Unique equilibria and substitution effects in a stochastic model of the marriage market

Choo-Siow (2006) propose a behavioral model for the marriage market which incorporates random identically distributed noise into the preferences of each of the participants — à la McFadden — to fit virtually any observed distribution of matches. We prove existence and uniqueness of the resulting equilibrium marriage distribution, and find a representation of it in closed form. This allows us to derive smooth dependence of this distribution on exogenous preference and population parameters, and establish sign, symmetry, and size of the various substitution effects, making comparative statics possible. For example, we show that an increase in the population of men of any given type in this model leads to an increase in unmarried men of each type, and a decrease in unmarried women of each type. We show that an increase in the number of men of a given type increases the equilibrium transfer paid by such men to their spouses, and also increases the percentage of men of that type who choose to remain unmarried. While the above trends may not seem surprising, the verification of such properties helps to substantiate the validity of the model. Moreover, we make unexpected predictions which could be tested: namely, the percentage change of type i unmarrieds with respect to fluctuations in the total number of type j men or women turns out to form a symmetric positive-definite matrix rij = rji in this model, and thus to satisfy bounds such as |rij | ≤ (riirjj) . Our existence result follows from a variational principle and a simple estimate, rather than a fixed point theorem. Fixed point approaches to the existence part of our result have been explored by others [6] [8] [12], but are much more complicated and yield neither uniqueness, nor comparative statics, nor an explicit representation of the solution.