Long‐term forecasting with innovation diffusion models: The impact of replacement purchases

Abstract

The model presented in this paper integrates two distinct components of the demand for durable goods: adoptions and replacements. The adoption of a new product is modeled as an innovation diffusion process, using price and population as exogenous variables. Adopters are expected to eventually replace their old units of the product, with a probability which depends on the age of the owned unit, and other random factors such as overload, stylechanges etc.It is shown that the integration of adoption and replacement demand components in our model yields quality sales forecasts, not only under conditions where detailed data on replacement sales is available, but also when the forecaster's access is limited to total sales data and educated guesses on certain elements of the replacement process. KEY WORDS Long term forecasting Diffusion modelsDurable goods Sales forecasting Innovation diffusion models have been widely applied in the forecasting of demand for consumer durables, especially in the early years of the product's life. The most popular model in these applications is the one proposed by Bass (1969). This model assumes that the adoption of an innovation depends on a communication process: the new product is first adopted by a few innovators, who in turn generate word-of-mouth, influencing others to imitate. In recent years, several variants of this basic model have been proposed (Dodds, 1973; Sharif and Kabir 1976; Dodson and Muller, 1978; Easingwood, Mahajan and Muller, 1983) in order to obtain a more realistic representation of the adoption process. Nevertheless, as Heeler and Hustad (1980) point out, certain limitations of this modeling approach appear to persist. These authors studied a large number of innovations in several countries and their findings could not match the predictive validity results obtained in previous applications of the model in the U.S.A. One of the limitations of the model as a forecasting tool stems from the fact that i t provides forecasts of adoptions (first time buyers) only, while in general the forecaster's predictive focus centers on the total sales of the product (which include adoptions and replacement purchases). Long term sales forecasting with diffusion models is especially inappropriate, since adoption sales as a proportion of total sales progressively decreases with time. Industry statistics underscore this point emphatically (see Table 1).

An aggregate sales model for consumer durables incorporating a time-varying mean replacement age

“…Olson and Choi (1985) use the Rayleigh distribution, Kamakura and Balasubramanian (1987) introduce the truncated normal distribution and Bayus et al (1989) use the Weibull distribution. The discrete, deterministic form of this approach may be written as:…”

“…More speci®cally, following the comparison of distributions by Kamakura and Balasubramanian (1987) and Bayus (1988), we choose the truncated normal distribution as the functional form of the distribution. At any given time, FaY t is assumed to follow the truncated normal distribution with mean replacement age, L, and shape parameter, h. However, we assume that the mean replacement age, L, may vary over time such that:…”

“…This condition is analysed in the Appendix. Further, by treating the cumulative probability as a proportion of the aggregate population (see Kamakura and Balasubramanian, 1987), the replacement purchases may be determined by…”

“…Note, however, there is no trend evident in terms of the shape parameter, nor any apparent relationship with the mean aggregate replacement age. There is no theoretical or empirical evidence to suggest it will vary, and previous experience shows it falls within a small range (Kamakura and Balasubramanian, 1987). Since it is not possible to reliably estimate both L and h from sales data alone, in the time-varying model the parameter h is constrained to be constant over time.…”

“…Hence, rather than directly estimating the replacement distributions from the census data, the entire model of replacement sales is estimated using sales data. Previous research has jointly estimated the ®rst purchase diusion model and replacement model using total sales data (Olson and Choi, 1985;Kamakura and Balasubramanian, 1987). Since data for the aggregate replacement component are available in this application, it is possible to estimate the replacement model independently.…”