Paper:

# Mixed Oligopoly: Analysis of Consistent Equilibria

## Vyacheslav V. Kalashnikov^{*1,*2,*3}, Vladimir A. Bulavsky^{*2},

Nataliya I. Kalashnykova^{*3,*4}, Junzo Watada^{*5},

and Diego de Jesús Hernández-Rodríguez^{*4}

^{*1}Tecnológico de Monterrey (ITESM), Campus Monterrey, 64849, Mexico

^{*2}Central Economics & Mathematics Institute (CEMI), Russian Academy of Sciences (RAS), Moscow 117418, Russia

^{*3}Sumy State University, Sumy 40007, Ukraine

^{*4}Department of Physics & Maths (FCFM), Universidad Autónoma de Nuevo León (UANL), San Nicolás de los Garza, 66450, Mexico

^{*5}Graduate School of Information, Production and Systems, Waseda University, 2-7 Hibikino, Wakamatsuku, Kitakyushu, Fukuoka 808-0135, Japan

In this paper, a model of mixed oligopoly with conjectured variations equilibrium (CVE) is examined, in which one of the agents maximizes a convex combination of its net profit with the domestic social surplus. The agents’ conjectures concern the price variations, which depend on the variations in their production outputs. Using the established existence and uniqueness results for the CVE (the *exterior equilibrium*) for any fixed set of feasible conjectures, the notion of the interior equilibrium is introduced by developing a conjecture consistency criterion. Then, the existence theoremfor the interior equilibrium (defined as a CVE state with *consistent conjectures*) is proven. When the convex combination coefficient tends to 1 (thus transforming the model into the mixed oligopoly in its extreme form), two trends are apparent. First, for private companies, the equilibrium with consistent conjectures becomes more proficient than the Cournot-Nash equilibrium. Second, there exists a (unique) value of the convex combination coefficient such that the private agent’s aggregate profit is the same in both the above-mentioned equilibria, which makes subsidies to producers or consumers unnecessary.

Nataliya I. Kalashnykova, Junzo Watada, and

and Diego de Jesús Hernández-Rodríguez, “Mixed Oligopoly: Analysis of Consistent Equilibria,”

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.18, No.6, pp. 971-984, 2014.

- [1] R.C. Cornes and M. Sepahvand, “Cournot vs Stackelberg equilibria with a public enterprise and international competition,” Discussion Paper No.03/12, University of Nottingham, School of Economics, United Kingdom, 2003.
- [2] C. Fershtman, “The interdependence between ownership status and market structure: The case of privatization,” Economica, Vol.57, pp. 319-328, 1990.
- [3] T.Matsumura, “Stackelberg mixed duopoly with a foreign competitor,” Bulletin of Economics Research, Vol.55, pp. 275-287, 2003.
- [4] N. Matsushima and T. Matsumura, “Mixed oligopoly and spatial agglomeration,” Canadian J. of Economics, Vol.36, pp. 62-87, 2003.
- [5] T. Matsumura and O. Kanda, “Mixed oligopoly at free entry markets,” J. of Economics, Vol.84, pp. 27-48, 2005.
- [6] N. J. Ireland and P. J. Law, “The Economics of Labour-Managed Enterprises,” Croom Helm, London, 1982.
- [7] J. P. Bonin and L. Putterman, “Economics of Cooperation and the Labor-Managed Economy,” Harwood Academic Publisher, Chur, Switzerland, 1987.
- [8] F.H. Stephan (Ed.), “The Performance of Labour-Managed Firms,” Macmillan Press, London, 1982.
- [9] L. Putterman, “Labour-managed firms,” In S. N. Durlauf and L. E. Blume (Ed.), The New Palgrave Dictionary of Economics, Vol.4, pp. 791-795, Palgrave Macmillan, Basingstoke, Hampshire, 2008.
- [10] B. Saha and R. Sensarma, “State ownership, credit risk and bank competition: A mixed oligopoly approach,” Working Paper, University of Hertfordshire Business School, Hatfield, England, 2009.
- [11] A. Mumcu, S. Oğur, and Ü. Zenginobuz, “Competition between regulated and non-regulated generators on electric power networks,”

online at http://mpra.ub.uni-muenchen.de/376/ MPRA Paper No.376, posted November 7, 2007/00:59. - [12] A. L. Bowley, “TheMathematical Groundwork of Economics,” Oxford University Press, Oxford, 1924.
- [13] R. Frisch, “Monopoly, polypoly: The concept of force in the economy,” Int. Economics Papers, Vol.1, pp. 23-36, 1951. (“Monopole, polypole – La notion de force en économie,” Nationaløkonomisk Tidsskrift, Vol.71, pp. 241-259, 1933.)
- [14] J. Laitner, ““Rational” duopoly equilibria,” Quarterly J. of Economics, Vol.95, pp. 641-662, 1980.
- [15] C. Figuières, A. Jean-Marie, N. Quérou, and M. Tidball, “Theory of Conjectural Variations,” World Scientific, Singapore, Taibei, 2004.
- [16] N. Giocoli, “The escape from conjectural variations: The consistency condition in duopoly theory from Bowley to Fellner,” Cambridge J. of Economics, Vol.29, pp. 601-618, Oxford University Press, 2005.
- [17] T. Lindh, “The inconsistency of consistent conjectures, Coming back to Cournot,” J. of Economic Behavior and Organization, Vol.18:c, pp. 69-90, 1992.
- [18] V. V. Kalashnikov, V. A. Bulavsky, N. I. Kalashnykova, and F. J. Castillo, “Consistent conjectures in mixed oligopoly,” European J. of Operational Research, Vol.210, pp. 729-735, 2011.
- [19] V. A. Bulavsky, “Structure of demand and equilibrium in a model of oligopoly,” Economics and Mathematical Methods (Ekonomika i Matematicheskie Metody), Vol.33, pp. 112-134, Central Economics and Mathematics Institute, Moscow, 1997 (in Russian).
- [20] N. I. Kalashnykova, V. A. Bulavsky, V. V. Kalashnikov and F. J. Castillo-Pérez, “Consistent conjectural variations equilibrium in a mixed duopoly,” J. of Advanced Computational Intelligence and Intelligent Informatics, Vol.15, pp. 425-432, 2011.
- [21] V. V. Kalashnikov, N. I. Kalashnykova, and J. F. Camacho, “Partially mixed duopoly and oligopoly: Consistent conjectural variations equilibrium (CCVE), Part 1,” Juan Carlos Leyva López et al. (Eds.), Studies on Knowledge Discovery, Knowledge Management and Decision Making, Fourth Int. Workshop Proc. EUREKA2013, Mazatlán, November 4-8, 2013, Atlantis Press, Amsterdam-Pars-Beijing, pp. 198-206, 2013.
- [22] V. V. Kalashnikov, N. I. Kalashnykova, and J. F. Camacho, “Partially mixed duopoly and oligopoly: Consistent conjectural variations equilibrium (CCVE). Part 2,” Juan Carlos Leyva López et al. (Eds.), Studies on Knowledge Discovery, Knowledge Management and Decision Making, Fourth Int. Workshop Proc. EUREKA2013, Mazatlán, November 4-8, 2013, Atlantis Press, Amsterdam-Paris-Beijing, pp. 207-217, 2013.
- [23] Y. F. Liu, Y. X. Ni, F. F. Wu, and B. Cai, “Existence and uniqueness of consistent conjectural variation equilibrium in electricity markets,” Int. J. of Electrical Power and Energy Systems, Vol.29, pp. 455-461, 2007.
- [24] E. J. Dockner, “A dynamic theory of conjectural variations,” J. of Industrial Economics, Vol.40, pp. 377-395, 1992.
- [25] V. A. Bulavsky and V. V. Kalashnikov, “One-parametric method to study equilibrium,” Economics and Mathematical Methods (Ekonomika i Matematicheskie Metody), Vol.30, pp. 129-138, Central Economics and Mathematics Institute, Moscow, 1994 (in Russian).
- [26] V. A. Bulavsky and V. V. Kalashnikov, “Equilibrium in generalized Cournot and Stackelberg models,” Economics and Mathematical Methods (Ekonomika i Matematicheskie Metody), Vol.31, pp. 164-176, Central Economics and Mathematics Institute, Moscow, 1995 (in Russian).