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Cournot II: Returns to Scale and Stability

  • Tönu Puu
Chapter

Abstract

Cournot did not just invent quantity competition duopoly; he considered the entire scale of steps from monopoly to perfect competition and described the proper rules for monopoly pricing and perfect competition pricing, assuming oligopoly price to be between these extremes. He considered the route in terms of adding new competitors to an already existent market, which, by the way, explained why he focused on quantity competition. Given the simple setups for Cournot duopoly—linear demand and constant marginal cost—it came as a shock when it was discovered that the models became unstable if the competitors exceeded some rather small number. What use is it to know that an increasing number of competitors has the perfect competitive state as target if both the asymptotic state and the route are destabilized? It occurred to the present author that the problem is on the side of production. Constant marginal costs, or constant returns to scale, means that we deal with firms of potentially infinite size. Any small profit margin between price and cost could be multiplied up any number of times through expanding production. Once this is realized, destabilization is no longer surprising. Further, it was probably not such a comparison Cournot was after; he might have wanted to compare a small number of big firms to a large number of small firms. But small and big cannot be defined without decreasing returns and capacity limits. Such can be introduced through the use of a nonstandard variant of the Constant Elasticity of Substitution production function. One just has to fix capital inputs through an act of investment, and production with the remaining variable inputs is automatically provided with a capacity limit for the lifetime of invested capital. This actually removes the destabilization problem. Once one is that far, it is natural to assume that capital wears out, and one may try to formulate an endogenous process for capital wear and regeneration, which is, in fact, done. However, many alternatives to this part of modelling are conceivable. The stability issue is bound to become complicated with capital renewal included, and there are interesting scenarios with, for instance, never ending oscillations and spontaneous formation of synchronized competitor groups.

References

  1. Ahmed E, Agiza NH (1998) Dynamics of a Cournot game with n competitors. Chaos, Solitons Fractals 10:1179–1184CrossRefGoogle Scholar
  2. Cánovas J, Puu T, Ruiz M (2008) The Cournot–Theocharis problem revisited. Chaos, Solitons Fractals 37:1025–1039CrossRefGoogle Scholar
  3. Cánovas J, Panchuk A, Puu T (2015) Asymptotic dynamics of a piecewise smooth map modelling a competitive market. Math Comput Simul 117:20–38CrossRefGoogle Scholar
  4. Edgeworth FY (1897) La teoria pura del monopolio. Giornale degli Economisti 15:13–31Google Scholar
  5. Frisch R (1965) Theory of production. D. Reidel Publishing Company, DordrechtCrossRefGoogle Scholar
  6. Heathfield FH, Wibe S (1987) An introduction to cost and production functions. MacMillan, LondonCrossRefGoogle Scholar
  7. Johansen L (1972) Production functions. North-Holland, AmsterdamGoogle Scholar
  8. Palander TF (1936) Instability in competition between two sellers. In: Abstracts of papers presented at the research conference on economics and statistics held by the Cowles Commission at Colorado College, Colorado College Publications, General Series No. 208, Studies Series No. 21Google Scholar
  9. Palander TF (1939) Konkurrens och marknadsjämvikt vid duopol och oligopol. Ekonomisk Tidskrift 41:124–145, 222–250CrossRefGoogle Scholar
  10. Puu T (2007) Layout of a new industry: from oligopoly to competition. Pure Math Appl 16:475–492Google Scholar
  11. Puu T (2008) On the stability of Cournot equilibrium when the number of competitors increases. J Econ Behav Organ 66:445–456CrossRefGoogle Scholar
  12. Puu T, Norin A (2003) Cournot duopoly when the competitors operate under capacity constraints. Chaos, Solitons Fractals 18:577–592CrossRefGoogle Scholar
  13. Puu T, Panchuk A (2009) Oligopoly and stability. Chaos, Solitons Fractals 41:2505–2516CrossRefGoogle Scholar
  14. Puu T, Rúiz Marin M (2006) The dynamics of a triopoly Cournot game when the competitors operate under capacity constraints. Chaos, Solitons Fractals 28:403–413CrossRefGoogle Scholar
  15. Theocharis RD (1959) On the stability of the Cournot solution on the oligopoly. Rev Econ Stud 27:133–134CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

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

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