In this paper, we propose a diffusion model for a subscription service. The evolution over time of the number of subscribers is governed by a differential equation combining two processes, namely, a customer acquisition process and a customer attrition process. Assuming profit-maximization behavior of the firm, we use dynamic programming to optimize the customer equity and determine optimal customer relationship marketing expenditures. We implement an augmented Kalman filter with continuous state and discrete observations to estimate the model's parameters using market data of two well-known companies in the telecommunications sector.

To the best of our knowledge, this is the first paper to model acquisition and retention efforts in the context of a diffusion model. By doing so, we extend the literature on product diffusion to services, that is, beyond its traditional area of durable (and occasionally non-durable) products. By the same token, we contribute to the literature on customer relationship marketing (CRM) where social interactions have been overlooked. Our analytical and numerical results provide a better understanding of the relationship between the optimal customer equity, the customer lifetime value, the prospect lifetime value and the optimal acquisition and retention spending. Our model and estimation approach give the tools for assessing empirically the role of CRM spending, social interactions and other factors in the service subscription dynamics. Our empirical results show indeed that CRM spending and external incentives have a significant effect on acquisition and retention processes, and that this effect is market specific.