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Lagrange Multipliers as Marginal Rates of Substitution in Multi‐Constraint Optimization Problems
Author(s) -
Weber Christian E.
Publication year - 2001
Publication title -
metroeconomica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.256
H-Index - 29
eISSN - 1467-999X
pISSN - 0026-1386
DOI - 10.1111/1467-999x.00106
Subject(s) - lagrange multiplier , constraint (computer aided design) , substitution (logic) , mathematical optimization , dual (grammatical number) , economics , constraint algorithm , portfolio , function (biology) , mathematical economics , rationing , karush–kuhn–tucker conditions , optimization problem , mathematics , computer science , finance , art , health care , geometry , literature , evolutionary biology , biology , programming language , economic growth
This paper shows that, when a function is optimized subject to several binding constraints, some of the Lagrange multipliers in the dual problems can be interpreted as marginal rates of substitution among certain arguments in the generalized indirect objective function for the primal problem. It also shows how to calculate these Lagrange multipliers from observable price–quantity data. Three particular examples are discussed: a firm that minimizes costs subject to both fixed output and rationing constraints, a household that maximizes utility subject to both income and time constraints, and portfolio choice under uncertainty treated as a multiple constraint optimization problem.