  We consider the theoretical properties of a model which encompasses
bi-partite matching under transferable utility on the one hand, and hedonic
pricing on the other. This framework is intimately connected to tripartite
matching problems (known as multi-marginal optimal transport problems in the
mathematical literature). We exploit this relationship in two ways; first, we
show that a known structural result from multi-marginal optimal transport can
be used to establish an upper bound on the dimension of the support of stable
matchings. Next, assuming the distribution of agents on one side of the market
is continuous, we identify a condition on their preferences that ensures purity
and uniqueness of the stable matching; this condition is a variant of a known
condition in the mathematical literature, which guarantees analogous properties
in the multi-marginal optimal transport problem. We exhibit several examples of
surplus functions for which our condition is satisfied, as well as some for
which it fails.
