Schedule of the Hausdorff School "Optimal Transport meets Economic Theory"

Abstracts

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.

Course material
The lecture slides will be available before each lecture.

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

http://alfredgalichon.com/hausdorff-lectures/

Lecture 1  Optimal Transportation Between Unequal Dimensions

In the last few decades,  the theory of optimal transportation has blossomed into a powerful tool for exploring applications both within and outside mathematics. Its impact is felt in such far flung areas as geometry, analysis, dynamics, partial differential equations, economics, machine learning, weather prediction, and computer vision.   The basic problem is to transport one probability density onto other,  while minimizing a given cost c(x,y) per unit transported.  In the vast majority of applications,  the probability densities live on spaces with the same (finite) dimension.  After briefly surveying a few highlights from this theory,  we focus our attention on what can be said when the densities instead live on spaces with two different (yet finite) dimensions.    Although the answer can still be characterized as the solution to a fully nonlinear differential equation,  it now becomes badly nonlocal in general.  Remarkably however,  one can identify conditions under which the equation becomes local, elliptic, and amenable to further analysis.

See [62] [63] [65] and [66] at http://www.math.toronto.edu/mccann/publications

Lecture 2 On Concavity of the Monopolist's Problem Facing Consumers with Nonlinear Price Preferences

The principal-agent problem is an important paradigm in economic theoryor studying the value of private information;  the nonlinear pricing problem faced by a monopolist is a particular example.  In this lecture,  we identify structural conditions on the consumers' preferences and the monopolist's profit functions which guarantee either concavity or convexity of the monopolist's profit maximization.  Uniqueness and stability of the solution are particular consequences of this concavity.  Our conditions are closely related to criteria given by Trudinger and others for prescribed Jacobian equations to have smooth solutions,  while being simpler in many respects.   By allowing for different dimensions of agents and contracts,  nonlinear dependence of agent preferences on prices, and of the monopolist's profits on agent identities,   it improves on the literature in a number of ways.   The same mathematics can also be adapted to the maximization of societal welfare by a regulated monopoly,   This is based on joint work with PhD student Shuanjian Zhang.

See [41] and  [64] at http://www.math.toronto.edu/mccann/publications

Lecture 3 The Dynamics of Multisectorial Matching

Economists are interested in studying who matches with whom (and why) in the educational, labour, and marriage sectors.  With Aloysius Siow, Xianwen Shi, and Ronald Wolthoff,  I proposed a toy model for this process, which is based on the assumption that production in any sector requires completion of two complementary tasks.  Individuals are assumed to have both social and cognitive skills,  which can be modified through education, and which determine what they choose to specialize in and with whom they choose to partner.  Our model predicts variable, endogenous, many-to-one matching.  Given a fixed initial distribution of characteristics,  the steady-state equilibrium of this model is the solution to an (infinite dimensional) linear program,  for which we developed a duality theory exhibiting a phase transition depending on the number of students who can be mentored.  If this number is two or more, then a continuous distributions of skills leads to formation of a pyramid in the education market with a small number of gurus at the top of the market having unbounded wage gradients. With recent PhD student Rosemonde Lareau-Dussault,  I have developed a corresponding overlapping generations model for the multisectorial matching problem described above, and have identified conditions under which the steady-state solution becomes a stable fixed-point of the resulting dynamics.

See [50] [58] [59] at www.math.toronto.edu/mccann/publications and www.math.toronto.edu/mccann/papers/Lareau-Dussault17.pdf

A large part of stochastic portfolio theory, as initiated by Robert Fernholz in the 1990s, is concerned with construction of practical equity portfolios that can beat the stock market index by active rule-based trading. The truly remarkable part of the theory is that it requires no probabilistic modeling on the future behavior of stock prices. There is a Monge-Kantorovich optimal transport problem that naturally arises in the construction of such portfolios. The transport problem brings a host of powerful notions such as duality which have practical uses. But the problem is also of independent mathematical interest and related ideas have appeared elsewhere in the optimal transport literature. Apart from the theory we will discuss data and open problems both in theory and in applications.

We will follow this outline:
Lecture 1: Introduction to portfolios, relative arbitrage, and optimal transport
Lecture 2: Information geometry and optimal frequency of trading
Lecture 3: Transportation cost and the multiplicative Schrodinger problem

Remarkably, it remains an open challenge to quantify the effects of model uncertainty in derivative pricing in a coherent way. From a mathematical perspective, this is a delicate issue which touches on deep classical problems of stochastic analysis.
We will review how this question can be linked to a martingale version of the transport problem. Indeed, this yields a powerful geometric approach to the problem of model uncertainty and, more generally, the theory of stochastic processes.

In many economic applications, optimal transport plans together with the solution of the associated dual problem have the interpretation of an equilibrium in a decentralized market and may thus be viewed as the solution to an “equilibrium transport problem.” Standard results from optimal transport theory provide conditions for the existence and uniqueness of solutions to such equilibrium transport problems when utility is perfectly transferable. In this lecture, I will indicate how such results can be generalized to the economically relevant case of equilibrium transport problems with imperfectly transferable utility. The main tool for doing so is a generalization of the conjugacy of generalized convex functions, which is called a Galois connection.

This lecture is based on joint work with Larry Samuelson (Yale University): https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2579057