Cournot: The Conjectural Variations Approach to Duopoly Theory

“Conjecturizing” Cournot: The Conjectural Variations Approach to Duopoly Theory Nicola Giocoli

It is a well-recognized peculiarity of the so-called years of “high theory” that for the first time in the history of economics the mental variables— that is, the expectations, conjectures, and beliefs—of the agents featuring in economic models gained the spotlight and became the explicit object of formal analysis. This happened as a consequence of the effort to extend the notion of static equilibrium to a multiperiod setup. Thus, elements such as an agent’s knowledge, uncertainty, and foresight became absolutely central, so much so that, following G. L. S. Shackle’s 1967 classic, it became commonplace to single them out as the leading topics during the period ranging from the mid-1920s to the early 1940s. Indeed, it was deemed impossible even to define the notion of economic equilibrium without relating it to the time dimension and to planning behavior, that is, without specifying how an agent’s mental variables are formed and how economic interactions are framed in a multiperiod horizon. Crucial notions in contemporary economics such as temporary and intertemporal equilibrium or the ex ante–ex post distinction bear witness to the achievements of the “high theorists.” In a single paper one cannot even briefly sketch the main events and characters leading to the rise of mental variables in interwar economics. Luckily, we already have some brilliant accounts of these developments Correspondence may be addressed to Nicola Giocoli, Department of Economics, University of Pisa, Via Curtatone e Montanara 15, 56126 Pisa, Italy; e-mail: [email protected]. I wish to thank Marco Dardi, Maria Cristina Marcuzzo, Salvatore Rizzello, Annalisa Rosselli, and the referees for their useful comments and suggestions. Any remaining errors are mine. History of Political Economy 35:2 © 2003 by Duke University Press.

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as they took place in the key fields of general equilibrium and business cycle theory.1 However, there does not exist an equivalent amount of work covering the similar developments that occurred in the same period in another crucial field, that of imperfect competition theory. We have indeed a rich literature dedicated to, say, the Sraffian critiques of Marshallian cost curves, the debate over Clapham’s empty boxes, and Robinson’s and Chamberlin’s books, to name a few, but the focus is always upon either the mechanical or the empirical side of imperfect competition. This is rather peculiar, since the interaction between oligopolists offered the interwar economists the most natural ground for the first theoretical investigations of the formation and revision of the agents’ expectations and multiperiod plans. While an exhaustive reconstruction of the development of oligopoly theory during the years of “high theory” would be a daunting task, in what follows I will tackle a rather limited topic, namely, the interwar debate over the determinateness of the solution for the standard Cournot duopoly model with quantity competition. As I show in this essay, the main outcome of the debate was the acknowledgment of the necessity to explicitly take into account each firm’s conjectures in order to provide such a solution. Thus, 1920s–1930s duopoly theory should be recognized as another instance of the rise of mental variables—something that seems to have escaped existent narratives. Starting from A. L. Bowley’s 1924 Mathematical Groundwork and from the 1924 edition of A. C. Pigou’s Economics of Welfare, the idea began to spread that the Cournot-style reaction functions of the duopolists had to be given a conjectural interpretation. The problem was that of reconciling the reaction functions—an allegedly dynamic concept—with the static setup of the Cournot model.2 The latter ambiguously mixed a static formalization of what today would be called a one-shot simultaneous game with a dynamic story in terms of each firm’s actions, reactions, and counterreactions that would be better modeled as a sequential game. In particular, Cournot’s reaction functions were formulated as a static equilibrium notion but had as argument the actual output produced by the other firm, a piece of information that each firm could possess only 1. See, for example, Shackle 1967, Hansson 1982, Donzelli 1986, Ingrao 1989, Ingrao and Israel 1990, Currie and Steedman 1990, Weintraub 1991, and Zappia 2001—not to mention the vast literature on individual economists such as Friedrich Hayek, John R. Hicks, Erik Lindahl, and John Maynard Keynes. For an assessment of the role of mental variables in interwar economics and of their relation with early modern game theory, see Giocoli 2003. 2. If not specified otherwise, by the term Cournot model I mean the standard interpretation of Cournot’s 1838 contribution and not his own original presentation. See below, section 1.

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in a sequential game. Yet, as is well known, the feature of the Cournot model that raised the most objections was another one, namely, the assumption that each firm behaved as if its rival would not react to its own move. This assumption was really puzzling, because in Cournot’s pseudodynamic story each firm would inevitably realize that its rival was not passive at all but reacted to its moves. Bowley and Pigou offered a chance to overcome the problems created both by the assumption and by the tricky relation between statics and dynamics. The two authors emphasized the conjectural nature of the reaction functions: the former by generalizing the Cournot assumption with the new notion of conjectural variations, the latter by stressing the role of psychological features in any oligopolistic setup. Yet this double opening to mental variables had one major drawback, namely, the indeterminacy of the duopoly equilibrium, since it entailed the existence of a different solution for every possible state of the firms’ conjectures. The success of the conjectural approach was immediate and for almost a decade it dominated the scene of oligopoly theory. The indeterminacy of the duopoly equilibrium caused no particular problems, since it confirmed the result reached by F. Y. Edgeworth and the other early critics of the Cournot model. Moreover, the approach made it possible to tell a more convincing story about how duopolists competed in a static setup. What featured in each firm’s reaction function was in fact the expectation of the rival’s output, not the rival’s actual choice. Such expectation descended from a sort of instantaneous mental experiment run by each duopolist, not from a trial-and-error process involving a sequence of interactions.3 The goal of this essay is to present the main steps that led the conjectural approach to its dominant position in the literature. I leave instead to future research the investigation of its quick demise in the second half of the 1930s. 1. A Basic Duopoly Model In this section I present a simple model of duopolistic competition that serves as a benchmark in the following sections.4 Consider two firms 3. In the mid-1930s also the Stackelberg model of oligopolistic competition underwent a similar process of “conjecturization.” This time the transformation of the original model, which was formulated in terms of the tangible reactions to the firms’ actual output choices, into a conjectural one was almost immediate, as it was carried out by the reviewers of Stackelberg’s 1934 book. 4. The model is taken from Varian 1992, chap. 16.

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that produce a homogeneous product with output levels q1 and q2 , and an aggregate output of Q = q1 +q2 . Provided the invertibility conditions are met, the market price associated with this output may be expressed in terms of the inverse demand function: p(Q) ≡ p(q1 + q2 ). Each firm i is supposed to have a cost function given by ci (qi ), i = 1, 2. Assuming that the strategic variable for both firms is the output level, firm 1’s maximization problem is maxq1 π1 (q1 , q2 ) = p(q1 + q2 )q1 − c1 (q1 ). This shows that firm 1’s profit depends on the output choice of firm 2. In order to make an informed decision, firm 1 must therefore forecast firm 2’s choice. A similar problem can be formulated for firm 2. According to the standard version of the Cournot assumption, each firm expects the other not to modify its behavior as the market price changes.5 The first-order conditions (FOCs) of the firms’ maximization problems are ∂πi (q1 , q2 )/∂qi = p(Q)+p (Q)qi −ci (qi ) = 0, i = 1, 2. These two FOCs characterize what I call the basic Cournot duopoly model. In general, we require that in order for the firms’ choices to constitute an equilibrium, two conditions need to be satisfied. The first condition is that neither firm, on the basis of its own beliefs, must desire to modify its choice. The second condition is that the equilibrium actions of the firms are consistent with the beliefs upon which they act. Thus, in the duopoly model an equilibrium is given by every pair of output levels (qˆ1 , qˆ2 ) such that (1) each firm is choosing its profit-maximizing output given the beliefs about the other firm’s choice; and (2) each firm’s beliefs are correct at equilibrium. In our model such an equilibrium pair identifies the Cournot equilibrium. The FOC for firm 1 determines the optimal choice of output as a function of its beliefs about firm 2’s output choice. This defines the reaction function (RF) of firm 1, that is, it depicts how firm 1 will modify its output choice according to the various beliefs it might have about q2 . Firm 1’s RF f1 (q2 ) is implicitly defined by ∂π1 (f1 (q2 ), q2 )/∂q1 = 0. The slope of this RF indicates how firm 1 optimally reacts to a change in its beliefs about the rival’s q2 : f1 (q2 ) = −(∂ 2 π1 /∂q1 ∂q2 )/(∂ 2 π1 /∂q12 ). A similar result holds for firm 2’s RF f2 (q1 ). Instead of characterizing the RFs in terms of one firm’s beliefs about its rival’s choice, we may follow an alternative approach, authoritatively 5. Another conjectural element features in the model, namely, that both firms must know the market (i.e., the expected reaction of the buyers to any variation in the price must coincide with the actual reaction).


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supported by James Friedman. He defines the RF as a function that determines a firm’s action in a given period in terms of the other firm’s action during the preceding period. This view requires that the duopoly game be played sequentially, while the one based upon the firm’s beliefs fits well also in a one-shot simultaneous choice setup (Friedman 1977, 149). With his definition, Friedman (1992, 363) aims at stressing the difference between the role of the Cournot equilibrium in a simultaneous choice model and that played by the same concept in a dynamic, multistage context. Taken as a solution to the former, the Cournot equilibrium is in fact quite convincing. Since both firms make only one choice and do so simultaneously, it is apparent that the decision of, say, firm 1 cannot affect the choice of firm 2, and vice versa; this validates the Cournot assumption of taking the rival’s action as given. Note, however, that what one firm expects the other to choose does influence its own decision, even in the simultaneous game, since an equilibrium is characterized by both firms’ having their expectations fulfilled. Yet these expectations can be safely constructed even in a static, one-shot setup, thereby eliminating the confusion between statics and dynamics typical of the naive versions of the Cournot model.6 It follows that we can obtain the Cournot equilibrium as a proper solution for a simultaneous game independently of the RFs, the latter—when defined à la Friedman—being the main culprit for the above-mentioned confusion. Crucial in Friedman’s approach is the idea of the Cournot solution to the duopoly game as a static Nash equilibrium. Yet, this idea is quite a recent one, being based upon the reinterpretation of the Cournot model as a fixed-point problem offered by Martin Shubik in the late 1950s.7 Throughout the 1920s and 1930s, instead, the standard (but not exclusive) interpretation of the model focused upon the dynamic convergence process to equilibrium, so that the study of the RFs was (almost) never separated from the analysis of the equilibrium. This implied that most oligopoly theorists of the period had to come to terms with the inevitable mixture of statics and dynamics that the orthodox reading of Cournot entailed, as well as with the attribution to the firms of a seemingly irrational behavior. To summarize, two views of the RFs have emerged in the literature. One considers them as based upon the firms’ beliefs, the other upon the 6. An example of a static method for constructing a pair of conjectures capable of supporting the Cournot equilibrium in a simultaneous game is offered in Daughety 1988. 7. See Shubik 1959, 63; Leonard 1994; and Giocoli 2003, chap. 5.

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firms’ actual choices in the preceding period. In the former view, the RFs are meaningful also in the static setup of a simultaneous game. In the latter view, the RFs are meaningful only in a sequential game, so that the static problem of the Cournot equilibrium can be dealt with separately and independently of the RFs themselves. In other words, if we stick to a definition of the RFs in terms of the firms’ actual choices, the use of these functions in a simultaneous game must necessarily be just a technical device, since each firm selects its own action before observing the rival’s move and has no chance at all to react. No real economic meaning can be attached to the RFs in such a setup. If instead the RFs are defined as a decision rule giving the optimal choice for a firm as a function of the conjectured action of the rival, then even in a simultaneous setting the RFs do have a meaning. The clear-cut formulation of the two views is a recent achievement but, as I show in the following sections, the contrast first emerged in the period under scrutiny. 2. Cournot Duopoly: The Indeterminacy Issue In a recent paper, Jean Magnan de Bornier (1992, sec. 4) argued that at least until the 1930s the issue in duopoly theory was not that of contrasting two different competitive strategies, namely, Cournot’s quantityadjustment versus Bertrand’s price-adjustment, but rather that of investigating whether the duopoly model had a determinate solution. This observation is quite convincing but not exhaustive. Another missing theme in the literature was in fact that of the firms’ conjectures. Given that Cournot did not explicitly mention them, the duopolists’ conjectures over the rival’s behavior were either attributed to Cournot himself 8 or simply ignored by most commentators. 2.1 Edgeworth and Pigou on Duopolistic Indeterminacy Several objections were raised against Cournot’s conclusion that the solution could be perfectly determined. Building upon Bertrand’s critique (Bertrand [1883] 1992), Edgeworth ([1897] 1968) brought the most significant attack with a well-known paper in the Giornale degli economisti. 8. As noted by Mary Ann Dimand and Robert Dimand (1996, 26–28), early reviewers of the 1838 book, such as P.-G. Fauveau (1867) and Bertrand ([1883] 1992), believed that it was Cournot, not the duopolists, who assumed that should either firm change its output, the rival’s quantity could nonetheless remain constant.


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By showing that the imposition of a quantity constraint caused the market price to oscillate inside an interval, he concluded that the equilibrium of the duopoly model was hopelessly indeterminate. Edgeworth’s argument was so convincing that still in the 1920s his result was taken as the benchmark on the matter. Edgeworth himself, in an unusual boast of confidence, claimed in the introduction to the 1925 English translation of his 1897 paper that “now the demolition of Cournot’s theory is generally accepted.”9 This position was shared, among many others, by Pigou, who in the 1924 edition of his highly influential Economics of Welfare stated that “[Edgeworth’s] view is now accepted by all mathematical economists” (237). Pigou’s endorsement of the indeterminacy result is particularly important for us, since it was to reinforce Edgeworth’s argument that he introduced the theme of the duopolists’ conjectures. After the above-quoted sentence, Pigou continued in fact by arguing that in a duopoly situation the firms’ investments10 were interdependent and that each duopolist’s investment choice depended upon the “judgment of the policy which the other will pursue, and this judgment may be anything according to the mood of each and his expectation of success from a policy of bluff. As in a game of chess, each player’s move is related to his reading of the psychology of his opponent and his guess as to that opponent’s reply. Hence the investment of each separately and of the two jointly is indeterminate” (238; emphases added). What Pigou failed to realize was that the cause of indeterminacy of the duopoly equilibrium that he suggested in the passage differed from the one that Edgeworth suggested. Pigou referred in fact to the conjectures that each firm held about its rival’s behavior; yet these had been hardly mentioned in the previous debate over the Cournot model: they played no role either in Bertrand’s critique (see note 8) or in the most well known part of Edgeworth’s model, although they featured prominently in a neglected section of the latter’s analysis, namely, where Edgeworth ([1897] 1968, 212–13) presented the metaphor of the two explorers dragging their sledges over the Arctic plains. 9. Edgeworth [1897] 1968, 201. Remember also that in his classic review of Amoroso 1921, Edgeworth (1922, 405) had recommended that in case of an English translation just one section ought to be cut out, precisely the one in which Amoroso defended Cournot’s solution. On Edgeworth’s own duopoly model, see Nichol 1935 and Casarosa 1984. 10. The choice variables in Pigou’s argument were the firms’ investments, not the quantities or the prices.

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In these pages Edgeworth pioneered no less than the saddle point condition11 for an equilibrium point, the latter being defined as a position from which none of the explorers had an interest in moving. Yet he believed this position to be an unstable one: he foreran in fact both outcomes of the Stackelberg model, that is, both the leadership equilibrium, where one of the explorers anticipates the rival’s RF while the latter behaves à la Cournot, and the absence of equilibrium in the so-called double-leader case, where both explorers believe the rival to play à la Cournot.12 These outcomes were taken by Edgeworth as a validation of his belief that the result of the duopoly model was indeterminate. They showed, in fact, first, that the saddle point solution was not robust to the unilateral deviation made by any of the explorers who believed he could anticipate the rival’s RF, and, then, that the ensuing leadership equilibrium was not robust to a similar deviation made by the second explorer as well. The whole argument depended of course upon the agents’ conjectures as to the rival’s behavior. Yet, despite their exceptional content, these pages by Edgeworth failed to entice the interests of economists. What consolidated in the literature was instead a rather mechanistic version of Edgeworth’s analysis, where the indeterminacy of the equilibrium followed from the imposition of the quantity constraint and manifested itself through the endless sequence of price adjustments made by the two firms. As a consequence, the conjectural nature of the duopolistic indeterminacy had to be “rediscovered” by Pigou in 1924. Exactly in the same year another British economist, A. L. Bowley, made a formal step in the same direction, by explicitly introducing the firms’ conjectures in the Cournot model through the notion of conjectural variations. This was an ambiguous operation. On the one hand, it could be read as a validation of the indeterminacy result: it gave in fact a mathematical form to Pigou’s idea that each firm’s conjecture about its rival’s behavior could be anything, so that the duopoly equilibrium was hopelessly indeterminate. On the other hand, it could also be taken as an invitation to attach a specific value to the firms’ conjectures, so that a series of special cases could be investigated where the outcome was indeed well determinate. Actually, both interpretations were advanced in the literature. A general consensus arose instead on the fact that, thanks to Bowley’s idea, the firms’ conjectures had eventually gained the center 11. “Hog’s back” in Edgeworth’s words. 12. See Stackelberg 1934, [1943] 1952; and Varian 1992, chap. 16.


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stage in duopoly theory. Before exposing the details of Bowley’s proposal, let me briefly refer to Cournot’s original analysis, in order to validate the claim that no explicit reference to the duopolists’ conjectures could be found in it. 2.2 Cournot’s No-Conjecture Model It is hard to read today Cournot’s famous chapter 7 “Of the Competition of Producers” (Cournot [1838] 1971, 79–89) without being influenced by the modern notions of fixed point and Nash equilibrium. Yet, the association of Cournot with noncooperative game theory is just fortyyears old. Before Shubik’s 1959 reinterpretation, in fact, the Cournot duopoly model was viewed as a dynamic analysis involving a sequential adjustment process undertaken by two firms that ignored that their own choices could influence the rival’s behavior.13 Thus, the outcome of the process—the equilibrium position—was not considered as a static fixed point (namely, a Nash equilibrium), but as the end state of a trialand-error path. This reading of Cournot is not surprising at all if we look at his model through the eyes of a pre-Nash economist living in the early decades of the twentieth century. The reading is in fact fairly consistent with what Cournot actually said. The key remark comes at the very beginning of Cournot’s analysis in chapter 7. He argues that in his model the firms are supposed to act independently, that is, they are not to reach any formal agreement to collaborate (79–80). In the light of this assumption, let us consider Cournot’s comments on the possibility of a joint monopoly position, that is, one in which the duopolists maximize their total profits by jointly producing the monopoly output and selling it at the monopoly price. He claims that such a position should constitute a stable solution for duopolistic competition because any unilateral deviation is only temporarily profitable for the deviating firm since the rival’s retaliation soon arrives to punish it. Thus, any deviation from the joint monopoly solution is a mistake because it overlooks the rival’s reaction (83). Yet he adds that the actual self-policing of this solution would require that each firm be capable of anticipating the rival’s reaction. He believes, however, that economic agents can never be supposed to be free from errors or wrong forecasts, 13. Note that the fixed-point reinterpretation of Cournot was, to say the least, far from explicit in Shubik’s previous forays in duopoly theory, namely, Mayberry, Nash, and Shubik 1953 and Shubik 1955.

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that is, cannot be taken to be perfectly rational decision-makers (83). As a consequence, the joint monopoly outcome is not a stable equilibrium, because it is implemented by “imperfect” agents who will always have an incentive to deviate. Only a formal agreement can enforce and maintain it, but this is excluded by assumption. In short, Cournot’s idea is that while the joint monopoly is the outcome that would be reached by “perfect” decision-makers who anticipate the rival’s reaction, actual oligopolistic competition is the realm of “imperfect” agents, subject to errors and lack of foresight. These are the same agents that feature in the trial-and-error adjustment process to Cournot equilibrium, namely, agents who never learn from their mistakes and myopically go on neglecting the rival’s reaction. As far as each duopolist’s view over the other firm’s behavior is concerned, the textual evidence of Cournot’s original model brings no support to the idea that the conjectural element is involved. The basic feature is again that firms act independently, that is, that no firm can have any direct influence on the decision of the other firm. It follows that all that firm 1 can do when q2 has been determined is to set q1 at the best possible value, and vice versa. This is formally represented in the original model by a pair of RFs that are based upon the actual preceding choice of each firm’s rival, so that the “game” is explicitly given a sequential structure (80–81). Both firms are therefore myopic “only” in the sense that they fail to take into account the indirect effect of their decisions upon the rival’s behavior. In other words, Cournot’s duopolists are never attributed the conjecture or belief that the rival’s offer is given, but only the myopic idea that only direct influence is relevant. Thus, the adjustment process to equilibrium does not require the firms to stick to a constantly disconfirmed conjecture as to what will happen to the rival’s output after one’s own action, but rather to formulate no conjectures at all on such reaction. It is only when they evaluate—through a static mental experiment—the profitability of the joint monopoly outcome that firms are modeled by Cournot as formulating conjectures on the rival’s reaction to a possible deviation. This, however, is an activity that can be safely undertaken only by “perfect” decision-makers, while a real firm would fail to foresee the reaction or would anticipate it wrongly. It is because of their inability to perform such a mental experiment that Cournot’s firms neglect the collusive opportunity and engage in a sequential “game” in which they abstain from formulating any conjecture on the rival’s behavior. In short, the conjectural element is not a part of Cournot’s original model: the


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mythical duopolists who assume that the rival’s quantity is given are just a product of later interpretations.14 Of course, the distinction between a myopic duopolist who ignores the indirect effect of his actions and one who explicitly assumes that the rival will not react may appear rather subtle, if not fictional.Yet, a precise identification of the assumptions underlying the Cournot model entails great historical significance. This is because one of the most important demarcations in the history of economic modeling is that between the “mental” and the “mechanical” models, that is, between the models in which the decision-makers’ reasoning process is explicitly considered, so that some hypotheses are offered as to what expectations and beliefs are held and how they are formed and revised, and the models in which no such features are taken into account, so that decision-making is viewed as a “black box” producing choices given the objective data of the choice environment. Very few models of the former kind were available in the economic literature before the rise of mental variables during the interwar years. They were indeed so rare—and the demarcation is so crucial—that when analyzing a pre-1930 model it is hard to satisfy ourselves with a mere “as if” argument, whereby a past economist’s attribution of specific conjectures to his model’s agents can benevolently be granted as tacit or implicit. As a consequence, it seems not so trivial to distinguish between the case of duopolists who are modeled as formulating a precise, although simple, conjecture and the case of duopolists who make their decisions without conjecturing anything at all. My point— admittedly, nothing more than a working hypothesis formulated only on a textual basis—is that the mathematician Cournot seemed to believe that only the conjectures of “perfect” agents can be modeled, while formal analysis is powerless with respect to the imperfect thought processes of real-world agents, thereby forcing the rigorous analyst to avoid altogether the recourse to mental variables.15 14. My view is confirmed by the careful reconstruction in Nichol 1934b. Another legend concerning Cournot’s model is that of the firms choosing the quantities as their strategic variable. For a convincing argument that Cournot’s firms are indeed competing in prices (so that two different prices temporarily coexist at each stage of the competitive process), see Magnan de Bornier 1992, 625–31, and Zanni 1995, 2–4. On the same issue, see also the recent exchange between Clarence Morrison (2001) and Magnan de Bornier (2001). 15. One of the referees suggested that further support for my hypothesis may lie in the fact that the 1838 book extended to economics some of the optimization techniques that Cournot had previously developed for application to mechanical phenomena (see Prekopa 1980, 535– 36).

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To reiterate the point, Cournot’s original model is a rudimentary dynamic model showing how the market equilibrium can emerge in a situation where the firms formulate no conjecture at all on the rival’s behavior but simply adjust their output to the rival’s offer in the preceding period. As the number of firms goes up, the mistake committed by ignoring the indirect influence is ever smaller, until in the limiting case the market becomes perfectly competitive (84).16 Of course the Cournot model also shows that the equilibrium obtains by solving the simultaneous system of the two RFs (81). This goes in the direction of the static interpretation of the model and its solution. Yet the system is just a device to show how to calculate the equilibrium that emerges from the dynamic process. As I explained above, any static fixed-point view of the Cournot equilibrium would require that an interpretation in terms of the firms’ expectations be attached to the RFs (see section 1). This interpretation would, however, betray the original formulation of Cournot’s RFs in which mental variables are not mentioned at all. As argued by Magnan de Bornier, it was Irving Fisher who put forward the legend that Cournot’s major fault in duopoly theory was the idea that each firm acted following the assumption that the rival’s output was given. He judged both this assumption and the twin one that each firm took the rival’s price as given to be clearly in contrast with what happened in real markets. He argued in fact that “as a matter of fact, no business man assumes either that his rival’s output or price will remain constant any more than a chess player assumes that his opponent will not interfere with his effort to capture a knight. On the contrary, his whole thought is to forecast what move the rival will make in response to one of his own” (Fisher 1898, 126). Yet, despite Fisher’s invitation to deal with the psychological side of what he considered a very complex dynamic problem,17 most oligopoly theorists in the early twentieth century remained faithful to Cournot’s text and so avoided referring to the duopolists’ conjectures or beliefs. Duopolistic competition was modeled as a sequential process whereby 16. Note that the limiting result implies that in a perfectly competitive market firms can legitimately avoid formulating conjectures on their rivals’ behavior, but not that they must necessarily do so. This point was raised by Oskar Morgenstern ([1935] 1976, 181) and George Stigler (1940, 525). 17. One of the referees pointed out the apparent contrast between Fisher’s 1898 view (which could be coupled with his explicit consideration of expectations in the well-known formula for the determination of real interest rates) and his previous 1892 critiques against the use of psychology in value theory; see Fisher [1892] 1925, 5, 23.


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each firm reacted to what the rival had done in the previous period while neglecting the influence of its own action. Thus, the theorists committed the same mistake as Cournot’s duopolists, that is, they ignored the indirect influence (namely, they modeled the firms as ignoring it). This attitude was so widespread that it was defined as the “usual way” to view oligopoly models (Chamberlin 1929, 66). This is why Bowley’s 1924 proposal raised so much interest and discussion. 3. Bowley’s Conjectural Variations Approach In his Mathematical Groundwork, Bowley (1924, 38) argued that in order to solve the FOCs of a standard duopoly problem with quantitycompetition “we should need to know [q2 ] as a function of [q1 ], and this depends on what each producer thinks the other is likely to do. There is then likely to be an oscillation in the neighborhood of the price given by the equation marginal price = selling price, unless they combine and arrange what each shall produce so as to maximize their combined profit.” Replacing the original notation with that introduced in section 1 we can formalize Bowley’s proposal as follows. Let vij = ∂qj /∂qi be firm i’s arbitrary conjecture about how j will respond to a small variation of i’s output. Firm 1’s FOC becomes ∂π1 (q1 , q2 )/∂q1 = p(Q)+p (Q)[1+ v12 ]q1 − c1 (q1 ) = 0. A similar FOC holds for firm 2, whose conjectural parameter is v21 . The term vij in the FOCs was a novelty with respect to the usual representation of the duopoly problem. This term, which was later called conjectural variation (CV), meant that the solution of the duopoly model depended upon the exact value of each firm’s conjecture over the rival’s reaction. For example, if v12 = v21 = 0, the FOCs turned into those of the standard Cournot model in which each firm believed that the rival would not react to one’s own choice. As I said before, this on the one hand stressed the indeterminacy of the result, as no constraint was placed upon the conjectures, but on the other hand it allowed the theorists to devise values for the CV terms that could warrant a determinate solution. In short, Bowley’s proposal showed that the duopoly model was indeed indeterminate, but only up to a proper assignment of a value to the CV term. As to Bowley himself, he was probably more inclined to highlight the indeterminacy of the result. This is at least what can be argued from the second part of the previous quotation, where he seemed to claim that the

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dependence of the solution upon the conjectures entailed that the system would “oscillate,” unless an explicit collusive agreement was reached. Hence collusion represented for Bowley the way out from the indeterminacy caused by the conjectural term. Bowley’s proposal opened a new direction for the debate on oligopoly theory, one in which the firms’ conjectures were explicitly at the center of attention. As Cournot had anticipated, the most immediate effect of allowing the mental variables into the duopoly model was that of enhancing the plausibility of the joint monopoly solution. In a review of Mathematical Groundwork, Allyn Young argued that if we allowed the conjectural element to enter the analysis, we had also to concede that each firm could anticipate the ultimate consequences of the rival’s chain of adjustments and thus discover that they were less profitable than the joint monopoly outcome. Collusion would then turn out to be the stable solution of the duopoly model, since each perfectly rational duopolist would understand that deviating from it would cause losses to both firms (Young 1925, 134). Similarly, in a 1928 paper Joseph Schumpeter argued that intelligent duopolists could not fail to realize all the implications of their situation, so that “they will hit upon, and adhere to, the price which maximises monopoly revenue for both taken together. . . . The case will not differ from the case of conscious combination—in principle—and be just as determinate” (370 n). Schumpeter added that similar results had been “independently arrived at by Dr. Chamberlin.” Chamberlin’s analysis went to press one year later in the Quarterly Journal of Economics. 4. Chamberlin on Conjectural Variations An influential paper in the literature on duopoly theory was published in 1929 by Edward Chamberlin in the QJE, with the title “Duopoly: Value Where Sellers Are Few.” The paper was later reproduced with slight variations as chapter 3 of Chamberlin’s well-known book, The Theory of Monopolistic Competition (1933). 4.1 Making Conjectures Explicit The central statements of the paper were as follows. First, Chamberlin claimed that all the essential principles of oligopolistic competition could be discovered through the analysis of the duopoly case (1929,


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64–65).18 Second, the duopoly problem was not one problem, but several. Hence the indeterminacy of the equilibrium was only with respect to the choice of the best set of possible assumptions underlying the various problems, while inside the given “rules of the game” the duopoly problem had always a determinate solution (91). Third, he claimed that if the duopolists were able to consider the total influence of their own actions upon the market situation, the solution would always be the joint monopoly outcome (92). And fourth, he argued that the only element that could make the solution indeterminate, thereby blocking the formation of a joint monopoly, was the uncertainty affecting the firms’ conjectures (92–93). These four statements represented a major breakthrough in the literature, since they clashed with the consolidated viewpoint according to which the duopoly equilibrium was indeterminate, and marked the “revenge” of those who believed in the determinateness of the solution. Moreover, they brought forward the bold thesis that whenever the firms were intelligent enough to read beforehand the whole structure of their interaction, they would always independently realize that their mutual convenience lay in the joint monopoly outcome. This thesis had obvious policy implications and so raised an intense debate.19 To formulate the previous statements, Chamberlin moved from recognizing that the interdependence of outcomes was the essential feature of the duopoly situation. Each firm was forced to take into account the policy of the rival, so that it had to acknowledge the indirect influence that its own actions exercised upon the profit through the mediation of the rival’s reaction (65). He observed that the standard approach to duopoly was instead that of viewing each firm as determining its own actions under the hypothesis that the rival was unaffected by them. This meant that each firm took into account only the direct influence of its actions. Yet, according to Chamberlin, the only solution consistent with the hypothesis of profit-maximizing behavior was one in which both firms recognized the total influence of their actions (66). 18. This idea, which was almost universally accepted in the period, implied the neglect of coalitional issues such as those raised by Alfred Marshall ([1890] 1925). 19. A pair of contributions that focused on themes other than the conjectural issues deserve to be mentioned here: the paper by Ronald Coase (1934), where it was argued that if the speed of the duopolists’ reactions was accounted for, then joint monopoly could not be the normal outcome, since it could be established only through an explicit agreement; and the article by A. J. Nichol (1934a), which argued that the firms’ cost structure and production conditions were much more relevant for the validity of Chamberlin’s thesis than the accuracy of the firms’ foresight.

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Much of Chamberlin’s paper was dedicated to examining the traditional models in the literature: the Cournot model under the hypothesis of the quantity-passivity of the rival; the Cournot model under the hypothesis of the price-passivity of the rival; the critiques made against Cournot by Bertrand, Marshall, and Pareto, all based upon the refutation of the price-passivity assumption; and the Edgeworth price-oscillation model. Then Chamberlin advanced his objections to Edgeworth’s analysis and proposed a new model of price competition that was strikingly similar to the one developed in the same period by Richard Kahn at Cambridge University.20 Yet it was Chamberlin himself who recognized that a new phase in the analysis of duopoly opened as soon as we realized that in all the abovementioned models the firms did not behave truly rationally. A fully rational behavior required in fact the consideration of the total influence of one’s own actions, while in all the previous models each firm was assumed to ignore the rival’s reaction (83). Chamberlin (1929, 83 n. 6) acknowledged Fisher’s merit for having been the first to underline the relevance of this point in his 1898 paper. Thus, in order to devise a model of rational duopolistic behavior, we had to “let each seller, then, in seeking to maximize his profit, reflect well, and compass the total consequences of his move. He must consider not merely what his competitor is doing now, but also what he will be forced to do in the light of the change which he himself is contemplating” (83–84). As even some of his critics did not fail to notice,21 the explicit introduction of the conjectural element was Chamberlin’s most significant contribution in the paper, as well as in chapter 3 of the 1933 book. It should be noted that while the previous passage seems to open the door to an explicit dynamic approach to duopoly, where competition is a multiperiod process in which firms formulate their expectations and then revise them according to the market’s outcome, what Chamberlin was actually thinking of was instead a static mental experiment, in which each duopolist was intelligent enough to exploit all the available information to anticipate the whole chain of reactions and counterreactions that would follow from any of his actions. This explains why Chamberlin endorsed Young’s 1925 suggestion that, given such mental processes, the only stable solution to the duopoly case was the joint monopoly one (84). 20. See Kahn [1929] 1989, chap. 7. 21. See, for example, Nichol 1934a, 318.


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The idea was in fact that if each duopolist was seeking “his maximum profit rationally and intelligently, he will realize that when there are only two or a few sellers, his own move has a considerable effect upon his competitors, and that this makes it idle to suppose that they will accept without retaliation the losses he forces upon them. Since the result of a [price] cut by anyone is inevitably to decrease his own profits, no one will cut, and, altho the sellers are entirely independent, the equilibrium result is the same as tho there were a monopolistic agreement between them” (85).22 An implication of Chamberlin’s argument was that, contrary to what Cournot had claimed, there was no gradual convergence to the perfectly competitive outcome as the number of firms went up. Regardless of the number of firms, if each firm’s action did have some influence on the rivals’ behavior and if the firms were intelligent enough to anticipate the outcome of their total interaction, the solution would always be the joint monopoly outcome. A discontinuity arose only when each firm’s influence upon the others became negligible (and everyone realized this). According to Chamberlin, if what disappeared was just the indirect influence, we had the Cournot solution, whereas if also the direct influence vanished, we had the competitive outcome (84).23 Another implication was that a certain degree of competition in a duopoly could be maintained only by the absence of the firms’ perfect knowledge and foresight, that is, by the uncertainty affecting a firm’s expectations as to what its rival was going to do. As remarked by Chamberlin, this was the same kind of uncertainty that Pigou had envisaged in 1924 (see above, section 2.1), in the sense that it was not based upon the agents’ neglect of the indirect influence of their actions, but only upon their actual inability to foresee the rival’s reactions. It was the uncertainty about what belief to hold on the rival’s conduct (87). Chamberlin indicated various sources of this uncertainty. For example, there may be uncertainty over what variable was kept fixed, if any at all, by the rival, or uncertainty over the rival’s intelligence and 22. The idea that the joint monopoly outcome is the result of a static mental experiment neutralizes the critique by Richard Kahn ([1929] 1989, chap. 7, sec. 31) and William Fellner (1949, 67) against Chamberlin’s solution, namely, that no explanation was given as to how the duopolists reached in concrete the monopoly outcome. No convergence process is actually involved, but just the immediate recognition by the two perfectly intelligent duopolists that this is the mutually most beneficial result. 23. This discontinuity was criticized by many commentators of Chamberlin’s paper and book; see, for example, Kahn [1929] 1989, chap. 7, sec. 31; and Harrod 1933, 663–64.

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farsightedness, or uncertainty over the real effect of one’s own move upon the rival’s profit (87–88). Moreover, the rival’s reaction could be delayed in time, so that an interval of positive gain might exist after a firm’s deviation from the joint monopolistic action. This complicated the problem, since each firm should weigh the costs and benefits of the deviation and formulate another conjecture as to the timing of the rival’s retaliation (90–91). Given all these possible sources of uncertainty, Chamberlin concluded that to support the determinate outcome in the duopoly situation, no hypothesis on the firms’ intelligence short of omniscience would suffice (91). This admission paved the way for the revenge of the indeterminacy tradition just a few years later, as well as for Chamberlin’s own indeterminacy result presented in the appendix to his paper. 4.2 Assessing Bowley The mathematical appendix of the 1929 paper (93–100)24 contains, among other things, Chamberlin’s assessment of Bowley’s CV approach. His main idea, although admittedly put forward “only with hesitation” (99), was that this approach was quite similar to that of Cournot, since it did not encompass a kind of uncertainty different from that considered in the Cournot model. Thus, although his works helped popularize the use of CV terms, a fair account of Chamberlin’s position cannot neglect that he did not view Bowley’s approach as a general method to tackle duopoly theory. Behind this idea lay Chamberlin’s denial that firm 1’s offer q1 could really be a function of q2 ,25 as seemingly implied by the CV term. He argued in fact that when no uncertainty over the rival’s behavior existed, the solution to the duopoly model presented by Bowley coincided with Cournot’s, that is, the solution arose when the CV terms disappeared (93). To understand this statement we have to take into account that Chamberlin modeled the Cournot RFs as if they featured the actual output choice of the rival. He believed in fact that Cournot’s hypothesis was just that each firm set its own supply in the light of the rival’s present offering, which was assumed not to change, since all indirect influences were neglected (66). Hence, in both the Cournot and the Bowley model, if no uncertainty over the rival’s offer existed, each firm would simply 24. Reproduced with some modifications as appendix A in Chamberlin 1933, 179–87. 25. Namely, that it could univocally depend upon the value of q2 . Chamberlin used the term function quite loosely here.


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seek to maximize its profits by setting its own offer at the most advantageous level given the amount simultaneously offered by the rival. This meant to Chamberlin that q1 was not really a function of q2 , and vice versa, but rather that the two were a pair of optimal offers resulting from the simultaneous resolution of a static maximization problem where firm 1 knew what firm 2 was offering because it knew that firm 2 optimized on the basis of what firm 1 itself was offering, and vice versa. This reasoning reveals that Chamberlin was struggling to distinguish between a functional relationship and a fixed-point condition. In terms of James Friedman’s distinction (see above, section 1), what he had in mind was a static fixed-point view of Cournot equilibrium. According to Chamberlin, the real difference between Bowley and Cournot emerged when, due to the presence of uncertainty, none of the firms knew what the rival was offering. He believed that in such a situation the duopoly was hopelessly indeterminate: the two firms would stand opposite to each other indefinitely, each waiting for the other to move first (100).26 It was only in this situation that the two quantities were really one function of the other. Thus, the CV term was nothing but the mathematical expression of the fact that the two offers were mutually interdependent and that under uncertainty no firm knew what the rival was offering. In short, the CV term reflected the duopolist’s uncertainty over the rival’s behavior. This term, however, could not offer any way out from the standstill: to assign a specific value to a firm’s CV would imply, in fact, that the uncertainty had disappeared at least as far as the firm in question was concerned. Thus, Chamberlin endorsed the interpretation of Bowley’s approach that stressed its nature as a mathematical representation of the indeterminacy of the duopoly outcome, and not the possibility that it offered to solve the indeterminacy via the assignment of specific values to the CV terms. Chamberlin added that as soon as one of the firms broke the standstill and made its offer, both CV terms canceled because both offers became perfectly known (100).27 The Bowley model turned out again to be equivalent to the Cournot one, in that they both identified a determinate 26. A few years later a similar standstill will be attributed by Morgenstern not to the presence of uncertainty but to that of perfect foresight; see Morgenstern [1935] 1976, esp. 174, 181–82. 27. Note, casual as it may be, the lexical choice: the CV terms are said to “cancel,” instead of being “set equal to zero.” This might reflect the difference between knowing the rival’s action and formulating a conjecture about the rival’s passivity.

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solution by assuming that the duopolists ignored the indirect influence. Thus, he concluded that the real difference between Bowley and Cournot was quite small since they both neglected that the truly dividing line in oligopoly theory was whether the firms were assumed to take into account the total influence of their actions. To Chamberlin’s view, the only reasonable conjecture that a rational firm might hold and that might be represented by a CV term would be the one capturing such a total influence, thereby leading to the joint monopoly outcome. Beyond this case, the only role of the Bowley model was to represent the firms’ interdependence in a situation of uncertainty. Hence, the CVs did not constitute a general approach to the oligopoly problem, but just a partial way to depict it. 5. The Conjectural Approach Gains Ground In this section I overview two of the most significant contributions to the spread of conjectural variations, namely, those by A. C. Pigou and Ragnar Frisch. Such authoritative endorsements warranted the new approach a spell of dominance in the literature on oligopoly theory from the early 1930s. 5.1 Pigou’s New Attitude One of the most significant reactions to Chamberlin’s paper was Pigou’s decision to revise his presentation of duopoly theory in the fourth edition of Economics of Welfare. Chamberlin (1929, 80–82) had criticized the way Pigou had dealt with Edgeworth’s interval of oscillation in the 1924 edition of the book. Pigou seemed to concede the point, since he eliminated this analysis from the 1932 edition and replaced it with some generic remarks over the relation between the extent of the indeterminacy and the degree of market perfection or the number of firms. What mattered most, however, was Pigou’s choice to cut out also the reference to the fact that Edgeworth’s indeterminacy result was accepted by all mathematical economists, as well as the passage where he had argued that the firms’ conjectures “may be anything according to the mood of each etc.” (see above, section 2.1). In the revised version Pigou (1932, 236) claimed instead that “in more recent discussions there is apparent some measure of return towards Cournot,” that is, toward a determinate outcome. He exemplified the statement by showing how two special


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assumptions, namely, the quantity-passivity and the price-passivity rules, could lead to a determinate equilibrium. These were two of the cases examined in Chamberlin’s paper. It is remarkable, however, that Pigou preserved the chess analogy and expressed his skepticism as to the actual possibility of the duopolists’ formulating a correct conjecture over the rival’s behavior (236–37). Thus he maintained his general conclusion, albeit this time in a hesitating form, that the equilibrium of the duopoly model was basically indeterminate because attributing to the firms the knowledge of the objective elements of the market (the cost and demand functions) did not suffice to identify it (236–37). Pigou dealt one more time with the theme of duopoly in chapter 17 of his 1935 book, The Economics of Stationary States. He started with a simple two agents–two goods model of bilateral monopoly and noted that to determine the exchange rate we needed to know the agents’ beliefs about the effects upon the exchange rate of any change in one’s own offer. Following Chamberlin’s criterion of total influence, Pigou added that each belief depended in turn on the firm’s opinion as to how the rival would react. These beliefs were not given a priori among the conditions of the model, and the latter did not even suffice to determine them, “for how the other will, in fact, respond depends in turn on his opinion of what further response the first will make; and so on indefinitely” (92). In mathematical terms the problem was indeterminate, since we had two behavioral equations and four unknowns: the actual moves and the beliefs (92). A similar problem arose in the duopoly case. Again the key question was whether the conditions of the problem contained any further circumstance capable of enforcing upon each duopolist some definite view about the rival’s reaction (93). According to Pigou, two possibilities had been suggested in the literature: one was Cournot’s hypothesis, the other was the joint monopoly hypothesis. Both gave a determinate solution to the duopoly case (93–94). The influence of Chamberlin’s lesson was apparent once more. Yet Pigou argued that none of these possibilities captured real-world competition: the former was simply untenable if agents were rational, while the latter clashed with the observed short life of cartels and agreements (94). Thus a basic indeterminacy existed in the duopoly model, “due to the fact that our equations contain as elements beliefs on the part of some [firm] about the effect on the rate of exchange that will take place if [it] increases [its] output; while the conditions of the problem do not enable us to infer what these beliefs are”

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(96). Yet “the data may enable us to find limits within which the beliefs lie” (96). It was to accomplish the latter task that Bowley’s CVs entered into play. In the mathematical appendix to the book (appendix 5, 276–79), Pigou argued that the duopoly model was determinate for any pair of v12 and v21 , where these variables represented the firms’ conjectures about the rival’s reaction. He then showed the solution resulting from the attribution of some special values to the CV terms and determined the limiting values that the CVs might take. These were also the limiting beliefs that the firms might hold and the boundaries of the problem’s indeterminacy. Thus, contrary to Chamberlin’s view that Bowley’s approach could at best constitute a partial representation of duopolistic indeterminacy, Pigou endorsed the other end of the interpretative spectrum and gave a precise operational character to the approach: all we had to do was to assign an ultra-definite value to the pair of CVs and we would get an ultradeterminate solution for the duopoly problem. The real puzzle was given by the fact that real-world competition seemed not to warrant any of the ultra-definite values. The reason was of course the presence of an unavoidable element of uncertainty in the firms’ conjectures. This explains why, Bowley’s approach notwithstanding, Pigou did not renege his belief that the duopoly problem was basically indeterminate. Moreover, given the limited scope of the 1935 book—the properties of stationary states— no analysis of the process through which the firms revised their beliefs when these were disconfirmed by actual events could ever find a place in Pigou’s inquiry. 5.2 Frisch’s Generalization A major contribution to the development of Bowley’s CV approach came from Ragnar Frisch’s famous 1933 paper “Monopoly-Polypoly-The Concept of Force in the Economy” ([1933] 1951). The paper owes its prominent status in the history of twentieth-century economics to the introduction of the technique of the phase diagram for the analysis of dynamic phenomena. What matters for us, however, is that it contained a very general model of oligopolistic competition that analyzed in mechanical terms the interplay of the forces of a finite number of nonatomic agents ([1933] 1951, 24). One of the key elements of the model was represented by a generalization of Bowley’s CV terms.


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In Frisch’s terminology, each oligopolist had one or more parameters of action that he could set according to his own free will, although he had to take into account the actions of the other oligopolists, so that each parameter was subjected to both a direct and an indirect influence (28– 29). The model was absolutely general in that no assumption was made as to both the action parameters and the channels through which the indirect influence could manifest itself. Each oligopolist was taken to be aware that his profit depended not only on his own parameters but also on the other oligopolists’ ones. This implied that each oligopolist should form an opinion as to how a change in his own parameters was going to affect his rivals’ choices. Frisch recognized that the way oligopolists formed their opinions was absolutely crucial for the functioning of the model (30). He proposed three profiles of action that the oligopolists might follow. The first profile was called autonomous action: the agents behaved à la Cournot, that is, as if the change in their own parameters did not affect the others (30). The second profile was called continuous reaction: the agents behaved as if the possible change in the other oligopolists’ parameters were a continuous function of the change in their own ones (31). To tie down this kind of behavior, Frisch defined the notion of conjectural hk = (∂zk /∂zk )(zk /zh ), that is, the change elasticity of parameters:28 zij i j j i h in i’s parameter zi that oligopolist j believed to be caused by the change in his own parameter zjk . It is apparent that this elasticity was nothing but a generalization of Bowley’s CV term. Frisch underlined that these coefficients did not express what actually happened to i’s parameter but only what firm j expected to happen (31). Finally, the third profile of action was called superior adaptation: the agents were divided into two groups, the first featuring those who adhered to the first profile (the autonomous players), the second containing the agents who followed a new pattern, that is, who were assumed both to know that the oligopolists in the first group behaved autonomously and to be able to exploit such knowledge to maximize their profits. These agents were called conjectural players. Their superiority was showed by the fact that their own elasticity coefficients, as long as they concerned the behavior of the autonomous players, were not merely conjectural but real. The only conjectures formulated by the players in the second group

28. Once more I have made the original notation uniform with that of section 1.

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referred to the expected reactions of the other members of the group (31–32).29 Frisch then defined a parameter’s force of attraction as the intensity of the motive inducing each oligopolist to change the parameter itself (33). Two reasons might cause such a change: either events objectively known by the oligopolist, or conjectural modifications in the rivals’ parameters. By combining these two reasons the oligopolist could estimate the total effect on his profits of a small change in one of his parameters under the assumption that the rivals’ parameters changed according to the conjectural elasticities. The elasticity of i’s profits πi with respect to parameter zih gave the parameter’s force of attraction: ωih = (∂πi /∂zih )(zih /πi ). The rest of Frisch’s analysis was conducted only in terms of the forces of attraction, so that, although the ωih depended upon the conjectural coefficients, the conjectural character of the whole argument might escape the attention of those who limited themselves to studying the phase diagrams beginning on page 34 of the paper.30 Frisch’s paper gave strong formal support to the CV approach. His model was praised in John Hicks’s influential survey on monopoly theory as the best general presentation of Bowley’s approach (Hicks [1935] 1953, 375). This survey represented in a sense the zenith of the conjectural variation method, since Hicks explicitly acknowledged it as the approach to oligopoly theory “that appears to be gaining ground at present” (375). As I will briefly argue in the concluding section, while this statement gave a fairly accurate representation of the state of the art of oligopoly analysis at the time of Hicks’s survey, it did not take long for the use of conjectural terms in oligopoly models to be seriously questioned.31 6. Concluding Remarks Bowley’s conjectural variations seemed to provide the Holy Grail of oligopoly theory, namely, the unitary approach to the topic that had been 29. Note that although Frisch’s paper predates Stackelberg’s 1934 book, the autonomous players clearly behave like Stackelberg followers, while the conjectural players behave like Stackelberg leaders. On Frisch’s oligopoly model, see also Dimand and Dimand 1996, 30–31. 30. This, however, was not the case of A. Smithies and L. J. Savage who, inspired by Frisch’s paper, presented in 1940 a sequential duopoly model in which each firm formed an initial conjecture on the rival’s output choice and then revised it in the following stages. 31. Note that Hicks himself was skeptical about Bowley’s approach; see Hicks [1935] 1953, 381.


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romanced by more than one generation of economists. Unfortunately, the CVs do not constitute a satisfactory method of tackling the oligopolists’ behavior. The reason is again the confusion they induce between statics and dynamics. Each CV term, in fact, indicates that one of the firms expects the other to react in some specific way to its own choice. But how can the rival react if the game is played just once and simultaneously? Turning again to Friedman’s distinction outlined in section 1, either we have an explicitly sequential or repeated game or we have to admit that the only reasonable CV is the Cournot one: the rival is expected not to react simply because the game ends with the simultaneous moves. Moreover, the CVs are quasi-dynamic concepts whose meaning and use in a dynamic setup together with static equilibrium notions like the reaction functions are highly questionable. The theoretical drawbacks of the CV approach were soon reckoned also by the interwar economists. In the mid-1930s, that is, exactly when the approach could legitimately be mentioned by Hicks as the orthodox way to tackle the analysis of oligopolistic competition, a number of questions were raised against it. To start with, where did Bowley’s CV terms come from? How did a firm form them? Moreover, if a firm was capable of performing a mental experiment in order to formulate a definite expectation as to the rival’s choice, why should it not be assumed that it was able also to anticipate the ultimate consequences of the experiment and thus to realize that the most profitable action for both duopolists was to produce the joint monopoly output? And if this was indeed the case, was it so reasonable for a firm to trust the reasoning ability of the rival and conclude that he too would reach such a conclusion? Finally, how could a quasi-dynamic concept like a CV term be reconciled with a static equilibrium notion like a reaction function? The history of how these questions emerged in the literature and what kinds of answers were given to them goes beyond the limited scope of this essay, especially because it would require an analysis of the development of Stackelberg’s variant of the standard duopoly model.32 A few words can be added instead linking the objections against the CV approach to the more general debate going on in the interwar period concerning the role of mental variables in economic theory. 32. To mention just a few references, see Nichol 1934a, 1934b, 1935; Morgenstern [1935] 1976; Kaldor 1936; Kahn 1937; and Stigler 1940. Note in particular that the last question in the text (“Finally, how could . . . ”) was first raised in Kahn’s paper. The ultimate critique against Bowley’s approach came after World War II in chapter 2 of Fellner 1949.

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That link is provided by a common necessity of any economic model featuring a role for mental variables, namely, that of tying down the agents’ conjectures and beliefs in terms of an equilibrium notion. This necessity brought to light the two different meanings of the term equilibrium. In any stable dynamic system, in fact, the term can be viewed “as a state of no motion, and as an attractor of arbitrary motions of the underlying dynamic process” (Weintraub 1991, 18). In the first sense, the equilibrium is characterized by the satisfaction of a set of static conditions: there is no mechanism through which the equilibrium is established, and the idea is that of every individual’s actions and plans being in harmony with everyone else’s (Weintraub 1991, 102)—what can be called the consistency view of equilibrium. In the second sense, instead, the equilibrium is closely associated with the mechanical idea of reaching a balance of forces, that is, with the playing out of some kind of economic behavior, like the dampening of the oscillations of a pendulum. This requires the existence of an equilibration process by virtue of which the equilibrium is actually reached or asymptotically tended to (Weintraub 1991, 102)—thereby justifying the name process view attached to this second meaning of equilibrium. The so-called years of “high theory” contributed in two fundamental directions. First, thanks in particular to the works of Frisch and, later, of Paul Samuelson, the tools of dynamic systems theory were used to make a precise distinction between the two meanings. Second, thanks to Friedrich Hayek, Erik Lindahl, and Hicks, the consistency view of equilibrium replaced the traditional process view as the dominant interpretation of equilibrium. As a consequence, the equilibrium state was separated from the adjustment process and the latter was reinterpreted in terms of the agents’ expectations and plans. The characterization of equilibrium as a situation of mutual consistency of plans permitted the association of equilibrium itself with an instant of time, thereby making it possible to describe the dynamic evolution of the system in terms of a sequence of equilibrium positions—the so-called temporary equilibrium method. Moreover, the explicit role assigned to the agents’ mental variables allowed the definition (not the analysis) of disequilibrium in terms of the disappointment of plans, thereby linking the absence of equilibrium to the behavioral or forecasting mistakes made by the agents. However, the consistency view demanded a price be paid, namely, the abandonment of the analysis of the equilibration process, that is, of the process through which the system achieved the equilibrium. A


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serious theoretical stalemate emerged, since in order to obtain a proper definition of disequilibrium as a situation of mutually inconsistent plans the interwar economists had to give up the possibility of investigating the disequilibrium itself in terms of a process of revision of the agents’ expectations and plans. This deadlock—which clearly transpired from the writings of authors such as Hayek, Lindahl, Oskar Morgenstern, and Hicks—was felt also by the oligopoly theorists. Bowley’s CV approach had formally brought the mental variables into duopoly analysis. It did not take long for the consistency condition to be envisaged and called forth as the necessary condition to tie down each firm’s conjectures about its rival’s reaction.33 Yet, the condition sufficed only to define the duopoly equilibrium, not to explain how the firms actually reached it. No assumption about the duopolists’ forecasting ability— short of the limiting case of perfect foresight—could properly justify the achievement of equilibrium as a pair of consistent output choices. These difficulties, which reproduced in the field of oligopoly analysis the same puzzles that were blocking the efforts of both the general equilibrium and the business cycle theorists, led to questions like the ones listed at the beginning of this section. What matters most, they created the proper conditions for the post–World War II prevalence of an empirical, almost atheoretical, approach to imperfect competition—the wellknown structure-conduct-performance approach. References Amoroso, L. 1921. Lezioni di economia matematica. Bologna: Zanichelli. Bertrand, J. [1883] 1992. Review of Théorie mathematique de la richesse sociale, by Léon Walras, and Recherches sur les principes mathématiques de la théorie des richesses, by Augustin Cournot. In Magnan de Bornier 1992, 646–53. Bowley, A. L. 1924. The Mathematical Groundwork of Economics. Oxford: Oxford University Press. Casarosa, C. 1984. Il prezzo minimo del modello di Edgeworth. Economia politica 1.1:63–78. Chamberlin, E. H. 1929. Duopoly: Value Where Sellers Are Few. Quarterly Journal of Economics 44:63–100.

33. First formulated by Roy Harrod (1934) and then fully analyzed by Wassily Leontief (1936), the consistency condition required that in order for the duopoly equilibrium to be determinate, each firm’s conjectured reaction had to be equal to its actual reaction. For a discussion of this condition, see the references in the previous footnote.

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