1Among the critical remarks made by Antoine Rebeyrol in his comment on my paper in the current issue of Cahiers d’économie politique the most important are the following two. First, he criticises my reconstruction of Edgeworth’s “theory of recontracting” as overly generous for the following reason: after duly recalling Edgeworth’s distinction between the so-called “theory of simple contract,” concerning two traders only in the context of an un-replicated Edgeworth-Box economy, and the so-called “theory of exchange” proper, concerning more than two traders in the context of replicated Edgeworth-Box economies, I would fail to recognise the “contradictory” character of such two theories. Secondly, he reproaches my critical reading of Negishi’s 1982 contribution as overly severe for, while overstressing a few minor drawbacks in Negishi’s discussion, I would fail to appreciate the true strength of his case: according to Rebeyrol, in fact, by cleverly exploiting the properties of the Cournot-Walras theory of arbitrage, Negishi would have succeeded in vindicating the use of the competitive equilibrium apparatus not only, as originally argued by Edgeworth, in the framework of unboundedly large economies, but also in the framework of small economies with a few traders only (just four traders, according two Negishi, who can be further reduced to two traders only, according to Rebeyrol’s reinterpretation of Negishi’s contribution). In this reply, I will try to show that both objections are misplaced.
2Let’s tackle the first issue first. According to Rebeyrol, Edgeworth’s “theory of simple contract” (henceforth, TSC) and his “theory of exchange” proper (henceforth, TEP) “are not only different but even contradictory”: for, while the former “excludes recontracting,” the latter is based on it; moreover, while in the former “there is a process” whose “dynamical study…is easy to clarify, no temporality is involved in [the] latter model: there is no path, no process at work.” Yet, both the distinctions drawn by Rebeyrol can be shown to be unfounded.
3As to the use (or rejection) of the recontracting apparatus, the first point to be stressed is that, while Edgeworth’s writing style is usually far from crystal-clear, the rules of recontracting, including the rules governing coalition formation and dissolution, represent an agreeable exception in this respect: for such rules are precisely stated right at the beginning of the part of Mathematical Psychics devoted to so-called problem of “Economical Calculus” [Edgeworth, 1881: 16–9], well in advance of the introduction and formal discussion of the distinction between TSC and TEP; it should be clear, therefore, that such rules apply to both “theories.” And, in effect, when Rebeyrol states that in “the theory of ‘simple contract’ […] Edgeworth no longer thinks in terms of coalitions,” he makes a mistake: for Edgeworth’s “set of final settlements,” i.e., that portion of the “contract curve” that represents, in modern terms, the core of an un-replicated Edgeworth-Box economy (henceforth, EB economy), is nothing but the set of the allocations that are not blocked by any one of the only three coalitions that can possibly be formed in such an economy (namely, the grand coalition of the two traders and the two individual coalitions, each consisting of one trader only). In conclusion, contrary to Rebeyrol’s claim, the process of recontracting, in Edgeworth’s sense, underlies TSC no less than TEP.
4This last statement naturally leads us to discuss the second issue raised by Rebeyrol with reference to Edgeworth’s distinction between TSC and TEP, namely, the alleged existence of a striking contrast between the temporality (dynamical character) of the former and the atemporality (statical character) of the latter. As will be shown, however, such supposed contrast is unsubstantiated, too.
5In Edgeworth’s view, in fact, recontracting is a process (preferably, but not exclusively, interpreted as a purely verbal, mentalistic process) which characterises both “theories” alike. Though insisting that recontracting is a process, however, Edgeworth is adamant in claiming that no general dynamic deterministic theory of such process is available nor, he suggests, can be made available in the foreseeable future. What the theorist can only do, in this respect, is to provide some illustrations of the process, i.e., he can describe possible sequences of allocations tentatively reached and subsequently discarded by the traders and their coalitions during the process. In principle, this can be done whatever the number of traders in the economy. Yet, in practice, the few illustrations of the recontracting process which are provided by Edgeworth over all his long scientific career concern the un-replicated EB economy only. Such illustrations are purely verbal in Mathematical Psychics (1881) and in Edgeworth’s 1904 paper on the theory of distribution; the only geometrical illustration of a possible path followed by the traders during the recontracting process is put forward in Edgeworth’s 1891 paper on Marshall’s theory of barter, where the path leading from the endowment allocation to a core allocation is represented by means of a broken line in an EB diagram. But this, contrary to what Rebeyrol seems to suggest, is not due to any fundamental theoretical difference between TSC and TEP, but only to the trivial fact that, with two traders only, a simple geometrical method is at hand to depict the path in an EB diagram, a method that cannot be employed with any number of agents greater than two.
6Edgeworth’s 1891 geometrical illustration of a possible path leading to a core allocation in an un-replicated EB economy deserves some additional comments, for it has to do with a few further misunderstandings possibly underlying Rebeyrol’s note. When Edgeworth develops his geometrical illustration of a possible path in his 1891 paper, he is very careful in specifying a sequence of trade moves that strictly abide by the rules of recontracting. Among other things, such rules also imply that any allocation in the interior of the area of exchange which the two traders should happen to reach by way of appropriate trade moves at some step in the process cannot be blocked by the decision of an individual trader alone: with only two traders in the economy, in fact, there is no third party who can be “left out in the cold” [1881: 37], so that the dissolution of the grand coalition would worsen the situation of both traders alike with respect to the situation attained by them at any allocation in the interior of the area of exchange (this is so because, in this case, the dissolution of the grand coalition caused by the unilateral decision of either trader would bring both traders back to their original endowments, to which lower utility levels are associated). This means that any allocation in the interior of the area of exchange, but not belonging to the core, can only be blocked by the grand coalition, provided that the two traders agree upon a trade move producing a reallocation of the aggregate resources bringing about an improvement for them both. Due to this reason, all the steps in a path illustrating the recontracting process in an un-replicated EB economy are necessarily “irreversible,” in the sense that, since both traders’ utilities must increase (or at least not decrease) along the sequence, no trade move connecting any two allocations along the path can be “reversed.” Rebeyrol rightly hints at such “irreversibility,” but wrongly associates it with a totally unrelated phenomenon, the so-called phenomenon of non-tâtonnement, which has nothing to do with the “irreversibility” under question: the latter, in fact, is simply due to the strict application of the rules of recontracting to the situation at hand, once again proving that Rebeyrol’s claim that “Edgeworth’s ‘theory of simple contract’ excludes recontracting” is groundless.
7Let us turn now to the objections raised by Rebeyrol against my critique of Negishi’s approach. Rebeyrol is right in stressing that “[a]rbitrage is an essential feature of competition and as such cannot be neglected.” In this regard, though recognising that “economists [still] miss an appropriate formulation of the arbitrage process,” he appears to appreciate Negishi’s attempt to show that, contrary to Edgeworth’s original viewpoint, “the action of…a single arbitrageur is enough to dismiss core allocations which are not Walrasian equilibrium, whatever the total number of agents.” If accomplished, this task would indeed represent a remarkable achievement: for it would provide an extremely powerful underpinning for the competitive (Walrasian) equilibrium construct, an underpinning that would be free from Edgeworth’s demanding requirement of the existence of an unboundedly large number of traders in the economy.
8Yet, as I think I have been able to conclusively show in Sections 5 and 6 of my paper, Negishi is far from attaining his target: his attempt, in fact, is so full of ambiguities and mistakes (about the timing and nature of the trading process, the character of the so-called “open” coalitions, the credibility of such coalitions’ threats, etc.) as to make his endeavour a complete failure. On top of its shortcomings, however, what is really paradoxical in Negishi’s argument is that, while paying lip service to the effectiveness of the arbitraging process in bringing about the competitive equilibrium outcome, he makes no use whatsoever of anything like an arbitraging activity in developing his alleged explanation of the convergence of any economy, even a small one, to a Walrasian equilibrium allocation.
9As a matter of fact, Negishi’s explanation relies on a crucial example so constructed as to show that, even in a simple 2-replica EB economy with four traders only, all core allocations but the Walrasian ones can be blocked by appropriately specified “open” coalitions, consisting of two ‘winners’ (presumably viewed by Negishi as the arbitrageurs) and two ‘losers.’ In Negishi’s example, the two ‘winners’ are supposed to be able to persuade the ‘losers’ to sign with them, to the ‘losers’’ own detriment, downsized versions of the same contracts as the ones they had already signed before, at the same rates of exchange as before. The ‘winners,’ in Negishi’s own words, “exploit” the ‘losers,’ taking advantage of their “unawareness,” which might be better called ‘foolishness.’ Rebeyrol resents my use of the word “cheating” to qualify the ‘winners’’ behaviour. But how should one denote the behaviour of somebody who is able to convince somebody else to ‘voluntarily’ sign a contract which is bound to positively worsen the latter’s situation with respect to the status quo? Yet, with regard to Negishi’s example, it is even more important to raise the following questions: How can one meaningfully speak of an arbitraging process, when no change in the rates of exchange is involved in passing from the old to the new contracts? Where does such a alleged arbitraging activity lie?
10Now, Rebeyrol seems to agree that Negishi’s approach is not entirely satisfactory. Yet, instead of rejecting it in view of its shortcomings, he takes the opposite route: namely, he proposes to strengthen Negishi’s claims further by showing that, even in a simple un-replicated EB economy with two traders only, all core allocations but the Walrasian equilibrium ones can be blocked by appropriate coalitions consisting of one individual trader only.
11Rebeyrol’s idea can be spelled out as follows. Let x = (x1, x2) be a core allocation in an un-replicated EB economy, supposedly reached by the two traders participating in the economy at the end of an unspecified bargaining process; the two traders are indexed by i = 1, 2. In the spirit of the II Fundamental Theorem of Welfare Economics, for any such Pareto-efficient allocation one can identify a shadow relative price, say p12, coinciding with the common Marginal Rate of Substitution between the two commodities (henceforth, MRS) characterising the two traders at x. According to Rebeyrol, the normalised shadow price system implicitly defined by the traders’ preferences at x, say p = (p12, 1), can be regarded as an “objective” measuring rod, by means of which trader i’s core allocation, xi, and initial endowment, ei, can be evaluated. By making use of such assessments, each trader can compute his individual gain index, gi = p · (xi – ei), the two indexes having the following properties: ?igi = 0, for all p, xi, and ei, and gi = 0 if and only if xi is trader i’s Walrasian equilibrium allocation, for i = 1, 2. By means of such index, each trader can “objectively” determine whether he is a ‘winner’ (if gi > 0) or a ‘loser’ (if gi < 0) at the end of the bargaining process leading to x. Finally, since any ‘loser’ would block the corresponding core allocation, all core allocations which are not Walrasian equilibrium ones would be blocked in this way.
12Yet Rebeyrol’s argument is defective in at least two respects. First, since each trader’s MRS is a local concept, whose value varies with the allocation with which it is associated, there is no reason why the common MRS associated with any specific core allocation should be regarded by any trader as a global, “objective” measuring rod for evaluating whatever allocation, hence for determining which trader is a ‘winner’ or a ‘loser’ at the end of the bargaining process concerned; as a matter of fact, in the applications of the II Fundamental Theorem of Welfare Economics, from which Rebeyrol draws his inspiration, there is nothing “objective” in the price system selected by the planner to support any specific Pareto-efficient allocation as a price equilibrium with transfers, nor would the traders spontaneously employ those prices in order to compute the value of their endowments and to determine the required transfers if such prices and transfers were not imposed upon them by the planner. Secondly, any alleged ‘loser’ who were to block the core allocation for which his gain index turns out to be negative would fall back to his endowment allocation, incurring a utility loss; but this, according to both Edgeworth’s rules of recontracting and common sense, is precisely the reason why the trader concerned would never block such allocation. In support of his argument, Rebeyrol quotes a passage drawn from Edgeworth’s 1904 article on the theory of distribution. Yet, contrary to what Rebeyrol seems to suggest, in that passage Edgeworth is by no means concerned with the possibility that core allocations be blocked by some unsatisfied trader; rather, what he is really concerned with are the first few steps in a recontracting process supposedly leading the two traders to a core allocation in an un-replicated EB economy. Therefore, in Edgeworth’s illustration, the allocation blocked by the grand coalition at the second step of the process, when the two traders eventually realise that “the system of bargains entered into on the first occasion does not fit the real dispositions of the parties” [Edgeworth, 1904: 40], is certainly not a core allocation, as in Rebeyrol’s argument, but an allocation in the area of exchange where the mutually advantageous opportunities offered by trade have not yet been completely exploited, i.e., an allocation where the MRSs of the two traders still differ from one another.
