The impact of changes in relative weights on the optimal solution of a maximization problem
type: Item
instance of: scholarly article
title: The impact of changes in relative weights on the optimal solution of a maximization problem
zbMATH Open document ID: 0605.90102
DOI: 10.1016/0304-4068(86)90005-4
published in: Journal of Mathematical Economics
publication date: 1986-01-01T00:00:00Z
review text: Starting point for the analysis is the standard maximization problem under constraints: \[ \max_{x_ 1,...,x_ n}\sum^{n}_{1}w_ iu_ i(x_ i)\quad s.t.\quad f(x_ 1,...,x_ n)\leq 0, \] where \(u_ i(\cdot)\) denotes a concave, monotonically increasing and twice differentiable utility function defined for a positive variable, \(w_ i\) denotes a positive weight attached to \(u_ i(\cdot)\) and f(\(\cdot)\) denotes a twice differentiable function which is increasing and convex. The main subject of the paper is the analysis of the effects of a variation of the weights on the optimal solution of the optimization problem cited above. As most interesting results the author is able to prove the following propositions. Result A: Let \(x^ o\) be the optimal solution for \(w^ o\) and x' be the optimal solution for w'. Then we have \(\sum^{n}_{1}(w^ o_ i-w_ i')(u_ i(x^ o_ i)-u_ i(x_ i'))\geq 0\). This result obviously implies that w and \(x^*\) are positively correlated. Result B: Suppose for example that f(\(\cdot)\) is both convex and Schur- convex and \(u_ i(.)=u(.)\). Then we have \(w_ n\geq w_{n-1}\geq...\geq w_ 1\Leftrightarrow x^*_ n\geq x^*_{n-1}\geq...\geq x^*_ 1\). This result is especially interesting for economic applications. Let for instance the \(w_ i\) be interpreted as discount factors in an intertemporal consumer choice problem. Then result B implies that the optimal consumption stream is decreasing over time.
Mathematics Subject Classification ID: 90C30
Mathematics Subject Classification ID: 91B62
Mathematics Subject Classification ID: 90C90
Mathematics Subject Classification ID: 49M37
zbMATH DE Number: 3978835
zbMATH Keywords: constrained maximization
zbMATH Keywords: weighted objective functions
zbMATH Keywords: variation of the weights
author: Leonard K. Cheng
MaRDI profile type: Publication
full work available at URL: 0304-4068(86)90005-4
OpenAlex ID: W1988789794
cites work: Q5514010
cites work: Solution of a system of functional-differential equations arising from an optimization problem
cites work: Q3888784
cites work: Power and Taxes
cites work: Q3854875
cites work: Multiple objective decision making - methods and applications. A state- of-the-art survey. In collaboration with Sudhakar R. Paidy and Kwangsun Yoon
cites work: Q3950274
cites work: Q4058428
cites work: Another Proof that Convex Functions are Locally Lipschitz
cites work: Portfolio Theory, Asset Demand and Taxation: Comparative Statics with Many Assets
description: scientific article; zbMATH DE number 3978835