The Dirichlet: A Comprehensive Model of Buying Behaviour

Abstract

The model describes how frequently-bought branded consumer products like instant coffee or toothpaste are purchased when the market is stationary and unsegmented. This is the common situation where, over the time-periods analysed, (A) the sales of each brand show little variation, (B) the different brands show no special groupings. We introduce the empirical data in Section 1 and the model itself in Section 2. The model assumes a mixture of distributions at four levels (Section 2.1): (i) Purchasing of the product-class takes the form of a Poisson process for each consumer, (ii) The purchasing rates of different consumers follow a Gamma distribution, (iii) Each consumer's choices among the available brands follow a multinomial distribution, and (iv) These choice probabilities follow a multivariate Beta or Dirichlet distribution across different consumers. In Section 2.2 we give justifications for these assumptions. More importantly, the model has successfully described the patterns observed in more than 40 product-fields (Section 3) and therefore provides interpretative norms (Section 3.1). As input the model only requires the sales level of each brand and two parameters. These can be identified as two aspects of consumer diversity, namely how much people differ from each other in (a) their purchasing rates and (b) their brand-choice preferences (Section 3.2). The model encompasses earlier, more limited formulations which often remain easier to use, as noted in Section 4.1. We briefly review the literature on other models in Section 4.2 and comment on aspects of model-building and on some future areas of work in Section 5.