Application of Coupled Fixed (or Best Proximity) Points in Market Equilibrium in Oligopoly Markets

TL;DR

This paper generalizes coupled fixed point theory and applies it to model equilibrium in oligopoly markets, accommodating real-world constraints and different competitive strategies using response functions.

Contribution

It introduces a unified formal method based on response functions to model oligopoly equilibria, extending classical fixed point approaches to more realistic market scenarios.

Findings

Established existence and uniqueness conditions for fixed points in Banach spaces.

Developed error estimates for iterative fixed point approximation.

Created a flexible model capturing various market structures and barriers.

Abstract

We present a possible kind of generalization of the notion of ordered pairs of cyclic maps and coupled fixed points and its application in modelling of equilibrium in oligopoly markets. We have obtained sufficient conditions for the existence and uniqueness of fixed (or best proximity) points in complete metric spaces (uniformly convex Banach spaces). We get an error estimates of the fixed (or best proximity), provided that we have used sequences of successive iterations. We illustrate one possible application of the results by building 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 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.