Finite purchasing power and computations of Bertrand-Nash equilibrium prices

This article considers the computation of Bertrand-Nash equilibrium prices when the consumer population has finite purchasing power. The literal KKT conditions for equilibria contain spurious solutions that are not equilibria but can be computed by existing software, even with prominent regularization strategies for ill-posed problems. We prove a reformulated complementarity problem based on a fixed-point representation of equilibrium prices improves computational reliability and provide computational evidence of its efficiency on an empirically-relevant problem. Scientific inferences from empirical Bertrand competition models with explicit limits on individual purchasing power will benefit significantly from our proposed methods for computing equilibrium prices. An analysis of floating-point computations also implies that any model will have finite purchasing power when implemented on existing computing machines, and thus the techniques discussed here have general value. We discuss a heuristic to identify, and potentially mitigate, the impact of computationally-imposed finite purchasing power on computations of equilibrium prices in any model.