UNCONSTRAINED OPTIMIZATION

Vasileios Zikos

Chapter 4 in Essential Mathematics for Economics:A Student's Guide to Key Mathematical Techniques, 2026, pp 57-71 from World Scientific Publishing Co. Pte. Ltd.

Abstract: After completing this chapter, students will be able to:Understand the distinction between necessary and sufficient conditions for maximum and minimum points.Identify the optimal point of a function with one or more variables.Perform a comparative static analysis using the envelope theorem.Apply calculus to identify a firm’s profit-maximizing output or a consumer’s demand for goods.This chapter is concerned with unconstrained optimization problems, where an objective function is optimized in the absence of any constraint. The chapter presents necessary and sufficient conditions for solving such problems. As we will discuss, the first-order conditions are derived from the partial derivatives of the objective function with respect to each choice variable, and these derivatives are set to zero. The first-order conditions are necessary for optimality and provide candidate points for a local optimum. The next step involves identifying the exact type of each candidate point. To do this, second-order conditions are employed based on the signs of the principal minors of the Hessian matrix.

Keywords: Microeconomics; Applied Microeconomics; Optimization; First-order Conditions; Second-order Conditions; Equality Constraints; Inequality Constraints; Concavity; Convexity; Matrix Algebra; Determinants; Differential Calculus; Integral Calculus; First-order Linear Differential Equations; Utility Maximization; Profit Maximization; Research Applications (search for similar items in EconPapers)
JEL-codes: C01 C02 C61 D00 (search for similar items in EconPapers)
Date: 2026
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