Abstract
We generalize the classical equilibrium existence theorems by dispensing with the assumption of continuity of preferences. Our new existence results allow us to dispense with the interiority assumption on the initial endowments. Furthermore, we allow for non-ordered, interdependent and price-dependent preferences.
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Notes
See the symposium of Carmona (2011) for additional references.
It should be mentioned that independently of our work, Reny (2013) has also obtained related results.
For a correspondence \(F\), \(\text{ co }F\) denotes the convex hull of \(F\).
If the sub-correspondence \(F_x\) has a closed graph and \(X\) is finite dimensional, then \(\text{ co }F_x\) still has a closed graph since the convex hull of a closed set is closed in finite dimensional spaces. However, this may not be true if one works with infinite dimensional spaces. One can easily see that assuming the sub-correspondence \(F_x\) is convex valued and has a closed graph would suffice for our aim.
Reny (2013) proposed a similar condition called “correspondence security” in the setting of discontinuous games and proved an equilibrium existence theorem for an abstract game.
If \(U_i = \emptyset \) for all \(i\), then the correspondence \(\overline{A} = \prod _{i\in I} \overline{A_i}\) is nonempty, convex valued and upper hemicontinuous. As a result, there exists a fixed-point \(x^*\) of \(\overline{A}\) which is an equilibrium.
It should be noted that using the existence of maximal element theorem for L-majorized correspondences (see Yannelis and Prabhakar 1983), it is known that the metrizability assumption is not needed. Indeed, the proof of Borglin and Keiding (1976) remains valid if one replaces the KF-majorization by L-majorization. The existence of maximal element theorem for correspondences having the continuous inclusion property can be used to show that the metrizability in our Theorem 1 is not needed, see Footnote 10.
Suppose that \(X\) is a compact and convex subset of a Hausdorff locally convex linear topological space. Let \(P:X\rightarrow 2^X\) be a correspondence such that \(x \notin \text{ co }P(x)\) for all \(x\in X\). If \(P\) has the continuous inclusion property at each \(x\in X\) such that \(P(x) \ne \emptyset \), then there exists a point \(x^* \in X\) such that \(P(x^*) = \emptyset \).
We allow for very general preferences, which can be interdependent and price-dependent. See McKenzie (1955) and Shafer and Sonnenschein (1975) for more discussions. For agent \(i\), \(y_i \in P_i(x,p)\) means that \(y_i\) is strictly preferred to \(x_i\) provided that all other components are unchanged at the price \(p\in \triangle \).
The commodity space \(X_i\) can be sufficiently large. For example, we can let \(X_i = \{x_i \in \mathbb {R}^l_+ :x_i \le K\cdot \sum _{i \in I} e_i\}\), where \(K\) is an arbitrarily large positive number.
As suggested by an anonymous referee, one could allow \(X_i = \mathbb {R}^l\) by assuming that if \(x_i \in X_i\) and \(x'_i \in P_i(x)\), then also \((1 - \lambda )x_i + \lambda x'_i \in P_i(x)\) for all \(0 < \lambda < 1\). With this assumption one needs to consider only one truncation of the consumption sets (any truncation which contains the feasible consumption points as interior points).
The function \(q(x)\) is viewed as the inner product \(q \cdot x\) when \(q\) is a price vector and \(x\) is an allocation.
If \(z \ne 0\), then there exists a point \(q \in \triangle '\) such that \(q(z) < 0\), which implies that \(-q(z) > 0\). However, \(-q \in \triangle '\), a contradiction.
Such a remark has been also made by Carmona and Podczeck (2015).
The pair \((p^*, x^*)\) is called a free (non-free) disposal quasi-equilibrium if: (1) for each \(i\in I\), \(x^*_i \in B_i(p^*)\); (2) \(x_i \in P_i(x^*, p^*)\) implies that \(p^* \cdot x_i \ge p^* \cdot e_i\); (3) \(\sum _{i\in I} x_i^* \le \sum _{i\in I} e_i\) (\(\sum _{i\in I} x_i^* = \sum _{i\in I} e_i\)).
Given an allocation \(x = (x_1,x_2) = ((x_1^1,x_1^2), (x_2^1,x_2^2))\) in the edgeworth box, the set of allocations which is preferred to \(x\) for agent \(1\) is the set of all points above the curve \(y_1^1 \cdot y_1^2 = x_1^1 \cdot x_1^2\) such that the segment \(\{(y_1^1,y_1^2):y_1^1-x_1^1 = y_1^2 - x_1^2, x_1^1 \le y_1^1<\frac{3}{2} x_1^1\}\) is removed.
For example, one can choose the point \(z_i = (y_i^1 - \epsilon , y_i^2 - 2\epsilon )\), where \(\epsilon \) is a positive number. It is easy to see that if \(\epsilon \) is sufficiently small, then \(z_i\) is an interior point of \(P_i(x)\).
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We are grateful to Conny Podczeck, Pavlo Prokopovych and a referee for comments and suggestions. We also thank Phil Reny for sharing his result in Reny (2013).
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He, W., Yannelis, N.C. Existence of Walrasian equilibria with discontinuous, non-ordered, interdependent and price-dependent preferences. Econ Theory 61, 497–513 (2016). https://doi.org/10.1007/s00199-015-0875-x
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DOI: https://doi.org/10.1007/s00199-015-0875-x