Abstract
The existence of competitive equilibrium for a large exchange economy over the commodity space ℓ ∞ will be discussed. We define the economy as a distribution on the space of consumers’ characteristics following Hart and Kohlberg (J. Math. Econ. 1:167–174, 1974), and prove the theorem without the assumption of convexity of preferences.
Earlier versions of the paper were presented at seminars held at Kobe University and Keio University. I thank participants of the seminars, in particular, Toru Maruyama and Nobusumi Sagara. I also want to thank Mitsunori Noguchi. At each stage of the research, his comments have been most helpful. An anonymous referee of this journal pointed out several mistakes of the paper. Of course, remaining errors are my own.
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- 1.
Since the space ℓ ∞ is not separable, these results are not considered as generalizations of Bewley [3]. However, Khan–Yannelis pointed out that their result includes the space L ∞(Σ), the space of essentially bounded measurable functions on a finite measure space Σ.
- 2.
Bewley [5] also used this approach.
- 3.
\(\nu _{\mathcal{P}\times \Omega }\) is the marginal distribution of ν on \(\mathcal{P}\times \Omega \).
- 4.
Note that Mas–Colell [12] could not use the individual form, since he included the indivisible commodities in his model, hence actually he could not assume the convexity. Jones [9] showed that the indivisibility and bounded assumption on the consumption set are not necessary for the equilibrium existence theorem for the economies of the coalitional form with the commodity space ca(K). Ostroy and Zame [15] further developed the works of Mas–Colell and Jones, and proved the existence and the core equivalence theorems for an economy of the individual form with the commodity space ca(K) in which the consumption set is the positive orthant of ca(K). However, they also had to assume the convexity on the preferences. For existence theorems without the convexity of preferences for the individual form, see Rustichini–Yannelis [19] and Podczeck [16].
- 5.
An anonymous referee of this journal kindly informed the author that Theorem A1 is a special case of Proposition 2 in Khan and Yamazaki [10]. However, he decided to keep this appendix, since the proof of Theorem A1 is much simpler than that of Khan–Yamazaki (since their result is more general), and it makes the paper completely self-contained.
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Suzuki, T. (2013). Competitive equilibria of a large exchange economy on the commodity space ℓ∞ . In: Kusuoka, S., Maruyama, T. (eds) Advances in Mathematical Economics Volume 17. Advances in Mathematical Economics, vol 17. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54324-4_4
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