Econometrics, Quantitative Economics, Data Science

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equiprice

EQUIPRICE: Equilibrium methods for Resource Allocation and Dynamic Pricing

European Research Council consolidator grant (ERC-CoG) No. 866274, 2020-2025

Project description

This project seeks to build an innovative economic toolbox (ranging from modelling, computation, inference, and empirical applications) for the study of equilibrium models with gross substitutes, with applications to models of matching with or without transfers, trade flows on networks, multinomial choice models, as well as hedonic and dynamic pricing models. While under-emphasized in general equilibrium theory, equilibrium models with gross substitutes are very relevant to these problems as each of these problems can be recast as such.
Thus far, almost any tractable empirical model of these problems typically required making the strong assumption of quasi-linear utilities, leading to a predominance of models with transferable utility in applied work. The current project seeks to develop a new paradigm to move beyond the transferable utility framework to the imperfectly transferable utility one, where the agent’s utilities are no longer quasi-linear.
The mathematical structure of gross substitutes will replace the structure of convexity underlying in models with transferable utility.
To investigate this class of models, one builds a general framework embedding all the models described above, the “equilibrium flow problem.” The gross substitute property is properly generalized and properties (existence of an equilibrium, uniqueness, lattice structure) are derived. Computational algorithms that rely on gross substitutability are designed and implemented. The econometrics of the problem is addressed (estimation, inference, model selection). Applications to various fields such as labor economics, family economics, international trade, urban economics, industrial organization, etc. are investigated.
The project touches upon other disciplines. It will propose new ideas in applied mathematics, offer new algorithms of interest in computer science and machine learning, and provide new methods in other social sciences (like sociology, demography and geography).
This project is hosted by Sciences Po.

Seasonal events

Summer: Alfred’s ‘math+econ+code’ masterclass

Summer 2020: masterclass on equilibrium transport and matching models in economics.
Summer 2021: masterclass on linear programming in economics.
Summer 2022: masterclass on optimal transport methods in economics.
Summer 2023: masterclass on network economics: trade, social networks, causality.
Summer 2024: masterclass on algorithms for economics.

Fall: multidisciplinary retreat on open investigations

Fall 2020: Lattices, discrete convexity, and supermodularity
Fall 2021: Parallel computing
Fall 2022: Handling of large datasets
Fall 2023: Variational inequalities
Fall 2024: Lemke-Howson and Scarf lemma

Winter: guest lecture series

Winter 2021 (a): General equilibrium with gross substitutes
Winter 2021 (b): Nonlinear complementarity problems
Winter 2022 (a): New trends in empirical industrial organization
Winter 2022 (b): Generated prescribed Jacobian equations
Winter 2023 (a): Discrete convexity, polymatroids and submodularity
Winter 2023 (b): Recent developments in optimal transport
Winter 2024 (a): Minimax-regret estimation
Winter 2024 (b): Scarf’s algorithm and its posterity
Winter 2025 (a): Schrodinger-Bernstein systems
Winter 2025 (b): Asynchronous Parallel Computing

Spring: Paris research workshop

Spring 2021: Advances in computational economics.
Spring 2022: Universal gravity.
Spring 2023: Inference in models of matching with and without transferable utility.
Spring 2024: Optimal and equilibrium transport.
Spring 2025: Structural models of the labor market: inference, computation and model selection.

SOFTWARE

Project TraMEpy

The TraME software (Transportation Methods for Econometrics, http://www.trame-project.com/) is a collection of libraries for solving problems of equilibrium computation and estimation in consumer demand and matching frameworks via the Mass Transportation Approach. It will be completely revamped to accommodate for the equilibrium flow problem in its full generality (beyond bipartite networks), novel equilibrium algorithms added, enhanced HPC capabilities, and a new Python implementation.

Open positions

OPEN: One post-doctoral researcher (TBA)

The post-doctoral researcher will be expected to contribute to the intellectual advancement of the project, interact with the PhD students, and participate in the organization of the seasonal events described below. The post-doctoral position is envisioned for four years; however, depending on individual circumstances, it may also be two consecutive two-year positions.

OPEN: Two doctoral fellows (TBA)

One doctoral fellow will have a focus on computational economics. The other doctoral fellow will focus on empirical economics.

OPEN: research interns (TBA)

Research intern positions are project-specific and will be announced later.

Econ-coding challenge

To be announced lated.

PSE lectures

Short course

Optimal transport and economic applications

PSE summer school, June 22-26, 2020

Content

This short course is focused on optimal transport theory and matching models and their applications to economics, in various fields such as labor markets, economics of marriage, industrial organization, matching platforms, networks, and international trade. It will provide the crossed perspectives of theory, empirics and computation. A particular emphasis will be given on computation (R and Python). This course is partly based on Galichon’s 2016 monograph, Optimal Transport Methods in Economics. Princeton University Press.

Schedule

Wednesday, June 24, 9am-12:30pm
Thursday, June 25, 9am-12:30pm
Friday, June 26, 9am-11am

Course material

The lecture slides will be available before each lecture.

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

• intro to optimal transport (3h)
• multinomial choice (3h)
• matching models (3h)

mec_equil_archive_2018-05

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_optim_archive_2019_06

ECON-GA 3503

‘math+econ+code’ in Paris: A masterclass on optimal transport, choice and matching models

NYU Paris (57 bd St-Germain, 75005 Paris), June 17-21, 2019 (30 hours)

Instructor: A. Galichon (NYU Econ+Math). Email: alfred.galichon@nyu.edu.

** Note: Interested students must contact the instructor ahead of time.**

Description

This intensive course, part of the ‘math+econ+code’ series, is focused on models of demand, matching models, and optimal transport methods, with various applications pertaining to labor markets, economics of marriage, industrial organization, matching platforms, networks, and international trade, from the crossed perspectives of theory, empirics and computation. It will introduce tools from economic theory, mathematics, econometrics and computing, on a needs basis, without any particular prerequisite other than the equivalent of a first year graduate sequence in econ or in applied math.
Because it aims at providing a bridge between theory and practice, the teaching format is somewhat unusual: each teaching “block” will be made of 50 minutes of theory followed by 1 hour of coding, based on an empirical application related to the theory just seen. Students are expected to write their own code, and we will ensure that it is operational at the end of each block. This course is therefore closer to cooking lessons than to traditional lectures.
The course is open to graduate students in the fields of economics and applied mathematics, but also in other quantitative disciplines. Students need to bring a laptop with them to the lectures. The knowledge of a particular programming language is not required; students are however expected to have some experience with programming. The course can be taken for credit or as a registered auditor.
The lecturer is Alfred Galichon (professor of economics and of mathematics at NYU). Pauline Corblet (graduate economics student at Sciences Po) and Octavia Ghelfi and James Nesbit (graduate economics students at NYU) have helped prepare the course material. The course is partly based on Galichon’s book, Optimal Transport Methods in Economics.

Support from NSF grant DMS-1716489 is acknowledged.

Course material

Available on Github here.

Practical information

• Schedule: Monday to Friday, 8am-12noon and 2pm-4pm. Location: NYU Paris, 57 boulevard Saint-Germain, 75005 Paris, Room TBA.
• Credits: 2, assessed through a take-home exam or a short final paper, at the student’s option.
• A syllabus is available at http://alfredgalichon.com/mec_optim/.
• NYU Students need to register on Albert (code ECON-GA 3503). Non-NYU student can be allowed to audit the course. All students need to contact the instructor (alfred.galichon@nyu.edu) ahead of time.

Outline

• Monday: linear programming, dynamic programming, network flows
• Tuesday: optimal transport toolbox 1 (discrete, one-dimensional, semi-discrete cases)
• Wednesday: optimal transport toolbox 2 (continuous transport, convex analysis, entropic regularization)
• Thursday: static and dynamic multinomial choice
• Friday: statistical estimation of models of matching with transfers

Synopsis

SYNOPSIS
Part I: Tools
Day 1: linear programming (Monday)
Block 1. Basics of linear programming (morning 1st half)
• Theory: linear programming duality; complementary slackness; minimax formulation
• Coding: How to eat optimally? Dataset: Stigler’s original diet data (1945).
Block 2. Network flow problems (morning 2nd half)
• Theory: directed graphs and min-cost flow problem
• Coding: How to find the shortest path through a network? Dataset: Paris subway; New York City street network.
Block 3. Dynamic programming as linear programming (afternoon)
• Theory: Bellman’s equation; interpretation of duality; forward induction, backward induction
• Coding: When to repair mechanical engines? Dataset: Rust’s bus maintenance data (1994).

Day 2: optimal transport I (Tuesday)
Block 4. Discrete matching (morning 1st half)
• Theory: Shapley-Shubik duality; stability; decentralized equilibrium
• Coding: How to solve it? Dataset from Dupuy and Galichon (JPE 2014).
Block 5. Positive assortative matching (morning 2nd half)
• Theory: Becker’s model; compensating differentials; comonotonicity
• Coding: What is a CEO worth? Dataset: Gabaix-Landier’s (QJE 2008) CEO pay data.

Block 6. Hotelling’s characteristics model (afternoon)
• Theory: power diagrams, Aurenhammer’s method
• Coding: How to infer the unobservable quality of a car model? Dataset: Feenstra-Levinsohn (Restud 1994) car data.

Day 3: optimal transport II (Wednesday)
Block 7. Continuous multivariate matching (morning 1st half)
• Theory: Knott-Smith criterion; Brenier’s map; McCann’s theorem
• Applications: exercises.
Block 8. A short tutorial on convex analysis (morning 2nd half)
• Theory: convex duality; Fenchel’s inequality; subdifferentials and their inverses
• Application: exercises.
Block 9. Regularized optimal transport (afternoon)
• Theory: optimal transport with entropic regularization, and with other regularizations.
• Coding: coordinate descent and the IPFP algorithm.

Part II. Models
Day 4: models of static and dynamic multinomial choice (Thursday)
Block 10. Basics of static discrete choice (morning 1st half)
• Theory: Dary-Zachary-Williams theorem, generalized entropy of choice, the inversion theorem
• Coding: How to solve it? simulation methods; AR, SARS, and GHK. Dataset: Greene and Hensher (1997) data on choice of travel mode.
Block 11. Demand models, old and new (morning 2nd half)
• Theory: the GEV model; the random coefficient logit model and the pure characteristics models
• Coding: How to estimate demand for automobiles? Dataset: BLP.
Block 12. Dynamic discrete choice methods (afternoon)
• Theory: Rust’s model; estimation; normalization issues
• Coding: maintenance choice.

Day 5: empirical matching models, the quasilinear case (Friday)
Block 13. Separable models of matching (morning 1st half)
• Theory: matching with unobservable heterogeneity
• Coding: Did Roe vs. Wade decrease the value of marriage? Dataset: Choo and Siow (JPE 2006).
Block 14. The gravity equation (morning 2nd half)
• Theory: optimal transport and the gravity equation; generalized linear models and pseudo-Poisson maximum likelihood estimation
• Coding: How to forecast international trade flows? estimating the gravity equation based on WTO international trade data.
Block 15. High-dimensional matching models (afternoon)
• Theory: estimation of rank-constrained models
• Application: Does physical appearance have a price? matching on socioeconomic and anthropomorphic characteristics. Dataset: Chiappori, Oreffice and Quintana-Domeque’s (JPE 2012).

mec_optim_archive_2019-01

ECON-GA 3503.1410

‘math+econ+code’ masterclass on optimization in economics: optimal transport, demand models and matching models

NYU Courant Institute, January 14-18, 2019 (30 hours)

Instructor: A. Galichon (NYU Econ+Math)

Description

This intensive course, part of the ‘math+econ+code’ series, is focused on models of demand, matching models, and optimal transport methods, with various applications pertaining to labor markets, economics of marriage, industrial organization, matching platforms, networks, and international trade, from the crossed perspectives of theory, empirics and computation. It will introduce tools from economic theory, mathematics, econometrics and computing, on a needs basis, without any particular prerequisite other than the equivalent of a first year graduate sequence in econ or in applied math.
Because it aims at providing a bridge between theory and practice, the teaching format is somewhat unusual: each teaching “block” will be made of 50 minutes of theory followed by 1 hour of coding, based on an empirical application related to the theory just seen. Students are expected to write their own code, and we will ensure that it is operational at the end of each block. This course is therefore closer to cooking lessons than to traditional lectures.
The course, jointly offered by NYU Econ and the Courant Institute, is open to graduate students in the fields of economics and applied mathematics, but also in other quantitative disciplines. Students need to bring a laptop with them to the lectures. The knowledge of a particular programming language is not required; students are however expected to have some experience with programming. The course can be taken for credit or as a registered auditor.
The lecturer is Alfred Galichon (professor of economics and of mathematics at NYU). James Nesbit (graduate Economics student at NYU) helped prepare the course material. The course is partly based on Galichon’s book, Optimal Transport Methods in Economics.

Support from NSF grant DMS-1716489 is acknowledged.

Course material

Available on Github here.

Practical information

• Schedule: Mon 1/14 — Fri 1/18, 2019, 9am-1pm and 2pm-4pm. Location: WWH 101 in the Courant building (251 Mercer St)
• Credits: 2, assessed through a take-home exam or a short final paper, at the student’s option.
• A syllabus is available at http://alfredgalichon.com/matheconcode/.
• Students need to register on Albert (code ECON-GA 3503.1410). For more information please contact: galichon@cims.nyu.edu.

Outline

• Monday: linear programming, dynamic programming, network flows
• Tuesday: optimal transport toolbox
• Wednesday: convex analysis, nonlinear inverse problems, and multivariate quantiles
• Thursday: static and dynamic multinomial choice
• Friday: statistical estimation of models of matching with transfers

Synopsis

SYNOPSIS
Part I: Tools
Day 1: linear programming (Monday)
Block 1. Basics of linear programming (morning 1st half)
• Theory: linear programming duality; complementary slackness; minimax formulation
• Coding: How to eat optimally? Dataset: Stigler’s original diet data (1945).
Block 2. Network flow problems (morning 2nd half)
• Theory: directed graphs and min-cost flow problem
• Coding: How to find the shortest path through a network? Dataset: Paris subway; New York City street network.
Block 3. Dynamic programming as linear programming (afternoon)
• Theory: Bellman’s equation; interpretation of duality; forward induction, backward induction
• Coding: When to repair mechanical engines? Dataset: Rust’s bus maintenance data (1994).

Day 2: optimal transport I (Tuesday)
Block 4. Discrete matching (morning 1st half)
• Theory: Shapley-Shubik duality; stability; decentralized equilibrium
• Coding: How to solve it? Dataset from Dupuy and Galichon (JPE 2014).
Block 5. Positive assortative matching (morning 2nd half)
• Theory: Becker’s model; compensating differentials; comonotonicity
• Coding: What is a CEO worth? Dataset: Gabaix-Landier’s (QJE 2008) CEO pay data.

Block 6. Hotelling’s characteristics model (afternoon)
• Theory: power diagrams, Aurenhammer’s method
• Coding: How to infer the unobservable quality of a car model? Dataset: Feenstra-Levinsohn (Restud 1994) car data.

Day 3: optimal transport II (Wednesday)
Block 7. Continuous multivariate matching (morning 1st half)
• Theory: Knott-Smith criterion; Brenier’s map; McCann’s theorem
• Applications: exercises.
Block 8. A short tutorial on convex analysis (morning 2nd half)
• Theory: convex duality; Fenchel’s inequality; subdifferentials and their inverses
• Application: exercises.
Block 9. Regularized optimal transport (afternoon)
• Theory: optimal transport with entropic regularization, and with other regularizations.
• Coding: coordinate descent and the IPFP algorithm.

Part II. Models
Day 4: models of static and dynamic multinomial choice (Thursday)
Block 10. Basics of static discrete choice (morning 1st half)
• Theory: Dary-Zachary-Williams theorem, generalized entropy of choice, the inversion theorem
• Coding: How to solve it? simulation methods; AR, SARS, and GHK. Dataset: Greene and Hensher (1997) data on choice of travel mode.
Block 11. Demand models, old and new (morning 2nd half)
• Theory: the GEV model; the random coefficient logit model and the pure characteristics models
• Coding: How to estimate demand for automobiles? Dataset: BLP.
Block 12. Dynamic discrete choice methods (afternoon)
• Theory: Rust’s model; estimation; normalization issues
• Coding: maintenance choice.

Day 5: empirical matching models, the quasilinear case (Friday)
Block 13. Separable models of matching (morning 1st half)
• Theory: matching with unobservable heterogeneity
• Coding: Did Roe vs. Wade decrease the value of marriage? Dataset: Choo and Siow (JPE 2006).
Block 14. The gravity equation (morning 2nd half)
• Theory: optimal transport and the gravity equation; generalized linear models and pseudo-Poisson maximum likelihood estimation
• Coding: How to forecast international trade flows? estimating the gravity equation based on WTO international trade data.
Block 15. High-dimensional matching models (afternoon)
• Theory: estimation of rank-constrained models
• Application: Does physical appearance have a price? matching on socioeconomic and anthropomorphic characteristics. Dataset: Chiappori, Oreffice and Quintana-Domeque’s (JPE 2012).

mec_optim_archive_2018-01

ARCHIVE

Current version to be found here.

ECON-GA 3002.015 and MATH.GA 2840.002

‘math+econ+code’ masterclass on optimization in economics: optimal transport, demand models and matching models

NYU Courant Institute, January 15-20, 2018 (36 hours)

Instructors: A. Galichon (NYU Econ+Math), Keith O’Hara (NYU Econ) and Yifei Sun (NYU Math)

Description

This intensive course, part of the ‘math+econ+code’ series, is focused on models of demand, matching models, and optimal transport methods, with various applications pertaining to labor markets, economics of marriage, industrial organization, matching platforms, networks, and international trade, from the crossed perspectives of theory, empirics and computation. It will introduce tools from economic theory, mathematics, econometrics and computing, on a needs basis, without any particular prerequisite other than the equivalent of a first year graduate sequence in econ or in applied math.
Because it aims at providing a bridge between theory and practice, the teaching format is somewhat unusual: each teaching “block” will be made of 50 minutes of theory followed by 1 hour of coding, based on an empirical application related to the theory just seen. Students are expected to write their own code, and the teaching staff will ensure that it is operational at the end of each block. This course is therefore closer to cooking lessons than to traditional lectures.
The course, jointly offered by NYU Econ and the Courant Institute, is open to graduate students in the fields of economics and applied mathematics, but also in other quantitative disciplines. Students need to bring a laptop with them to the lectures. The knowledge of a particular programming language is not required; students are however expected to have some experience with programming. The course can be taken for credit or as a registered auditor.
The teaching staff is Alfred Galichon (professor of economics and of mathematics at NYU), Keith O’hara (economics graduate student at NYU and R and C++ guru), and Yifei Sun (mathematics graduate student at NYU focusing on machine learning). They are authors of the TraME library (https://github.com/TraME-Project), a toolbox for inference and simulation of discrete choice and matching problems. The course is based on Galichon’s popular graduate classes previously taught at NYU, and on his recent book, Optimal Transport Methods in Economics.

Support from NSF grant DMS-1716489 is acknowledged.

Course material

Available on Github here.

Practical information

• Schedule: Mon 1/15 — Sat 1/20, 2018, 8am-12 noon and 1pm-3pm. Location: WWH 102 in the Courant building (251 Mercer St)
• Credits: 3, assessed through a take-home exam or a short final paper, at the student’s option.
• A syllabus is available at http://alfredgalichon.com/matheconcode/.
• Students need to register on Albert (MATH-GA 2840.002 or ECON-GA 3002.015). For more information please contact: galichon@cims.nyu.edu.

Outline

• Monday 1/15: linear programming, dynamic programming, network flows
• Tuesday 1/16: optimal transport toolbox
• Wednesday 1/17: convex analysis, nonlinear inverse problems, and multivariate quantiles
• Thursday 1/18: static and dynamic multinomial choice
• Friday 1/19: statistical estimation of models of matching with transfers
• Saturday 1/20: more general models of matching

Synopsis

SYNOPSIS
Part I: Tools
Day 1: linear programming (Monday Jan 15)
Block 1. Basics of linear programming (8am-9:50am)
• Theory: linear programming duality; complementary slackness; minimax formulation
• Coding: How to eat optimally? Dataset: Stigler’s original diet data (1945).
Block 2. Network flow problems (10:10am-noon)
• Theory: directed graphs and min-cost flow problem
• Coding: How to find the shortest path through a network? Dataset: Paris subway; New York City street network.
Block 3. Dynamic programming as linear programming (1pm-2:50pm)
• Theory: Bellman’s equation; interpretation of duality; forward induction, backward induction
• Coding: When to repair mechanical engines? Dataset: Rust’s bus maintenance data (1994).

Day 2: optimal transport I (Tuesday Jan 16)
Block 4. Discrete matching (8am-9:50am)
• Theory: Shapley-Shubik duality; stability; decentralized equilibrium
• Coding: How to solve it? Dataset from Dupuy and Galichon (JPE 2014).
Block 5. Positive assortative matching (10:10am-noon)
• Theory: Becker’s model; compensating differentials; comonotonicity
• Coding: What is a CEO worth? Dataset: Gabaix-Landier’s (QJE 2008) CEO pay data.

Block 6. Hotelling’s characteristics model (1pm-2:50pm)
• Theory: power diagrams, Aurenhammer’s method
• Coding: How to infer the unobservable quality of a car model? Dataset: Feenstra-Levinsohn (Restud 1994) car data.

Day 3: optimal transport II (Wednesday Jan 17)
Block 7. Continuous multivariate matching (8am-9:50am)
• Theory: Knott-Smith criterion; Brenier’s map; McCann’s theorem
• Coding: How to solve it? the iterated proportional fitting procedure (IPFP). Dataset from Dupuy and Galichon (JPE 2014).
Block 8. Convex analysis and nonlinear inverse problems (10:10am-noon)
• Theory: convex duality; Fenchel’s inequality; subdifferentials and their inverses
• Coding: How to optimize with big data? Proximal gradient algorithms; LASSO; stochastic gradient algorithms.
Block 9. Quantiles methods (1pm-2:50pm)
• Theory: Rosenblatt’s quantiles; vector quantiles; vector quantile regression
• Coding: How to predict demand? vector quantile regression. Dataset: Engel’s (1857) original food expenditure data.

Part II. Models
Day 4: models of static and dynamic multinomial choice (Thursday Jan 18)
Block 10. Basics of static discrete choice (8am-9:50am)
• Theory: Dary-Zachary-Williams theorem, generalized entropy of choice, the inversion theorem
• Coding: How to solve it? simulation methods; AR, SARS, and GHK. Dataset: Greene and Hensher (1997) data on choice of travel mode.
Block 11. Demand models, old and new (10:10am-noon)
• Theory: the GEV model; the random coefficient logit model and the pure characteristics models
• Coding: How to estimate demand for automobiles? Dataset: BLP.
Block 12. Dynamic discrete choice methods (1pm-2:50pm)
• Theory: Rust’s model; estimation; normalization issues
• Coding: career choice.

Day 5: empirical matching models, the quasilinear case (Friday Jan 19)
Block 13. Separable models of matching (8am-9:50am)
• Theory: matching with unobservable heterogeneity
• Coding: Did Roe vs. Wade decrease the value of marriage? Dataset: Choo and Siow (JPE 2006).
Block 14. The gravity equation (10:10am-noon)
• Theory: optimal transport and the gravity equation; generalized linear models and pseudo-Poisson maximum likelihood estimation
• Coding: How to forecast international trade flows? estimating the gravity equation based on WTO international trade data.
Block 15. High-dimensional matching models (1pm-2:50pm)
• Theory: estimation of rank-constrained models
• Application: Does physical appearance have a price? matching on socioeconomic and anthropomorphic characteristics. Dataset: Chiappori, Oreffice and Quintana-Domeque’s (JPE 2012).

Day 6: empirical matching models beyond quasilinearity (Saturday Jan 20)
Block 16. Matching with imperfectly transferable utility (8am-9:50am)
• Theory: Galois connections, distance-to-frontier function, nonlinear complementary slackness, equilibrium transport
• Application: How do taxes affect matching patterns and wages? Dataset: Football coach data from Dupuy, Galichon, Jaffe and Kominers (2017).
Block 17. Integrating matching models and collective models of intrahousehold bargaining (10:10am-noon)
• Theory: collective models, Pareto weights, sharing rule
• Application: Do people marry for consumption or companionship? Dataset: Galichon, Kominers and Weber (2017).
Block 18. Matching with nontransferable utility (1pm-2:50pm)
• Theory: the Dagsvik-Menzel model; nonprice rationing and the NTU-logit separable model
• Application: Revisiting Choo and Siow’s data.

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 XXXX

‘math+econ+code’ part deux: equilibrium transport and matching models in economics

NYU Paris, 57 boulevard Saint-Germain, June 8-12, 2020

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.

Instructors

A. Galichon (NYU Econ+Math)

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 five 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. The course relies on scientific context-based learning: 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

‘math+econ+code’ masterclass on optimal transport and economic applications. Next edition: NYU New York, Jan 20-24, 2020. Past editions: NYU Paris, June 17-21, 2019, NYU NY, Jan 14-18, 2019, NYU NY, Jan 15-20, 2018.

‘math+econ+code’ masterclass on equilibrium transport and matching models in economics. Next edition: NYU Paris, June 8-12, 2020. Past editions: NYU NY, May 21-26, 2018.

‘math+econ+code’ masterclass on linear programming in economics (TBA).

‘math+econ+code’ masterclass on networks economics (TBA).

‘math+econ+code’ masterclass on big data in economics (TBA).

Diffusion list

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