Manifolds of Preferences and Equilibria

This work is concerned with the study of local and global properties of spaces of preferences and their applications to general equilibrium, utility and demand analysis. Differential topology and global analysis tools are used to study the mathematical aspects of these problems, adding further results and techniques to an approach introduced in mathematical economics by Gerard Debreu in 1970. Spaces of smooth, not necessarily convex or increasing, preferences are shown here to be representable as differentiable Hilbert manifolds. These structures on spaces of preferences are then used to extend major results on the regularity of equilibria, based on Sard's theorem and Abraham-Thom transversality theory, to economies where agents are described by their preferences and endowments. Certain topological properties of these manifolds of preferences are studied. Applications are also given to the study of the utilities and the demands of the agents in relation to the underlying preferences. The results point to extensions and indicate new branches of research of both economic and mathematical interest; these are written as suggestions or conjectures in the text.