A New Product Growth Model for Consumer Durables Frank M. Bass

Overview A growth model for the timing of initial purchase of new products (durable products context) Explicit behavioral rationale and assumptions based on Roger’s (1962) innovation diffusion theory Good predictive accuracy based on historical data on 11 products (actual sales, sales peak and timing of the peak) A theoretical framework for long-range forecasting in this case, durable products so sales = initial purchase; but the theory goes beyond and applies to a broad range of product categories and settings) Basically, it captures how new product growth (based on initial purchase) is determined by the timing Made very precise and explicit assumptions based on behavioral theory!

Theoretical Rationale Diffusion of Innovation (Rogers 1962): innovation diffuses through interpersonal communication over time in a social system The diffusion process heavily relies on human capital (e.g., social pressures from other adopters) However, the relationship between this “human capital” (people who have adopted) and “diffusion” (the likelihood to adopt for non-adopters) has not been explicitly and precisely specified in previous diffusion and adoption models Main contribution: make the explicit and precise behavioral assumption based on Rogers’ literary behavioral theory

Theoretical Rationale Arbitrary five classes of adopters: innovators (2.5%); early adopters (13.5%); early majority (34%); late majority (34%); and laggards (16%) Innovators: adopt an innovation independently of other people (i.e., adoption is not influenced by adoption decisions of others) Imitators (last 4 group): adoption is influenced by adoption decisions of other members in the social system (i.e., social pressure) Fundamental behavioral assumption: the probability that an initial purchase will be made at T given that no purchase has yet been made is a linear function of the number of previous buyers Another burning question is that – based on the theory, if the likelihood to adopt is mainly driven by interpersonal communication in a social system, then a T=0 when no such social pressure and influence is available, how does the innovation diffuse (new product get adopted?) Percentage is based on the potential market (m)

Model Assumptions P(T) – the probability of an initial purchase at T given that no purchase has yet been made p – innovation coefficient (the probability of an initial purchase at T=0) q – imitation coefficient m – total number of initial purchase in the selected period of interest (market potential) Y(T) – number of previous adopters (i.e., previous initial purchases) p – the magnitude reflects the importance of innovators in the social system (q/m) Y(T) – the social pressure on imitators to adopt as a liner function of previous adopters Unit of measure of time is selected such that p can map onto the 2.5% innovator segment Period of interest is selected to excluded replacement sales (initial purchase = total sales, captured by m) Probability is referred to as a likelihood (conditional likelihood of adoption at time T is a linear function of the number of previous adoptions

Derivation Sales at T = S(T) = mf(T) = P(T) [m-Y(T)] f(T) – probability of initial purchase at T (unconditional) F(T) – probability of initial purchase in the (0,T) interval Sales at T = S(T) = mf(T) = P(T) [m-Y(T)] Final growth rate expression: Differentiate S to find the time T that sales/growth rate reaches the peak Sales can be expressed as a function of the three parameters p, q, m and known variable Y(T) Initial purchase consists of both innovators and imitators Innovator influence will diminish with time p is innovation coefficient and q is imitator coefficient

Optimization A maximum only exists when q > p, which is usually the case for successful new products (when q < p, the new product is likely to fail) Sales rate will attain the maximum value when cumulative sales Y(T) is approximately m/2 Figure 1 (in this research) – sales grow to a peak and level off and stabilize at lower level (accounted for by replacement purchase and decline of initial purchase) Not the concern of this paper as the focus is only on the timing of initial purchase (leads to one of the model assumption: period of interest is constrained) Prior literature (Haines1964; Fourt and Woodlock 1960) on growth model for new brands or products – exponential growth to some asymptote.

Discrete Analogue Basic Model: To estimate p, q, and m from discrete time series data, the following analogue is used: ST – sales at T YT-1 – cumulative sales through (0,T-1) a – estimates pm b – estimates q – p c – estimates –q/m herefore, the maximum value of S as a func- tion of time coincides with the maximum value of S as a function of cumulative sales

Regression Analysis Annual time series sales data for eleven different durable products Period of analysis restricted such that repeat purchase is not a concern Good model fit (R square) Total number of initial purchases to be made over the product life cycle can be estimated Good fit with both magnitude and timing of the peak All the estimated parameters seem reasonable (p,q and m) A weak performance test (ex post comparison)

Regression Analysis

Regression Analysis

Basic Model Performance Performance of Basic Model

Basic Model Performance Over Long Range Time Interval

Basic Model Performance Over Long Range Time Interval In poor R square situation, the general trend is captured by the model.

Long Range Forecasting For new product context, oftentimes there is no data (before the product launch) or limited data (a short period of time since new product introduction) When there is no data available Q: which one is easier to guess? Sales curve or model parameters? A: Model parameters! Potential market m – expert intuition or consumer survey Innovation and imitation coefficient p and q – buying motives based on survey Parameter estimates based on historical data on similar products

Long Range Forecasting When limited data is available: only three observations are needed to objectively estimate the three parameters Caution: parameter estimates become very sensitive even to small variations of observations Solution: perform plausibility test based on intuition, customer survey, etc.

Long Range Forecasting Sales did peak in 1968!

Summary A simple and parsimonious growth model to explain/predict the timing of initial purchase of new products Explicit and precise behavioral assumptions based on a simple theory Good empirical generalizability (explaining and predictive power) Informative for long range forecasting even with limited or no observations Underlying behavioral assumption has an intuitive appeal Parameters can have intuitive interpretations Peres et al. (2010) two implicit assumptions: 1) social system is homogeneous and 2) fully connected (i.e., does not account for heterogeneity and it is assumed that consumers who have not yet adopted the new product are fully aware of the number of adopters)

Some Extensions Diffusion of successive generations of new products (generational diffusion model of sales, Norton and Bass 1987). Adding decision variables into diffusion models (e.g., price, ads, etc.) Conceptualize diffusion beyond interpersonal communication by including social interdependence (e.g., network externalities and social signals) Incorporate customer heterogeneity Peres et al. (2010) implicit assumption of Bass model and its extensions: social system is homogeneous and fully connected (i.e., does not account for unobservable heterogeneity and it is assumed that consumers who have not yet adopted the new product are fully aware of the number of adopters)

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