New ideas, products, and practices take time to diffuse, a fact that is often attributed to some form of heterogeneity among potential adopters. This paper analyzes the effect of incorporating heterogeneity into three broad classes of models — contagion, social influence, and social learning. Each type of model leaves a characteristic ‘footprint’ on the shape of the adoption curve that amounts to a restriction on the pattern of acceleration with very few restrictions on the distribution of parameters. These restrictions provide a basis for discriminating empirically between different models, and have potential application to marketing, technological change, fads, and epidemics.
1. Introduction
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.[1] 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.[2]
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 unavailable.
