Some applications of set-valued mappings in mathematical economics
type: Item
instance of: scholarly article
title: Some applications of set-valued mappings in mathematical economics
zbMATH Open document ID: 0714.90006
DOI: 10.1016/0304-4068(91)90018-O
published in: Journal of Mathematical Economics
publication date: 1991-01-01T00:00:00Z
review text: By a preference on a separable metrizable space X we mean any set-valued mapping R: \(X\to 2^ X\) (y is said to be preferred to x iff \(y\in R(x))\). A real-valued function u on X is said to be a utility function for R if \(y\in R(x)\Rightarrow u(y)\geq u(x)\) and \(y\in R(x)\), \(x\not\in R(y)\Rightarrow u(y)>u(x)\). A characterization of preferences that admit d-Lipschitz utility functions is given, where d is a metric on X. Also characterized are choice functions that can be rationalized by d- Lipschitz or continuous utility functions. The asymptotic behavior of a dynamic system determined by R is another subject studied in the paper. The trajectories of such a system are sequences \(\chi =(\chi (t))^{\infty}_{t=0}\) with \(\chi\) (t)\(\in R(\chi (t-1))\), \(t=1,2,...\), where R is supposed to be non-void-compact- valued. Let H(R) denote the set of continuous isotone functions on X, \[ W_ u:=\{x:\;u(x)=\min_{y\in R(x)}u(y)\},\text{ and } W:=\cap_{u\in H(R)}W_ u. \] The notion of a d-quasi-periodic point is defined and the set W is characterized by the equality \(W=\cap_{d\in D}Q_ d\), where \(Q_ d\) is the set of all d-quasi-periodic points, and D is the set of all bounded metrics compatible with the topology of X. Yet another characterization of W as a global structure of the dynamic system is given.
Mathematics Subject Classification ID: 91B16
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 93A10
Mathematics Subject Classification ID: 91B62
Mathematics Subject Classification ID: 91B08
Mathematics Subject Classification ID: 93C55
Mathematics Subject Classification ID: 34D45
zbMATH DE Number: 4176769
zbMATH Keywords: set-valued mapping
zbMATH Keywords: d-Lipschitz utility functions
zbMATH Keywords: d-quasi-periodic point
zbMATH Keywords: global structure
author: Vladimir L. Levin
MaRDI profile type: Publication
cites work: Q3324260
cites work: Q3251751
cites work: Stability theory of dynamical systems.
cites work: Q5614853
cites work: Q5832752
cites work: Q3688668
cites work: An equilibrium existence theorem for a general model without ordered preferences
cites work: Nontransitive-nontotal consumer theory
cites work: Q3217870
cites work: Lipschitz pre-orders and Lipschitz utility functions
cites work: Q3724046
cites work: Q3768707
cites work: Stable Sets and Stable Points of Set-Valued Dynamic Systems with Applications to Game Theory
cites work: Duality and rationality
cites work: Q3918614
cites work: Equilibrium in abstract economies without ordered preferences
full work available at URL: 0304-4068(91)90018-o
OpenAlex ID: W2077005992
description: scientific article; zbMATH DE number 4176769