1 Our comment on Franco Donzelli’s article focuses on two points . First, the ‘theory of exchange’ and the ‘theory of simple contract’ proposed by Edgeworth are shown to be contradictory. Second, the importance of arbitrage and the relevance of Negishi’s objection to Edgeworth’s competitive views are stressed.
I - ‘Theory of Exchange’ versus ‘Theory of Simple Contract’
2 Donzelli’s way of representing Edgeworth’s contribution deserves credit as it clearly distinguishes two cases: on the one hand, isolated exchange of two goods between two agents (called “Edgeworth’s theory of simple contract”), and on the other hand, the replications of this basic economy (named “Edgeworth’s theory of exchange”). According to Donzelli, they are “two separate theories,” but he unfortunately goes back on this view when he states that both theories “rest on the same ‘fundamental’ principle, namely, the ‘principle of recontract.'...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.” In that case, one would actually have one theory and not two; i.e. the theory of exchange generalizing the theory of simple contract, which is only a particular case of the former one. Nonetheless, when it comes to Edgeworth’s works, such a homogeneity is questionable.
3 Let’s consider these two cases through a comparison with the Walrasian tâtonnement. In the theory of exchange, the contracts considered by a coalition are cancelled when at least one agent finds a better arrangement in this coalition or in another one. Seen in that light, the Edgeworthian recontracting is similar to the Walrasian tâtonnement, where agents recalculate their supplies and demands when new prices are given: in so doing, agents reevaluate the same initial endowments and cancel their previous plans. Yet the comparison doesn’t extend any further than this, because the tâtonnement calls for a dynamical equation relating price changes to excess demands, even if the process rests on a purely virtual time and excludes disequilibrium transactions. On the contrary, any idea of a series of coalitions and of allocations is meaningless in Edgeworth’s theory of exchange, as rightly shown by Donzelli. What matters is completeness: any possible allocation in any possible coalition must be considered. By contrast with Walrasian tâtonnement, Edgeworth’s theory is often praised for the weakness of its institutional hypotheses, but it is also considered as heavily information consuming. Between Edgeworth and Walras, there would be a trade-off between institutional generality and informational efficiency; however, our main point is different. The law of supply and demand gives a dynamical structure to the Walrasian tâtonnement, while the exploration of the whole range of possibilities in the Edgeworthian recontracting must be complete, whatever its order. No temporality is involved in this latter model: there is no path, no process at work. There is only a large set of comparative statics operations (things working the same way in the proof of the so called ‘convergence’ of the core towards the set of Walrasian allocations). As is clear from Debreu and Scarf , there is no temporality in the recontract model.
4 However, when he turns his attention to the theory of ‘simple contract’ or isolated exchange, Edgeworth no longer thinks in terms of coalitions, as shown in Mathematical Psychics or in his article on Marshall, for instance. “Why, indeed, should an isolated couple exchange every portion of their respective commodities at the same rate of exchange?” [1881: 109] asks Edgeworth, raising a question which is ignored by modern cooperative game theorists. On these grounds, he was able to formulate his well-known and deep criticism of Walras, stating that the law of one price can’t be assumed in disequilibrium. In his view, both isolated traders reach a core allocation following a path that resembles a broken line. Along such a path, there exists a sequence of nested lens-shaped areas, delimited by the agents’ indifference curves, which intersect one another at the point they just attained. To make sure that such a process is indeed at stake, let’s imagine switches in the order of the segments connecting the initial allocation to the final one: such rearrangements would not ensure each elementary move to be Pareto improving. Moreover, in order to maintain Edgeworth’s fundamental indetermination result, it is necessary to assume that agents are committed to past exchanges that cannot be questioned. This is true whatever the interpretation of the theory of simple contract, whether as a sequence of actual exchanges in real time, or as a sequence of purely verbal agreements. Even in this latter case, negotiation never moves backwards and the indetermination of the ‘final settlement’ crucially depends on the irreversibility of time. Edgeworth’s ‘theory of simple contract’ excludes recontracting: it is simply a non-tâtonnement process.
5 Edgeworth asserts the existence of paths impossible to be theorized: “[We] have no general dynamical theory determining the path of the economic system.” [1925: 311] According to him, “la direction que suit le système pour arriver à la position d’équilibre ne rentre pas dans la sphère de la science.” [1891a: 364] With regard to his own analysis, Edgeworth’s suggestion is here deeply ambiguous, since the absence of a path in his theory of exchange in a replicated economy basically makes any dynamical analysis groundless. By contrast, in his theory of simple contract, there is a process and its dynamical study, which is easy to clarify, and is not about the stability of one predetermined equilibrium but is rather concerned with the stability of the way leading to a path-dependent final settlement. The two theories, a static one and a dynamic one, are not only different but contradictory, as one is based on the recontracting the other rejects.
II - Arbitrage and Negishi’s Objection
6 The critical significance of this dehomogenization of Edgeworth has to be assessed. Takashi Negishi  addresses an explicit criticism of this theory of exchange in replicated economies. His point is obviously not the theorem showing the shrinking of the core: one may always define the core of an economy as the set of allocations upon which no coalition (à la Edgeworth or à la Debreu-Scarf) can improve, and show that it shrinks to the set of competitive equilibria when the replication of the economy is indefinitely reproduced. Nevertheless, we have to question the relevance of this representation Edgeworth describes as “the process of recontract [which] is believed to be a more elementary manifestation of the propensity to truck then even the effort to buy in the cheapest and sell in the dearest market.” [1925: 311–2] He thinks he has discovered the very ground of competition: according to him, recontract “is a sort of higgling which may be regarded as more fundamental than [the Walrasian] conception.” [1891a] Firstly, this view is difficult to understand because there is no process in his theory of exchange (as we already mentioned). Secondly, Negishi proposes that the simple acknowledgement of this “propensity to buy in the cheapest and sell in the dearest market,” as it appears in the eminently competitive behavior of arbitrageurs, is sufficient to challenge the indetermination of the end of the process when the number of agents is finite.
7 Arbitrageurs are agents aiming at price exploitation. They inquire into prevailing implicit prices, which are hidden in the black box of multilateral trading (or in the opacity of financial markets), to see if they are consistent-or alternatively, if they can generate a gain through the pure game of exchange, when arbitrage opportunities appear. Such a gain may only appear in trades with nonarbitrageurs (defined as agents who did not identify these exchange opportunities) because this is a zero-sum game: the nonarbitrageurs would refuse the trade if they were aware that they merely stand to lose what arbitrageurs win. Formalizing an arbitrage process is difficult, for several reasons. The first one is the existence of nonarbitrageurs: why have they not discovered the exchange opportunities, if they exist? The second one is the potential unboundedness of the arbitrageurs’ gains, measured in any numéraire they may want to adopt: no equivalent of the Fisher separation theorem (about the independence between the production decisions in terms of net present value of the firm and the consumption decisions based on preferences) holds, because in the presence of arbitrage opportunities, the maximization of any arbitrageur’s wealth leads to an unbounded result. A chrematistics dedicated to arbitrages and linked up to the economics of needs is still missing from economic theory. Such a piece of theory should feature the equilibrium formation process, exhibiting how price moves can at the same time ensure their mutual consistency and balance markets. How do economists react to the lack of such a theory of equilibration involving arbitrages? All of them agree on the following point: one may sometimes accept the assumption of an institutional device which immediately eliminates arbitrage opportunities, thanks to the introduction of money, or the introduction of a numéraire for prices in a situation involving more than two commodities. Otherwise, an equilibrium condition is required in order to define the value of things; and such a condition does exclude every arbitrage opportunity. For any market equilibrium to be conceivable, an arbitrage-free market is needed. So even if economists miss an appropriate formulation of the arbitrage process, they agree that equilibrium, as a lasting situation, cannot coexist with remaining arbitrage opportunities.
8 Negishi points out that Edgeworth’s position is very singular, as he shows that arbitrage opportunities are involved in the way leading to one of Edgeworth’s “final settlements” (Walrasian equilibrium excepted). This criticism is justified: in the simple case of two goods, it is easy to see that reallocations inside a coalition are generally implemented by inconsistent implicit prices, providing incentives to buy and sell the same good. These prices will also differ between coalitions, with the same result. Could the indefinite replication of the economy mimic the arbitrageurs’ activity? That would be deceptive, because the action of at least a single arbitrageur is enough to dismiss core allocations which are not part of the Walrasian equilibrium, whatever the total number of agents. Donzelli’s critique of Negishi is irrelevant. Arbitrageurs gaining at the expense of other agents are not cheats: the terms of quid pro quo and voluntary exchange are always satisfied. Moreover, they don’t transgress any budget constraint, since a consistent price system is required to formulate such a constraint. It would be a mistake to assert that arbitrageurs “could destabilize the [Walrasian] equilibrium allocation.” On the contrary, Walras  showed after Cournot  that no arbitrage gain is possible in equilibrium. According to Walras, arbitrage activity leads to its own vanishing, and this is the usual view among economists.
9 We nonetheless concur with Donzelli that Negishi’s formulation of arbitrage is partly disappointing because he “tries to exploit Edgeworth’s analytical weaponry against Edgeworth’s own conclusion.” Negishi indeed analyzes arbitrages in terms of coalitions, while an arbitrageur does not need to be part of any coalition. He is not involved in any cooperative behavior and instead acts as an opportunist, taking advantage of price inconsistencies. This does not mean that Negishi’s critical assessment regarding Edgeworth’s replicated economies is not useful: on the contrary, it should be extended beyond replicated economies, to the simple case of isolated exchange.
10 Is it conceivable for a core allocation resulting from a non-tâtonnement process to be an enduring issue between two agents? In the line of what Donzelli calls “Berry’s ‘stationary’ interpretation” of recontracting, let’s quote Edgeworth [1904: 40]:
“On the first day in our example a set of hirings are made which prove not to be in accordance with the disposition of the parties. These contracts terminating with the day, the parties encounter the following day, with disposition the same as on the first day,—like combatants armis animisque refecti,—in all respects as they were at the beginning of the first encounter, except that they have obtained by experience the knowledge that the system of bargains entered into on the first occasion does not fit the real disposition of the parties.”
12 Let’s consider the agents after they reached some degree of core allocation by the evening of the first day: in the terms used by Edgeworth in his paper on Marshall, they have experienced “shortages or plenties,” insofar as the core allocation is not a Walrasian equilibrium. Starting from an initial position displaying divergent subjective appreciations (their unequal MRS), they reached some degree of core allocation, coming finally to an agreement about an objective appreciation of goods (their equal MRS being an implicit consistent price system). Using this final common knowledge appreciation, each of them is able to compute the gain index gi=p(xi − ei), where p is the vector of these objective appreciations, xi is agent i's final allocation in the core and ei is his initial endowment. This measure, which is nil for every i at the Walrasian equilibrium, is equal to the transfer an agent would obtain from a planner wanting to implement the allocation reached in the core. Indeed the index gi is not measured in utility but in numéraire: it is a measure of the noncooperative part of exchange (with only two agents, each one can calculate the same index for the other, as the sum of these values is necessarily nil.) Any agent can know, according to the sign of this magnitude, if he recorded a loss or a gain through the plurality of prices, i.e., if he actually behaved as an arbitrageur or as a nonarbitrageur in the sequence followed during the previous non-tâtonnement. Now, let’s suppose that the initial situation is reproduced the day after. The agent who “obtained by experience the knowledge” of a loss would have learned enough about the “real disposition of the parties” to refuse the repetition of the final allocation of the previous day. The indefinite reproduction of this allocation can only rely on the irrational acceptance, by one of the agents, of a loss he is yet able to objectively measure. Only the Walrasian general equilibrium can resist the test of indefinite repetition in stationary conditions.
13 One can question the pertinence of Edgeworth’s representation of perfect competition, resting on an infinite number of agents involved in coalitions and excluding the activity of arbitrage. Do arbitrageurs not accurately picture the competitive agent? Arbitrage is an essential feature of competition and, as such, cannot be neglected. Yet it is inconsistent with Edgeworth’s theories since, thanks to arbitrages, non-Walrasian core allocations cannot persist, whatever the number of agents. Negishi was right to stress this point. He should also be praised for his attempt to develop the analysis of arbitrages, even if it may appear as unsatisfactory. Since Walras , it seems like no significant progress has been made in the analysis of the processes performing the condition of absence of arbitrage opportunity. Yet this question deserves a new consideration, if we want to reach a deeper understanding of competitive processes.
EconomiX, University of Paris Ouest Nanterre La Defense. E-mail: email@example.com
I would like to thank Carlo Benetti and Fabrice Tricou for helpful comments and suggestions.