On the existence of the competitive equilibrium in Grossman and Shapiro (1984)

In their seminal paper, Grossman and Shapiro (1984) assume that it is not profitable for a firm to deviate to the supercompetitive price of Salop (1979). In this paper, 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 and Shapiro (1984) 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 more than four firms in the market, at least 96% of the consumers must receive at least one ad, and the percentage increases with the number of firms.