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The Economics of Hilbert's Hotel: An Expository Note

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  • 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
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    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)

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    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

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