A Note on Optimal Product Pricing
Maximilian Schaller, Stephen Boyd
TL;DR
The paper tackles profit-maximizing pricing for multiple products under self- and cross-elasticities with convex constraints, formulating the problem as a nonconvex PPP that is well-approximated by near-convex structure. It develops and compares three solution approaches: CCP, quadratic MM (QMM), and general NLP, all leveraging convex optimization techniques. Across large-scale numerical experiments, the methods consistently converge to the same local maximum regardless of initialization, hinting at (but not proving) near-global optimality, and the authors provide open-source code for reproducibility. These findings offer practical, scalable tools for constrained multi-product pricing in settings with complex demand interdependencies.
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 minorize-maximize 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.