# mta

# The mass transport approach to demand inversion in multinomial choice models

## Description:

Multinomial choice models constitute a fundamental toolbox of microeconomic analysis. Although this classification is a bit arbitrary, they usually divide into discrete choice models, in which the choice set is finite (e.g. a consuming choosing a model of car), and hedonic models, in which the choice set is continuous (e.g. a consumer choosing the quality of a wine). An important problem in these models is the problem of demand inversion, namely how to recover the payoffs associated with each alternative based on the corresponding market shares. We have developed a methodology called the “mass transport approach” to perform demand inversion in choice models using matching theory.

Multinomial choice models are usually thought of as conceptually distinct from matching models. The traditional wisdom is that matching models are “two-sided” (on the labor market, workers and firms choose each other), while demand models are “one-sided” (consumers choose yoghurts, but yoghurts don’t choose consumers). In work with Bonnet and Shum, we build on the findings of earlier papers with Salanié and with Chiong and Shum to show that this distinction has no bite, and that in fact, a model where consumers choose yoghurts is observationally equivalent to a (hypothetical) dual model where yoghurts choose consumers, or to a model where consumers “match” with yoghurts. At the heart of the “mass transport” approach to demand inversion lies our equivalence theorem: identifying the systematic payoffs in a multinomial choice model is equivalent to the determining a stable pair in a matching model. We use this reformulation to make use of matching theory in order to provide new theoretical results and new computational techniques in demand models. This finding gives rise to a novel class of efficient computational algorithms to invert multinomial choice models, that are based on matching algorithms. In ongoing work with Chernozhukov, Henry and Pass, we extend these methods to the case when the alternative are continuous, i.e. hedonic models.

See a brief description in my book, *Optimal transport methods in economics*, chap. 9.2.

## My co-authors:

Odran Bonnet, Khai Chiong, Victor Chernozhukov, Marc Henry, Brendan Pass, and Bernard Salanié.

## Presentation slides:

Available here.

## Code:

See arum routines of the TraME library.

References:

Alfred Galichon, and Bernard Salanié (2012). Cupid’s Invisible Hand: Social Surplus and Identification in Matching Models. Revision requested (2nd round), * Review of Economic Studies*. Available here.

Khai Chiong, Alfred Galichon, and Matt Shum (2016). Duality in dynamic discrete choice models.

**7(1), pp. 83—115. Available here.**

*Quantitative Economics*Odran Bonnet, Alfred Galichon, and Matt Shum (2017). Yogurts choose consumers? Identification of Random Utility Models via Two-Sided Matching. Available here.

Victor Chernozhukov, Alfred Galichon, Marc Henry, and Brendan Pass (2017). Single market nonparametric identification of multi-attribute hedonic equilibrium models. Available here.