Lesson 3 Marshall vs' Walras on Equilibrium and Disequilibrium

1
Lesson 3Marshall vs. Walras on Equilibrium and
Disequilibrium
Franco Donzelli Topics in the History of
Equilibrium Analysis

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

2
Introduction 1

• The problem
• Do Walrass and Marshalls 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

3
Introduction 2

• The working hypothesis
• Walrass and Marshalls 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

4
Introduction 3

• The structure of the presentation
• A common ground for the analysis the
  pure-exchange, two-commodity economy
• Walrass approach
• basic assumptions about the trading process
• the model of a pure-exchange, two-commodity
  economy
• interpretation and textual evidence
• limitations and extensions
• Marshalls approach
• 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

• Walrass Eléments déconomie 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)
• Marshalls 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 (Marshalls stance ambiguous
  on this)

6
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 ?i (?1i, ?2i) ? R2 \ 0
• Let
• x (x1,, xI) ? X xi Xi ? R2I be an
  allocation
• ? ? ?i ? R2 be the aggregate endowments
• Ape2xI x ? X ? xi ? be the set of
  feasible allocations
• Assume
• ui() twice continuously differentiable, with

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The pure-exchange, two-commodity economy 3

• Let
• Epe2xI (Xi, ui(), ?i)Ii1 be a
  pure-exchange, two-commodity economy
• EEB Epe2x2 (R2, ui(), ?i)2i1 be an
  Edgeworth Box economy
• Given Epe2xI, ?x ? Ape2xI, let
• MRSi21(xi) dx2i/dx1iui(xidxi)u(xi)
  (?u(xi)/?u(x1i) / (?u(xi)/?u(x2i)
• be consumer is marginal rate of substitution
  of commodity 2 for commodity 1 when his
  consumption is xi
• Let
• zi(xi) (z1i, z2i)(xi) xi - ?i (x1i - ?1i,
  x2i - ?2i) ? R2
• be consumer is excess demand, when his
  consumption is xi
• If zli(xi) gt 0, then zli(xi) is called consumer
  is net demand proper for commodity I and
  consumer i is said to be a net buyer
• If zli(xi) lt 0, then zli(xi) is called consumer
  is net supply for commodity I and consumer i is
  said to be a net seller

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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 lt MRSi21(xi) ,
• then consumer is 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 gt MRSi21(xi) ,
• then consumer is utility decreases (resp.,
  increases) if he is a net buyer (resp., a net
  seller) of commodity 1

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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 is 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) (v1i(x1i), v2i(x2i)) gtgt
  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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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)) vli(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 Walrass or
  Marshalls approach, however, requires further
  assumptions, which are specific to either
  economist.

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Walrass 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 Walrass original discussion,
  ambiguities not excepted.
• The three assumptions underlie not only the model
  of a pure-exchange, two-commodity economy, but
  all of Walrass 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.

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Walrass 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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Walrass model of a pure-exchange, two-commodity
economy 1

• Let Epe2xI (Xi, ui(), ?i)Ii1 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.

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Walrass 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

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Walrass model of a pure-exchange, two-commodity
economy 3

• From that system one gets consumer is Walrasian
  direct demand and excess demand functions, for i
  1, , I
• xi(p12,?i) and zi(p12, ?i) xi(p12,?i) - ?i
• 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, ?) ?i zi(p12, ?i) ?i xi(p12,?i) - ?i
• be the aggregate demand function, where ?
  (?1,, ?I).
• The market-clearing conditions can be written as
• where p12W is a Walrasian equilibrium price of
  commodity 1 in terms of commodity 2.

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Walrass 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
• 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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Walrass 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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Walrass 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, Walrass 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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Walrass 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 Marshalls 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 Walrass original
  example in his 1874 first theoretical
  contribution, the mémoire entitled Principe
  dune théorie de léchange).
• On this point, however, Walras is very clear.
  For, a few lines after the securities example, he
  adds

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Walrass 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
• 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.

21
Walrass 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 Walrass 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
  Walrass approach.

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Walrass 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 Walrass 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.
• Bertrands critique (1883) and Walrass 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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Walrass 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

24
Walrass 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.

25
Marshalls basic assumptions about the trading
process 1

• Marshall does not assume traders to behave
  competitively (in Walrass 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.
• Marshalls 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 MRSs 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.

26
Marshalls basic assumptions about the trading
process 2

• Let MRS21i(?1i,?2i) (?u(?1i,?2i)/?u(x1i) /
  (?u(?1i,?2i)/?u(x2i) be the initial value of
  consumer is marginal rate of substitution of
  commodity 2 for commodity 1
• Supposing ?j s.t. j ? i and MRS21j(?1j,?2j) ?
  MRS21i(?1i,?2i), let
• kij(?) min MRS21i(?1i,?2i), MRS21j(?1j,?2j)
• and
• Kij(?) max MRS21i(?1i,?2i), MRS21j(?1j,?2j)
• 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(?),Kij(?)
• Marshall assumes that any weakly advantageous
  bargain will be exploited.

27
Marshalls 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(), ?i)2i1 .

28
Marshalls 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.

29
Marshalls 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 Marshalls discussion. Let i
  1,2. Assuming MRS211(?11,?21) ? MRS212(?12,?22),
  let
• k12(?) min MRS211(?11,?21), MRS212(?12,?22)
  lt
• lt max MRS211(?11,?21), MRS212(?12,?22) K12
  (?)
• The Pareto set of EEB is the set

30
Marshalls model of an Edgeworth Box economy 3

• while the contract curve of EEB is the set
• CEB ? Ø. 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
• ,
• being constant over the trading process, would
  qualify as a true equilibrium rate.

31
Marshalls model of an Edgeworth Box economy 4

• Finally, since
• MRSi21(xi) dx2i/dx1iui(xidxi)u(xi)
  (?u(xi)/?u(x1i) / (?u(xi)/?u(x2i),
• for i 1,2, in Marshalls 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.

32
Marshalls 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)

33
Marshalls 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

34
Marshalls model of an Edgeworth Box economy 7

• Let consumer is utility function be quasi-linear
  in commodity 2, that is
• where (?ui(x1i, x2i)/(?x1i)) v1i(x1i),
  assumed positive and decreasing for x1i ? 0,
  ?1, 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)) v1i(x1i)
• depends only on the quantity consumed of
  commodity 1.
• Let
• d1i(x1i,?1i) max 0, x1i - ?1i
• be consumer is net demand proper for commodity
  1 for x1i ? 0, ?1

35
Marshalls model of an Edgeworth Box economy 8

• s1i(x1i,?1i) min 0, x1i - ?1i
• be consumer is net supply of commodity 1 for
  x1i ? 0, ?1
• If x1i gt ?1i, then d1i(x1i,?1i) gt 0 and consumer
  i is a net buyer of commodity 1 hence MRS21i(xi)
  v1i(x1i) can be interpreted as a buyers
  reservation price, or demand price.
• If x1i lt ?1i, then s1i(x1i,?1i) gt 0 and consumer
  i is a net seller of commodity 1 hence
  MRS21i(xi) v1i(x1i) can be interpreted as a
  sellers reservation price, or supply price.
• Consumer is Marshallian inverse supply
  correspondence of commodity 1, p1is 0,?1i ?
  R, is defined as follows
• p1is(s1i) 0,v1i(x1i) for s1i 0
• p1is(s1i) v1i(?1i s1i) for s1i ? (0,?1i) (a
  continuous increasing function)
• p1is(s1i) v1i(0),8) for s1i ?1i

36
Marshalls model of an Edgeworth Box economy 9

• Consumer is Marshallian inverse demand
  correspondence for commodity 1, p1id 0,?1 -
  ?1i ? R, is defined as follows
• p1id(d1i) (8, v1i(?1i) for d1i 0
• p1id(d1i) v1i(?1i d1i) for d1i ? (0, ?1 -
  ?1i) (a continuous decreasing function)
• p1id(d1i) (v1i(?1), 0 for d1i ?1 - ?1i
• 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 is Marshallian direct supply function
  of commodity 1 is the continuous function s1i R
  ? 0,?1i defined as follows
• s1i(p1is) 0 for p1is ? 0, v1i(?1i))
• s1i(p1is) ?1i (v1i)-1(p1is) for p1is ?
  v1i(?1i), v1i(0))
• s1i(p1is) ?1i for p1is ? v1i(0), 8)

37
Marshalls model of an Edgeworth Box economy 10

• Consumer is Marshallian direct demand function
  for commodity 1 is the continuous function d1i
  R ? 0, ?1 - ?1i defined as follows
• d1i(p1id) 0 for p1id ? (8, v1i(?1i)
• d1i(p1is) (v1i)-1(p1id) - ?1i for p1id ?
  (v1i(?1i), v1i(?1)
• d1i(p1id) ?1 - ?1i for p1id ? (v1i(?1),0
• Let p1mind mini v1i(?1) and p1maxd maxi
  v1i(?1i)
• let p1mins mini v1i(?1i) and p1maxs maxi
  v1i(0).
• If the consumers tastes are not identical, then
  p1maxd gt p1mins
• Let d1(p1d) ?i d1i(p1id), for p1d p1id, for i
  1, 2 and p1d ? 0, 8)
• let s1(p1s) ?i s1i(p1is), for p1s p1is, for
  i 1, 2 and p1s ? 0, 8).

38
Marshalls 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
  ?1 ?1i, i 1,2
• s1() is nondecreasing in p1s, and strictly
  increasing for
• p1s ? p1mins, p1maxs, except possibly when s1i
  ?1i, i 1,2
• Hence, supposing p1maxd v1i(?1i) and p1mins
  v1j(?1j), i, j 1, 2 and i ? j, and assuming
  v1i(?1i) lt v1j(0), there must exist a unique
  price
• p1M p1dM p1sM ? (p1mins,p1maxd) s.t.
• or

39
Marshalls 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).

40
Marshalls 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 -
  ?1i) - ½bi(x1i - ?1i)2 x2i, i 1,2.
• Hence MRS21i(xi) v1i(x1i) ai - bi(x1i -
  ?1i) , i 1,2.
• Assumption 2.
• K12(?) MRS211(?1) a1 gt a2 MRS212(?2)
  k12(?)
• Assumption 3.
• ?11 lt ?12 v11(?1) lt v12(0)
• Equilibrium
• p11d(d11) p11s(s12) and d11 s12 . Hence

41
Marshalls model of an Edgeworth Box economy 14
p11d, p11s
p12d, p12s
p1d, p1s
p12s
p11s
p1maxs
s1(p1s)
p1maxd
v11(?11)
p1M
v12(?12)
p1mins
d1(p1d)
p1mind
p11d
p12d
?11
?12
?1
?1 -?11
?1 - ?12
d1, s1
d12, s12
q1M
d11, s11
42
Marshalls model of an Edgeworth Box economy 15

• When the two consumers have already cumulatively
  traded a quantity q1 of commodity 1, such that
  q1 ? 0, q1(p1M)), there still exists a positive
  difference between the demand and the supply
  price of commodity 1 corresponding to q1, that
  is p1d(q1) - p1s(q1) gt 0.
• Hence there still is room for a weakly
  advantageous marginal trade between the two
  consumers, at any rate of exchange
• p1 ? p1d(q1),p1s(q1)
• 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.

43
Marshalls 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
• 
  ,
• where i, j 1,2, i ? j i, j are s.t. v1i(?1i)
  p1mins and v1j(?1j) p1maxd.
• Hence in Marshalls model there exists no
  counterpart of equation (2) in Walrass model,
  where it provides the market-clearing condition
  for commodity 2.
• Further, in Marshalls 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.

44
Marshalls 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(), ?i)Ii1 with I gt2,
• 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
  Marshalls 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
  traders resources (meaningless)

45
Marshalls 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 gt 2,
  is immediate.

46
Marshalls temporary equilibrium model 3

• 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(p1d) ?i d1i(p1id), for p1d p1id, for i
  1, , I, and p1d ? 0, 8),
• and
• s1I(p1s) ?i s1i(p1is), for p1s p1is, for i
  1, , I, and p1s ? 0, 8).
• 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.

47
Marshalls temporary equilibrium model 4

• Marshalls interpretation of equations (8) and
  (9) is essentially the same as that of equations
  (6) and (7). Yet Marshalls claims are not
  entirely justified.
• Marshalls 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)

48
Marshalls 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.

49
Marshalls 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 Igt2, 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.

50
Marshalls 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.

51
Marshalls pure-exchange models limitations and
extensions 1

• Marshalls Edgeworth Box model is propedeutical
  to his temporary equilibrium model, which is in
  turn propedeutical to Marshalls normal
  equilibrium models.
• But, even if propedeutical, Marshalls
  pure-exchange models still play a fundamental
  role in the overall structure of Marshalls
  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.

52
Marshalls 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.

53
Conclusions 1

• In this paper we have contrasted the received
  view on price theory, according to which Walrass
  and Marshalls 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.

54
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.