  We present a general analysis of multidimensional matching problems with
transferable utility, paying particular attention to the case in which the
dimensions of heterogeneity on the two sides of the market are unequal. A
particular emphasis is put on problems where agents on one side of the market
are multidimensional and agents on the other side are uni-dimensional, we
describe a general approach to solve such problems. Lastly, we analyze several
examples, including an hedonic model with differentiated products, a marriage
market model where wives are differentiated in income and fertility, and a
competitive variation of the Rochet-Chon\'e problem. In the latter example, we
show that the bunching phenomena, observed by Rochet and Chon\'e in the
monopoly context, do not occur in the competitive context

Economics;Analysis of PDEs;Optimization and Control