Stable matching in a common generalization of the marriage and assignment models

In the theory of two-sided matching markets there are two well-known models: the marriage model (where no money is involved) and the assignment model (where payments are involved). Roth and Sotomayor (1990) asked for an explanation for the similarities in behavior between those two models. We address this question by introducing a common generalization that preserves the two important features: the existence of a stable outcome and the lattice property of the set of stable outcomes.