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NEGISHI ON EDGEWORTH ON JEVONS’S LAW OF INDIFFERENCE, WALRAS’S EQUILIBRIUM, AND THE ROLE OF LARGE NUMBERS:

A CRITICAL ASSESSMENT

FRANCO DONZELLI

Working Paper n. 2011-23

OTTOBRE 2011

Negishi on Edgeworth on Jevons�s Law ofIndi¤erence, Walras�s Equilibrium, and the Roleof Large Numbers: A Critical Assessment

Franco Donzelli�

Department of Economics, Business and StatisticsUniversità degli Studi di Milano

October 3, 2011

1 Introduction

In 1881 Francis Ysidro Edgeworth publishes his path-breaking booklet Math-ematical Psychics (henceforth MP), thereby initiating a tradition that wouldlead, about eighty years later, to the development of a theory of decentralisedexchange based on multilateral interactions, a special branch of the broaderapproach known nowadays as �coalitional�or �cooperative game theory�. In de-veloping his analysis, Edgeworth extensively relies on Jevons�s and Walras�stheoretical systems, as put forward in the former�s Theory of Political Econ-omy (henceforth TPE ; 1st ed. 1871, 2nd ed. 1879) and the latter�s Elémentsd�économie politique pure (henceforth EEPP ; 1st ed. 1874-77). From Chapter 4of Jevons�s TPE Edgeworth derives the model of a two-trader, two-commodity,pure-exchange economy, as well as the equilibrium conditions, i.e., Jevons�s cel-ebrated "equations of exchange", and a "general law of the utmost importancein economics" (Jevons 1970, p. 137), called the "principle of uniformity" in the1st edition of TPE (1871, p. 99) and the "law of indi¤erence" in the 2nd one(1879, p. 90); from Walras�s EEPP, instead, he derives the fully-�edged gener-alisation of the Jevonsian notion of equilibrium to a multi-agent, multi-marketeconomy (Walras 1988, pp. 137-8, 170-3).Yet, though building upon his predecessors�concepts and models, Edgeworth

embeds them in a wider analytical framework, characterised by a seemingly moregeneral solution concept, which encompasses the Jevonsian or Walrasian equi-librium notion as special cases: the "set of �nal settlements", representing thesolution concept of Edgeworth�s MP model of a pure-exchange economy, can be

�I thank Claire Pignol and Antoine Rebeyrol for their insightful comments on preliminarydrafts of this paper and their invaluable advice in talks and correspondence. The usualdisclaimer applies.

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be viewed as the �rst instance of the fundamental solution concept underlyingmodern coalitional game theory, namely, the �core�of a coalitional game. As iswell-known, Edgeworth is able to prove that the Walrasian equilibrium of a pure-exchange economy, assumed unique, belongs to the contract curve; moreover,by jointly relying on his �recontracting�and �replication�mechanisms, he is alsoable to prove that the contract curve shrinks to the Walrasian equilibrium as thenumber of traders in the economy grows unboundedly large. This limiting resultcan indeed be viewed as buttressing the Walrasian or competitive equilibriumconcept, for it provides yet another game-theoretic foundation for it; however,the same result is also viewed by Edgeworth as rigorously limiting the possibleuse of the Walrasian equilibrium machinery, which includes not only the equilib-rium concept as such, but also the law of indi¤erence or, what is the same, theprinciple of price uniformity or also the �law of one price�, as that principle isoccasionally called in the Walrasian context. According to Edgeworth, in fact,the scope of application of the Walrasian or competitive equilibrium appara-tus should be strictly con�ned to unboundedly large economies, i.e., economieswhere the traders�number is "practically in�nite", for only such economies canbe legitimately regarded as "perfectly competitive". On the contrary, �niteeconomies are characterised by limited competition; moreover, whenever thetraders� number is short of the "practically in�nite", the law of indi¤erencecannot be supposed to hold and the Walrasian equilibrium price and allocationcannot be taken to faithfully describe the actual state of the economy.In 1982, almost exactly one century after the appearance of MP, Tagashi

Negishi, one of the most productive general equilibrium theorists of the sec-ond half of the last century, publishes the unconventional paper "A Note onJevons�s Law of Indi¤erence and Competitive Equilibrium", where he tries toexploit Edgeworth�s analytical weaponry against Edgeworth�s own conclusions,as summarised above. Precisely, Negishi endeavours to prove that, by dulyrevising and extending Edgeworth�s concept of coalition and by consequentlyreinterpreting the associated recontracting and replication mechanisms, one canuncover the driving force tacitly underlying Jevons�s law of indi¤erence, namely,that arbitrage mechanism that must be taken to be at work behind any ten-dency towards price uniformity. In Negishi�s opinion, such mechanism, far fromrequiring, as supposed by Edgeworth, the existence of in�nitely many tradersto carry its e¤ects through, is fully e¤ective in any �nite economy, even in avery small one, provided that there exists some room for arbitraging; but anumber of traders greater than two is the only requirement on which the chanceof undertaking arbitraging activities depends - a simple requirement apparentlyneglected by Jevons in his original discussion in TPE, where the law of indi¤er-ence is supposed to hold true even in a two-trader economy.Negishi�s conclusions are indeed far-reaching and even surprising, for, if

proven true, they would subvert the tenet, well-established in the literature,that assuming perfect competition and relying on the Walrasian equilibrium ma-chinery are really justi�ed in large economies only. Yet, in spite of the sweepingcharacter of Negishi�s claims, his argument has been essentially ignored for al-most two decades after its appearance; only at the turn of the century Negishi�s

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viewpoint has been approvingly resumed by a few scholars (Rebeyrol 1999,Pignol 2000). Yet, in spite of such revival, Negishi�s critique of Edgeworth�sapproach has not yet been thouroughly assessed in the literature. The aim ofthe present paper is to �ll this gap.The paper is structured as follows. Section 2 will introduce the analytical

and diagrammatical apparatus required for later purposes, that is, the modelof a pure-exchange, two-commodity, two-trader economy with cornered traders,which is the simple model actually employed by Jevons, Edgeworth, and Negishito develop their respective ideas about the law of indi¤erence and competitiveequilibrium; the same model, though simpler than the simplest pure-exchangemodel put forward by Walras in EEPP, can nevertheless be so adjusted as toaccount for Walras�s equilibrium apparatus, too. In Section 3, after discussingEdgeworth�s �recontracting�mechanism, we shall examine his "theory of simplecontract", that is, his model of a pure-exchange, two-commodity, two-tradereconomy. In Section 4, after discussing Edgeworth�s replication mechanism, weshall analyse his reinterpretation of both Jevons�s law of indi¤erence and Wal-ras�s equilibrium concept as properties of unboundedly large economies. Section5 will summarise Negishi�s critique of Edgeworth�s approach. In Section 6 weshall critically discuss Negishi�s conjecture that no large number of traders isrequired to elicit price uniformity and competive equilibrium, provided that ar-bitrage opportunities are allowed for and duly taken into account. Section 7concludes.

2 The model of an Edgeworth-Box economy

Let us consider a pure-exchange economy with a �nite number L = 2 of com-modities, denoted by l = 1; 2, and a �nite number I = 2 of consumers-traders(henceforth indi¤erently referred to as either consumers or traders), denotedby i = A;B. Each consumer i is characterized by a consumption set Xi =fxi � (x1i; x2i)g = R2+, a utility function ui : Xi ! R, and endowments!i � (!1i; !2i; ) 2 R2+n f0g. Let x = (xA; xB) 2 X = �i=A;BXi � R2�2+

be an allocation;_! = (�!1; �!2) =

Pi=A;B !i 2 R2++ be the aggregate endow-

ments; A2�2 =nx 2 X j

Pi=A;B xi =

_!obe the set of feasible, non-wasteful

allocations. A pure-exchange, two-commodity, two-consumer economy with theabove characteristics will be called an Edgeworth-Box economy, so named byBowley (1924) after Edgeworth (1881), and will be denoted in the following by

E2�2 =n�R2+; ui (�) ; !i

�i=A;B

o. The consumers�characteristics (consumption

sets, utility functions, endowments) represent the data of the Edgeworth-Boxeconomy; they are assumed to be �xed up until the exchange problem is solvedand an equilibrium (or, more generally, a solution) is established. The periodover which the data remain �xed will be referred to as a �trade round�. In confor-mity with standard usage in contemporary models of an Edgeworth-Box econ-omy, the traders�utility functions are assumed to be continuously di¤erentiable,strongly monotonic, and strictly quasi-concave. Further, in accordance with

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Jevons�s and Edgeworth�s original assumptions, the traders are supposed to becornered, that is, they are supposed to hold a positive quantity of one commodityonly; speci�cally, in the following we shall assume: !A = (!1A; !2A) = (�!1; 0)and !B = (!1B ; !2B) = (0; �!2). (The assumption of cornered traders applies toWalras�s most elementary pure-exchange model, too.)The evolution of the economy E2�2 over continuous time must be viewed

as a sequence of disconnected, discrete-time trade rounds, each characterisedby its own data and, hopefully, its own equilibrium (or solution). Since eachtrade round must be viewed as self-contained, there cannot be any carry-overof endowments from one trade round to the next; at the same time, since thedata must not change over each trade round, the endowments must not wearo¤ or be consumed up until an equilibrium (or a solution) is reached; so that,in the end, the endowments must be perfectly durable over a trade round andperfectly perishable when the trade round is over (see Hicks 1989, pp. 7-11).The model of an Edgeworth-Box economy speci�ed above can be graphi-

cally represented by means of the homonymous diagram, where the lengths ofthe sides of the rectangle are respectively given by the aggregate endowments,�!1 and �!2, and each point in the rectangle represents a feasible, non-wastefulallocation x = (xA; xB) (see Figure 1 below).

Figure1

By employing the analytical and diagrammatical apparatus that has just

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been introduced, let us now brie�y review the solutions respectively providedby Jevons and Walras to the equilibrium determination problem.Starting with Jevons, let us �rst consider what he calls an "act of exchange"

between the two traders (Jevons 1970, pp. 138-9). Such an act involves thetrade of either "in�nitely small" or "�nite" quantities of the two commodities:it may be called �di¤erential� in the former case and ��nite� in the latter. Ineither case the quantity of the commodity given in exchange will be taken tobe negative (that is, dxli < 0 or �xli < 0, if commodity l is given by traderi, for i = A;B and l = 1; 2), while the quantity of the commodity received inexchange will be taken to be positive (that is, dxli > 0 or �xli > 0, if com-modity l is received by trader i, for i = A;B and l = 1; 2). Since an act ofexchange is necessarily bilateral, the vectors of the quantities traded satisfy thefollowing conditions: (dx1A; dx2A) = �(dx1B ; dx2B), if the act is di¤erential;(�x1A;�x2A) = �(�x1B ;�x2B), if the act is �nite. Finally, let dxl = jdxlij(resp., �xl = j�xlij) be the absolute value of the quantity exchanged of com-modity l in a di¤erential (resp., �nite) act of exchange. Then, still following

Jevons (1970, pp. 138), a ratio of the type dx2dx1

=���dx2Adx1A

��� = ���dx2Bdx1B

��� > 0 (resp.,

�x2�x1

=����x2A�x1A

��� = ����x2B�x1B

��� > 0) will be called a di¤erential (resp., �nite) "ratio

of exchange". As we shall see, a special kind of �nite "ratio of exchange" playsa fundamental role in Jevons�s theory of exchange: it is the �nite "ratio" x2

x1,

where x1 = ��x1A = �(x1A � !1A) = �!1 � x1A = �x1B = x1B � !1B = x1Band x2 = ��x2B = �(x2B � !2B) = �!2 � x2B = �x2A = x2A � !2A = x2A.Before moving to the equilibrium characterisation issue, we need to introduce

one further concept, which will prove useful in the following discussion not onlyof Jevons�s theoretical model, but also of Walras�s, Edgeworth�s, and Negishi�s.Given an Edgeworth-Box economy satisfying the above assumptions on endow-ments and utilities, let us de�ne consumer i�s marginal rate of substitution ofcommodity 2 for commodity 1 when i�s consumption is xi, MRSi21(xi), as thequantity of commodity 2 that consumer i would be willing to exchange for oneunit of commodity 1 at the margin, in order to keep his utility unchanged atthe original level ui(xi). From this de�nition it follows that:

MRSi21(xi) �����dx2idx1i

����ui(xi+dxi)=ui(xi)

=

@ui(xi)@x1i

@ui(xi)@x2i

; i = A;B.

In their writings Jevons, Walras, and Edgeworth ignore the notion of the mar-ginal rate of substitution. Yet, they do know and systematically employ thenotion of the marginal utility of commodity l for consumer i, which, under thestated assumptions on the properties of the utility functions, is well-de�ned andbounded away from zero everywhere in the consumption set. Moreover, thoughnot explicitly discussing the concept of the marginal rate of substitution as such,they do implicitly make use of it in their analyses, since they compute the ratiosof the values of the marginal utility functions of each consumer correspondingto speci�c consumption bundles and examine the role of such ratios in solvingthe exchange equilibrium problem.

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Now, in facing the equilibrium characterisation issue, Jevons (1970, pp. 139-140) starts by asking "at what point the exchange will cease to be bene�cial"for the traders. His answer is as follows: for each trader i, given an arbitrarily

"established [di¤erential] ratio of exchange", say

����� ^dx2idx1i

�����, the point at which theexchange ceases to be bene�cial for i, called the "point of equilibrium" by Jevons,is identi�ed by the condition that the given di¤erential ratio be equal to traderi�s marginal rate of substitution evaluated at the equilibrium point,MRSi21(x̂i),that is:

������^

dx2idx1i

������ =MRSi21(x̂i) =@ui(x̂i)@x1i

@ui(x̂i)@x2i

; i = A;B. (1)

It should be noted that Jevons, after deriving the individual optimisationconditions (1) for either trader separately (1970, p. 142), never thinks of com-bining them into a single equation, so that, in spite of Edgeworth�s overgenerous,yet unfounded, acknowledgement of Jevons�s priority (Edgeworth 1881, p. 21),he never obtains an equation of the following type:

MRSA21(xCA) =

@uA(xCA)@x1A

@uA(xCA)@x2A

=

@uB(xCB)@x1B

@uB(xCB)@x2B

=MRSB21(xCB), (2)

which, together with the feasibility conditions,

xCA + xCB = !A + !B = �!, (3)

would de�ne the "contract curve" (or the �Pareto set�) of the Edgeworth-Boxeconomy concerned. In Fig. 1 the "contract curve", so named by Edgeworth(1881, p. 21), as will be seen in the next Section, is the curve connecting OAand OB .Now, equations (1) and (3) are not su¢ cient to make the model determinate,

for one has just four equations to determine six unknowns�x1A; x1B ; x2A; x2B ;

dx2Adx1A

; dx2Bdx1B

�.

It is precisely at this point that the law of indi¤erence comes to rescue, for itprovides the two further equations that are needed to close Jevons�s model.In this regard, it is worth recalling that Jevons had concluded the section ofChapter 4 of TPE on "The Law of Indi¤erence" with the following sentence:

Thus, from the self-evident principle, stated on p. 137 [i.e., the lawof indi¤erence], that there cannot, in the same market, at the samemoment, be two di¤erent prices for the same uniform commodity,it follows that the last increments in an act of exchange must beexchanged in the same ratio as the whole quantities exchanged. [...]

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This result we may express by stating that the increments concernedin the process of exchange must obey the equation

dx2dx1

=x2x1. [�]

The use that we shall make of this equation will be seen in the nextsection. (Jevons 1970, p. 139; Jevons�s italics)

Now, the only analytical use made by Jevons of the law of indi¤erence, i.e., ofequation [�], is to allow the theorist to replace the di¤erential ratio of exchange,dx2dx1, by the �nite one, x2x1 , thereby obtaining the missing equations needed to

close his model. In our formalism, account being taken of (3), such equationscan be written as follows:

����dx2Adx1A

���� = ����dx2Bdx1B

���� = dx2dx1

=x2x1=�!2 � x2Bx1B

=x2A

�!1 � x1A. (4)

By substituting (4) into (1) and simplifying, one �nally obtains:

@uA(�!1�xJ1 ;xJ2 )

@x1A

@uA(�!1�xJ1 ;xJ2 )@x2A

=xJ2xJ1

=

@uB(xJ1 ;�!2�xJ2 )

@x1B

@uB(xJ1 ;�!2�xJ2 )@x2B

, (5)

which are Jevons�s "equations of exchange" (1970, p. 143), de�ning theequilibrium allocation, xJ =

�xJA; x

JB

�=���!1 � xJ1 ; xJ2

�;�xJ1 ; �!2 � xJ2

��, as well

as the equilibrium "ratio of exchange", or relative price of commodity 1 in terms

of commodity 2, xJ2

xJ1.

Under the stated assumptions, a Jevonsian equilibrium�xJ2xJ1; xJ

�can be

proven to exist, even if it is not necessarily unique. Yet, in order to simplify theexposition, let us suppose the equilibrium to be unique. Then the equilibriumact of exchange can be graphically represented in Fig. 1 by means of the thickstraight line segment connecting the endowment allocation ! with Jevons�s equi-librium allocation xJ . The equilibrium "ratio of exchange", in turn, is given bythe absolute value of the constant slope of such straight line segment (tan�J).Let us now move to Walras�s pure-exchange, two-commodity model. Unlike

Jevons, Walras need not restrict his analysis to an economy with only twotraders: as a matter of fact, the traders� number is supposed from the startto be any arbitrary integer (Walras 1988, pp. 74, 136-7). Yet, in order tomake Walras�s model more easily comparable with Jevons�s and Edgeworth�s,let us suppose that only two cornered traders, once again denoted by i = A;B,participate in the pure-exchange economy under discussion. The traders aresupposed to have the same charcteristics as before.

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Walras assumes the traders to behave competitively: precisely, they are sup-posed to take prices as given parameters and to choose their optimal trade plans,called "dispositions à l�enchère" or "au rabais" (Walras 1988, pp. 70, 83), insuch a way as to maximize their respective utility functions, under their respec-tive Walrasian budget constraints. Then let p = (p12; 1) 2 R2++ be the pricesystem, expressed in terms of commodity 2 taken as the numeraire. Since inthe economy there are only two commodities, there can only exist one indepen-dent relative price, p12 � 1

p21. The assumed existence of uniform prices, the

same for all the traders and made known to all of them under all circumstances,expresses the particular version of the law of indi¤erence that is supposed torule in the Walrasian theoretical system; as already recalled, such version ofthe law is often referred to as the �law of one price�in this context. It shouldbe stressed, however, that while Jevons�s law of indi¤erence is assumed to holdonly at equilibrium, of which it is a distinctive feature, Walras�s law of one priceis assumed to hold both at equilibrium and out of equilibrium.For all p = (p12; 1), the optimization problem to be solved by trader i can

be formally written as:

MRSi21(x1i; x2i) =

@ui(xi)@x1i

@ui(xi)@x2i

= p12, (6)

p � xi = p � !i, (7)

where equation (6) is Walras�s "condition de satisfaction maxima" for traderi and equation (7) is the Walrasian budget constraint of the same trader, fori = A;B (Walras 1988, pp. 111-7).By solving this system, one obtains for each trader the individual gross de-

mand functions for both commodities, from which one can derive the individualnet demand and supply functions for either commodity, that is:

xA(p12) = (x1A(p12); x2A(p12)) ,

s1A(p12) = �!1A � x1A(p12),d2A(p12) = x2A(p12),

and

xB(p12) = (x1B(p12); x2B(p12)) ,

d1B(p12) = x1B(p12),

s2B(p12) = �!2B � x2B(p12).

Then, by aggregating over the traders, one obtains the aggregate gross de-mand functions, as well as the aggregate net demand and supply functions,from which one can derive the aggregate excess demand functions for eithercommodity, that is:

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z1(p12) = �i=A;B (x1i(p12)� !1i) = x1(p12)� �!1 = d1B(p12)� s1A(p12)z2(p12) = �i=A;B (x2i(p12)� !2i) = x2(p12)� �!2 = d2A(p12)� s2B(p12)

Finally, by setting the aggregate excess demand functions equal to zero, oneobtains the market clearing equations, one for each commodity:

zl(pW12) = xl(p

W12)� �!l = 0, l = 1; 2. (8)

Yet, given Walras� Law, that is, p � z = p12z1(p12) + z2(p12) � 0, 8p =(p12; 1) 2 R++ � f1g, only one equation provides an independent equilibriumcondition. By solving either equation, therefore, we obtain the Walrasian equi-librium relative price, pW12 . TheWalrasian equilibrium allocation, x

W =�xWA ; x

WB

�=��

xW1A; xW2A

�;�xW1B ; x

W2B

��, can then be obtained by plugging the Walrasian equi-

librium price into the individual gross demand functions, that is,�xWi�=�

xi�pW12��, for i = A;B (Walras 1988, pp. 136-7). Under the stated condi-

tions, a "solution" exists, even if it is not necessarily unique (Walras 1988, p.97). Yet, for simplicity, we shall assume uniqueness in the following. The Wal-rasian equilibrium allocation xW can be plotted in the Edgeworth-Box diagramdrawn in Fig. 1 above. The ray from ! through xW is the equilibrium budgetline for either trader, the absolute value of its slope (tan�W ) representing theWalrasian equilibrium price pW12 .The method of "solution" suggested above is "analytique", for it consists

in �nding the roots of a system of ordinary algebraic equations. But one canalso restate the problem in geometrical terms, that is, one can give it "la formegéométrique", by drawing for each commodity the demand and supply curvesand �nding their intersection. Alternatively, one can plot the graphs of the grossdemand functions of the two traders, xA(�) and xB(�), in the Edgeworth-Boxdiagram of the economy under question; such graphs, called "o¤er curves" afterJohnson (1913, p. 487), are labelled oA and oB in Fig. 1 above. Then �ndingthe intersection of the two graphs provides another geometrical method foridentifying the Walrasian equilibrium allocation and, implicitly, the Walrasianequilibrium price, too.As can be seen, in order to identify the equilibrium allocation and price

of the Edgeworth-Box economy, Walras follows a path that signi�cantly di¤ersfrom the path followed by Jevons in pursuing the same purpose: in particular,while prices and excess demand functions (or o¤er curves) play a fundamentalrole in Walras�s theoretical system, nothing comparable can be found in Jevons�stheory of exchange, as Walras himself does not fail to point out in both an openletter he addresses to Jevons in 1874 (Walras 1993, p. 50) and a well-knownpassage of EEPP, from the second edition onwards (1988, pp. 251-3, 2-5).Yet, in spite of the di¤erent routes taken, the �nal outcome is the same, for,assuming uniqueness, Walras�s equilibrium allocation coincides with Jevons�s,i.e., xW = xJ , and Walras�s equilibrium relative price coincides with Jevons�s

equilibrium ratio of exchange, i.e., pW12 = tan�W = tan�J =

xJ2xJ1.

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3 Edgeworth�s theory of simple contract

The point departure for Edgeworth�s discussion of the exchange problem inMP is precisely represented by Jevons�s "theory of exchange", as developed inChapter 4 of TPE (Edgeworth 1881, pp. 20, 39, and App. V, pp. 104-16).Yet, instead of accepting Jevons�s basic idea that an exchange model involv-ing just two "trading bodies" is su¢ cient to cover the whole range of a prioripossible trade situations, Edgeworth clearly distinguishes from the very starttwo separate theories: the "theory of simple contract", which concerns just twoindividual traders, and the "theory of exchange" proper, which deals insteadwith a greater number of traders, more or less competing with each other (1881,pp. 29, 31, 109). Edgeworth�s fundamental conjecture is that the intensityof competition among the traders increases with their number; moreover, thegreater the competition among the traders, the more determinate is the outcomeof the trading process; as a consequence, "perfect competition" and "perfectlydeterminate" outcome can only obtain when the traders�number is "practicallyin�nite". In between the two extremes of an "isolated couple" of traders, on theone hand, and a "perfect" market populated by a "practically in�nite" num-ber of traders, on the other, there is of course a whole range of intermediatesituations (1881, pp. 20, 39, 146-7). So, right at the beginning of MP, Edge-worth raises his fundamental question: "How far contract is indeterminate".The "general answer" he is ready to o¤er is as follows:

(�) Contract without competition is indeterminate, (�) Contractwith perfect competition is perfectly determinate, ( ) Contract withmore or less perfect competition is less or more indeterminate. (Edge-worth, 1881, p. 20)

The above statement might appear to suggest that, from Edgeworth�s view-point, the "theory of simple contract", the "theory of exchange under less thanperfect competition", and the "theory of exchange under perfect competition"should be regarded as altogether di¤erent theories, based on alternative princi-ples. Yet, this conclusion would be wholly unfounded, for all the above men-tioned theories rest on the same "fundamental" principle, namely, the "principleof recontract" (Edgeworth 1925b, p. 312), whose general operating rules are setat the beginning of MP (1881, pp. 16-20), well before the detailed discussion ofeither the "theory of simple contract" or the "theory of exchange" under "moreor less perfect competition" is developed. In Edgeworth�s view, it is preciselythe recontracting mechanism which ensures the convergence of any economy,irrespective of its size, to some allocation in a suitably de�ned solution set. Thesize of the solution set, hence the determinateness of the solution, is inverselyrelated to the size of the economy, as measured by the number of traders; butthe de�ning properties of the solution are always the same, for they hinge onthe universal principle of recontract.By resorting to the language of modern coalitional game theory, let us now

introduce a few concepts that help formalising the working of the "process of

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recontract". A �coalition�is a non-empty subset of traders. A coalition �blocks�an allocation if, by re-allocating the aggregate endowments owned by the coali-tion members among the coalition members themselves, it obtains a coalitionallocation, called �blocking allocation�, which ensures a utility level at least asgreat as that granted by the original allocation to all the coalition members,and a greater utility level to at least one of them. Finally, the �core� of aneconomy is the set of all the allocations that cannot be blocked by any coali-tion. According to Edgeworth�s rules of recontracting, coalitions can freely formand dissolve without cost or penalty. When the traders�number increases, thenumber of coalitions that can possibly be formed, hence the number of blockingallocations, grows exponentially; in the last analysis, this is the reason why thecore shrinks and the determinateness of the solution increases with the traders�number.Let us start by considering the �theory of simple contract�. In developing

this theory, Edgeworth employs the very same model of a two-commodity econ-omy with two cornered traders as the one originally discussed by Jevons. Hence,Edgeworth�s pure-exchange, two-trader, two-commodity economy, E2�2, can bedescribed by means of an Edgeworth-Box diagram as before1 . Unlike Jevons,however, Edgeworth (1881, pp. 21) precisely de�nes the contract curve, which,as already explained, is the set of the allocations satifying equations (2) above,that is, the set of Pareto-optimal allocations, denoted by

�xC =

�xCA; x

CB

�.

Edgeworth (pp. 19, 29) also identi�es the "set of �nal settlements", which isthe subset of the contract curve contained in the �area of exchange�, i.e., inthe lens-shaped region of the plane enclosed between the traders� indi¤erencecurves passing through the endowment allocation. The allocations belonging tothe area of exchange are the feasible allocations that also satisfy the individualrationality constraints. Letting

�xE =

�xEA; x

EB

�j xEA + xEB = !A + !B = �!;uA

�xEA�� uA (!A) ; uB

�xEB�� uB (!B)

denote the set of the allocations belonging to the area of exchange, the set

of �nal settlements coincides with the intersection of the contract curve andthe area of exchange,

�xC =

�xCA; x

CB

�\�xE =

�xEA; x

EB

�. Edgeworth�s set of

�nal settlements corresponds to what, in the current terminology of coalitionalgame theory, would be called the core of the Edgeworth-Box economy: for itis the set of all the allocations that cannot be blocked by any of the threecoalitions that can possibly be formed in a two-trader economy, namely, the�grand coalition� or �coalition of the whole�, fA;Bg, and the two individualcoalitions, i.e., the singletons fAg and fBg. In Fig. 2 below, the contract curve

1The diagrams of the pure-exchange, two-trader, two-commodity economy actually appear-ing in MP (1881, pp. 28, 114) are drawn in the four-coordinate space of the traders�net trades,rather than in the four-coordinate space of the traders�allocations, as is common nowadays,in the wake of Pareto�s introduction of the latter kind of diagram in 1906. In this paper wehave preferred to keep to the currently prevailing representation. In any case, the choice ofwhich representation to adopt is immaterial, since there exists a one-to-one correspondencebetween the two.

11

is the locus connecting the origins OA and OB , while the set of �nal settlementsis the portion of the contract curve going from x1r to x1l, where x1r and x1l

are respectively the right and the left point of intersection between the contractcurve and the traders�indi¤erence curves through the endowment allocation2 .

Figure 2

According to Edgeworth, in the model of an Edgeworth-Box economy, the�[�nal] settlements [...] are represented by an inde�nite number of points�(Edgeworth, 1881, p. 29). Such indeterminateness is the hallmark of the theoryof simple contract. The reasons for this can be easily spelled out. Accordingto Edgeworth�s trading rule, the two traders will trade if and only if their actsof exchange are (weakly) mutually advantageous, that is, if they increase theutility of at least one of them, without decreasing the utility of either trader.As before, the acts of exchange may be �nite or di¤erential. In the wake ofFeldman (1973, p. 465), let us call such acts "bilateral trade moves". Now,if MRSA21 (!A) 6= MRSB21 (!B), there are in�nitely many bilateral trade movesthat can be carried out by the traders at the start of the bargaining process.A similar situation, however, would arise over and over again at each alloca-tion in the area of exchange that might be reached after any move is e¤ected,

2The superscript 1 is there to remind the reader that the Edgeworth-Box economy underquestion is an unreplicated two-trader economy. In the next Section, in discussing Edgeworth�sreplication mechanism, we shall �nd it convenient to identify the unreplicated Edgeworth-Boxeconomy, E2�2, with the �ctitious 1-replica economy, E2�21 .

12

as long as the traders�marginal rates of substitution evaluated at that alloca-tion diverge. In view of this, lacking any further detailed information aboutthe traders�knowledge, attitudes, skills, etc., the actual path followed by thetwo traders during the bargaining process cannot be speci�ed: for it dependson idiosyncratic circumstances which are speci�c to either trader and, as such,irreducible to general rules or systematic theorizing (Edgeworth 1881, pp. 29-30). What can be stated for sure, under the trading rule speci�ed above, isthat: 1) the process will come to an end (perhaps in the limit), and 2) its endpoint, that is, the �nal settlement eventually arrived at through the bargainingprocess, will be such that the traders�marginal rates of substitution evaluatedat that point are the same. Yet, if the path cannot be speci�ed, its end pointcannot be identi�ed either: the path will certainly end up somewhere in thecore of the Edgeworth-Box economy, namely in the set

�xC\�xE, but which

speci�c allocation is eventually reached cannot be theoretically predicted.The idea that no general deterministic dynamical theory of the path followed

by the trading process can possibly be put forward in the short run, or evenhoped for in the long run, is reiterated by Edgeworth over and over again, frombeginning to end of his long scienti�c life, not only with reference to the theoryof simple contract presently under discussion, but also, as we shall see, withreference to the theory of exchange proper, dealing with any large economy,characterised by any number of traders short of the "practically in�nite" (see,e.g., Edgeworth 1889a, p. 268; 1891a, pp. 364, 365-6; 1904, pp. 39-40; 1925, p.311).Now, granting that the theorist can produce no general theory of the path

from the initial endowment to the �nal settlement in the core of the economy,where the process eventually ends, the most she can do is to indicate "par-ticular paths [...] by way of illustration, to ��x the ideas�, as mathematicianssay" (Edgeworth 1904, pp. 39-40). Following Edgeworth�s lead, let us thenundertake to provide an "illustration" of a "particular path" described by thetrading process taking place in the Edgeworth-Box economy E2�2 (see Fig. 2above). In order to ��x the ideas�, let us suppose that the bilateral trade movesare �nite, so that a �nal settlement is reached after a �nite sequence of moves,say after � 2 N moves, where � > 1. Let t0 2 Z denote the trade round (a�nite time interval) over which the trading process is supposed to take place.Each move is indexed by � = 1; :::;�. The �rst move takes the economy fromthe endowment allocation, x0t0 = !t0 , to the allocation x

1t0 ; the generic move �

carries the economy from x��1t0 to x�t0 ; the last move takes the economy fromx��1t0 to x�t0 . The broken line connecting the successive allocations in the �nite

sequence�x�t0���=0

describes the path followed by the traders during the tradingprocess. The absolute value of the slope of the straight line segment connectingany two successive allocations in the sequence

�x�t0���=0

de�nes the rate of ex-change implicit in the move leading to the allocation located at the end of thesegment: so, for the generic pair of allocations x��1t0 and x�t0 , the rate of exchange

implicit in the �-th move is given by�x�2t0�x�1t0

=

����x�2At0�x��12At0

x�1At0�x��11At0

���� = ����x�2Bt0�x��12Bt0

x�1Bt0�x��11Bt0

���� � 0.13

Similar results would obtain in case the bilateral trade moves were in�nitesimal,rather than �nite: in that case, assuming the path followed during the tradingprocess to be described by a smooth curve with a continuously changing slope,

everywhere well-de�ned, then the in�nitesimal rate of exchangedx�2t0dx�1t0

, ruling at

each allocation x�t0 reached during the process, would coincide with the absolutevalue of the slope of the curve at that point (Edgeworth 1881, pp. 39, 105).According to Edgeworth, the rate of exchange implicit in any trade move can,and generally does, vary along the path, in a way that is largely unpredictable.The only property of the rate of exchange which can be precisely predictedis that the rate of exchange implicit in the last move �, that is, in the moveleading to the �nal settlement x�t0 , must equal the common value of the mar-ginal rates of substitution of the two traders evaluated at that point: namely, if

the last move is �nite (resp., in�nitesimal), one must have�x�2t0�x�1t0

(resp.,dx�2t0dx�1t0

)

=MRSA21t0(x�t0) =MRS

B21t0(x

�t0) (1881, pp. 21-3, 109).

Referring to the above illustration, the following questions naturally arise:Are the bilateral trade moves plotted in Fig. 2 observable or unobservable? Aresuch moves e¤ectively carried out or just discussed by the two traders in a sortof preliminary �cheap talk�session? Are they successive in time or, being part ofa verbal session preceding any implementation, essentially simultaneous? Thethree dilemmas are of course related. Moreover, they have all to do with theinterpretation of the time index �: Does such index run over the entire traderound t0, so that � must be interpreted as a �real� time index, recording thesequence of observable transactions, or does it run over a separate, preliminarysession, so that � must be interpreted as a �logical�time index, recording thesequence of mental ruminations and verbal exchanges?In MP, nature and timing of the trading process are not discussed in any

detail, as Edgeworth himself will openly acknowledge ten years later, at least asfar as the �time� factor is concerned (1891a, p. 367). Moreover, Edgeworth�soccasional remarks about these characteristic features of the bargaining processdo not generally distinguish between the theory of simple contract and thetheory of exchange proper, making the interpreter�s task even more di¢ cult.But even the few passages scattered across MP which seem unambiguously torefer to the theory of simple contract only (1881, pp. 24-5, 28-30, 109, 115-6)do not help solving the interpretative dilemmas listed above.All doubts left open by MP, however, are dispelled ten years later by a

paper in Italian, published by Edgeworth in 18913 , where the author criticallydiscusses Marshall�s theory of barter, as stated in the Appendix entitled �ANote on Barter�, placed at the end of Book V, chapter II, of the �rst edition of

3See Edgeworth (1891b). A partial English "translation" of the 1891b paper in Italian(most probably taken from Edgeworth�s original English version of his 1891 paper) appearsin Volume II of Edgeworth�s Papers Relating to Political Economy (see 1925d). Yet, for ourpresent concerns, what really matters is the full Italian version, seemingly ignored by latercritics and interpreters, rather than the abridged English re-issue.

14

Marshall�s Principles of Economics, which had appeared the year before4 .In his "Note", only three-page long, Marshall had been able to develop his

theory of barter by making use, as is customary for him, of an illustration basedon a numerical example: in this case, the example concerns two traders, A andB, trading "apples" for "nuts". He had then employed a special case of suchtheory - a case arising when one of the two commodities, say nuts, is supposedto share what Marshall believes to be the most characteristic trait of money,namely, the constancy of its marginal utility - as a starting point to deal with themore general issue of the establishment of the so-called "temporary" or "marketequilibrium of demand and supply", which is precisely the subject-matter of thechapter of the Principles to which the "Note" was originally appended (the titleof that chapter being in fact "Temporary Equilibrium of Demand and Supply").Once again, Marshall�s discussion had been couched in terms of a numericalillustration concerning the "corn market in a country town", where an arbitrary�nite number of traders (greater than two) is supposed to participate in theexchanges (Marshall 1961a, pp. 331-6).In his 1891 paper Edgeworth carefully analyses Marshall�s theory of barter,

focussing especially on Marshall�s conclusion that, under the special assump-tion of constancy of the marginal utility of one of the two traded commodities(nuts), the bargaining process ends up in a perfectly determinate outcome: thisconclusion, in fact, is apparently so much at variance with Edgeworth�s conjec-ture about the indeterminateness of the theory of simple contract as to stir hiscritical reaction. Edgeworth�s paper gives rise to a lively controversy, turningparticularly on the issue of the determinateness of equilibrium and involving, atMarshall�s instigation, the Cambridge mathematician Berry, to whose criticism(1891) Edgeworth immediately replies with a short rejoinder (1891c)5 . Here weare only interested in explaining how the barter controversy helps Edgeworthclarifying his own ideas about the nature and timing of the bargaining processin the Edgeworth-Box model.At the very start of his 1891b paper, Edgeworth is ready to praise Marshall�s

theory of barter for avoiding

the common error of attributing to two persons who are bargainingwith each other a �xed rate of exchange governing the whole trans-action. A uniform rate of exchange, he remarks, is applicable onlyto the case of perfect market. (Edgeworth 1925d, p. 315; see also1891b, p. 234)

He then proceeds to discuss the numerical illustration of the barter processput forward by Marshall in his �Note�, formalising it by means of an Edgeworth-

4As explained by Guillebaud�s editorial notes to the ninth (variorum) edition of the Prin-ciples (Marshall, 1961b, p. 790-1), from the �fth edition (1907) onwards, the subject-matterof the Appendix "A Note on Barter" was moved, without any substantial alteration, from theend of Book V, chapter II, to the �nal section of the Principles, grouping all the Appendices,where it became "Appendix F. Barter" (Marshall, 1961a, p. 791-3).

5The exchanges of letters between Marshall, Berry and Edgeworth, reprinted in Marshall(1961b, pp. 791-8), should also be consulted. On the controversy see Newman (1990), Creedy(2006), and Donzelli (2008, pp. 15-30, and 2009, pp. 32-6).

15

Box diagram plotted in the space of net trades (Edgeworth, 1925d, p. 316, Fig.1; 1891b, p. 236, Fig. 1), where there appears a �series of short lines�, thatis, a �broken line�, exactly corresponding to the broken line drawn in Fig. 2above. Commenting upon the meaning of such drawing, Edgeworth claims thatit �corresponds to successive barters (at di¤erent rates of exchange) of a few nutsfor a few apples� (1925d, p. 316; 1891b, p. 2366). Now, as the last sentenceconclusively shows, Edgeworth, when directly referring to Marshall�s illustra-tion, conceives of the trading process as a sequence of observable, irreversible,piecemeal exchanges, taking place at successive time instants distributed over agiven trade round7 . For further reference, let us label this �rst interpretation ofthe trading process in Edgeworth�s theory of simple contract as the �observable�one.Yet, in a footnote appended to a passage by Marshall concerning the po-

tential multiplicity of end points in the barter process concerned, points whichcorrespond to Edgeworth�s "�nal settlements" and are labelled as "equilibria"by Marshall, Edgeworth o¤ers a signi�cantly di¤erent interpretation of both thetime structure and the very nature of the process under question:

It should be noted that such a multiplicity of equilibria occurs if wesuppose that A and B, without actually trading successive doses ofapples and nuts, and therefore without actually experiencing eitherany shortage or plenty of the traded goods, only think of such trades,or discuss them, while bargaining with one another, and mentallyappreciate their hedonistic e¤ects. (Edgeworth, 1891b, p. 235, fn.1; our translation8)

According to the �mentalistic�interpretation of the bargaining process sug-gested in the above passage, no observable trades are supposed to actually takeplace during the preliminary bargaining phase between the traders. But the tim-ing of trades which are only "thought of" or "discussed" by the traders becomesunclear; moreover, under the �mentalistic� interpretation, the very representa-tion of the process by means of the "broken line" drawn in Fig. 2, corresponding"to successive barters (at di¤erent rates of exchange) of a few nuts for a few

6The sentence quoted in the text, drawn from the English version of the paper, as re-issued in the 1925 collection of Edgeworth�s economic papers, does not exactly coincide withthe corresponding sentence in the original Italian version of the paper: in the latter, in fact,the sentence is interspersed with references to a passage from Marshall�s "Note", which isquoted in full in the Italian version, but is suppressed in the English one.

7Edgeworth�s formalisation of Marshall�s theory of barter by means of an Edgeworth-Boxdiagram and the associated conceptual apparatus exactly corresponds to what Marshall hadhad in mind, but had only been able to express by means of a numerical example. As amatter of fact, Marshall himself explicitly recognised the faithfulness of "Prof. Edgeworth�smathematical version of [Marshall�s own] theory of barter" by incorporating Edgeworth�sdiagram and formalism in Note XII bis of the Mathematical Appendix of Marshall�s Principles(1961a, pp. 844-5). This Note, added to the second edition of the Principles appearing in1891, soon after the publication of Edgeworth�s 1891b paper, will be kept in all the followingeditions without substantial alterations (Marshall 1961b, pp. 834).

8This footnote belongs to a part of the 1891 paper in Italian (1891b) that is not reproducedin the later abridged English re-issue of the paper (1925d).

16

apples�, becomes questionable: for, strictly speaking, given that no observabletrades are assumed to take place during the bargaining process, and given thatthe talks and discussions between the traders are just a private a¤air of theirown, lacking any public evidence, strictly speaking there is nothing to be repre-sented in the Edgeworth-Box diagram, even for purely illustrative purposes, ex-cept for the �nal settlement (denoted xMt0 in Fig. 2 above, where the superscriptM stands for �mentalistic�), that is, except for the point in the core eventuallyarrived at, which is the only observable outcome of the process. As a matterof fact, in MP, where the �observable� interpretation of the trading process isnowhere discussed in any detail, one can �nd no geometrical description of thepath followed by the bargaining process by means of a "broken line" of the typeemployed in the 1891b paper with the purpose of formalising Marshall�s theoryof barter, which certainly presupposes �observable�piecemeal trades; the onlypoints plotted in the Edgeworth-Box diagrams appearing in MP are points inthe core of the economy. This might seem to support the idea that, in Edge-worth�s view, the "broken line" representation of the trading process only �tsthe �observable� interpretation, being inconsistent with the �mentalistic� one.We should keep this in mind in discussing the nature and timing of the tradingprocess in Edgeworth�s theory of exchange proper, to which we now turn.

4 Edgeworth�s theory of exchange

As will be recalled, Edgeworth�s basic conjecture is that both the extent of thecompetition among the traders and the determinateness of the outcome follow-ing from their interaction increase with the number of traders participating inthe economy. In order to prove his conjecture, Edgeworth contrives the deviceof evenly increasing the traders�number by repeatedly replicating the originalEdgeworth-Box economy. Namely, given an Edgeworth-Box economy with twotraders, E2�2, where the traders, each with speci�ed characteristics (i.e., con-sumption set, endowment, and utility function), are respectively labelled A andB, given any positive integer n 2 Z+, an n-replica economy, E2�2n , is an econ-omy where there exist 2n traders, of which n, having the same characteristicsas individual A, may be called type-A traders, while the remaining n, havingthe same characteristics as individual B, may be called type-B traders. As al-ready explained, in discussing the replication mechanism it is convenient to setE2�21 � E2�2.Within this framework, Edgeworth is able to (almost) prove three funda-

mental theorems which, in accordance with current usage, may be called the�Walrasian Equilibrium Is in the Core�, the �Equal Treatment in the Core�,and the �Shrinking Core�theorems (Varian, 1992, pp. 389-92); they respectivelycorrespond to Debreu and Scarf�s generalised versions of the original theorems,labelled as Theorems 1, 2, and 3 at pp. 240-3 of their 1963 paper.As far as the �Walrasian Equilibrium Is in the Core� theorem, Edgeworth

(1881, pp. 38-40, 112-6) can easily prove it with reference to the Edgeworth-Boxeconomy from which he starts. It is interesting to note that Edgeworth�s proof

17

heavily relies on the Walrasian apparatus of the "demand curves", appearing as�o¤er curves�in the Edgeworth�s-Box diagram (see Fig. 2 above): for, given thatthe Walrasian equilibrium (assumed unique) is identi�ed by the intersection ofthe o¤er curves of the two traders, it necessarily satis�es both the conditionsfor Pareto optimality (equality of the marginal rates of substitution) and theindividual rationality conditions, hence it belongs to the core of E2�21 . Theextension of this result to an arbitrary n-replica economy with n > 1, however,must be postponed until when the �Equal Treatment in the Core�theorem hasbeen proven.This, in e¤ect, is the �rst theorem to be discussed by Edgeworth in the

framework of replicated economies. In this regard, the formal proof put forwardin MP (1881, p. 35) with reference to a 2-replica economy, E2�22 , can easilybe generalised to an arbitrary n-replica economy, E2�2n . Though not providingthe required generalisation, Edgeworth takes anyhow for granted the result.By so doing, he feels justi�ed in extending the use of both the contract-curveequation and the geometrical apparatus referring to the original EdgeworthBox economy, E2�21 , to any n-replica economy with n > 1, E2�2n : to this end,in fact, it is enough to reinterpret the two traders of the original economy, Aand B, as the representatives of their replicated sets or types, each type owningn identical traders, and similarly to reinterpret each two-trader allocation inthe contract-curve of the original economy as the average two-type allocationin the contract-curve of the n-replica economy. It should be stressed that theproposed reinterpretation is only justi�ed with reference to core allocations: foronly in the core (of the appropriate n-replica economy) does the equal treatmentproperty hold, so that it is only in the core that one can legitimately confoundindividual traders and trader types. By virtue of this reinterpretation, the samesuitably generalised Edgeworth-Box diagram can be used to represent both coreallocations of individual traders and average core allocations of trader types.Further, since the conditions de�ning a competitive equilibrium in the two-trader economy are also satis�ed in any replicated economy, the competitiveequilibrium (assumed unique) of the original Edgeworth-Box economy, E2�21 ,can be identi�ed with the competitive equilibrium of any n-replica economy,E2�2n , so that the �Walrasian Equilibrium Is in the Core�theorem automaticallygeneralises to all replicated economies.Yet, while the competitive equilibrium remains essentially the same, except

for the size of the economy and, as a consequence, the dimensionality of theequilibrium allocation, the core of the economy does not remain unchanged un-der the replication mechanism, but, as stated by the "Shrinking Core" theorem,it shrinks to the Walrasian equilibrium as the traders�number increases withoutlimits. By proving this theorem, one would also succeed in supporting Edge-worth�s conjecture as to the relation between determinateness and size of theeconomy. Edgeworth�s own sketch of a proof (1881, pp. 35-7) is as follows.First, by means of a geometrical argument based on a simple 2-replication

of the original Edgeworth-Box economy, he proves that �the points of thecontract-curve in the immediate neighbourhood of the limits [of that portion ofthe contract-curve that corresponds to the core of the original Edgeworth-Box

18

economy] cannot be �nal settlements�, hence cannot belong to the core of the2-replica economy, E2�22 . Let us take, e.g., the rightmost allocation in the coreof the Edgeworth-Box economy, E2�21 ; in Fig. 3 below this allocation is denotedby x1r =

�x1rA ; x

1rB

�. Let us now consider the 2-replica economy, E2�22 , which

owns 4 traders in the whole: precisely, 2 identical type-A traders, denoted A1and A2, and 2 identical type-B traders, denoted B1 and B2. The x1r allocationcan now be reinterpreted as a type-representative, equal-treatment allocationin the 2-replica economy, that is, x1r =

�x1rA ; x

1rB

�, with x1rA = x1rA1

= x1rA2and

x1rB = x1rB1= x1rB2

. Yet, by exploiting the strict quasi-concavity of the util-ity functions, Edgeworth is able to show that a coalition formed by the twotype-B traders, B1 and B2, who are relatively �disadvantaged� in the x1r al-location, and one of the relatively �advantaged� type-A traders, say A1, canblock the x1r allocation by reallocating the coalition endowments in such away as to get the blocking allocation

�x2bA1

; x2bB1; x2bB2

�, satisfying x2bB1

= x2bB2and

x2bA1+ x2bB1

+ x2bB2= !A1 + !B1 + !B2 , where the superscripts 2 and b stand

for �2-replica economy� and �beginning of the blocking process�, respectively;the other type-A trader, A2, would be simply left with his endowment, that is,x2bA2

= !A2.

Figure 3

Yet, the resulting 4-trader allocation in the 2-replica economy,�x2bA1

; x2bA2; x2bB1

; x2bB2

�,

which does not satisfy the equal treatment property (as far as type-A tradersare concerned) and does not lie on the contract-curve, would be blocked by

19

some other 3-trader coalition; the recontracting process would go on until a newallocation located in the portion of the contract-curve to the left of x1r wouldbe reached. However, all the allocations lying on the portion of the contract-curve between x1r and x2r would be eliminated by means of the same blockingprocedure as the one by means of which x1r was initially eliminated. Thereason why x2r =

�x2rA ; x

2rB

�, with x2rA = x2rA1

= x2rA2and x2rB = x2rB1

= x2rB2,

can no longer be eliminated in this way should be clear from the diagramin Fig. 3. For consider the allocation

�x2eA1

; x2eA2; x2eB1

; x2eB2

�, with x2eA1

= x2rA1,

x2eA2= !A2

, and x2eB1= x2eB2

, where the superscript e in stands for �end of theblocking process�; since uA1

�x2eA1

�= uA1

�x2rA1

�, uB1

�x2eB1

�= uB1

�x2rB1

�, and

uB2

�x2eB2

�= uB2

�x2rB2

�, no 3-trader coalition consisting of one type-A trader,

say A1, and the two type-B traders, B1 and B2, can improve upon the allocation�x2rA1

; x2rB1; x2rB2

�; hence the allocation x2r =

�x2rA ; x

2rB

�, cannot be blocked and re-

mains in the core of the 2-replica economy, representing its rightmost allocation.A similar process would have taken place if one had started from the leftmostallocation x1l: in a 2-replica economy, all the allocations lying in the portionof the contract-curve between x1l and x2l would be eliminated by the sameblocking procedure as before. The set of the allocations in the contract-curvewhich cannot be eliminated in this way, that is, the set of the allocations lyingin the portion of the contract-curve between x2l and x2r, extremes included, isthe core of the 2-replica economy, or 2-core, which is strictly contained in theoriginal core, or 1-core.Consider now an allocation in the contract curve such that the straight line

segment connecting it with the endowment allocation (the South-East cornerof the Edgeworth-Box diagram) cuts the indi¤erence curves passing throughit. Any such allocation can be eliminated by a su¢ ciently large blocking coali-tion, by means of a suitably generalised version of the procedure suggested byEdgeworth with reference to a 2-replica economy. As the replication processproceeds, larger and larger blocking coalitions can be formed, so that more andmore allocations in the contract-curve can be eliminated by means of Edge-worth�s blocking procedure. Hence, the replication process generates a nestedsequence of cores converging towards the Walrasian (or Jevonsian) equilibriumallocation xW (or, what is the same, xJ): in fact, since "the common tangent toboth indi¤erence-curves at the point [xW or xJ ] is the vector from the [endow-ment allocation]", the Walrasian (or Jevonsian) equilibrium allocation is theonly allocation in the contract-curve which survives Edgeworth�s eliminationprocess when the number of replications, hence of traders, grows unboundedlylarge. This is enough to prove the "Shrinking core" theorem (Edgeworth 1881,p. 38).Edgeworth�s last remark also suggests which are, in his opinion, the truly

distinguishing features of the Jevonsian (or Walrasian) equilibrium allocation:among all the �nal settlements in the original core, the equilibrium allocationis the only one that can be reached in one single step, through one single tradecarried out at a uniform rate of exchange, coinciding with Jevons�s equilibrium"ratio of exchange" or Walras�s equilibrium "price". According to Edgeworth,

20

therefore, price uniformity is an equilibrium phenomenon; as such, it can onlyhold when the number of traders is "practically in�nite", so that "perfect com-petition" rules. This conclusion allows one to assess Edgeworth�s stance towardsJevons�s law of indi¤erence and Walras�s law of one price.As far as Jevons�s law of indi¤erence is concerned, Edgeworth�s attitude is

ambivalent: one the one hand, he rediscovers and endorses the law, making of ita central ingredient of his equilibrium theory, as Jevons had already done beforehim; on the other, he reproaches Jevons for not being su¢ ciently clear aboutthe range of application of the law. Edgeworth devotes an entire Appendixof MP (Appendix V, pp. 104-116) to the elucidation of the di¤erences andsimilarities existing between his approach and Jevons�s. The following passageis particularly revealing:

It has been prominently put forward in these pages that the Jevonian�Law of Indi¤erence�has place only where there is competition, and,indeed, perfect competition. Why, indeed, should an isolated coupleexchange every portion of their respective commodities at the samerate of exchange? Or what meaning can be attached to such a lawin their case? The dealing of an isolated couple would be regulatednot by the theory of exchange (stated p. 31), but by the theory ofsimple contract (stated p. 29).

This consideration has not been brought so prominently forward inProfessor Jevons�s theory of exchange, but it does not seem to be lostsight of. His couple of dealers are, I take it, a sort of typical couple,clothed with the property of �Indi¤erence�[...]; an individual dealeronly is presented, but there is presupposed a class of competitors inthe background. (Edgeworth 1881, p. 109)

As regards the use made by Walras of the law of one price, Edgeworth�sposition is highly critical in two related respects. First, as we have repeatedlystressed, price uniformity can hold true for Edgeworth only in the frameworkof a perfectly competitive equilibrium; therefore, in his view, no justi�cationwhatsoever can be o¤ered for assuming, as Walras does, that the law of oneprice holds not only at equilibrium, but also in a disequilibrium framework, asis the case with Walras�s tâtonnement (Edgeworth 1881, pp. 30-1, 47-8, 109,116 fn. 1, 143 fn. 1; 1891a, p. 367, fn. 1; 1910, p. 374; 1915, p. 453; 1925b,p. 312). Secondly, price uniformity ought not to be directly "postulated", asWalras does (Edgeworth 1881, p. 40), but it should be made to descend fromsome more basic principle, such as Edgeworth�s principle of recontract:

Walras�s laboured description of prices set up or �cried�in the mar-ket is calculated to divert attention from a sort of higgling which maybe regarded as more fundamental than his conception, the process ofrecontract as described in these pages and in an earlier essay [Edge-worth 1925d, partial English re-issue of 1891b]. It is believed tobe a more elementary manifestation of the propensity to truck than

21

even the e¤ort to buy in the cheapest and sell in the dearest mar-ket. The proposition that there is only one price in a perfect marketmay be regarded as deducible from the more axiomatic principle ofrecontract (Mathematical Psychics, p. 40 and context). (Edgeworth1925b, pp. 311-2).

This last remark brings us back to an issue which, having already beendiscussed with reference to Edgeworth�s theory of simple contract, needs nowto be taken up again with reference to his theory of exchange: namely, the issueof the nature and timing of the recontracting process. As will be recalled, inanalysing the theory of simple contract, we have discovered the existence inEdgeworth�s writings of two alternative interpretations of recontracting, bothconsistent with the requirements of the theory: the �observable�interpretationand the �mentalistic�one. The question that naturally arises at this point is thefollowing: Are the two interpretations still consistent with the requirements ofthe broader theory of exchange?To answer this question one has to consider that the Edgeworth-Box econ-

omy, with just two traders, is peculiar in a number of respects, so that manyof its characteristics do not survive, when the traders�number increases to anyinteger n > 2. First of all, when n = 2, bilateral trade moves are the onlykind of trading activities that can possibly be conceived of and implementedin the economy; on the contrary, when n > 2, multilateral trading, involvingmore than 2 traders at a time, opens up as a new opportunity. Now, in anEdgeworth-Box economy, a sequence of �nite bilateral trade moves, which canbe geometrically represented by means a broken line in the Edgeworth-Box di-agram (such as the path from x0t0 = ! to x�t0 in Fig. 2 above), necessarilyconverges to an allocation in the core of the economy, provided that each piece-meal trade in the sequence is weakly improving for either trader. But in anylarger pure-exchange, two-commodity economy, if only bilateral trading were tobe allowed for, then there would be no guarantee that the trading process wouldconverge to a Pareto-optimal allocation (see Feldman 1973): in order to ensureconvergence to an allocation in the suitably generalised Pareto set of the largereconomy, multilateral trading must be allowed to take place. This, however, hasfurther consequences that must be spelled out.Consider a straight-line segment representing a �nite bilateral trade move in

the broken line that, according to the �observable�interpretation of the tradingprocess, describes the path from the endowment allocation to a core allocationin the Edgeworth-Box economy. As has been explained, the absolute value ofthe slope of such segment represents the rate of exchange (or relative price)implicit in such trade move. Consider now any two allocations in the space ofallocations appropriate to a larger economy with n > 2. One can always connectsuch two allocations with a straight-line segment; but, barring special cases, suchsegment cannot in general be interpreted as describing a bilateral trade move, formultilateral trading is generally required to go from one allocation to another;moreover, no precise meaning could be attached to the geometrical properties ofthe segment, for the multilateral trading activities required to bring about the

22

assumed shift might involve an unspeci�ed set of bilateral trade moves betweenany trader in the economy and some other traders or all of them, simultaneouslytaking place at di¤erent rates of exchange.Finally, consider the elimination procedure employed by Edgeworth in the

proof of the "Shrinking core" theorem: here an allocation in the Pareto setof an economy with n > 2 traders is blocked by a coalition of (n� 1) traderswho, by reallocating their own resources among themselves, are able to weaklyimprove their own welfare, at the expenses of the n-th trader, whose conditionnecessarily deteriorates. Now, let us compare the original allocation with then-dimensional allocation obtained by combining the (n� 1)-dimensional allo-cation of the blocking coalition with the endowment allocation of the excludedtrader. In the �rst place, the latter n-dimensional allocation does not weaklyimprove upon the original one, due to the worsening of the n-th trader�s sit-uation; but then the move from the former to the latter allocation does notsatisfy the rule that only weakly advantageous shifts should be implemented,a rule underlying the �observable�interpretation of the recontracting process inthe Edgeworth-Box economy. In the second place, no such move could actuallytake place if the original allocation had been concretely arrived at through anobservable sequence of irreversible trading activities, for, in such case, the n-th trader would stick to his original allocation, without accepting any changethat would worsen his condition. But this implies that, for the purposes of theelimination procedure put forward by Edgeworth in the context of an economywith a number of traders greater than 2, all trading and bargaining activitiesshould be regarded as purely conceptual and fully reversible; moreover, Edge-worth�s elimination procedure should not be supposed to generate any sequenceor path: the blocking allocations ought not to be viewed as points in a sequenceor trajectory, but as tentative proposals whose only purpose is to destabilisealternative allocations.The above considerations lead us to conclude that the �observable�interpre-

tation of the recontracting process does not generalise from the Edgeworth-Boxeconomy to any larger economy with a number of traders greater than 2; onlythe �mentalistic�interpretation survives the increase in the number of traders9 .

9 It is worth noting that the �mentalistic�interpretation of the recontracting process is theonly one to be explicitly accounted for by the few economists rediscovering and discussingEdgeworth�s contribution during the 1930s, after half a century of generalised neglect (see, inparticular, Kaldor 1934, p. 127, and Hicks 1934, p. 342, and 1939, p. 128). Many decadesbefore, however, Berry, both in his reply to Edgeworth�s 1891b paper in Italian (Berry 1891,p. 552) and in a private letter addressed to Edgeworth (published in Marshall 1961b, p. 794),had suggested yet another interpretation of the recontracting process, according to which therecontracting process, rather than taking place over one single isolated trade round, wouldtake place over a sequence of trade rounds under unchanging economic conditions. Berry�s�stationary� interpretation di¤ers from the �mentalistic� one, for the contracts stipulated ineach trade round may give rise to consequences which are observable, at least in principle.Yet it preserves the essential trait of the �mentalistic�interpretation: for, due to the assumedstationarity of the economic data over the sequence of trade rounds involved in the recontract-ing process and the assumed impermanence of both the contracts and their e¤ects beyond theboundaries of the trade round in which the contracts themselves are stipulated, in the lastanalysis it secures that revocability of contracts which is the hallmark of the �mentalistic�

23

Yet, in spite of this conclusion, one might be tempted, for the purposes of il-lustrating the recontracting process, to construct paths connecting sequences ofallocations located in a space of allocations of appropriate dimensionality. How-ever, for the reasons given above, such attempts, dangerous in themselves, wouldalso prove pointless, for no information (concerning implicit rates of exchangeor prices or other variables of economic interest) could be inferred from the pro-posed illustrations. It is convenient to bear this remark in mind in examiningNegishi�s contribution, to which we eventually turn.

5 Negishi critique of Edgeworth�s approach

In his 1982 paper, Negishi aims at vindicating Jevons�s law of indi¤erence byshowing that, though not so powerful as implicitly suggested by Jevons himself,such law is more powerful than conceded by Walras and Edgeworth, providedthat its underlying arbitrage mechanism is duly taken into account. Negishi�sstance on Jevons�s and Walras�s contributions on this issue is similar to thattaken by Edgeworth, with a di¤erence: while for Edgeworth, as we have seen,Jevons�s law of indi¤erence holds only in large economies, so that a "practicallyin�nite" number of traders is required for price uniformity and equilibrium es-tablishment, for Negishi no large numbers are necessary for the law to hold, auniform price to rule, and a Jevonsian or Walrasian equilibrium to obtain. Yet,unlike Jevons, for whom two traders only are apparently enough for price uni-formity and equilibrium establishment, an Edgeworth-Box economy is not su¢ -cient for Negishi, for in such an economy there would be no room for arbitrage;but a slightly larger economy, such as an Edgeworthian 2-replica economy, withonly four, pairwise identical traders, would do, for with more than two tradersarbitrage becomes possible.In developing his argument, Negishi largely relies on Edgeworth�s conceptual

system, even if he freely builds upon the foundations laid by his predecessor.Let us then consider a 2-replica economy, plotting its curves and variables ina standard Edgeworth-Box diagram (Fig. 4 below). Let x1l, x1r, x2l, x2r, xJ ,and xW be interpreted as before. Further, let x22 be an allocation lying in theportion of the contract curve between x2l and xJ (or xW ), with x22 6= xJ = xW .According to Edgeworth, such allocation would belong to the 2-core, for it couldnot be blocked by means of the standard mechanism of coalition formation anddissolution at work in a 2-replica economy of the Edgeworthian type: as anumber of traders greater than 4 would be required to block an allocation likex22, in a 2-replica economy it would tend to persist, once arrived at by whateverroute, for want of an e¤ective elimination mechanism.

interpretation. While, in his rejoinder to Berry, Edgeworth (1891c, p. 317) displays his pref-erence for the �mentalistic�interpretation over the �stationary�one, in his later papers (1904,p. 40; 1907, pp. 526 ¤., 1925c, p. 313) he appears to regard the two interpretations asessentially equivalent and interchangeable.

24

Figure 4

Edgeworth�s result is quite standard. Yet Negishi disputes it, for, accordingto him, it follows from Edgeworth�s excessively conservative interpretation of thecoalition concept and the associated recontracting construct. For Edgeworth,as we have seen, a coalition is simply a non-empty subset of traders; a coalitionis said to block an allocation when, by reallocating the resources of its ownmembers only, can produce an outcome that is weakly preferable for all itsmembers, and them only, to that implied by the original allocation. For Negishi(1982, p. 228), however, Edgeworth�s coalitions are just "closed" or "pure"coalitions, for they take into accont resources and welfare of their members only;but since the real world is full of instances of "open" or "impure" coalitions,where the boundaries between coalition members and non-members are blurred,the theory should draw its inspiration from reality and learn to employ a widerconcept of coalition, too.Keeping this in mind, let us now consider how the allocation x22 might have

been arrived at. Let us approach the problem step by step. To start with, letus reinterpret the economy under discussion, supposing that it is not a 2-replicaeconomy, as imagined up to now, but an Edgeworth-Box economy, as in Edge-worth�s theory of simple contract; furthermore, let us endorse the �observable�interpretation of the trading process, that is, let us view it as a sequence ofpiecemeal �nite bilateral trades taking place over the same trade round. Insuch a case, we would be forced to conclude that the allocation x22, lying in the

25

contract-curve, but di¤erent from the Walrasian equilibrium, must be reachedby a path involving at least two partial piecemeal trades associated with di¤erentrates of exchange. Even if there exists an in�nite number of paths potentiallyleading the two-trader economy to x22, let us suppose, for simplicity, that theactual path consists of just two piecemeal successive bilateral trades, the �rstleading the traders from ! to x21 and the second from x21 to x

22; the two trades

take place at the respective rates of exchange r112 and r212, with r

112 < r

212, where

r212 =MRSA21

�x22�=MRSB21

�x22�(see Fig. 4 above).

Up to this point, we would just be following in Edgeworth�s steps, borrow-ing especially from his 1891b paper in Italian on Marshall�s theory of barter.Yet, at this point, Negishi parts company with Edgeworth, for he suggests tointerpret the same broken line from ! to x22, via x

21, as the path travelled by

the four, pairwise identical traders (A1 and A2, B1 and B2) belonging to the2-replica economy from which we started: Negishi�s idea is that A1 trades withB1, A2 with B2, and that the respective successive piecemeal trades of the twotrading pairs are the same, so that !, x21 and x

22 should now be interpreted as

the two-component allocations of the representatives of the two trader-types,that is, ! = (!Ai ; !Bi), x

21 =

�x21Ai

; x21Bi

�and x22 =

�x22Ai

; x22Bi

�, for i = 1; 2.

However, Negishi�s proposed switch from Edgeworth�s theory of simple contract(two traders only) to his theory of exchange proper (more than two traders) isunwarranted, in so far as the suggested change is brought about by means ofa mechanical reinterpretation of the same geometrical apparatus: as has beenseen in the previous Section, in fact, when there are more than two traders, the�observable�interpretation of the trading process as a sequence of irrevocable,piecemeal, bilateral trade moves within the same trade round ought to be aban-doned in favour of the �mentalistic�one, where the �nal allocation is supposed tobe arrived at through an unobservable bargaining process, made up of tentative,revocable, multilateral trading proposals, whose speci�cation or geometrical de-scription ought not to be attempted. Hence, with his suggested interpretationof the broken line in Fig. 4, Negishi is trying to mix together some featuresof Edgeworth�s theory of simple contract with other aspects pertaining to thelatter�s theory of exchange proper. Yet, the resulting mixture is really a muddle,as can be gathered from Negishi�s hesitations and ambiguities about the natureand timing of the contracts appearing in his story: on some occasions they aresupposed to be "provisional" and revocable, hence presumably simultaneous(or anyhow preliminary) and unobservable; on other occasions, however, theyare supposed to be "successive" and capable of being speci�ed in detail, hencepresumably observable (Negishi 1982, pp. 225-6).Putting provisionally aside these di¢ culties, and temporarily accepting, for

the discussion�s sake, Negishi�s suggested interpretation of the diagram in Fig.4, let us now see how allocation x22, unblockable in a 2-replica economy accordingto Edgeworth, can instead be blocked according to Negishi. The basic idea isthat what cannot be done by Edgeworth�s "closed" coalitions, can instead beaccomplished by Negishi�s "open" coalitions. For example, let us suppose thatA2 and B1 form an "open" coalition fAo2; Bo1g, meaning by this that, beyondfully relying on their own resources, Ao2 and B

o1 also keep some "links" with the

26

other two traders, A1 and B2, into whose resources they also have some limitedchance to tap. In order to block the allocation x22, A

o2 and B

o1 proceed as follows:

Bo1 cancels part of his �second�contract with A1 at the unfavourable (for him)rate of exchange r212, while keeping the rest of this contract and the whole ofhis ��rst�contract with A1 at the favourable (for him) rate of exchange r112; A

o2

cancels part of her ��rst�contract with B2 at the unfavourable (for her) rateof exchange r112, while accepting to o¤set the suppressed part of her contractwith B2 by stipulating a new contract with Bo1 at identical terms, but in theopposite direction (this last assumption being made only for simplicity, for anyrate of exchange in the open interval

�r112; r

212

�would make both members of the

"open" coalition better o¤). The outcome of this sort of revised recontractingproposal is shown in Fig. 4, where the arrows denote the displacements dueto the proposed partial cancellations and new stipulations of contracts, andxNA1

, xNA2, xNB1

, and xNB2are the �nal consumption bundles of the four traders,

measured with respect to the appropriate individual coordinate axes, at the endof the process imagined by Negishi in his example (the superscript N standingfor Negishi). The �nal allocation of the "open" coalition,

�xNA2

; xNB1

�, is weakly

advantageous for its members with respect to the original one,�x22A2

; x22B1

�,

for uA2

�xNA2

�= uA2

�x22A2

�and uB1

�xNB1

�> uB1

�x22B1

�. Hence, in Negishi�s

opinion, the open coalition fAo2; Bo1g can block the 2-core allocation x22.This sort of revised elimination procedure, based on Negishi�s wider coalition

concept and enlarged recontracting construct, applies to all allocations in thecontract-curve di¤erent from the Walrasian or Jevonsian equilibrium allocation,xW = xJ . The reason for this, according to Negishi, is simple: the eliminationprocedure rests on the exploitation of arbitrage opportunities, whose very exis-tence depends on the co-existence of di¤erent rates of exchange at one and thesame time; but while all the allocations in the contract-curve di¤erent from theequilibrium allocation must be reached in at least two steps, each associated witha speci�c rate of exchange, so that arbitrage opportunities do exist in this case,the equilibrium allocation, instead, is arrived at in one single step, associatedwith one single rate of exchange, so that no arbitrage opportunity can arise here.Hence, while all the allocations other than the equilibrium one would succumbto the revised elimination procedure, the Walrasian or Jevonsian equilibrium al-location would live through it. Contrary to Edgeworth�s conclusions, therefore,price uniformity and equilibrium establishment would not require a "practicallyin�nite" number of traders: Jevons�s law of indi¤erence would assert itself in a�nite economy, too, provided that the traders�number were greater than two,so that arbitrage could display all its strength.

6 A critique of Negishi�s critique

The above story, however, is seriously faulty. In the �rst place, the issue of thetiming of the trading process, provisionally set aside above, must now be takenup again. In Negishi�s example four piecemeal bilateral trades are supposed totake place to start with, involving two pairs of traders (either A1 and B1, or A2

27

and B2) and two steps (say, step 1 and step 2): for each pair of traders thereare two trades at di¤erent rates of exchange, one for each step; likewise, foreach step there are two trades at the same rate of exchange, one for each pair oftraders. Now, at each step there can be no arbitrage, since the rate of exchangeis the same for the two trades occurring at the same step10 . So, one is led tothink of arbitraging activities between di¤erent steps, since the rates of exchangedi¤er as between trades occurring at step 1 and 2, respectively. Yet, if the stepsare thought of as "successive" in �real� time and the contracts are regardedas irrevocable, how can arbitrage occur? So the steps must be conceived of as"successive" only in the �logical�time characterising a preliminary verbal sessiontaking place among the traders before any observable actions are carried out; bythe same token, the contracts under discussion must be viewed as "provisional"and revocable without cost or penalty.Now, with reference to a �nite economy as the one described by Negishi,

Edgeworth would readily acknowledge that di¤erent rates of exchange typicallycharacterise di¤erent trades among di¤erent pairs of traders: for, in his opinion,such multiplicity of rates of exchange is the distinguishing feature of any �niteeconomy; arbitrage goes as far as it can, but is unable to produce price unifor-mity unless the number of traders is unboundedly large. The reason for thisis that, in Edgeworth�s view, arbitrage can only operate through recontracting,that is, through the mechanism of coalition formation and dissolution. But,in a �nite economy, the number of coalitions that can form to the advantageof all their members is bounded, so that the power of arbitrage in smoothingout price di¤erences is limited, too. Negishi believes that this constraint canbe easily by-passed, by allowing for a simple extension of the coalition concept,from Edgeworth�s "closed" to his own "open" coalition concept. Yet, a num-ber of fundamental objections can be raised against Negishi�s use of the "open"coalition concept for his purposes.Let�s go back to Negishi�s example. One can easily check that the true mem-

bers of the "open" coalition, Ao2 and Bo1 , can (weakly) increase their utility levels

only by "exploiting" the other two traders, A1 and B2, with whom they keepsome "links": in e¤ect, the utility levels of the other two traders necessarilydecrease as a result of the revised recontracting proposal of the "open" coalitionmembers, namely, uA1

�xNA1

�< uA1

�x22A1

�and uB2

�xNB2

�< uB2

�x22B2

�. Yet,

since an "open" coalition can improve the welfare of its members only at theexpenses of the welfare of non-members, Negishi�s concept of an "open" coali-tion immediately appears as much weaker than Edgeworth�s "closed" coalition

10The assumption that the rate of exchange and the quantities exchanged are the samefor the two trading pairs at each step, implying that arbitrage has already carried its e¤ectsthrough at the same step, is made by Negishi in order to simplify his illustration: for, underNegishi�s assumption, the same broken line representing a simple two-step �observable�tradingprocess taking place in a two-trader economy can be employed to represent a two-step tradingprocess taking place in a 2-replica economy with four, pairwise identical traders. RelaxingNegishi�s simplifying assumption would indeed enrich the picture, re-opening the issue ofarbitrage at each step. Yet, as will be seen in a moment, the di¢ culties arising from suchextension would not di¤er in kind from those to be discussed in the text with reference to thetreatment of arbitrage across di¤erent steps.

28

concept, for in a "closed" coalition all improvement can only come from thereallocation of the resources owned by the coalition members.Negishi (1882, p. 228) recognises that an "open" coalition is less stable than

a "closed" one, but he adds that "the stability of the realised coalition itselfis not required to block an allocation in Edgeworth�s exchange game". Yet, ifNegishi is right in pointing out that the stability of any realised (or, better,proposed) coalition is not necessary for blocking, he is certainly wrong when heforgets that the credibility of the threat made by a blocking coalition, whichrests precisely on the autarchic character of Edgeworth�s "closed" coalitions, isessential for blocking. Moreover, "open" coalitions are supposed to rely on theunawareness of the cheated partners (Negishi 1882, p. 228). Yet, if cheatingis allowed for, and assumed not to be spotted, provided that the extent of theswindle is limited, then even a Walrasian or Jevonsian equilibrium allocationwould not be immune from the destabilising power of "open" coalitions. In theend, therefore, Negishi�s approach appears to oscillate between proving too little(for the threats of "open" coalitions are not credible, so that no allocation couldbe blocked in this way) and proving too much (for, if cheating is allowed for anddeemed e¤ective, then any allocation, including the equilibrium one, could bedestabilised in this way).

7 Concluding remarks

Jevons, Walras, and Edgeworth develop their pure-exchange equilibrium mod-els over the decade 1871-1881. All three of them make use of some version of alaw, called law of indi¤erence (or principle of uniformity) by Jevons and Edge-worth and often referred to as the law of one price in connection with Walrasianeconomics. Writing after Jevons and Walras, Edgeworth can and does explic-itly refer to their theories, either to praise or to criticise them. In particular,Edgeworth (1881) shares with Jevons the idea that the law of indi¤erence mustbe regarded as an equilibrium property. Unlike Jevons, however, he denies thevalidity of the law in all the economies where the traders�number is short ofthe "practically in�nite". In this regard, he tells a plausible story, based on thejoint operation of his replication and recontracting mechanisms, a story meantto support the emergence of the law, as well as the convergence of the economytowards a Jevonsian or Walrasian equilibrium, as the traders� number growsunboundedly.About one century later, Negishi (1882) resumes the time-honoured discus-

sion about the law, taking an unconventional stance. Precisely, Negishi strivesto prove that, contrary to Edgeworth�s original conjecture, a competitive equi-librium can be attained even in small economies, provided that the true drivingforce underlying Jevons�s law of indi¤erence, namely, its implicit arbitrage mech-anism, is allowed to operate and carry its e¤ects through. Yet, in the paper wehave shown that Negishi�s proof is unconvincing, for he tries to exploit Edge-worth�s machinery without preserving the latter�s carefully chosen assumptionsand quali�cations. Moreover, Negishi�s concept of an "open" coalition, meant

29

to generalise Edgeworth�s concept of a "closed" coalition, is so weak and incon-sistent that its associated blocking mechanism is wholly untrustworthy.The failure of Negishi�s attempt con�rms, in the last analysis, that Edge-

worth�s results cannot be improved upon, so long as one accepts, as Negishi does,to keep the analysis at the level of abstraction and generality chosen by Edge-worth. Stronger results may be hoped for only by further specifying the tradingtechnology, the bargaining machinery, the information transmission mechanism,or any other property of the trading or contracting process. But this, of course,would make the hopefully stronger results depend on the special assumptionsadopted.

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