The Equilibrium Existence Duality: Equilibrium with Indivisibilities & Income Effects

TL;DR

This paper introduces an Equilibrium Existence Duality showing that, with indivisible goods, equilibrium existence hinges on substitution effects rather than income effects, enabling transfer of results across different utility settings.

Contribution

It establishes a duality framework that links equilibrium existence conditions between transferable utility and income effect settings, extending valuation classifications.

Findings

Net substitutability suffices for equilibrium existence.

The duality framework transfers results between utility settings.

Necessary and sufficient conditions for substitution effects are provided.

Abstract

We show that, with indivisible goods, the existence of competitive equilibrium fundamentally depends on agents' substitution effects, not their income effects. Our Equilibrium Existence Duality allows us to transport results on the existence of competitive equilibrium from settings with transferable utility to settings with income effects. One consequence is that net substitutability---which is a strictly weaker condition than gross substitutability---is sufficient for the existence of competitive equilibrium. We also extend the ``demand types'' classification of valuations to settings with income effects and give necessary and sufficient conditions for a pattern of substitution effects to guarantee the existence of competitive equilibrium.