Abstract
Hedonic pricing with quasi-linear preferences is shown to be equivalent to stable matching with transferable utilities and a participation constraint, and to an optimal transportation (Monge–Kantorovich) linear programming problem. Optimal assignments in the latter correspond to stable matchings, and to hedonic equilibria. These assignments are shown to exist in great generality; their marginal indirect payoffs with respect to agent type are shown to be unique whenever direct payoffs vary smoothly with type. Under a generalized Spence-Mirrlees condition (also known as a twist condition) the assignments are shown to be unique and to be pure, meaning the matching is one-to-one outside a negligible set. For smooth problems set on compact, connected type spaces such as the circle, there is a topological obstruction to purity, but we give a weaker condition still guaranteeing uniqueness of the stable match.
Similar content being viewed by others
References
Ahmad, N.: The geometry of shape recognition via a Monge–Kantorovich optimal transport problem. PhD thesis, Brown University (2004)
Ahmad, N., Kim, H.K., McCann, R.J.: External doubly stochastic measures and optimal transportation (2009). Preprint at www.math.toronto.edu/mccann
Anderson E.J., Nash P.: Linear Programming in Infinite-Dimensional Spaces. Wiley, Chichester (1987)
Brenier Y.: The dual least action problem for an ideal, incompressible fluid. Arch Ration Mech Anal 122, 323–351 (1993)
Caffarelli, L.A., McCann, R.J.: Free boundaries in optimal transport and Monge-Ampère obstacle problems. Ann Math (2) to appear (2009)
Carlier G.: Duality and existence for a class of mass transportation problems and economic applications. Adv Math Econ 5, 1–21 (2003)
Carlier, G., Ekeland, I.: Matching for teams. Econ Theory, this volume (2009)
Dudley R.M.: Real Analysis and Probability. Revised reprint of the 1989 original. Cambridge University Press, Cambridge (2002)
Ekeland I.: An optimal matching problem. Control Optim Calc Var 11(1), 57–71 (2005)
Ekeland, I.: Existence, uniqueness and efficiency of equilibrium in hedonic markets with multidimensional types. Econ Theory, this volume (2009)
Gangbo, W.: (1995) Habilitation thesis. Université de Metz
Gangbo W., McCann R.J.: The geometry of optimal transportation. Acta Math 177, 113–161 (1996)
Gangbo W., McCann R.J.: Shape recognition via Wasserstein distance. Q Appl Math 58, 705–737 (2000)
Gangbo W., Świȩch A.: Optimal maps for the multidimensional Monge–Kantorovich problem. Commun Pure Appl Math 51, 23–45 (1998)
Gretsky N.E., Ostroy J.M., Zame W.R.: The nonatomic assignment model. Econ Theory 2, 103–127 (1992)
Gretsky N.E., Ostroy J.M., Zame W.R.: Perfect competition in the continuous assignment model. J Econ Theory 88, 60–118 (1999)
Heckman, J.J., Matzkin, R., Nesheim, L.P.: Nonparametric estimation of nonadditive hedonic models. CEMMAP Working Paper CWP03/05 (2005)
Hestir K., Williams S.C.: Supports of doubly stochastic measures. Bernoulli 1, 217–243 (1995)
Hildenbrand W.: Core and Equilibria of a Large Economy. Princeton University Press, Princeton (1974)
Kantorovich L.: On the translocation of masses. CR (Doklady) Acad Sci URSS (NS) 37, 199–201 (1942)
Kellerer H.G.: Duality theorems for marginal problems. Z Wahrsch Verw Gebiete 67, 399–432 (1984)
Kim, Y.-H., McCann, R.J.: Continuity, curvature, and the general covariance of optimal transportation. J Eur Math Soc (JEMS), to appear (2009)
Ma X.-N., Trudinger N., Wang X.-J.: Regularity of potential functions of the optimal transportation problem. Arch Ration Mech Anal 177, 151–183 (2005)
McCann R.J.: Existence and uniqueness of monotone measure-preserving maps. Duke Math J 80, 309–323 (1995)
McCann R.J.: Exact solutions to the transportation problem on the line. R Soc Lond Proc Ser A Math Phys Eng Sci 455, 1341–1380 (1999)
Monge, G.: Mémoire sur la théorie des déblais et de remblais. Histoire de l’ Académie Royale des Sciences de Paris, avec les Mémoires de Mathématique et de Physique pour la même année, pp. 666–704 (1781)
Nürnberger G.: Approximation by Spline Functions. Springer, Berlin (1980)
Plakhov A.Yu.: Exact solutions of the one-dimensional Monge–Kantorovich problem (Russian). Mat Sb 195, 57–74 (2004)
Rachev S.T., Rüschendorf L.: Mass Transportation Problems. Probability Applications. Springer, New York (1998)
Roth A., Sotomayor M.: Two-sided Matching: A study in Game-theoretic Modeling and Analysis. Econometric Society Monograph Series. Cambridge University Press, Cambridge (1990)
Shapley L.S., Shubik M.: The assignment game I: The core. Int J Game Theory 1, 111–130 (1972)
Smith C, Knott M: On Hoeffding-Fréchet bounds and cyclic monotone relations. J Multivar Anal 40, 328–334 (1992)
Tinbergen J.: On the theory of income distribution. Weltwirtschaftliches Archiv 77, 155–173 (1956)
Uckelmann L.: Optimal couplings between onedimensional distributions. In: Beneš, V., Štěpán, J. (eds) Distributions with Given Marginals and Moment Problems, pp. 261–273. Kluwer Academic Publishers, Dordrecht (1997)
Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence (2003)
Villani, C.: Cours d’Intégration et Analyse de Fourier (2006). Preprint at http://www.umpa.ens-lyon.fr/~cvillani/Cours/iaf-2006.html
Villani C.: Optimal transport. Old and new. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 338. Springer, Berlin (2009)
von Neumann, J.: A certain zero-sum two-person game equivalent to the optimal assignment problem. In: Contributions to the Theory of Games, vol. 2, pp. 5–12. Princeton University Press, Princeton (1953)
Author information
Authors and Affiliations
Corresponding author
Additional information
It is a pleasure to thank Ivar Ekeland, whose conferences at the Banff International Research Station (BIRS) in 2004 and 2005 brought us together, and whose work on hedonic pricing (Ekeland 2005, 2009) inspired our investigation. RJM also wishes to thank Najma Ahmad, Wilfrid Gangbo, and Hwa Kil Kim, for fruitful conversations related to the subtwist criterion for uniqueness, and Nassif Ghoussoub for insightful remarks. The authors are pleased to acknowledge the support of Natural Sciences and Engineering Research Council of Canada Grant 217006-03 and United States National Science Foundation Grants 0241858, 0433990, 0532398 and DMS-0354729. They also acknowledge support through CEMMAP from the Leverhulme Trust and the UK Economic and Social Research Council grant RES-589-28-0001. ©2008 by the authors.
Rights and permissions
About this article
Cite this article
Chiappori, PA., McCann, R.J. & Nesheim, L.P. Hedonic price equilibria, stable matching, and optimal transport: equivalence, topology, and uniqueness. Econ Theory 42, 317–354 (2010). https://doi.org/10.1007/s00199-009-0455-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00199-009-0455-z
Keywords
- Hedonic price equilibrium
- Matching
- Optimal transportation
- Spence-Mirrlees condition
- Monge–Kantorovich
- Twist condition