Optimization of Functions of Several Variables

Zrinka Lukač ()
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Zrinka Lukač: University of Zagreb, Faculty of Economics & Business

Chapter 18 in Economic Analysis Through Mathematics, 2026, pp 413-456 from Springer

Abstract: Abstract We have already seen the importance of optimization for functions of one variable. Since relations between economic variables are often described by functions of several variables, in economic theory the optimization of such functions is even more important. For example, a consumer chooses the basket of goods which maximizes his/her utility, a company selects the combination of inputs which produces a required level of output at minimal costs, etc. In this chapter we derive the necessary and sufficient conditions, also known as the first order and the second order conditions, for the existence of unconstrained extrema of functions of several variables. These conditions are generalizations of conditions that apply to functions of one variable. We state the algorithm for finding the unconstrained extrema of functions of several variables. Finally, we consider the problem of finding the extrema of functions of several variables with one constraint in the form of an equation and present the substitution method and the Lagrange multipliers method for solving it. Also, we derive the interpretation of the Lagrange multiplier.

Keywords: Local extrema; Global extrema; Unconstrained optimization; The first order conditions; The second order condition; Constrained optimization; Substitution method; Lagrange multipliers method; Lagrange multiplier (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sptchp:978-3-032-08812-3_18

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DOI: 10.1007/978-3-032-08812-3_18

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