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On an “Important Principle” of Arrow and Debreu

Author

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  • Vilks Arnis

    (HHL Leipzig Graduate School of Management, Leipzig, Germany)

Abstract

In their seminal 1954 paper on the existence of competitive equilibrium, Arrow and Debreu state what they call an “important principle”, namely that it is necessary for the existence of equilibrium that every consumer has some asset or can supply some labour service which has a positive price at equilibrium. It does not seem to have been noticed that this claim is incorrect. We provide a very simple model of a private ownership economy with three goods where a competitive equilibrium exists, but consumers who have nothing to sell but their labour end up with zero wealth in equilibrium. As zero wealth must be taken to mean non-survival, and the Arrow–Debreu model is frequently interpreted as assuming that all consumers can survive without trade, we also briefly discuss the issue of non-survival in equilibrium. We finally point out that our example illustrates the possibility that technological progress may result in a situation where the value of work becomes negligible.

Suggested Citation

  • Vilks Arnis, 2022. "On an “Important Principle” of Arrow and Debreu," The B.E. Journal of Theoretical Economics, De Gruyter, vol. 22(2), pages 621-627, June.
    • Handle: RePEc:bpj:bejtec:v:22:y:2022:i:2:p:621-627:n:2
    DOI: 10.1515/bejte-2021-0074
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    More about this item

    Keywords

    Arrow–Debreu model; competitive equilibrium; survival without trade; irreducibility; robots; working poor;
    All these keywords.

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • D41 - Microeconomics - - Market Structure, Pricing, and Design - - - Perfect Competition
    • D51 - Microeconomics - - General Equilibrium and Disequilibrium - - - Exchange and Production Economies
    • J64 - Labor and Demographic Economics - - Mobility, Unemployment, Vacancies, and Immigrant Workers - - - Unemployment: Models, Duration, Incidence, and Job Search

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