Paper:

# Consistent Conjectural Variations Equilibrium in a Semi-Mixed Duopoly

## Vitaliy V. Kalashnikov^{*1}, José G. Flores-Muñiz^{*2}, Vyacheslav V. Kalashnikov^{*3,*4,*5}, and Nataliya I. Kalashnykova^{*2}

^{*1}Department of Economics, Universidad Autónoma de Nuevo León (UANL)

Campus Mederos, Av. Lázaro Crdenas 4600, Monterrey, Nuevo León, Mexico

^{*2}Department of Physics and Mathematics, Universidad Autónoma de Nuevo León (UANL)

Av. Universidad S/N, Ciudad Universitaria, San Nicolás de los Garza, Nuevo León, Mexico

^{*3}Tecnológico de Monterrey (ITESM)

Campus Monterrey, Ave. Eugenio Garza Sada 2501 Sur, Monterrey, Nuevo León 64849, Mexico

^{*4}Central Economics and Mathematics Institute (CEMI), Russian Academy of Sciences

Nakhimovsky pr. 47, Moscow 117418, Russia

^{*5}Sumy State University

Rimsky-Korsakov st. 2, Sumy 40007, Ukraine

This paper considers conjectural variations equilibrium (CVE) in the one item market with a mixed duopoly of competitors. The duopoly is called *semi-mixed* because one (semi-public) company’s objective is to maximize a convex combination of her net profit and domestic social surplus (DSS). The two agents make conjectures about fluctuations of the equilibrium price occurring after their supplies having been varied. Based on the concepts of the *exterior* and *interior equilibrium*, as well as the existence theorem for the interior equilibrium (a.k.a. the consistent CVE, or the exterior equilibrium with *consistent conjectures*) demonstrated in the authors’ previous papers, we analyze the behavior of the interior equilibrium as a function of the semi-public firm’s level of socialization. When this parameter reflected by the convex combination coefficient tends to 1, thus transforming the semi-public company into a completely public one, and the considered model into the classical mixed duopoly, two trends are apparent. First, for the private company, the equilibrium with consistent conjectures (CCVE) becomes more attractive (lucrative) than the Cournot-Nash equilibrium. Second, there exists a (unique in the case of an affine demand function) value of the convex combination coefficient such that the private agent’s profit is the same in both of the above-mentioned equilibrium types, thus making no subsidy to the producer or to the consumers necessary. Numerical experiments with various mixed duopoly models confirm the robustness of the proposed algorithm for finding the optimal value of the above-mentioned combination coefficient (a.k.a. the semi-public company’s socialization level).

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.21, No.7, pp. 1125-1134, 2017.

- [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 (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 U. Zenginobuz, “Competition between regulated and non-regulated generators on electric power networks,” MPRA Paper No.376, Online at http://mpra.ub.uni-muenchen.de/376/, MPRA Paper No.376, posted Nov. 7, 2007.
- [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, pp. c69-90, 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. Adv. Comput. Intell. Intell. Inform., 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, 4th Int. Workshop Proc. EUREKA2013, Mazatlán, pp. 198-206, Atlantis Press, Amsterdam-Paris-Beijing, 2013.
- [22] V. V. Kalashnikov, V. A. Bulavsky, N. I. Kalashnykova, J. Watada, and D. J. Hernández-Rodríguez, “Analysis of consistent equilibria in a mixed duopoly,” J. Adv. Comput. Intell. Intell. Inform., Vol.18, No.6, pp. 962-970, 2014.
- [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, No.455-461, 2007.