Abstract: Although it is not generally remembered or known, Cournot applied his equilibrium concept to both quantity rivalry and price rivalry. This makes some of the nomenclature in modern game theory seriously inappropriate. Several critics, past and present, have treated Cournot's quantity rivalry case as only conveniently veiled price rivalry. If one pursues Cournot's mathematics far enough, it is clear that he has a method that applies symmetrically to both quantity and price rivalry. (JEL B10, C72, L13)

 * The author wishes to thank Michael R. Baye, Horst Raff, and Joaquim Silvestre for assistance without implicating them in any shortcomings the paper may have.

In 1838 Augustin Cournot published his now famous Recherches sur les Principes Mathematiques de la Theorie des Richesses. In this small volume Cournot set forth explicitly, and with mathematical precision, much of the modern day theory of competition, monopoly, and oligopoly. In 1883 J. Bertrand undertook a joint review of Cournot's book and Leon Walras's Theorie Mathematique de la Richesse Sociale which had just appeared. In this review Bertrand argued that Cournot's equilibrium for duopoly was not a true equilibrium because, "..., whatever the common price adopted, if one of the owners, alone, reduces his price, he will, ignoring any minor exceptions, attract all of the buyers, and thus double his revenue if his rival lets him do so"   [Bertrand, 1883, as translated by Margaret Chevaillier and appended to Magnan de Bornier, 1992] .  It is now a textbook commonplace that for homogeneous products, if each rival assumes that the other rival will let him do so, this type of rivalry would lead to the competitive result of price set equal to marginal cost.

Cournot arrived at the equilibrium criticized by Bertrand by assuming that each rival took the other rivals' quantities as given and then put what would be its profit maximizing quantity on the market. Stating each rival's profit function in terms of the quantities that all of the rivals place on the market, Cournot differentiated each rival's profit function partially with respect to that rival's own quantity and equated each of the resulting expressions to zero. (Cournot used the "d" notation for both partial and ordinary differentiation. See pp. 79-86 of the Bacon [1897] translation of Cournot [1838] as reprinted by Augustus M. Kelly [1960]. All page citations for Cournot will be for this volume.) For the duopoly case, Cournot plotted the resulting equations in rectangular coordinates and pointed out that it is evident that an equilibrium can only be established where the curves intersect [Cournot, p. 81]. Figure 2 gives the plotted curves and illustrates the sequential algorithm (given on p. 81) for finding this equilibrium. (All of the figures are given on a fold out sheet at the back of the book.)  In the more general case of n proprietors, equilibrium is given by the simultaneous solution of the equations [Cournot, pp. 84-85].

In plotting the respective first order conditions (for maximizing the profit of each rival given the other rivals' quantities) Cournot is implicitly solving for functions giving the reactions of each rival to the other rivals' strategies. In modern game theory these functions are called "best response functions." Where the curves intersect (for the two dimensional case) or for the simultaneous solution of Cournot's equations (in general), it turns out that all of the rivals conjectures about strategies are actually correct. No rival changes its strategy in reaction to the observed strategies of the other rivals. In the early 1950's J. F. Nash [1950, 1951] extended this basic idea to noncooperative games in general and provided sufficient conditions for the existence of such equilibria. In modern game theory best response solutions with mutually correct conjectures are referred to as Nash equilibria.

The above thumbnail sketch is provided as background for discussing the nomenclature that has evolved in the application of modern game theory to the analysis of market structures. Almost without exception in the current industrial organization literature, market rivalry involving quantity strategies is referred to as "Cournot competition" and market rivalry involving price strategies is referred to as "Bertrand competition." The corresponding equilibria are referred to as "Cournot equilibria" and "Bertrand equilibria." Where the equilibria are best response solutions with mutually correct conjectures, they are respectively described as being "Cournot-Nash" and "Bertrand-Nash." In light of the thumbnail sketch, this would seem to be convenient nomenclature that is firmly rooted in the historical evolution of economic ideas. In point of fact, this nomenclature actually does great violence to the history of economic thought.

What has been forgotten or never learned is that Cournot in his 1838 classic treated both quantity rivalry and price rivalry and treated them symmetrically (in the sense of analyzing both in terms of best response functions with equilibrium given where conjectures are mutually correct). The most glaring example of the problem arises in the analysis of oligopoly with differentiated products. With differentiated products it is no longer the case that if one rival were able to get its price lower than all of the other rivals, then that rival would take the entire market (at its price). In this case there is a best response solution in terms of prices with mutually correct conjectures. In modern IO literature this solution is uniformly referred to as the "Nash" or the "Bertrand-Nash" equilibrium. But whereas the duopoly best response curves for the quantity rivalry case with homogeneous goods are negatively sloped, the duopoly best response curves for price rivalry with heterogeneous goods are positively sloped. This is because one rival raising price enhances the profit opportunities of the other rivals.

The differing slopes of the best response curves might be thought to be justification of not attaching Cournot's name to the modern solution for differentiated oligopoly, but this would be false justification. On pp. 100-101, for a duopoly case, Cournot states proprietor profit functions in terms of the prices of both rivals and partially differentiates each of the profit functions with respect to their own prices. He then equates both of the resulting expressions to zero [p. 101] and indicates that the same method of reasoning that was used in the quantity rivalry case applies. Then positively sloped and intersecting best response curve are given in Figure 7 with prices measured on the coordinate axes. Thus the derivation of figure 7 is identical with the derivation of the modern theory of differentiated oligopoly. It then follows that if proper names are used to identify this case, one of the names must be Cournot rather than Bertrand. There is nothing in Bertrand's review that would justify the use of his name in this context. (There is one small caveat that must be entered here and that is that on pp. 100-101 Cournot is not discussing differentiated oligopoly but rather the case of a composite commodity whose components are supplied by rival monopolists. The analytical structures are essentially similar nonetheless.)

When proper names are used to identify concepts, the names used should be those of the individuals who originated the concepts. Where another individual has significantly extended an idea and/or made especially significant applications of an idea, it is legitimate to add that name with a hyphen as for example "Marshall-Lerner" in the case of the sum of elasticities theorem. Thus the equilibrium of competitive pricing for homogeneous goods under oligopoly is properly identified as being "Bertrand" or "Bertrand-Nash." But these labels should never be applied to the case of oligopolistic price rivalry with heterogeneous goods. The equilibria in both this case and the usual quantity rivalry case should be identified as "Cournot" or "Cournot-Nash." Justification of the present practice on convenience grounds is spurious. "Quantity competition" and "price competition" trip off the tongue just as easily as "Cournot competition" and "Bertrand competition." To resist the required change in nomenclature because the current usage is firmly entrenched would be an unseemly surrender to the status quo.
 

REFERENCES
Bertrand, J. "Theorie Mathematique de la Richesse Sociale," Journal des Savants, 67, 1883, pp. 499-508.

Cournot, Augustin. Recherches sur les Principes Mathematiques de la Theorie des Richesses, Paris: Hachette, 1838. (Italian translation in Biblioteca Dell'Econ., 1875. English translation by N. T. Bacon published in Economic Classics  [Macmillan, 1897] and reprinted in 1960 by Augustus M. Kelly.)

Magnan de Bornier, Jean. "The 'Cournot-Bertrand Debate': A  Historical Perspective," History of Political Economy, 24,  3, 1992, pp. 623-54.

Nash, J. F., Jr. "Equilibrium Points in n-Person Games,"  Proceeding of the National Academy of Science U.S.A., 36, 1950, pp. 48-49.
_____. "Non-Cooperative Games. Annals of Mathematics," 54, 1951, pp. 289-95.

Walras, Leon. Theorie Mathematiques de la Richesse Sociale.  Lausanne: Corbaz, 1883.