# 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),

*. Available here.*

**Journal of Political Economy**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.