Abstract
We provide a generalization of the Gale–Nikaido–Debreu’s lemma for discontinuous excess demand in the light of recent work on the existence of equilibria in games with discontinuous payoffs. The standard upper hemicontinuity property of the excess demand is replaced by the weaker concept of “continuous inclusion property” introduced by He and Yannelis (J Math Anal Appl 450(2):1421–1433, 2017) and we allow for the cone P of admissible prices to be general enough to cover cases for which commodities cannot be freely disposed of.
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Notes
Assume, for example, that the total production set \(Y \subseteq {\mathbb {R}}^\ell \) satisfies the free disposal assumption \(Y - P^0 \subseteq Y\), for some closed convex cone \(P\subseteq {\mathbb {R}}^\ell \); then profit maximization leads to prices belonging to the given cone P.
When P is degenerate, that is, \(P=\{0\}\), then \(\mathrm{co}\,[P\cap S ]=\emptyset \) and the result cannot hold.
Take \(P={\mathbb {R}}^\ell \) and consider the (single-valued) correspondence \(Z:B\rightarrow {\mathbb {R}}^\ell \) defined by \(Z(p):=\{-p\}\). Then \(\emptyset \ne Z(p^*) \cap P^0 \) if and only if \(p^*=0\), which is the unique equilibrium point of Z in the unit ball \(B= \mathrm{co}\,[P\cap S ]\).
The first idea to get a global selection is to define \({\bar{Z}}\) as follows: \({\bar{Z}}(p) := \cup _{\{i\in I: p\in O_i\}}Z_i(p)\). However, the correspondence \({\bar{Z}}\) may not be upper hemicontinuous.
Note that this step is not needed in the proof of Theorem 2, which only needs to consider Step 2 with Z satisfying the Strong Walras’ law.
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Many valuable discussions with Nicholas Yannelis at an earlier stage allowed to improve the paper.
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Cornet, B. The Gale–Nikaido–Debreu lemma with discontinuous excess demand. Econ Theory Bull 8, 169–180 (2020). https://doi.org/10.1007/s40505-019-00181-5
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DOI: https://doi.org/10.1007/s40505-019-00181-5