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).
