Title:

General equilibrium theory in infinite dimensions : an application of Fredholm Index Theory

This thesis deals with generic determinacy and the number of equilibria for infinite dimensional economies. Our work could be seen as an infinitedimensional analogue of Dierker and Dierker (1972) by characterising equilibria of an economy as a zero of the aggregate excess demand and studying its transversality. In this case, we can use extensions of the SardSmale theorem. Assuming separable utilities we give a new proof of generic determinacy of equilibria. We define regular price systems in this setting and show that an economy is regular if and only if its associated excess demand function only has regular equilibrium prices. We also define the infinite equilibrium manifold à la Balasko and show that it has the structure of a Banach manifold. We provide conditions that guarantee global uniqueness of equilibria for smooth infinite economies. We do this by introducing to the economic literature the notion of ZRothe vector fields that will allow us to construct an index theorem à la Dierker (1972); this shows that the number of equilibria is odd and in particular gives a new proof of existence. Extending the finite dimensional results of Balasko (1988), we characterise the equilibrium manifold as a covering space of the set of economies and we study global conditions under which the natural projection map is a diffeomorphism. We finally study the effects that critical equilibria have on the global invertibility of the natural projection map.
