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Regularity properties of optimal transportation problemsarising in hedonic pricing models

Published online by Cambridge University Press:  28 March 2013

Brendan Pass
Brendan Pass*
Affiliation:
Department of Mathematical and Statistical Sciences, 632 CAB, University of Alberta, Edmonton, T6G 2G1 Alberta, Canada. pass@ualberta.ca.
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Abstract

We study a form of optimal transportation surplus functions which arise in hedonicpricing models. We derive a formula for the Ma–Trudinger–Wang curvature of thesefunctions, yielding necessary and sufficient conditions for them to satisfy(A3w). We use this to give explicit new examples of surplus functionssatisfying (A3w), of the formb(x,y) = H(x + y)where H is a convex function on ℝn. We alsoshow that the distribution of equilibrium contracts in this hedonic pricing model isabsolutely continuous with respect to Lebesgue measure, implying that buyers are fullyseparated by the contracts they sign, a result of potential economic interest.

Keywords

Optimal transportationhedonic pricingMa–Trudinger–Wang curvaturematchingMonge–Kantorovichregularity of solutions

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Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 3 , July 2013 , pp. 668 - 678
Copyright
© EDP Sciences, SMAI, 2013

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References

Brenier, Y., Decomposition polaire et rearrangement monotone des champs de vecteurs. C. R. Acad. Sci. Paris Ser. I Math. 305 (1987) 805–808. Google Scholar
Caffarelli, L.A., The regularity of mappings with a convex potential. J. Amer. Math. Soc. 5 (1992) 99–104. Google Scholar
Caffarelli, L.A., Boundary regularity of maps with convex potentials. Comm. Pure Appl. Math. 45 (1992) 1141–1151. Google Scholar
Caffarelli, L.A., Boundary regularity of maps with convex potentials-II. Ann. of Math. 144 (1996) 453–496. Google Scholar
L. Caffarelli, Allocation maps with general cost functions, in Partial Differential Equations and Applications, edited by P. Marcellini, G. Talenti and E. Vesintini. Lect. Notes Pure Appl. Math. 177 (1996) 29–35.
Carlier, G. and Ekeland, I., Matching for teams. Econ. Theory 42 (2010) 397–418. Google Scholar
Chiappori, P.-A., McCann, R. and Nesheim, L., Hedonic price equilibria, stable matching and optimal transport, equivalence, topology and uniqueness. Econ. Theory 42 (2010) 317–354. Google Scholar
Delanoë, P., Classical solvability in dimension two of the second boundary-value problem associated with the Monge-Ampere operator. Ann. Inst. Henri Poincaré, Anal. Non Lineaire 8 (1991) 442–457. Google Scholar
Delanoë, P., Gradient rearrangement for diffeomorphisms of a compact manifold. Differ. Geom. Appl. 20 (2004) 145–165. Google Scholar
Delanoë, P. and Ge, Y., J. Reine Angew. Math. 646 (2010) 65–115.
Ekeland, I., An optimal matching problem. ESAIM: COCV 11 (2005) 5771. Google Scholar
Ekeland, I., Existence, uniqueness and efficiency of equilibrium in hedonic markets with multidimensional types. Econ. Theory 42 (2010) 275–315. Google Scholar
Figalli, A. and Rifford, L., Continuity of optimal transport maps on small deformations of S2. Commun. Pure Appl. Math. 62 (2009) 1670–1706. Google Scholar
Figalli, A., Kim, Y.-H. and McCann, R.J., When is multidimensional screening a convex program? J. Econ. Theory 146 (2011) 454–478. Google Scholar
A. Figalli, Y.-H. Kim and R.J. McCann, Höelder continuity and injectivity of optimal maps. Preprint available at http://www.math.toronto.edu/mccann/papers/C1aA3w.pdf.
A. Figalli, Y.-H. Kim and R.J. McCann, Regularity of optimal transport maps on multiple products of sphere. To appear in J. Eur. Math. Soc. Currently available at http://www.math.toronto.edu/mccann/papers/sphere-product.pdf.
Figalli, A., Rifford, L. and Villani, C., Necessary and sufficient conditions for continuity of optimal transport maps on Riemannian manifolds. Tohoku Math. J. 63 (2011) 855–876. Google Scholar
Figalli, A., Rifford, L. and Villani, C., Nearly round spheres look convex. Amer. J. Math. 134 (2012) 109–139. Google Scholar
Figalli, A., Rifford, L. and Villani, C., On the Ma–Trudinger–Wang curvature on surfaces. Calc. Var. Partial Differ. Equ. 39 (2010) 307–332. Google Scholar
W. Gangbo, Habilitation thesis, Universite de Metz (1995).
Gangbo, W. and McCann, R.J., The geometry of optimal transportation. Acta Math. 177 (1996) 113–161. Google Scholar
Y.-H. Kim, Counterexamples to continuity of optimal transportation on positively curved Riemannian manifolds. Int. Math. Res. Not. 2008 (2008) doi:10.1093/imrn/rnn120. CrossRef
Kim, Y.-H. and McCann, R.J., Continuity, curvature and the general covariance of optimal transportation. J. Eur. Math. Soc. 12 (2010) 1009–1040. Google Scholar
Y.-H. Kim and R.J. McCann, Towards the smoothness of optimal maps on Riemannian submersions and Riemannian products (of round spheres in particular). To appear in J. Reine Angew. Math. Currently available at http://www.math.toronto.edu/mccann/papers/RiemSub.pdf.
Levin, V., Abstract cyclical monotonicity and Monge solutions for the general Monge-Kantorovich problem. Set-Val. Anal. 7 (1999) 7–32. Google Scholar
J. Liu, Hölder regularity in optimal mappings in optimal transportation. To appear in Calc. Var. Partial Differ. Equ.
Loeper, G., On the regularity of maps solutions of optimal transportation problems. Acta Math. 202 (2009) 241–283. Google Scholar
Loeper, G., Regularity of optimal maps on the sphere: The quadratic cost and the reflector antenna. Arch. Rational Mech. Anal. 199 (2011) 269–289. Google Scholar
Loeper, G. and Villani, C., Regularity of optimal transport in curved geometry: the nonfocal case. Duke Math. J. 151 (2010) 431–485. Google Scholar
Ma, X.-N., Trudinger, N. and Wang, X.-J., Regularity of potential functions of the optimal transportation problem. Arch. Rational Mech. Anal. 177 (2005) 151–183. Google Scholar
McCann, R.J., Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal. 11 (2001) 589–608. Google Scholar
McCann, R., Pass, B. and Warren, M., Rectifiability of optimal transportation plans. Can. J. Math. 64 (2012) 924–933. Google Scholar
B. Pass, Ph.D. thesis, University of Toronto (2011).
Pass, B., Regularity of optimal transportation between spaces with different dimensions. Math. Res. Lett. 19 (2012) 291–307. Google Scholar
Trudinger, N. and Wang, X.-J., On the second boundary value problem for Monge-Ampere type equations and optimal transportation. Ann. Sc. Norm. Super. Pisa Cl. Sci. 8 (2009) 143–174. Google Scholar
Trudinger, N. and Wang, X.-J., On strict convexity and C 1-regularity of potential functions in optimal transportation. Arch. Rational Mech. Anal. 192 (2009) 403–418. Google Scholar
Urbas, J., On the second boundary value problem for equations of Monge-Ampere type. J. Reine Angew. Math. 487 (1997) 115–124. Google Scholar
C. Villani, Optimal transport: old and new, in Grundlehren der mathematischen Wissenschaften. Springer, New York 338 (2009).
Wang, X.-J., On the design of a reflector antenna. Inverse Probl. 12 (1996) 351–375. Google Scholar