Abstract
This paper derives necessary and sufficient conditions for allocations to be incentive compatible in multidimensional screening problems that satisfy a generalized single crossing property. We then devise a numerical method based on these results to solve multidimensional screening problems. Importantly, our numerical method can be applied to multidimensional screening problems for which existing approaches cannot be applied. We apply this method to several numerical examples in the context of multidimensional optimal taxation. In addition to illustrating how to apply our theoretical results and implement our numerical method, our simulations highlight the importance of bunching in optimal multidimensional taxation. Finally, we prove that bunching must occur in multidimensional optimal taxation problems when the social planner has sufficiently redistributive preferences.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability
The datasets analyzed in the current study are available from IPUMS at https://www.ipums.org/
References
Aguilera, N.E., Morin, P.: Approximating optimization problems over convex functions. Numer. Math. 111(1), 1–34 (2008)
Armstrong, M.: Multiproduct nonlinear pricing. Econometrica 64, 51–75 (1996)
Baron, D.P., Myerson, R.B.: Regulating a monopolist with unknown costs. Econometrica 50(4), 911 (1982)
Basov, S.: Hamiltonian approach to multi-dimensional screening. J. Math. Econ. 36(1), 77–94 (2001). https://doi.org/10.1016/s0304-4068(01)00070-2
Bergstrom, K., Dodds, W.: Optimal taxation with multiple dimensions of heterogeneity. J. Public Econ. 200, 104442 (2021). https://doi.org/10.1016/j.jpubeco.2021.104442
Blomquist, S., Selin, H.: Hourly wage rate and taxable labor income responsiveness to changes in marginal tax rates. J. Public Econ. 94, 878–889 (2010)
Boadway, R., Jacquet, L.: Optimal marginal and average income taxation under maximin. J. Econ. Theory 143(1), 425–441 (2008). https://doi.org/10.1016/j.jet.2008.01.003
Boerma, J., Tsyvinski, A., Zimin, A.: Bunching and taxing multidimensional skills. SSRN Electron. J. (2022) http://www.nber.org/papers/w30015
Carlier, G.: A general existence result for the principal-agent problem with adverse selection. J. Math. Econ. 35(1), 129–150 (2001)
Chetty, R.: A new method of estimating risk aversion. Am. Econ. Rev. 96(5), 1821–1834 (2006)
Chiappori, P.A., McCann, R., Pass, B.: Multidimensional matching (2017)
Figalli, A., Kim, Y.H., McCann, R.J.: When is multidimensional screening a convex program? J. Econ. Theory 146(2), 454–478 (2011). https://doi.org/10.1016/j.jet.2010.11.006
Gale, D., Nikaido, H.: The Jacobian matrix and global univalence of map**s. Math. Annalen 159, 81–93 (1965)
Golosov, M., Tsyvinski, A., Werquin, N.: A variational approach to the analysis of tax systems. NBER Working Paper 20780 48 (2014)
Grabovsky, Y., Truskinovsky, L.: Roughening instability of broken extremals. Arch. Ration. Mech. Anal. 200(1), 183–202 (2010). https://doi.org/10.1007/s00205-010-0377-8
Heinonen, J.: Lectures on Lipschitz Analysis. Number 100. University of Jyväskylä, Jyväskylä (2005)
Hencl, S., Vejnar, B.: Sobolev homeomorphism that cannot be approximated by diffeomorphisms in \(w^{1,1}\). Arch. Ration. Mech. Anal. 219(1), 183–202 (2015)
Hirsch, M.W.: Differential Topology. Springer, Berlin (1988)
Kinderlehrer, D., Stampacchia, G.: An introduction to variational inequalities and their applications. SIAM (1980)
Kleven, H.J., Kreiner, C.T., Saez, E.: The optimal income taxation of couples. Econometrica 77(2), 537–560 (2009)
Koliha, J.J.: Approximation of convex functions. Real Anal. Exch. 29(1), 465 (2004). https://doi.org/10.14321/realanalexch.29.1.0465
Krasikov, I., Golosov, M.: Multidimensional screening in public finance: the optimal taxation of couples (2022)
Laffont, J.J., Tirole, J.: A Theory of Incentives in Procurement and Regulation. MIT Press, Cambridge (1994)
Lockwood, B.B., Weinzierl, M.: De gustibus non est taxandum: heterogeneity in preferences and optimal redistribution. J. Public Econ. 124, 74–80 (2016)
Matkowski, J.: Mean-value theorem for vector-valued functions. Math. Bohem. 137(4), 415–423 (2012)
McAfee, R., McMillan, J.: Multidimensional incentive compatibility and mechanism design. J. Econ. Theory 46(2), 335–354 (1988). https://doi.org/10.1016/0022-0531(88)90135-4
Merigot, Q., Oudet, E.: Handling convexity-like constraints in variational problems. SIAM J. Numer. Anal. 52(5), 2466–2487 (2014)
Milgrom, P., Segal, I.: Envelope theorems for arbitrary choice sets. Econometrica 70, 583–601 (2002)
Miller, J.D.: Some global inverse function theorems. J. Math. Anal. Appl. 100(2), 375–384 (1984). https://doi.org/10.1016/0022-247X(84)90087-8
Mirrlees, J.: An exploration in the theory of optimal income taxation. Rev. Econ. Stud. 38, 175–208 (1971)
Mirrlees, J.: Optimal tax theory: a synthesis. J. Public Econ. 6(4), 327–358 (1976)
Mirrlees, J.: The Theory of Optimal Taxation. Handbook of Mathematical Economics 3, 1197–1249 (1986). https://doi.org/10.1016/S1573-4382(86)03006-0
Mussa, M., Rosen, S.: Monopoly and product quality. J. Econ. Theory 18(2), 301–317 (1978). https://doi.org/10.1016/0022-0531(78)90085-6
Oberman, A.M.: A numerical method for variational problems with convexity constraints. SIAM J. Sci. Comput. 35(1), A378–A396 (2013)
Rochet, J.C.: A necessary and sufficient condition for rationalizability in a quasi-linear context. J. Math. Econ. 16(2), 191–200 (1987). https://doi.org/10.1016/0304-4068(87)90007-3
Rochet, J.C., Chone, P.: Ironing, swee**, and multidimensional screening. Econometrica 66(4), 783 (1998). https://doi.org/10.2307/2999574
Saez, E.: Using elasticities to derive optimal income tax rates. Rev. Econ. Stud. 68, 205–229 (2001)
Spiritus, K., Lehmann, E., Renes, S., Zoutman, F.: Optimal taxation with multiple incomes and types. SSRN Electron. J. (2022). https://doi.org/10.2139/ssrn.4016499
Tsatsomeros, M.J.: Generating and Detecting Matrices with Positive Principal Minors, pp. 115–132. Nova Science Publishers, Inc, Hauppauge (2004)
Udriste, C.: Simplified multitime maximum principle. Balkan J. Geom. Appl. 14(1), 102–119 (2009)
Urbinati, N.: The Walrasian objection mechanism and Mas-Colell’s bargaining set in economies with many commodities. Econ. Theor. 76(1), 45–68 (2022). https://doi.org/10.1007/s00199-022-01454-0
Villani, C.: Optimal Transport Old and New. Springer, Berlin (2009)
Acknowledgements
The findings, interpretations, and conclusions expressed in this paper are entirely those of the author. Thank you to two anonymous referees and the editor, Nicholas Yannelis, for providing many helpful comments that improved the paper. Thanks to various seminar participants at Tulane University and Virginia Tech. Thanks to Stefano Barbieri for providing helpful comments on this paper. This paper also benefited significantly from conversations with Ilia Krasikov and Mike Golosov. Finally, a special thanks to Katy Bergstrom who has been immensely helpful in providing feedback on this project.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author has no competing interests to declare that are relevant to the content of this article.
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.
About this article
Cite this article
Dodds, W. Solving multidimensional screening problems using a generalized single crossing property. Econ Theory 77, 1025–1084 (2024). https://doi.org/10.1007/s00199-023-01519-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00199-023-01519-8