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

Accurately predicting the sales trajectory of a product is critically important for firms’ medium- and long-term planning. However, reliable sales prediction models are very difficult to find when repeat purchases or subscription renewals account for a large proportion of product sales, which is often the case for technological products. This study introduces a new sales growth model, the generalized diffusion model with repeat purchases (GDMR), to address this problem. The GDMR draws upon a branch of mathematics called fractional calculus and formulates a product’s sales growth rate using a novel noninteger-order integral equation. Compared with benchmark methods, the GDMR is simple and easy to implement, is suitable for a wide variety of products, and predicts better than benchmark models such as time series and machine learning models. Furthermore, the GDMR can reliably recover a product’s progress of adoptions even when only sales data are available. Because of these important advantages, the GDMR can help firms better understand their products’ market positions and, subsequently, make more informed decisions in production and inventory planning, transportation and logistics, and sales and marketing, thus improving the effectiveness and efficiency of their business operations.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.