Regularity properties of optimal transportation problems arising in hedonic pricing models
ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 3, pp. 668-678.

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

DOI : 10.1051/cocv/2012027
Classification : 35J15, 49N60, 58E17, 91B68
Mots clés : optimal transportation, hedonic pricing, Ma-Trudinger-Wang curvature, matching, Monge-Kantorovich, regularity of solutions
@article{COCV_2013__19_3_668_0,
     author = {Pass, Brendan},
     title = {Regularity properties of optimal transportation problems arising in hedonic pricing models},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {668--678},
     publisher = {EDP-Sciences},
     volume = {19},
     number = {3},
     year = {2013},
     doi = {10.1051/cocv/2012027},
     mrnumber = {3092356},
     zbl = {1271.91053},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2012027/}
}
TY  - JOUR
AU  - Pass, Brendan
TI  - Regularity properties of optimal transportation problems arising in hedonic pricing models
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2013
SP  - 668
EP  - 678
VL  - 19
IS  - 3
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2012027/
DO  - 10.1051/cocv/2012027
LA  - en
ID  - COCV_2013__19_3_668_0
ER  - 
%0 Journal Article
%A Pass, Brendan
%T Regularity properties of optimal transportation problems arising in hedonic pricing models
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2013
%P 668-678
%V 19
%N 3
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2012027/
%R 10.1051/cocv/2012027
%G en
%F COCV_2013__19_3_668_0
Pass, Brendan. Regularity properties of optimal transportation problems arising in hedonic pricing models. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 3, pp. 668-678. doi : 10.1051/cocv/2012027. http://archive.numdam.org/articles/10.1051/cocv/2012027/

[1] Y. Brenier, Decomposition polaire et rearrangement monotone des champs de vecteurs. C. R. Acad. Sci. Paris Ser. I Math. 305 (1987) 805-808. | MR | Zbl

[2] L.A. Caffarelli, The regularity of mappings with a convex potential. J. Amer. Math. Soc. 5 (1992) 99-104. | MR | Zbl

[3] L.A. Caffarelli, Boundary regularity of maps with convex potentials. Comm. Pure Appl. Math. 45 (1992) 1141-1151. | MR | Zbl

[4] L.A. Caffarelli, Boundary regularity of maps with convex potentials-II. Ann. of Math. 144 (1996) 453-496. | MR | Zbl

[5] 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. | MR | Zbl

[6] G. Carlier and I. Ekeland, Matching for teams. Econ. Theory 42 (2010) 397-418. | MR | Zbl

[7] P.-A. Chiappori, R. Mccann and L. Nesheim, Hedonic price equilibria, stable matching and optimal transport, equivalence, topology and uniqueness. Econ. Theory 42 (2010) 317-354. | MR | Zbl

[8] P. Delanoë, 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. | EuDML | Numdam | MR | Zbl

[9] P. Delanoë, Gradient rearrangement for diffeomorphisms of a compact manifold. Differ. Geom. Appl. 20 (2004) 145-165. | MR | Zbl

[10] P. Delanoë and Y. Ge, J. Reine Angew. Math. 646 (2010) 65-115. | MR | Zbl

[11] I. Ekeland, An optimal matching problem. ESAIM: COCV 11 (2005) 5771. | Numdam | MR | Zbl

[12] I. Ekeland, Existence, uniqueness and efficiency of equilibrium in hedonic markets with multidimensional types. Econ. Theory 42 (2010) 275-315. | MR | Zbl

[13] A. Figalli and L. Rifford, Continuity of optimal transport maps on small deformations of S2. Commun. Pure Appl. Math. 62 (2009) 1670-1706. | MR | Zbl

[14] A. Figalli, Y.-H. Kim and R.J. Mccann, When is multidimensional screening a convex program? J. Econ. Theory 146 (2011) 454-478. | MR | Zbl

[15] 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. | MR | Zbl

[16] 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. | MR | Zbl

[17] A. Figalli, L. Rifford and C. Villani, Necessary and sufficient conditions for continuity of optimal transport maps on Riemannian manifolds. Tohoku Math. J. 63 (2011) 855-876. | MR | Zbl

[18] A. Figalli, L. Rifford and C. Villani, Nearly round spheres look convex. Amer. J. Math. 134 (2012) 109-139. | MR | Zbl

[19] A. Figalli, L. Rifford and C. Villani, On the Ma-Trudinger-Wang curvature on surfaces. Calc. Var. Partial Differ. Equ. 39 (2010) 307-332. | MR | Zbl

[20] W. Gangbo, Habilitation thesis, Universite de Metz (1995).

[21] W. Gangbo and R.J. Mccann, The geometry of optimal transportation. Acta Math. 177 (1996) 113-161. | MR | Zbl

[22] 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. | MR | Zbl

[23] Y.-H. Kim and R.J. Mccann, Continuity, curvature and the general covariance of optimal transportation. J. Eur. Math. Soc. 12 (2010) 1009-1040. | MR | Zbl

[24] 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. | MR | Zbl

[25] V. Levin, Abstract cyclical monotonicity and Monge solutions for the general Monge-Kantorovich problem. Set-Val. Anal. 7 (1999) 7-32. | MR | Zbl

[26] J. Liu, Hölder regularity in optimal mappings in optimal transportation. To appear in Calc. Var. Partial Differ. Equ. | MR | Zbl

[27] G. Loeper, On the regularity of maps solutions of optimal transportation problems. Acta Math. 202 (2009) 241-283. | MR | Zbl

[28] G. Loeper, Regularity of optimal maps on the sphere: The quadratic cost and the reflector antenna. Arch. Rational Mech. Anal. 199 (2011) 269-289. | MR | Zbl

[29] G. Loeper and C. Villani, Regularity of optimal transport in curved geometry: the nonfocal case. Duke Math. J. 151 (2010) 431-485. | MR | Zbl

[30] X.-N. Ma, N. Trudinger and X.-J. Wang, Regularity of potential functions of the optimal transportation problem. Arch. Rational Mech. Anal. 177 (2005) 151-183. | MR | Zbl

[31] R.J. Mccann, Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal. 11 (2001) 589-608. | MR | Zbl

[32] R. Mccann, B. Pass and M. Warren, Rectifiability of optimal transportation plans. Can. J. Math. 64 (2012) 924-933. | MR | Zbl

[33] B. Pass, Ph.D. thesis, University of Toronto (2011).

[34] B. Pass, Regularity of optimal transportation between spaces with different dimensions. Math. Res. Lett. 19 (2012) 291-307. | MR | Zbl

[35] N. Trudinger and X.-J. Wang, 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. | Numdam | MR | Zbl

[36] N. Trudinger and X.-J. Wang, On strict convexity and C1-regularity of potential functions in optimal transportation. Arch. Rational Mech. Anal. 192 (2009) 403-418. | MR | Zbl

[37] J. Urbas, On the second boundary value problem for equations of Monge-Ampere type. J. Reine Angew. Math. 487 (1997) 115-124. | MR | Zbl

[38] C. Villani, Optimal transport: old and new, in Grundlehren der mathematischen Wissenschaften. Springer, New York 338 (2009). | MR | Zbl

[39] X.-J. Wang, On the design of a reflector antenna. Inverse Probl. 12 (1996) 351-375. | MR | Zbl

Cité par Sources :