Discrete Choices under Social Influence: Generic Properties

We consider a model of socially interacting individuals that make a binary choice in a context of positive additive endogenous externalities. It encompasses as particular cases several models from the sociology and economics literature. We extend previous results to the case of a general distribution of idiosyncratic preferences, called here Idiosyncratic Willingnesses to Pay (IWP).Positive additive externalities yield a family of inverse demand curves that include the classical downward sloping ones but also new ones with non constant convexity. When $j$, the ratio of the social influene strength to the standard deviation of the IWP distribution, is small enough, the inverse demand is a classical monotonic (decreasing) function of the adoption rate. Even if the IWP distribution is mono-modal, there is a critical value of $j$ above which the inverse demand is non monotonic, decreasing for small and high adoption rates, but increasing within some intermediate range. Depending on the price there are thus either one or two equilibria.Beyond this first result, we exhibit the {\em generic} properties of the boundaries limiting the regions where the system presents different types of equilibria (unique or multiple). These properties are shown to depend {\em only} on qualitative features of the IWP distribution: modality (number of maxima), smoothness and type of support (compact or infinite).The main results are summarized as {\em phase diagrams} in the space of the model parameters, on which the regions of multiple equilibria are precisely delimited.