topics history equilibrium 083

Lesson 3

Marshall vs. Walras on

Equilibrium and Disequilibrium

Ph.D. Program in Economics

University of York

February-March 2008

Franco Donzelli

Topics in the History of Equilibrium Analysis

Lesson 3 - Marshall vs. Walras 2 Franco Donzelli

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


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


Introduction 3

The structure of the presentation:

1. A common ground for the analysis: the pure-exchange, two-

commodity economy

2. Walras’s approach:

1. basic assumptions about the trading process

2. the model of a pure-exchange, two-commodity economy

3. interpretation and textual evidence

4. limitations and extensions

3. Marshall’s approach:

1. basic assumptions about the trading process

2. the model of an Edgeworth Box economy

3. the “temporary equilibrium” model

4. limitations and extensions

4. Comparison between the two approaches and conclusions


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)



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

ω ̅ = ∑ ωi ∈ ℝ2++ be the aggregate endowments

Ape2xI = {x ∈ X| ∑ xi = ω ̅} 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



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 ∈ Ape

2xI, 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



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



Hence the marginal rate of substitution of commodity 2 for commodity 1, MRSi

21(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.



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.


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 pure-exchange, 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, two-commodity economy, and then discussed with reference to the whole Walrasian approach.


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.


Walras’s model of a pure-exchange, two-commodity economy 1

Let ℰpe2xI = {(Xi, ui(∙), ωi)

Ii=1} be the pure-exchange, two-

commodity 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.



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:

ui x1i,x2i

x1i

ui x1i,x2i

x2i

p12 1

p12x1i x2i p121i 2i , 1



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:

where p12W is a Walrasian equilibrium price of commodity 1 in

terms of commodity 2.

z1p12

W,0 2

z2p12

W,0 , 2



From budget equations, by rearranging terms and summing over

consumers, we get the so-called Walras’ Law:

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(p12

W),…, xi(p12W),…, xI(p12

W)) is

associated.

i1

Ip12z1ip12 ,iz2ip12 ,ip12z1p12 ,z2p12 ,0,p12 0 .


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.



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.



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:



“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:

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.



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.



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.


Walras’s model: limitations and extensions 1

Walras strenuously resists the generalized adoption of the no-

trade-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


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.


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.



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) ≠ MRS21

i(ω1i,ω2i), let

kij(ω) = min {MRS21i(ω1i,ω2i), MRS21

j(ω1j,ω2j)}

and

Kij(ω) = max {MRS21i(ω1i,ω2i), MRS21

j(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.



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} .


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.



“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) ≠ MRS21

2(ω12,ω22), let

k12(ω) = min {MRS211(ω11,ω21), MRS21

2(ω12,ω22)} <

< max {MRS211(ω11,ω21), MRS21

2(ω12,ω22)} = K12 (ω)

The Pareto set of ℰEB is the set

PEB x

P Ape

22MRS

21

1x

1

PMRS

21

2x

2

P,



while the contract curve of ℰEB is the set

CEB ≠ ∅. Any xC ∈ CEB is an “equilibrium” allocation and any

MRS21i(xj

C) = 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*) = MRS21

2(x2*) satisfying

the additional condition

,

being constant over the trading process, would qualify as a “true

equilibrium” rate.

CEB xC PEB u1x1

Cu11,u2x2

Cu22.

p1

x21

21

x11

11

x22

22

x12

12



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:

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”.

u ix i

x1i

/u ix i

x2i

dx2i

dx1i

x2i

2i

x1i

1i

,



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”)



“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 = ℰpe

2x2,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



Let consumer i’s utility function be quasi-linear in commodity 2, that is:

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]

uix1i,x2iv1ix1ix2i , i 1,2 ,



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 MRS21

i(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 MRS21

i(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, p1i

s: [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



Consumer i’s Marshallian inverse demand correspondence for commodity 1, p1i

d: [0,ω ̅1 - ω1i] → ℝ+, is defined as follows:

p1id(d1i) = (∞, v’1i(ω1i)] for d1i = 0

p1id(d1i) = v’1i(ω1i + d1i) for d1i ∈ (0, ω ̅1 - ω1i) (a continuous decreasing

function)

p1id(d1i) = (v’1i(ω ̅1), 0] for d1i = ω ̅1 - 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 p1i

s ∈ [0, v’1i(ω1i))

s1i(p1is) = ω1i – (v’1i)

-1(p1is) for p1i

s ∈ [v’1i(ω1i), v’1i(0))

s1i(p1is) = ω1i for p1i

s ∈ [v’1i(0), ∞)



Consumer i’s Marshallian direct demand function for commodity 1 is

the continuous function d1i: ℝ+ → [0, ω ̅1 - ω1i] defined as follows:

d1i(p1id) = 0 for p1i

d ∈ (∞, v’1i(ω1i)]

d1i(p1is) = (v’1i)

-1(p1id) - ω1i for p1i

d ∈ (v’1i(ω1i), v’1i(ω ̅1)]

d1i(p1id) = ω ̅1 - ω1i for p1i

d ∈ (v’1i(ω ̅1),0]

Let p1mind = mini {v’1i(ω ̅1)} and p1max

d = maxi {v’1i(ω1i)};

let p1mins = mini {v’1i(ω1i)} and p1max

s = maxi {v’1i(0)}.

If the consumers’ tastes are not identical, then p1maxd > p1min

s

Let d1(p1d) = ∑i d1i(p1i

d), for p1d = p1i

d, for i = 1, 2 and p1d ∈ [0, ∞);

let s1(p1s) = ∑i s1i(p1i

s), for p1s = p1i

s, for i = 1, 2 and p1s ∈ [0, ∞).



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 ∈ [p1min

d, p1maxd], except possibly when d1i = ω ̅1 – ω1i, i = 1,2

s1(∙) is nondecreasing in p1s, and strictly increasing for

p1s ∈ [p1min

s, p1maxs], except possibly when s1i = ω1i, i = 1,2

Hence, supposing p1maxd = v’1i(ω1i) and p1min

s = v’1j(ω1j), i, j = 1, 2 and i ≠ j,

and assuming v’1i(ω ̅1i) < v’1j(0), there must exist a unique price

p1M = p1

dM = p1sM ∈ (p1min

s,p1maxd) s.t.

or

d1p1

Ms1p1

M 6

d1p1

Ms1p1

M0 7



where p1M is the Marshallian equilibrium price of commodity 1 in

terms of commodity 2, while the common value d1(p1M) = s1(p1

M) =

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 p12

W, 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).



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 = MRS21

2(ω2) = k12(ω)

Assumption 3.

ω11 < ω12 ; v’11(ω ̅1) < v’12(0)

Equilibrium:

p11d(d11) = p11

s(s12) and d11 = s12 . Hence:

p1M = (a1b2 + a2b1) / (b1 + b2) ;

q1M = (a1 - a2) / (b1 + b2)



p11d,

p11s

p12d,

p12s

p1d,

p1s

d11, s11 d12, s12 d1, s1

p1M

q1M

p11s

p11d p12

d

p12s

s1(p1s)

d1(p1d)

p1maxs

p1mins

p1mind

p1maxd

ω11 ω12 ω ̅1 ω̅1 -ω11 ω ̅1 - ω12

v’11(ω11)

v’12(ω12)



When the two consumers have already cumulatively traded a quantity q̂1 of commodity 1, such that q̂1 ∈ [0, q1(p1

M)), there still exists a positive difference between the demand and the supply price of commodity 1 corresponding to q̂1, that is p1

d(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),p1

s(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.



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

,

where i, j = 1,2, i ≠ j; i, j are s.t. v’1i(ω1i) = p1mins and v’1j(ω1j) = p1max

d.

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.

0

q1p1M

p1i

ss1ids1i,

0

q1p1M

p1j

dd1jdd1j


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)



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, ℰEB

m = ℰpe

2x2,m, to the "temporary equilibrium" model of a pure-exchange economy with I consumers, ℰpe

2xI,m, with I > 2, is immediate.



We shall rewrite equations (6) and (7) as:

it being understood that, in deriving equations (8) and (9), the Marshallian aggregate demand and supply functions for commodity 1 are, respectively:

d1I(p1

d) = ∑i d1i(p1id), for p1

d = p1id, for i = 1, …, I, and p1

d ∈ [0, ∞),

and

s1I(p1

s) = ∑i s1i(p1is), for p1

s = p1is, for i = 1, …, I, and p1

s ∈ [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(p1

M) is the Marshallian “temporary equilibrium” quantity of commodity 1.

d1

Ip

1

I,Ms

1

Ip

1

I,M 8

d1

Ip

1

I,Ms

1

Ip

1

I,M0 , 9



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

willing to sell willing to buy

37s. 1000 quarters 600 quarters

36s. 700 " 700 "

35s. 600 " 900 "



“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.



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, ℰpe

2xI,m with I>2, as it was in an Edgeworth Box economy with a money-like commodity, ℰEB

m?

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 p1

M, the Marshallian equilibrium price of commodity 1 in terms of commodity 2, crucially depends on the existence of only two traders in the economy.



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(p1

M)), and the marginal supply price of the only marginal

seller, p1s(q1(p1

M)).

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.


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 quasi-linear in one of the two commodities, and the nature of the commodities themselves, one of which is interpreted as a money-like commodity or money tout court.


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.


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.


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.

topics history equilibrium 083

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