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 equilibrium from transferable utility economies 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 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.