Comparative static and computational methods for an empirical one-to-one transferable utility matching model

I show that the equilibrium distribution of matches associated with the empirical transferable utility (TU) one-to-one matching model introduced by Choo and Siow (2006a,b) 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. In the context of a single agent discrete choice problem, the assumption of utility maximization provides a tight link between the observed population distribution of choices and the unobserved population distribution of preferences (McFadden, 1974; Manski, 1975; Matzkin, 2007). In contrast the mapping from an observed distribution of matches between two sets of heterogenous agents, say men and women in a marriage market, and total match utility or surplus is less well-understood. In one-to-one matching problems two agents must agree to form a match. Rivalry is important, constraining choice: an individual’s utility maximizing match partner may be unavailable, herself preferring to match with someone else. Rivalry, a consequence of the two-sided aspect of the problem, makes the problem of inferring the distribution of match surplus from information of who matches with whom difficult (Fox, 2010; Graham, 2011; Echenique, Lee, Shum and Yenmez, 2013). I study a variant of the structural matching model introduced by Choo and Siow (2006,a,b) (henceforth the ‘CS model’).1 Abstractly the CS model is a two-sided model of discrete choice subject to a market clearing, or adding-up, restriction (cf., Graham, 2011). The general equilibrium nature of the CS model makes a complete understanding of its economic properties difficult. In his ∗Department of Economics, University of California Berkeley, 530 Evans Hall #3380, Berkeley, CA 947203888, e-mail: [email protected], web: http : //emlab.berkeley.edu/ bgraham/. I would like to thank Gary Chamberlain, Eugene Choo, Alfred Galichon, Michael Jansson, Konrad Menzel, Ariel Pakes, and numerous participants in the 2013 Advances in Econometrics Conference at Louisiana State University and a seminar at HarvardMIT for helpful discussions. A special thanks to Gary Chamberlain, Eugene Choo and Alfred Galichon for written feedback on an earlier draft. All the usual disclaimers apply. See Dagsvik (2000) for an earlier, and related, contribution.