对伯川德博弈的正式数学证明与扩展运用

A CONTRIBUTION TO THE FORMAL PROOF,AMENDMENT AND EXPAND APPLICATION OF BERTRAND MODEL

十九世纪法国经济学家伯川德提出了伯川德寡头模型,尽管中外学者对之给出了各种证明方法,但到目前为止还没有一个正式的数学证明.本文引入广义函数中的冲击函数和阶跃函数,很好地刻画两个寡头基于同质产品的间断需求函数,以代替不是很严密的用直观图形推演或分情况讨论式的证明,从而对伯川德博弈进行了严密的数学证明.本文证明:原始伯川德博弈的结论是近似的,严格上的纳什均衡点是双方都定价于比边际成本高的一个相同位置上,这是由于基于完全信息的理性行为,只有市场需求曲线为水平线时,均衡结果才收敛于伯川德博弈.因此,从完全信息出发,可以证明所谓的"伯川德博弈"只在有限条件下存在.另外,作为一个扩展应用,本文还运用冲击函数和阶跃函数的性质对斯威齐模型进行了数学证明,并演示了如何运用该方法对此类问题进行动态分析.

J.Bertrand who was an economist of France has put forward the Bertrand Paradox Model in 19th century. And some interrelated theories about the model have provided much useful advices for the developing of enterprise, but it＇s a pity that the model hasn＇t been formally proved through mathematics. We introduced lash function and step function in this article,and depicted the finite demand function of the two oligopohes which is based on homogeneity product exactly by the characters of the two functions, instead of the proof of intuitionistic graphic supposition or discussion based on different situation.And sequentially,make a formal proof for the model to fetch on the pity.And as a expand application, we also make a proof of the Sweezy Model by the characters of the two functions in the article.