Paper:

# Analysis of Consistent Equilibria in a Mixed Duopoly

## 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

This paper examines a model of a mixed duopoly with conjectural variations equilibrium (CVE), in which one of the agents maximizes a convex combination of his/her net profit and domestic social surplus. The agents’ conjectures concern the price variations, which depend on their production output variations. Based on the already established existence and uniqueness results for the CVE (called the *exterior equilibrium*) for any set of feasible conjectures, the notion of interior equilibrium is introduced by developing a consistency criterion for the conjectures (referred to as influence coefficients), and the existence theorem for the interior equilibrium (understood as a CVE state with *consistent conjectures*) is proven. When the convex combination coefficient tends to 1, thus transforming the model into the mixed duopoly in its extreme form, two trends are apparent. First, for the private company, the equilibrium with consistent conjectures becomes more proficient than the Cournot-Nash equilibrium. Second, there exists a (unique) value of the combination coefficient such that the private agent’s profit is the same in both of the above-mentioned equilibria, which makes subsidies to the producer or to consumers unnecessary.

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.18, No.6, pp. 962-970, 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, Vol36, 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,” S. N. Durlauf and L. E. Blume (Eds.), 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 on November 7, 2007 [Accessed February 14, 2014] - [12] A. L. Bowley, The Mathematical 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, No.1, 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 (JACIII), Vol.15, No.4, 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, 4th Int. Workshop Proceedings EUREKA2013, Mazatlán, November 4-8, 2013, Atlantis Press, Amsterdam-Paris-Beijing, pp. 198-206, 2013.
- [22] 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.
- [23] E. J. Dockner, “A dynamic theory of conjectural variations,” J. of Industrial Economics, Vol.40, pp. 377-395, 1992.
- [24] 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).
- [25] 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).