## Final Report Summary - ECOMATCH (Economics of Matching Markets: Theoretical and Empirical Investigations)

The goal of the project is to build a tractable model aiming at explaining what makes people match on diverse markets such as the labor market, the marriage market, the market for certain services, etc. What explains matching patterns is two features: complementaries, which may be stronger or weaker according to both agents' types; and the relative scarcity of each agent. Broadly speaking, the project aims at measuring complementarities in order to be able to predict how people would match is the number of agents of each types were to change; this will allow to make predictions on marital outcome, inequalities, the level of the wages on the marriage market, etc.

Affinity matrix estimation, in the logit case and beyond – with Dupuy, the PI has developed a methodology to estimate matching surplus in matching models. Assuming transferable utility, bilinear surplus, and continuous logit heterogeneities, they are able to work out a particularly tractable method for estimating the matrix which determines the matching complementarities, which they call ''affinity matrix''. The resulting procedure is a convex optimization problem. This research has appeared in the Journal of Political Economy in 2014. A more particular version of this model where the marginal distributions are Gaussian, and all computations are explicit, has been investigated with Bojilov and is forthcoming in Economic Theory. With Ciscato and Gousse, a version of this model analyzing same-sex households has been developed. With Salanie, the PI has extended this methodology to the case when the interactions are not necessarily logit. The result estimation procedure is then flexible enough that the unobserved heterogeneity may be parameterized. The corresponding paper has passed a second round of revision at the Review of Economic Studies.

Quantile methods in higher dimensions – the PI has worked on the theory of optimal transportation, which are the mathematical foundations for one of the main class of matching models, which are models with transferable utility. One of the by-products of these models is to provide a useful notion of multivariate quantiles, which is flexible enough as it allows to obtain tools such as quantile regression, and a measure of centrality. Two papers on this topics have appeared in the Annals of Statistics. The first one is joint with Carlier and Chernozhukov, and deals with multivariate quantile regression; the second one is joint with Chernozhukov, Hallin and Henry, and focuses on multivariate centrality.

The Mass Transport Approach to demand inversion – the PI has worked on how to use tools from linear programming, optimal transportation, and more generally, matching theory, in order to perform identification in discrete choice models. With Shum and Chiong, the PI has written a paper published in Quantitative Economics demonstrating the power of techniques from convex analysis and linear programming in order to perform demand inversion in discrete choice models. These techniques are extremely promising and the PI has an ongoing project with Shum and Bonnet which considers an extension to nonadditive random utility models.

A framework for ITU matching – with Kominers and Weber, the PI has developed an empirical framework for matching models beyond the case of transferable utility. The framework is flexible enough to capture cases of fully transferable utility (the Becker model of marriage), non-transferable utility (the Gale-Shapley model of college admission), and a wide range of intermediate situations where the transfers are imperfectly transferable, including taxes, public goods, deadweight losses, etc.

Estimation and identification in hedonic models – the PI has developed estimation procedures for hedonic models. With Dupuy and Henry, he has written a short paper demonstrating how maximum entropy methods could apply to identify preferences and technology in hedonic models, which was published in Mathematical and Financial Economics. The PI has several projects which aim at taking this kind of model to data.

TraME software - In the winter of 2014, PI has initiated a software project which aims at developing algorithm for estimation and counterfactuals in matching models. This software, which is publicly and freely available under a GNU 2.0 license (https://github.com/TraME-Project/TraME) incorporates a number of the algorithms and estimation procedures which the PI has developed in his work on matching.

Affinity matrix estimation, in the logit case and beyond – with Dupuy, the PI has developed a methodology to estimate matching surplus in matching models. Assuming transferable utility, bilinear surplus, and continuous logit heterogeneities, they are able to work out a particularly tractable method for estimating the matrix which determines the matching complementarities, which they call ''affinity matrix''. The resulting procedure is a convex optimization problem. This research has appeared in the Journal of Political Economy in 2014. A more particular version of this model where the marginal distributions are Gaussian, and all computations are explicit, has been investigated with Bojilov and is forthcoming in Economic Theory. With Ciscato and Gousse, a version of this model analyzing same-sex households has been developed. With Salanie, the PI has extended this methodology to the case when the interactions are not necessarily logit. The result estimation procedure is then flexible enough that the unobserved heterogeneity may be parameterized. The corresponding paper has passed a second round of revision at the Review of Economic Studies.

Quantile methods in higher dimensions – the PI has worked on the theory of optimal transportation, which are the mathematical foundations for one of the main class of matching models, which are models with transferable utility. One of the by-products of these models is to provide a useful notion of multivariate quantiles, which is flexible enough as it allows to obtain tools such as quantile regression, and a measure of centrality. Two papers on this topics have appeared in the Annals of Statistics. The first one is joint with Carlier and Chernozhukov, and deals with multivariate quantile regression; the second one is joint with Chernozhukov, Hallin and Henry, and focuses on multivariate centrality.

The Mass Transport Approach to demand inversion – the PI has worked on how to use tools from linear programming, optimal transportation, and more generally, matching theory, in order to perform identification in discrete choice models. With Shum and Chiong, the PI has written a paper published in Quantitative Economics demonstrating the power of techniques from convex analysis and linear programming in order to perform demand inversion in discrete choice models. These techniques are extremely promising and the PI has an ongoing project with Shum and Bonnet which considers an extension to nonadditive random utility models.

A framework for ITU matching – with Kominers and Weber, the PI has developed an empirical framework for matching models beyond the case of transferable utility. The framework is flexible enough to capture cases of fully transferable utility (the Becker model of marriage), non-transferable utility (the Gale-Shapley model of college admission), and a wide range of intermediate situations where the transfers are imperfectly transferable, including taxes, public goods, deadweight losses, etc.

Estimation and identification in hedonic models – the PI has developed estimation procedures for hedonic models. With Dupuy and Henry, he has written a short paper demonstrating how maximum entropy methods could apply to identify preferences and technology in hedonic models, which was published in Mathematical and Financial Economics. The PI has several projects which aim at taking this kind of model to data.

TraME software - In the winter of 2014, PI has initiated a software project which aims at developing algorithm for estimation and counterfactuals in matching models. This software, which is publicly and freely available under a GNU 2.0 license (https://github.com/TraME-Project/TraME) incorporates a number of the algorithms and estimation procedures which the PI has developed in his work on matching.