  We consider the matching with contracts framework of Hatfield and Milgrom
when one side (a firm or hospital) can make monetary transfers (offer wages) to
the other (a worker or doctor). In a standard model, monetary transfers are not
restricted. However, we assume that each hospital has a fixed budget; that is,
the total amount of wages allocated by each hospital to the doctors is
constrained. With this constraint, stable matchings may fail to exist and
checking for the existence is hard. To deal with the nonexistence, we focus on
near-feasible matchings that can exceed each hospital budget by a certain
amount, and We introduce a new concept of compatibility. We show that the
compatibility condition is a sufficient condition for the existence of a
near-feasible stable matching in the matching with contracts framework. Under a
slight restriction on hospitals' preferences, we provide mechanisms that
efficiently return a near-feasible stable matching with respect to the actual
amount of wages allocated by each hospital. By sacrificing strategy-proofness,
the best possible bound of budget excess is achieved.
