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Optimization of Functions of Several Variables

In: Economic Analysis Through Mathematics

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  • Zrinka Lukač

    (University of Zagreb, Faculty of Economics & Business)

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.

Suggested Citation

  • Zrinka Lukač, 2026. "Optimization of Functions of Several Variables," Springer Texts in Business and Economics, in: Economic Analysis Through Mathematics, chapter 18, pages 413-456, Springer.
    • Handle: RePEc:spr:sptchp:978-3-032-08812-3_18
    DOI: 10.1007/978-3-032-08812-3_18
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