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Convexity and multi-dimensional screening for spaces with different dimensions

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  • Pass, Brendan

Abstract

We study the principal–agent problem. We show that b-convexity of the space of products, a condition which appears in a recent paper by Figalli, Kim and McCann (2011) [9], is necessary to formulate the problem as a maximization over a convex set. We then show that when the dimension m of the space of types is larger than the dimension n of the space of products, this condition implies that the extra dimensions do not encode independent economic information. When m is smaller than n, we show that under b-convexity of the space of types, it is always optimal for the principal to offer goods only from a certain prescribed subset. We show that this is equivalent to offering an m-dimensional space of goods.

Suggested Citation

  • Pass, Brendan, 2012. "Convexity and multi-dimensional screening for spaces with different dimensions," Journal of Economic Theory, Elsevier, vol. 147(6), pages 2399-2418.
  • Handle: RePEc:eee:jetheo:v:147:y:2012:i:6:p:2399-2418
    DOI: 10.1016/j.jet.2012.05.004
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    References listed on IDEAS

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    1. Figalli, Alessio & Kim, Young-Heon & McCann, Robert J., 2011. "When is multidimensional screening a convex program?," Journal of Economic Theory, Elsevier, vol. 146(2), pages 454-478, March.
    2. Dewatripont,Mathias & Hansen,Lars Peter & Turnovsky,Stephen J. (ed.), 2003. "Advances in Economics and Econometrics," Cambridge Books, Cambridge University Press, number 9780521818728.
    3. Baron, David P & Myerson, Roger B, 1982. "Regulating a Monopolist with Unknown Costs," Econometrica, Econometric Society, vol. 50(4), pages 911-930, July.
    4. Spence, Michael, 1974. "Competitive and optimal responses to signals: An analysis of efficiency and distribution," Journal of Economic Theory, Elsevier, vol. 7(3), pages 296-332, March.
    5. Dewatripont,Mathias & Hansen,Lars Peter & Turnovsky,Stephen J. (ed.), 2003. "Advances in Economics and Econometrics," Cambridge Books, Cambridge University Press, number 9780521524131.
    6. Mussa, Michael & Rosen, Sherwin, 1978. "Monopoly and product quality," Journal of Economic Theory, Elsevier, vol. 18(2), pages 301-317, August.
    7. Basov, Suren, 2001. "Hamiltonian approach to multi-dimensional screening," Journal of Mathematical Economics, Elsevier, vol. 36(1), pages 77-94, September.
    8. Dewatripont,Mathias & Hansen,Lars Peter & Turnovsky,Stephen J. (ed.), 2003. "Advances in Economics and Econometrics," Cambridge Books, Cambridge University Press, number 9780521818742.
    9. Dewatripont,Mathias & Hansen,Lars Peter & Turnovsky,Stephen J. (ed.), 2003. "Advances in Economics and Econometrics," Cambridge Books, Cambridge University Press, number 9780521524117.
    10. Monteiro, Paulo K. & Page Jr., Frank H., 1998. "Optimal selling mechanisms for multiproduct monopolists: incentive compatibility in the presence of budget constraints," Journal of Mathematical Economics, Elsevier, vol. 30(4), pages 473-502, November.
    11. Armstrong, Mark, 1996. "Multiproduct Nonlinear Pricing," Econometrica, Econometric Society, vol. 64(1), pages 51-75, January.
    12. Carlier, Guillaume, 2001. "A general existence result for the principal-agent problem with adverse selection," Journal of Mathematical Economics, Elsevier, vol. 35(1), pages 129-150, February.
    13. J. A. Mirrlees, 1971. "An Exploration in the Theory of Optimum Income Taxation," Review of Economic Studies, Oxford University Press, vol. 38(2), pages 175-208.
    14. Jean-Charles Rochet & Philippe Chone, 1998. "Ironing, Sweeping, and Multidimensional Screening," Econometrica, Econometric Society, vol. 66(4), pages 783-826, July.
    15. Dewatripont,Mathias & Hansen,Lars Peter & Turnovsky,Stephen J. (ed.), 2003. "Advances in Economics and Econometrics," Cambridge Books, Cambridge University Press, number 9780521818735.
    16. Dewatripont,Mathias & Hansen,Lars Peter & Turnovsky,Stephen J. (ed.), 2003. "Advances in Economics and Econometrics," Cambridge Books, Cambridge University Press, number 9780521524124.
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    Cited by:

    1. Moser, Christian & Olea de Souza e Silva, Pedro, 2019. "Optimal Paternalistic Savings Policies," MPRA Paper 95383, University Library of Munich, Germany.

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    More about this item

    Keywords

    Principal–agent problem; Multi-dimensional screening; Monopoly; Asymmetric information; Convexity; Unequal dimensions; Optimal transportation; Exclusion;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • D42 - Microeconomics - - Market Structure, Pricing, and Design - - - Monopoly
    • D82 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Asymmetric and Private Information; Mechanism Design
    • H21 - Public Economics - - Taxation, Subsidies, and Revenue - - - Efficiency; Optimal Taxation

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