  Becker (1973) presents a bilateral matching model in which scalar types
describe agents. For this framework, he establishes the conditions under which
positive sorting between agents' attributes is the unique market outcome.
Becker's celebrated sorting result has been applied to address many economic
questions. However, recent empirical studies in the fields of health,
household, and labor economics suggest that agents have multiple
outcome-relevant attributes. In this paper, I study a matching model with
multidimensional types. I offer multidimensional generalizations of concordance
and supermodularity to construct three multidimensional sorting patterns and
two classes of multidimensional complementarities. For each of these sorting
patterns, I identify the sufficient conditions which guarantee its optimality.
In practice, we observe sorting patterns between observed attributes that are
aggregated over unobserved characteristics. To reconcile theory with practice,
I establish the link between production complementarities and the aggregated
sorting patterns. Finally, I examine the relationship between agents' health
status and their spouses' education levels among U.S. households within the
framework for multidimensional matching markets. Preliminary analysis reveals a
weak positive association between agents' health status and their spouses'
education levels. This weak positive association is estimated to be a product
of three factors: (a) an attraction between better-educated individuals, (b) an
attraction between healthier individuals, and (c) a weak positive association
between agents' health status and their education levels. The attraction
channel suggests that the insurance risk associated with a two-person family
plan is higher than the aggregate risk associated with two individual policies.
