Bertrand Competition under Network Externalities

Two sellers engage in price competition to attract buyers located on a network. The value of the good of either seller to any buyer depends on the number of neighbors on the network who consume the same good. For a generic specification of consumption externalities, we show that an equilibrium price equals the marginal cost if and only if the buyer network is complete or cyclic. When the externalities are approximately linear in the size of consumption, we identify the classes of networks in which one of the sellers monopolizes the market, or the two sellers segment the market.