  We study two-sided many-to-one matching markets with transferable utilities,
e.g., labor and rental housing markets, in which money can exchange hands
between agents, subject to distributional constraints on the set of feasible
allocations. In such markets, we establish the efficiency of equilibrium
arrangements, specified by an assignment and transfers between agents on the
two sides of the market, and study the conditions on the distributional
constraints and agent preferences under which equilibria exist and can be
computed efficiently. To this end, we first consider the setting when the
number of institutions (e.g., firms in a labor market) is one and show that
equilibrium arrangements exist irrespective of the nature of the constraint
structure or the agents' preferences. However, equilibrium arrangements may not
exist in markets with multiple institutions even when agents on each side have
linear (or additively separable) preferences over agents on the other side.
Thus, for markets with linear preferences, we study sufficient conditions on
the constraint structure that guarantee the existence of equilibria using
linear programming duality. Our linear programming approach not only
generalizes that of Shapley and Shubik (1971) in the one-to-one matching
setting to the many-to-one matching setting under distributional constraints
but also provides a method to compute market equilibria efficiently.
