CONCAVITY AND CONVEXITY OF FUNCTIONS

Vasileios Zikos

Chapter 3 in Essential Mathematics for Economics:A Student's Guide to Key Mathematical Techniques, 2026, pp 39-56 from World Scientific Publishing Co. Pte. Ltd.

Abstract: After completing this chapter, students will be able to:Understand the difference between a convex function and a convex set.Derive the Hessian matrix and the bordered Hessian matrix.Use the Hessian matrix to identify concave and convex functions.Use the bordered Hessian matrix to identify quasi-concave and quasi-convex functions.The present chapter focuses on the shape of functions. In doing so, it aims to take the reader one step closer to the study of optimization problems. An objective is to introduce and discuss the concept of a convex set, which is essential as concave or convex functions typically need their domains to be a convex set. The chapter then describes conditions under which a function is concave or convex. For a function of one variable, a straightforward condition for concavity and convexity is provided by the second derivative, whereas for functions of more than one variable the Hessian matrix is employed. Finally, the notion of concavity and convexity is generalized by describing conditions under which a function is quasi-concave and quasi-convex.

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
References: Add references at CitEc
Citations:

Downloads: (external link)
https://www.worldscientific.com/doi/pdf/10.1142/9789819814114_0003 (application/pdf)
https://www.worldscientific.com/doi/abs/10.1142/9789819814114_0003 (text/html)
Ebook Access is available upon purchase.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:wsi:wschap:9789819814114_0003

Ordering information: This item can be ordered from

Access Statistics for this chapter

More chapters in World Scientific Book Chapters from World Scientific Publishing Co. Pte. Ltd.
Bibliographic data for series maintained by Tai Tone Lim ().