We study a system where many identical customers use a single service. Each customer experiences both positive externalities (a positive "network effect") and negative externalities (a "congestion effect") from other customers using the service. Such a model arises frequently in practice: application services on wireless networks, peer-to-peer networks, and social networks are examples.

We characterize the social optimum, where a social planner determines the usage level of each customer. We also characterize the Nash equilibrium achieved when the usage levels are determined by the customers themselves, in their self-interest. We study the ratio of the welfare in Nash equilibrium to that in the social optimum. We demonstrate that there is an *optimal scale*, i.e., a number of customers at which this ratio is maximized; further, the optimal ratio is unity. We also show that this same optimal scale maximizes the Nash welfare of a single individual. We interpret our results in terms of club formation, and study the size of the club as the impact of the positive externality grows.

Joint work with Sunil Kumar, Stanford GSB.