The existence of equilibrium when excess demand obeys the weak axiom

This paper gives a non-fixed point theoretic proof of equilibrium existence when the excess demand function of an exchange economy obeys the weak axiom. The proof is simple and geometrically intuitive, and it also permits a weakening of the continuity assumption on the excess demand function. (This paper is a more developed and (one hopes) improved version of an earlier working paper called An Elementary Equilibrium Existence Theorem, Working Paper W6 (2000), Nuffield College, Oxford.)