This paper studies a general model of matching with constraints. Observing that a stable matching typically does not exist, we focus on feasible, individually rational, and fair matchings. We characterize such matchings by fixed points of a certain function. Building on this result, we characterize the class of constraints on individual schools under which there exists a student-optimal fair matching, the matching that is the most preferred by every student among those satisfying the three desirable properties. We study the numerical relevance of our theory using data on government-organized daycare allocation.

In markets for indivisible resources such as workers and objects, subsidy and taxation for an agent may depend on the set of acquired resources and prices. This paper investigates how such transfer policies interfere with the substitutes condition, which is critical for market equilibrium existence and auction mechanism performance among other important issues. For environments where the condition holds in the absence of policy intervention, we investigate which transfer policies preserve the substitutes condition in various economically meaningful settings, establishing a series of characterisation theorems. For environments where the condition may fail without policy intervention, we examine how to use transfer policies to re-establish it, finding exactly when transfer policies based on scales are effective for that purpose. These results serve to inform policymakers, market designers, and market participants of how transfer policies may impact markets, so more informed decisions can be made.