Existence, Uniqueness and Stability of Market Equilibrium in Oligopoly Markets

Abstract

In this paper we build a pragmatic model on competition in oligopoly markets. To achieve this goal, we use an approach based on studying the response functions of each market participant, thus making it possible to address both Cournot and Bertrand industrial structures with a unified formal method. In contrast to the restrictive theoretical constructs of duopoly equilibrium, our study is able to account for real-world limitations like minimal sustainable production levels and exclusive access to certain resources. We prove and demonstrate that by using carefully constructed response functions it is possible to build and calibrate a model that reflects different competitive strategies used in extremely concentrated markets. The response functions approach makes it also possible to take into consideration different barriers to entry. By fitting to the response functions rather than the profit maximization of the payoff functions problem we alter the classical optimization problem to a problem of coupled fixed points, which has the benefit that considering corner optimum, corner equilibria and convexity condition of the payoff function can be skipped.

A variational principle, coupled fixed points and market equilibrium

“…Empty intersection of production sets -Such a case may seem extreme, but it is not impossible. For example, if one of the companies is working at a very large scale, it may simply be impractical to sustain a low level of output [14]. On the other hand, if the company is too small to undertake large projects, it may also happen that expanding its production beyond certain limit is not feasible.…”

“…Results about existence, uniqueness and stability in duopoly markets when the response functions do not satisfy the mixed monotone property are obtained in [14] and illustrated with number of different kinds of examples.…”

“…As it is shown in a number of articles, there holds x = y. A generalization of coupled fixed or best proximity points (x, y) when x = y is presented in [14].…”

Existence of Coupled Best Proximity Points of p-Cyclic Contractions

“…It turns out that the last 10 years there is a great interest on coupled fixed points, both in fundamental results and their applications [2][3][4][5]. We would like to mention a new kind of applications in the theory of equilibrium in duopoly markets [6,7].…”

“…The notion of best proximity points [9] actually generalizes the notion of cyclic maps from [8], as far as if A ∩ B = ∅, then any best proximity point is a fixed point, too. It turns out that best proximity points are interesting not only as a pure mathematical results, but also as a possibility for a new approach in solving of different types of problems [2][3][4][5][6][7].…”