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On an “Important Principle” of Arrow and Debreu

  • Arnis Vilks ORCID logo EMAIL logo

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.

JEL Classifications: C62; D41; D51; J64

Corresponding author: Arnis Vilks, HHL Leipzig Graduate School of Management, Leipzig, Germany, E-mail:

Acknowledgement

Helpful comments by Tony Bryant and Harald Wiese on an earlier version of this paper are gratefully acknowledged.

References

Acemoglu, D., and P. Restrepo. 2020. “Robots and Jobs: Evidence from US Labor Markets.” Journal of Political Economy 128: 2188–244. https://doi.org/10.1086/705716.Search in Google Scholar

Aliprantis, C., D. Brown, and O. Burkinshaw. 1989. Existence and Optimality of Competitive Equilibria. Berlin: Springer.10.1007/978-3-662-21893-8Search in Google Scholar

Arrow, K., and G. Debreu. 1954. “Existence of an Equilibrium for a Competitive Economy.” Econometrica 22: 265–90. https://doi.org/10.2307/1907353.Search in Google Scholar

Arrow, K., and F. Hahn. 1971. General Competitive Analysis. Amsterdam: North-Holland.Search in Google Scholar

Brady, D., A. Fullerton, and J. M. Cross. 2010. “More Than Just Nickels and Dimes: A Cross-National Analysis of Working Poverty in Affluent Democracies.” Social Problems 57: 559–85. https://doi.org/10.1525/sp.2010.57.4.559.Search in Google Scholar

Bryant, W. D. A. 2010. General Equilibrium. Theory and Evidence. New Jersey: World Scientific Publishing.10.1142/6875Search in Google Scholar

Coles, J. L., and P. J. Hammond. 1995. “Walrasian Equilibrium without Survival: Existence, Efficiency, and Remedial Policy.” In Development, Welfare and Ethics: A Festschrift for Amartya Sen, edited by K. Basu, P. Pattanaik, and K. Suzumura, Oxford: Oxford University Press.Search in Google Scholar

Collier, P. 2018. The Future of Capitalism. London: Penguin.Search in Google Scholar

Debreu, G. 1959. Theory of Value. An Axiomatic Analysis of Economic Equilibrium. New York: Wiley.Search in Google Scholar

Gale, D. 1976. “The Linear Exchange Model.” Journal of Mathematical Economics 3: 205–9. https://doi.org/10.1016/0304-4068(76)90029-x.Search in Google Scholar

Koopmans, T. C. 1957. “Allocation of Resources and the Price System.” In Three Essays on the State of Economic Science, 1–126. New York: McGraw-Hill.Search in Google Scholar

MasColell, A., M. D. Whinston, and J. R. Green. 1995. Microeconomic Theory. Oxford: Oxford University Press.Search in Google Scholar

McKenzie, L. W. 1959. “On the Existence of General Equilibrium for a Competitive Market.” Econometrica 27: 54–71. https://doi.org/10.2307/1907777.Search in Google Scholar

McKenzie, L. W. 1961. “On the Existence of General Equilibrium: Some Corrections.” Econometrica 29: 247–8.Search in Google Scholar

Moore, J. C. 1975. “The Existence of “Compensated Equilibrium” and the Structure of the Pareto Efficiency Frontier.” International Economic Review 16: 267–300. https://doi.org/10.2307/2525812.Search in Google Scholar

Moore, J. C. 2007. General Equilibrium and Welfare Economics. An Introduction. Berlin: Springer.Search in Google Scholar

OECD 2020. Unemployment Rates. OECD - Updated November 2020. http://www.oecd.org/sdd/labour-stats/unemployment-rates-oecd-update-november-2020.htm (accessed January 15, 2020).Search in Google Scholar

Piketty, T. 2014. Capital in the Twenty-First Century. Cambridge: Belknap Press.10.4159/9780674369542Search in Google Scholar

Sen, A. K. 1981. “Ingredients of Famine Analysis: Availability and Entitlements.” Quarterly Journal of Economics 96: 433–64.10.2307/1882681Search in Google Scholar

Stiglitz, J. E. 2015. The Great Divide. London: Penguin.Search in Google Scholar

Susskind, D. 2020. A World without Work: Technology, Automation, and How We Should Respond. New York: Metropolitan Books.Search in Google Scholar

Received: 2021-05-26
Accepted: 2021-07-18
Published Online: 2021-08-17

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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