On the orientability of the asset equilibrium manifold

Abstract

This paper addresses partly an open question raised in the Handbook of Mathematical economics about the orientability of the pseudo-equilibrium manifold in the basic two-period General Equilibrium with Incomplete markets (GEI) model. For a broad class of explicit asset structures, it is proved that the asset equilibrium space is an orientable manifold if S − J is even. This implies, under the same conditions, the orientability of the pseudo-equilibrium manifold. By a standard homotopy argument, it also entails the index theorem for S − J even. A particular case is Momi's result, i.e the index theorem for generic endowments and real asset structures if S − J is even.JEL classification : D52

Failure of the index theorem in an incomplete market economy

“…and consider the functionK 1 Bich (2006) and Predtetchinski (2006) for detailed investigations of the asset (pseudo-) equilibrium manifold. We can focus on the functionK 1 F |V × [0, 1] because of the next lemma whose proof is given in the Appendix.…”

“…A natural question to follow is whether or not the index theorem holds for the incomplete market model. Recent works by Momi (2003), Bich (2006), and Predtetchinski (2006) gave a partial answer to this question: the index theorem typically holds when the degree of incompleteness, which is defined as the difference between the number of states and the number of securities, is an even number. The purpose of this paper is to study the case where the degree of incompleteness is an odd number.…”