Skip to main content
SpringerLink
Log in

A Spatial Pareto Exchange Economy Problem

  • Published:
Applied Mathematics & Optimization Aims and scope Submit manuscript

Abstract

We use convex duality techniques to study a spatial Pareto problem with transport costs and derive a spatial second welfare theorem. The existence of an integrable equilibrium distribution of quantities is nontrivial and established under general monotonicity assumptions. Our variational approach also enables us to give a numerical algorithm à la Sinkhorn and present simulations for equilibrium prices and quantities in one-dimensional domains and a network of French cities.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Notes

  1. by nondecreasing we mean that \(\beta -\beta ' \in {\mathbb {R}}_+^N\) implies \(U(x, \beta ) \ge U(x, \beta ')\).

  2. It is worth noticing here that the soft max approximation can be given a micro foundation by considering a Gumbel random term affecting transport costs, see [3, 14].

  3. for simplicity, we take U independent of the location but there is no extra difficulty in having a dependence in y in a multiplicative prefactor or even in the exponents \(a_i\).

  4. given by the website https://www.coordonnees-gps.fr/distance.

References

  1. Villani, C.: Topics in optimal transportation. Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence, RI (2003)

  2. Santambrogio, F.: Optimal transport for applied mathematicians. Progress in Nonlinear Differential Equations and their Applications, 87. Birkhäuser/Springer, Cham (2015). Calculus of variations, PDEs, and modeling

  3. Galichon, A.: Optimal Transport Methods in Economics. Princeton University Press, Princeton (2016)

    Book  Google Scholar 

  4. Jordan, R., Kinderlehrer, David, Otto, F.: The variational formulation of the Fokker–Planck equation. SIAM J. Math. Anal. 29(1), 1–17 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich, 2nd edn. Birkhäuser, Basel (2008)

  6. Marco, C.: Sinkhorn distances: Lightspeed computation of optimal transport. In Advances in Neural Information Processing Systems, pp 2292–2300 (2013)

  7. Benamou, J. D., Carlier, G., Cuturi, M., Nenna, L., & Peyré, G.: Iterative Bregman projections for regularized transportation problems. SIAM J. Sci. Comput. 37(2), A1111–A1138 (2015)

  8. Peyré, G., Cuturi, M.: Computational optimal transport: with applications to data science. Found. Trends Mach. Learn. 11(5–6), 355–607 (2019)

    Article  MATH  Google Scholar 

  9. Peyré, G.: Entropic approximation of wasserstein gradient flows. SIAM J. Imaging Sci. 8(4), 2323–2351 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Komlós, J.: A generalization of a problem of Steinhaus. Acta Math. Acad. Sci. Hungar. 18, 217–229 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  11. Rockafellar, R.T.: Integrals which are convex functionals. Pacific J. Math. 24, 525–539 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  12. Rockafellar, R.T.: Integrals which are convex functionals. II. Pacific J. Math. 39, 439–469 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ivar, E. and Roger, T.: Convex analysis and variational problems, volume 28 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, english edition (1999)

  14. Alfred, G., Bernard, S.: Cupid’s invisible hand: social surplus and identification in matching models. Rev. Econ. Stud. 89(5), 2600–2629 (2021)

    MathSciNet  MATH  Google Scholar 

  15. Beck, A., Tetruashvili, L.: On the convergence of block coordinate descent type methods. SIAM J. Optim. 23(4), 2037–2060 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

GC is grateful to the Agence Nationale de la Recherche for its support through the projects MAGA (ANR-16-CE40-0014) and MFG (ANR-16-CE40-0015-01). GC also acknowledges the support of the Lagrange Mathematics and Computation Research Center.

Funding

The authors have not disclosed any funding.

Author information

Authors and Affiliations

  1. CEREMADE, Université Paris Dauphine, PSL, and SAMM, Université Paris 1 Panthéon Sorbonne, Paris, France

    X. Bacon

  2. CEREMADE, Université Paris Dauphine, PSL, and INRIA-Paris, MOKAPLAN, Paris, France

    G. Carlier

  3. SAMM, Université Paris 1 Panthéon-Sorbonne, Paris, France

    B. Nazaret

Authors
  1. X. Bacon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Carlier
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. Nazaret
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to X. Bacon.

Ethics declarations

Conflict of interest

The authors have not disclosed any competing interests..

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bacon, X., Carlier, G. & Nazaret, B. A Spatial Pareto Exchange Economy Problem. Appl Math Optim 87, 45 (2023). https://doi.org/10.1007/s00245-022-09947-z

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00245-022-09947-z

Keywords

Mathematics Subject Classification

Search

Navigation