A differential game of a duopoly with network externalities

In this work, we develop a differential game of a duopoly where two firms compete for market share in an industry with Network Externalities. Here the evolution of the market share is modeled in such a way that the effects of advertising efforts that both firms make are a function of the share itself. This means that the efficacy of marketing efforts are diminished with low market share and enhanced when it is higher. We show that Network Externalities can influence the decision a firm makes about marketing expenditures. Particularly, when a firm is large enough, the creation of a monopoly is easier when this market structure is present. For this, we obtain the optimal strategies for the firms and test them on a simulation, where we compare the market with and without this kind of externalities. We find that the value of the market share in proportion with the cost of obtaining it by advertising efforts is the key to know the long term equilibrium market share.