1 - Introduction
1 According to Stigler, and somewhat stating the obvious, , “competition is a rivalry between individuals...and it arises whenever two or more parties strive for something that all cannot obtain” [Stigler 1987: 531]. Market competition is indeed a rivalry, among producers on one side and among consumers on the other, but also among producers and consumers for the exchange gains they must share. In other words, competition can more widely induce distribution conflicts. In this latter sense, two isolated traders negotiating in an Edgeworthian context can be said to be in competition . Stressing the struggle in the sharing of exchange gains, Makowski and Ostroy , , on this issue, relate competitive valuation to individual appropriation. Increasing returns notably generate exchange gains comparable to a collective good whose appropriation is subject to a costly and thus inefficient negotiation among agents.
2 Surprisingly, the established benchmark of competition, so-called perfect competition, neutralizes these strategic behaviors and conflictual interactions, resulting in the elimination of these distribution conflicts . As shown by Makowski and Ostroy, perfect competition (associated with constant returns) dissolves these conflicts of appropriation, since every individual can be paid according to his marginal contribution. In the perfectly competitive framework, competition is a pacified interaction grounded on a strict organization; it is not a conflicting interaction operated by resourceful opportunistic individuals.
3 Exposed by Makowski and Ostroy, this divergence between passive “market takers” and creative market makers sharpens the well-known difference between gentle price takers and the strategic price makers who attempt to exploit their market powers. In other words, Makowski and Ostroy deepen the gap between perfect competition (or the Walrasian rule) and imperfect competition (or the Cournotian revival).
4 More generally, these links between perfect and imperfect competitions are not straightforward, both competitions being related in two ways. Firstly, perfect competition remains the benchmark from which theories of imperfect competition are built: in this first way, the definition of perfect competition is taken for granted—mainly the law of one price and the price taking behavior—and each model of imperfect competition then tries to amend some assumption(s) usually associated with it. This is the way—from perfect to imperfect competition—adopted in applied or industrial economics (section 3). Secondly and conversely, as perfect competition rather postulates what it is supposed to explain, game theoretic imperfect competition is also used to provide foundations to perfect competition, to derive competitive behaviors from strategic behaviors. In that second way, from imperfect competition to perfect competition, the definition of perfect competition is no longer used as an unquestioned starting point, but rather as the limit case of imperfect competition. From perfect competition as the standard definition of competition, one switches to perfect competition as a degenerate case of competition (section 4).
5 Before developing these two kinds of relations, we consider (without being exhaustive) some notions of competition developed in the history of economics, including the marginalist notion of competition (section 2).
2 - The General Notion of Competition
6 As a phenomenon of rivalry, competition may be compared to a race between conflicting individuals and may also be connected with Darwinian selection, a process that displays the survival of the fittest (and the elimination of others). In this line of thought, competition is qualified as an efficient mode of selection, the “best ones” being the former winners and the present survivors. Unleashed competition may thus appear as a self-defeating process: when the most efficient competitor has prevailed, a monopoly arises and competition vanishes. To ensure the robustness of competition, many industrial economists recommend a public intervention to protect and to organize the competitive interaction: some “rules of the game” should be defined and enforced to eliminate abuses of dominant position (antitrust laws) and to ensure the independence of behaviors (anticollusion laws) in markets. Austrian economists like Kirzner  defend an opposite view. For them, dominant firms will stay dominant as long as they maintain their competitive advantage. Competition is a never-ending race and temporary winners can't stop struggling. So monopoly may be legitimized, beyond the justification given by Schumpeter  in terms of innovation, as long as they don't build barriers to entry . For Hayek , competition policy should wisely stay modest and not disrupt the precious “spontaneous order.” Furthermore, the market system not only displays competition: it emerged through a historical competitive process as the most effective system (and as a system of individual freedom). So Austrian economists generalize the principle of competition and praise it for its virtues. They also underline its operation as a dynamical process while classical and neoclassical economists tend to emphasize the effects of competition .
7 The views on competition by Smith , Ricardo  and Marx  are pretty close, even if they may disagree on other topics . In the classical framework, competition is essentially connected to the logic of capital and its general effects are balancing, but also unbalancing. Firstly, each capitalist compares the rate of return he gets in his current allocation to the rates he would get in alternative activities. Capital is moved from low profitability sectors to high profitability ones. So production decreases in the former sectors, where prices and profits are raised (as the commodity is scarcer); and supply increases in the latter ones, where prices and profits are reduced (as the commodity is more abundant). Such a competitive process, called gravitation, is equilibrating and tends to make all rates of return equal . The leveling of the rates of profit among sectors is thus explained by free competition, a mechanism that determines not only the formation of prices but also the distribution of income . Secondly, many classical economists predicted a long-run decrease of the general rate of profit: Ricardo feared a gloomy future and Marx foresaw a global crisis for capitalism. Competition plays a major part in this systemic trend, pushing capitalists to keep on investing in order to survive the competitive race. For Marx, this process leads to a rise in the organic composition of capital and eventually to a fall of the general rate of profit.
8 Marginalist economists changed the economic conceptual framework, shifting from a conception of society made of socioeconomic groups (or even struggling classes) to a conception of society made of homogeneous individuals. To solve the general problem of economic coordination, neoclassical political economists such as Walras relied on competition . Market competition is not simple rivalry and opens the possibility of a general settlement, a unanimous agreement on a situation that can be determined in two logical steps. First, to put Jevons' law of indifference in Mill's terms: “there cannot be two prices in the same market,” because the poorly paid supplier and the heavily charged demander would want to change their deals and could do it (simply by recontracting together) . Second, to put the law of supply and demand in Mill's terms: “Demand and supply…will be made equal. If unequal at any moment, competition equalizes them.” There cannot be a disequilibrium price on the market, because the rationed demanders/suppliers would consent to a price change with which all the suppliers/demanders would agree . So, market coordination is successful if transactions are all done at the same price, the price equating supply and demand, which determines market equilibrium.
3 - From Perfect Competition to Imperfect Competition: Definitions
9 Stigler  recalled the way the informal notion of perfect competition emerged in relation with the development of imperfect competition in the 1930s. More precisely, he showed how these features today usually associated to perfect competition were adopted after World War II. These features cover large themes: large number of agents (also related to the absence of collusion or coalition), perfect information, free entry and homogeneity of commodities (sometimes also associated with divisibility) .
10 These intuitive features or conditions are used to justify what competition as a process is supposed to achieve: first, the homogenization of price embodied in the ‘law of one price' and second, the absence of strategic behavior embodied in the price taking behavior. Indeed, perfect information is supposed to guarantee that the price is unique, while homogeneity is supposed to ensure that this price is the only signal required for individual choices. Besides, the large number of agents is used to justify the price-taking behavior, while the large number of agents is justified by free entry. This definition of perfect competition is mainly used in applied or industrial economics. Models of imperfect competition are then built upon the amendment of at least one of these features. These theories—dealing with monopoly, duopoly, information asymmetry, product differentiation and so on—then constitute a nonunified myriad of partial equilibrium models displaying a kind of strategic behavior and raising a social welfare issue.
11 Debreu  broke with these previous intuitive features of perfect competition—large number, free entry and so on—providing the formal and rigorous definition that had become canonical in general equilibrium theory: each good is associated with a single price expressed in a numéraire and each rational agent (either a producer or a consumer) is a quantity maker who behaves as a price-taker .
12 Debreu, and more widely the axiomatization of general equilibrium, had the advantage of clarifying the assumptions formally required by perfect competition. And these clarifications brought two consequences. First, it clearly set the conditions required for the existence and the Paretian optimality of the general equilibrium and thus strictly drew the normative scope of the competitive framework: “General competitive equilibrium above all teaches the extent to which a social allocation of resources can be achieved by independent private decisions coordinated through market.” [Arrow 1972: 128]
13 Second, axiomatization provided a clarification of the status of competitive traits. Indeed, the relation between these intuitive features and Arrow-Debreu's formal definition is sometimes ambiguous. Assumptions of homogeneity and divisibility are perfectly embodied in the theory, to such a point that they may seem redundant. The strict definition of goods—as physically specified, dated and localized commodities—includes homogeneity since two commodities which would not be perfectly identical would necessarily constitute two different commodities; and the definition of the commodity space as IRn also allows to avoid indivisibility. “Strikingly however, free entry and large number play no explicit role in this theory: all the theorems would hold if there were but a single potential buyer and seller of any commodity.” [Roberts 1987: 838] First, the number of agents is fixed in Arrow-Debreu's framework; second, again from a formal point of view, the large number assumption is not necessary, even if it is usually closely related to perfect competition. The way the price-taking behavior can be based on the large number of agents is a subject of intense discussions since Edgeworth's conjecture  and Debreu-Scarf's proof .
14 So perfect competition—embodied in the law of one price and the price-taking behavior—can be seen as a framework that “can really prevail only at equilibrium,” [Arrow 1959: 41] for it is rather postulated than endogenously set and does not allow any satisfactory theory of price and equilibrium formation . Despite these well-known critiques (regarding the setting of the price and beyond about the stability of the tâtonnement), perfect competition remains the benchmark from which neoclassical imperfect competitive theories are built, probably because of its strong normative content and its unified theoretical framework.
15 Imperfect competition has since experienced a rebirth: one does not start from the given perfect competition assumptions, and then try to amend them; but, conversely, one grounds these assumptions thanks to a game theoretic rationale. In particular, imperfect competition is used to found competitive behaviors from strategic behaviors, in order to endogenize the previous postulated competitive framework.
4 - From Imperfect to Perfect Competition: Foundations
16 Three different lines of research were developed in a general equilibrium framework to give a rational foundation to perfect competition, or rather to competitive equilibrium. In addition, they all share the common perspective to provide game theoretic foundations for perfect competition by the construction of a game whose equilibrium coincides with the competitive equilibrium in specific conditions (typically for large economies) . Without being exhaustive, let us sketch these three lines of research.
17 The Edgeworthian line studies the relation between the core and the competitive allocations [Hildenbrand, 1974]. The core provides a foundation of the competitive equilibrium. Two kinds of approaches may be invoked: the asymptotic approach and the atomless approach . Both approaches are based on cooperative game theory. The asymptotic approach was performed through replication procedure for finite economies by Debreu and Scarf . They notably show that, provided that the initial endowments are uniformly bounded, the core of an economy replicated an infinite number of times converges toward the set of competitive equilibria. The other one is the atomless approach, where the set of agents is indexed by a continuum with an atomless measure space . Aumann  considers a market with a continuum of traders. According to him: “Though writers on economic equilibrium have traditionally assumed perfect competition, they have, paradoxically, adopted a mathematical model that does not fit this assumption. Indeed, the influence of an individual participant on the economy cannot be mathematically negligible, as long as there are only finitely many participants. Thus a mathematical model appropriate to the intuition notion of perfect competition must contain infinitely many participants. We submit that the most natural model for this purpose contains a continuum of participants, similar to the continuum of points on a line or the continuum of particles in a fluid.” [Aumann 1964: 39] In this Edgeworthian perspective, the price-taking and the coalitional strategic behaviors can be identified under individual negligibility. The cooperative approach has been criticized by advocates of the noncooperative approach, especially because it did not explain the formation of coalitions, which notably questions how a core allocation is reached.
18 The second line of research introduces noncooperative behaviors under Cournotian competition [Mas-Colell, 1982]. The allocations are implemented through interrelated markets, and competitive behaviors in markets can be obtained from an environment in which strategic interactions occur . Two ways were followed. The first one is the strategic market games which were developed to circumvent the auctioneer [Shubik, 1973; Shapley and Shubik, 1977; Dubey and Shubik, 1978; Postlewaite and Roberts, 1976; Postlewaite and Schmeidler, 1978; Sahi and Yao, 1989; Amir et al., 1991] among others . Some results of convergence in terms of limit theorems are obtained. The second way, opened by Gabszewicz and Vial  in an exchange economy with production, and pursued by Codognato and Gabszewicz , , d'Aspremont, Dos Santos Ferreira and Gérard-Varet  for pure exchange economies, features the working and the consequences of market power in interrelated markets. Different Cournot-Walras equilibrium concepts can be modelled, each depending on the way strategic behavior is introduced [Gabszewicz and Michel, 1997] . In these Nashian perspectives, the Cournot-Walras equilibria can coincide with the competitive equilibrium in the case of large economies [Lahmandi-Ayed, 2001]. More recently, the Cournot-Walras equilibrium concept has been developed within the strategic market games approach in market economies with a continuum of traders [Busetto et al., 2008; 2011] to endogenize the type of behavior as either competitive or strategic.
19 The third line is recent and throws a different light on the strategic foundations of perfect competition, in bringing together matching and bargaining games [Gale, 2000]. Conversely to the two preceding lines of research, these dynamic models aim at specifying the economic institutions within which agents interact. Rubinstein and Wolinsky  consider a set of traders flowing through the market over time and searching at random for trading opportunities. When they meet, the terms of trades are determined by bargaining. It can be shown that the dynamic matching and the bargaining implement the competitive equilibrium [Gale, 1986]. Some extensions are given by McLennan and Sonnenschein . These finite game models generate limits theorems, but they are essentially assessed on period length or discount factor and not on the number of traders. So the coincidence between perfect and imperfect competitions does not here depend on the (large) size of the economy, contrary to the basic idea developed in the Edgeworthian and Cournotian perspectives.
5 - Contribution Highlights
Competition, Appropriation and Valuation
20 Neoclassical economics essentially envisions the price system as “the barometer of valuation” involving price taking reactive agents. In “Appropriation for a competitive theory of value,” L. Makowski and J.M. Ostroy contrast this view with the alternative conception of the price system as “the means of appropriation” moved by price-making opportunistic agents. There is actually no strict mutual exclusion between valuation and appropriation, but each paradigm or theory tends favor one theme and to downgrade the other one. On one side, from Walras to Debreu, neoclassical economics focused on valuation, appropriation being either an independent concern or a secondary issue. In this line, appropriation is a property of valuation: “Appropriation is the standard by which the efficiency of valuation is measured.” On the other side, in the view studied and developed by Makowski and Ostroy, appropriation is promoted as the main question and as the driving force of valuation. In that line, “appropriation is a generic description of an individual's efforts to extract gains from others,” so distribution leads (and is not absorbed by) price determination. The duality between valuation and appropriation proposed by Makowski and Ostroy is fruitful and innovative, as it allows them to develop a modified version of mainstream economics on the basis of a new light thrown on the history of economic thought. Under this perspective, five different themes can be revisited: rent as an appropriation phenomenon in classical political economy; the primacy of valuation in neoclassical political economy; valuation and appropriation in planned or market economic systems; property rights as the bound for appropriation in institutional economics and finally, of course, incentive and appropriation in new microeconomics.
The Edgeworth-Walras Convergence
21 A. Álvarez's paper is entitled “Price taking vs great numbers: A critique of the Edgeworth-Walras convergence à la Debreu Scarf.” This limit theorem is used to give foundations to the Walrasian equilibrium, arguing that Edgeworth's theory of recontract is a suitable way to reach competitive allocations. It is seen as a generalization of Edgeworth's conjecture which dictates that Walrasian equilibrium is a limit case of Edgeworthian equilibrium when the number of agents is infinite. The paper challenges this usual interpretation of the Edgeworth-Walras convergence. First, it shows that the Walrasian framework avoids questions that are unavoidable in an Edgeworthian framework: the price-taking assumption rests on institutional assumptions—the numéraire, the presence of the auctioneer and of brokers—which all try to neutralize strategic behaviors, to avoid price manipulation. Second, it shows that the strategic behaviors behind Edgeworth's conception of recontracting are misleading in the cooperative game framework of Debreu Scarf and that the core is not a good representation of Edgeworth's ‘final settlements'. The assimilation of the ‘final settlements' and the coalitions of the game theory is based on a well-known but very particular example—the replica economy— of Edgeworth's writings and involves an incorrect interpretation of Edgeworth's notions of coalition, combination or cooperative association.
22 F. Donzelli's “critical assessment” about “Negishi on Edgeworth on Jevons's Law of indifference, Walras' equilibrium, and the role of large numbers” also deals with the Edgeworth-Walras convergence with an infinite number of agents and focuses on the Edgeworthian concept of coalition. However, the point of view adopted by F. Donzelli is different: he questions the objection addressed by Negishi  to the so-called Edgeworth's conjecture. First, the paper sets a unified reformulation—with the contemporary tools of microeconomics—of Walrasian and Edgeworthian equilibria, clarifying then the status of Jevons's ‘law of indifference' and of the large number assumption for both authors. Second, using his reformulation of Edgeworth's process of recontract, F. Donzelli addresses objections to Negishi's result. The latter has shown, using some open coalitions that the arbitrage mechanisms supporting the law of one price are at work as soon there are two agents: in other words, reaching Walrasian equilibria would not require the assumption of a large number of agents. F. Donzelli's paper criticizes this position, stressing that Negishi uses assumptions which are not compatible with Edgeworth's recontracting. These interpretations of Edgeworth's recontracting and coalitions and the role of arbitrage mechanisms have then been the subject of a discussion with A. Rebeyrol in this issue.
Cournot, Bertrand and Edgeworth
23 L. Koutsougeras’ article, entitled “Price Taking as the Asymptotic Limit of Strategic Behavior,” provides a generalization of the idea that Nash equilibria tend towards competitive equilibria as the number of agents increases without limit. The author develops an asymptotic approach to explain the emergence of price taking behavior in the framework of strategic market games. This framework enables us to deal with price formation and trade process in line with Dubey-Shubik , Peck-Shell  and Sahi-Yao  among others. He notably provides a criterion—a rate of convergence—to measure the individual behavior's departure from price taking, using the wedge between the price vector's hyperplane and the supporting hyperplane of the indifference surface in equilibrium . It is shown that, under assumptions on the distribution of traders' characteristics, this wedge (the gap indicator) becomes arbitrarily small for all traders as the number of agents tends toward infinity. So, strategic behavior is approximately the same as price taking in an economy with a large number of agents. The approximation becomes finer when the number of traders increases.
24 In his article, entitled “On price-making contracts and economic theory: rethinking Bertrand and Edgeworth,” R. Routledge outlines an alternative approach to contract formation which combines Bertrand's  insight regarding price competition with Edgeworth's  notion of recontracting. In the first part of the paper, the author criticizes the inability of the standard models (related to the Bertrand paradox) to provide a convincing price-making foundation for competitive equilibrium. He provides a richer approach to contract formation which may give a more credible foundation for perfect competition. The main result is that a price-making analog of the Debreu-Scarf  approach can be provided to show that, in large markets, price-making contracts are ‘close' to the competitive equilibrium. Walrasian allocations are contained in the Edgeworth core and price-taking equilibria are contained in the Bertrand core. Hence, as the set of market traders becomes large, the only prices which remain in the Bertrand core are competitive prices. Therefore a new price-making foundation for the competitive equilibrium emerges from the Bertrand core.
Final Synthetic Notes
25 In his analytical note, “Competition: the ways to perfection,” R. Dos Santos Ferreira distinguishes two (times two) types of foundations for perfect competition. The first way to perfection is the absence of market power found in large economies. This “small size” argument can be found in a noncooperative context à la Cournot or in a cooperative context à la Edgeworth. Beyond the monopoly case, Cournot sees competition as a noncooperative game between producers. When the number of competitors goes up, the market power of each regularly goes down. “Indefinite competition” is the limit case where the market price has become immune to each producer's action and tends to equal the marginal cost. Edgeworth develops a similar idea in a different context, when he sees pure exchange as a cooperative game between contractors. When the number of traders goes up, the bargaining power of each regularly goes down. In a large economy, the shrunk core is nothing more than the set of Walrasian allocations. The second way to perfection is a given normal conduct, either peaceful or aggressive. That “standard of behavior” argument can be found in Walras (among others) for the quiet mode or in Bertrand for the hostile mode. Price-taking behavior may be seen as a norm of conduct voluntarily observed by the market agents, whatever their sizes. Under this vision, the presence of price takers has to be completed by the intervention of market maker(s): tâtonnement operators in Walras and a “market participant” for Debreu. Bertrand advances another norm of conduct for producers: price undercutting. Such an “aggressive” form of price setting may lead to a market result similar to the “peaceful” perfectly competitive equilibrium, as soon as there are only two producers. So, perfect competition is here conceived of as an outcome that can be obtained through opposite conducts.
26 To start his epistemological and ethical comment, “The perfect competition paradigm: evolving from its ambiguities,” M. Dardi states that this paradigm is not a reference for the working of competition, but a benchmark for the alleged outcome of competition: equilibrium. This entails issues about the epistemological status of the paradigm, which is intensified by the numerous interpretations of the normative meaning at stake. The universal construction of the competitive equilibrium is fascinating, as it joins individual pursuit of happiness and efficient social order; but two shortcomings may be identified. First, there are persisting epistemological ambiguities. The assumed competitive result relies on a peculiar kind of market organization, the tâtonnement, which displays neither robust stability nor empirical accuracy. Debreu accepted perfect competition to be reduced to an algebraic structure isolated from facts, far from both Walras's idealism and Pareto's positivism. Recently, many forms of competitive outcomes have been studied thanks to complex dynamics approaches and game theory models, which challenge the perfect competition paradigm’s position as the only benchmark. Second, there are deeper ethical ambiguities, about the reduction of social happiness to its individual and tradable components. Behind the questioning of Pareto optimality, individual preferences are challenged as the only basis for social good: with Sen, democratic values are brought back on the economic field and economics takes back its shape of political economy, embedded in ethics and political philosophy.
27 The workshop Foundations, definitions and usages of perfect competition took place at the University of Paris Ouest Nanterre la Défense on January 13th and 14th, 2011. We are grateful to the research center EconomiX (UMR CNRS 7235), the department of economics of the University of Paris Ouest Nanterre la Défense (thanks to C. Bidard), and the Region Île-de-France for providing financial and organizational support. We also acknowledge A. Béraud, G. Codognato, M De Vroey, B. Ingrao and A. Rubinstein for their report activities, and also N. Yannelis for his participation in the round table.
See Edgeworth .
How can the benchmark of competition prelude strategic behaviors and distribution conflicts? Some advance the heuristic necessity of a simplified model prior to complex models of competition. Others cite the persistence of a Walrasian utopia (a project of justice) or denounce the ideological denial of distribution issues.
According to the Austrian school, “free competition” means “free entry on markets” above all.
See McNulty .
See Duménil and Lévy , who also mention that the classical field of competition is not homogeneous: nonreproducible resources (whose appropriation generates rents) are distinguished from reproducible commodities (for which the theory of production price applies).
See Eatwell  and Semmler .
See Arena , pages 133 to 137.
They also distinguished between aspects of perfect competition: its just virtues (Walras) and its efficient properties (Pareto).
The notion of recontracting is due to Edgeworth .
Or just some of them, depending on the prevailing rationing scheme.
These features are not normalized. Stigler  does not evoke homogeneity but introduces mobility of factors. That is why one can also find five features in the literature. In contrast, Roberts  does not evoke mobility but homogeneity and divisibility, and also the absence of externality and increasing returns. The way these features fluctuate among authors reveals loose relations with the general equilibrium framework.
One must add to the definition of perfect competition the condition of “universality of markets” [Arrow, 1969] for it avoids the presence of externalities and more widely of all missing markets.
As stressed in the famous quotation from Arrow, “each individual participant is supposed to take price as given and determine his choices as to purchases and sales accordingly. There is no one left over whose job it is to make a decision on price.” [Arrow 1959: 43]
The “no-surplus” characterization of perfect competition in terms of “valuation without appropriation” [Ostroy 1980; Makowski and Ostroy, 2001] can be seen as an alternative way to found perfect competition.
Khan  reviews several methods for modeling perfect competition in terms of core equivalence and existence theorems. The seminal theoretical reference is Edgeworth .
Shitovitz  assumes an economy in which atoms face an atomless sector. He notably obtains the counterintuitive result that when atoms have the same endowments and preferences the core coincides with the competitive allocations.
The seminal reference is Cournot .
On this point, see and Postlewaite-Schmeidler  who show that the price-taking behavior may be a dominant strategy.
Another approach, based on conjectural variations, takes into account the perceptions by agents of their market environment and concerns the agent’s beliefs about the rivals’ responses to change in his own strategy [Bowley, 1924]. The degree of market power may be parameterized by conjectural variations [Dixit, 1986]. It can also be shown that the competitive equilibrium is a locally consistent conjectural equilibrium [Julien, 2010], sustaining the idea that perfect competition is only defined in equilibrium.
The marginal rate of substitution in case the utility function is twice differentiable.