Economics is fundamentally the study of how economic actors decide to organize their activities and to allocate their joint production while each actor is pursuing his own interests. Mathematical modeling of these interactions requires the use of sophisticated techniques from the mathematical programming literature; in particular, the growing field of mathematical programming with equilibrium constraints. Standard optimization theory and numerical software assumes that the solution satisfies a constraint qualification, whereas many problems in economics and engineering do not satisfy this assumption. Moreover, many problems have nonconvexities that cause problems for algorithms focused on finding local solutions. Therefore, new algorithms must be developed to solve the optimization problems arising in modern economic analysis. This study will develop new methods that are more robust than current techniques for solving these problems. To facilitate further research by economists and others, the project will implement the new methods in public domain software and circulate illustrative examples that will help teach users how to use the software.
The new mathematical techniques will have substantial value for the study of many important economic problems. The project will study problems in optimal taxation where the population is made up of individuals of widely varying abilities, education, age, and family status. Current analyses look at far simpler examples and must ignore real world complexity due to mathematical limitations. This study also will show how to use these techniques to address problems in the design of electricity markets, health insurance, and other complex economic institutions. The numerical algorithms developed in this study will allow many other researchers to examine a wide variety of problems that currently are impossible to study using existing methods. This award was supported as part of the fiscal year 2006 Mathematical Sciences priority area special competition on Mathematical Social and Behavioral Sciences (MSBS).
