D. Combining Positive and Negative Feedback

Having explored the behavior of simple positive and negative feedback systems, lets turn to a more complex setting where multiple loops interact nonlinearly. The growth of new products often follows S-shaped growth patterns. As discussed in Chapter 4, S-shaped growth results from a system in which positive feedbacks dominate early on, but, as the state of the system grows relative to its limits, negative loops begin to dominate. Consider the growth of a new product or product category, such as mobile phones, DVD, or the world wide web. If successful, the installed base (number of cell phones, DVD units, or websites) usually grows exponentially for a while. Eventually, however, as the market begins to saturate, growth slows, and the installed base reaches a maximum determined by the size of the potential market for the product category.

What are the positive feedbacks that generate the initial exponential growth in the diffusion of a successful innovation, and what are the negative feedbacks that limit its growth? The spread of rumors and new ideas, the adoption of new technologies, and the growth of new products can all be viewed as epidemics in which the innovation spreads by positive feedback as those who have adopted it infect those who have not. The concept of positive feedback as a driver of adoption and diffusion is very general and can be applied to many domains of social contagion. As early adopters of a new product expose their friends, families, acquaintances and colleagues to it, some are persuaded to try it or buy it themselves. In all these cases, those who have already adopted the product come into contact with those who have not, exposing them to it, infecting some of them with the desire to buy the new product, and further increasing the population of adopters. Any situation in which people imitate the behavior, beliefs, or purchases of others, any situation in which people jump on the bandwagon, describes a situation of positive feedback by social contagion. Of course, once the population of potential adopters has been depleted, the adoption (infection) rate falls to zero.

The total potential market for the product is divided into two stocks: those who have already purchased the product (the Adopters) and those who have not yet adopted it (the Potential Adopters). As people adopt the product, the Adoption Rate rises, moving people from the Potential Adopter pool to the Adopter pool.

The Vensim equations for the stock and flow structure are therefore:

Adopters = INTEG(Adoption Rate, Initial Adopters) .(1)

Potential Adopters = INTEG(Adoption Rate, Total Market Size Adopters).. (2)

he number of Initial Adopters is modeled as a parameter so you can easily change it for sensitivity tests without editing the model equations. Note that there are no inflows to or outflows from either stock from outside the system boundary (the adoption rate moves people from one category to the other). Therefore, the Total Market Size is constant since Potential Adopters + Adopters = Total Market Size. The Total Market Size will be set as a parameter (constant) in your simulation.

Question 1. What determines the Adoption Rate? In this model, adoption occurs only through word of mouth:

Adoption Rate = Adoption from Word of Mouth (3)

Question 2. What then determines Adoption from Word of Mouth? Potential adopters come into contact with adopters through social interactions. However, not every encounter with an adopter results in infection (that is, adoption and purchase of the product). The persuasiveness of the word of mouth and the attractiveness of the product affect the fraction of word of mouth encounters that result in adoption of the product. The fraction of contacts that are sufficiently persuasive to induce the potential adopter to buy the new product, called the Adoption Fraction, is the probability that a given encounter between an adopter and potential adopter results in purchase by the potential adopter. Hence:

Adoption from Word of Mouth = Adoption Fraction * Contacts with Adopters .(4)

The rate at which potential adopters come in contact with adopters depends on the total rate at which social interaction occurs and the probability of encountering an adopter.

Contacts with Adopters = Social Contacts * Probability of Contact with Adopters.(5)

People in the relevant community are assumed to interact with one another at some rate. The Contact Frequency is the number of other people each person in the community comes in contact with, on average, per time period. Therefore, the total number of contacts experienced by all potential adopters per time period is

Social Contacts = Contact Frequency * Potential Adopters .(6)

Question 3: What fraction of these contacts is with a person who already has the product and can generate word of mouth? That is, what is the probability of contact with adopters? The most common assumption is that people in the community interact randomly, so that the probability of contact with an adopter is simply equal to the fraction of adopters in the total community:

Probability of Contact with Adopters = Adopters/Total Market Size .(7)

Question 4. Finally, the revenue generated by the product is given by the price and purchase rate; the purchase rate, in turn, is the product of the adoption rate and the number of units bought by each adopter:

Revenue = Purchase Rate * Price ..(8)

Purchase Rate = Adoption Rate * Units Purchased per Person .(9)