1 – Introduction
1In 1881 Francis Ysidro Edgeworth publishes his pathbreaking booklet Mathematical Psychics (henceforth MP), thereby initiating a tradition that would lead, about eighty years later, to the development of a theory of decentralised exchange based on multilateral interactions, a special branch of the broader approach known nowadays as ‘coalitional’ or ‘cooperative game theory.’ In developing his analysis, Edgeworth extensively relies on Jevons’s and Walras’s theoretical systems, as put forward in the former’s Theory of Political Economy (henceforth TPE; 1^{st} ed. 1871, 2^{nd} ed. 1879) and the latter’s Éléments d’économie politique pure (henceforth EEPP; 1^{st} ed. 187477). From Chapter 4 of Jevons’s TPE Edgeworth derives the model of a twotrader, twocommodity, pureexchange economy, as well as the equilibrium conditions, i.e., Jevons’s celebrated “equations of exchange,” and a “general law of the utmost importance in economics” [Jevons, 1970: 137], called the “principle of uniformity” in the 1^{st} edition of TPE [1871: 99] and the “law of indifference” in the 2^{nd} one [1879: 90]; from Walras’s EEPP, instead, he derives the fullyfledged generalisation of the Jevonsian notion of equilibrium to a multiagent, multimarket economy [Walras, 1988: 1378, 1703].
2Yet, though building upon his predecessors’ concepts and models, Edgeworth embeds them in a wider analytical framework, characterised by a seemingly more general solution concept, which encompasses the Jevonsian or Walrasian equilibrium notion as special cases: the “set of final settlements,” representing the solution concept of Edgeworth’s MP model of a pureexchange economy, can be be viewed as the first instance of the fundamental solution concept underlying modern coalitional game theory, namely, the ‘core’ of a coalitional game. As is wellknown, Edgeworth is able to prove that the Walrasian equilibrium of a pureexchange economy, assumed unique, belongs to the contract curve; moreover, by jointly relying on his ‘recontracting’ and ‘replication’ mechanisms, he is also able to prove that the contract curve shrinks to the Walrasian equilibrium as the number of traders in the economy grows unboundedly large. This limiting result can indeed be viewed as buttressing the Walrasian or competitive equilibrium concept, for it provides yet another gametheoretic foundation for it; however, the same result is also viewed by Edgeworth as rigorously limiting the possible use of the Walrasian equilibrium machinery, which includes not only the equilibrium concept as such, but also the law of indifference or, what is the same, the principle of price uniformity or also the ‘law of one price,’ as that principle is occasionally called in the Walrasian context. According to Edgeworth, in fact, the scope of application of the Walrasian or competitive equilibrium apparatus should be strictly confined to unboundedly large economies, i.e., economies where the traders’ number is “practically infinite,” for only such economies can be legitimately regarded as “perfectly competitive.” On the contrary, finite economies are characterised by limited competition; moreover, whenever the traders’ number is short of the “practically infinite,” the law of indifference cannot be supposed to hold and the Walrasian equilibrium price and allocation cannot be taken to faithfully describe the actual state of the economy.
3In 1982, almost exactly one century after the appearance of MP, Tagashi Negishi, one of the most productive general equilibrium theorists of the second half of the last century, publishes the unconventional paper “A Note on Jevons’s Law of Indifference and Competitive Equilibrium,” where he tries to exploit Edgeworth’s analytical weaponry against Edgeworth’s own conclusions, as summarised above. Precisely, Negishi endeavours to prove that, by duly revising and extending Edgeworth’s concept of coalition and by consequently reinterpreting the associated recontracting and replication mechanisms, one can uncover the driving force tacitly underlying Jevons’s law of indifference, namely, that arbitrage mechanism that must be taken to be at work behind any tendency towards price uniformity. In Negishi’s opinion, such mechanism, far from requiring, as supposed by Edgeworth, the existence of infinitely many traders to carry its effects through, is fully effective in any finite economy, even in a very small one, provided that there exists some room for arbitraging; but a number of traders greater than two is the only requirement on which the chance of undertaking arbitraging activities depends  a simple requirement apparently neglected by Jevons in his original discussion in TPE, where the law of indifference is supposed to hold true even in a twotrader economy.
4Negishi’s conclusions are indeed farreaching and even surprising, for, if proven true, they would subvert the tenet, wellestablished in the literature, that assuming perfect competition and relying on the Walrasian equilibrium machinery are really justified in large economies only. Yet, in spite of the sweeping character of Negishi’s claims, his argument has been essentially ignored for almost two decades after its appearance; only at the turn of the century Negishi’s viewpoint has been approvingly resumed by a few scholars [Rebeyrol, 1999; Pignol, 2000]. Yet, in spite of such revival, Negishi’s critique of Edgeworth’s approach has not yet been thouroughly assessed in the literature. The aim of the present paper is to fill this gap.
5The paper is structured as follows. Section 2 will introduce the analytical and diagrammatical apparatus required for later purposes, that is, the model of a pureexchange, twocommodity, twotrader economy with cornered traders, which is the simple model actually employed by Jevons, Edgeworth, and Negishi to develop their respective ideas about the law of indifference and competitive equilibrium; the same model, though simpler than the simplest pureexchange model put forward by Walras in EEPP, can nevertheless be so adjusted as to account for Walras’s equilibrium apparatus, too. In Section 3, after discussing Edgeworth’s ‘recontracting’ mechanism, we shall examine his «theory of simple contract», that is, his model of a pureexchange, twocommodity, twotrader economy. In Section 4, after discussing Edgeworth’s replication mechanism, we shall analyse his reinterpretation of both Jevons’s law of indifference and Walras’s equilibrium concept as properties of unboundedly large economies. Section 5 will summarise Negishi’s critique of Edgeworth’s approach. In Section 6 we shall critically discuss Negishi’s conjecture that no large number of traders is required to elicit price uniformity and competive equilibrium, provided that arbitrage opportunities are allowed for and duly taken into account. Section 7 concludes.
2 – The model of an EdgeworthBox economy
6Let us consider a pureexchange economy with a finite number L = 2 of commodities, denoted by I = 1,2, and a finite number I = 2 of consumerstraders (henceforth indifferently referred to as either consumers or traders), denoted by i = A,B. Each consumer i is characterized by a consumption set X_{i} = {x_{i} ? (x_{1i}, x_{2i})} = ?^{2}_{+}, a utility function u_{i} : X_{i} ? ?, and endowments ?_{i} ? (?_{1i}, ?_{2i}) ? ?^{2}_{+}{0}.
7Let be an allocation; be the aggregate endowments; be the set of feasible, nonwasteful allocations. A pureexchange, twocommodity, twoconsumer economy with the above characteristics will be called an EdgeworthBox economy, so named by Bowley (1924) after Edgeworth (1881), and will be denoted in the following by . The consumers’ characteristics (consumption sets, utility functions, endowments) represent the data of the EdgeworthBox economy; they are assumed to be fixed up until the exchange problem is solved and an equilibrium (or, more generally, a solution) is established. The period over which the data remain fixed will be referred to as a ‘trade round’. In conformity with standard usage in contemporary models of an EdgeworthBox economy, the traders’ utility functions are assumed to be continuously differentiable, strongly monotonic, and strictly quasiconcave. Further, in accordance with Jevons’s and Edgeworth’s original assumptions, the traders are supposed to be cornered, that is, they are supposed to hold a positive quantity of one commodity only; specifically, in the following we shall assume: and . (The assumption of cornered traders applies to Walras’s most elementary pureexchange model, too.)
8The evolution of the economy ?^{2×2} over continuous time must be viewed as a sequence of disconnected, discretetime trade rounds, each characterised by its own data and, hopefully, its own equilibrium (or solution). Since each trade round must be viewed as selfcontained, there cannot be any carryover of endowments from one trade round to the next; at the same time, since the data must not change over each trade round, the endowments must not wear off 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 and perfectly perishable when the trade round is over (see Hicks [1989: 711)].
9The model of an EdgeworthBox economy specified above can be graphically represented by means of the homonymous diagram, where the lengths of the sides of the rectangle are respectively given by the aggregate endowments, and , and each point in the rectangle represents a feasible, nonwasteful allocation x = (x_{A}, x_{B}) (see Figure 1 below).
10By employing the analytical and diagrammatical apparatus that has just been introduced, let us now briefly review the solutions respectively provided by Jevons and Walras to the equilibrium determination problem.
11Starting with Jevons, let us first consider what he calls an “act of exchange” between the two traders [Jevons, 1970: 1389]. Such an act involves the trade of either ‘infinitely small’ or ‘finite’ quantities of the two commodities: it may be called ‘differential’ in the former case and ‘finite’ in the latter. In either case the quantity of the commodity given in exchange will be taken to be negative (that is, dx_{li} < 0 or ?x_{li} < 0, if commodity l is given by trader i, for i = A, B and I = 1,2), while the quantity of the commodity received in exchange will be taken to be positive (that is, dx_{li} > 0 or ?x_{li} > 0, if commodity l is received by trader i, for i = A, B and l = 1,2). Since an act of exchange is necessarily bilateral, the vectors of the quantities traded satisfy the following conditions: (dx_{1A}, dx_{2A}) = ? (dx_{1B}, dx_{2B}), if the act is differential; (?x_{1A}, ?x_{2A}) = ? (?x_{1B}, ?x_{2B}), if the act is finite. Finally, let dx_{l} = dx_{li} (resp., ?x_{l} = ?x_{li}) be the absolute value of the quantity exchanged of commodity l in a differential (resp., finite) act of exchange.
12Then, still following Jevons [1970: 138], a ratio of the type (resp., ) will be called a differential (resp., finite) « ratio of exchange ».
13As we shall see, a special kind of finite « ratio of exchange » plays a fundamental role in Jevons’s theory of exchange: it is the finite “ratio” , where and .
14Before moving to the equilibrium characterisation issue, we need to introduce one further concept, which will prove useful in the following discussion not only of Jevons’s theoretical model, but also of Walras’s, Edgeworth’s, and Negishi’s. Given an EdgeworthBox economy satisfying the above assumptions on endowments and utilities, let us define consumer i’s marginal rate of substitution of commodity 2 for commodity 1 when i’s consumption is x_{i}, MRS^{i}_{21}(x_{i}), as the quantity of commodity 2 that consumer i would be willing to exchange for one unit of commodity 1 at the margin, in order to keep his utility unchanged at the original level u_{i}(x_{i}). From this definition it follows that:
16In their writings, Jevons, Walras, and Edgeworth ignore the notion of the marginal rate of substitution. Yet, they do know and systematically employ the notion of the marginal utility of commodity l for consumer i, which, under the stated assumptions on the properties of the utility functions, is welldefined and bounded away from zero everywhere in the consumption set. Moreover, though not 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 ratios of the values of the marginal utility functions of each consumer corresponding to specific consumption bundles and examine the role of such ratios in solving the exchange equilibrium problem.
17Now, in facing the equilibrium characterisation issue, Jevons [1970: 13940] starts by asking “at what point the exchange will cease to be beneficial” for the traders. His answer is as follows: for each trader i, given an arbitrarily “established [differential] ratio of exchange,” say , the point at which the exchange ceases to be beneficial for i, called the “point of equilibrium” by Jevons, is identified by the condition that the given differential ratio be equal to trader i’s marginal rate of substitution evaluated at the equilibrium point, that is:
19It should be noted that Jevons, after deriving the individual optimisation conditions (1) for either trader separately [1970: 142], never thinks of combining them into a single equation, so that, in spite of Edgeworth’s overgenerous, yet unfounded, acknowledgement of Jevons’s priority [Edgeworth, 1881: 21], he never obtains an equation of the following type:
21which, together with the feasibility conditions,
23would define the “contract curve” (or the ‘Pareto set’) of the EdgeworthBox economy concerned. In Fig. 1 the “contract curve,” so named by Edgeworth [1881: 21], as will be seen in the next Section, is the curve connecting O_{A} and O_{B}.
24Now, equations (1) and (3) are not sufficient to make the model determinate, for one has just four equations to determine six unknowns . It is precisely at this point that the law of indifference comes to rescue, for it provides 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 of Chapter 4 of TPE on “The Law of Indifference” with the following sentence:
Thus, from the selfevident principle, stated on p. 137 [i.e., the law of indifference], that there cannot, in the same market, at the same moment, be two different prices for the same uniform commodity, it follows that the last increments in an act of exchange must be exchanged in the same ratio as the whole quantities exchanged…This result we may express by stating that the increments concerned in the process of exchange must obey the equation
The use that we shall make of this equation will be seen in the next section.
28Now, the only analytical use made by Jevons of the law of indifference, i.e., of equation [*], is to allow the theorist to replace the differential ratio of exchange, , by the finite one, , thereby obtaining the missing equations needed to close his model. In our formalism, account being taken of (3), such equations can be written as follows:
30By substituting (4) into (1) and simplifying, one finally obtains:
32which are Jevons’s “quations of exchange” [1970: 143], defining the equilibrium allocation, , as well as the equilibrium “ratio of exchange,” or relative price of commodity 1 in terms of commodity 2, .
33Under the stated assumptions, a Jevonsian equilibrium can be proven to exist, even if it is not necessarily unique. Yet, in order to simplify the exposition, let us suppose the equilibrium to be unique. Then the equilibrium act of exchange can be graphically represented in Fig. 1 by means of the thick straight line segment connecting the endowment allocation ? with Jevons’s equilibrium allocation x^{J}. The equilibrium “ratio of exchange,” in turn, is given by the absolute value of the constant slope of such straight line segment (tan?^{J}).
34Let us now move to Walras’s pureexchange, twocommodity model. Unlike Jevons, Walras need not restrict his analysis to an economy with only two traders: as a matter of fact, the traders’ number is supposed from the start to be any arbitrary integer [Walras, 1988: 74, 1367]. Yet, in order to make 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 pureexchange economy under discussion. The traders are supposed to have the same characteristics as before.
35Walras assumes the traders to behave competitively: precisely, they are supposed to take prices as given parameters and to choose their optimal trade plans, called “dispositions à l’enchère” or “au rabais” [Walras, 1988: 70, 83], in such a way as to maximize their respective utility functions, under their respective Walrasian budget constraints. Then let p = (p_{12}, 1) ? ?^{2}_{++} be the price system, expressed in terms of commodity 2 taken as the numeraire. Since in the economy there are only two commodities, there can only exist one independent relative price, . 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 indifference that is supposed to rule in the Walrasian theoretical system; as already recalled, such version of the law is often referred to as the ‘law of one price’ in this context. It should be stressed, however, that while Jevons’s law of indifference is assumed to hold only at equilibrium, of which it is a distinctive feature, Walras’s law of one price is assumed to hold both at equilibrium and out of equilibrium.
36For all p = (p_{12}, 1), the optimization problem to be solved by trader i can be formally written as:
38where equation (6) is Walras’s «condition de satisfaction maxima» for trader i and equation (7) is the Walrasian budget constraint of the same trader, for i = A, B [Walras, 1988: 1117].
39By solving this system, one obtains for each trader the individual gross demand functions for both commodities, from which one can derive the individual net demand and supply functions for either commodity, that is:
41and
43Then, by aggregating over the traders, one obtains the aggregate gross demand functions, as well as the aggregate net demand and supply functions, from which one can derive the aggregate excess demand functions for either commodity, that is:
45Finally, by setting the aggregate excess demand functions equal to zero, one obtains the market clearing equations, one for each commodity:
47Yet, given Walras’ Law, that is, p · z = p_{12}z_{1}(p_{12}) + z_{2}(p_{12}) ? 0, ?p = (p_{12}, 1) ? ?_{++} × {1}, only one equation provides an independent equilibrium condition. By solving either equation, therefore, we obtain the Walrasian equilibrium relative price, p^{W}_{12}. The Walrasian equilibrium allocation, x^{W} = (x^{W}_{A}, x^{W}_{B}) = ((x^{W}_{1A}, x^{W}_{2A}), (x^{W}_{1B}, x^{W}_{2B})), can then be obtained by plugging the Walrasian equilibrium price into the individual gross demand functions, that is, (x^{W}_{i}) = (x_{i}(p^{W}_{12}), for i = A, B [Walras, 1988: 1367]. Under the stated conditions, a “solution” exists, even if it is not necessarily unique [Walras, 1988: 97]. Yet, for simplicity, we shall assume uniqueness in the following. The Walrasian equilibrium allocation x^{W} can be plotted in the EdgeworthBox diagram drawn in Fig. 1 above. The ray from UC netratat through x^{W} is the equilibrium budget line for either trader, the absolute value of its slope (tan?^{W}) representing the Walrasian equilibrium price p^{W}_{12}.
48The method of “solution” netratat suggested above is ‘analytique,’ for it consists in finding the roots of a system of ordinary algebraic equations. But one can also restate the problem in geometrical terms, that is, one can give it «la forme géométrique», by drawing for each commodity the demand and supply curves and finding their intersection. Alternatively, one can plot the graphs of the gross demand functions of the two traders, x_{A}(·) and x_{B}(·), in the EdgeworthBox diagram of the economy under question; such graphs, called “offer curves” after Johnson [1913: 487], are labelled o_{A} and o_{B} in Fig. 1 above. Then finding the intersection of the two graphs provides another geometrical method for identifying the Walrasian equilibrium allocation and, implicitly, the Walrasian equilibrium price, too.
49As can be seen, in order to identify the equilibrium allocation and price of the EdgeworthBox economy, Walras follows a path that significantly differs from the path followed by Jevons in pursuing the same purpose: in particular, while prices and excess demand functions (or offer curves) play a fundamental role in Walras’s theoretical system, nothing comparable can be found in Jevons’s theory of exchange, as Walras himself does not fail to point out in both an open letter he addresses to Jevons in 1874 (Walras 1993: 50) and a wellknown passage of EEPP, from the second edition onwards (1988: 2513, 25). Yet, in spite of the different routes taken, the final outcome is the same, for, assuming uniqueness, Walras’s equilibrium allocation coincides with Jevons’s, i.e., x^{W} = x^{J}, and Walras’s equilibrium relative price coincides with Jevons’s equilibrium ratio of exchange, i.e.,
3 – Edgeworth’s theory of simple contract
50The point departure for Edgeworth’s discussion of the exchange problem in MP is precisely represented by Jevons’s “theory of exchange,” as developed in Chapter 4 of TPE [Edgeworth 1881: 20, 39, and App. V: 10416]. Yet, instead of accepting Jevons’s basic idea that an exchange model involving just two “trading bodies” is sufficient to cover the whole range of a priori possible trade situations, Edgeworth clearly distinguishes from the very start two separate theories: the “theory of simple contract,” which concerns just two individual traders, and the “theory of exchange” proper, which deals instead with a greater number of traders, more or less competing with each other [1881: 29, 31, 109]. Edgeworth’s fundamental conjecture is that the intensity of competition among the traders increases with their number; moreover, the greater the competition among the traders, the more determinate is the outcome of the trading process; as a consequence, “perfect competition” and “perfectly determinate” outcome can only obtain when the traders’ number is “practically infinite.” In between the two extremes of an “isolated couple” of traders, on the one hand, and a “perfect” market populated by a “practically infinite” number of traders, on the other, there is of course a whole range of intermediate situations [1881: 20, 39, 1467]. So, right at the beginning of MP, Edgeworth raises his fundamental question: “How far contract is indeterminate.” The “general answer” he is ready to offer is as follows:
(?) Contract without competition is indeterminate, (?) Contract with perfect competition is perfectly determinate, (?) Contract with more or less perfect competition is less or more indeterminate.
52The above statement might appear to suggest that, from Edgeworth’s viewpoint, the “theory of simple contract,” the “theory of exchange under less than perfect competition,” and the “theory of exchange under perfect competition” should be regarded as altogether different theories, based on alternative principles. Yet, this conclusion would be wholly unfounded, for all the above mentioned theories rest on the same “fundamental” principle, namely, the “principle of recontract” [Edgeworth, 1925b: 312], whose general operating rules are set at the beginning of MP [1881: 1620], well before the detailed discussion of either the “theory of simple contract” or the “theory of exchange” under “more or less perfect competition” is developed. In Edgeworth’s view, it is precisely the recontracting mechanism which ensures the convergence of any economy, irrespective of its size, to some allocation in a suitably defined solution set. The size of the solution set, hence the determinateness of the solution, is inversely related to the size of the economy, as measured by the number of traders; but the defining properties of the solution are always the same, for they hinge on the universal principle of recontract.
53By 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 recontract.” A ‘coalition’ is a nonempty subset of traders. A coalition ‘blocks’ an allocation if, by reallocating the aggregate endowments owned by the coalition members among the coalition members themselves, it obtains a coalition allocation, called ‘blocking allocation’, which ensures a utility level at least as great 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 an economy is the set of all the allocations that cannot be blocked by any coalition. According to Edgeworth’s rules of recontracting, coalitions can freely form and dissolve without cost or penalty. When the traders’ number increases, the number of coalitions that can possibly be formed, hence the number of blocking allocations, grows exponentially; in the last analysis, this is the reason why the core shrinks and the determinateness of the solution increases with the traders’ number.
54Let us start by considering the “theory of simple contract.” In developing this theory, Edgeworth employs the very same model of a twocommodity economy with two cornered traders as the one originally discussed by Jevons. Hence, Edgeworth’s pureexchange, twotrader, twocommodity economy, ?^{2×2}, can be described by means of an EdgeworthBox diagram as before [3]. Unlike Jevons, however, Edgeworth [1881: 21] precisely defines the contract curve, which, as already explained, is the set of the allocations satisfying equations (2) above, that is, the set of Paretooptimal allocations, denoted by {x^{C} = (x^{C}_{A}, x^{C}_{B})}. Edgeworth [ibid.: 19, 29] also identifies the UC netratatset of final settlements,UC netratat which is the subset of the contract curve contained in the ‘area of exchange’, i.e., in the lensshaped region of the plane enclosed between the traders’ indifference curves passing through the endowment allocation. The allocations belonging to the area of exchange are the feasible allocations that also satisfy the individual rationality constraints. Letting
56denote the set of the allocations belonging to the area of exchange, the set of final settlements coincides with the intersection of the contract curve and the area of exchange, {x^{C} = (x^{C}_{A}, x^{C}_{B})} ? {x^{E} = (x^{E}_{A}, x^{E}_{B})}. Edgeworth’s set of final settlements corresponds to what, in the current terminology of coalitional game theory, would be called the core of the EdgeworthBox economy: for it is the set of all the allocations that cannot be blocked by any of the three coalitions that can possibly be formed in a twotrader economy, namely, the ‘grand coalition’ or ‘coalition of the whole’, {A, B}, and the two individual coalitions, i.e., the singletons {A} and {B}. In Fig. 2 below, the contract curve is the locus connecting the origins O_{A} and O_{B}, while the set of final settlements is the portion of the contract curve going from x^{1r} to x^{1l}, where x^{1r} and x^{1l} are respectively the right and the left point of intersection between the contract curve and the traders’ indifference curves through the endowment allocation [4].
57According to Edgeworth, in the model of an EdgeworthBox economy, the “[final] settlements…are represented by an indefinite number of points” [Edgeworth, 1881: 29]. Such indeterminateness is the hallmark of the theory of simple contract. The reasons for this can be easily spelled out. According to Edgeworth’s trading rule, the two traders will trade if and only if their acts of exchange are (weakly) mutually advantageous, that is, if they increase the utility of at least one of them, without decreasing the utility of either trader. As before, the acts of exchange may be finite or differential. In the wake of Feldman [1973: 465], let us call such acts “bilateral trade moves.” Now, if MRS^{A}_{21}(?_{A}) ? MRS^{B}_{21}(?_{B}), there are infinitely many bilateral trade moves that 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 allocation in the area of exchange that might be reached after any move is effected, as long as the traders’ marginal rates of substitution evaluated at that allocation diverge. In view of this, lacking any further detailed information about the traders’ knowledge, attitudes, skills, etc., the actual path followed by the two traders during the bargaining process cannot be specified: for it depends on idiosyncratic circumstances which are specific to either trader and, as such, irreducible to general rules or systematic theorizing [Edgeworth, 1881: 2930]. What can be stated for sure, under the trading rule specified above, is that: 1) the process will come to an end (perhaps in the limit), and 2) its end point, that is, the final settlement eventually arrived at through the bargaining process, will be such that the traders’ marginal rates of substitution evaluated at that point are the same. Yet, if the path cannot be specified, its end point cannot be identified either: the path will certainly end up somewhere in the core of the EdgeworthBox economy, namely in the set {x^{C}} ? {x^{E}}, but which specific allocation is eventually reached cannot be theoretically predicted.
58The 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 even hoped for in the long run, is reiterated by Edgeworth over and over again, from beginning to end of his long scientific life, not only with reference to the theory of simple contract presently under discussion, but also, as we shall see, with reference to the theory of exchange proper, dealing with any large economy, characterised by any number of traders short of the «practically infinite» (see, e.g., Edgeworth [1889a: 268; 1891a: 364, 3656; 1904: 3940; 1925: 311]).
59Now, granting that the theorist can produce no general theory of the path from the initial endowment to the final settlement in the core of the economy, where the process eventually ends, the most she can do is to indicate “particular paths…by way of illustration, to ‘fix the ideas’, as mathematicians say” [Edgeworth, 1904: 3940]. Following Edgeworth’s lead, let us then undertake to provide an “illustration” of a “particular path” described by the trading process taking place in the EdgeworthBox economy ?^{2×2} (see Fig. 2 above). In order to ‘fix the ideas’, let us suppose that the bilateral trade moves are finite, so that a final settlement is reached after a finite sequence of moves, say after ? ? ? moves, where ? > 1. Let t_{0} ? ? denote the trade round (a finite time interval) over which the trading process is supposed to take place. Each move is indexed by ? = 1,…, ?. The first move takes the economy from the endowment allocation, x^{?}_{t0} = ?_{t}_{0}, to the allocation x^{1}_{t0}; the generic move ? carries the economy from to ; the last move takes the economy from to . The broken line connecting the successive allocations in the finite sequence describes the path followed by the traders during the trading process. The absolute value of the slope of the straight line segment connecting any two successive allocations in the sequence defines the rate of exchange implicit in the move leading to the allocation located at the end of the segment: so, for the generic pair of allocations and , the rate of exchange implicit in the ?th move is given by . Similar results would obtain in case the bilateral trade moves were infinitesimal, rather than finite: in that case, assuming the path followed during the trading process to be described by a smooth curve with a continuously changing slope, everywhere welldefined, then the infinitesimal rate of exchange , ruling at each allocation reached during the process, would coincide with the absolute value of the slope of the curve at that point [Edgeworth, 1881: 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 predicted is that the rate of exchange implicit in the last move UC netratat, that is, in the move leading to the final settlement , must equal the common value of the marginal rates of substitution of the two traders evaluated at that point: namely, if the last move is finite (resp., infinitesimal), one must have (resp., ) [1881: 213, 109].
60Referring to the above illustration, the following questions naturally arise: are the bilateral trade moves plotted in Fig. 2 observable or unobservable? Are such moves effectively carried out or just discussed by the two traders in a sort of preliminary ‘cheap talk’ session? Are they successive in time or, being part of a verbal session preceding any implementation, essentially simultaneous? The three dilemmas are of course related. Moreover, they have all to do with the interpretation of the time index ?: does such index run over the entire trade round t_{0}, so that ? must be interpreted as a ‘real’ time index, recording the sequence of observable transactions, or does it run over a separate, preliminary session, so that ? must be interpreted as a ‘logical’ time index, recording the sequence of mental ruminations and verbal exchanges?
61In 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 as far as the “time” factor is concerned [1891a: 367]. Moreover, Edgeworth’s occasional remarks about these characteristic features of the bargaining process do not generally distinguish between the theory of simple contract and the theory of exchange proper, making the interpreter’s task even more difficult. But even the few passages scattered across MP which seem unambiguously to refer to the theory of simple contract only [1881: 245, 2830, 109, 1156] do not help solving the interpretative dilemmas listed above.
62All doubts left open by MP, however, are dispelled ten years later by a paper in Italian, published by Edgeworth in 1891, [5] where the author critically discusses Marshall’s theory of barter, as stated in the Appendix entitled “A Note on Barter,” placed at the end of Book V, chapter II, of the first edition of Marshall’s Principles of Economics, which had appeared the year before. [6]
63In his “Note,” only threepage long, Marshall had been able to develop his theory of barter by making use, as is customary for him, of an illustration based on a numerical example: in this case, the example concerns two traders, A and B, trading “apples” for “nuts.” He had then employed a special case of such theory—a case arising when one of the two commodities, say nuts, is supposed to 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 the more general issue of the establishment of the socalled “temporary” or “market equilibrium of demand and supply,” which is precisely the subjectmatter of the chapter of the Principles to which the “Note” was originally appended (the title of that chapter being in fact “Temporary Equilibrium of Demand and Supply”). Once again, Marshall’s discussion had been couched in terms of a numerical illustration concerning the “corn market in a country town,” where an arbitrary finite number of traders (greater than two) is supposed to participate in the exchanges [Marshall, 1961a: 3316].
64In his 1891 paper Edgeworth carefully analyses Marshall’s theory of barter, focussing especially on Marshall’s conclusion that, under the special assumption of constancy of the marginal utility of one of the two traded commodities (nuts), the bargaining process ends up in a perfectly determinate outcome: this conclusion, in fact, is apparently so much at variance with Edgeworth’s conjecture about the indeterminateness of the theory of simple contract as to stir his critical reaction. Edgeworth’s paper gives rise to a lively controversy, turning particularly on the issue of the determinateness of equilibrium and involving, at Marshall’s instigation, the Cambridge mathematician Berry, to whose criticism [1891] Edgeworth immediately replies with a short rejoinder [1891c]. [7] Here we are only interested in explaining how the barter controversy helps Edgeworth clarifying his own ideas about the nature and timing of the bargaining process in the EdgeworthBox model.
65At 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 bargaining with each other a fixed rate of exchange governing the whole transaction. A uniform rate of exchange, he remarks, is applicable only to the case of perfect market.
67He then proceeds to discuss the numerical illustration of the barter process put forward by Marshall in his “Note,” formalising it by means of an EdgeworthBox diagram plotted in the space of net trades [Edgeworth, 1925d: 316, Fig. 1; 1891b: 236, Fig. 1], where there appears a “series of short lines,” that is, a “broken line,” exactly corresponding to the broken line drawn in Fig. 2 above. Commenting upon the meaning of such drawing, Edgeworth claims that it “corresponds to successive barters (at different rates of exchange) of a few nuts for a few apples” [1925d: 316; 1891b: 236]. [8] Now, as the last sentence conclusively shows, Edgeworth, when directly referring to Marshall’s illustration, conceives of the trading process as a sequence of observable, irreversible, piecemeal exchanges, taking place at successive time instants distributed over a given trade round. [9] For further reference, let us label this first interpretation of the trading process in Edgeworth’s theory of simple contract as the ‘observable’ one.
68Yet, in a footnote appended to a passage by Marshall concerning the potential multiplicity of end points in the barter process concerned, points which correspond to Edgeworth’s “final settlements” and are labelled as “equilibria” by Marshall, Edgeworth offers a significantly different interpretation of both the time structure and the very nature of the process under question:
It should be noted that such a multiplicity of equilibria occurs if we suppose that A and B, without actually trading successive doses of apples and nuts, and therefore without actually experiencing either any shortage or plenty of the traded goods, only think of such trades, or discuss them, while bargaining with one another, and mentally appreciate their hedonistic effects.
70According to the ‘mentalistic’ interpretation of the bargaining process suggested in the above passage, no observable trades are supposed to actually take place during the preliminary bargaining phase between the traders. But the timing of trades which are only “thought of” or “discussed” by the traders becomes unclear; moreover, under the ‘mentalistic’ interpretation, the very representation of the process by means of the «broken line» drawn in Fig. 2, corresponding “to successive barters (at different rates of exchange) of a few nuts for a few apples,” becomes questionable: for, given that no observable trades are assumed to take place during the bargaining process, and given that the talks and discussions between the traders are just a private affair of their own, lacking any public evidence, strictly speaking there is nothing to be represented in the EdgeworthBox diagram, even for purely illustrative purposes, except for the final settlement (denoted in Fig. 2 above, where the superscript M stands for ‘mentalistic’), that is, except for the point in the core eventually arrived at, which is the only observable outcome of the process. As a matter of fact, in MP, where the ‘observable’ interpretation of the trading process is nowhere discussed in any detail, one can find no geometrical description of the path followed by the bargaining process by means of a “broken line” of the type employed in the 1891b paper with the purpose of formalising Marshall’s theory of barter, which certainly presupposes ‘observable’ piecemeal trades; the only points plotted in the EdgeworthBox diagrams appearing in MP are points in the core of the economy. This might seem to support the idea that, in Edgeworth’s view, the «broken line» representation of the trading process only fits the ‘observable’ interpretation, being inconsistent with the ‘mentalistic’ one. We should keep this in mind in discussing the nature and timing of the trading process in Edgeworth’s theory of exchange proper, to which we now turn.
4 – Edgeworth’s theory of exchange
71As will be recalled, Edgeworth’s basic conjecture is that both the extent of the competition among the traders and the determinateness of the outcome following from their interaction increase with the number of traders participating in the economy. In order to prove his conjecture, Edgeworth contrives the device of evenly increasing the traders’ number by repeatedly replicating the original EdgeworthBox economy. Namely, given an EdgeworthBox economy with two traders, ?^{2×2}, where the traders, each with specified characteristics (i.e., consumption set, endowment, and utility function), are respectively labelled A and B, given any positive integer n ? ?_{+}, an nreplica economy, ?^{2×2}_{n}, is an economy where there exist 2n traders, of which n, having the same characteristics as individual A, may be called typeA traders, while the remaining n, having the same characteristics as individual B, may be called typeB traders. As already explained, in discussing the replication mechanism it is convenient to set ?^{2×2}_{1} ? ?^{2×2}.
72Within this framework, Edgeworth is able to (almost) prove three fundamental 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: 38992]; they respectively correspond to Debreu and Scarf’s generalised versions of the original theorems, labelled as Theorems 1, 2, and 3 at p. 2403 of their 1963 paper.
73As far as the “Walrasian Equilibrium Is in the Core” theorem, Edgeworth [1881: 3840, 1126] can easily prove it with reference to the EdgeworthBox economy from which he starts. It is interesting to note that Edgeworth’s proof heavily relies on the Walrasian apparatus of the “demand curves,” appearing as ‘offer curves’ in the Edgeworth’sBox diagram (see Fig. 2 above): for, given that the Walrasian equilibrium (assumed unique) is identified by the intersection of the offer curves of the two traders, it necessarily satisfies both the conditions for Pareto optimality (equality of the marginal rates of substitution) and the individual rationality conditions, hence it belongs to the core of ?^{2×2}_{1}. The extension of this result to an arbitrary nreplica economy with n > 1, however, must be postponed until when the “Equal Treatment in the Core” theorem has been proven.
74This, in effect, is the first theorem to be discussed by Edgeworth in the framework of replicated economies. In this regard, the formal proof put forward in MP [1881: 35] with reference to a 2replica economy, ?^{2×2}_{2}, can easily be generalised to an arbitrary nreplica economy, ?^{2×2}_{n}. Though not providing the required generalisation, Edgeworth takes anyhow for granted the result. By so doing, he feels justified in extending the use of both the contractcurve equation and the geometrical apparatus referring to the original Edgeworth Box economy, ?^{2×2}_{1}, to any nreplica economy with n > 1, ?^{2×2}_{n}: to this end, in fact, it is enough to reinterpret the two traders of the original economy, A and B, as the representatives of their replicated sets or types, each type owning n identical traders, and similarly to reinterpret each twotrader allocation in the contractcurve of the original economy as the average twotype allocation in the contractcurve of the nreplica economy. It should be stressed that the proposed reinterpretation is only justified with reference to core allocations: for only in the core (of the appropriate nreplica economy) does the equal treatment property hold, so that it is only in the core that one can legitimately confound individual traders and trader types. By virtue of this reinterpretation, the same suitably generalised EdgeworthBox diagram can be used to represent both core allocations of individual traders and average core allocations of trader types. Further, since the conditions defining a competitive equilibrium in the twotrader economy are also satisfied in any replicated economy, the competitive equilibrium (assumed unique) of the original EdgeworthBox economy, ?^{2×2}_{1}, can be identified with the competitive equilibrium of any nreplica economy, ?^{2×2}_{n}, so that the «Walrasian Equilibrium Is in the Core» theorem automatically generalises to all replicated economies.
75Yet, while the competitive equilibrium remains essentially the same, except for the size of the economy and, as a consequence, the dimensionality of the equilibrium allocation, the core of the economy does not remain unchanged under the replication mechanism, but, as stated by the «Shrinking Core» theorem, it shrinks to the Walrasian equilibrium as the traders’ number increases without limits. By proving this theorem, one would also succeed in supporting Edgeworth’s conjecture as to the relation between determinateness and size of the economy. Edgeworth’s own sketch of a proof [1881: 357] is as follows.
76First, by means of a geometrical argument based on a simple 2replication of the original EdgeworthBox economy, he proves that “the points of the contractcurve in the immediate neighbourhood of the limits [of that portion of the contractcurve that corresponds to the core of the original EdgeworthBox economy] cannot be final settlements,” hence cannot belong to the core of the 2replica economy, ?^{2×2}_{2}. Let us take, e.g., the rightmost allocation in the core of the EdgeworthBox economy, ?^{2×2}_{1}; in Fig. 3 below this allocation is denoted by x^{1r} = (x^{1r}_{A}, x^{1r}_{B}). Let us now consider the 2replica economy, ?^{2×2}_{2}, which owns 4 traders in the whole: precisely, 2 identical typeA traders, denoted A_{1} and A_{2}, and 2 identical typeB traders, denoted B_{1} and B_{2}. The x^{1r} allocation can now be reinterpreted as a typerepresentative, equaltreatment allocation in the 2replica economy, that is, x^{1r} = (x^{1r}_{A}, x^{1r}_{B}), with and . Yet, by exploiting the strict quasiconcavity of the utility functions, Edgeworth is able to show that a coalition formed by the two typeB traders, B_{1} and B_{2}, who are relatively ‘disadvantaged’ in the x^{1r} allocation, and one of the relatively ‘advantaged’ typeA traders, say A_{1}, can block the x^{1r} allocation by reallocating the coalition endowments in such a way as to get the blocking allocation , satisfying and , where the superscripts 2 and b stand for ‘2replica economy’ and ‘beginning of the blocking process’, respectively; the other typeA trader, A_{2}, would be simply left with his endowment, that is, .
77Yet, the resulting 4trader allocation in the 2replica economy, , which does not satisfy the equal treatment property (as far as typeA traders are concerned) and does not lie on the contractcurve, would be blocked by some other 3trader coalition; the recontracting process would go on until a new allocation located in the portion of the contractcurve to the left of x^{1r} would be reached. However, all the allocations lying on the portion of the contractcurve between x^{1r} and x^{2r} would be eliminated by means of the same blocking procedure as the one by means of which x^{1r} was initially eliminated. The reason why x^{2r} = (x^{2r}_{A}, x^{2r}_{B}), with and , can no longer be eliminated in this way should be clear from the diagram in Fig. 3. For consider the allocation , with , and , where the superscript e in stands for ‘end of the blocking process’; since , , and , no 3trader coalition consisting of one typeA trader, say A_{1}, and the two typeB traders, B_{1} and B_{2}, can improve upon the allocation ; hence the allocation x^{2r} = (x^{2r}_{A}, x^{2r}_{B}), cannot be blocked and remains in the core of the 2replica economy, representing its rightmost allocation. A similar process would have taken place if one had started from the leftmost allocation x^{1l}: in a 2replica economy, all the allocations lying in the portion of the contractcurve between x^{1l} and x^{2l} would be eliminated by the same blocking procedure as before. The set of the allocations in the contractcurve which cannot be eliminated in this way, that is, the set of the allocations lying in the portion of the contractcurve between x^{2l} and x^{2r}, extremes included, is the core of the 2replica economy, or 2core, which is strictly contained in the original core, or 1core.
78Consider now an allocation in the contract curve such that the straight line segment connecting it with the endowment allocation (the SouthEast corner of the EdgeworthBox diagram) cuts the indifference curves passing through it. Any such allocation can be eliminated by a sufficiently large blocking coalition, by means of a suitably generalised version of the procedure suggested by Edgeworth with reference to a 2replica economy. As the replication process proceeds, larger and larger blocking coalitions can be formed, so that more and more allocations in the contractcurve can be eliminated by means of Edgeworth’s blocking procedure. Hence, the replication process generates a nested sequence of cores converging towards the Walrasian (or Jevonsian) equilibrium allocation x^{W} (or, what is the same, x^{J}): in fact, since “the common tangent to both indifferencecurves at the point [x^{W} or x^{J}] is the vector from the [endowment allocation],” the Walrasian (or Jevonsian) equilibrium allocation is the only allocation in the contractcurve which survives Edgeworth’s elimination process when the number of replications, hence of traders, grows unboundedly large. This is enough to prove the “Shrinking Core” theorem [Edgeworth, 1881: 38].
79Edgeworth’s last remark also suggests which are, in his opinion, the truly distinguishing features of the Jevonsian (or Walrasian) equilibrium allocation: among all the final settlements in the original core, the equilibrium allocation is the only one that can be reached in one single step, through one single trade carried out at a uniform rate of exchange, coinciding with Jevons’s equilibrium “ratio of exchange” or Walras’s equilibrium “price.” According to Edgeworth, therefore, price uniformity is an equilibrium phenomenon; as such, it can only hold when the number of traders is “practically infinite,” so that “perfect competition” rules. This conclusion allows one to assess Edgeworth’s stance towards Jevons’s law of indifference and Walras’s law of one price.
80As far as Jevons’s law of indifference is concerned, Edgeworth’s attitude is ambivalent: one the one hand, he rediscovers and endorses the law, making of it a central ingredient of his equilibrium theory, as Jevons had already done before him; on the other, he reproaches Jevons for not being sufficiently clear about the range of application of the law. Edgeworth devotes an entire Appendix of MP [Appendix V: 10416] to the elucidation of the differences and similarities existing between his approach and Jevons’s. The following passage is particularly revealing:
It has been prominently put forward in these pages that the Jevonian ‘Law of Indifference’ has place only where there is competition, and, indeed, perfect competition. Why, indeed, should an isolated couple exchange every portion of their respective commodities at the same rate of exchange? Or what meaning can be attached to such a law in their case? The dealing of an isolated couple would be regulated not by the theory of exchange (stated p. 31), but by the theory of simple contract (stated p. 29).
This consideration has not been brought so prominently forward in Professor Jevons’s theory of exchange, but it does not seem to be lost sight of. His couple of dealers are, I take it, a sort of typical couple, clothed with the property of ‘Indifference’ …an individual dealer only is presented, but there is presupposed a class of competitors in the background.
82As regards the use made by Walras of the law of one price, Edgeworth’s position is highly critical in two related respects. First, as we have repeatedly stressed, price uniformity can hold true for Edgeworth only in the framework of a perfectly competitive equilibrium; therefore, in his view, no justification whatsoever can be offered for assuming, as Walras does, that the law of one price holds not only at equilibrium, but also in a disequilibrium framework, as is the case with Walras’s tâtonnement [Edgeworth, 1881: 301, 478, 109, 116 fn. 1, 143 fn. 1; 1891a: 367, fn. 1; 1910: 374; 1915; 453; 1925b; 312]. Secondly, price uniformity ought not to be directly “postulated,” as Walras does [Edgeworth, 1881: 40], but it should be made to descend from some more basic principle, such as Edgeworth’s principle of recontract:
Walras’s laboured description of prices set up or “cried” in the market is calculated to divert attention from a sort of higgling which may be regarded as more fundamental than his conception, the process of recontract as described in these pages and in an earlier essay [Edgeworth, 1925d, partial English reissue of 1891b]. It is believed to be a more elementary manifestation of the propensity to truck than even the effort to buy in the cheapest and sell in the dearest market. The proposition that there is only one price in a perfect market may be regarded as deducible from the more axiomatic principle of recontract (Mathematical Psychics, p. 40 and context).
84This last remark brings us back to an issue which, having already been discussed with reference to Edgeworth’s theory of simple contract, needs now to be taken up again with reference to his theory of exchange: namely, the issue of the nature and timing of the recontracting process. As will be recalled, in analysing the theory of simple contract, we have discovered the existence in Edgeworth’s writings of two alternative interpretations of recontracting, both consistent with the requirements of the theory: the ‘observable’ interpretation and the ‘mentalistic’ one. The question that naturally arises at this point is the following: Are the two interpretations still consistent with the requirements of the broader theory of exchange?
85To answer this question one has to consider that the EdgeworthBox economy, with just two traders, is peculiar in a number of respects, so that many of its characteristics do not survive, when the traders’ number increases to any integer n > 2. First of all, when n = 2, bilateral trade moves are the only kind of trading activities that can possibly be conceived of and implemented in the economy; on the contrary, when n > 2, multilateral trading, involving more than 2 traders at a time, opens up as a new opportunity. Now, in an EdgeworthBox economy, a sequence of finite bilateral trade moves, which can be geometrically represented by means of a broken line in the EdgeworthBox diagram (such as the path from to in Fig. 2 above), necessarily converges to an allocation in the core of the economy, provided that each piecemeal trade in the sequence is weakly improving for either trader. But in any larger pureexchange, twocommodity economy, if only bilateral trading were to be allowed for, then there would be no guarantee that the trading process would converge to a Paretooptimal allocation [Feldman, 1973]: in order to ensure convergence to an allocation in the suitably generalised Pareto set of the larger economy, multilateral trading must be allowed to take place. This, however, has further consequences that must be spelled out.
86Consider a straightline segment representing a finite bilateral trade move in the broken line that, according to the ‘observable’ interpretation of the trading process, describes the path from the endowment allocation to a core allocation in the EdgeworthBox economy. As has been explained, the absolute value of the 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 of allocations appropriate to a larger economy with n > 2. One can always connect such two allocations with a straightline segment; but, barring special cases, such segment cannot in general be interpreted as describing a bilateral trade move, for multilateral trading is generally required to go from one allocation to another; moreover, no precise meaning could be attached to the geometrical properties of the segment, for the multilateral trading activities required to bring about the assumed shift might involve an unspecified set of bilateral trade moves between any trader in the economy and some other traders or all of them, simultaneously taking place at different rates of exchange.
87Finally, consider the elimination procedure employed by Edgeworth in the proof of the “Shrinking Core” theorem: here an allocation in the Pareto set of an economy with n > 2 traders is blocked by a coalition of (n ? 1) traders who, by reallocating their own resources among themselves, are able to weakly improve their own welfare, at the expenses of the nth trader, whose condition necessarily deteriorates. Now, let us compare the original allocation with the ndimensional allocation obtained by combining the (n ? 1)dimensional allocation of the blocking coalition with the endowment allocation of the excluded trader. In the first place, the latter ndimensional allocation does not weakly improve upon the original one, due to the worsening of the nth trader’s situation; but then the move from the former to the latter allocation does not satisfy the rule that only weakly advantageous shifts should be implemented, a rule underlying the ‘observable’ interpretation of the recontracting process in the EdgeworthBox economy. In the second place, no such move could actually take place if the original allocation had been concretely arrived at through an observable sequence of irreversible trading activities, for, in such case, the nth trader would stick to his original allocation, without accepting any change that would worsen his condition. But this implies that, for the purposes of the elimination procedure put forward by Edgeworth in the context of an economy with a number of traders greater than 2, all trading and bargaining activities should be regarded as purely conceptual and fully reversible; moreover, Edgeworth’s elimination procedure should not be supposed to generate any sequence or path: the blocking allocations ought not to be viewed as points in a sequence or trajectory, but as tentative proposals whose only purpose is to destabilise alternative allocations.
88The above considerations lead us to conclude that the ‘observable’ interpretation of the recontracting process does not generalise from the EdgeworthBox economy to any larger economy with a number of traders greater than 2; only the ‘mentalistic’ interpretation survives the increase in the number of traders. [11] Yet, in spite of this conclusion, one might be tempted, for the purposes of illustrating the recontracting process, to construct paths connecting sequences of allocations located in a space of allocations of appropriate dimensionality. However, for the reasons given above, such attempts, dangerous in themselves, would also prove pointless, for no information (concerning implicit rates of exchange or prices or other variables of economic interest) could be inferred from the proposed illustrations. It is convenient to bear this remark in mind in examining Negishi’s contribution, to which we eventually turn.
5 – Negishi critique of Edgeworth’s approach
89In his 1982 paper, Negishi aims at vindicating Jevons’s law of indifference by showing that, though not so powerful as implicitly suggested by Jevons himself, such law is more powerful than conceded by Walras and Edgeworth, provided that its underlying arbitrage mechanism is duly taken into account. Negishi’s stance on Jevons’s and Walras’s contributions on this issue is similar to that taken by Edgeworth, with a difference: while for Edgeworth, as we have seen, Jevons’s law of indifference holds only in large economies, so that a «practically infinite» number of traders is required for price uniformity and equilibrium establishment, for Negishi no large numbers are necessary for the law to hold, a uniform price to rule, and a Jevonsian or Walrasian equilibrium to obtain. Yet, unlike Jevons, for whom two traders only are apparently enough for price uniformity and equilibrium establishment, an EdgeworthBox economy is not sufficient for Negishi, for in such an economy there would be no room for arbitrage; but a slightly larger economy, such as an Edgeworthian 2replica economy, with only four, pairwise identical traders, would do, for with more than two traders arbitrage becomes possible.
90In 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 2replica economy, plotting its curves and variables in a standard EdgeworthBox diagram (Fig. 4 below). Let x^{1l}, x^{1r}, x^{2l}, x^{2r}, x^{J}, and x^{W} be interpreted as before. Further, let x^{2}_{2} be an allocation lying in the portion of the contract curve between x^{2l} and x^{J} (or x^{W}), with x^{2}_{2} ? x^{J} = x^{W}. According to Edgeworth, such allocation would belong to the 2core, for it could not be blocked by means of the standard mechanism of coalition formation and dissolution at work in a 2replica economy of the Edgeworthian type: as a number of traders greater than 4 would be required to block an allocation like x^{2}_{2}, in a 2replica economy it would tend to persist, once arrived at by whatever route, for want of an effective elimination mechanism.
91Edgeworth’s result is quite standard. Yet Negishi disputes it, for, according to him, it follows from Edgeworth’s excessively conservative interpretation of the coalition concept and the associated recontracting construct. For Edgeworth, as we have seen, a coalition is simply a nonempty subset of traders; a coalition is said to block an allocation when, by reallocating the resources of its own members only, can produce an outcome that is weakly preferable for all its members, and them only, to that implied by the original allocation. For Negishi [1982: 228], however, Edgeworth’s coalitions are just “closed” or “pure” coalitions, for they take into account 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 nonmembers are blurred, the theory should draw its inspiration from reality and learn to employ a wider concept of coalition, too.
92Keeping this in mind, let us now consider how the allocation x^{2}_{2} might have been arrived at. Let us approach the problem step by step. To start with, let us reinterpret the economy under discussion, supposing that it is not a 2replica economy, as imagined up to now, but an EdgeworthBox economy, as in Edgeworth’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 of piecemeal finite bilateral trades taking place over the same trade round. In such a case, we would be forced to conclude that the allocation x^{2}_{2}, lying in the contractcurve, but different from the Walrasian equilibrium, must be reached by a path involving at least two partial piecemeal trades associated with different rates of exchange. Even if there exists an infinite number of paths potentially leading the twotrader economy to x^{2}_{2}, let us suppose, for simplicity, that the actual path consists of just two piecemeal successive bilateral trades, the first leading the traders from ? to x^{2}_{1} and the second from x^{2}_{1} to x^{2}_{2}; the two trades take place at the respective rates of exchange r^{1}_{12} and r^{2}_{12}, with r^{1}_{12} < r^{2}_{12}, where r^{2}_{12} = MRS^{A}_{21}(x^{2}_{2}) = MRS^{B}_{21}(x^{2}_{2}) (see Fig. 4 above).
93Up to this point, we would just be following in Edgeworth’s steps, borrowing 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 to interpret the same broken line from ? to x^{2}_{2}, via x^{2}_{1}, as the path travelled by the four, pairwise identical traders (A_{1} and A_{2}, B_{1} and B_{2}) belonging to the 2replica economy from which we started: Negishi’s idea is that A_{1} trades with B_{1}, A_{2} with B_{2}, and that the respective successive piecemeal trades of the two trading pairs are the same, so that ?, x^{2}_{1} and x^{2}_{2} should now be interpreted as the twocomponent allocations of the representatives of the two tradertypes, that is, , and , 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) is unwarranted, in so far as the suggested change is brought about by means of a mechanical reinterpretation of the same geometrical apparatus: as has been seen 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 abandoned in favour of the ‘mentalistic’ one, where the final allocation is supposed to be arrived at through an unobservable bargaining process, made up of tentative, revocable, multilateral trading proposals, whose specification or geometrical description ought not to be attempted. Hence, with his suggested interpretation of the broken line in Fig. 4, Negishi is trying to mix together some features of Edgeworth’s theory of simple contract with other aspects pertaining to the latter’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 nature and timing of the contracts appearing in his story: on some occasions they are supposed to be «provisional» and revocable, hence presumably simultaneous (or anyhow preliminary) and unobservable; on other occasions, however, they are supposed to be “successive” and capable of being specified in detail, hence presumably observable [Negishi, 1982: 2256].
94Putting provisionally aside these difficulties, and temporarily accepting, for the discussion’s sake, Negishi’s suggested interpretation of the diagram in Fig. 4, let us now see how allocation x^{2}_{2}, unblockable in a 2replica economy according to Edgeworth, can instead be blocked according to Negishi. The basic idea is that what cannot be done by Edgeworth’s UC netratatclosedUC netratat coalitions, can instead be accomplished by Negishi’s UC netratatopenUC netratat coalitions. For example, let us suppose that A_{2} and B_{1}form an “open” coalition {A^{o}_{2}, B^{o}_{1}}, meaning by this that, beyond fully relying on their own resources, A^{o}_{2} and B^{o}_{1} also keep some «links» with the other two traders, A_{1} and B_{2}, into whose resources they also have some limited chance to tap. In order to block the allocation x^{2}_{2}, A^{o}_{2} and B^{o}_{1} proceed as follows: B^{o}_{1} cancels part of his ‘second’ contract with A_{1} at the unfavourable (for him) rate of exchange r^{2}_{12}, while keeping the rest of this contract and the whole of his ‘first’ contract with A_{1} at the favourable (for him) rate of exchange r^{1}_{12}; A^{o}_{2} cancels part of her ‘first’ contract with B_{2} at the unfavourable (for her) rate of exchange r^{1}_{12}, while accepting to offset the suppressed part of her contract with B_{2} by stipulating a new contract with B^{o}_{1} at identical terms, but in the opposite direction (this last assumption being made only for simplicity, for any rate of exchange in the open interval (r^{1}_{12}, r^{2}_{12}) would make both members of the «open» coalition better off). The outcome of this sort of revised recontracting proposal is shown in Fig. 4, where the arrows denote the displacements due to the proposed partial cancellations and new stipulations of contracts, and , and are the final consumption bundles of the four traders, measured with respect to the appropriate individual coordinate axes, at the end of the process imagined by Negishi in his example (the superscript N standing for Negishi). The final allocation of the «open» coalition, , is weakly advantageous for its members with respect to the original one, , for and . Hence, in Negishi’s opinion, the open coalition {A^{o}_{2}, B^{o}_{1}} can block the 2core allocation x^{2}_{2}.
95This sort of revised elimination procedure, based on Negishi’s wider coalition concept and enlarged recontracting construct, applies to all allocations in the contractcurve different from the Walrasian or Jevonsian equilibrium allocation, x^{W} = x^{J}. The reason for this, according to Negishi, is simple: the elimination procedure rests on the exploitation of arbitrage opportunities, whose very existence depends on the coexistence of different rates of exchange at one and the same time; but while all the allocations in the contractcurve different from the equilibrium allocation must be reached in at least two steps, each associated with a specific rate of exchange, so that arbitrage opportunities do exist in this case, the equilibrium allocation, instead, is arrived at in one single step, associated with one single rate of exchange, so that no arbitrage opportunity can arise here. Hence, while all the allocations other than the equilibrium one would succumb to the revised elimination procedure, the Walrasian or Jevonsian equilibrium allocation would live through it. Contrary to Edgeworth’s conclusions, therefore, price uniformity and equilibrium establishment would not require a “practically infinite” number of traders: Jevons’s law of indifference would assert itself in a finite 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
96The above story, however, is seriously faulty. In the first place, the issue of the timing of the trading process, provisionally set aside above, must now be taken up again. In Negishi’s example four piecemeal bilateral trades are supposed to take place to start with, involving two pairs of traders (either A_{1} and B_{1}, or A_{2} and B_{2}) and two steps (say, step 1 and step 2): for each pair of traders there are two trades at different rates of exchange, one for each step; likewise, for each step there are two trades at the same rate of exchange, one for each pair of traders. Now, at each step there can be no arbitrage, since the rate of exchange is the same for the two trades occurring at the same step [12]. So, one is led to think of arbitraging activities between different steps, since the rates of exchange differ as between trades occurring at step 1 and 2, respectively. Yet, if the steps are thought of as “successive” in ‘real’ time and the contracts are regarded as irrevocable, how can arbitrage occur? So the steps must be conceived of as «successive» only in the ‘logical’ time characterising a preliminary verbal session taking place among the traders before any observable actions are carried out; by the same token, the contracts under discussion must be viewed as “provisional” and revocable without cost or penalty.
97Now, with reference to a finite economy as the one described by Negishi, Edgeworth would readily acknowledge that different rates of exchange typically characterise different trades among different pairs of traders: for, in his opinion, such multiplicity of rates of exchange is the distinguishing feature of any finite economy; arbitrage goes as far as it can, but is unable to produce price uniformity unless the number of traders is unboundedly large. The reason for this is that, in Edgeworth’s view, arbitrage can only operate through recontracting, that is, through the mechanism of coalition formation and dissolution. But, in a finite economy, the number of coalitions that can form to the advantage of all their members is bounded, so that the power of arbitrage in smoothing out price differences is limited, too. Negishi believes that this constraint can be easily bypassed, by allowing for a simple extension of the coalition concept, from Edgeworth’s “closed” to his own “open” coalition concept. Yet, a number of fundamental objections can be raised against Negishi’s use of the “open” coalition concept for his purposes.
98Let’s go back to Negishi’s example. One can easily check that the true members of the “open” coalition, A^{o}_{2} and B^{o}_{1}, can (weakly) increase their utility levels only by “exploiting” the other two traders, A_{1} and B_{2}, with whom they keep some “links”: in effect, the utility levels of the other two traders necessarily decrease as a result of the revised recontracting proposal of the “open” coalition members, namely, and . Yet, since an “open” coalition can improve the welfare of its members only at the expenses of the welfare of nonmembers, Negishi’s concept of an “open” coalition immediately appears as much weaker than Edgeworth’s “closed” coalition concept, for in a “closed” coalition all improvement can only come from the reallocation of the resources owned by the coalition members.
99Negishi [1982: 228] recognises that an “open” coalition is less stable than a “closed” one, but he adds that “the stability of the realised coalition itself is not required to block an allocation in Edgeworth’s exchange game.” Yet, if Negishi 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 he forgets that the credibility of the threat made by a blocking coalition, which rests precisely on the autarchic character of Edgeworth’s “closed” coalitions, is essential for blocking. Moreover, “open” coalitions are supposed to rely on the unawareness of the cheated partners [Negishi, 1982: 228]. Yet, if cheating is allowed for, and assumed not to be spotted, provided that the extent of the swindle is limited, then even a Walrasian or Jevonsian equilibrium allocation would not be immune from the destabilising power of “open” coalitions. In the end, therefore, Negishi’s approach appears to oscillate between proving too little (for the threats of “open” coalitions are not credible, so that no allocation could be blocked in this way) and proving too much (for, if cheating is allowed for and deemed effective, then any allocation, including the equilibrium one, could be destabilised in this way).
7 – Concluding remarks
100Jevons, Walras, and Edgeworth develop their pureexchange equilibrium models over the decade 187181. All three of them make use of some version of a law, called law of indifference (or principle of uniformity) by Jevons and Edgeworth and often referred to as the law of one price in connection with Walrasian economics. Writing after Jevons and Walras, Edgeworth can and does explicitly refer to their theories, either to praise or to criticise them. In particular, Edgeworth [1881] shares with Jevons the idea that the law of indifference must be regarded as an equilibrium property. Unlike Jevons, however, he denies the validity of the law in all the economies where the traders’ number is short of the «practically infinite». In this regard, he tells a plausible story, based on the joint operation of his replication and recontracting mechanisms, a story meant to support the emergence of the law, as well as the convergence of the economy towards a Jevonsian or Walrasian equilibrium, as the traders’ number grows unboundedly.
101About one century later, Negishi [1982] resumes the timehonoured discussion about the law, taking an unconventional stance. Precisely, Negishi strives to prove that, contrary to Edgeworth’s original conjecture, a competitive equilibrium can be attained even in small economies, provided that the true driving force underlying Jevons’s law of indifference, namely, its implicit arbitrage mechanism, is allowed to operate and carry its effects through. Yet, in the paper we have shown that Negishi’s proof is unconvincing, for he tries to exploit Edgeworth’s machinery without preserving the latter’s carefully chosen assumptions and qualifications.Moreover, Negishi’s concept of an “open” coalition, meant to generalise Edgeworth’s concept of a “closed” coalition, is so weak and inconsistent that its associated blocking mechanism is wholly untrustworthy.
102The failure of Negishi’s attempt confirms, in the last analysis, that Edgeworth’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 Edgeworth. Stronger results may be hoped for only by further specifying the trading technology, 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 assumptions adopted.
Notes

[1]
I thank Claire Pignol and Antoine Rebeyrol for their insightful comments on preliminary drafts of this paper and their invaluable advice in talks and correspondence. The usual disclaimer applies.

[2]
Department of Economics, Management and Quantitative Methods, UniversitUC netratat degli Studi di Milano. Email : franco.donzelli@unimi.it

[3]
The diagrams of the pureexchange, twotrader, twocommodity economy actually appearing in MP [1881: 28, 114] are drawn in the fourcoordinate space of the traders’ net trades, rather than in the fourcoordinate 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 we have preferred to keep to the currently prevailing representation. In any case, the choice of which representation to adopt is immaterial, since there exists a onetoone correspondence between the two.

[4]
The superscript 1 is there to remind the reader that the EdgeworthBox economy under question is an unreplicated twotrader economy. In the next Section, in discussing Edgeworth’s replication mechanism, we shall find it convenient to identify the unreplicated EdgeworthBox economy, ?^{2×2}, with the fictitious 1replica economy, ?_{1}^{2×2}.

[5]
See 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) appears in Volume II of Edgeworth’s Papers Relating to Political Economy [1925d]. Yet, for our present concerns, what really matters is the full Italian version, seemingly ignored by later critics and interpreters, rather than the abridged English reissue.

[6]
As explained by Guillebaud’s editorial notes to the ninth (variorum) edition of the PrinciplesMarshall, 1961b: 790791], from the fifth edition [1907] onwards, the subjectmatter of the Appendix “A Note on Barter” was moved, without any substantial alteration, from the end of Book V, chapter II, to the final section of the Principles, grouping all the Appendices, where it became “Appendix F. Barter” [Marshall, 1961a: 791793].

[7]
The exchanges of letters between Marshall, Berry and Edgeworth, reprinted in Marshall [1961b: 791798], should also be consulted. On the controversy see Newman [1990], Creedy [2006], and Donzelli [2008: 1530, and 2009: 3236].

[8]
The sentence quoted in the text, drawn from the English version of the paper, as reissued in the 1925 collection of Edgeworth’s economic papers, does not exactly coincide with the 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 is quoted in full in the Italian version, but is suppressed in the English one.

[9]
Edgeworth’s formalisation of Marshall’s theory of barter by means of an EdgeworthBox diagram and the associated conceptual apparatus exactly corresponds to what Marshall had had in mind, but had only been able to express by means of a numerical example. As a matter of fact, Marshall himself explicitly recognised the faithfulness of “Prof. Edgeworth’s mathematical version of [Marshall’s own] theory of barter” by incorporating Edgeworth’s diagram and formalism in Note XII bis of the Mathematical Appendix of Marshall’s Principles [1961a: 844845]. This Note, added to the second edition of the Principles appearing in 1891, soon after the publication of Edgeworth’s 1891b paper, will be kept in all the following editions without substantial alterations [Marshall, 1961b: 834].

[10]
This footnote belongs to a part of the 1891 paper in Italian [1891b] that is not reproduced in the later abridged English reissue of the paper [1925d].

[11]
It is worth noting that the ‘mentalistic’ interpretation of the recontracting process is the only one to be explicitly accounted for by the few economists rediscovering and discussing Edgeworth’s contribution during the 1930s, after half a century of generalised neglect; see, in particular [Kaldor, 1934: 127; Hicks, 1934: 342; 1939: 128]. Many decades before, however, Berry, both in his reply to Edgeworth’s 1891b paper in Italian [Berry, 1891: 552] and in a private letter addressed to Edgeworth published in Marshall [1961b: 794], had suggested yet another interpretation of the recontracting process, according to which the recontracting process, rather than taking place over one single isolated trade round, would take place over a sequence of trade rounds under unchanging economic conditions. Berry’s ‘stationary’ interpretation differs from the ‘mentalistic’ one, for the contracts stipulated in each 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 assumed stationarity of the economic data over the sequence of trade rounds involved in the recontracting process and the assumed impermanence of both the contracts and their effects beyond the boundaries of the trade round in which the contracts themselves are stipulated, in the last analysis it secures that revocability of contracts which is the hallmark of the ‘mentalistic’ interpretation. While, in his rejoinder to Berry, Edgeworth [1891c: 317] displays his preference for the ‘mentalistic’ interpretation over the ‘stationary’ one, in his later papers [1904:40; 1907: 526 ff., 1925c: 313] he appears to regard the two interpretations as essentially equivalent and interchangeable.

[12]
The assumption that the rate of exchange and the quantities exchanged are the same for the two trading pairs at each step, implying that arbitrage has already carried its effects through at the same step, is made by Negishi in order to simplify his illustration: for, under Negishi’s assumption, the same broken line representing a simple twostep ‘observable’ trading process taking place in a twotrader economy can be employed to represent a twostep trading process taking place in a 2replica economy with four, pairwise identical traders. Relaxing Negishi’s simplifying assumption would indeed enrich the picture, reopening the issue of arbitrage at each step. Yet, as will be seen in a moment, the difficulties arising from such extension would not differ in kind from those to be discussed in the text with reference to the treatment of arbitrage across different steps.