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I show that the equilibrium distribution of matches associated with the empirical transferable utility one-to-one matching (TUM) model introduced by Choo and Siow (2006a, 2006b) corresponds to the fixed point of system of K + L nonlinear equations; with K and L respectively equal to the number of discrete types of women and men. I use this representation to derive new comparative static results, showing how the match distribution varies with match surplus and the marginal distributions of agent types.