IDEAS home Printed from https://ideas.repec.org/p/mpg/wpaper/2023_05.html
   My bibliography  Save this paper

The Economics of Hilbert's Hotel: An Expository Note

Author

Listed:
  • Martin F. Hellwig

    (Max Planck Institute for Research on Collective Goods, Bonn)

Abstract

This expository note uses Hilbert's "infinite hotel", a hotel where one can always find place for another guest even if the hotel is already full, to illustrate the failure of the First Welfare Theorem in "large-square" economies that have infinitely many participants as well as infinitely many goods. Hilbert's hotel with infinitely many guests has a similar mathematical structure as the overlapping-generations model of Allais (1947) and Samuelson (1958). The phenomenon of "dynamic inefficiency" in such models represents a failure of the First Welfare Theorem in "large-square" economies, rather than frictions from the sequential nature of markets.

Suggested Citation

  • Martin F. Hellwig, 2023. "The Economics of Hilbert's Hotel: An Expository Note," Discussion Paper Series of the Max Planck Institute for Research on Collective Goods 2023_05, Max Planck Institute for Research on Collective Goods.
    • Handle: RePEc:mpg:wpaper:2023_05
    as

    Download full text from publisher

    File URL: https://www.coll.mpg.de/pdf_dat/2023_05online.pdf
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Balasko, Yves & Shell, Karl, 1980. "The overlapping-generations model, I: The case of pure exchange without money," Journal of Economic Theory, Elsevier, vol. 23(3), pages 281-306, December.
    2. Martin F. Hellwig, 2023. "Overlapping-Generations Economies under Uncertainty: Dynamic Inefficiency/Efficiency with Multiple Assets and no Labour," Discussion Paper Series of the Max Planck Institute for Research on Collective Goods 2023_04, Max Planck Institute for Research on Collective Goods.
    3. Tirole, Jean, 1985. "Asset Bubbles and Overlapping Generations," Econometrica, Econometric Society, vol. 53(6), pages 1499-1528, November.
    4. Bewley, Truman F., 1972. "Existence of equilibria in economies with infinitely many commodities," Journal of Economic Theory, Elsevier, vol. 4(3), pages 514-540, June.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Julia, Knolle, 2014. "An Empirical Comparison of Interest and Growth Rates," MPRA Paper 59520, University Library of Munich, Germany.
    2. Zhigang Feng & Jianjun Miao & Adrian Peralta‐Alva & Manuel S. Santos, 2014. "Numerical Simulation Of Nonoptimal Dynamic Equilibrium Models," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 55(1), pages 83-110, February.
    3. Charalambos Aliprantis & Kim Border & Owen Burkinshaw, 1996. "Market economies with many commodities," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 19(1), pages 113-185, March.
    4. Stefano Bosi & Cuong Le Van & Ngoc-Sang Pham, 2017. "Rational Land and Housing Bubbles in Infinite-Horizon Economies," Studies in Economic Theory, in: Kazuo Nishimura & Alain Venditti & Nicholas C. Yannelis (ed.), Sunspots and Non-Linear Dynamics, chapter 0, pages 203-230, Springer.
    5. Florenzano Monique, 1991. "Quasiequilibria in abstract economies application to the overlapping generations models," CEPREMAP Working Papers (Couverture Orange) 9117, CEPREMAP.
    6. Guerrieri, V. & Uhlig, H., 2016. "Housing and Credit Markets," Handbook of Macroeconomics, in: J. B. Taylor & Harald Uhlig (ed.), Handbook of Macroeconomics, edition 1, volume 2, chapter 0, pages 1427-1496, Elsevier.
    7. Mark Pingle & Leigh Tesfatsion, 1993. "``Active Intermediation in a Monetary Overlapping Generations Economy''," Macroeconomics 9312001, University Library of Munich, Germany, revised 04 Dec 1993.
    8. Chattopadhyay, Subir, 2008. "The Cass criterion, the net dividend criterion, and optimality," Journal of Economic Theory, Elsevier, vol. 139(1), pages 335-352, March.
    9. Aiyagari, S. Rao, 1987. "Optimality and monetary equilibria in stationary overlapping generations models with long-lived agents: Growth versus discounting," Journal of Economic Theory, Elsevier, vol. 43(2), pages 292-313, December.
    10. Tomohiro Hirano & Alexis Akira Toda, 2025. "Bubble Necessity Theorem," Journal of Political Economy, University of Chicago Press, vol. 133(1), pages 111-145.
    11. James Sprigg & Mark Ehlen, 2007. "Comparative dynamics in an overlapping-generations model: the effects of quasi-rational discrete choice on finding and maintaining Nash equilibrium," Computational Economics, Springer;Society for Computational Economics, vol. 29(1), pages 69-96, February.
    12. Tomohiro Hirano & Alexis Akira Toda, 2023. "Housing Bubbles with Phase Transitions," Papers 2303.11365, arXiv.org, revised Jul 2024.
    13. Barbie, Martin & Hagedorn, Marcus & Kaul, Ashok, 2000. "Dynamic Efficiency and Pareto Optimality in a Stochastic OLG Model with Production and Social Security," IZA Discussion Papers 209, Institute of Labor Economics (IZA).
    14. Martin F. Hellwig, 2021. "Safe Assets, Risky Assets, and Dynamic Inefficiency in Overlapping-Generations Economies," Discussion Paper Series of the Max Planck Institute for Research on Collective Goods 2021_10, Max Planck Institute for Research on Collective Goods.
    15. Barbie, Martin & Hagedorn, Marcus & Kaul, Ashok, 2001. "Government Debt as Insurance against Macroeconomic Risk," IZA Discussion Papers 412, Institute of Labor Economics (IZA).
    16. Philippe Weil, 2008. "Overlapping Generations: The First Jubilee," Journal of Economic Perspectives, American Economic Association, vol. 22(4), pages 115-134, Fall.
    17. Harald Uhlig, 1998. "Capital Income Taxation and the Sustainability of Permanent Primary Deficits," Palgrave Macmillan Books, in: Steven Brakman & Hans Ees & Simon K. Kuipers (ed.), Market Behaviour and Macroeconomic Modelling, chapter 12, pages 309-337, Palgrave Macmillan.
    18. Brito, Paulo & Dilão, Rui, 2010. "Equilibrium price dynamics in an overlapping-generations exchange economy," Journal of Mathematical Economics, Elsevier, vol. 46(3), pages 343-355, May.
    19. Bertocchi, Graziella & Kehagias, Athanasios, 1995. "Efficiency and optimality in stochastic models with production," Journal of Economic Dynamics and Control, Elsevier, vol. 19(1-2), pages 303-325.
    20. Pingle, Mark & Tesfatsion, Leigh, 1998. "Active Intermediation In Overlapping Generations Economies With Production And Unsecured Debt," Macroeconomic Dynamics, Cambridge University Press, vol. 2(2), pages 183-212, June.

    More about this item

    Keywords

    HilbertÂ’s hotel; overlapping-generations models; dynamic inefficiency; First Welfare Theorem;
    All these keywords.

    JEL classification:

    • D15 - Microeconomics - - Household Behavior - - - Intertemporal Household Choice; Life Cycle Models and Saving
    • D61 - Microeconomics - - Welfare Economics - - - Allocative Efficiency; Cost-Benefit Analysis
    • E62 - Macroeconomics and Monetary Economics - - Macroeconomic Policy, Macroeconomic Aspects of Public Finance, and General Outlook - - - Fiscal Policy; Modern Monetary Theory

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:mpg:wpaper:2023_05. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Marc Martin (email available below). General contact details of provider: https://edirc.repec.org/data/mppggde.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.