Dimand-Dore on Cournot-Bertrand: A Reply and More

Atlantic
Economic
Journal

VOLUME 27
NUMBER 3

September 1999

Pages 334-339

This note refutes the novel thesis of Jean Magnan de Bornier [1992] that all of Cournot's analysis in his 1838 classic was about price rivalry. It also reiterates the proposition that Cournot treated both quantity and price rivalry and that Cournot was clearly aware of the symmetry between quantity conjectures and price conjectures in 1838. It is conjectured that the many misinterpretations of Cournot derive from failure to work through Cournot's simple mathematics and a mental truncation of the Recherches into just Chapter VII. (JEL B10, C72, L13)

Robert Dimand and Mohammed Dore [1992] are to be commended for their efforts in publicizing the fact that Cournot's 1838 masterpiece was favorably reviewed on this side of the Atlantic (by John Cherriman [1857] of the University of Toronto) before it was so reviewed in Europe. They are also to be commended for calling attention to the fact that Bertrand made more substantive contributions to game theory than those contained in his famous joint review of Cournot and Walras. However, their "Further Note" raises issues that require a reply and further discussion.

Dimand and Dore [1999] indicate that "Morrison [1998, p. 174] thus concedes too much to Bertrand by stating that 'the equilibrium of competitive pricing for homogenous goods under oligopoly is properly identified as being Bertrand or Bertrand-Nash.'" Admittedly, attaching Bertrand's name solo to this case may be inappropriate, but this concession is based on reasons entirely different from those given by Dimand and Dore. Indeed, it is surprising that they would bring this up since, after giving Bertrand's description of Cournot's Chapter VII adjustment mechanism, they indicate that "Bertrand attributed to Cournot an analysis of price rivalry leading to the zero-profit competitive results." If this in fact was the case, there would be no qualms about attaching Bertrand's name to the competitive case.

Clearly, Dimand and Dore [1999] are following a different canon than Morrison [1998] for associating names with models. Morrison's [p. 174] canon clearly states that "when proper names are used to identify concepts, the names used should be of those individuals who originated the concepts. Where another individual has significantly extended an idea or made especially significant applications of an idea, it is legitimate to add that name with a hyphen." It can only be inferred as to what canon Dimand and Dore are following. Apparently, proper names can be used to identify models only when the originator has sanctified the model and originators are allowed to sanctify only one model per situation. This inference is based on the fact that Dimand and Dore allege that Bertrand "recognized only the coalitional solution" and then cite the fact that this solution is not Nash as a reason not to identify the competitive solution as Bertrand or Bertrand-Nash. How else could the fact that the collusion solution is not Nash have any bearing on assigning names to the competitive solution? Bertrand stated that "it would be in their interest to join together in partnership...but this solution is rejected (by Cournot)" [Friedman, 1988, p. 7]. Presumably, this statement sanctifies the collusion solution to Bertrand, although Dimand and Dore attempt to base it on another statement by Bertrand, analyzed in detail below.

It is proposed that the Dimand-Dore canon be tested with a conceptual experiment. Suppose that Dimand and Dore derive, and publish for the first time, three models. For identification purposes, they will be labeled A, B, and C. Model B is incorrectly described as a formalization of a theory propounded by Morrison. (Careful examination of Morrison's recorded blatherings shows no basis for this attribution.) Dimand and Dore sanctify model C by venturing the opinion that, of the three models, model C will be the best predictor. (Presumably, the derivation of model A was just a warmup exercise.) Subsequently, models A and C recede into oblivion but model B catches on in a big way and even becomes a textbook commonplace. Now, if model B came to be known as the Dimand-Dore model, would this be inappropriate?

Actually, the beginning of Morrison [1998] was very careful not to attribute the competitive solution to Bertrand. Margaret Chevaillier [Magnan de Bornier 1992, p. 649] translated that "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." Then Morrison [1998] observed, "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" [p. 172] The reason for caution then (and concession now) derives from Bertrand's statement that "...there is no limit to the downward movement..." [Friedman 1988, p. 77] of prices in Cournot's model. However, rather than viewing this as Bertrand's sanctification of the collusion solution, this is viewed as creating uncertainty as to where Bertrand assumed the price cutting would end. In the calculus of Newton and Leibniz, to "decrease without limit" would be rendered as "approaching negative infinity" [Taylor, 1955, pp. 62-3]. No one would seriously consider this a possibility or think that Bertrand was attributing this solution to Cournot. Thus, it is not possible to take literally Bertrand's statement of unlimited downward movement. However, the statement does introduce uncertainty as to what (if anything) he actually had in mind. Dimand and Dore raise another question which, after considerable development, will have some bearing on this question.

Jean Magnan de Bornier's 1992 article on Cournot and Bertrand is extensively cited by Dimand and Dore. They indicate that "Magnan de Bornier made clear that Cournot's analysis was symmetric in price rivalry and quantity rivalry," and included him among those who established that "it would be appropriate to refer to the models as Cournot in quantities and Cournot in prices." Actually, Magnan de Bornier argued quite strenuously (and incorrectly) that Cournot's analysis was all price rivalry.

First, "consulting the chapters on oligopoly and competition, we find that the same hypothesis is maintained throughout, with the (quantitatively small and only apparent exception of section 43 where the classical duopoly model is presented..." [Magnan de Bornier 1992, p. 626]. "So only one section out of forty (26-65) in the book that deal with price theory (and only one out of seven in the chapter on oligopoly) is written with quantity as the apparently strategic variable" [Magnan de Bornier 1992, p. 630]. In a footnote, "when dealing with 'unlimited competition,' Cournot still seems to treat price as a decision variable. The equilibrium condition for the competitive firm is written deriving each firm's profit relative to p, as if small firms could manipulate it. This is obviously inconsistent, and the demonstration that Cournot gives...of the condition price being equal to marginal cost is incorrect (see 90)." [Magnan de Bornier 1992, p. 626]

In all of these assertions Magnan de Bornier is obviously wrong. All of the optimization in Chapter VII is in terms of quantities. The first order conditions in this chapter consist of (1) and (2) for the costless duopoly case, system (5) for the costless n-rival case, and system (6) for the n-rival case with production costs. All of these equations were obviously obtained by differentiating with respect to quantities. For the competitive case, the first unnumbered equation in Chapter VIII was obtained by applying the inverse derivative rule to the k-th equation in system (6) (in Chapter VII) with n large so that the output of one producer is negligible relative to the market. Thus, Cournot's proof that competitive firms will produce to the point where price equals marginal cost is elegant rather than incorrect.

It is true that Cournot used the inverse derivative rule to shift from quantity derivatives to price derivatives, but he did so for a reason other than the one attributed to him by Magnan de Bornier. In Section 45 he attempts to demonstrate that the solution for his model approaches the competitive solution as the number of rivals increases. Here, there is great irony in that, in terms of modern price theory, his argument would have been easier to understand if he had remained with quantity derivatives.⁽²⁾ Consider the market demand curve and the curve marginal to it (which would be the monopoly marginal revenue curve). Aggregating Cournot's (5) (Chapter VII) yields:

Now consider verticals between the demand (AR) curve and the monopoly marginal revenue curve. Since this is the costless case, with one rival, MR is zero at equilibrium. With two rivals, the positive part of the vertical equals the negative part. With three rivals, two thirds of the vertical is below the quantity axis and one third above, and so on. Obviously, if the marginal revenue curve decreases monotonically until the market is saturated, then the quantity increases and the price decreases as rivals are added.

The reason that Magnan de Bornier argues that the alleged exception of Section 43 is only apparent is because Cournot states that "proprietor (1) can have no direct influence on the determination of D₂: all that he can do, when D₂ has been determined by proprietor (2), is to choose for D₁ the value which is best for him. This he will be able to accomplish by properly adjusting his price, except as proprietor (2), who, seeing himself forced to accept this price and this value of D₁, may adopt a new value for D₂" [Cournot 1838, p. 80]. However, since Cournot says, "...choose for D₁ the value which is best...," then quantity is indeed the strategic variable and price is a tactical variable used to effect the quantity strategy. Thus, all of the arguments that Magnan de Bornier gives to support his novel thesis are bogus. If Bertrand made the same mistake as Magnan de Bornier, then Dimand and Dore could be correct in saying that Bertrand incorrectly attributed the zero profit competitive solution to Cournot. However, if Bertrand made this mistake, it would be clear that the price reductions he describes would proceed into negative profit territory. On the other hand, if Bertrand correctly understood that Cournot was using price as a tactical variable, then he would possibly qualify for canonization, even by Dimand and Dore, since the Cournot quantity solution is at prices above the zero profit level. These questions will remain unresolved due to Bertrand's careless statement that the price reduction would have no limit.

Although Magnan de Bornier argued that Cournot only dealt with price rivalry, it is true that he did take up the question of symmetry between price and quantity rivalry when discussing the "Bertrand myth." In his paper [1992, p. 638], he states that "it is, then, clear that Fisher suggested the symmetry between quantity and price conjectures and was the first to do so...." Actually, it was Cournot himself who pointed out this symmetry.⁽³⁾ In Chapter VII he equates to zero the partial derivatives of the rivals' profit functions with respect to their own quantities and solves the resulting equations to get the equilibrium point. In Chapter IX he equates to zero the partial derivatives of the rivals' profit functions with respect to their own prices and solves the resulting equations to get the equilibrium point. When he does this in Chapter IX, he states, "...we can show, as in Chapter VII, and, by the same construction,...that the coordinates of the point of intersection i (or the roots of equations (1) and (2)) are the only values of p₁ and p₂ compatible with stable equilibrium" [Cournot 1838, p. 101]. How could Cournot have stated it more clearly?

Although questions of Bertrand's intentions must remain unresolved, Cournot's exposition is clear and available to anyone taking a course in principles of economics and two good semesters of calculus.⁽⁴⁾ Which raises the question as to how Cournot could come to be so misrepresented. In Magnan de Bornier's case, it is clear that he simply did not work through Cournot's mathematics although, presumably, he was capable of doing so.⁽⁵⁾ For the profession, more generally, there may be an additional explanation. Dimand and Dore refer to a collection of papers edited by Andrew Daughety [1988]. Included with the papers is a translation of Bertrand by James Friedman and a reprint of Chapter VII from Bacon's [1897] translation of Cournot. Perhaps, in the collective mind of the profession, Cournot's masterpiece has been truncated into just Chapter VII.

Bertrand, Joseph. "Theorie Mathematique de la Richesse Sociale," Journal des Savants, 67, 1883, pp. 499-508; translated by James W. Friedman in Andrew F. Daughety, ed., Cournot Oligopoly, Cambridge, United Kingdom: Cambridge University Press, 1988, pp. 73-81; translated by Margaret Chevaillier in the Appendix to Jean Magna de Bornier, History of Political Economy, 1992, pp. 646-53; reprinted in M. Dimand and R., Dimand, eds., The Foundations of Game Theory (Volumes I-III), Cheltenham, United Kingdom: Edward Elgar, 1997.

Cherriman, J. B. "Review of Cournot (1838)," Canadian Journal of Industry, Science and Art, 2, 1857.

Cournot, Augustin. Recherches sur les Principes Mathematiques de la Theorie des Richesses, Paris, France: L. Hachette, 1838; Italian translation in Biblioteca Dell'Econ., 1875; English translation by N. T. Bacon in Economic Classics, New York, NY: Macmillan, 1897; reprinted by Augustus M. Kelly, 1960.

Daughety, Andrew F., ed. Cournot Oligopoly: Characterization and Applications, Cambridge, United Kingdom: Cambridge University Press, 1988.

Dimand, Robert; Dore, Mohammed H. I. "Cournot, Bertrand, and Game Theory: A Further Note," Atlantic Economic Journal, 27, 3, 1999.

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

Marshall, Alfred. Principles of Economics, London, United Kingdom: Macmillan, 1890.

Morrison, Clarence C. "Cournot, Bertrand, and Modern Game Theory," Atlantic Economic Journal, 26, 2, 1998, pp. 172-4.

2. This may be because Alfred Marshall [1890] followed Cournot in making quantity the independent variable.
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3. This is what was attempted when Morrison [1998] wrote, "what has been forgotten (or never learned) is that, in his 1838 classic, Cournot symmetrically treated both quantity rivalry and price rivalry (in the sense of analyzing both (in terms of) best-response functions with equilibrium given where conjectures are mutually correct)." Due solely to careless proofing, the crucial phrase "in terms of" did not make it into print, thus changing the meaning of the sentence. Fortunately, with the kind permission of the editor, the manuscript was posted on the Internet prior to publication. It is still available at http://www.indiana.edu/~econweb/faculty/morrpap1.html.
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4. Here, good simply means not watered down. It does not necessarily refer to the more rigorous advanced calculus inspired by Cournot's contemporary, Augustin-Louis Cauchy (1769-1857).
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5. Magnan de Bornier [p. 626] correctly points out that Cournot's monopoly model is equivalent to the usual modern formulation. This equivalence is demonstrated by loosely applying the inverse derivative rule.
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