Abstract
We study the assignment of discrete resources in a general model encompassing a wide range of applied environments, such as school choice, course allocation, and refugee resettlement. We allow single-unit and general multi-unit demands and any linear constraints. We prove the Second Welfare Theorem for these environments and a strong version of the First Welfare Theorem. In this way, we establish an equivalence between strong efficiency and decentralization through prices in discrete environments. Showing that all strongly efficient outcomes can be implemented through pseudomarkets, we provide a foundation for using pseudomarkets in market design.
Highlights
Efficiency is the key objective in assignment of discrete resources, or bundles of resources, in environments such as school choice, course assignment, and refugee resettlement
The two main questions the present paper addresses are: how flexible is the pseudomarket approach? in particular, can all efficient assignments be implemented via pseudomarkets? By answering these two questions—and establishing a positive answer to the second one—we provide a foundation for the market design literature’s focus on pseudomarkets: in market design contexts, our characterization of efficient assignments allows one to restrict attention to pseudomarkets at least in settings, such as large markets, where pseudomarket price mechanisms are incentive compatible.[3]
We prove that strong efficiency is sufficient and necessary for the Second Welfare Theorem, that is we prove the analogue of the First Welfare Theorem for strong efficiency: every Walrasian equilibrium is efficient in the strong sense.[7]
Summary
Efficiency is the key objective in assignment of discrete resources, or bundles of resources, in environments such as school choice, course assignment, and refugee resettlement. Our paper contributes to the literatures on constraints in market design—e.g., Budish, Che, Kojima, and Milgrom (2013) and He, Miralles, Pycia, and Yan (2018)—and on multi-unit assignment—e.g., Sonmez and Unver (2010), Budish (2011), and Budish and Cantillon (2012)—that extended the idea of using token money to allocate objects beyond the canonical Hylland and Zekchauser setting.[12] Our Second Welfare Theorem is complementary to these papers and provides a microfoundation for their focus on pseudomarkets; none of these earlier papers provided such a microfoundation. We improve upon the First Welfare Theorems established in these papers by showing that pseudomarket equilibria are Pareto efficient and strongly efficient, and our general multi-unit demand setting goes beyond the settings studied in these papers: our analysis allows arbitrary utility profiles over bundles of objects and arbitrary linear constraints. They proved a Second Welfare Theorem for the primitive equilibrium concept; when preferences are strictly monotone, their primitive equilibrium concept corresponds to the standard equilibrium concept; when specialized to our setting, this equilibrium concept becomes equivalent to the quasi-equilibrium discussed above.[15]
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