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Choo-Siow (2006) proposed a model for the marriage market which allows forrandom identically distributed noise in the preferences of each of theparticipants. The randomness is McFadden-type, which permits an explicitresolution of the equilibrium preference probabilities. The purpose of thisnote is to prove uniqueness of the resulting equilibrium marriage distribution,and find a representation of it in closed form. This allows us to derive smoothdependence of this distribution on exogenous preference and populationparameters, and establish sign, symmetry, and size of the various substitutioneffects, facilitating comparative statics. For example, we show that anincrease in the population of men of any given type in this model leads to anincrease in single men of each type, and a decrease in single women of eachtype. We show that an increase in the number of men of a given type increasesthe equilibrium transfer paid by such men to their spouses, and also increasesthe percentage of men of that type who choose to remain unmarried. Theverification of such properties helps to substantiate the validity of themodel. Moreover, we make unexpected predictions which could be tested: thepercentage change of type $i$ unmarrieds with respect to fluctuations in thetotal number of type $j$ men or women turns out to form a symmetricpositive-definite matrix $r_{ij}=r_{ji}$ in this model, and thus to satisfybounds like $|r_{ij}| \le (r_{ii}r_{jj})^{1/2}$. Along the way, we give a newproof for the existence of an equilibrium, based on a strictly convexvariational principle and a simple estimate, rather than a fixed point theorem.Fixed point approaches to the existence part of our result have been exploredby others \cite{CSS} \cite{Dag} \cite{Fox}, but are much more complicated andyield neither uniqueness, nor comparative statics, nor an explicitrepresentation of the solution.

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