Mathematical Economic Model of the Complex System in International Economics

在属于被称为复杂系统经济学和经济物理学的复杂系统科学的数学社会科学领域中，许多数学模型是利用统计力学技术来分析社会经济现象的。然而，在复杂系统科学的一个新领域中，几乎还不存在与统计力学理论相同的数学基础。因此，我们用函数分析的方法对一个数学经济模型进行了分析。我们建立了一个数学模型，描述了欧盟市场一体化中劳动力流动与经济增长相互作用所产生的自组织现象。在统计力学中，人口迁移理论中的一个主方程的解被证明与福克-普朗克方程的解非常接近，这是利用物理上允许的克拉默斯-莫亚尔展开，当需要的迁移成本是…时更多的国际劳动力流动是足够大的。此外，我们还能够从数学上精确地证明Hotling的人口迁移理论和Weidlich-Haag的人口迁移理论的一致性。在一篇文章《人类人口迁移和生物颗粒扩散之间的几何相似性》中，我们建立了一个基于代理的模型，该模型描述了当表示某种局部经济优势的效用函数是代理密度的线性函数或二次函数时的人口迁移现象。此外，在传统的研究中，对于Kramers-MoYal展开式和高次项展开式的收敛的数学精确证明是极其困难的，因此没有提供几乎有意义的结果。然而，我们证明了当人口迁移的代价足够大时，利用我们修正的有限Kramers-MoYal展开式，人口迁移理论中的主方程的解与Fokker-Planck方程的解非常接近。较少

In a field of mathematical social science that belongs to a science of the complex system to be called economics of complex system and economic physics, many mathematical models are built by using technique of statistical mechanics for analysis of a social economic phenomenon. However, in a new field of a science of the complex system, a mathematical foundation equal to that for the theory of statistical mechanics does not almost exist yet. Therefore, we analyzed a mathematical economic model with a method of functional analysis. We built a mathematical model that expresses a self-organization phenomenon that caused by interaction of labor movement and economic growth like that happened in the EU market unification. In statistical mechanics a solution of a master equation in a population movement theory proved to be very close to a solution of the Fokker-Planck equation by using the Kramers-Moyal expansion in closeness of degree permitted in physics when the moving cost that is necessa … More ry for international labor mobility is large enough. In addition, we were able to prove mathematically and precisely the consistency of the theory of population movement of Hotelling's with that of Weidlich-Haag's.In an article, "A geometrical similarity between migration of human population and diffusion of biological particles", we built an Agent-based Model that describes a population movement phenomenon when a utility function expressing a certain local economic advantage is a linear function or a quadratic function of agent density. Furthermore, in conventional studies, mathematically precise proofs for the convergence of the Kramers-Moyal expansion and an expansion of high degree terms were extremely difficult, so an almost meaningful result was not provided. However, we proved that a solution for a master equation in a theory of population movement by using the limited Kramers-Moyal expansion that modified by us was very close to a solution for the Fokker-Planck equation when the cost of population movement was large enough. Less