Excess demand approach with non-convexity and discontinuity: a generalization of the Gale–Nikaido–Kuhn–Debreu lemma

This paper provides a three-fold generalization of the Gale–Nikaido–Kuhn–Debreu lemma, a fundamental result for classical Walrasian general equilibrium theory. It weakens the (i) convexity assumption on the given (excess demand) correspondence, and (ii) the continuity assumption by synthesizing both the majorization and inclusion approaches, and uses these results to derive a (iii) theorem on the existence of a Walrasian equilibrium in an exchange economy with externalities, and non-ordered, non-convex, price-dependent preferences under a weaker continuity assumption that is assumed in the literature. It also highlights a trade-off between the weakened continuity and convexity assumptions.