6.2. The Bass Model – An Elegant Special Case of a Diffusion Model
The Bass model offers a particular view of a socially-driven adoption process and elegant algebraic formulations with which to express a 'theory' or dynamic hypothesis for growth and saturation. Developed by Frank Bass, the model was first published in Management Science in 1969. The article has since been voted among the top 10 most influential papers by INFORMS (Institute for Operations Research and the Management Sciences) members. Figure 6.3 shows how the principal influences on adoption are operationalised.[] Essentially, it is a contagion model. Adopters 'infect' potential adopters through word-of-mouth thereby causing them to adopt. Note there is no influence whatever from the innovating firm on the adoption rate. In the full Bass model, this stark assumption is relaxed, but for now let's concentrate on the structure and formulation of contagion. There are two interacting feedback loops – a positive reinforcing loop representing word-of-mouth and a balancing loop representing market saturation. The more adopters, the greater the adoption from word-of-mouth. This contagion is limited as the number of potential adopters falls (since each new adopter comes from the pool of potential adopters and people are conserved!!). A contagion model is a good way to capture the interplay of word-of-mouth and market saturation. Adoption from word-of-mouth is a function both of adopters and potential adopters. The contact rate and adoption fraction tell us something about the strength of the word-of-mouth effect. Next, we examine the algebraic formulation of these concepts.
[] The model developed in Figures 6.3–6.8 of this chapter are the system dynamics equivalent of the original continuous time differential equations specified by Bass. However, Bass did not explicitly discuss the feedback loop structure of his model or identify the processes controlling adoption in terms of word-of-mouth and market saturation. The translation of the Bass model into the stocks, flows and feedback loops shown here was carried out by John Sterman 2000 and is described in Business Dynamics (Chapter 9).
Figure 6.3. Stocks, flows and feedback loops in a contagion model of new product adoption
Figure 6.4. Equations for adoption through word-of-mouth – a social contagion formulation
Figure 6.4 shows the equations for adoption from word-of-mouth. Consumers interact socially. The total population is assumed to be one million people. They meet each other from time to time and may talk about their latest purchase. This propensity to socialise and chat is captured in the 'contact rate', set at a value 100 and defined in terms of people contacted per person per year (which is dimensionally equivalent to 1/year). Hence, the number of encounters per year by adopters with the rest of the population is the product of the contact rate and adopters, which is the first term in the equation. Some of these encounters, the ones with potential adopters, lead to adoption (whereas an encounter between two adopters cannot, by definition, result in adoption). The probability that any randomly selected encounter is an encounter between an adopter and a potential adopter is equal to the proportion of potential adopters in the total population, which is the ratio in the second term of the equation. Note that this ratio naturally declines as adoption proceeds, reaching zero when the market is fully saturated. Not every encounter between an adopter and potential adopter results in adoption. The fraction of successful encounters is called the adoption fraction and is set at a value 0.02, meaning that two per cent of encounters lead to adoption.
Figure 6.5 shows the equations for the stock and flow network of adopters and potential adopters. By now the formulation of these standard, yet vital, accumulation equations should be familiar. Incidentally, the model denotes time in years and the computation step or delta time dt is 1/16th of a year, small enough to ensure numerical accuracy. The number of adopters at a given point in time t is equal to the previous number of adopters at time (t−1) plus the adoption rate over the interval dt. Conversely, the number of potential adopters at time t is the previous number minus the adoption rate over the interval. These symmetrical formulations ensure that the total population of customers remains constant. Initially, there are 10 adopters among a total population of one million. So the remaining 999 990 people are potential adopters. The adoption rate is equal to adoption from word-of-mouth.
Figure 6.5. Stock accumulation equations for adopters and potential adopters
6.2.1. The Dynamics of Product Adoption by Word-of-mouth
Open the model called 'Bass Word-of-Mouth' in the CD folder for Chapter 6 and browse the diagram and equations. Go to the graph where there is a time chart that displays adopters and potential adopters over a 10-year horizon. There is also a slider for the adoption fraction to test different assumptions about the strength of social contagion. The slider is initially set at 0.02 which is the value supplied in the equations above. Run the model to obtain the two charts shown in Figure 6.6. The top chart shows adopters (line 1) and potential adopters (line 2) plotted on a scale from zero to one million. At the start there are only 10 adopters, a drop in the ocean. These few enthusiasts begin to spread the word. For four years, however, they and their growing band of followers appear to have almost no impact on the huge mass of potential adopters who have never heard of the product. That is the nature of social contagion. For a long time, the enthusiasts are lone voices and the adoption rate in the bottom chart remains close to zero, though in fact steady exponential growth is taking place but on a scale dwarfed by the total population of one million people. During year 4, the number of adopters swells from only 7 000 to more than 40000 and they are beginning to be heard. Over the next two years, the adoption rate rises from 14 000 per year to a peak of almost 500 000 per year. The bulk of the million-person market is converted in the interval between years five and eight, by which time the adoption rate is much reduced as market saturation sets in.
Figure 6.6. Dynamics of product adoption by word-of-mouth
6.2.2. The Need to Kick-start Adoption
Adoption peaks at 500 000 per year and yet four years after product launch the adoption rate is still less than one-tenth of its peak value. This barely perceptible build-up is typical of exponential growth that starts from a very small base. In the logical extreme where the initial number of adopters is zero then, even with one million potential adopters, there is no growth at all because the product is totally unknown and remains that way. Although such an extreme is unrealistic, it demonstrates the need to 'seed' the market with early adopters in order to kick-start word-of-mouth. That's where the firm has an important role to play through marketing of various kinds, such as advertising, promotions and trade shows. In practice, it would be necessary to investigate how marketing really works, starting from a rough picture like Figure 6.2 and adding the necessary operating detail inside the firm to discover new feedback loops that augment word-of-mouth. First, however, we consider the Bass model for a specific feedback representation of advertising and its effect on product adoption.
6.2.3. The Complete Bass Diffusion Model With Advertising
Advertising is represented as a separate influence on the adoption rate as shown in Figure 6.7. It is a subtle formulation (and once again elegant and compact) because advertising is cleverly woven into a feedback loop, a balancing loop in this case. The obvious part of the formulation is that advertising is capable of generating adoption separately from word-of-mouth. There are now two influences on the adoption rate: adoption from word-of-mouth and adoption from advertising. The clever part of the formulation is that adoption from advertising itself depends on potential adopters thereby providing the link that closes the new loop on the left. The basic idea is that well-targeted advertising converts a proportion of potential adopters in each period, where the proportion depends on advertising effectiveness. Hence, advertising has its biggest impact early in the adoption process when there are lots of potential adopters to reach and convert. At least that's what the Bass formulation assumes. If advertising really works this way then it dovetails neatly with word-of-mouth by providing exactly the early kick-start needed.
Figure 6.7. The complete bass diffusion model with advertising
Figure 6.8. Bass equations for adoption with advertising
The corresponding equations are shown in Figure 6.8. The adoption rate is the sum of adoption from advertising and adoption from word-of-mouth. Intuitively the equation makes sense because there are now two independent additive effects on the adoption rate. They are independent in the sense that advertising does not change the nature of social contagion (it just boosts the number of product advocates) and contagion does not influence advertising. Adoption from advertising is expressed as the product of potential adopters and advertising effectiveness and captures the ability of advertising to reach the masses early on, when word-of-mouth cannot. Advertising effectiveness is a constant fraction per year set at 0.01. The two equations together mean that advertising converts a fixed proportion of potential adopters in each period, at an annualised rate of one per cent per year. It is useful to think about the numerical implications of the formulation at the start of the simulation when, as we have already seen, word-of-mouth is feeble. There are one million potential adopters (less 10 initial active adopters). Advertising reaches all of them and converts them at a rate of one per cent annually, which initially is equal to 10 000 people per year. This number is far, far more than the early word-of-mouth adoption (which hovers between zero and one per year), but interestingly far, far less than the peak word-of-mouth adoption (which can number hundreds of thousands per year). Hence, advertising dwarfs word-of-mouth in the early years but is subsequently dwarfed by it. Such a shift of power between drivers of adoption is both dynamically and strategically important as simulations will clearly show.
6.2.4. The Dynamics of Product Adoption by Word-of-mouth and Advertising
The new trajectories of product adoption, boosted by advertising, are shown in Figure 6.9. There are striking differences by comparison with Figure 6.6. The adoption rate (bottom chart) rises much sooner and reaches a peak shortly before year three, four years earlier than adoption by word-of-mouth alone. The peak rate in both cases is roughly 500 000 customers per year, increased only slightly to 505 000 with advertising. This similar scaling confirms the point that word-of-mouth dominates adoption once it gets going. The trajectories for adopters and potential adopters (top chart) show that the whole adoption process is moved forward in time and is complete by year five, whereas before it was only just beginning in year five. The simulation shows just how strategically important advertising can be. It creates a huge timing advantage by unleashing the power of word-of-mouth much earlier.
Figure 6.9. Dynamics of product adoption by word-of-mouth and advertising
To further investigate the dynamics of new product adoption let's conduct a simulation experiment. Open the model called 'Bass Diffusion with Advertising' in the CD folder for Chapter 6. You will see the same time charts as in Figure 6.9. Also, there are two sliders to create what-ifs. Advertising effectiveness moves in the range between zero and 0.02 and adoption fraction moves between zero and 0.03. Now imagine launching a different product whose appeal is less obvious to consumers and so word-of-mouth is muted. The adoption fraction is halved from 0.02 to 0.01 meaning that only one per cent of contacts between adopters and potential adopters result in adoption. Move the slider accordingly. Meanwhile advertising effectiveness is unchanged. Before simulating try to imagine the new trajectory of adopters. You can sketch your expectation by turning to page three of the graph pad (click the 'page tab' in the bottom left of the chart). There you will find the simulated trajectory of adopters (line 1) from the previous run. There is also another variable called 'sketch of adopters' (line 2) that runs along the time axis. To create your own sketch just click, hold and drag, starting anywhere along the line. A new trajectory appears that can be any shape you want. When you are finished, click run to simulate and check out your intuition.
The result of muted word-of-mouth is shown in Figure 6.10. The adoption rate (bottom chart) is noticeably flatter than before and reaches a peak of some 255 000 customers per year. Nevertheless, adopters (line 1, top chart) still show a clear pattern of S-shaped growth that rises more gradually. Interestingly, the adoption process is almost complete by the middle of year 7, a whole year earlier than was achieved in Figure 6.6 for an appealing product without advertising.