Stability and dynamic behaviors of a limited monopoly with a gradient adjustment mechanism

Li, Xiaoliang; Li, Bo; Liu, Li

doi:10.1016/j.chaos.2023.113106

Cited by 7 publications

(4 citation statements)

References 33 publications

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“…Figure 3 depicts two-dimensional bifurcation diagrams to show more complex dynamics with respect to the two parameters λ and r (r 1 = r 2 = r). Readers can refer to [43] for additional details regarding two-dimensional bifurcation diagrams. In the numerical simulations producing the two-dimensional bifurcation diagrams, we fix the parameters λ = 0, r 1 = r 2 = 1, and d 1 = d 2 = 0.…”

Section: Numerical Simulationsmentioning

confidence: 99%

Effects of Competition Intensities and R&D Spillovers on a Cournot Duopoly Game of Digital Economies

Li,

Su,

Wang

2023

Fractal Fract

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In this paper, we introduce a Cournot duopoly game that can characterize fierce competition in digital economies and employ it to examine the effects of research and development (R&D) spillovers while considering various competition intensities. We obtain the analytical solution of the Nash equilibrium and the expression of commodity price, firm production, and variable profit under some key competition intensities. Furthermore, we analyze the local stability of the Nash equilibrium and derive that the equilibrium may lose its stability only through a 1:4 resonance bifurcation. Numerical simulations are conducted, through which we find that the Nash equilibrium transitions to complex dynamics through a cascade of period-doubling bifurcations. Phase portraits are also provided to illustrate more details of the dynamics, which confirm the previous theoretical finding that the Nash equilibrium loses its stability through a 1:4 resonance bifurcation.

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Section: Numerical Simulationsmentioning

confidence: 99%

Effects of Competition Intensities and R&D Spillovers on a Cournot Duopoly Game of Digital Economies

Li,

Su,

Wang

2023

Fractal Fract

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“…In this strand of studies, extensive work has been done by Cavalli and Naimzada [ 20 ], Cavalli et al [ 21 ], Tramontana et al [ 22 ], Li and Su [ 23 ]. Furthermore, modern tools based on symbolic computations (see, e.g., [ 24 – 26 ] for additional details) have greatly improved analytical studies on the stability of dynamic oligopoly games. Also, many other related studies on the dynamics of oligopoly models should be mentioned, including [ 27 – 32 ].…”

Section: Introductionmentioning

confidence: 99%

Dynamics of a Cournot game with bounded rational firms and various scale effects

Li,

2024

PLoS ONE

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In this paper, we focus on a dynamic Cournot game in the market with a nonlinear (isoelastic) demand function. In our model, there are N competing firms featured by nonlinear cost functions, which enhances our model’s resemblance to real-world scenarios. Firstly, we focus on the homogeneous case where firms’ marginal costs change at equal rates. We establish analytical expressions of the market supply at equilibrium and perform comparative static analysis. In addition, we investigate the local stability under different economies of scale and show that there could be multiple stable equilibria if firms face economies of scale. The heterogeneous case where firms’ marginal costs change at distinct rates is much more complex, thus we investigate the duopoly game with only two firms involved. Methods of symbolic computations such as triangular decomposition and partial cylindrical algebraic decomposition are employed in the analytical investigations of the equilibrium, which is nearly impossible by using the pencil-and-paper approach since the closed-form equilibrium is quite complicated. According to the computational results, we derive that two stable positive equilibria may coexist if both firms face economies of scale. Additionally, we conduct preliminary numerical simulations and find two different types of complex dynamics of the model considered in this paper: complex trajectories such as periodic and chaotic orbits may appear; the topological structure of the basins of attraction may be complex.

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“…Since decades ago, many economists have been making efforts to model and investigate oligopolistic competition by discrete dynamical systems of form with non-monotonic reaction functions. See, e.g., [8,19,25,34,38]. There are two strands of establishing non-monotonic reaction functions.…”

Section: Introductionmentioning

confidence: 99%

Stability and chaos of the duopoly model of Kopel: A study based on symbolic computations

Kongyan

Wei

et al. 2023

Preprint

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Since Kopel's duopoly model was proposed about three decades ago, there are almost no analytical results on the equilibria and their stability in the asymmetric case. The first objective of our study is to fill this gap. This paper analyzes the asymmetric duopoly model of Kopel analytically by using several tools based on symbolic computations. We discuss the possibility of the existence of multiple positive equilibria and establish necessary and sufficient conditions for a given number of positive equilibria to exist. The possible positions of the equilibria in Kopel's model are also explored. Furthermore, in the asymmetric model of Kopel, if the duopolists adopt the best response reactions or homogeneous adaptive expectations, we establish rigorous conditions for the local stability of equilibria for the first time. The occurrence of chaos in Kopel's model seems to be supported by observations through numerical simulations, which, however, is challenging to prove rigorously. The second objective is to prove the existence of snapback repellers in Kopel's map, which implies the existence of chaos in the sense of Li–Yorke according to Marotto's theorem.

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Stability and dynamic behaviors of a limited monopoly with a gradient adjustment mechanism

Cited by 7 publications

References 33 publications

Effects of Competition Intensities and R&D Spillovers on a Cournot Duopoly Game of Digital Economies

Effects of Competition Intensities and R&D Spillovers on a Cournot Duopoly Game of Digital Economies

Dynamics of a Cournot game with bounded rational firms and various scale effects

Stability and chaos of the duopoly model of Kopel: A study based on symbolic computations

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