Abstract
This paper provides four theorems on the existence of a free-disposal equilibrium in a Walrasian economy: the first with an arbitrary set of agents with compact consumption sets, the next highlighting the trade-offs involved in the relaxation of the compactness assumption, and the last two with a countable set of agents endowed with a weighting structure. The results generalize theorems in the antecedent literature pioneered by Shafer–Sonnenschein in 1975, and currently in the form taken in He–Yannelis 2016. The paper also provides counterexamples to the existence of non-free-disposal equilibrium in cases of both a countable set of agents and an atomless measure space of agents. One of the examples is related to one Chiaki Hara presented in 2005. The examples are of interest because they satisfy all the hypotheses of Shafer’s 1976 result on the existence of a non-free-disposal equilibrium, except for the assumption of a finite set of agents. The work builds on recent work of the authors on abstract economies, and contributes to the ongoing discussion on the modelling of “large” societies.
Highlights
The authors are grateful to Hugo Sonnenschein for stimulating correspondence and Irfan Alam, Liuchun Deng, Rich McLean and Nicholas Yannelis for conversation and collaboration
There is no aggregation of consumptions and endowments, while in exchange economies, aggregation is central to the definitions of allocation and Walrasian equilibrium
In a model containing goods like food, shelter, ..., and oxygen, society is perfectly happy with a free-disposal equilibrium in which the demand for oxygen is strictly less than the supply, and the price of oxygen is zero
Summary
For the necessary background to nonstandard analysis we refer the reader to Arkeryd et al (1996), Cutland et al (1995) and Loeb and Wolff (2000). We begin this section by a brief introduction to the relevant notation from nonstandard analysis. We use ∗ to denote the nonstandard extension map taking elements, sets, functions, relations, etc., to their nonstandard counterparts. Topological space (Y , T ) and let y ∈ Y , the monad of ∗ y, denoted by μ(∗ y) is defined to be the set {∗U : y ∈ U ∈ T }. We use NS(∗Y ) to denote the subset of near-standard elements in ∗Y. We let st : NS(∗Y ) → Y denote the standard part map taking near-standard elements to their standard parts. In both cases, the notation elides the underlying space Y and the topology T , because the space and topology will always be clear from context. X is compact if every point of ∗ X is near-standard
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