Magnan de Bornier on Cournot-Bertrand

Recently, Jean Magnan de Bornier (1992) undertook a clarification of the debate between Augustin Cournot and Joseph Bertrand from a historical perspective. In that article he argues strongly for a new and novel interpretation of Cournot’s Recherches (1838)—that Cournot considered only price rivalry and that the traditional interpretation of Cournot’s treatment of duopoly and oligopoly, in terms of quantity rivalry, has resulted from a superficial reading of the Recherches. The purpose of this note is to argue that Magnan de Bornier’s thesis will not survive working through the Recherches, as opposed to merely reading it. We will examine Magnan de Bornier’s arguments supporting his interpretation and demonstrate that those arguments are inconsistent with Cournot’s mathematics.

First we have, “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…. So only one section out of forty (26 to 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, 626, 630). This is incorrect. The systems of equations (5) in section 46 and (6) in section 47 (as well as equations (1) and (2)) could only have been obtained by differentiating with respect to quantities. Since these equations are the only first-order conditions in the chapter on oligopoly, it follows that quantities are the only optimizing variables in the chapter.

Magnan de Bornier believes that Cournot only apparently indicated quantity as the strategic variable because of the following statement: “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] 1960, 80). But since Cournot says “choose for D₁ the value which is best,” quantity is indeed the choice or strategic variable. (In general, the optimization variable(s) is (are) the strategic variable(s).) That this choice is implemented by lowering price makes price a tactical variable rather than a strategic variable. And Magnan de Bornier’s insistence (627 n) on a literal versus a free translation of “le” does not alter the situation. In the quote above, proprietor 1 lowers his price (which he is free to do), and when (or if) proprietor 2 follows suit, it is the price until proprietor 2 then lowers his price. A proprietor’s price is the price posted by that proprietor and may or may not be the market price. It matters not whether anything is sold at that price.

Returning to the question of how frequently or infrequently Cournot used quantity as a strategic variable, once you get into Cournot’s way of doing things you see that Magnan de Bornier is correct that Cournot is consistent throughout chapter 7, but the consistency is in terms of always differentiating with respect to quantities. As previously noted, the first-order conditions for all of the maximizations in chapter 7 are (1) and (2), (5), and (6). These were all obtained by differentiating with respect to quantities. It is true that, after obtaining first-order conditions, Cournot introduces price derivatives using the inverse derivative rule, but this is done for a reason other than to show that price is really the strategic (choice) variable.

What other reason could Cournot possibly have had? Section 45 would suggest that the purpose for changing to price derivatives is to try to demonstrate his conjecture that increasing the number of firms in an oligopolistic market...