An Aggregative Approach to Price Equilibrium Among Multi-Product Firms With Nested Demand

Abstract

In this paper, I take advantage of the aggregative properties of the nested logit, and nested CES demands to show equilibrium existence for a price-setting game among multi-product firms. As opposed to previous existence results for this class of demand schedules, I allow for a fully flexible nesting structure (i.e., firms can offer products in multiple nests and nests can host products of various firms). This significantly relaxes the model’s implications for markups and substitution patterns, making the result relevant for empirical research. Furthermore, I show that in the presence of multiple equilibria, there is a profit-maximal equilibrium that is most preferred by firms and least preferred by consumers, and a profit-minimal equilibrium that is least preferred by firms and most preferred by consumers. Finally, as a by-product of the existence result, I show that FOCs are sufficient for profit maximization. This implies that the standard approach to marginal cost estimation in the empirical literature, i.e., inverting FOCs, correctly identifies the costs that rationalize pricing decisions.