基于非线性理论的古诺—伯川德混合模型研究及应用
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摘要
寡头垄断博弈模型以其复杂性和多样性吸引着数学家、经济学家和管理学家的研究目光。在实际的经济活动中，除了以产量作为决策变量的古诺模型和以价格作为决策变量的伯川德模型外，还会同时存在分别以产量和价格作为竞争手段的模型，即古诺-伯川德混合模型。本论文在国内外相关领域的研究成果的基础上，运用博弈理论、动态经济学理论和非线性动力学方法，通过与古诺模型、伯川德模型的对比，研究了古诺-伯川德混合模型的复杂非线性行为，为更好的理解寡头市场的竞争行为和演化机制提供参考。本论文的主要研究内容及创新性结果如下：
1.假设不同厂商提供的产品具有一定的差异性，基于有限理性假设，构建了双寡头古诺模型、伯川德模型和古诺-伯川德混合模型，证明了纳什平衡点的存在性，并求出了其局部稳定区域；研究了模型的复杂动态行为，包括系统随参数变化所表现出的分岔现象、最大李雅普诺夫指数、奇异吸引子、蝴蝶效应等动力学特征，并对系统特性所代表的经济学含义作出了解释。
2.对比了古诺模型、伯川德模型和古诺-伯川德混合模型在纳什平衡点处利润、价格、产量、消费者剩余和社会福利的关系，在周期解和混沌状态时平均产量、平均价格和平均利润的关系，总结了古诺-伯川德混合模型区别于古诺、伯川德模型的特征；同时研究了产品差异性水平对三种模型稳定性的影响。
3.建立了具有延迟的古诺-伯川德混合模型，分析了延迟对系统纳什平衡点局部稳定性的影响，并与不考虑延迟的系统进行对比，通过数值仿真发现合理的选取延迟权重可以将系统从混沌状态控制到稳定或者周期解状态。
4.根据古诺-伯川德混合模型的特点，构造了一个三寡头古诺-伯川德混合模型，通过对系统的平衡点的稳定性分析和系统的分岔分析，研究了系统的复杂非线性演化行为，同时研究了系统的初值对系统的影响，给出了系统在稳定和周期解的吸引域。通过分析决策参数和初值对系统动力学演化行为的影响，为企业的管理者和决策者提供决策参考。
Oligopoly game model, with its complexity and diversity, attracts mathematicians,economists and management scientists’ research attention. In real economic activity, be-sides the Cournot model, in which firms choose the output as the decision variable, andthe Bertrand model, in which firms choose the price as the tool of competition, there arealso the situations, in which there coexist two groups of firms, the first of which optimallyadjust output and the other one optimally adjust their prices to ensure maximum profit,which is known as the Cournot-Bertrand mixed model. In this paper, based on the researchat home and abroad, the Cournot-Bertrand mixed models are analyzed by comparing withCournot and Bertrand models, using the game theory, dynamic economics and nonlineardynamics methods. This research can help us to better understand the competition andevolutionary mechanism of the oligopoly market. The main content of this dissertation isas follows:
1. With the assumption that there is a certain degree of diferentiation betweenthe products ofered by firms, whose decision-making principles are bounded rational,duopoly Cournot, Bertrand and Cournot-Bertrand mixed models are built. The existenceand local stability of Nash equilibrium point are investigated and the stability regionis plotted. The complex dynamical behaviors of the Cournot, Bertrand and Cournot-Bertrand mixed models are further studied, including bifurcation scenarios, the largestLyapunov exponent, strange attractor, butterfly efect, etc. The influences of the parame-ters on the systems’ performance are discussed from the perspective of economics.
2. The profit, price, output, consumer surplus and social welfare of the Cournot,Bertrand and Cournot-Bertrand mixed models in the Nash equilibrium point are com-pared investigated. The average output, average price and average profits in the periodicsolution and chaos are also studied. The diferent characters of the three kind of modelsare summarized. The efects of the product diferentiation on systems’ stability are alsoanalyzed.
3. A Cournot-Bertrand mixed model with delayed bounded rationality expectationis established. The influences of the weight of delay on the stability of Nash equilibriumpoint are studied. Through comparing with the system which does not contain the delay, the results show that the system with delay can be controlled from chaos to stability orperiod states by reasonably choosing the weight of delay.
4. A triopoly Cournot-Bertrand mixed model which is composed of one state-ownedenterprise and two private enterprises is constructed. The existence and local stability ofNash equilibrium point and the bifurcations of the system with variety of the decisionparameters are investigated. The influences of initial values on the system are also con-sidered and the basins of attraction are given. The influences of the parameters and ini-tial values on the system dynamics evolution are analyzed, and these results can providedecision-making reference for the enterprises’ managers and decision makers.
1.假设不同厂商提供的产品具有一定的差异性，基于有限理性假设，构建了双寡头古诺模型、伯川德模型和古诺-伯川德混合模型，证明了纳什平衡点的存在性，并求出了其局部稳定区域；研究了模型的复杂动态行为，包括系统随参数变化所表现出的分岔现象、最大李雅普诺夫指数、奇异吸引子、蝴蝶效应等动力学特征，并对系统特性所代表的经济学含义作出了解释。
2.对比了古诺模型、伯川德模型和古诺-伯川德混合模型在纳什平衡点处利润、价格、产量、消费者剩余和社会福利的关系，在周期解和混沌状态时平均产量、平均价格和平均利润的关系，总结了古诺-伯川德混合模型区别于古诺、伯川德模型的特征；同时研究了产品差异性水平对三种模型稳定性的影响。
3.建立了具有延迟的古诺-伯川德混合模型，分析了延迟对系统纳什平衡点局部稳定性的影响，并与不考虑延迟的系统进行对比，通过数值仿真发现合理的选取延迟权重可以将系统从混沌状态控制到稳定或者周期解状态。
4.根据古诺-伯川德混合模型的特点，构造了一个三寡头古诺-伯川德混合模型，通过对系统的平衡点的稳定性分析和系统的分岔分析，研究了系统的复杂非线性演化行为，同时研究了系统的初值对系统的影响，给出了系统在稳定和周期解的吸引域。通过分析决策参数和初值对系统动力学演化行为的影响，为企业的管理者和决策者提供决策参考。
Oligopoly game model, with its complexity and diversity, attracts mathematicians,economists and management scientists’ research attention. In real economic activity, be-sides the Cournot model, in which firms choose the output as the decision variable, andthe Bertrand model, in which firms choose the price as the tool of competition, there arealso the situations, in which there coexist two groups of firms, the first of which optimallyadjust output and the other one optimally adjust their prices to ensure maximum profit,which is known as the Cournot-Bertrand mixed model. In this paper, based on the researchat home and abroad, the Cournot-Bertrand mixed models are analyzed by comparing withCournot and Bertrand models, using the game theory, dynamic economics and nonlineardynamics methods. This research can help us to better understand the competition andevolutionary mechanism of the oligopoly market. The main content of this dissertation isas follows:
1. With the assumption that there is a certain degree of diferentiation betweenthe products ofered by firms, whose decision-making principles are bounded rational,duopoly Cournot, Bertrand and Cournot-Bertrand mixed models are built. The existenceand local stability of Nash equilibrium point are investigated and the stability regionis plotted. The complex dynamical behaviors of the Cournot, Bertrand and Cournot-Bertrand mixed models are further studied, including bifurcation scenarios, the largestLyapunov exponent, strange attractor, butterfly efect, etc. The influences of the parame-ters on the systems’ performance are discussed from the perspective of economics.
2. The profit, price, output, consumer surplus and social welfare of the Cournot,Bertrand and Cournot-Bertrand mixed models in the Nash equilibrium point are com-pared investigated. The average output, average price and average profits in the periodicsolution and chaos are also studied. The diferent characters of the three kind of modelsare summarized. The efects of the product diferentiation on systems’ stability are alsoanalyzed.
3. A Cournot-Bertrand mixed model with delayed bounded rationality expectationis established. The influences of the weight of delay on the stability of Nash equilibriumpoint are studied. Through comparing with the system which does not contain the delay, the results show that the system with delay can be controlled from chaos to stability orperiod states by reasonably choosing the weight of delay.
4. A triopoly Cournot-Bertrand mixed model which is composed of one state-ownedenterprise and two private enterprises is constructed. The existence and local stability ofNash equilibrium point and the bifurcations of the system with variety of the decisionparameters are investigated. The influences of initial values on the system are also con-sidered and the basins of attraction are given. The influences of the parameters and ini-tial values on the system dynamics evolution are analyzed, and these results can providedecision-making reference for the enterprises’ managers and decision makers.
引文
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