Cupid’s Invisible Hand: Social Surplus and Identification in Matching Models

Abstract

We investigate a matching game with transferable utility when some of the characteristics of the players are unobservable to the analyst. We allow for a wide class of distributions of unobserved heterogeneity, subject only to a separability assumption that generalizes Choo and Siow (2006). We first show that the stable matching maximizes a social gain function that trades off two terms. The first term is simply the average surplus due to the observable characteristics; and the second one can be interpreted as a generalized entropy function that reflects the impact of the unobserved characteristics. We use this result to derive simple closed-form formulae that identify the joint surplus in every possible match and the equilibrium utilities of all participants, given any known distribution of unobserved heterogeneity. Moreover, we show that if transfers are observed, then the pre-transfer utilities of both partners are also identified. We conclude by discussing some empirical approaches suggested by these results for the study of marriage markets, hedonic prices, and the market for CEOs.

Latent indices in assortative matching models

“…Most of the recent literature on identification and estimation of matching games studies the transferable utility (TU) model, in which the equilibrium governs the matches as well as the surplus split between the agents with quasilinear preferences for money (Choo and Siow (2006), Sorensen (2007), Fox (2010a), Gordon and Knight (2009), Galichon and Salanie (2012), Chiappori, Salanié, and Weiss (2017), among others). The equilibrium transfers are such that no two unmatched agents can find a profitable transfer in which they would like to match with each other.…”

“…A branch of this literature, following the work of Choo and Siow (2006), proposes identification and estimation of a transferable utility model based on the assumption that each agent's utility depends only on observed characteristics and an unobserved taste shock drawn from a specified distribution. Using this assumption, the papers propose estimation and identification of group-specific surplus functions (Choo and Siow (2006), Galichon and Salanie (2012), Chiappori, Salanié, and Weiss (2017)). A different approach to identification in transferable utility models, due to Fox (2010a), is based on assuming that the structural unobservables are such that the probability of observing a particular match is higher if the total systematic, observable component of utility is larger than an alternative match.…”

“…Identification of the total surplus function in the transferable utility case, Φ(u v), or a cardinal measure of utilities in the nontransferable utility case will require additional assumptions. For example, one may simply interpret the latent index as a utility measure in the NTU case or assume a particular 7 For instance, Galichon and Salanie (2012) generalize the model by Choo and Siow (2006) and show that Φ x i z j is identified for a separable surplus function of the form Φ ij =Φ x i z j + ε i (z j ) + η j (x i ) with known distributions of ε i (z j ) and η j (x i ). These models therefore allow for unobserved preferences for observed characteristics, but do not allow for unobserved characteristics.…”

Competitive search equilibrium with multidimensional heterogeneity and two-sided ex-ante investments

“…In particular, Galichon and Salanié (2012), Dupuy andGalichon (2014), andFox (2010) use these techniques for the identi cation and estimation of frictionless matching models with transferable utility. This suggests that a similar methodology could be used to take competitive search models to the data at a high degree of disaggregation.…”

Entropy methods for identifying hedonic models

“…A revised version may be available directly from the author. Galichon and Salanié (2014), this paper proposes a new identification strategy for hedonic models in a single market. This methodology allows one to introduce heterogeneities in both consumers' and producers' attributes and to recover producers' profits and consumers' utilities based on the observation of production and consumption patterns and the set of hedonic prices.…”