Investigation of Equilibrium in Oligopoly Markets with the Help of Tripled Fixed Points in Banach Spaces

Atanas Ilchev, Vanya Ivanova, Hristina Kulina (), Polina Yaneva and Boyan Zlatanov
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Atanas Ilchev: Department of Mathematical Analysis, Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 4000 Plovdiv, Bulgaria
Vanya Ivanova: Department of Computer Systems, Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 4000 Plovdiv, Bulgaria
Hristina Kulina: Department of Mathematical Analysis, Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 4000 Plovdiv, Bulgaria
Polina Yaneva: Department of Mathematical Analysis, Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 4000 Plovdiv, Bulgaria
Boyan Zlatanov: Department of Mathematical Analysis, Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 4000 Plovdiv, Bulgaria

Econometrics, 2024, vol. 12, issue 2, 1-24

Abstract: In the study we explore an oligopoly market for equilibrium and stability based on statistical data with the help of response functions rather than payoff maximization. To achieve this, we extend the concept of coupled fixed points to triple fixed points. We propose a new model that leads to generalized triple fixed points. We present a possible application of the generalized tripled fixed point model to the study of market equilibrium in an oligopolistic market dominated by three major competitors. The task of maximizing the payout functions of the three players is modified by the concept of generalized tripled fixed points of response functions. The presented model for generalized tripled fixed points of response functions is equivalent to Cournot payoff maximization, provided that the market price function and the three players’ cost functions are differentiable. Furthermore, we demonstrate that the contractive condition corresponds to the second-order constraints in payoff maximization. Moreover, the model under consideration is stable in the sense that it ensures the stability of the consecutive production process, as opposed to the payoff maximization model with which the market equilibrium may not be stable. A possible gap in the applications of the classical technique for maximization of the payoff functions is that the price function in the market may not be known, and any approximation of it may lead to the solution of a task different from the one generated by the market. We use empirical data from Bulgaria’s beer market to illustrate the created model. The statistical data gives fair information on how the players react without knowing the price function, their cost function, or their aims towards a specific market. We present two models based on the real data and their approximations, respectively. The two models, although different, show similar behavior in terms of time and the stability of the market equilibrium. Thus, the notion of response functions and tripled fixed points seems to present a justified way of modeling market processes in oligopoly markets when searching whether the market has reached equilibrium and if this equilibrium is unique and stable in time

Keywords: fixed point; tripled fixed point; market equilibrium; response functions; oligopoly market (search for similar items in EconPapers)
JEL-codes: B23 C C00 C01 C1 C2 C3 C4 C5 C8 (search for similar items in EconPapers)
Date: 2024
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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