Consistent Conjectural Variations Equilibrium in a Mixed Duopoly

Nataliya I. Kalashnykova; Vladimir A. Bulavsky; Vyacheslav V. Kalashnikov; Felipe J. Castillo Pérez

doi:10.20965/jaciii.2011.p0425

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JACIII Vol.15 No.4 pp. 425-432

doi: 10.20965/jaciii.2011.p0425

(2011)

Paper:

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Consistent Conjectural Variations Equilibrium in a Mixed Duopoly

Nataliya I. Kalashnykova^1,2, Vladimir A. Bulavsky^3, Vyacheslav V. Kalashnikov^2,3,4, and Felipe J. Castillo Pérez^*1

^*1FCFM, UANL, San Nicolás de los Garza, N.L. 66450, Mexico

^*2Sumy State University, Sumy, Ukraine 40007

^*3Central Economics & Mathematics Institute (CEMI), Moscow 117418, Russia

^*4ITESM, Campus Monterrey, Monterrey, N.L. 64849, Mexico

Received:

July 23, 2010

Accepted:

November 18, 2010

Published:

June 20, 2011

Keywords:

conjectural variations equilibrium (CVE), mixed duopoly, consistent conjectures

Abstract

In this paper, we consider a model of mixed duopoly with Conjectured Variations Equilibrium (CVE). The agents’ conjectures concern the price variations depending on the increase or decrease in their production outputs. We establish existence and uniqueness results for the conjectured variations equilibrium (called an exterior equilibrium) for any set of feasible conjectures. To introduce the notion of an interior equilibrium, we develop a consistency criterion for the conjectures (referred to as influence coefficients) and prove the existence theorem for the interior equilibrium (understood as a CVE with consistent conjectures). To prepare the base for the extension of our results to the case of non-differentiable demand functions, we also investigate the behavior of the consistent conjectures in dependence upon a parameter representing the demand function derivative with respect to the market price.

Cite this article as:

N. Kalashnykova, V. Bulavsky, V. Kalashnikov, and F. Pérez, “Consistent Conjectural Variations Equilibrium in a Mixed Duopoly,” J. Adv. Comput. Intell. Intell. Inform., Vol.15 No.4, pp. 425-432, 2011.

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References

[1] N. Matsushima and T. Matsumura, “Mixed oligopoly and spatial agglomeration,” Canadian J. Econ., Vol.36, No.1, pp. 62-87, 2003.
[2] C. Fershtman, “The interdependence between ownership status and market structure: The case of privatization,” Economica, Vol.57, No.3, pp. 319-328, 1990.
[3] T. Matsumura and O. Kanda, “Mixed oligopoly at free entry markets,” J. Econ., Vol.84, No.1, pp. 27-48, 2005.
[4] T. Matsumura, “Stackelberg mixed duopoly with a foreign competitor,” Bull. Econ. Res., Vol.55, No.2, pp. 275-287, 2003.
[5] 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.
[6] C. Figuières, A. Jean-Marie, N. Quérou, and M. Tidball, “Theory of Conjectural Variations,” World Scientific, 2004.
[7] A. L. Bowley, “TheMathematical Groundwork of Economics,” Oxford University Press, 1924.
[8] R. Frisch, “Monopoly, polypoly: The concept of force in the economy,” Int. Econ. Papers, Vol.1, No.1, pp. 23-36, 1951. (“Monopole, polypole – La notion de force en économie,” Nationaløkonomisk Tidsskrift, Vol.71, No.2, pp. 241-259, 1933.)
[9] V. A. Bulavsky and V. V. Kalashnikov, “One-parametric method to study equilibrium,” Economics and Mathematical Methods (Ekonomika i Matematicheskie Metody), Vol.30, No.3, pp. 129-138, 1994. (in Russian)
[10] V. A. Bulavsky and V. V. Kalashnikov, “Equilibrium in generalized Cournot and Stackelberg models,” Economics and Mathematical Methods (Ekonomika i Matematicheskie Metody), Vol.31, No.4, pp. 164-176, 1995. (in Russian)
[11] G. Isac, V. A. Bulavsky, and V. V. Kalashnikov, “Complementarity, Equilibrium, Efficiency and Economics,” Kluwer Academic Publishers, 2002.
[12] V. V. Kalashnikov, C. Kemfert, and V. V. Kalashnikov-Jr., “Conjectural variations equilibrium in a mixed duopoly,” European J. Oper. Res., Vol.192, No.3, pp. 717-729, 2009.
[13] V. V. Kalashnikov, E. Cordero, and V. V. Kalashnikov-Jr., “Cournot and Stackelberg equilibrium in mixed duopoly models,” Optimization, Vol.59, No.5, pp. 689-706, 2010.
[14] V. V. Kalashnikov, V. A. Bulavsky, N. I. Kalashnykova, and F. J. Castillo, “Consistent conjectures in mixed oligopoly,” European J. Oper. Res., Vol.210, No.3, pp. 729-735, 2011.
[15] V. A. Bulavsky, “Structure of demand and equilibrium in a model of oligopoly,” Economics and Mathematical Methods (Ekonomika i Matematicheskie Metody), Vol.33, No.3, pp. 112-134, 1997. (in Russian).
[16] 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. Electrical Power Energy Sys., Vol.29, No.4, pp. 455-461, 2007.

This article is published under a Creative Commons Attribution-NoDerivatives 4.0 Internationa License.

[1] [1] N. Matsushima and T. Matsumura, “Mixed oligopoly and spatial agglomeration,” Canadian J. Econ., Vol.36, No.1, pp. 62-87, 2003.

[2] [2] C. Fershtman, “The interdependence between ownership status and market structure: The case of privatization,” Economica, Vol.57, No.3, pp. 319-328, 1990.

[3] [3] T. Matsumura and O. Kanda, “Mixed oligopoly at free entry markets,” J. Econ., Vol.84, No.1, pp. 27-48, 2005.

[4] [4] T. Matsumura, “Stackelberg mixed duopoly with a foreign competitor,” Bull. Econ. Res., Vol.55, No.2, pp. 275-287, 2003.

[5] [5] 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.

[6] [6] C. Figuières, A. Jean-Marie, N. Quérou, and M. Tidball, “Theory of Conjectural Variations,” World Scientific, 2004.

[7] [7] A. L. Bowley, “TheMathematical Groundwork of Economics,” Oxford University Press, 1924.

[8] [8] R. Frisch, “Monopoly, polypoly: The concept of force in the economy,” Int. Econ. Papers, Vol.1, No.1, pp. 23-36, 1951. (“Monopole, polypole – La notion de force en économie,” Nationaløkonomisk Tidsskrift, Vol.71, No.2, pp. 241-259, 1933.)

[9] [9] V. A. Bulavsky and V. V. Kalashnikov, “One-parametric method to study equilibrium,” Economics and Mathematical Methods (Ekonomika i Matematicheskie Metody), Vol.30, No.3, pp. 129-138, 1994. (in Russian)

[10] [10] V. A. Bulavsky and V. V. Kalashnikov, “Equilibrium in generalized Cournot and Stackelberg models,” Economics and Mathematical Methods (Ekonomika i Matematicheskie Metody), Vol.31, No.4, pp. 164-176, 1995. (in Russian)

[11] [11] G. Isac, V. A. Bulavsky, and V. V. Kalashnikov, “Complementarity, Equilibrium, Efficiency and Economics,” Kluwer Academic Publishers, 2002.

[12] [12] V. V. Kalashnikov, C. Kemfert, and V. V. Kalashnikov-Jr., “Conjectural variations equilibrium in a mixed duopoly,” European J. Oper. Res., Vol.192, No.3, pp. 717-729, 2009.

[13] [13] V. V. Kalashnikov, E. Cordero, and V. V. Kalashnikov-Jr., “Cournot and Stackelberg equilibrium in mixed duopoly models,” Optimization, Vol.59, No.5, pp. 689-706, 2010.

[14] [14] V. V. Kalashnikov, V. A. Bulavsky, N. I. Kalashnykova, and F. J. Castillo, “Consistent conjectures in mixed oligopoly,” European J. Oper. Res., Vol.210, No.3, pp. 729-735, 2011.

[15] [15] V. A. Bulavsky, “Structure of demand and equilibrium in a model of oligopoly,” Economics and Mathematical Methods (Ekonomika i Matematicheskie Metody), Vol.33, No.3, pp. 112-134, 1997. (in Russian).

[16] [16] 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. Electrical Power Energy Sys., Vol.29, No.4, pp. 455-461, 2007.

Consistent Conjectural Variations Equilibrium in a Mixed Duopoly

Nataliya I. Kalashnykova*1,*2, Vladimir A. Bulavsky*3, Vyacheslav V. Kalashnikov*2,*3,*4, and Felipe J. Castillo Pérez*1

Nataliya I. Kalashnykova^1,2, Vladimir A. Bulavsky^3, Vyacheslav V. Kalashnikov^2,3,4, and Felipe J. Castillo Pérez^*1