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We show that, with indivisible goods, the existence of competitiveequilibrium fundamentally depends on agents' substitution effects, not theirincome effects. Our Equilibrium Existence Duality allows us to transportresults on the existence of competitive equilibrium from settings withtransferable utility to settings with income effects. One consequence is thatnet substitutability---which is a strictly weaker condition than grosssubstitutability---is sufficient for the existence of competitive equilibrium.We also extend the ``demand types'' classification of valuations to settingswith income effects and give necessary and sufficient conditions for a patternof substitution effects to guarantee the existence of competitive equilibrium.

The Equilibrium Existence Duality: Equilibrium with Indivisibilities & Income Effects