Multidimensional matching∗ Pierre-Andre ́ Chiappori†, Robert McCann‡and Brendan

We present general mathematical methods used in the study 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. We then describe an approach to solve a new subclass of these problems: problems where agents on one side of the market are multidimen-sional and agents on the other side are uni-dimensional. Lastly, we analyze several examples, including an hedonic model with dif-ferentiated products, a marriage market model where wives are differentiated in income and fertility, and a competitive variation of the Rochet-Chone ́ problem. In the latter example, we show that the bunching phenomena, observed by Rochet and Chone ́ in the monopoly context, do not occur in the competitive context.