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.
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://www.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://www.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://www.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://www.numdam.org/articles/10.1051/cocv/2012027/
[1] , Decomposition polaire et rearrangement monotone des champs de vecteurs. C. R. Acad. Sci. Paris Ser. I Math. 305 (1987) 805-808. | MR | Zbl
[2] , The regularity of mappings with a convex potential. J. Amer. Math. Soc. 5 (1992) 99-104. | MR | Zbl
[3] , Boundary regularity of maps with convex potentials. Comm. Pure Appl. Math. 45 (1992) 1141-1151. | MR | Zbl
[4] , Boundary regularity of maps with convex potentials-II. Ann. of Math. 144 (1996) 453-496. | MR | Zbl
[5] , 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] and , Matching for teams. Econ. Theory 42 (2010) 397-418. | MR | Zbl
[7] , and , Hedonic price equilibria, stable matching and optimal transport, equivalence, topology and uniqueness. Econ. Theory 42 (2010) 317-354. | MR | Zbl
[8] , 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] , Gradient rearrangement for diffeomorphisms of a compact manifold. Differ. Geom. Appl. 20 (2004) 145-165. | MR | Zbl
[10] and , J. Reine Angew. Math. 646 (2010) 65-115. | MR | Zbl
[11] , An optimal matching problem. ESAIM: COCV 11 (2005) 5771. | Numdam | MR | Zbl
[12] , Existence, uniqueness and efficiency of equilibrium in hedonic markets with multidimensional types. Econ. Theory 42 (2010) 275-315. | MR | Zbl
[13] and , Continuity of optimal transport maps on small deformations of S2. Commun. Pure Appl. Math. 62 (2009) 1670-1706. | MR | Zbl
[14] , and , When is multidimensional screening a convex program? J. Econ. Theory 146 (2011) 454-478. | MR | Zbl
[15] , and , Höelder continuity and injectivity of optimal maps. Preprint available at http://www.math.toronto.edu/mccann/papers/C1aA3w.pdf. | MR | Zbl
[16] , and , 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] , and , Necessary and sufficient conditions for continuity of optimal transport maps on Riemannian manifolds. Tohoku Math. J. 63 (2011) 855-876. | MR | Zbl
[18] , and , Nearly round spheres look convex. Amer. J. Math. 134 (2012) 109-139. | MR | Zbl
[19] , and , On the Ma-Trudinger-Wang curvature on surfaces. Calc. Var. Partial Differ. Equ. 39 (2010) 307-332. | MR | Zbl
[20] , Habilitation thesis, Universite de Metz (1995).
[21] and , The geometry of optimal transportation. Acta Math. 177 (1996) 113-161. | MR | Zbl
[22] , 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] and , Continuity, curvature and the general covariance of optimal transportation. J. Eur. Math. Soc. 12 (2010) 1009-1040. | MR | Zbl
[24] and , 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] , Abstract cyclical monotonicity and Monge solutions for the general Monge-Kantorovich problem. Set-Val. Anal. 7 (1999) 7-32. | MR | Zbl
[26] , Hölder regularity in optimal mappings in optimal transportation. To appear in Calc. Var. Partial Differ. Equ. | MR | Zbl
[27] , On the regularity of maps solutions of optimal transportation problems. Acta Math. 202 (2009) 241-283. | MR | Zbl
[28] , 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] and , Regularity of optimal transport in curved geometry: the nonfocal case. Duke Math. J. 151 (2010) 431-485. | MR | Zbl
[30] , and , Regularity of potential functions of the optimal transportation problem. Arch. Rational Mech. Anal. 177 (2005) 151-183. | MR | Zbl
[31] , Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal. 11 (2001) 589-608. | MR | Zbl
[32] , and , Rectifiability of optimal transportation plans. Can. J. Math. 64 (2012) 924-933. | MR | Zbl
[33] , Ph.D. thesis, University of Toronto (2011).
[34] , Regularity of optimal transportation between spaces with different dimensions. Math. Res. Lett. 19 (2012) 291-307. | MR | Zbl
[35] and , 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] and , On strict convexity and C1-regularity of potential functions in optimal transportation. Arch. Rational Mech. Anal. 192 (2009) 403-418. | MR | Zbl
[37] , On the second boundary value problem for equations of Monge-Ampere type. J. Reine Angew. Math. 487 (1997) 115-124. | MR | Zbl
[38] , Optimal transport: old and new, in Grundlehren der mathematischen Wissenschaften. Springer, New York 338 (2009). | MR | Zbl
[39] , On the design of a reflector antenna. Inverse Probl. 12 (1996) 351-375. | MR | Zbl
Cité par Sources :
