Some results on the existence of utility functions on path connected spaces
type: Item
instance of: scholarly article
title: Some results on the existence of utility functions on path connected spaces
zbMATH Open document ID: 0642.90018
DOI: 10.1016/0304-4068(87)90004-8
author: Paulo Monteiro
published in: Journal of Mathematical Economics
publication date: 1987-01-01T00:00:00Z
review text: This paper concerns the existence of utility representations for preferences defined on a path connected space X. This includes any convex set. A classical result of \textit{S. Eilenberg} [Amer. J. Math. 63, 39-45 (1941)] proves the existence of utility representations when the consumption set is connected and separable. In an infinite dimensional space the above result may not be useful, because we lack, in general, the separability of the space. The non-separable spaces \(L^{\infty}\) and ca(K) are typical examples in mathematical economics. In this paper we show that a continuous preference relation \(\succcurlyeq\), on X, has a continuous utility representation if and only if it is countably bounded, i.e., there is some countable subset F of X such that for all x in X there exist y and z in F with \(y\succcurlyeq x\succcurlyeq z\). An easy corollary states that any continuous preference which has a best and a worst point has a continuous representation. We also obtain a convex continuous preference on a Banach lattice that has not a utility representation, because it is not countably bounded.
Mathematics Subject Classification ID: 91B16
zbMATH DE Number: 4047386
zbMATH Keywords: existence of utility representations
zbMATH Keywords: path connected space
zbMATH Keywords: continuous preference relation
zbMATH Keywords: countably bounded
zbMATH Keywords: Banach lattice
MaRDI profile type: Publication
full work available at URL: 0304-4068(87)90004-8
OpenAlex ID: W2042799260
cites work: Continuity Properties of Paretian Utility
cites work: Utility functions on ordered convex sets
cites work: The Price Equilibrium Existence Problem in Topological Vector Lattices
cites work: Q5555984
cites work: Q5613067
Wikidata QID: Q126657316
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description: scientific article; zbMATH DE number 4047386