The Cournot duopoly Kopel Model

1. The Cournot duopoly Kopel Model • In the recent literatures, it is also demonstrated that the oligopolistic markets • may become chaotic. Oligopoly theory is one of the oldest branches of mathematical economics dated back to 1838 when its basic model was proposed by French economist Cournot • Among the first who have shown the Cournot model may lead to complex • behavior, such as periodic and chaotic behavior, was Puu [9][10] . [9] T. Puu, Chaos in duopoly pricing, Chaos Solitons Fractals 1 (1991) 573-581. [10] T. Puu, The chaotic duopolists revisited, J. Econom, Behav. Organ. 33 (1998) 385-394 • In this work, we consider a general case of a duopoly model: The Cournot duopoly Kopel Model [11] [11] Kopel, M., ”Simplex and Complex Adjustment Dynamics in Cournot Duopoly Models,” Chaos, Solitions and Fractals, 7, 2031-2048, 1996.

2. The Cournot duopoly Kopel Model ☺Previous work: Analyzing chaotic behavior numerically ☺Our work: • A rigorous proof for existence of chaos from mathematical • point of view is given. • Two different types of intermittent chaos in this model are found • and analyzed.

3. 混沌经济学 ● Day于1982年将非线性动态引入到经济学中， 引发了人们对传统经济学的反思，为人们提供 了崭新的视角 ● 宏观经济中存在混沌现象 ● 在微观经济学领域，厂商或者其他经济个体 所经营产品的价格、生产或销售的产品数量都 可能产生波动，呈现出混沌动态

4. 4.1 一个古诺双寡头经济模型描述 ●X和Y代表两个寡头厂商 ●厂商X和厂商Y在t+1时间段生产的产品数量分别用x(t+1)和y(t+1)表示 古诺双寡头 Kopel经济模型[7] ●Nash平衡点： [7] Kopel M. Chaos, Solitons & Fractals, 1996, 7: 2031~2048

5. 4.2分形分析 ， ● 分形图： 混沌 混沌 周期 平衡点 周期 平衡点

6. 4.2分形分析 ● 由光滑经济周期演变为混沌： 非光滑经济周期 光滑经济周期 混沌

7. 4.2分形分析 ● 混沌吸引子共存现象 两个共存的混沌吸引子 吸引域 由于对称性，混沌吸引子共存现象普遍存在

8. 4.3混沌吸引子的计算机辅助证明 ● 将双寡头Kopel模型改写为向量形式： 其中： ● 研究映射 的动态（ ）： 定义为，以此类推得到：

9. 4.3混沌吸引子的计算机辅助证明 定理 Kopel经济模型具有如下性质：关于四边形 的映射 存在一个闭的不变集 ，使得 与2个符号的移位映射半共轭，且 因此，当 时，古诺双寡头Kopel经济 模型有正拓扑熵。

10. 4.4 间歇混沌特性分析 • PM-I型间歇混沌： 分形图

11. 4.4 间歇混沌特性分析 • 分岔前后x的时间序列 过渡混沌 分岔前， 6倍周期点 分岔后，

12. 4.4 间歇混沌特性分析 • PM-I型间歇混沌：层流态平均持续时间分布 幂指数特征值: -0.5 结论：服从幂指数为-0.496的幂律分布

13. 4.4 间歇混沌特性分析 发生激变前， 发生激变后， 发生激变后， 发生激变前， • 诱发激变导致的间歇混沌： 分形图

14. 4.4 间歇混沌特性分析 • 诱发激变导致的间歇混沌： 层流态平均持续时间分布 幂指数特征值: [-3/2,-1/2] 结论：服从幂指数为-0.65的幂律分布

15. 4.4 间歇混沌特性分析 • 在经济学系统中出现的间歇混沌现象可以 解释为系统本身具有调节机制，不借助于任 何外力，系统总是能够将混乱的市场调整回 (相对)平稳状态 • 或者解释为系统有记忆机制，总是能够记 住混乱前的状态并恢复

16. 4.5 长期平均利润分析 • 混沌能否带来更多的利润？ 混沌动态的平均利润: 非零平衡点:

17. 4.5 长期平均利润分析 结论：混沌市场并不是完全有害的

18. 4.6 控制混沌到Nash平衡点 定理 • 稳健的经济市场仍然是人们最需要的 考虑受控的古诺双寡头Kopel经济模型，平衡点 是 局部渐近稳定的，当且仅当

19. 4.6 控制混沌到Nash平衡点 • 仿真结果 （ ） 令

20. 四、 经济系统中的混沌动态研究 小结： ●重点研究了一个古诺双寡头经济模型中的各种 混沌动态，从理论上证明了混沌存在性，并分 析了混沌对利润的影响，得到了混沌并非完全 有害的结论

21. The Cournot duopoly Kopel Model • Model description Consider two firms X and Y : (1) Where, denote the goods quantities that firm X and firm Y produce in period t, respectively. Remarks: (1). Both firms must consider the actions and reactions of the competitor (2). The competitors have choose their actions simultaneously (3). Each firm forms the expectation on the quantity of the other firm, which depend on their own quantity and the quantity of the other firm both produced in the previous period, in order to determine a profit maximizing quantity to produce in the next period.

22. The Cournot duopoly Kopel Model • Nash-equilibria of the Kopel model The fixed points (Nash-equilibrium ) of system (1) satisfy the equations: (2) The solutions of Eq. (2) give four equilibria: ⅰ ⅱ ⅲ for ⅳ for Remark: The fixed points depend on . In case ⅱ, we should have (positive solution). Also, in case ⅲ and ⅳ, we should have (real solution).

23. The Cournot duopoly Kopel Model • Bifurcation analysis • bifurcation diagram: rich and complex dynamics bifurcation diagram provides a general view of the evolution process of the dynamical behaviors by plotting a state variable with the abscissa being one parameter (a) (b) Fig. 1 Bifurcation diagram. (a) Fix , and . (b) Fix and

24. The Cournot duopoly Kopel Model • Observation of chaotic attractors and basins of attraction (b) (c) (a) Smooth Cycle Lost of Smoothness Chaotic Fig. 2 Different attractors in Kopel model. (a) One smooth invariant cycle with , . (b) Invariant cycle loses its smoothness when , . (c) Chaotic attractor with .

25. The Cournot duopoly Kopel Model • Coexistence of two chaotic attractors: (a) (b) Fig. 3 Two chaotic attractors coexist with different initial conditions when . (a) Phase portraits of the two chaotic attractors and the four Nash equilibria; (b) The basins of attractions.

26. The Cournot duopoly Kopel Model • Horseshoe Chaos in the model • A convenient expression for the Kopel model is described as follows: (3) where After a great many trial attempts, we will discuss the dynamics of system (3) with under the map and obtain that there exists a horseshoe in this attractor.

27. The Cournot duopoly Kopel Model Fig. 4 The attractor when and the proper quadrangle . We take a proper quadrangle |ABCD| to be a subset in the plane with its four vertices being

28. The Cournot duopoly Kopel Model Fig. 5. The quadrangle and its image . Theorem 1. For the map corresponding to the quadrangle , there exists a closed invariant set for which is semi-conjugate to the 2-shift map. Hence, .

29. The Cournot duopoly Kopel Model To prove the above theorem, we should find two mutually disjointed subsets of such that there exists a -connected family with respect to them. Proof. We select two appropriate subsets in . The first one is denoted by a as shown the yellow quadrangle in Fig. 6, with and be its left and right edge, respectively. Then under the map , is mapped to which is on the right side of the edge , and is mapped to which is on the left side of the edge . Fig. 6 The subset and its image under the map with .

30. The Cournot duopoly Kopel Model The second subset is the purple quadrangle, denoted by b, as shown in Fig. 7. Take and to be the left and right edge of b, respectively. Then the image ison the left side of the edge , and the image is on the right side of the edge. Fig. 7 The subset and its image under the map with .

31. The Cournot duopoly Kopel Model Upon the abovesimulation results, it is easy to see that the subset a and b are disjointed and it follows that for every connection v with respect to a and b, the images and lie wholly across the quadrangles , that is to say,the images and are still connections with respect to a and b. According to topological horseshoe Theorem, there exists a -connected family, which means that is semi-conjugate to the 2-shift map. Hence,based on the Lemma, we know that the entropyof is not less than , which implies system (1) has positive topological entropy with . The proof is thuscompleted.

32. The Cournot duopoly Kopel Model • Intermittency chaos dynamics Laminar state Burst state A typical intermittency phenomenon Intermittent events are characterized by time series that display time intervals with low variabilities interrupted by bursts of very high variabilities

33. The Cournot duopoly Kopel Model • Type of intermittency: PM type-I Intermittency PM type-II Intermittency PM type-III Intermittency (1) Pomeau-Manneville intermittency (2) Crisis-induced intermittency: (3) Attractor widening Attractor merging In-out Intermittency On-off Intermittency