A Note on Optimal Product Pricing

TL;DR

This paper explores methods for optimal product pricing considering elasticity effects, comparing three algorithms that reliably find consistent local maxima, suggesting near-global optimality.

Contribution

It introduces and compares three optimization techniques for pricing problems involving convex and concave functions, demonstrating their effectiveness in numerical examples.

Findings

All three methods converge to the same local maximum.

The methods reliably find consistent solutions regardless of starting prices.

Numerical results suggest the solutions are likely globally optimal.

Abstract

We consider the problem of choosing prices of a set of products so as to maximize profit, taking into account self-elasticity and cross-elasticity, subject to constraints on the prices. We show that this problem can be formulated as maximizing the sum of a convex and concave function. We compare three methods for finding a locally optimal approximate solution. The first is based on the convex-concave procedure, and involves solving a short sequence of convex problems. Another one uses a custom minorization-maximization method, and involves solving a sequence of quadratic programs. The final method is to use a general purpose nonlinear programming method. In numerical examples all three converge reliably to the same local maximum, independent of the starting prices, leading us to believe that the prices found are likely globally optimal.