A Note on the Existence of the Competitive Equilibrium in Grossman and Shapiro (1984)

In their seminal paper, Grossman, G. M., and C. Shapiro. 1984. "Informative Advertising with Differentiated Products." The Review of Economic Studies 51: 63-81 assume that it is not profitable for a firm to deviate to the supercompetitive price of Salop, S. C. 1979. "Monopolistic Competition with outside Goods." The Bell Journal of Economics 10: 141-56. In this note, it is shown that this assumption is violated if, roughly, each firm reaches less than half of all consumers unless it is a duopoly. This implies that most of the simulations in Grossman, G. M., and C. Shapiro. 1984. "Informative Advertising with Differentiated Products." The Review of Economic Studies 51: 63-81 are not actually equilibria. More importantly, this implies that for their equilibrium to exist nearly all consumers must receive at least one ad. For example, with just four firms in the market, at least 96% of the consumers must receive at least one ad, and this percentage increases with the number of firms in the market.