Regular economies with non-ordered preferences

We consider exchange economies with non-ordered preferences and externalities. Under continuity and boundary conditions, we prove an index formula, which implies the existence of equilibria for all initial endowments, and the upper semi-continuity of the Walras correspondence. We then posit a differentiability assumption on the preferences. It allows us to prove that the equilibrium manifold is a nonempty differentiable sub-manifold of an Euclidean space. We define regular economies as the regular values of the natural projection. We show that the set of regular economies is open, dense and of full Lebesgue measure. From the index formula, one deduces that a regular economy has a finite odd number of equilibria, and, for each of them, the implicit function theorem implies that there exists a local differentiable selection. So, even with only non-ordered preferences and externalities, exchange economies have nice properties for equilibrium analysis.