Abstract
In an exchange economy with a continuum of traders, we establish the equivalence theorem on the core and the set of competitive allocations without assuming monotonicity of traders’ preferences. Under weak assumptions we provide two alternative core equivalence theorems. The first one is for irreducible economies under Debreu’s assumption on quasi-equilibria. The second one is an extension of Aumann’s theorem under weaker assumptions than monotonicity.
Received: June 2, 2009
Revised: October 23, 2009
JEL classification: C71, D41, D51
Mathematics Subject Classification (2000): 28A20, 91A12, 91A13, 91B50
We are grateful to Prof. Ezra Einy for his valuable comments and pointing out some related references, and would like to thank an anonymous referee and the editor for their helpful comments.
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Notes
- 1.
Hildenbrand [6] mentioned that the monotonicity is not essential for Aumann’s proof.
- 2.
The notion of quasi-equilibrium was first defined by Debreu [2].
- 3.
This claim is obtained by a simple modification of Lemma 4.1 in Aumann [1].
- 4.
For x and y in \({\mathbb{R}}^{n}\), x ≫ y means that x i > y i for all coordinate i.
- 5.
The integral \({\int \nolimits \nolimits }_{S}\{x \in {\mathbb{R}}_{+}^{n}\mid x {\succ }_{t}f(t)\}\) denotes a set defined by
$$\begin{array}{rcl} \left \{{\int \nolimits \nolimits }_{S}h\mid h : T \rightarrow {\mathbb{R}}_{+}^{n},h(t) {\succ }_{t}f(t)\mathrm{a.e.}t \in T\right \}.& & \\ \end{array}$$ - 6.
“Some traders” means that the set of such traders has a positive measure.
- 7.
By 1 m we denote a vector whose m-th coordinate is 1 and whose other coordinates are 0.
- 8.
S m is the non-null coalition defined in Assumption 6 for each \(m = 1, 2,\ldots, n\).
- 9.
In Problem 9 of Hildenbrand [5, p. 143], he claimed that it is possible to prove an equivalence theorem for an irreducible exchange economy by using an analogous argument to Theorem 1 in Hildenbrand [5, p. 133], but it seems that the monotonicity assumption is indispensable in the argument. However, there is no problem with using the method of proof in Hildenbrand [4].
References
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Honda, J., Takekuma, SI. (2010). A note on Aumann’s core equivalence theorem without monotonicity. In: Kusuoka, S., Maruyama, T. (eds) Advances in Mathematical Economics. Advances in Mathematical Economics, vol 13. Springer, Tokyo. https://doi.org/10.1007/978-4-431-99490-9_2
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