Modeling Product Diffusion and Repeat Purchases: A Fractional Calculus-Based Approach（CEIBS Working Paper, No. 040/2020/MKT, 2020）

Classic product diffusion models such as the Bass Model typically consider only initial product purchases. For products that experience frequent upgrades or have multiple versions, repeat product purchases can constitute a significant proportion of sales. Despite the long tradition of product diffusion research, there exists no viable model option when repeat purchases are inseparable from initial ones in sales data. The present study proposes a new sales growth model, termed the Generalized Diffusion Model with Repeat Purchases (GDMR), to fill this void. The GDMR treats the sales process as an economic process with memory and formulates the growth rate of sales using a non-integer-order integral equation rather than an integer-order differential equation typically used 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, thus making it a suitable model for a wide variety of products. We also demonstrate that the GDMR can effectively recover adoption (i.e., initial sales) curves when only sales data is available, thus underscoring its theoretical validity. Furthermore, the GDMR can be extended to incorporate marketing mix variables, further enhancing its value in managerial decision-making.