This paper gives a new proof of the index formula established by [momi, t., 2003. The index theorem for a gei economy when the degree of incompleteness is even. Journal of mathematical economics 39, 273–297] for an economy with incomplete asset markets where the difference between the number of states (ss) and the number of assets (jj) is an even number. The proof uses a single globally defined homotopy function on the asset pseudo-equilibrium manifold connecting the excess demand of a given economy to the individual excess demand of the unconstrained agent. We show that the asset pseudo-equilibrium manifold is orientable if the number s-js-j is even and deduce the index formula from the homotopy invariance theorem for the degree of a map.