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Modeling Bass Diffusion with Rivalry

This is the last of a three-part series on the Limits to Growth Archetype.  The first part can be accessed here and the second part here.

Last time, we explored the effects of Type 1 rivalry (rivalry between different companies in a developing market) on the Bass diffusion model by replicating the model structure.  This part will generalize this structure and add Type 2 rivalry (customers switching between brands).

Bass Diffusion with Type 1 Rivalry

To model the general case of an emerging market with multiple competitors, we can return to the original single company case and use arrays to add additional companies.  In this case, everything except Potential Customers needs to be arrayed, as shown below (and available by clicking here).


For this example, three companies will be competing for the pool of Potential Customers.  Each array has one-dimension, named Company, and that dimension has three elements, named A, B, and C, one for each company.  Although each different parameter, wom multiplier, fraction gained per $K, and marketing spend in $K, can be separately specified for each company, all three companies use the same values initially.  All three companies, however, do not enter the market at the same time.  Company A enters the market at the start of the simulation, company B enters six months later, and company C enters six months after that.

Recall that the marketing spend is the trigger for a company to start gaining customers.  Thus, the staggered market entrance can be modeled with the following equation for marketing spend in $K:


The STEP function is used to start the marketing spend for each company at the desired time.  The ARRAYIDX function returns the integer index of the array element, so it will be 1 for company A, 2 for company B, and 3 for company C.  Thus, the offsets from the start of the simulation for the launch of each company’s marketing campaign are 0, 6, and 12, respectively.

This leads to the following behavior:


Note that under these circumstances, the first company to enter the market retains a leadership position.  However, companies B and C could anticipate this and market more strongly.  What if company B spent 50% more and company C spent 100% more than company A on marketing that is similarly effective?  This could be modeling by once again changing the equation for marketing spend in $K, this time to:

STEP(10 + (ARRAYIDX() – 1)*5, STARTTIME + (ARRAYIDX() – 1)*6)

The effect in this case is quite different.  All three companies end up with approximately equal market shares.


Bass Diffusion with Type 1 and Type 2 Rivalry

Once a market is established, companies will also make an effort to win competitor’s customers away.  A previous post showed how to model switching between brands using arrays.  Adding that structure to this model gives a model (available by clicking here) that implements both type 1 and type 2 rivalry.


The specific mechanism that convinces customers to switch brands is not explicitly modeled here so that the structure matches that in the earlier post.  Instead, this model uses constant switching rates; the earliest entrant, company A is given the most favorable values.  However, to make the model more dynamic and also useful, there should be factors that influence the relative attractiveness of the two brands, for example, marketing spend, product quality, product features, and customer service.

Giving an advantage to the early entrant to the market has the expected effect of boosting their customer base at the expense of the others.  This another example of Success to the Successful.


Note the scale of this graph had to be changed relative to the earlier graphs to show all of company A’s customers.  The effect of the customer switching is so strong that even increasing marketing for company’s B and C does not change the end result.  Of course, the switching factors in this model are constant and increased marketing should indeed change them.

This does, however, show that it is important to intervene in the correct place to reach your desired outcome.  Once the market is mature, the dynamics are dominated by the switching effects, not the process of gaining customers.  Thus, efforts to change market share by changing the rate of gaining customers are ineffective.  Energy should instead be focused on reducing the number of customers who switch to other brands and increasing the number of customers gained from other brands.

This model does not model the decline of the market, through customer loss, since we are mostly concerned with growth and maturation.  If desired, the loss would be modeled with an outflow from Customers as shown in the earlier brand switching post.

Finally, this model assumes that Customers belong to one company alone, i.e., no one uses the product from more than one company at one time.  Sharing customers is known as type 3 rivalry and is beyond the scope of this post.

A Note About Revenues

Gaining customers is, of course, not the entire story.  The purpose of gaining customers is to create revenue for your business.  There are two opportunities to generate revenue in this model:

    1. From your existing customer base.  This stock-driven revenue stream is typical for manufacturers of non-durable goods such as bleach and razors, or for subscription-based or service businesses, such as magazines, cable TV, or beauty salons.  If your customer base is large, your revenues will be high.  The Bass diffusion model shows a growing customer base and therefore a growing revenue stream (see curve 1 below).
    2. When a person is converted from a Potential Customer to a Customer or a customer switches from one brand to another.  This flow-driven revenue stream is the only source of revenue for manufacturers of durable goods, such as cars, computers, and refrigerators.  However, it often also applies to software companies.  Since the flow rate determines the revenue stream, if the rate at which you gain customers is large, your revenue will be high.  Note, however, that the Bass diffusion model shows us this rate falls sharply as the market matures, as the number of Customers saturates (see curve 2 below).  To keep revenues high in a mature market for this kind of business, it is necessary to entice customers away from competitors, build in product obsolescence (so your customers have to buy again), or enter new markets.


A Final Aside on the Bass Diffusion Model

In his excellent book Strategic Modelling and Business Dynamics, John Morecroft explains (on page 173) that the effect of marketing in the Bass diffusion model has a stronger purpose than just to start the ball rolling.  Without the marketing effect, market growth through word-of-mouth alone is much slower than any business would desire, or typically experience.  The following graph compares the growth with and without the marketing effect (Customers is initialized to one in both cases).


Without the marketing expenditure, the conversion of Potential Customers to Customers does not peak until the month 91, whereas, with marketing expenditure, it peaks in month 50, in almost half the time.  This is also reflected in the time it takes the market to mature:


available by clicking here)
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  • Eric

    Karim, I really enjoy these posts. You pick real world issues and explain them in a model construct that is sufficient to grasp the dynamics, without overburdening it with details (we in corporate America) often get buried under!

  • kchichakly

    Thank you, Eric! I will try to keep that in mind in my future posts. We all have a tendency to allow our models to grow and grow and grow, sometimes beyond our ability to comprehend them!

  • well, This part introduces the Bass diffusion model, an effective way to … expand the above model to include Type…

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