This paper develops a theory of competitive equilibrium with indivisible goods based entirely on economic conditions on demand. The key idea is to analyze complementarity and substitutability between bundles of goods, rather than merely between goods themselves. This approach allows us to formulate sufficient and essentially necessary conditions for equilibrium existence—which unify settings with complements and settings with substitutes. Our analysis has implications for auction design.

Designing marketplaces in complex settings requires both novel economic theory and real-world engineering, often drawing upon ideas from fields such as computer science and operations research. In Discovering Prices: Auction Design in Markets with Complex Constraints, Milgrom (2017) explains the theory and design of the United States’ “incentive auction” that reallocated wireless spectrum licenses from television broadcasters to telecoms. Milgrom’s account teaches us how economic designers can grapple with complexity both in theory and in practice. Along the way, we come to understand several different types of complexity that can arise in marketplace design.

We show how frictions and continuous transfers jointly affect equilibria in a model of matching in trading networks. Our model incorporates distortionary frictions such as transaction taxes and commissions. When contracts are fully substitutable for firms, competitive equilibria exist and coincide with outcomes that satisfy a cooperative solution concept called trail stability . However, competitive equilibria are generally neither stable nor Pareto‐efficient.

Current refugee resettlement processes account for neither the preferences of refugees nor the priorities of hosting communities. We introduce a new framework for matching with multidimensional knapsack constraints that captures the (possibly multidimensional) sizes of refugee families and the capacities of communities. We propose four refugee resettlement mechanisms and two solution concepts that can be used in refugee resettlement matching under various institutional and informational constraints. Our theoretical results and simulations using refugee resettlement data suggest that preference-based matching mechanisms can improve match efficiency, respect priorities of communities, and incentivize refugees to report where they would prefer to settle.

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. Further applications give new existence results beyond the case of (net) substitutes. Our results have implications for auction design.