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Doubling Back on Double Marginalization

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Abstract

“Double marginalization” and “Elimination of Double marginalization” are catch-phrases commonly used in the IO literature. In this article, I trace back the origin of the idea to Cournot (1838, Ch. IX) on complementary goods monopolies. Through the years Cournot’s contribution remained a reference but ended being viewed as a special case of the bilateral monopoly model. Yet, it is worth wondering why the most cited paper on this issue is nowadays (Spengler in J Polit Econ 58(4):347–352, 1950) which contains only an informal treatment of the question. In addition to retracing the origin of the idea, I emphasize the elegant proof of Cournot for the simultaneous game and extend it to the sequential game. I also show that prices are usually higher in the sequential game but that they could be lower if demand is very convex.

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Notes

  1. The document is available online: 2020 U.S. Vertical Merger Guidelines. These Guidelines were withdrawn by the U.S. Federal Trade Commission in September 2021.

  2. Among others, two recent contributions on EDM are Slade and Kwoka (2020) and Slade (2021).

  3. Searching for ‘double mark-up’ in Google Scholar gives 1,520 results but none before their article!

  4. For which a 1982 working paper exists: #82ll, Institute for Policy Analysis, University of Toronto. It is also revealing that in the same issue of the Journal of Business, Bittlingmayer (1983) is a related article (vertical restrictions with stores along a Hotelling line) but “double”, “mark-up”, and “marginalization” are absent.

  5. Unfortunately, one can easily lose track of oral traditions. I have unsuccessfully searched in Bork (1954, 1978) and Posner (1976). McGee and Bassett (1976, fn. 7), corroborates somewhat Mathewson and Winter’s memories: “Bork was not the only one to argue that some or all received arguments against vertical integration were wrong: M. A. Adelman, Aaron Director, J. J. Spengler, and others had each analyzed at least part of the problem. For example, see Adelman (1949a), Spengler (1950). Vertical integration was one subject of Aaron Director’s socratic analysis, which contributed much to an oral tradition in and around the University of Chicago. Some evidences of this tradition with respect to integration are seen, for example, in a student note: Editors (1952). For a review of still older analyses, see Machlup and Taber (1960).”

  6. By contrast, in Perry (1989) the expressions DM and EDM are absent. Yet, in the same Handbook, Katz (1989) includes the association of DM/EDM with Spengler.

  7. This is illustrated by Fig. 2 (see Sect. 4.3) which shows the Google Scholar citations received by Spengler (1950) for every five-year periods since its publication. Most of them start after 1985.

  8. For example, microprocessors and hard drives are complementary items. The same holds for avionics and jet engines. For many goods, consumer service is a key complement. And, indeed, a distribution service (whether Amazon or Wallmart) is a required complement for many consumer goods.

  9. I am grateful to L. White for this insight.

  10. Admittedly, the sign of the derivative could change with the level of prices.

  11. In the case of substitute goods, the pecuniary externality is positive; independent sellers choose lower prices.

  12. The 1925 version appeared in a collection of Edgeworth’s articles, it is a translation of Edgeworth (1897) published in Italian.

  13. There is a nice analogy with quantity competition between producers of a homogenous good. In the absence of cost, Sonnenschein (1968) shows that both models are formally equivalent. Yet, with cost functions, and in particular with convex costs, the analogy is less formal. For duality results under constant return to scale, see Bergstrom (1978).

  14. Chapter IX is organized as follows. First, §55–58, the case of two complementary inputs (the case of n inputs is briefly treated) is considered in the absence of production cost. Asymmetric cost functions (still for two inputs) are introduced in §59. In §60 the cost of assembling the two inputs is introduced. In §61, Cournot takes into account that each input seller has two demands: one for the composite good, which depends on the prices of both complementary goods, and another one which depends only on its own price. But the analysis is cut short, as Cournot finds the f.o.c. too complicated to interpret. From §62 to the end, Cournot turns to perfect competition (which is the topic of his chapter VIII) for the production of both copper and zinc.

  15. This is another addition to the long list of multiple independent discoveries.

  16. Ellet was an engineer who built bridges. See https://en.wikipedia.org/wiki/Charles_Ellet_Jr.

  17. Cournot did not use the Lerner (1934) index, and stopped at \(D(p) + \frac{1}{2}D'(p)\left[ p - C'\left( D(p)\right) \right] =0\).

  18. An English translation was provided in 1897 by Nathaniel T. Bacon: “An association of monopolists, working for their own interest, in this instance will work for the interest of consumers, which is exactly the opposite of what happens with competing producers.” I am grateful to a referee for this reference.

  19. Slightly before, Ellet (§ 59, p. 77) also wrote: “This fact on a little reflection will convince us of the importance to the community as well as to the stockholders of having the great lines of improvement in the country put under the control of the same interest. That is to say it would result unfavorably to the public as well as generally to one or other of the works to have the trade carried a portion of the distance by one line and then taken up and transported the balance of the way by another.”

  20. This review, which was probably not widely read, is reprinted in Dimand (1995). I am grateful to a referee for this reference.

  21. Indeed, one of Edgeworth’s criticisms of Cournot was about the timing and he made this remark to both the model of Chapter VII and that of Chapter IX.

  22. Either there would be eternal barter between the two monopolists or an equilibrium is reached but it is not possible to anticipate which one. The indeterminacy result of bargaining already appears in Edgeworth (1881) where Robinson Crusoe and Friday bargain over a wage.

  23. Indeed, there is a caveat: Cournot equilibrium does not exist when \(1+1/n\le \theta <2\) because in that case the slopes of the reaction functions are larger than 1 and they do not intersect.

  24. See Ekelund and Hébert (2002) for a broader view on the origins of Microeconomics.

  25. It was published after the first English translation in 1897 of Cournot (1838) by Nathaniel T. Bacon.

  26. Bertrand and Edgeworth proposed different assumptions (possibly implicitly in the case of Bertrand) that lead to different games, which extended our comprehension of imperfect competition but which certainly did not refute Cournot’s concept.

  27. de Bornier (1992) explains why Bertrand’s review of Cournot was superficial. He also underlines Edgeworth’s acrimony towards Cournot. It is clear that Game Theory has been instrumental in structuring the economic debate.

  28. In his review of Cournot, Fisher (1898, p. 128) writes: “But, although Cournot’s conclusions are in the main consonant with facts, his analysis of motives in the minds of the two monopolists is subject to much the same objection as above expressed in the case of two competitors.” He then refers to Edgeworth. I am grateful to a referee for pointing out this passage.

  29. Moore (1905) had published a biographical account of Cournot’s life.

  30. Around the centenary of Cournot’s book, at least four articles—Roy (1933), Nichol (1938), Fisher (1938), and Roy (1939)—were published in Econometrica.

  31. Nichol was at Duke University as was Spengler. In the collection of articles on the History of Economic Thought, Spengler and Allen (1960), the only article on Cournot is Nichol (1934), which does not mention Cournot’s chapter IX.

  32. See Machlup and Taber (1960) for a more complete list of references and how they are related. See also Harsanyi (1956) for a critical discussion of Zeuthen’s, Hicks’s, and Nash’s theories of bargaining.

  33. Ellet (1839) was hardly cited. Calsoyas (1950)—which is an article on Ellet’s achievements—starts with “The cult of Cournot and the lack of extensive research in the early mathematical writings on economic topics have kept important writers in virtual oblivion.”

  34. See his Chapter IX. Burns does not cite Cournot (nor Edgeworth) in this book but he does cite Chamberlin (1933) (in which Chamberlin 1929 is reproduced) and Hotelling (1929), so he was certainly aware of Cournot –although nothing indicates he knew in detail Cournot’s chapter IX. Burns (1936) was an important book at the time. It had many reviews, notably in the Economic Journal, the Journal of Political Economy, the Quarterly Journal of Economics, and the American Economic Review.

  35. I am grateful to the Editor for this reference. This book quickly became a reference and it was reviewed in many journals, including the Economic Journal, the Journal of Political Economy, the Annals of the American Academy of Political and Social Science, and the Journal of the American Statistical Association.

  36. Demand is linear and the goods are not necessarily perfect complements. Linear demands for a differentiated goods set-up were already introduced by Edgeworth (1897, 1925) who solved the model only for perfect complements. In von Stackelberg (1934) a more extensive solution is given (for both price or quantity as strategic variables and for both simultaneous and sequential choices), see the Mathematical Appendix. von Stackelberg (1934) and Allen (1938) (who uses a more general form) are therefore precursors of the work of Shubik and Levitan (1980). See Choné and Linnemer (2020). As neither Stackelberg nor Allen derive the demands from a quadratic utility, however, a crucial relationship between the coefficients is missed.

  37. Allen cites Roos (1934) as his main source for this section. In Roos’s book, Cournot is cited nine times but not in Roos’s Chapter VIII which Allen refers to.

  38. For example, in Adams (1953) it is Hoffman who is cited, not Spengler. Bork (1954) does not cite Spengler but only Adelman (1949a).

  39. Under some restrictive assumptions they also showed that the upstream price is the same whether there is perfect competition downstream or a monopoly.

  40. A twist is that Bronfenbrenner was a colleague of Spengler at Duke.

  41. Frederic (1970), indeed, gives a formal proof (for linear demand and constant marginal cost) but unfortunately in the “Bilateral monopoly” section. He refers to this proof when discussing Spengler’s paper in the section “Are the Benefits of Countervailing Power Passed On?”.

  42. Greenhut and Ohta (1978) reply concedes (almost) nothing! The most novel part in Greenhut and Ohta (1976) is the section where the downstream level is assumed to be a Cournot oligopoly. But Perry (1978) also finds a mistake there.

  43. Salinger’s result has been successfully tested empirically by Luco and Marshall (2020).

  44. They have a very instructive introduction and they cite an empirical study on the Silk Road between China and Italy, Karni and Chakrabarti (1997).

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Acknowledgements

I am grateful to Philippe Choné, Larry White, and Ralph Winter for constructive comments. I was able to improve the quality of the paper thanks to the valuable input of the Editor and two referees. A special thanks goes to Magali Noël-Linnemer. This research is supported by a grant of the French National Research Agency (ANR), “Investissements d’Avenir” (LabEx Ecodec/ANR-11-LABX-0047).

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Linnemer, L. Doubling Back on Double Marginalization. Rev Ind Organ 61, 1–19 (2022). https://doi.org/10.1007/s11151-022-09872-z

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