Estimating Life Cycle Sales of Technology Products with Frequent Repeat Purchases: A Fractional Calculus-Based Approach

Accurately predicting the sales trajectory of a product in its life cycle is critically important for firms’ medium- and long-term planning. Because classic product-diffusion models such as the Bass model consider only initial product purchases, they are ill-fitted for sales prediction for today’s technology products with a shorter life cycle and frequent repeat purchases or subscription renewals. Despite the long tradition of product diffusion research, there exists no viable model option when repeat purchases constitute a large proportion of product sales. The present study introduces a new sales growth model, termed the generalized diffusion model with repeat purchases (GDMR), to fill this void. The GDMR formulates the growth rate of sales using a noninteger-order integral equation rather than the integer-order differential equation typically adopted in existing diffusion models. The GDMR is parsimonious and easy to implement. Empirical results show that the GDMR fits sales data with varying proportions of repeat purchases quite well, making it suitable for predicting sales of a wide variety of products. In addition, the GDMR can be extended to incorporate marketing mix variables, thus enhancing its applicability in business decision making. Furthermore, using both real and simulated data, we show that the GDMR can reliably recover a product’s adoption trend using only sales data, thus cementing its theoretical validity and empirical effectiveness. Finally, we show that the GDMR is superior to generic time series and machine learning models in predicting future product sales.