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The 1975 Nobel Prize winner in Economics was Kantorovich who reformulated the optimal transportation theory of Monge and applied it to optimal resource allocation. The Wasserstein distance is central to this optimization model.

Can anyone familiar with optimal transport give an overview of some of its applications to economics? maybe beyond just moving mass/products to separate destination points. It's hard to understand how production, consumption and pricing comes into the picture compared to non-economics optimal transport.

If they even are still in use, how have these methods been developed and improved for economists in recent publications?

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Optimal transport methods are very much still in use in economics. The show up in two-sided matching with side-payments, contract theory, hedonic pricing, partial identification in econometrics, and a couple of other areas. You can find an excellent overview of economic applications in the 2016 book Optimal Transport Methods in Economics by Alfred Galichon.

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  • $\begingroup$ I have it but don't get how production and consumption enter the Wasserstein formula. could you explain that, and the applications you listed $\endgroup$
    – develarist
    Nov 7 '20 at 15:55
  • $\begingroup$ I'm not sure where consumption and production come in. I guess by "Wasserstein formula" you mean the Wasserstein distance? In many applications, one uses more general functions. Sketching out the application needs a lot of space though. $\endgroup$ Nov 7 '20 at 19:20
  • $\begingroup$ What is more general than the Wasserstein distance? I think how consumption, production and pricing come in is important for the answer, versus non-econ transport problems $\endgroup$
    – develarist
    Nov 8 '20 at 9:22
  • $\begingroup$ The Wasserstein distance minimizes the integral of a metric over all couplings. But you can solve the minimization problem for much more general functions of two variables than a metric and that is important for applications. I don't know of applications of optimal transport to consumption and production theory. In matching, pricing comes in via Kantorovich duality. That is all in the book. $\endgroup$ Nov 8 '20 at 9:26
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I am very (very) much not familiar with optimal transport, but there are applications in mechanism design.

For example, Daskalakis, Deckelbaum & Tzamos (ECTA 2017, "Strong Duality for a Multiple-Good Monopolist") characterize optimal mechanisms to sell multiple-goods. This is a very, very hard problem that has troubled economists for years (at least since Adams & Yellen QJE 1976). They approach it by setting up a minimization problem that is dual to the monopolist's profit maximization problem. This dual is in the form of an optimal transport problem, and strong duality holds. Establishing the latter is quite involved "following the the proof of Monge-Kantorovivich duality in Villani (2008)" (which I have not read). The conncection of mechanism design and optimal transportation they employ is explained in Ekeland (2010).

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