An abstract extremal principle with applications to welfare economics
type: Item
instance of: scholarly article
title: An abstract extremal principle with applications to welfare economics
zbMATH Open document ID: 1016.91078
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-08-05T00:00:00Z
review text: The author considers a model of welfare economics with a commodity space being a Banach space \(E\). The model involves \(n\) consumers with consumption sets \(C_i\subset E\), \(i=1,\dots,n\). Each consumer has a preference set \(P_i(x)\) that consists of elements in \(C_i\) preferred to \(x_i\) by this consumer at the consumption plan \(x=(x_i)\in C_1\times\cdots\times C_n\). The generalized preference relation is given by \(n\) multifunctions \(P_i\colon C_1\times\cdots\times C_n\to C_i\). The model also involves \(m\) firms, technological posibilities of which are represented by sets \(S_j\subset E\), \(j=1,\dots,m\). Given a net demand constraint set \(W\subset E\), \(x=(x_i)\in C_1\times\cdots\times C_n\) and \(y=(y_j)\in S_1\times\cdots\times S_m\), \((x,y)\) is feasible allocation if \(\sum_ix_i-\sum_jy_j\in W\). In this context the author considers three generalized notions of Pareto optimal allocations which reduce to conventional concepts of Pareto optimality for economic models with preference relations given by some preorder and/or utility functions. The main results are necessary conditions for all three notions of Pareto optimal allocations. The proof is based on reducing the three notions to locally extremal points of some systems of sets, and applying a new general extremal principle obtained in terms of abstract prenormal and normal structures in Banach spaces.
Mathematics Subject Classification ID: 91B54
Mathematics Subject Classification ID: 58E17
Mathematics Subject Classification ID: 90B50
zbMATH DE Number: 1545042
zbMATH Keywords: Pareto optimality
zbMATH Keywords: welfare economics
zbMATH Keywords: prenormal and normal structures
zbMATH Keywords: normal cones and subdifferentials
author: Boris S. Mordukhovich
reviewed by: Rimas Norvaiša
MaRDI profile type: Publication
full work available at URL: jmaa.2000.7041
OpenAlex ID: W2074772521
cites work: Positive operators on Krein spaces
cites work: Q4070950
cites work: Valuation equilibrium and Pareto optimum in non-convex economies
cites work: A nonconvex separation property in Banach spaces
cites work: Compactly epi-Lipschitzian convex sets and functions in normed spaces
cites work: On the equivalence of some basic principles in variational analysis
cites work: Tangential approximations
cites work: Viscosity Solutions and Viscosity Subderivatives in Smooth Banach Spaces with Applications to Metric Regularity
cites work: Q4000274
cites work: Q5588318
cites work: Prices and Pareto optima
cites work: Pareto Optimality in Non-Convex Economies
cites work: Approximate subdifferentials and applications 3: the metric theory
cites work: Fuzzy principles and characterization of trustworthiness
cites work: Q4524521
cites work: Q4524523
cites work: A nonconvex separation property and some applications
cites work: Ioffe's normal cone and the foundations of welfare economics: The infinite dimensional theory
cites work: Q3896139
cites work: Maximum principle in the problem of time optimal response with nonsmooth constraints
cites work: Q3785473
cites work: Generalized differential calculus for nonsmooth and set-valued mappings
cites work: The extremal principle and its applications to optimization and economics
cites work: Extremal characterizations of asplund spaces
cites work: Nonsmooth sequential analysis in Asplund spaces
cites work: Nonconvex differential calculus for infinite-dimensional multifunctions
cites work: Convex functions, monotone operators and differentiability.
cites work: Generalized Directional Derivatives and Subgradients of Nonconvex Functions
cites work: Variational Analysis
cites work: The equivalence of several basic theorems for subdifferentials
DOI: 10.1006/JMAA.2000.7041
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description: scientific article; zbMATH DE number 1545042