Cournot I: Constant Returns

  • Tönu Puu


Cournot duopoly has now been an inspiration for mathematical modelling in economics for almost two Centuries. As the principles are very simple to grasp one needs not have a lifetime of indoctrination with economic theory to be convinced. Once nonlinear science took the stage it was mathematicians such as Rand, and Poston and Stewart, who started using applications to duopoly. Not altogether surprising, after all Cournot was a mathematician. The aim then was to create iterative processes in the style of the logistic map. Cases where both reaction functions were in this style were suggested, which, of course, could produce up to four different fixed points. Unfortunately, these were not based on economic principles, i.e., were not derived from demand functions that emerged from utility maximization. In 1991 the present author proposed the case of Cobb-Douglas utility where the resulting demand functions are always reciprocal to commodity price. Budget shares for the consumers, and hence also aggregate revenues for the producers then became constant. Dana and Montrucchio had proposed a similar case as an example 5 years before, though without fully appreciating the potential of the idea. This “isoelastic” demand function easily lends itself to explorations of dynamics. Yet, it also has its snags, above all the constancy of revenues, which leads to the absurd situation that a single supplier can reduce output to zero and sell it at an infinite price, which does not affect revenues at all. On the other hand costs can be reduced to zero by cancelling production, so the best choice is to produce nothing. This, of course, is absurd and makes the model unsuitable to deal with monopoly or collusion. The problem persists in duopoly, despite the fact that the intersection of the reaction curves in the origin is totally unstable; the curves at this intersection even have infinite slopes. Nevertheless we need some mechanism that pushes the system away from the origin if it lands there. This may sound simple, but it is not! Any such repulsion mechanism tends to take over the show, and blur the essential dynamics we want to analyze. As the reaction curves intersect the axes whereas outputs cannot be negative, it is inevitable that an occasional dropping out from production makes the system land in the origin. Once one firm chooses zero output, the other will do it as well. We need something to prevent the system from ever visiting the origin, which will be done through assuming adaptation. This means that the firms never move to the calculated best reply, just part of the way, which may be 99.99%. Given this, the assumption seems to be quite innocuous.


  1. Abraham RH, Gardini L, Mira C (1997) Chaos in discrete dynamical systems. Springer, BerlinCrossRefGoogle Scholar
  2. Agiza NH (1998) Explicit stability zones for Cournot game with 3 and 4 competitors. Chaos, Solitons Fractals 9:1955–1966CrossRefGoogle Scholar
  3. Agliari A, Gardini L, Puu, T (2000) The dynamics of a triopoly Cournot game. Chaos, Solitons Fractals 11:2531–2560CrossRefGoogle Scholar
  4. Agliari A, Gardini L, Puu T (2005a) Global bifurcations in duopoly when the Cournot point is destabilized through a Subcritical Neimark bifurcation. Int Game Theory Rev 8:1–20CrossRefGoogle Scholar
  5. Agliari A, Gardini L, Puu T (2005b) Some global bifurcations related to the appearance of closed invariant curves. Math Comput Simul 68:201–219CrossRefGoogle Scholar
  6. Ahmed E, Agiza NH (1998) Dynamics of a Cournot game with n competitors. Chaos, Solitons Fractals 10:1179–1184CrossRefGoogle Scholar
  7. Bertrand J (1883) Théorie mathématique de la richesse sociale. J des Savants 48:499–508Google Scholar
  8. Bischi GI, Mammana C, Gardini L (2000) Multistability and cyclic attractors in duopoly games. Chaos, Solitons Fractals 11:543–564CrossRefGoogle Scholar
  9. Cournot A (1838) Récherces sur les principes mathématiques de la th éorie des richesses (Paris)Google Scholar
  10. Dana RA, Montrucchio L (1986) Dynamic complexity in duopoly games. J Econ Theory 40:40–56CrossRefGoogle Scholar
  11. Edgeworth FY (1897) La teoria pura del monopolio. Giornale degli Economisti 15:13–31Google Scholar
  12. Guckenheimer J, Holmes P (1986) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer, BerlinGoogle Scholar
  13. Hotelling H (1929) Stability in competition. Econ J 39:41–57CrossRefGoogle Scholar
  14. Kuznetsov YA (1995) Elements of applied bifurcation theory. Springer, BerlinCrossRefGoogle Scholar
  15. Mira C, Gardini L, Barugola A, Cathala JC (1996) Chaotic dynamics in two-dimensional noninvertible maps. World Scientific, SingaporeCrossRefGoogle Scholar
  16. Palander TF (1939) Konkurrens och marknadsjämvikt vid duopol och oligopol. Ekonomisk Tidskrift 41:124–145, 222–250CrossRefGoogle Scholar
  17. Puu T (1991) Chaos in duopoly pricing. Chaos, Solitons Fractals 1:573–581. Republished in Rosser JB (ed) Complexity in economics (Edward Elgar Publishing Inc., 2004)Google Scholar
  18. Puu T (1996) Complex dynamics with three oligopolists. Chaos, Solitons Fractals 7:2075–2081CrossRefGoogle Scholar
  19. Puu T (1998) The chaotic duopolists revisited. J Econ Behav Organ 33:385–394CrossRefGoogle Scholar
  20. Puu T (2000, 2003) Attractors, bifurcations, & chaos - nonlinear phenomena in economics. Springer, Berlin. ISBN 3-540-66862-4. Second revised and enlarged edition (Springer-Verlag, ISBN 3-540-40226-8)Google Scholar
  21. Puu T (2005) Complex oligopoly dynamics. In: Lines M (ed) Nonlinear dynamical systems in economics. CISM-Springer lecture notes in economics and mathematical systems, vol. 796. Springer, Berlin, pp 165–186CrossRefGoogle Scholar
  22. Rand D (1978) Exotic phenomena in games and duopoly models. J Math Econ 5:173–184CrossRefGoogle Scholar
  23. Robinson JV (1933) The economics of imperfect competition. Cambridge University Press, CambridgeGoogle Scholar
  24. Tramontana F, Gardini L, Puu T (2010) Global bifurcations in a piecewise smooth Cournot duopoly. Chaos, Solitons Fractals 43:15–23CrossRefGoogle Scholar

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Authors and Affiliations

  • Tönu Puu
    • 1
  1. 1.Centre for Regional Science (CERUM)Umeå UniversityUmeåSweden

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