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

该项目分为三部分。第一个目标是扩展最近在群体行为的经济分析方面取得的进展，这些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，偏微分方程组通常是“高度”非线性的，可以使用外微分学的工具来研究。这些技术将被扩展到目前悬而未决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。这些领域具有共同的数学性质，即在保持度量映射集上的最大化（在数学中称为“运输”问题）。对这些共同特征进行明确而仔细的形式化分析，将允许扩展和概括在每个领域引入的创新。特别地，运输数学在测度理论的基础上提供的一般方法，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，可用于交通问题）。最后，将特别强调与前两节课相关的特定类型的问题。在数学上，它们处理一般偏微分方程组解的唯一性，当方程的数量大于函数的数量；这些问题的经济重要性源于它们在非参数识别中的应用。这项研究将融合经济学和数学的各个领域，这些领域在过去经历了独立（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，而数学概念可能产生新的经济见解。更一般地说，项目旨在实现的相互促进本身是重要的，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强调对两个领域的博士生和/或初级研究人员的交叉培训。最后，所考虑的经济问题引发了重要的政策问题，同时也带来了新的数学挑战。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特别竞赛的一部分得到支持的。

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).