# Bass Diffusion Model – Definition

### Bass Diffusion Model Definition

The Bass diffusion model, or merely the Bass model, is a differential equation used to describe the process involved in the adoption of new products into the population. Frank Bass developed it.

### A Little More on the Bass Diffusion Model

This model is used to present a rationale of the interaction between current and potential adopters of a new product. It states that there are two types of adopters which are the innovators and imitators. The speed and timing of adoption are based on the adopters’ tendency of innovating and imitate. Both sales forecasting of new products and technology forecasting utilize the Bass model. This model is mathematically defined as a Riccati equation with constant coefficients.

In 1833, Everett Rogers published a highly influential study describing the different stages in product adoption that was titled ‘Diffusion of Innovations.’ Then in 1969, a paper detailing a new model of productivity growth for consumer durables was published by Frank Bass who contributed a few mathematical ideas to it.

This model was discovered to fit the data of a majority of product introductions even though there was a wide range of variables in terms of managerial decisions such as pricing and advertising. This means that the shape of the curve is always the same despite the decision variables shifting the Bass curve in time.

There is only one extension of the model that is applicable under ordinary circumstances. It was developed in 1994 by Frank Bass, Trichy Krishnan, and Dipak Jain.

F(t) / 1-F(t) = (p+qF(t)) x (t)

### Relationship with other S-curves

The Bass diffusion model has two special cases

•         In the first special case, q=0 when the model reduces to the exponential distribution.
•         In the second special case, p=0 when the model reduces to the logistic distribution.

Since it is utilized in online social networks, the use of the Bass diffusion model has increased due to the recent rapid growth in online social networks. The model is useful in estimating the size and growth rate of social networks. When compared with alternative models such as the Weibull distribution, the Bass model gives a more pessimistic view of the future according to Christian Bauchkage and his co-authors.

As of October 2018, the paper ‘A New Product Growth for Model Consumer Durables’ which is about the Bass model, was one of the most cited in Google Scholar with 8499 citations. In marketing and management science, this model has been widely influential. It was one of the ten most frequently cited papers back in 2004 in the history of management science.

### Academic Research on Bass Diffusion Model

• New product diffusion models in marketing: A review and directions for research, Mahajan, V., Muller, E., & Bass, F. M. (1990). Journal of marketing, 54(1), 1-26. This paper investigates how the research on the modeling of diffusion has led to a body of literature with numerous articles, books, and other publications since the publication of the Bass model.
• A diffusion theory model of adoption and substitution for successive generations of high-technology products, Norton, J. A., & Bass, F. M. (1987). Management science, 33(9), 1069-1086. In this study, the dynamic sales behavior of successive generations of high-technology products is examined.
• Determination of adopter categories by using innovation diffusion models, Mahajan, V., Muller, E., & Srivastava, R. K. (1990). Journal of Marketing Research, 37-50. This paper discusses the possibility of developing adopter categories for product innovation through established diffusion models like the Bass model using the rational logic behind the classical adopter categorization approach.
• The forecasting of the mobile Internet in Taiwan by diffusion model, Chu, C. P., & Pan, J. G. (2008). Technological Forecasting and Social Change, 75(7), 1054-1067.  While using the concepts of ‘technical substitution’ and ‘multi-product competition’ which suits the mobile internet characteristics in Taiwan, this paper proposes a diffusion model capable of revealing the growth pattern of the mobile internet subscriber in this country.
• Maximum likelihood estimation for an innovation diffusion model of new product acceptance, Schmittlein, D. C., & Mahajan, V. (1982). Marketing science, 1(1), 57-78. This article proposes a maximum likelihood approach which was initially considered by bass, to estimate an innovation diffusion model of new product acceptance.
• Why the Bass model fits without decision variables, Bass, F. M., Krishnan, T. V., & Jain, D. C. (1994). Marketing science, 13(3), 203-223. This paper generalizes the Bass model to incorporate the decision variables such as price and advertising and also reduces it to the Bass model as a special case and then explains why the Bass model well without the inclusion of decision variables.
• New-product diffusion models, Mahajan, V., Muller, E., & Bass, F. M. (1993). Handbooks in operations research and management science, 5, 349-408. In this paper, the contributions made by management and marketing science literature to the increasing understanding of the dynamics of innovation diffusion are examined.
• A meta-analysis of applications of diffusion models, Sultan, F., Farley, J. U., & Lehmann, D. R. (1990). Journal of marketing research, 70-77. This paper carries out a meta-analysis of 213 applications about diffusion models derived from 15 articles to relate the model parameters to the innovation nature, the country being studied, model specification and the procedure of estimation.
• A nonuniform influence innovation diffusion model of new product acceptance, Easingwood, C. J., Mahajan, V., & Muller, E. (1983). Marketing Science, 2(3), 273-295.  This article proposes a nonuniform influence (NUI) innovation diffusion model to forecast the first adoptions of a new product. The proposed model overcomes three limitations that affect single-adoption diffusion models.
• A diffusion model incorporating product benefits, price, income and information, Horsky, D. (1990). Marketing Science, 9(4), 342-365. This paper uses a household production framework to demonstrate how a utility maximizing individual will have a reservation price for the product which is a function of the product benefits and his wage rate.
• Modelling and forecasting the diffusion of innovation–A 25-year review, Meade, N., & Islam, T. (2006). International Journal of forecasting, 22(3), 519-545. This article reviews the research on modeling and forecasting the diffusion of innovations and emphasizes its importance in improving forecasting accuracy and the addition of insight into the problem of forecasting.