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).

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