# 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