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.

中文翻译：

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均衡存在二元性：具有不可分割性和收入效应的均衡

我们表明，对于不可分割的商品，竞争均衡的存在从根本上取决于代理人的替代效应，而不是他们的收入效应。我们的均衡存在二元性允许我们将竞争均衡存在的结果从具有可转移效用的设置传递到具有收入效应的设置。一个结果是净替代性——严格来说比总替代性更弱的条件——足以存在竞争均衡。我们还将估值的“需求类型”分类扩展到具有收入效应的环境，并为替代效应模式提供必要和充分条件以保证竞争均衡的存在。