Abstract
This paper studies a Cournot duopoly game in which firms produce homogeneous goods and adopt a bounded rationality rule for updating productions. The firms are characterized by an isoelastic demand that is derived from a simple quadratic utility function with linear total costs. The two competing firms in this game seek the optimal quantities of their production by maximizing their relative profits. The model describing the game’s evolution is a two-dimensional nonlinear discrete map and has only one equilibrium point, which is a Nash point. The stability of this point is discussed and it is found that it loses its stability by two different ways, through flip and Neimark–Sacker bifurcations. Because of the asymmetric structure of the map due to different parameters, we show by means of global analysis and numerical simulation that the nonlinear, noninvertible map describing the game’s evolution can give rise to many important coexisting stable attractors (multistability). Analytically, some investigations are performed and prove the existence of areas known in literature with lobes.
Highlights
Market structure has been characterized by a few interdependent firms that collectively dominate the market
There was the most popular utility function, known Cobb–Douglas utility. It is a particular functional form of production function that has been widely adopted to represent the technological relationship between the amounts of two or more inputs and the amount of output that can be produced by those inputs
There was the constant elasticity of substitution (CES) utility, Singh and Vives utility, and others
Summary
Market structure has been characterized by a few interdependent firms that collectively dominate the market. Bounded rationality mechanism has been ranked first in such studies and has been deeply adopted in the modeling process It has been considered as a gradient-rule mechanism as it depends on the marginal profit and requires competing firms to carry out an estimation on it whether it increases or decreases, so firms can update their output production time period. Adopting the bounded rationality rule and a nonlinear quadratic utility function that is considered as a particular case form Singh and Vives utility a Cournot duopoly game that belongs to those discussed above is introduced in the current manuscript.
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