Equilibrium size in network with indirect network externalities

I present a simple model of determination of the equilibrium size of a network with indirect network externalities. Indirect network externalities can generate complementarity between goods, and then the demand functions for the network good are those of the complementary goods: the result in the determination of the equilibrium size in a market with indirect network externalities is the “classical” result with complementary goods. I calculate the number of consumers of each group that should be optimal for firms in different market structures, as perfect competition, monopoly and duopoly. The result is that the equilibrium size of the network with indirect network externalities depends on the market structure; it is wider in perfect competition than in monopoly and duopoly; because of the externalities, perfect competition is inefficient, that is, the equilibrium size in this market structure is smaller than the equilibrium size chosen by a social planner; prices charged in a duopolistic market with indirect network externalities are greater than prices charged in a monopoly market, and then, the equilibrium size in duopoly is smaller than in monopoly; this is the classical result that we obtain with complementary goods, generated by the indirect network externalities, and it is different from the result in markets with direct network externalities in which the equilibrium size in duopoly is greater than monopoly because the competition with substitute goods.