Abstract
The existence of competitive equilibrium for a large production economy over the commodity space \(\ell ^{\infty }\) in which infinitely many indivisible commodities are present will be proved. When indivisible commodities exist in the market, we cannot assume that the consumption set or the preferences are convex. Hence, we will define the economy as a distribution on the space of consumers’ characteristics following (Hart and Kohlberg in J Math Econ 1:167–174, 1974) and prove the existence of equilibria without the convexity assumptions invoking the technique of the dispersed endowments (Mas-Colell in J Econ Theory 16:443–456, 1977) and (Yamazaki in Econometrica 46:541–555, 1978). The realization of the equilibria via the individual form of economies on a saturated measure space is also discussed.
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Notes
Here \(\bar{\mathcal {B}}([0,1])\) stands for the completion of the Borel \(\sigma \)-algebra, or the Lebesgue measurable \(\sigma \)-algebra, and \(\bar{\ell }\) is the Lebesgue measure.
References
Aumann, R.J.: Markets with a continuum of traders. Econometrica 32, 39–50 (1964)
Aumann, R.J.: Existence of competitive equilibria in markets with a continuum of traders. Econometrica 34, 1–17 (1966)
Bewley, T.F., A very weak theorem on the existence of equilibria in atomless economies with infinitely many commodities. In: Ali Khan, M., Yannelis, N. (eds.) Equilibrium theory in infinite dimensional spaces. Springer, Berlin (1991)
Bewley, T.F.: Existence of equilibria with infinitely many commodities. J. Econ. Theory 4, 514–540 (1972)
Boyd, J.H.III., McKenzie, L.W.: The existence of competitive equilibrium over an infinite horizon with production and general consumption sets. Int. Econ. Rev. 24, 1–20 (1991)
Carmona, G., Podczeck, K.: On the existence of pure-strategy equilibria in large games. J. Econ. Theory 144, 1300–1319 (2009)
Debreu, G.: New concepts and techniques for equilibrium analysis. Int. Econ. Rev. 3, 257–273 (1962)
Debreu, G., Scarf, H.: A limit theorem on the core of an economy. Int. Econ. Rev. 3, 257–273 (1963)
Diestel, J., Uhl, J.J.: Vector measures, mathematical surveys and monographs 15. American Mathematical Society (1977)
Hart, S., Kohlberg, E.: On equally distributed correspondences. J. Math. Econ. 1, 167–174 (1974)
Hildenbrand, W.: Core and equilibria of a large economy. Princeton University Press, Princeton (1974)
Keisler, H.J., Sun, Y.: Why saturated probability spaces are necessary. Adv. Math. 221, 1584–1607 (2009)
Khan, M.A., Yannelis, N.C.: Equilibria in markets with a continuum of agents and commodities. In: Equilibrium theory in infinite dimensional spaces. Ali Khan, M., Yannelis, N. (eds), Springer, Berlin (1991)
Khan, M.A., Rath, K.P., Yu, H., Zhang, Y.: Strategic representation and realization of large distributional games, forthcoming in Economic Theory (2013)
Khan, M.A., Sagara, N., Suzuki, T.: An exact Fatou’s lemma for Gelfand integrals: equivalence of the saturation and Fatou properties, mimeo (2014a)
Khan, M.A., Sagara, N., Suzuki, T.: An exchange economy with differentiated commodities and a saturated measure space of consumers, mimeo (2014b)
Khan, M.A., Sagara, N., Suzuki, T.: On a theorem of Mas–Colell for a model with differentiated commodities, mimeo (2014c)
Khan, M.A., Yamazaki, A.: On the cores of economies with indivisible commodities and a continuum of traders. J. Econ. Theory 24, 218–225 (1981)
Mas-Colell, A.: A model of equilibrium with differentiated commodities. J. Math. Econ. 2, 263–296 (1975)
Mas-Colell, A.: Indivisible commodities and general equilibrium theory. J. Econ. Theory 16, 443–456 (1977)
McKenzie, L.W.: On the existence of competitive equilibrium for a competitive market. Econometrica 27, 54–71 (1959)
McKenzie, L.W.: Classical general equilibrium theory. MIT Press, Cambridge (2002)
Noguchi, M.: Economies with a continuum of consumers, a continuum of suppliers, and an infinite dimensional commodity space. J. Math. Econ. 27, 1–21 (1997a)
Noguchi, M.: Economies with a continuum of agents with the commodity-price pairing \((\ell ^{\infty },\ell ^1)\). J. Math. Econ. 28, 265–287 (1997b)
Noguchi, M.: Existence of nash equilibria in large games. J. Math. Econ. 45, 168–184 (2009)
Podczeck, K.: On the convexity and compactness of the integral of a Banach space valued correspondence. J. Math. Econ. 44, 836–852 (2008)
Royden, H.W.: Real analysis 3-rd edition. Macmillan, London (1988)
Sagara, N., Suzuki, T.: Exchange economies with infinitely many commodities and a saturated measure space of consumers, mimeo (2014)
Scarf, H.E.: Notes on the core of a production economy, in contributions to mathematical economics. In: Hildenbrand W., Mas-Colell, A. (eds.) Contribution to mathematical economics: in honor of Gerald Debreu, North Holland (1986)
Sun, Y.N., Yannelis, N.C.: Saturation and the integration of Banach valued correspondences. J. Math. Econ. 44, 861–865 (2008)
Suzuki, T.: General equilibrium analysis of production and increasing returns. World Scientific, New Jersey (2009)
Suzuki, T.: Core and Competitive equilibria of a coalitional exchange economy with infinite time horizon. J. Math. Econ. 49, 234–244 (2013a)
Suzuki, T.: Competitive equilibria of a large exchange economy on the commodity space \(\ell ^{\infty }\). Adv. Math. Econ. 17, 121–138 (2013b)
Suzuki, T.: An elementary proof of an infinite dimensional Fatou’s lemma with an application to market equilibrium analysis. J. Pure Appl. Math. 10, 159–182 (2013c)
Thomas, G.E.F.: Integration of functions with values in locally convex suslin spaces. Trans. Am. Math. Soc. 212, 61–81 (1975)
Yamazaki, A.: An equilibrium existence theorem without convexity assumptions. Econometrica 46, 541–555 (1978)
Yamazaki, A.: Diversified consumption characteristics and conditionally dispersed endowment distribution: regularizing effect and existence of equilibria. Econometrica 49, 639–654 (1981)
Yannelis, N.: Integration of Banach-valued correspondences. In: Equilibrium theory in infinite dimensional spaces. Ali Khan, M., Yannelis, N. (eds.), Springer, Berlin (1991)
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Earlier versions of the paper were presented at a seminar held at Keio University. I thank participants of the seminars, in particular, T. Maruyama, H. Ozaki and S.-I. Suda. Discussions with M. Ali Khan and N. Sagara were also helpful. The comments and encouragement of Jean-Michel Grandmont are gratefully acknowledged. I am also indebted to an anonymous referee of this journal in order to clarify several ambiguous points of earlier versions of the paper. Of course, remaining errors are solely my own.
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Suzuki, T. A coalitional production economy with infinitely many indivisible commodities. Econ Theory Bull 4, 35–52 (2016). https://doi.org/10.1007/s40505-015-0067-7
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DOI: https://doi.org/10.1007/s40505-015-0067-7
Keywords
- Competitive equilibrium
- Large production economy
- Infinite-Dimensional commodity spaces
- Indivisible commodities