Econometrics, Quantitative Economics, Data Science

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ucla-2018

Lecture series

Optimal transport methods in economics: an introduction

UCLA, April 16 and 20, 2018

Content

These lectures will provide an introduction to optimal transport and its applications in economics.

Schedule

Monday, April 16, 2018, 9am-11am
Friday, April 20, 2018, 9am-11am

Course material

The lecture slides are available before each lecture from the following github repository.

References

These lectures will be based on my text:
Galichon, A. (2016). Optimal transport methods in economics. Princeton.

Other references include:
∙ For mathematical foundations:
– [OTON] C. Villani, Optimal Transport: Old and New, AMS, 2008.
– [OTAM] F. Santambrogio, Optimal Transport for Applied Mathematicians, Birkhäuser, 2015.
∙ For an introduction with a fluid mechanics point of view:
– [TOT] C. Villani, Topics in Optimal Transportation, AMS, 2003.
∙ With a computational focus:
– [NOT] G. Peyré, M. Cuturi (2018). Numerical optimal transport, Arxiv.
∙ With a family economics focus:
– [MWT] P.-A. Chiappori. Matching with Transfers: The Economics of Love and Marriage, Princeton, 2017.

Outline

L1. The labor market as an optimal transport problem: efficiency and equilibrium
L2. Multivariate quantile methods using optimal transport

hausdorff-2018

Mini-course

Matching models with general transfers

Summer school “Optimal transport and economics,” Hausdorff Center, Bonn, July 23-27, 2018 (4h30)

Content

These lectures will deal with optimal and equilibrium transport, and applications to matching models in economics.
In order to introduce the Monge-Kantorovich theorem of optimal transport theory in lecture 1, we consider a stylized assignment problem. Assume that a central planner (say, a plant manager) needs to assign workers to machines in order to maximize total output. Workers vary by their individual characteristics, and machines come in various sorts, where the set of characteristics of workers and firms may be either discrete or continuous. The output of a worker assigned to a machine depends on both the worker’s and the machine’s characteristics, so some workers may be better with some machines, and worse with some others. The central planner’s problem, which is the optimal transport problem, consists of assigning workers to machines in a way such that the total output is maximized. It will predict the equilibrium wages and the assignment of workers to machine.
Lecture 2 will introduce additional heterogeneity in preferences, so that the surplus of a match is the sum of a deterministic and a random term. We shall show that this leads to a regularized optimal transport problem, with an additional regularization term which is an entropy in the case when random utility belong in the logit specification, but can be characterized much more generally as a generalized entropy beyond that case. We will discuss implications for identification, comparative statics and the estimation of these models. This model can be used to estimate the structural parameters of the matching market, i.e. workers’ productivity and job amenity.
In lecture 3, we shall discuss a far-reaching extension of this setting called equilibrium transport. The classical theory of optimal transport relies on the assumption that the utilities should be quasi linear in payments, that is, everybody has a valuation expressed in the same monetary unit, which can be transferred without losses. That assumption is, of course, very strong as various nonlinearities may arise in practice, from taxes for example. Removing this strong assumption requires moving beyond optimal transport theory, to “Equilibrium transport theory”, which is strongly connected with the theory of “prescribed Jacobians equations”. We will see that this is the right framework to unify collective models of the households with matching models, and we provide a key technical tool to handle these, the distance-to-frontier (DTF) function, and we will study in detail a regularized version of this problem.

Schedule

L1: Monday 11am-12:30pm
L2: Monday 2pm-3:30pm
L3: Wednesday 9am-10:30am

Course material

The lecture slides will be available before each lecture on the following github repository.

References

Galichon, A. (2016). Optimal transport methods in economics. Princeton.

Outline

L1. The labor market as an optimal transport problem: Monge-Kantorovich duality
L2. Introducing unobserved heterogeneity among agents: regularized optimal transport
L3. Introducing taxes: equilibrium transport

mec_equil

ECON-GA 3503

‘math+econ+code’ masterclass on competitive equilibrium

NYU Courant Insitute, May 21-26, 2018 (30 hours)

Description

This very intensive course, part of the ‘math+econ+code’ series, is focused on the computation of competitive equilibrium, which is at the core of surge pricing engines and allocation mechanisms. It will investigate diverse applications such as network congestion, surge pricing, and matching platforms. It provides a bridge between theory, empirics and computation and will introduce tools from economics, mathematical and computer science. Mathematical concepts (such as lattice programming, supermodularity, discrete convexity, Galois connections, etc.) will be taught on a needs basis while studying various economic models. The same is true of computational methods (such as tatonnement algorithms, asynchronous parallel computation, mathematical programming under equilibrium constraints, etc.). Hence there are no prerequisite other than the equivalent of a first-year graduate sequence in econ, applied mathematics or other quantitative disciplines.
The teaching format is somewhat unusual: the course will be taught over six consecutive days, with lectures in the morning and individual assignments in the afternoon. This course is very demanding from students, but the learning rewards are high. The morning lectures will alternate between 1 hour of theory followed by 1 hour of coding. Students are expected to write their own code, and the teaching staff will ensure that it is operational. This course is therefore closer to cooking lessons than to traditional lectures.
This NYU course has no equivalent in peer universities. An earlier course with a different focus but sharing the same format and philosophy was taught at NYU in January 2018; it was very popular and was attended by students from a number of other universities.

Aim of the course

• Provide the conceptual basis of competitive equilibrium with gross substitutes, along with various computational techniques (optimization problems, equilibrium problems). Show how asynchronous parallel computation is adapted for the computation of equilibrium. Applications to hedonic equilibrium, multinomial choice with peer effects, and congested traffic equilibrium on networks.
• Describe analytical methods to analyze demand systems with gross substitutes (Galois connections, lattice programming, monotone comparative statics) and use them to study properties of competitive equilibrium with gross substitutes. Describe the Kelso-Crawford-Hatfield-Milgrom algorithm. Application to stable matchings, and equilibrium models of taxation.
• Derive models of bundled demand and analyze them using notions of discrete convexity and polymatroids. Application to combinatorial auctions and bundled choice.

Teaching staff

A. Galichon (NYU Econ+Math): morning lectures
K. O’Hara (NYU Econ): afternoon presentations (coding)
Y. Sun (NYU Math): afternoon presentations (machine learning)
O. Ghelfi (NYU Econ): research assistantship

Course material

A Github repository containing the course material (lecture slides, datasets, code) will made available at the start of the course. Coding assignments will be uploaded on a separate repository located here.

Practical information

• Schedule: Mon 5/21 — Sat 5/26, 2018, 9am-1pm (morning); 2pm-3pm (afternoon). Location: Courant building (251 Mercer St), room 202.
• Credits: 2, assessed through participation in the coding assignments or a short final paper, at the student’s option.
• NYU students will need to register on Albert. Non-NYU students need to contact the lecturer: galichon@cims.nyu.edu.

Course material

Available before the lectures.

Outline

• Monday 5/21: competitive equilibrium with gross substitutes
• Tuesday 5/22: demand beyond quasi-linearity
• Wednesday 5/23: bundled demand
• Thursday 5/24: empirical models of demand
• Friday 5/25: empirical models of matching
• Saturday 5/26: equilibrium on networks

Synopsis

Part I: Tools
Day 1: Competitive equilibrium with gross substitutes (Monday, 4 hours)
Walrasian equilibrium and gross substitutes. Gradient descent, Newton method; coordinate update method. Isotone convergence and Tarski’s theorem. Parallel computation (synchronous and asynchronous). Fisher-Eisenberg-Gale markets. Hedonic models beyond the quasilinear case.
References:
• Ortega and Rheinboldt (1970). Iterative Solution of Nonlinear Equations in Several Variables. SIAM.
• Heckman, Matzkin, and Nesheim (2010). “Nonparametric identification and estimation of nonadditive hedonic models,” Econometrica.
• Gul and Stacchetti (1999). “Walrasian equilibrium with gross substitutes”. Journal of Economic Theory.
• Jain and Vazirani (2010). “Eisenberg–Gale markets: Algorithms and game-theoretic properties”. Games and Economic Behaviour.
• Cheung and Cole (2016). “A Unified Approach to Analyzing Asynchronous Coordinate Descent and Tatonnement”. Arxiv.

Day 2: Demand beyond quasi-linearity (Tuesday, 4 hours)
Lattices and supermodularity. Veinott’s strong set ordering and Topkis’ theorem; Milgrom-Shannon theorem. Z-maps, P-maps and M-maps. Galois connections and generalized convexity. Equilibrium transport. Models of matching models with imperfectly transferable utility. Stable matchings and Gale and Shapley’s algorithm.
References:
• Veinott (1989). Lattice programming. Unpublished lecture notes, Johns Hopkins University.
• Topkis (1998). Supermodularity and complementarity. Princeton.
• Milgrom and Shannon (1994). “Monotone comparative statics.” Econometrica.
• Rheinboldt (1974). Methods for solving systems of nonlinear equations. SIAM.
• Noeldeke and Samuelson (2018). The implementation duality. Econometrica.
• Kelso and Crawford (1981). Job Matching, Coalition Formation, and Gross Substitutes. Econometrica.
• Roth and Sotomayor (1992). Two-Sided Matching. A Study in Game-Theoretic Modeling and Analysis. Cambdidge.

Day 3: Bundled demand (Wednesday, 4 hours)
Discrete convexity. Lovasz extension; Polymatroids. Hatfield-Milgrom’s algorithm. Combinatorial auctions.
References:
• Fujishige (1991). Submodular functions and optimization. Elsevier.
• Vohra (2011). Mechanism design. A linear programming approach. Cambridge.
• Bikhchandani, Ostroy (2002). The Package Assignment Model”. JET.
• Hatfield and Milgrom (2005). Matching with contracts. AER.
• Danilov, Koshevoy, and Murota (2001). Discrete convexity and equilibria in economies with indivisible goods and money. Mathematical Social Sciences.

Part II: Models
Day 4: Empirical models of demand (Thursday, 4 hours)
Nonadditive random utility models. Strategic complements; supermodular games. Brock-Durlauf’s model of demand with peer effect. Mathematical programming with equilibrium constraints (MPEC).
References:
• Vives, X. (1990). “Nash Equilibrium with Strategic Complementarities,” JME.
• Milgrom and Roberts (1994). “Comparing Equilibria,” AER.
• Berry, Gandhi, Haile (2013). “Connected Substitutes and Invertibility of Demand,” Econometrica.
• Brock, Durlauf (2001). Discrete choice with social interactions. JPE.
• Dubé, Fox and Su (2012), “Improving the Numerical Performance of BLP Static and Dynamic Discrete Choice Random Coefficients Demand Estimation,” Econometrica.
• Bonnet, Galichon, O’Hara and Shum (2018). Yoghurts choose customers? Identification of random utility models via two-sided matching.

Day 5: Empirical models of matching (Friday, 4 hours)
Distance-to-frontier function, matching function equilibrium. Matching models with taxes. Matching with public consumption. Surge pricing.
References:
• Menzel (2015). Large Matching Markets as Two-Sided Demand Systems. Econometrica.
• Legros, Newman (2007). Beauty is a Beast, Frog is a Prince. Assortative Matching with Nontransferabilities. Econometrica.
• Galichon, Kominers, and Weber (2018). Costly Concessions: An Empirical Framework for Matching with Imperfectly Transferable Utility.

Day 6: Equilibrium on networks (Saturday, 4 hours)
Equilibrium on networks. Traffic congestion; Wardrop equilibrium. Braess’ paradox. Price of anarchy.
References:
• Roughgarden, Tardos (2002). How bad is selfish routing? Journal of the ACM.
• Roughgarden (2005). Selfish Routing and the Price of Anarchy. MIT.
• Bourlès, Bramoullé, and Perez‐Richet (2017). Altruism in networks. Econometrica.
• Wardrop (1952). “Some theoretical aspects of road traffic research”. Proc. Inst. Civ. Eng.
• Dafermos (1980). “Traffic Equilibrium and Variational Inequalities.” Transportation Science.
• Nagurney (1993). Network Economics: A Variational Inequality Approach. Kluwer.

mec

‘math+econ+code’ masterclasses

Alfred Galichon (NYU Econ+Math)

MEC logo

Description

An immersive learning experience
The ‘math+econ+code’ masterclasses are very intensive classes in which the students are immersed over six consecutive days on a topic at the interaction between mathematics, economics and computation. These classes are innovative in several respects: the condensed format, the mix of theoretical, computational and empirical components, and the emphasis on coding. The classes focus on the acquisition of an operational knowledge: throughout the week, students will learn the mathematical structures, the economic models, and how to code them in practice. Mathematical concepts and computational methods are introduced on a needs basis while studying various economic models, and thus there are no prerequisite other than the equivalent of a first-year graduate sequence in econ, applied mathematics or other quantitative disciplines.

A very active area of research
The intersection between economics, mathematics and computation is coming back as a major area of current research. There are at least two reasons for this. The first one is the emergence of online platforms, which act as central planners and need to solve complex computational problems such as matching service providers with customers, introducing potential dating partners, performing dynamic pricing tasks, etc. The second reason is that econometric methods have been cross-fertilized by novel techniques from machine learning, that heavily rely on computational tools.

“Closer to cooking lessons than to traditional lectures”
The teaching format of a ‘math+econ+code’ series is somewhat unusual: a class is typically taught over six consecutive days, with a lesson in the morning (alternating between theory blocks and coding blocks); a guest lecture in the early afternoon; followed by individual work on computational assignments in the last part of the afternoon. These classes are very demanding from students, but the learning rewards are also very high. Students are expected to write their own code, and the teaching staff will ensure that it is operational at the end of the day. In complete opposition to the trend of ‘massively open online courses’ (MOOCS), these classes place instead a particular emphasis on personal interactions between teaching staff and students. They are therefore closer to cooking lessons than to traditional lectures. Without equivalent elsewhere, these series are experiencing growing popularity and draw graduate students across various quantitative disciplines and universities.

Classes offered

Upcoming classes
‘math+econ+code’ masterclass on linear programming in economics (expected fall 2018).

‘math+econ+code’ masterclass on networks economics (expected spring 2019).

‘math+econ+code’ masterclass on machine learning in economics (expected fall 2019).

Past classes
‘math+econ+code’ masterclass on competitive equilibrium: Walrasian equilibrium with gross susbsitutes. NYU, May 21-26, 2018.

‘math+econ+code’ masterclass on optimization in economics: optimal transport, demand models and matching models. NYU, Jan 15-20, 2018.

Classes in preparation
‘math+econ+code’ masterclass on games and algorithms.

‘math+econ+code’ masterclass on Monte-Carlo methods in econometrics.

‘math+econ+code’ masterclass on cryptocurrencies.

Diffusion list

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microeconometrics2018S

KECD 2195

Advanced topics in microeconometrics: Matching Models and their Applications

Sciences Po, Economics Department, PhD Course Spring 2018

Course information

Instructor: Alfred Galichon.

Schedule: Mondays, 8am-10am, starting January 29, 2016.

Class meets: Jan 29, Feb 5,12,19, Mar 5(+),12,19(+),26, Apr 9,16,23,30.

Location: 28 SP, H103.

Course material: Available on this github repository.

Texts
The first part of the course will be based on my (optional) text:
[OTME] A. Galichon (2016). Optimal Transport Methods in Economics, Princeton University Press.

Other textbooks used for reference (although not required) are:
[TSM] A. Roth and M. Sotomayor. Two-Sided Matching A study in Game-Theoretic Modeling and Analysis, Monographs of the Econometrics Society, 1990.
[DCMS] K. Train. Discrete Choice Methods with Simulation. 2nd Edition. Cambridge University Press, 2009.
[TOT] C. Villani, Topics in Optimal transportation, AMS, 2003.
Course material
Available after each lecture on the class webpage at url http://alfredgalichon.com/microeconometrics2018s.
Description of the Course
This course provides the mathematical and computational tools needed for an operational knowledge of discrete choice models, matching models, and network flow models. A number of economic applications of these concepts will be discussed.
The first part of the course will introduce basic results around Optimal Transportation theory: the Monge-Kantorovich duality, the Optimal Assignment Problem, basic results in Linear Programming, and Convex Analysis. Those concepts will serve as building blocks in the sequel.
The second part will cover discrete choice models, from the classical theory to more recent advances. The classical Generalized Extreme Value (GEV) specification will be recalled, as well as Maximum Likelihood estimation in the parametric case. Comparative statics results will be derived using tools from Convex Analysis, and nonparametric identification will be worked out using Optimal Transport theory. Simulation methods will be covered. A computationally intensive application will be demonstrated.
The third part will be devoted to matching models with stochastic utility, starting with the Transferable Utility (TU) case which is then generalized to Imperfectly Transferable Utility (ITU) including Non-transferable Utility (NTU). Equilibrium computation in the general case will be worked out using techniques from General Equilibrium. The more specific, but empirically relevant logit case, will be efficiently addressed using more the specific techniques or Iterative Fitting. Various algorithms will be described and compared in practice. Moment Matching Estimation and Maximum Likelihood Estimation will be worked out and compared. Several applications, to Collective Models of Family Economics, and to Labor Markets with taxes, will be described.
Time permitting, the fourth and last part will provide an introduction to problems on networks. The basic tools to describe the topology on a network will be described: discrete differential operators, diffusions on networks, shortest paths on networks. The Optimal Transport problem on networks will be formulated, along with its extension to stochastic utility.

Textbooks
The first part of the course will be based on my textbook:
• [OTME] Optimal Transport Methods in Economics (Princeton University Press, in press), a draft of which is available at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2699381.
Other textbooks used for reference (although not required) are:
• [TSM] A. Roth and M. Sotomayor, Two-Sided Matching A study in Game-Theoretic Modeling and Analysis, Monographs of the Econometrics Society, 1990.
• [DCMS] Train, K.. Discrete Choice Methods with Simulation. 2nd Edition. Cambridge University Press, 2009.
• [TOT] C. Villani, Topics in Optimal transportation, AMS, 2003.
Organization of the Course
Part I: An introduction to Optimal Transport theory
L1. Monge-Kantorovich duality
• Primal and dual formulations
• The Monge-Kantorovich theorem
• Equilibrium and Optimality
Reference: [OTME] chapters 1 and 2.
Complements: [TOT], chapter 1.
L2. The optimal assignment problem
• Linear programming duality
• Purity, Stability
• Computation
Reference: [OTME], ch. 3, [TSM], Ch 8.
Complements: Shapley & Shubik (1972).
L3. The Becker model
• Copulas and comonotonicity
• Positive Assortative Matching
• The Wage Equation
Reference: [OTME], ch. 4. [TOT], Ch. 2.2
L4. Convex conjugacy
• Basics of convex analysis: Convex conjugates, Subdifferential, Fenchel-Young inequality
• Brenier’s theorem
Reference: [OTME], ch. 6. [TOT], ch. 2.1.

Part II: Discrete Choice models
L5. The logit model and its extensions
• The Logit model and its parametric estimation
• The Generalized Extreme Value (GEV) model
• The Daly-Zachary-Williams theorem
Reference: [DCMS], ch. 2-4, Anderson, de Palma & Thisse, Ch. 3, Carlier (2010).
L6. Identification of discrete choice models
• Reformulation as an Optimal Transport problem
• Consequences on the structure of the identifed set
• The Random Scalar Coefficient Model
• Incorporating peer effects
Reference: Hotz and Miller (1993), Chiong et al. (2014), Galichon and Salanie (2015).
L7. Simulation methods
• Simulation methods for parametric estimation
• Probit and the GHK simulator
• Simulation methods for nonparametric estimation
Reference: [DCMS], ch. 5 and 9, Chiong et al. (2014).

Part III: Matching models
L8. Models with transferable utility
• The TU-logit model of Choo and Siow
• Beyond Logit: general heterogeneity
• Simulation methods
• Moment matching estimation; Maximum Likelihood Estimation
Reference: Choo and Siow (2006), Galichon and Salanie (2015).
L9. Estimation of complementarity
• Index models
• Affinity matrix estimation
• Application: marital preference estimation
Reference: Chiappori, Oreffice and Quintana-Domeque (2012), Dupuy and Galichon (2014).
L10. Models with imperfectly transferable utility
• Equilibrium: Existence and Uniqueness
• The ITU-logit model
• Computation
• Maximum Likelihood Estimation
Reference: Galichon, Kominers and Weber (2015).
L11. Models with non-transferable utility
• Models with no idiosyncratic utility shocks
• Models with idiosyncratic utility shocks
Reference: Dagsvik (2000), Menzel (2015), Galichon and Hsieh (2015).

Part IV: Network models
L12. Optimal flow problems
• Basic concepts
• Min-cost flow problem
• Incorporating Stochastic Utility
Reference: [OTME], ch. 8. Koopmans (1949).
L13. Equilibrium flow problems
• Traffic equilibrium with congestion
• The Equilibrium Flow Problem.
Reference: Carlier (2010).
L14. Hedonic models
• Hedonic Equilibrium: definition and existence
• Estimation
Chiappori, McCann & Nesheim (2010), Ekeland, Heckman & Nesheim (2004), Dupuy, Galichon & Henry (2014).

Bibliography
• Anderson, de Palma, and Thisse (1992). Discrete Choice Theory of Product Differentiation. MIT Press.
• Aurenhammer, F. (1987). “Power diagrams: properties, algorithms and applications,” SIAM Journal on Computing.
• Becker, G. (1973). “A theory of marriage, part I,” Journal of Political Economy.
• Carlier, G. (2010). Lecture notes on “Optimal Transportation and Economic Applications.”.
• Chiong, K, Galichon, A., Shum, M. “Duality in dynamic discrete choice models.” Quantitative Economics, forthcoming.
• Choo, E., and Siow, A. (2006). “Who Marries Whom and Why,” Journal of Political Economy.
• Chiappori, P.-A., McCann, R., and Nesheim, L. (2010). “Hedonic price equilibria, stable matching, and optimal transport: equivalence, topology, and uniqueness,” Economic Theory.
• Pierre-André Chiappori, Sonia Oreffice and Climent Quintana-Domeque, C. (2012). “Fatter Attraction: Anthropometric and Socioeconomic Matching on the Marriage Market,” Journal of Political Economy 120, No. 4, pp. 659-695.
• Dagsvik, J. (2000) “Aggregation in matching markets,” International Economic Review 41, 27-57.
• Dupuy, A., and Galichon, A. (2014). “Personality traits and the marriage market,” Journal of Political Economy.
• Dupuy, A., Galichon, A. and Henry, M. (2014). “Entropy Methods for Identifying Hedonic Models,” Mathematics and Financial Economics.
• Ekeland, I., J. Heckman, and L. Nesheim (2004): “Identification and estimation of hedonic models,” Journal of Political Economy.
• Galichon, A. (2016). Optimal Transport Methods for Economics. Princeton University Press, in press.
• Galichon, A., Hsieh, Y.-W. (2015). “Love and Chance: Equilibrium and Identification in a Large NTU matching markets with stochastic choice”.
• Galichon, A., Kominers, S., and Weber, S. (2015). Costly Concessions: An Empirical Framework for Matching with Imperfectly Transferable Utility.
• Galichon, A., and Salanié, B. (2014). “Cupid’s Invisible Hand: Social Surplus and Identification in Matching Models”. Working paper.
• Heckman, J., R. Matzkin, and L. Nesheim (2010). “Nonparametric identification and estimation of nonadditive hedonic models,” Econometrica.
• Hotz, V.J. and Miller, R.A. (1993). “Conditional Choice Probabilities and the Estimation of Dynamic Models”. Review of Economic Studies 60, No. 3 , pp. 497-529.
• Koopmans, T. C. (1949), “Optimum utilization of the transportation system”. Econometrica.
• Menzel, K. (2015). Large Matching Markets as Two-Sided Demand Systems. Econometrica 83 (3), pages 897–941.
• Roth, A., and Sotomayor, M. (1990). Two-Sided Matching A study in Game-Theoretic Modeling and Analysis.
• Shapley, L. and Shubik, M. (1972) “The assignment game I: the core”. International Journal of Game Theory.
• Train, K. (2009). Discrete Choice Methods with Simulation. Cambridge University Press.
• Villani, C. (2003). Topics in Optimal transportation. Lecture Notes in Mathematics, AMS.
• Vohra, R. (2011). Mechanism Design. A Linear Programming Approach. Cambridge University Press.

equilibrium-transport

Equilibrium transport

Description:

The classical theory of optimal transport relies on a very strong basic assumption: that the utilities should be quasi linear in payments, that is, everybody has a valuation expressed in the same monetary unit, which can be transferred without losses. In that case, if the firm pays the worker an extra dollar, the utility of the firm is decreased by one dollar, and the utility of the worker is increased by one dollar. That assumption is, of course, very strong as various nonlinearities may arise in practice; these might be induced by taxes, by regulations such as price caps, by risk aversion, or by other various inefficiencies. Removing this strong assumption requires moving beyond optimal transport theory, and moving into what I call “Equilibrium transport theory”, although this terminology is not standard; economists prefer “matching with imperfectly transferable utility”, and mathematicians usually refer to “prescribed Jacobians equations”. The problem is intrinsically an equilibrium problem, as opposed to an optimization problem; in fact, it has a natural formulation in terms of a nonlinear complementarity problem (NCP). Because this is no longer a linear programming problem, a large share of the insights of optimal transport — in particular, duality theory and everything that relates to optimization — no longer applies. However, an equally large part — in particular, the lattice structure and everything that relates to isotonicity — still applies. In work with Kominers and Weber, we have shown that this is the right framework to unify collective models of the households with matching models, and we provide a key technical tool to handle these, the distance-to-frontier (DTF) function; we also provide algorithms of the Jacobi type to compute an equilibrium in a regularized version of these models. In work with Hsieh, we investigate the case when there are no profitable transfers whatsoever; in this case, it may be necessary for either side of the market to destroy utility fully inefficiently but only for the purposes of sustaining a decentralized allocation. In work with Dupuy, Jaffe and Kominers, we analyse a model of matching with taxes. In the case of a linear tax, the models reformulates as an optimal transport model, which is no longer the case under nonlinear taxes.
A brief description of the equilibrium transport problem can be found in the concluding chapter of my book, Optimal transport methods in economics, chap. 10.4.

 

My co-authors:

Arnaud Dupuy, Yu-Wei Hsieh, Sonia Jaffe, Scott Kominers, and Simon Weber.

 

Presentation slides:

Presentation slides can be found here.

 

Code:

See transfers routines of the TraME library.

 

References:

Alfred Galichon, Scott Kominers, and Simon Weber (2015). The Nonlinear Bernstein-Schrodinger Equation in Economics. Proceedings of the Second Conference “Geometric Science of Information”, F. Nielsen and F. Barbaresco, eds. Springer Lecture Notes in Computer Sciences 9389, pp. 51-59. Available here.
Alfred Galichon, Scott Kominers, and Simon Weber (2017). Costly Concessions: An Empirical Framework for Matching with Imperfectly Transferable Utility. Revision requested (2nd round), Journal of Political Economy. Available here.
Alfred Galichon, and Yu-Wei Hsieh (2017). A theory of decentralized matching markets without transfers, with an application to surge pricing. Under review. Available here.
Arnaud Dupuy, Alfred Galichon, Sonia Jaffe, and Scott Kominers (2017). Taxation in matching markets. Available here.

martingale-transport

Optimal martingale transport

Description:

Optimal transport theory has important applications in finance, more specifically in option pricing theory. Financial derivatives may depend on several underlying assets; this is the case of spread options, for instance, or of basket options. The standard Black-Scholes-Merton theory of option pricing says that if there is a liquid market of vanilla options on a single underlying, then the risk-neutral distribution of the underlying can be recovered from the option prices; and we can therefore obtain a unique price associated with any more complicated single-underlying option. However, in the case of an option on two underlying, the market prices on the single-name options do not imply the joint distribution of two such assets, and one can then define no-arbitrage bounds, which corresponds to the cheapest and most expensive prices of the option that is consistent with the market. These bounds formulate as a Monge-Kantorovich problem, and the dual problem ensures that they correspond to the most expensive sub-replicating (lower bound) and the cheapest super-replicating portfolio (upper bound).
In a number of cases, the two underlying quantities are not the value of two assets at the same date in time, but the price of the same asset at two different dates in the future. There is then an important further restriction on the joint distribution of these assets: they should be the margins of a martingale. Computing the bounds of the option prices leads then to a variant of Monge-Kantorovich theory, where one looks the optimal coupling that is a martingale. This further constraint yields a supplementary term in the dual formulation, which has an interesting interpretation in terms of sub/super-replicating portfolio: the portfolio is not only made of calls and puts at the two maturities (static hedging), but also allows for rebalancing at the earlier maturity, allowing for dynamic hedging.
Moving beyond the static problem, there are interesting dynamic formulation of the problem. In particular, one may consider among the set of semi-martingales that start at a given distribution and end up at a given distribution, those who minimize a the time integral of the expectation of a Lagrangian that depends on the drift and diffusions parameters. This nicely extends the Benamou-Brenier dynamic formulation of optimal transport, and can provide interesting insights on particular solutions to the Skorohod embedding problem.

 

My co-authors:

Guillaume Carlier, Pierre Henry-Labordere, Nizar Touzi.

 

Presentation slides:

Presentation slides can be found here.

 

References:

Guillaume Carlier, and Alfred Galichon (2012). Exponential convergence for a convexifying equation (2012). Control, Optimization and Calculus of Variations 18(3), pp. 611–620. Available here.
Pierre Henry-Labordere, Alfred Galichon, and Nizar Touzi (2014). A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options. Annals of Applied Probability 24 (1), pp. 312-336. Available here.

affinity

Affinity estimation

Description:

Affinity estimation is a general method for estimating matching models. Economic agents match because the output they generate together is often higher than the sum of the outputs they would generate individually. This occurs in family economics (couple formation), labour economics (employment), industrial organizations (industrial alliances), finance (mergers), international trade (trade flows), school choice (class formation), among other fields. Complementarities are crucial: the economic approach to marriage, for instance, assumes that two individual decide to match only if the utility they get together if they team up is greater than the utility they get staying single. Matching patterns, i.e. who matches with whom, will be determined by (i) the strength of complementarities, and (ii) by the relative scarcity of a given characteristics on which matching occurs.
The empirical estimation of matching models has had a distinguished tradition in econometrics since the 1950’s with early work by Tinbergen and others, but was revived in the mid­-2000s in the field of family economics by a landmark paper by Choo and Siow on the identification of the “gains from marriage” in a matching model with transferable utility and logit-type heterogeneities. In somewhat general terms, the question is: given the observation of matching patterns (frequency of matches between each pair of types) and sometimes supplementary information such as observations on the surplus generated, or on the transfers between partners, what surplus functions are compatible with the observations?
Choo and Siow’s paper opened up a avenue of possibilities for the estimation of matching models, which we have pursued relentlessly. In initial work with Salanié, we showed that Choo and Siow’s framework can be formulated as a regularized optimal transport problem and we have provided a general estimation framework for the estimation of matching models with any separable heterogeneity structure using a moments-based procedure. With Dupuy, we have introduced affinity estimation which offers a tractable parameterization of the matching surplus function, in a setting which is flexible enough to accommodate both discrete and continuous observable characteristics. Affinity estimation consists of taking a bi-linear parameterization of the surplus function such that the terms of the matrix of complementarities (“affinity matrix”) indicate the amount of attraction (positive or negative) between a pair of characteristics of two agents to be matched. We have shown that the equilibrium conditions in this model can be expressed a gravity equation, so that one can use an approach based on Poisson pseudo-maximum likelihood (PPML) with two-way fixed effects, as is done in the trade literature. This allows us to estimate the affinity matrix which indicates the characteristics on which the sorting primarily occurs. While the classical setting of affinity estimation is bipartite (i.e. the matched partners are drawn from separate populations: men and women, workers and firms, buyers and sellers), it can be extended to the unipartite setting, as we have shown in work with Ciscato and Goussé. This is of particular interest in family economics as it permits to use affinity estimation to analyze same-sex unions. In practice, one often encounters cases when agents’ individual characteristics are high dimensional. In ongoing work with Dupuy and Sun we address this problem by rank-regularization techniques coupled with a cross validation criterion.

 

My co-authors:

Edoardo Ciscato, Arnaud Dupuy, Marion Goussé, Bernard Salanié, and Yifei Sun.

 

Presentation slides:

Slides can be found here.

 

Code:

See models 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.
Arnaud Dupuy, and Alfred Galichon (2014). Personality traits and the marriage market. Journal of Political Economy 122 (6), pp. 1271-1319. Winner of the 2015 Edmond Malinvaud prize. Available here.
Arnaud Dupuy, and Alfred Galichon (2015). Canonical Correlation and Assortative Matching: A Remark. Annals of Economics and Statistics 119-120, pp. 375—383. Available here.
Edoardo Ciscato, Alfred Galichon, and Marion Goussé (2017). Like Attract Like? A Structural Comparison of Homogamy Across Same-Sex and Different-Sex Households. Revision requested, Journal of Political Economy. Available here.
Arnaud Dupuy, Alfred Galichon, and Yifei Sun (2017). Estimating matching affinity matrix under low-rank constraints. Available here.

vq

Vector quantiles

Description:

In dimension one, the quantile is the inverse of the cumulative distribution function. Quantiles are extremely useful objects because they fully characterize the distribution of an outcome, and they allow to provide directly a number of statistics of interest such as the median, the extremes, the deciles, etc. They allow to express maximal dependence between two random variables (comonotonicity) and are used in decision theory (rank-dependent expected utility); in finance (value-at-risk and Tail VaR); in microeconomic theory (efficient risksharing); in macroeconomics (inequality); in biometric (growth charts) among other disciplines. Quantile regression, pioneered by Roger Koenker, allows to model the dependence of a outcome with respect to a set of explanatory variables in a very flexible way, and has become extremely popular in econometrics.
The classical definition of quantiles based on the cumulative distribution function, however, does not lend itself well to a multivariate extension, and given the number of applications of the notion of quantiles, many authors have suggested various proposals.
We have proposed a novel definition of multivariate quantiles called “Vector quantiles” based on optimal transport. The idea is that instead of viewing the quantile map as the inverse of the cumulative distribution function, it is more fruitful to view it as the map that rescales a distribution of interest to the uniform distribution over [0,1] in the least possible distortive way, in the sense that the average squared distance between an outcome and its preimage by the map should be minimized. While equivalent to the classical definition in the univariate case, this definition lends itself to a natural multivariate generalization using Monge-Kantorovich theory. We thus define the multivariate quantile map as the one that attains the L2-Wasserstein distance, which is known in the optimal transport literature as Brenier’s map. With Carlier and Santambrogio, we gave a precise connection between vector quantiles and a celebrated earlier proposal by Rosenblatt. We have shown that many of the desirable properties and uses of univariate quantiles extend to the multivariate case if using our definition. With Ekeland and Henry, we have shown that this notion is a multivariate analog of the notion of Tail VaR used in finance. With Henry, we have shown that it allows to construct an extension of Yaari’s rank-dependent utility in decision theory, and have provided an axiomatization for it. With Carlier and Dana, we have shown that this notion extends Landsberger and Meilijson’s celebrated characterization of efficient risksharing arrangements. With Charpentier and Henry, we have shown the connection with Machina’s theory of local utility. With Carlier and Chernozhukov, we have shown that Koenker’s quantile regression can be naturally extended to the case when the dependent variable is multivariate if one adopts our definition of multivariate quantile. With Chernozhukov, Hallin and Henry we have defined empirical vector quantiles and have studied their consistency.
A number of challenges remains. Among these, an empirical process theory for multivariate quantiles (extending the univariate theory of the empirical quantile process) is the obvious next step, although this is highly non-trivial. Invariance issues are also interesting and challenging questions. Finally, while the link with other multivariate quantiles (such as Rosenblatt’s) is by now well understood, the link with other related concepts, such as Tukey’s halfspace depth, remains to be explored.
A brief description of vector quantiles and vector quantile regression can be found in my book, Optimal transport methods in economics, chap. 9.4-9.5.

 

My co-authors:

Guillaume Carlier, Arthur Charpentier, Victor Chernozhukov, Rose-Anne Dana, Ivar Ekeland, Marc Hallin, Marc Henry, and Filippo Santambrogio.

 

Presentation slides:

A presentation on vector quantile regression can be found here.

 

Code:

The code for vector quantile regression can be found in the following Github repository.

 

References:

Guillaume Carlier, Alfred Galichon, and Filippo Santambrogio (2010). From Knothe’s transport to Brenier’s map. SIAM Journal on Mathematical Analysis 41, Issue 6, pp. 2554-2576. Available here.
Ivar Ekeland, Alfred Galichon, and Marc Henry (2012). Comonotonic measures of multivariate risks. Mathematical Finance 22 (1), pp. 109-132. Available here.
Alfred Galichon and Marc Henry (2012). Dual theory of choice with multivariate risks. Journal of Economic Theory 147(4), pp. 1501–1516. Available here.
Guillaume Carlier, Rose-Anne Dana, and Alfred Galichon (2012). Pareto efficiency for the concave order and multivariate comonotonicity. Journal of Economic Theory 147(1), pp. 207–229. Available here.
Arthur Charpentier, Alfred Galichon, and Marc Henry (2016). Local utility and risk aversion. Mathematics of Operations Research 41(2), pp. 466—476.
Guillaume Carlier, Victor Chernozhukov, and Alfred Galichon (2016). Vector quantile regression: an optimal transport approach.Annals of Statistics 44 (3), pp. 1165–1192. Available here. Software available here.
Victor Chernozhukov, Alfred Galichon, Marc Hallin, and Marc Henry (2016). Monge-Kantorovich Depth, Quantiles, Ranks and Signs. Annals of Statistics. Available here.
Guillaume Carlier, Victor Chernozhukov, and Alfred Galichon (2017). Vector quantile regression beyond correct specification. Journal of Multivariate Analysis. Available here.

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, O’Hara 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, Keith O’Hara, 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. Quantitative Economics 7(1), pp. 83—115. Available here.
Odran Bonnet, Alfred Galichon, Keith O’Hara, 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.