure-exchange, two-commodity economy: Book V, Ch. 2 and App. F
A very small part of overall market equilibrium theory, but relevant
(Marshall’s stance ambiguous on this)
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The pure-exchange, two-commodity economy 2





L = 2 commodities, indexed by l = 1, 2
I consumers-traders, indexed by i = 1, …, I (I ≧ 2)
∀i = 1,…, I,

a consumption set Xi = {xi ≡ (x1i, x2i)} = ℝ2+ ;

a “cardinal” utility function ui: Xi → ℝ, assumed additively
separable in its arguments, that is:
ui(x1i, x2i) = v1i (x1i) + v2i(x2i)

endowments ωi ≡ (ω1i, ω2i) ∈ ℝ2+ \ {0}
Let:

x = (x1,…, xI) ∈ X = xi Xi ⊂ ℝ2I+ be an allocation

ω̅ = ∑ ωi ∈ ℝ2++ be the aggregate endowments

Ape2xI = {x ∈ X| ∑ xi = ω̅} be the set of feasible allocations
Assume:

ui(∙) twice continuously differentiable, with
∇ui(xi) = (v’1i(x1i), v’2i(x2i)) >> 0 and (v’’1i(x1i), v’’2i(x2i)) << 0, ∀xi ∈
Xi
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The pure-exchange, two-commodity economy 3





Let:

ℰpe2xI = {(Xi, ui(∙), ωi)Ii=1} be a pure-exchange, two-commodity
economy

ℰEB = ℰpe2x2 = {(ℝ2+, ui(∙), ωi)2i=1} be an Edgeworth Box
economy
Given ℰpe2xI, ∀x ∈ Ape2xI, let
MRSi21(xi) ≡ |dx2i/dx1i|ui(xi+dxi)=u(xi) = (∂u(xi)/∂u(x1i) / (∂u(xi)/∂u(x2i)
be consumer i’s marginal rate of substitution of commodity 2 for
commodity 1 when his consumption is xi
Let
zi(xi) ≡ (z1i, z2i)(xi) ≡ xi - ωi ≡ (x1i - ω1i, x2i - ω2i) ∈ ℝ2
be consumer i’s excess demand, when his consumption is xi
If zli(xi) > 0, then zli(xi) is called consumer i’s net demand proper
for commodity I and consumer i is said to be a net buyer
If zli(xi) < 0, then |zli(xi)| is called consumer i’s net supply for
commodity I and consumer i is said to be a net seller
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Lesson 3 - Marshall vs. Walras
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The pure-exchange, two-commodity economy 4

Let us suppose that consumer i can trade commodity 2 for
commodity 1

If the marginal rate at which he can trade is
- dx2/dx1 = |dx2/dx1| = MRSi21(xi) ,
then his utility is unaffected by the trade, since in that case:
du(xi) = ∇ui(xi)dxi = (∂u(xi)/∂u(x1i)dx1i + (∂u(xi)/∂u(x2i)dx2i = 0


On the contrary, if the marginal rate of exchange is
- dx2/dx1 = |dx2/dx1| < MRSi21(xi) ,
then consumer i’s utility increases (resp., decreases) if he is a net
buyer (resp., a net seller) of commodity 1
If instead the marginal rate of exchange is
- dx2/dx1 = |dx2/dx1| > MRSi21(xi) ,
then consumer i’s utility decreases (resp., increases) if he is a net
buyer (resp., a net seller) of commodity 1
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Lesson 3 - Marshall vs. Walras
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The pure-exchange, two-commodity economy 5

Hence the marginal rate of substitution of commodity 2 for
commodity 1, MRSi21(xi), can also be interpreted as the maximum
(resp., minimum) quantity of commodity 2 that a utility maximizing
buyer (resp., seller) of commodity 1 is willing to pay (resp., to
receive) at the margin in exchange for one unit of commodity 1,
when his consumption is xi.

MRSi21(xi) represents consumer i’s “reservation price” of
commodity 1 in terms of commodity 2, when his consumption is xi.

Both Walras and Marshall do not exactly employ the above
conceptual apparatus


They do not make any strong monotonicity assumption, ∇ui(xi) =
(v’1i(x1i), v’2i(x2i)) >> 0; Walras explicitly allows for consumers to
become satiated at finite consumption bundles. But to assume
non-satiation is an unobtrusive simplifying assumption.
They both ignore both the notion of marginal rate of substitution
and that of reservation price.
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Lesson 3 - Marshall vs. Walras
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The pure-exchange, two-commodity economy 6

Yet, they do know and systematically employ the notion of
marginal utility of commodity l for consumer i, which, under the
stated assumptions on the properties of the utility functions, is:
(∂ui(xi)/(∂xli)) = v’li(xli), for l = 1, 2.

Moreover, though not explicitly discussing the notion of marginal
rate of substitution as such, they do implicitly make use of it in
their analyses, since they compute the ratio of any two marginal
utility functions and examine its role in the agents’ choices.

Hence the above conceptual apparatus, though slightly more
general than that originally employed by Walras or Marshall, can
legitimately be said to lie at the foundation of both economists'
demand-and-supply analyses.

Any further development of either Walras’s or Marshall’s
approach, however, requires further assumptions, which are
specific to either economist.
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Walras’s three basic assumptions about the trading process 1

The three assumptions are separately stated, even if they are
obviously interrelated, and often confused (occasionally by
Walras himself) or jointly formulated in the literature.

The wording of the assumptions is carefully chosen in order to
make their statement consistent with Walras’s original discussion,
ambiguities not excepted.

The three assumptions underlie not only the model of a pureexchange, two-commodity economy, but all of Walras’s models (in
their final form).

The undefined terms in the assumptions will be first defined with
specific reference to the model of a pure-exchange, twocommodity economy, and then discussed with reference to the
whole Walrasian approach.
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Lesson 3 - Marshall vs. Walras
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Walras’s three basic assumptions about the trading process 2

Assumption 1. (“Law of one price" or "Jevons' law of
indifference")
At each instant of the trading process, a price is quoted in the
market for each commodity. Moreover, if any transaction
concerning a given commodity takes place at any instant of the
trading process, then it takes place at the price quoted at that
instant.

Assumption 2. ("Perfect competition")
All traders behave competitively, that is, they take prices as given
parameters in making their optimizing choices.

Assumption 3. ("No trade out of equilibrium")
No transaction concerning any commodity is allowed to take
place out of equilibrium.
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Lesson 3 - Marshall vs. Walras
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Walras’s model of a pure-exchange, two-commodity economy 1

Let ℰpe2xI = {(Xi, ui(∙), ωi)Ii=1} be the pure-exchange, twocommodity economy under consideration.

Let p = (p1, p2) ∈ ℝ2++ be the price system, where prices are
expressed in terms of units of account and are positive in view of
the strong monotonicity of preferences.

In view of assumption 1, the price system ought to be referred to
a particular instant of the trading process; but dating the variables
is unnecessary at this stage: for the exogenous variables
(consumption sets, preferences, endowments) are constant, while
the endogenous (prices and traders’ choices) are all
simultaneous.

Under assumptions 1 and 2, consumers optimizing choices are
homogeneous of zero degree in prices.
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Lesson 3 - Marshall vs. Walras
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Walras’s model of a pure-exchange, two-commodity economy 2

Hence prices can be normalized without any effect on consumers’
behavior.

Let p12 ≡ p1/p2) ≡ p21-1 be the relative price of commodity 1 in
terms of commodity 2, where the latter is taken as the numeraire
of the economy (which implies p2 ≡ 1).

Solving the constrained maximization problem for consumer i
yields:

u i x 1i,x 2i

x 1i

u i x 1i,x 2i

x 2i
p 12
p 12 x 1i x 2i p 12 1i 2i ,
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Lesson 3 - Marshall vs. Walras

1

1 
14
Walras’s model of a pure-exchange, two-commodity economy 3

From that system one gets consumer i’s Walrasian direct demand
and excess demand functions, for i = 1, …, I:
xi(p12,ωi) and zi(p12, ωi) = xi(p12,ωi) - ωi

Under assumptions 1 and 2, aggregating demand and excess
demand functions over consumers is immediate, since they all
receive the same price signals (by assumption 1), which they take
as given parameters (by assumption 2). Hence let
z(p12, ω) = ∑i zi(p12, ωi) = ∑i xi(p12,ωi) - ωi

be the aggregate demand function, where ω =(ω1,…, ωI).
The market-clearing conditions can be written as:
0
z1 
pW
12 , 

2 
0 ,
z2 
pW
12 , 

2 
where p12W is a Walrasian equilibrium price of commodity 1 in
terms of commodity 2.
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Lesson 3 - Marshall vs. Walras
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Walras’s model of a pure-exchange, two-commodity economy 4

From budget equations, by rearranging terms and summing over
consumers, we get the so-called Walras’ Law:
I
p 12 z1i 
p 12 , i z2i 
p 12 , i p 12 z1 
p 12 , z2 
p 12 , 0, 
p 12 0 .
 i1 

Due to Walras’ Law, equation (2’’) is necessarily satisfied when
equation (2’) holds. Hence we can focus on equation (2’).

Equation (2’) has at least one solution, not necessarily unique
under the stated assumptions.

Each solution yields a Walrasian equilibrium price of commodity 1
in terms of commodity 2, p12W, to which a Walrasian equilibrium
allocation x(p12W) = (x1(p12W),…, xi(p12W),…, xI(p12W)) is
associated.
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Lesson 3 - Marshall vs. Walras
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Walras’s model: textual evidence and interpretation 1

Right at the beginning of Lesson 5 of the Eléments, one finds a
long illustrative passage, where the functioning of the market for
“3 per cent French Rentes” is described in detail:
“Let us take, for example, trading in 3 per cent French Rentes on
the Paris Stock Exchange and confine our attention to these
operations alone. The three per cent, as they are called, are
quoted at 60 francs. [...]
We shall apply the term effective offer to any offer made, in this
way, of a definite amount of a commodity at a definite price. [...]
We shall apply the term effective demand to any such demand for
a definite amount of a commodity at a definite price.
We have now to make three suppositions according as the
demand is equal to, greater than, or less than the offer.
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Walras’s model: textual evidence and interpretation 2
First supposition. The quantity demanded at 60 francs is equal to
the quantity offered at this same price. [...] The rate of 60 francs is
maintained. The market is in a stationary state or equilibrium.
Second supposition. The brokers with orders to buy can no longer
find brokers with orders to sell. [...] Brokers [...] make bids at 60
francs 05 centimes. They raise the market price.
Third supposition. Brokers with orders to sell can no longer find
brokers with orders to buy. [...] Brokers [...] make offers at 59
francs 95 centimes. They lower the price.”
(Walras, 1954, pp. 84-85)

As this passage reveals, Walras’s starting point is represented by
a very realistic picture of the trading process, a picture which
stands at a very great distance from the image of that same
process emerging from the basic assumptions and the formal
model.
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Lesson 3 - Marshall vs. Walras
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Walras’s model: textual evidence and interpretation 3

Which is the true Walras?

The first striking difference between the model and the securities
example lies in the moneyless character of the former as
contrasted with the monetary character of the latter.

This is particularly relevant when we consider the monetary
character of Marshall’s “temporary equilibrium” model, where
“corn” is traded for money on the daily market of a small town
(“corn”, instead of “securities”, is the commodity traded for money
in Walras’s original example in his 1874 first theoretical
contribution, the mémoire entitled “Principe d’une théorie de
l’échange”).

On this point, however, Walras is very clear. For, a few lines after
the securities example, he adds:
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Walras’s model: textual evidence and interpretation 4

“Securities, however, are a very special kind of commodity.
Furthermore, the use of money in trading has peculiarities of its
own, the study of which must be postponed until later, and not
interwoven at the outset with the general phenomenon of value in
exchange. Let us, therefore, retrace our steps and state our
observations in scientific terms. We may take any two
commodities, say oats and wheat, or, more abstractly, (A) and
(B).” (Walras, 1954, pp. 86-87)

Coming now to the three basic assumptions about the trading
process, we see that all three of them are apparently disconfirmed
in the securities example:



Franco Donzelli
traders “make” prices, so that assumption 2 is violated;
different price bids can apparently coexist in time, so that also
assumption 1 fails;
trades can actually occur at out-of-equilibrium prices, so that
assumption 3 is violated as well.
Lesson 3 - Marshall vs. Walras
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Walras’s model: textual evidence and interpretation 5

Also in this case Walras tries to sharply distinguish the informal
presentation of an issue by means of an example from the “scientific”
discussion of the same issue by means of a formal model.

As far as assumptions 1 and 2 are concerned, his line of defense is
not wholly convincing, but in the end they are vindicated.

What is really problematic is Walras’s attitude towards assumption 3:

it is very likely that Walras did not initially realize the need for such
assumption as far as the pure-exchange model is concerned;

it is certain that he did not make any similar assumption
concerning the production model in any one of the first three
editions of the Eléments, that is, up to at least 1896.

But to allow out-of-equilibrium trades to actually occur in the
economy, as Walras does at least as far as the production model up
to 1896 is concerned, is inconsistent with the requirements of
equilibrium determination in Walras’s approach.
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Walras’s model: textual evidence and interpretation 6



In the pure-exchange model the occurrence of disequilibrium
transactions would make the equilibrium indeterminate

by altering the data of the economy (individual endowments)

by altering such data in an unpredictable way, for while Walras’s
theory can predict the optimally chosen plans of action at both
equilibrium and disequilibrium, it can only predict the individual
actions when the economy is at equilibrium.
Bertrand’s critique (1883) and Walras’s reaction (1885)
In the second edition (1889), Walras changes the securities
example, by adding

the words "Exchange takes place" in the case of market
equilibrium;

the expressions "Theoretically, trading should come to a halt"
and "Trading stops" in the case of excess demand and excess
supply, respectively.
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Walras’s model: limitations and extensions 1

Walras strenuously resists the generalized adoption of the notrade-out-of-equilibrium assumption because, together with the
other two, it turns

the adjustment process towards equilibrium into a virtual,
unobservable process occurring in a “logical” time entirely
disconnected from the “real” time over which the economy
evolves;

the equilibrium concept into an “instantaneous” equilibrium
concept, instantaneously arrived at in one instant of “real” time.

All this appears to Walras overly unrealistic and potentially
undermining the empirical content of the theory

Yet there is a trade-off between unrealism and generality, which
eventually convinces Walras to endorse all the three basic
assumptions about the trading process
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Walras’s model: limitations and extensions 2

Concerning generality:

by assuming “perfect competition” and the “law of one price”, Walras
(unlike Jevons and Marshall) can immediately attack the problem of
equilibrium determination in a pure-exchange economy with any finite
number of traders, rather than just two;

by giving up the descriptively realistic hypothesis that one of the two
commodities be money, and by deciding to normalize prices by means
of a numeraire, he makes the transition from a two-commodity to a
multi-commodity economy easier: for, when all commodities are
symmetrical, and every one can indifferently play the role of the
numeraire, the dimensionality of the price system (two vs. many
prices) becomes irrelevant; moreover, the cardinality assumption is
irrelevant and can be dispensed with;

by making the “no-trade-out-of-equilibrium assumption”, on top of
assuming “perfect competition” and the “law of one price”, he arrives at
defining a concept of “instantaneous equilibrium” which can be easily
applied, without significant change, to economies that are more
general than the pure-exchange economy, such as economies with
production, capital formation, and even money.
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Marshall’s basic assumptions about the trading process 1

Marshall does not assume traders to behave “competitively” (in
Walras’s sense), that is, as price-takers and quantity-adaptors.

Hence, in Marshall one does not find individual and aggregate
demand functions of the Walrasian type, since the latter depend on
the “perfect competition” assumption and the “law of one price”.

Marshall’s fundamental ideas about the trading process are that:



the trading process should be viewed as a sequence of bilateral
bargains, each involving two consumers at a time
the conditions governing each individual bargain depend on the
MRS’s of the two traders participating in it, viewed as reservation
prices (of either a buyer or a seller, as the case may be).
Precisely, let us focus on consumer i.
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Lesson 3 - Marshall vs. Walras
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Marshall’s basic assumptions about the trading process 2


Let MRS21i(ω1i,ω2i) = (∂u(ω1i,ω2i)/∂u(x1i) / (∂u(ω1i,ω2i)/∂u(x2i) be the
initial value of consumer i’s marginal rate of substitution of
commodity 2 for commodity 1
Supposing ∃j s.t. j ≠ i and MRS21j(ω1j,ω2j) ≠ MRS21i(ω1i,ω2i), let
kij(ω) = min {MRS21i(ω1i,ω2i), MRS21j(ω1j,ω2j)}
and
Kij(ω) = max {MRS21i(ω1i,ω2i), MRS21j(ω1j,ω2j)}


A bilateral bargain involving a marginal trade (dx1i,dx2i) = - (dx1j,dx2j)
between traders i and j is weakly advantageous to both iff
| (dx2i/dx1i)| = |(dx2j/dx1j| ∈ [kij(ω),Kij(ω)]
Marshall assumes that any weakly advantageous bargain will be
exploited.
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Lesson 3 - Marshall vs. Walras
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Marshall’s basic assumptions about the trading process 3

Hence, if the traders’ initial endowments are not all alike, the initial
allocation will change. But, the direction of change cannot be
predicted.

Similarly, even if one can predict that the trading process will come
to an end, neither the final allocation nor the final rate of exchange
can be predicted, failing further assumptions.

According to Marshall, this sort of indeterminacy is characteristic of
any trading process involving two commodities proper, that is, to any
“system of barter”.

To discuss the problem of indeterminacy, as well as other aspects of
barter, Marshall focuses attention on an Edgeworth Box economy
ℰEB = ℰpe2x2 = {(ℝ2+, ui(∙), ωi)2i=1} .
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Marshall’s model of an Edgeworth Box economy 1

Marshall shows that the barter process between two consumers
trading “apples” for “nuts” may follow a number of alternative paths,
each of which eventually terminates
“because any terms that the one is willing to propose would be
disadvantageous to the other. Up to this point exchange has
increased the satisfaction on both sides, but it can do so no further.
Equilibrium has been attained; but really it is not the equilibrium, it is
an accidental equilibrium” (Marshall, 1961a, p. 791; Marshall's
italics).

So, any final allocation or rate of exchange is an equilibrium
allocation or rate of exchange. But, in general, any such equilibrium
is “accidental” or “arbitrary”

There is however a path, characterized by a constant rate of
exchange between the two commodities over the exchange process,
which stands apart from all the other possible paths, occupying a
position that, according to Marshall, is theoretically unique, though
practically irrelevant.
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Marshall’s model of an Edgeworth Box economy 2

“There is, however, one equilibrium rate of exchange which has
some sort of right to be called the true equilibrium rate, because if
once hit upon would be adhered to throughout. [...] This is then the
true position of equilibrium; but there is no reason to suppose that it
will be reached in practice” (Marshall, 1961a, p. 791)

Let us formalize Marshall’s discussion. Let i =1,2. Assuming
MRS211(ω11,ω21) ≠ MRS212(ω12,ω22), let
k12(ω) = min {MRS211(ω11,ω21), MRS212(ω12,ω22)} <
< max {MRS211(ω11,ω21), MRS212(ω12,ω22)} = K12 (ω)

The Pareto set of ℰEB is the set
P EB x P  A 2pe2 MRS 121 
x P1 MRS 221 
x P2 ,
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Lesson 3 - Marshall vs. Walras
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Marshall’s model of an Edgeworth Box economy 3

while the contract curve of ℰEB is the set
C EB x C  P EB u 1 
x C1 u 1 
1 
, u2 
x C2 u 2 
2 .


CEB ≠ ∅. Any xC ∈ CEB is an “equilibrium” allocation and any
MRS21i(xjC) = p1C, for i =1, 2 is an “equilibrium” rate of exchange, but
in general such equilibria would be “arbitrary”.
Only a rate of exchange p1* = MRS211(x1*) = MRS212(x2*) satisfying
the additional condition
p 1

x
21
21 
x
11
11 

x
22
22 
x
12
12 
,
being constant over the trading process, would qualify as a “true
equilibrium” rate.
Franco Donzelli
Lesson 3 - Marshall vs. Walras
30
Marshall’s model of an Edgeworth Box economy 4

Finally, since
MRSi21(xi) ≡ |dx2i/dx1i|ui(xi+dxi)=u(xi) = (∂u(xi)/∂u(x1i) / (∂u(xi)/∂u(x2i),
for i = 1,2, in Marshall’s “true equilibrium” the following condition also
holds:

u i
x
u i
x
i 
i

x
1i
/

x
2i

dx 
2i
dx 
1i

x
2i
2i 
x
1i
1i 
,
which is nothing but Jevons’ equilibrium condition, as expressed in
The Theory of Political Economy (1871, Ch. 4, pp. 142-143).

As can be seen, the extreme form of Jevons’ “law of indifference” is
interpreted by Marshall as an equilibrium condition, precisely, as a
condition for achieving a “true equilibrium”. But “but there is no
reason to suppose that it will be reached in practice”.
Franco Donzelli
Lesson 3 - Marshall vs. Walras
31
Marshall’s model of an Edgeworth Box economy 5

For Marshall, the indeterminacy of equilibrium in the “apple” and
“nuts” model depends on its being a model of barter:
“The uncertainty of the rate at which the equilibrium is reached
depends indirectly on the fact that one commodity is being bartered
for another instead of being sold for money. For, since money is a
general purchasing medium, there are likely to be many dealers who
can conveniently take in, or give out, large supplies of it; and this
tends to steady the market.” (Marshall, 1961a, p. 793)

As far as the indeterminacy problem is concerned, the fundamental
property of money is that “its marginal utility is practically constant”

This allows one to distinguish between the “theory of barter” (two
commodities) and the “theory of buying and selling” (“money and
commodity”)
Franco Donzelli
Lesson 3 - Marshall vs. Walras
32
Marshall’s model of an Edgeworth Box economy 6
“The real distinction then between the theory of buying and selling
and that of barter is that in the former it generally is, and in the latter
it generally is not, right to assume that the stock of one of the things
which is in the market and ready to be exchanged for the other is
very large and in many hands; and that therefore its marginal utility
is practically constant.” (Marshall, 1961a, p. 793)


According to Marshall, if one commodity (“nuts”) shared the
essential properties of money (“constant marginal utility”), then the
indeterminacy problem would not arise in the Edgeworth Box model
either.
Let ℰEBm = ℰpe2x2,m be an Edgeworth Box economy where the first
commodity (“apples”) still is a commodity proper, but the second one
(“nuts”) is a money-like commodity, with constant marginal utility
Franco Donzelli
Lesson 3 - Marshall vs. Walras
33
Marshall’s model of an Edgeworth Box economy 7

Let consumer i’s utility function be quasi-linear in commodity 2, that is:
ui 
x 1i , x 2i v 1i 
x 1i x 2i ,
i 1, 2 ,
where (∂ui(x1i, x2i)/(∂x1i)) = v’1i(x1i), assumed positive and decreasing
for x1i ∈ [0, ω̅1], depends only on the quantity consumed of commodity
1, while (∂ui(x1i, x2i)/(∂x2i)), the marginal utility of the money-like
commodity, is constant (normalized to 1).

Hence
MRS21i(xi) = (∂ui(x1i, x2i)/(∂x1i)) / (∂ui(x1i, x2i)/(∂x2i)) = v’1i(x1i)
depends only on the quantity consumed of commodity 1.

Let
d1i(x1i,ω1i) = max {0, x1i - ω1i}
be consumer i’s net demand proper for commodity 1 for x1i ∈ [0, ω̅1]
Franco Donzelli
Lesson 3 - Marshall vs. Walras
34
Marshall’s model of an Edgeworth Box economy 8
s1i(x1i,ω1i) = |min {0, x1i - ω1i}|
be consumer i’s net supply of commodity 1 for x1i ∈ [0, ω̅1]

If x1i > ω1i, then d1i(x1i,ω1i) > 0 and consumer i is a net buyer of
commodity 1; hence MRS21i(xi) = v’1i(x1i) can be interpreted as a
buyer’s reservation price, or demand price.

If x1i < ω1i, then s1i(x1i,ω1i) > 0 and consumer i is a net seller of
commodity 1; hence MRS21i(xi) = v’1i(x1i) can be interpreted as a
seller’s reservation price, or supply price.

Consumer i’s Marshallian inverse supply correspondence of
commodity 1, p1is: [0,ω1i] → ℝ+, is defined as follows:
p1is(s1i) = [0,v’1i(x1i)] for s1i = 0
p1is(s1i) = v’1i(ω1i – s1i) for s1i ∈ (0,ω1i) (a continuous increasing function)
p1is(s1i) = [v’1i(0),∞) for s1i = ω1i
Franco Donzelli
Lesson 3 - Marshall vs. Walras
35
Marshall’s model of an Edgeworth Box economy 9

Consumer i’s Marshallian inverse demand correspondence for
commodity 1, p1id: [0,ω̅1 - ω1i] → ℝ+, is defined as follows:
p1id(d1i) = (∞, v’1i(ω1i)] for d1i = 0
p1id(d1i) = v’1i(ω1i + d1i) for d1i ∈ (0, ω̅1 - ω1i) (a continuous decreasing
function)
p1id(d1i) = (v’1i(ω̅1), 0] for d1i = ω̅1 - ω1i

By taking the inverses of the above two functions, and suitably
extending them to cover the whole price domain, one gets the
Marshallian direct supply and demand functions.

Consumer i’s Marshallian direct supply function of commodity 1 is the
continuous function s1i: ℝ+ → [0,ω1i] defined as follows:
s1i(p1is) = 0 for p1is ∈ [0, v’1i(ω1i))
s1i(p1is) = ω1i – (v’1i)-1(p1is) for p1is ∈ [v’1i(ω1i), v’1i(0))
s1i(p1is) = ω1i for p1is ∈ [v’1i(0), ∞)
Franco Donzelli
Lesson 3 - Marshall vs. Walras
36
Marshall’s model of an Edgeworth Box economy 10

Consumer i’s Marshallian direct demand function for commodity 1 is
the continuous function d1i: ℝ+ → [0, ω̅1 - ω1i] defined as follows:
d1i(p1id) = 0 for p1id ∈ (∞, v’1i(ω1i)]
d1i(p1is) = (v’1i)-1(p1id) - ω1i for p1id ∈ (v’1i(ω1i), v’1i(ω̅1)]
d1i(p1id) = ω̅1 - ω1i for p1id ∈ (v’1i(ω̅1),0]

Let p1mind = mini {v’1i(ω̅1)} and p1maxd = maxi {v’1i(ω1i)};
let p1mins = mini {v’1i(ω1i)} and p1maxs = maxi {v’1i(0)}.


If the consumers’ tastes are not identical, then p1maxd > p1mins
Let d1(p1d) = ∑i d1i(p1id), for p1d = p1id, for i = 1, 2 and p1d ∈ [0, ∞);
let s1(p1s) = ∑i s1i(p1is), for p1s = p1is, for i = 1, 2 and p1s ∈ [0, ∞).
Franco Donzelli
Lesson 3 - Marshall vs. Walras
37
Marshall’s model of an Edgeworth Box economy 11

The functions d1(∙) and s1(∙), arrived at by aggregating the individual
demand and supply functions over consumers, are called the Marshallian
aggregate demand and supply functions for commodity 1, respectively.

d1(∙) is nonincreasing in p1d, and strictly decreasing for
p1d ∈ [p1mind, p1maxd], except possibly when d1i = ω̅1 – ω1i, i = 1,2

s1(∙) is nondecreasing in p1s, and strictly increasing for
p1s ∈ [p1mins, p1maxs], except possibly when s1i = ω1i, i = 1,2
Hence, supposing p1maxd = v’1i(ω1i) and p1mins = v’1j(ω1j), i, j = 1, 2 and i ≠ j,
and assuming v’1i(ω̅1i) < v’1j(0), there must exist a unique price
p1M = p1dM = p1sM ∈ (p1mins,p1maxd) s.t.

or
Franco Donzelli
M
d1 
pM
s

p
1
1
1 

6
d1 
pM
pM
1 s 1 
1 0

7
Lesson 3 - Marshall vs. Walras
38
Marshall’s model of an Edgeworth Box economy 12
where p1M is the Marshallian equilibrium price of commodity 1 in
terms of commodity 2, while the common value d1(p1M) = s1(p1M) =
q1(p1M) is the Marshallian equilibrium total traded quantity of
commodity 1, or, for short, the equilibrium quantity of commodity 1.

Equation (7) resembles the Walrasian equilibrium equation (2’), any
solution of which is a Walrasian equilibrium price of commodity 1 in
terms of commodity 2, p12W.

Yet, in spite of its appearance, and unlike equation (2’), equation (7)
is not a market-clearing equation; similarly, p1M, unlike p12W, is not a
market a market-clearing price.

In fact, in general, the two consumers will not carry out their trades
at the constant rate p1M; and yet, even if different trades take place
at different rates, at the end of the process the total quantity traded
of commodity 1 will still be equal to the common value q1(p1M).
Franco Donzelli
Lesson 3 - Marshall vs. Walras
39
Marshall’s model of an Edgeworth Box economy 13

An illustration:

Assumption 1. (Utility functions quadratic in commodity 1 and quasi-linear
in commodity 2)
ui(x1i, x2i) = v1i(x1i) + v2i(x2i) = ai(x1i - ω1i) - ½bi(x1i - ω1i)2 + x2i, i = 1,2.
Hence: MRS21i(xi) = v’1i(x1i) = ai - bi(x1i - ω1i) , i = 1,2.

Assumption 2.
K12(ω) = MRS211(ω1) = a1 > a2 = MRS212(ω2) = k12(ω)

Assumption 3.
ω11 < ω12 ; v’11(ω̅1) < v’12(0)
Equilibrium:
p11d(d11) = p11s(s12) and d11 = s12 . Hence:
p1M = (a1b2 + a2b1) / (b1 + b2) ;
q1M = (a1 - a2) / (b1 + b2)
Franco Donzelli
Lesson 3 - Marshall vs. Walras
40
Marshall’s model of an Edgeworth Box economy 14
p11d,
p11s
p12d,
p12s
p11s
p12s
p1d,
p1s
p1maxs
s1(p1s)
p1maxd
v’11(ω11)
p1M
v’12(ω12)
p1mins
d1(p1d)
p1mind
p11d
ω11
ω̅1 -ω11
Franco Donzelli
p12d
d11, s11
ω̅1 - ω12
ω12
Lesson 3 - Marshall vs. Walras
d12, s12
q1M
ω̅1 d , s
1
1
41
Marshall’s model of an Edgeworth Box economy 15

When the two consumers have already cumulatively traded a
quantity q̂1 of commodity 1, such that q̂1 ∈ [0, q1(p1M)), there still
exists a positive difference between the demand and the supply
price of commodity 1 corresponding to q̂1, that is p1d(q̂1) - p1s(q̂1) > 0.

Hence there still is room for a weakly advantageous marginal trade
between the two consumers, at any rate of exchange
p̂1 ∈ [p1d(q̂1),p1s(q̂1)]

The rate of exchange p1M ought to be interpreted as the final rate to
which the sequence of the rates at which the consumers have
traded during the trading process necessarily converges, along a
path which may exhibit no regularity other than the stated
convergence.

The total quantity of commodity 1 traded by the traders, q1(p1M)),
ought instead to be interpreted as the quantity of commodity 1 to
which the monotonically increasing sequence of the quantities
cumulatively traded by the consumers during the exchange process
necessarily converges.
Franco Donzelli
Lesson 3 - Marshall vs. Walras
42
Marshall’s model of an Edgeworth Box economy 14

The total quantity of the money-like commodity 2 cumulatively
traded by the consumers at the end of the trading process remains
undetermined, its final value being however necessarily confined to
the interval
q 1
pM
q 1
pM
1  s
1  d
p 1i 
s 1i 
ds 1i ,
p 1j 
d 1j 
dd 1j
0
0


,
where i, j = 1,2, i ≠ j; i, j are s.t. v’1i(ω1i) = p1mins and v’1j(ω1j) = p1maxd.

Hence in Marshall’s model there exists no counterpart of equation
(2’’) in Walras’s model, where it provides the market-clearing
condition for commodity 2.

Further, in Marshall’s model there is nothing comparable to Walras’
Law, even if, due to the bilateral character of any exchange, the total
value of sales must always equal that of purchases for each
consumer, hence for the whole economy.
Franco Donzelli
Lesson 3 - Marshall vs. Walras
43
Marshall’s “temporary equilibrium” model 1

Marshall's "temporary equilibrium" model actually consists in a limited
extension of his Edgeworth Box model with a money-like commodity to
a pure-exchange, two-commodity economy with an arbitrary finite
number of traders, that is, an economy
ℰpe2xI,m = {(ℝ2+, ui(∙), ωi)Ii=1} with I >2,
where commodity 1 is a consumers' good, commodity 2 is money, and
the marginal utility of commodity 2 is assumed to be constant.

An ambiguity of the model:



formally: an entire economy → general equilibrium analysis
substantially: a single market → partial equilibrium analysis
Effects on the possibility of formalizing Marshall’s empirical justifications
for assuming the “marginal utility of money” to be “constant”:
 money should be “in large supply and general use” (possible)
 the expenditure on the good for which money is traded should
represent “a small part of [each trader’s] resources” (meaningless)
Franco Donzelli
Lesson 3 - Marshall vs. Walras
44
Marshall’s “temporary equilibrium” model 2

Assumption 1 (new):
No strategic or game-theoretic considerations are allowed: each
bilateral bargain is regarded as a self-contained transaction by the
two traders involved in it, so that each trader, in deciding whether to
get engaged in a bargain, takes into account only the immediate
effects of that bargain on his utility. (Edgeworth and Berry)

Assumption 2:
An individual bargain can only take place if it is weakly
advantageous for the two consumers involved in it.

Assumption 3:
Each consumer will not stop trading as long as he can increase his
utility by so doing.

Under these assumptions, the generalization of the model of an
Edgeworth Box economy with a money-like commodity, ℰEBm =
ℰpe2x2,m, to the "temporary equilibrium" model of a pure-exchange
economy with I consumers, ℰpe2xI,m, with I > 2, is immediate.
Franco Donzelli
Lesson 3 - Marshall vs. Walras
45
Marshall’s “temporary equilibrium” model 3

We shall rewrite equations (6) and (7) as:
I,M
I,M
d I1 
p 1 sI1 
p1 
I,M
I,M
d I1 
p 1 sI1 
p 1 0 ,

8

9
it being understood that, in deriving equations (8) and (9), the
Marshallian aggregate demand and supply functions for commodity 1
are, respectively:
d1I(p1d) = ∑i d1i(p1id), for p1d = p1id, for i = 1, …, I, and p1d ∈ [0, ∞),
and
s1I(p1s) = ∑i s1i(p1is), for p1s = p1is, for i = 1, …, I, and p1s ∈ [0, ∞).

In equations (8) and (9) p1M is the Marshallian “temporary equilibrium”
money price of commodity 1, while the common value d1(p1M) =
s1(p1M) = q1(p1M) is the Marshallian “temporary equilibrium” quantity of
commodity 1.
Franco Donzelli
Lesson 3 - Marshall vs. Walras
46
Marshall’s “temporary equilibrium” model 4

Marshall’s interpretation of equations (8) and (9) is essentially the
same as that of equations (6) and (7). Yet Marshall’s claims are not
entirely justified.

Marshall’s “temporary equilibrium” model is developed by means of
an example, referring to “a corn market in a country town”, where
“corn” is traded for “money”. The illustration is based on the “facts”
summarized by the following “table” (Marshall, 1961a, pp. 332-333):
At the price Holders will be Buyers will be
Franco Donzelli
willing to sell
willing to buy
37s.
1000 quarters
600 quarters
36s.
700
"
700
"
35s.
600
"
900
"
Lesson 3 - Marshall vs. Walras
47
Marshall’s “temporary equilibrium” model 5
“Many of the buyers may perhaps underrate the willingness of the
sellers to sell, with the effect that for some time the price rules at
the highest level at which any buyers can be found; and thus 500
quarters may be sold before the price sinks below 37s. But
afterwards the price must begin to fall and the result will still
probably be that 200 more quarters will be sold, and the market will
close on a price of about 36s. For when 700 quarters have been
sold, no seller will be anxious to dispose of any more except at a
higher price than 36s., and no buyer will be anxious to purchase
any more except at a lower price than 36s.
In the same way if the sellers had underrated the willingness of the
buyers to pay a high price, some of them might begin to sell at the
lowest price they would take, rather than have their corn left on
their hands, and in this case much corn might be sold at a price of
35s.; but the market would probably close on a price of 36s. and a
total sale of 700 quarters.” (Marshall, 1961, p. 334)

Here we have a distinctly non-Walrasian equilibration process,
since out-of-equilibrium trades are explicitly allowed for.
Franco Donzelli
Lesson 3 - Marshall vs. Walras
48
Marshall’s “temporary equilibrium” model 6

And yet the process is said to converge to a well-determined price of
corn in terms of money (36s.) and a well-determined total traded quantity
of corn (700 quarters), where such price and quantity incidentally
coincide with the Walrasian equilibrium ones.

According to Marshall, also in this case the determinateness of
equilibrium crucially depends on the "constant marginal utility of money"
assumption.

But is Marshall justified in supposing that the "constant marginal utility of
money" assumption is sufficient for granting equilibrium determinateness
in a pure-exchange economy with many traders, ℰpe2xI,m with I>2, as it
was in an Edgeworth Box economy with a money-like commodity, ℰEBm?

The answer is: not quite.

In the model of an Edgeworth Box economy with a money-like
commodity, the sharp result which has been obtained concerning p1M,
the Marshallian equilibrium price of commodity 1 in terms of commodity
2, crucially depends on the existence of only two traders in the economy.
Franco Donzelli
Lesson 3 - Marshall vs. Walras
49
Marshall’s “temporary equilibrium” model 7

With only two traders, the marginal rate of exchange at which the last marginal
trade occurs necessarily coincides with both the marginal demand price of the only
marginal buyer, p1d(q1(p1M)), and the marginal supply price of the only marginal
seller, p1s(q1(p1M)).

Hence, assuming uniqueness of the equilibrium price, it also necessarily coincides
with p1M, which can therefore be legitimately interpreted as the final rate to which
the sequence of the rates at which the traders have traded during the exchange
process necessarily converges.

But in Marshall's "temporary equilibrium" model there are more than two traders in
the economy: hence, in general, not only there may exist more than one marginal
buyer or seller, but also there may be some buyers or sellers that are not marginal.

Due to this, in Marshall's "temporary equilibrium" model the total quantity of
commodity 1 traded in the market still converges to the Marshallian "temporary
equilibrium" quantity, q1(p1M), but the sequence of the money prices of commodity 1
at which the traders buy and sell that commodity during the trading process no
longer necessarily converges to the Marshallian "temporary equilibrium" price, p1I,M:
the outcome depends on the order of the matchings.
Franco Donzelli
Lesson 3 - Marshall vs. Walras
50
Marshall’s pure-exchange models: limitations and extensions 1

Marshall’s Edgeworth Box model is propedeutical to his “temporary
equilibrium” model, which is in turn propedeutical to Marshall’s
normal equilibrium models.

But, even if propedeutical, Marshall’s pure-exchange models still
play a fundamental role in the overall structure of Marshall’s
thought.

In his pure-exchange, two-commodity models Marshall wants to
show how an equilibrium comes to be established as the final
outcome of a realistic process of exchange in "real" time, where
trades can actually take place at out-of-equilibrium rates of
exchange or prices.

This program inevitably raises the issue of equilibrium determinacy.

Marshall's solution consists in imposing some related restrictions
on the traders' utility functions, which are assumed to be quasilinear in one of the two commodities, and the nature of the
commodities themselves, one of which is interpreted as a moneylike commodity or money tout court.
Franco Donzelli
Lesson 3 - Marshall vs. Walras
51
Marshall’s pure-exchange models: limitations and extensions 2

But, at the same time, Marshall inexorably restrains the scope of
his analysis.

His suggested solution of the equilibrium indeterminacy problem
only applies when no more than one commodity proper is explicitly
accounted for in the model, so that the only unknowns to be
determined boil down to the money price and the quantity traded of
that single commodity proper.

There is no way to extend to a multi-commodity world, made up of
many interrelated markets, the results achieved by Marshall within
his one-commodity world, consisting in the isolated market where
the only commodity proper explicitly contemplated by the model is
traded for money.

Marshall's analysis remains necessarily confined to the partial
equilibrium framework in which it is originally couched, even when
production is brought into the picture, as it happens with normal
equilibrium models.
Franco Donzelli
Lesson 3 - Marshall vs. Walras
52
Conclusions 1

In this paper we have contrasted the received view on price theory,
according to which Walras’s and Marshall’s approaches, while
differing in scope, are basically similar in their aims, presuppositions,
and results.

By focusing on the pure-exchange, two-commodity economy, which
has been formally studied by both Walras and Marshall with the help
of similar tools, we have been able to precisely identify the
differences between the two approaches.

First, the very basic assumptions underlying the analysis of the
trading process and shaping the conception of a competitive
economy have been shown to widely differ between the two
economists.

Secondly, it has been shown that, starting from such different sets of
assumptions, the two authors arrive at entirely different models of the
pure-exchange, two-commodity economy.
Franco Donzelli
Lesson 3 - Marshall vs. Walras
53
Conclusions 2

By reducing the trading process to a purely virtual process in "logical"
time, Walras arrives at a well-defined notion of "instantaneous"
equilibrium, which can be easily extended to more general contexts (such
as pure-exchange, multi-commodity economies and production
economies).

By making a few further assumptions on the characteristics of the traders
and the nature of the commodities involved, one of which must be money
or a money-like commodity, Marshall can indeed show that a determinate
(or almost determinate) equilibrium emerges from a process of exchange
in "real" time with observable out-of-equilibrium trades.

But his analysis cannot be significantly generalized beyond the partial
equilibrium framework in which it is necessarily couched from the
beginning.

There exists a trade-off between realism and scope of the analysis:
the more realistic the representation of the disequilibrium trading process,
the less comprehensive and general is equilibrium analysis.
Franco Donzelli
Lesson 3 - Marshall vs. Walras
54