1On what theoretical foundations does perfect competition rest? Under what conditions do we have a chance of observing it in practice? In fact, how can we be sure that it is perfect competition that we possibly observe?

2A frequent answer to these questions is the one already given by Cournot [1838]. Competition is perfect (or indefinite, in Cournot’s terminology) when competitors are too small to influence the market price, a circumstance he regards as a good simplifying approximation for many markets. This view associating perfection with pettiness was later adopted by Edgeworth [1881] in a cooperative, instead of non-cooperative, approach. The same type of foundations, in its two variants, persisted in the twentieth century literature. It will be briefly examined in section 1.

3This view is however by no means the only way to perfection. An alternative view, expressed by Jevons [1871] or Walras [1874], but also implicit in Arrow and Debreu [1954], has the same characterization of perfect competition in terms of price taking conduct, but makes this conduct independent of size, just the result of the earnest observance by the agents of a passive behavioural rule (the active “higgling and bargaining” of the market being left to market makers). Also relying on conduct rather than size is the perfectly competitive outcome implemented through fierce competition between two producers, each undercutting his rival: the so-called Bertrand [1883] paradox. Both ways to perfection through conduct will be briefly analyzed in section 2.

# 1 – Vanishing power in large economies: The importance of being petty

4The first way to perfect competition that has been formally considered in the literature is the one introduced in chapter VIII of Cournot’s *Recherches* [1838], devoted to *indefinite* competition. As the word suggests, indefinite competition is a limit case, the one that is attained when producers’ market shares become insignificant with respect to market size, so that the market price becomes *insensitive* to producers’ actions. In other words, competition becomes indefinite when the market power (the ability to influence or manipulate the market price) of a higher and higher number of producers eventually vanishes.

5The same idea is applied almost half a century later by Edgeworth in his *Mathematical Psychics* (1881), although to a cooperative approach which contrasts with Cournot’s non-cooperative model. In this approach, as the *field of competition* expands, that is to say, as the number of potential contractors grows, their bargaining power, namely the ability for them to affect the final settlement by threatening to recontract, eventually vanishes and perfectly competitive settlements can alone persist in the limit.

# 1.1 – Vanishing market power in a non-cooperative game

6Cournot [1838, ch. VII] models competition in the market for a homogeneous good as a non-cooperative game between profit maximizing producers, acting *each one on his own.* They do so, by taking as given the outputs targeted by their competitors and also by taking into account the fact that *the price is necessarily the same* for all of them. The first order condition for producer *i*’s profit maximization at equilibrium (any one of Cournot’s equations (6), stating the equality of marginal revenue and marginal cost for each producer) can be written as

8where *q _{i}* is producer

*i*’s output,

*s*?

_{i}*q*/?

_{i}*his market share and*

_{j}q_{j}*c*his cost function, and where

_{i}*p*is the common price set by all producers, such that

*D*(

*p*) = ?

*according to the demand function*

_{j}q_{j}*D*, with elasticity

*D?*(

*p*)

*p/D*(

*p*)? ? ?(

*p*). This equation expresses Lerner’s index

*?*of the degree of monopoly, a measure of producer

_{i}*i*’s market power, as a ratio of his market share to the Marshallian elasticity of market demand at equilibrium.

9Clearly, the producer’s market power, his ability to influence the market price by varying his output, eventually vanishes as his market share is reduced to zero, hence as he becomes insignificant with respect to market size. If it is rather the whole market that we have in view, we may refer to the arithmetic mean of the individual degrees of monopoly, weighted by the corresponding market shares:

11where *H* is the Herfindahl index of market concentration. An indefinitely competitive market is one which displays no market power for *any* producer, so that *all* producers must, each one on his own, take the market price as given. According to the preceding equation, it appears as a market with a zero index of concentration.

12In Cournot’s approach, perfect competition may be viewed as an approximation of competition between insignificant producers interacting in large markets. Formally, it may be attained in the limit of a hypothetical process of market replication. Cournot did not explicitly refer to such a process. Nor did he suggest the existence of an actual mechanism impelling competition to its limit, but such a mechanism, namely *free entry and exit*, is well-known and often viewed as a condition for competition to be perfect. The idea is simple: under free entry and exit, if profits are positive, there is an incentive for potential producers to enter the market and increase output, triggering a price decrease, and if profits are negative, there is an incentive for incumbent producers to exit the market and diminish output, raising the price. Walras [1874, §188] describes this mechanism, essential for competition to be *free*, but the implementation of his *absolute* free competition, corresponding to Cournot’s indefinite competition, does not completely rest on entry and exit, as we shall see in the next section.

13Now, the mechanism of free entry and exit is supposed to lead to zero profits, by equalizing for each producer his average cost to the market price, whereas the vanishing of market power associated with indefinite competition supposes the equality of *marginal* cost to the market price. Under increasing average cost functions and no barriers to entry, there will be entrants as long as profits remain positive, so that the market will end up with an infinite number of producers each supplying a zero output at a zero market price. More realistically if, for every producer, the optimal scale is positive but small with respect to market size, that is, with respect to the quantity demanded at a price equal to the lowest average cost, then the market equilibrium will be close to the perfectly competitive equilibrium [Novshek, 1980]. Thus, free entry and exit implements perfect competition, at least approximately, provided the producers are indeed *structurally* small relative to the market.

14What happens if they are not? If the optimal scale of the potential entrant attracted by positive profits is large, he cannot neglect the depressing effect on the market price of his own contemplated addition to the aggregate output, an effect which may be strong enough as to drive the price below his minimum average cost, should he persist in his decision to enter. Thus, under significant internal economies of scale, competition, in spite of free entry, may well be far from perfect, displaying non negligible market power and even positive profits at equilibrium [d’Aspremont et al., 2000; Dos Santos Ferreira and Dufourt, 2009].

15A last remark is in order. There is a source of vanishing market power, as measured by Lerner’s index in a Cournot equilibrium, other than the squeeze of market shares, namely the indefinitely raising elasticity of market demand, which might result from ever increasing substitutability of the supplied good to rival goods in other markets. This possible way to perfect competition is completely independent of market shares being small: even a monopolist is threatened by the existence of close substitutes to his output. However, one may legitimately wonder if such a tight relationship between products cannot induce, beyond Cournot competition in a market for a strictly homogeneous good, some other form of strategic interaction across markets. Notice that Cournot himself proposes an example of such interaction, in the case where producers of complementary intermediate goods *concur* to the production of a final good [1838: ch. IX].

# 1.2 – Vanishing bargaining power in a cooperative game

16Edgeworth [1881, part II] models exchange, in the simplest case, as a contract between two individuals, each endowed with some quantity of a specific good. The contract reallocates the collective endowments of the two goods, and the final allocation, assumed to be obtained by mutual consent of the two parties, must be such that no party is worse off than with the initial endowment, and such that no other feasible allocation would be preferred by both. Edgeworth shows that, without competition, this contract is indeterminate, in the sense that the final allocation is any point on a *contract curve.* The point to be actually selected on this curve will be chosen either on normative grounds, for instance by arbitration, or on a positive ground, according to the relative bargaining power of the parties.

17Now, suppose competition to be introduced through the duplication of the set {1,2} of individuals with their own characteristics (preferences and endowments), so that we now have two individuals of each type, 1 and 2. Clearly, feasible allocations, in particular those on the contract curve, can also be duplicated. However, if preferences are strictly convex, it will be possible for two individuals of, say, type 1 to improve upon the pair of contracts that are most unfavourable to them, by recontracting, together and on an equal footing, with one individual of type 2. Then, the other individual of type 2, left alone, will want to compete with his fellow by offering a more advantageous contract to one or both of the individuals of type 1. We see that the most unfavourable contracts for one of the two parties, at the extremes of the contract curve, will always meet an objection in the duplicated field of competition. In other words, the bargaining power of the individuals of both types is now restrained, and the contract curve shorter.

18If we carry on this hypothetical process of replication, the contract curve will continue to shrink, contracts will become less and less indeterminate, and bargaining power will be more and more restricted until it eventually vanishes. In the limit we are left with only *perfectly* competitive, determinate, final settlements. This analysis was extended many decades later to economies with arbitrary numbers of commodities and types of consumers, and in which production is possible [Debreu and Scarf, 1963]. The concept of contract curve was generalized into the concept of *core* of the economy, namely the set of allocations to which no coalition of individuals would object. In an indefinitely replicated economy, a (replica) allocation which remains in the core for any replication is a perfectly competitive allocation.

19A last fundamental step in this approach to perfect competition, as a regime associated in an essential way with large economies filled with insignificant individuals, was made by Aumann [1964], who suggested that “the most natural mathematical model for a market with ‘perfect competition’ is one in which there is a continuum of traders,” and showed that “the core of such a market coincides with the set of its ‘equilibrium allocations,’ i.e., allocations which constitute a competitive equilibrium when combined with an appropriate price structure.” [1964: 39] Aumann’s step is fundamental, not so much for technical as for conceptual reasons. To quote him again: “When the notion of perfect competition *is built into the model*, that is, in a continuous market, one may expect that the core equals the set of equilibrium allocations.” [Ibid.: 40] The coincidence of the core and the set of equilibrium allocations is nevertheless remarkable, because the concept of core, applying indifferently to large and small economies, does not make use of any reference to prices. Market prices appear in this context as just a subproduct of the analysis, by no means essential, and specific to the case of a continuum of traders.

# 2 – Normal conduct in economies of any sort: the importance of being earnest… or fierce

20The two ways to perfect competition that we have considered up to now have in common the fact that they set it in the *limit* of some hypothetical, possibly implicit, process. Eventually, this process makes all the individuals (producers and consumers) structurally insignificant relative to the market, depriving them from any power to influence their respective environments. From now on, we shall explore other ways to perfect competition, which do not rest on the evanescence, induced by pettiness, of any market or bargaining powers, but rather on the voluntary observance of some *norm* of conduct, be it peaceful price taking [Jevons, 1871; Walras, 1874] or on the contrary aggressive price undercutting [Bertrand, 1883].

# 2.1 – Peaceful conduct: price taking and market making

21Price taking is the essence of perfect competition, at least in the non-cooperative approach which goes back to Cournot. By this we mean that in perfect competition every agent abstains not so much from posting the prices to be applied to ongoing and future transactions but, above all, from manipulating anyhow the market prices that are about to be settled. This is done by necessity in large economies, where agents are assumed to lack any market power. By contrast, we are now considering the case where every agent spontaneously renounces any potential market power, and earnestly takes as given the prices set by market makers.

22This view of perfect competition is clearly developed by Jevons [1871]:

By the mediation of a body of brokers a completeconsensusis established, and the stock of every seller or the demand of every buyer brought into the market. It is of the very essence of trade to have wide and constant information. A market, then, is theoretically perfect only when all traders have perfect knowledge of the conditions of supply and demand, and the consequent ratio of exchange; and in such a market, as we shall now see, there can only be one ratio of exchange of one uniform commodity at any moment.

24This passage ends with the statement of what Jevons calls the *law of indifference*, taking up Cournot’s characterization of perfect markets by the prompt leveling of prices. But the significant point in the present context is the distinction between two categories of individuals: *traders* (sellers and buyers) and *brokers.* In a market for a uniform commodity, it is the role of brokers to establish a consensual price that equates supply and demand, as decided by traders on the basis of their rational expectation of such price, that is, based on their “perfect knowledge of the conditions of supply and demand.”

25In spite of this knowledge, traders are not supposed to take advantage of their possible ability to influence the price. Actually, Jevons does not ignore the importance of market size in this context:

We may, firstly, express the conditions of a great market where vast quantities of some stock are available, so that any one small trader will not appreciably affect the ratio of exchange. This ratio is, then, approximately a fixed number, and each trader exchanges at that ratio just so much as suits him.

27However, when defining a *trading body* as *any* set of traders, Jevons writes:

We must use the expression with this wide meaning, because the principles of exchange are the same in nature, however wide or narrow may be the market considered. Every trading body is either an individual or an aggregate of individuals, and the law, in the case of the aggregate, must depend upon the fulfilment of law in the individuals.

29His theory of exchange is consistent with this point of view. He analyzes exchange between two trading bodies endowed with two different commodities, but he implicitly supposes that the theory is the same whatever the size of the bodies. As a consequence, the fundamental indeterminateness of bilateral monopoly pointed out by Edgeworth is not recognized, and price taking remains in all cases the norm of conduct.

30The approach of Marshall is essentially the same, except as concerns the insistence on the unimportance of size. Every *buyer* and *seller* is supposed to “guess the state of the market and to govern his actions accordingly” [Marshall, 1890: 332], which amounts to decide the quantities he wants to buy and sell at any possible price he can expect to hold. Prices will then be tentatively set by *dealers*, and one will emerge as the (temporary) equilibrium price

because if it were fixed on at the beginning, and adhered to throughout, it would exactly equate demand and supply (i.e.the amount which buyers were willing to purchase at that price would be just equal to that for which sellers were willing to take that price); and because every dealer who has a perfect knowledge of the circumstances of the market expects that price to be established.

32Walras [1874, § 41] also refers to the intermediation of brokers during the open outcry process in the best organized markets. However, he does not insist on the distinction between traders and brokers, but rather on the competition between traders themselves when they grope for the equilibrium price in the course of the *tâtonnement* process:

Left to itself, the exchange value arises naturally in the market, under the influence ofcompetition.As buyers, tradersoutbidthe other demanders, as sellers theyunderbidthe other suppliers, thus concurring in the setting of some exchange value, sometimes rising, sometimes falling, and sometimes stationary.

34In some sense, Walras identifies competition with the interaction between traders on the long side of an outbalanced market, hence with market making rather than with price taking proper. This competitive interaction is not confined to bidding, open to all traders (landowners, workers, capitalists and entrepreneurs): the *tâtonnement* in prices on the exchange scene is indeed completed by a *tâtonnement* in quantities by entrepreneurs on the production scene, consisting in an output increase when profits are positive and an output decrease when profits are negative. This is in part made possible by free entry and exit, as we have already seen, but Walras stresses the fact that free competition may well apply to a single firm. He notices indeed

that, if the multiplicity of firms leads to equilibrium in production, it is in theory not the only means of achieving this goal, and that a unique entrepreneur who would demand services by outbidding and supply products by underbidding, and who would in addition always restrict his production in case of a loss and always expand it in case of a profit, would obtain the same result.

36Walras’ entrepreneurs are essentially market makers, “competing” in each market to implement the price that equates demand and supply *at that price* and the quantities that equate the price and the average cost.

37Modern general equilibrium theory has inherited this behavioural approach to perfect competition. Arrow and Debreu [1954] assume the existence of a finite number *m* of consumption units and of a finite number *n* of production units. Both categories of units are *price takers* by assumption even if *m* and *n*, which are not restricted, are both arbitrarily low, possibly equal to 1. For technical reasons, in order to apply Debreu’s extension of Nash’s theorem of equilibrium existence for a (generalized) game, a fictitious player is introduced, the *market participant*, whose strategy is a normalized vector of prices and whose payoff is the value, at these prices, of aggregate excess demand. Since this value increases when prices are augmented in seller’s markets and diminished in buyer’s markets, the market participant is just an abstract *market* wafer accomplishing alone a *tâtonnement* in prices.

# 2.2 – Aggressive conduct: undercutting competitors’ prices

38The last way to perfect competition is again independent of the agents’ size, although it supposes the existence of at least one competitor for each agent. However, rather paradoxically, it does not rest on price taking, whether by necessity or by choice. A perfectly competitive outcome is now a possible consequence of extremely aggressive price setting, consisting for a supplier in undercutting competitors’ prices in order to take over the whole market.

39Such conduct was first described by Bertrand [1883] in his critique of Cournot’s non-cooperative approach to duopoly. Cournot [1838, ch. VII] had shown that any of two producers, acting on his own, would want to deviate from the collusive solution, that which jointly maximizes their profits. The deviation would consist in increasing the output target and consequently accepting a lower price. To this analysis Bertrand raises “the peremptory objection” that

“no solution is possible, the [price] decrease would have no limit; whatever indeed the adopted common price, if one of the competitors decreases alone his price, he attracts, neglecting unimportant exceptions, the whole sale, and he will double his revenue if his competitor lets him do.”

41And he further criticizes Cournot for having treated “the quantities sold by the two competitors” as “independent variables,” which they cannot be.

42There is in this objection a serious misinterpretation. Cournot’s producers take into account the fact that “the price is necessarily the same” for both, an expression of the law of one price in perfect markets. When changing his output target, Cournot’s producer must “appropriately modify the price” by referring to the demand function and conjecturing his competitor’s own output target. By contrast, Bertrand’s producer does not respect his competitor’s output target. Nor does he anticipate the likely price matching that would be immediately effected by his rival, who would certainly not “let him do,” gladly accepting to be ejected from the market.

43Nevertheless, we do observe the kind of price wars described by Bertrand. And price wars do make sense in an asymmetric environment, when some producer has, for instance, a cost advantage which allows him to get rid of his rivals for good. If this is the case, the outcome of “Bertrand competition” is monopoly with *limit pricing*, designed to keep the competitors out of the market. However, in the case of a *symmetric* duopoly with a linear cost function, the outcome of Bertrand competition coincides with the perfectly competitive outcome, with zero profits and zero market power (equality to price of average and marginal costs). This is the so-called “Bertrand equilibrium,” even if Bertrand himself has denied the existence of a solution to duopoly, and would probably reject this particular solution as unreasonable.

44The point we want to stress is however that a perfectly competitive outcome can be enforced by aggressive price making, even in a duopoly where the producers’ market power may be quite large (the “Bertrand paradox”). Here, *conduct*, not market *structure*, is clearly the foundation of perfect competition. There is a natural extension to this analysis: each oligopolist *i* may have an *intermediate* degree of *competitive aggressiveness*, leading to the degree of monopoly

46The *new empirical industrial organization* (NEIO) literature has provided estimations of the (supposedly uniform) degree of competitiveness, the *conduct parameter*, for different industries. Notice that, if *? _{i}* =

*?*for any

*i*, we obtain as the two extreme cases Cournot competition for

*?*= 0 and perfect competition for

*?*= 1.

47A popular rationalization of this parameter is given by the conjectural variations model [Bowley, 1924], with the assumption that producer *i* conjectures his competitor *j* to respond by *R ^{i}_{j}*(

*q*) to his choice of output target

_{i}*q*. We then obtain ?

_{i}*?*= ?

_{i}_{j?i}(

*dR*(

^{j}_{j}*q*)/

_{i}*dq*) as the cumulated conjectural variation of all the competitors of

_{i}*i.*Cournot-Nash conjectures naturally correspond to zero conjectural variations. At the other extreme, if the cumulated conjectural variation is ? 1, any change in

*i*’s output target is offset by an opposite change in the sum of competitors’ targets, so that the price is actually taken as given, like in perfect competition. Other rationalizations, more in line with pure Nash conjectures, are however possible [d’Aspremont and Dos Santos Ferreira, 2009]. They may lead to quite different interpretations of the conduct parameter. As an illustration, just recall that extreme aggressiveness and extreme peacefulness can both lead to a perfectly competitive outcome.