How to Generate Good Profit Maximization Problems

Abstract

In this article, the author considers the merits of two classes of profit maximization problems: those involving perfectly competitive firms with quadratic and cubic cost functions. While relatively easy to develop and solve, problems based on quadratic cost functions are too simple to address a number of important issues, such as the use of second-order conditions and the short-run shutdown condition. Problems based on cubic cost functions are mathematically richer but often involve messy arithmetic; furthermore, many are not plausible representations of a firm's costs. Finding cubic functions that do not suffer from these drawbacks can be a time-consuming process. The author addresses this issue by providing a procedure to generate profit maximization problems that are theoretically interesting, economically plausible, and computationally simple.

Keywords:

• cost functions
• intermediate microeconomics
• profit maximization

JEL codes:

• A22
• C02
• D21

ACKNOWLEDGEMENTS

The author thanks Steven Schmidt, Caleb Stroup, Yufei Ren, Jeffry Perloff, David Besanko, and Walter Nicholson for comments on earlier drafts of this article, and also thanks Ran Wang for excellent research assistance and Peggy Bielecki for administrative support.

Notes

¹ In classifying profit maximization problems, I include problems with marginal and average variable cost functions derived from quadratic and cubic cost functions. For example, a problem with a quadratic marginal cost or average variable cost function is classified as being based on a cubic cost function, even if no total cost function is given.

² There are at least two approaches to teaching profit maximization that do not involve the use of examples with parametric cost functions. A number of texts do not use calculus to analyze profit maximization decisions, relying instead on graphical analysis and numeric examples (e.g., Mansfield and Yohe 2004). In addition, at least one text (Varian 2010) opts for the greater abstraction and generality of nonparametric cost functions.

³ Throughout the article, I use “shutdown condition” to refer to the short-run shutdown condition. Problems based on quadratic cost functions have a distinct advantage for teaching about long-run equilibrium under perfect competition. The (representative) firm's long-run equilibrium output is and the equilibrium market price is . In contrast, with a cubic cost function, solving for minimum AC is messy at best.