Bertrand duopoly with a captive uninformed consumer segment

Abstract

This paper studies the comparative statics of increased market transparency on the pricing and profits of a Bertrand duopoly selling differentiated goods. The market consists of informed consumers who are aware of both firms’ products, and uninformed consumers who are captive to one firm, and market transparency is defined as the proportion of the former. We reconsider the micro-economic foundations of the two types of consumers’ demand functions and derive minimally sufficient conditions to establish clear-cut comparative statics for two separate cases of gross complements and substitutes, each case further split according to the nature of the game, strategic complements or strategic substitutes. Whether the prices increase or decrease in market transparency depends on the elasticity comparison between the two demand functions, and we reveal its close connection to the natural condition of log-supermodularity of the informed consumers’ demand in the two firms’ prices. The conventional wisdom that the equilibrium prices fall with more transparency is mostly confirmed, but with some exceptional cases identified.