#### H. Young

Paper #: 07-09-038

A basic puzzle posed by innovation diffusion is why there is often a long lag between an innovation’s first appearance and the time when a substantial number of people have adopted it. There is an extensive theoretical and empirical literature on this phenomenon and the mechanisms that might give rise to it. A common feature of these explanations is that heterogeneity among the agents is the reason that they adopt at different times. However, most of the extant models incorporate heterogeneity in a very restricted fashion, say by considering two homogeneous populations of agents, or by assuming that the heterogeneity is described by a particular family of distributions. In this paper I show how to incorporate heterogeneity into some of the benchmark models in marketing, sociology, and economics without imposing any parametric restrictions on the distribution of the underlying parameters. The resulting dynamical systems turn out to be surprisingly tractable; indeed, some of them can be solved explicitly for any distribution of the parameter values. I then demonstrate that each class of models leaves a distinctive “footprint”; in particular, they exhibit noticeably different patterns of acceleration, especially in the start-up phase, with few or no assumptions on the distribution of the parameters. The reason is that the models themselves have fundamentally different structures that even large differences in the distributions cannot overcome. It follows that, given sufficient data on the aggregate dynamics of a diffusion process, one could assess the relative plausibility of different mechanisms that might be driving it with little or no prior knowledge about the distribution of parameters. While this type of analysis is not an identification strategy, and is certainly no substitute for having good micro-level data, it could be useful in situations where such data are available. I shall consider three basic types of innovation diffusion models, each arising from a different account of how innovations spread. 1. Contagion. People adopt an innovation when they come in contact with someone who has already adopted. 2. Social threshold. People adopt when enough other people in the group have adopted. 3. Social learning. People adopt once they see enough evidence among prior adopters to convince them that the innovation is worth adopting. For each type of model I show how to incorporate heterogeneity of the parameters in considerable generality without losing analytical tractability; moreover this can be done even though there are multiple sources of heterogeneity. Of the three types of models, social learning is perhaps the most interesting from an economic standpoint, since it is based on the assumption that agents use payoff information from prior adopters in order to make a decision. (The other two classes are based on the notion of exposure rather than on utility maximization, though as we shall see the social threshold model can be reinterpreted in a utility maximization framework.) While there is a substantial theoretical and empirical literature on social learning models, however, surprisingly little prior work has been done on the implications of such models for the shape of the adoption curve. One of the main contributions of the paper is to show how to express the aggregate dynamics of such a model in an analytically tractable form even when there are multiple sources of heterogeneity among agents, including different costs of adoption, different prior information, and different amounts of “connectedness” with the rest of the population. A limitation of the analysis is the use of a mean-field approach in which the population is assumed to be large and encounters purely random. If instead agents interact through a fixed social network, the aggregate dynamics are substantially more complex and depend on the network topology. The extension of the approach to this situation will be considered in future work.