Applications of Transportation Problems and Strongly Nonlinear Systems of Partial Differential Equations in Economics

运输问题和强非线性偏微分方程组在经济学中的应用

• 批准号: 0532398
• 负责人: Pierre Chiappori
• 金额: --
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-09-01 至 2010-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0532398&HistoricalAwards=false
• 关键词: Applications; Transportation; Problems; Strongly; Nonlinear

The project is in three parts. A first goal is to extend recent progresses in the economic analysis of group behavior, which can be applied to households, families, firms, committees, clubs, villages, and other organizations. Mathematically, these problems can be translated into systems of partial differential equations, which are in general 'highly' non linear and can be studied using the tools of exterior differential calculus. These techniques will be extended to currently open questions and to new types of problems. A second goal is to explore the links between three fields of microeconomic theory: hedonic demands, matching models, and multidimensional contract theory. These fields share a common mathematical nature, namely maximization over sets of measure-preserving mappings (known in mathematics as 'transportation' problems). An explicit and careful formal analysis of these common features will allow extending and generalizing the innovations that have been introduced in each field. In particular, the general approaches provided by transportation mathematics, based on measure theory, should lead to general existence theorem for a broad class of hedonic models; conversely, tools that have been developed by economists may have useful applications in mathematics (e.g., an extension of the so-called Gale-Shapley algorithm, used in matching models, can be used for transportation problems). Finally, a special emphasis will be put on a specific type of problems that are related to the two previous classes. Mathematically, they deal with uniqueness of the solution to general systems of partial differential equations when the number of equations is larger than the number of functions; the economic importance of these problems stems from their use in nonparametric identification. The research will merge various fields of economics and mathematics that, in the past, have experienced independent (and somewhat divergent) developments. One can expect that economic analysis will raise new mathematical issues and questions, while mathematical concepts may generate new economic insights. More generally, the cross-fertilization the project is aimed at achieving is important per se, in that it can promote conceptual and empirical innovations in both fields. A particularly important aspect will be the emphasis put on cross-training of PhD students and/or junior researchers in both fields. Finally, the economic problems under consideration raise important policy issues, while they generate new mathematical challenges. This award was supported as part of the fiscal year 2005 Mathematical Sciences priority area special competition on Mathematical Social and Behavioral Sciences (MSBS).

该项目分为三个部分。 第一个目标是扩展群体行为的经济分析的最新进展，可以应用于家庭，家庭，公司，委员会，俱乐部，村庄和其他组织。 在数学上，这些问题可以转化为偏微分方程组，这在一般情况下是“高度”非线性，可以使用外部微分的工具进行研究。 这些技术将被扩展到目前开放的问题和新类型的问题。 第二个目标是探索微观经济理论的三个领域之间的联系：享乐需求，匹配模型和多维契约理论。 这些领域有一个共同的数学性质，即最大化集的测量保持映射（在数学中称为“运输”问题）。 对这些共同特征的明确和仔细的正式分析将允许扩展和概括在每个领域中引入的创新。 特别是，运输数学提供的一般方法，基于测度理论，应该导致广泛的一类享乐模型的一般存在定理;相反，经济学家开发的工具可能在数学中有有用的应用（例如，在匹配模型中使用的所谓Gale-Shapley算法的扩展可用于运输问题）。 最后，将特别强调与前两节课有关的特定类型的问题。 在数学上，他们处理的唯一性的解决方案，一般系统的偏微分方程时，数量的方程是大于数量的功能;经济的重要性，这些问题源于他们的使用在非参数识别。 这项研究将融合经济学和数学的各个领域，在过去，这些领域经历了独立的（和有些分歧的）发展。 人们可以预期，经济分析将提出新的数学问题，而数学概念可能会产生新的经济见解。 更一般地说，该项目旨在实现的相互促进本身就很重要，因为它可以促进这两个领域的概念和经验创新。 一个特别重要的方面将是把重点放在交叉培训的博士生和/或初级研究人员在这两个领域。 最后，所考虑的经济问题提出了重要的政策问题，同时也产生了新的数学挑战。 该奖项是作为2005财政年度数学科学优先领域数学社会和行为科学（MSBS）特别竞赛的一部分获得支持的。