The Bass Diffusion Model

Total sales of a product in any given period = (Sales to Innovators) + (Sales to Imitators)

Innovators are people who are early adopters of the product. They are people who buy a product because they like it; not because other people are buying.

Imitators are people who have heard about a product from someone else and are likely buying it because other people are.

(Sales to Imitators) = (Total Market – Previous Buyers) * (% of Imitators) * ([Previous Buyers] / [Total Market])

The Bass Diffusion Model states that:

Qt = p(M – Nt-1) + q*(Nt-1/M)*(M – Nt-1)


Qt = the number of adopters during time t
M = ultimate number of adopters (market size)
Nt-1 = cumulative number of adopters at the beginning of time t
q = effect of each adopter on each non-adopter. This is the coefficient of imitation.
p = individual conversion rate absent adopters’ influence. This is the coefficient of innovation.

The innovation effect is described by:

p(M – Nt-1)

And the imitation effect is described by:

q*(Nt-1/M)*(M – Nt-1)

We can rewrite the Bass Diffusion Model as follows:

Qt = pM + (q – p)Nt-1 – (q/M)N2t-1

Notice that this is a quadratic equation. A regression analysis can be performed on the data to find coefficients such that:

Qt = a + bNt-1 – cN2t-1


a = pM
b = q – p
c = -(q/M)

M can be found by applying a variation of the Quadratic formula:

M = [-b +/- sqrt(b2 – 4ac)] / 2c

Finally, since a is known from regression and M has been calculated, p can be found. Then, p can be used to find q.

Researching “The Chronicles of Narnia: Price Caspian”

For MKT 402 at Simon, we have been asked to estimate the potential market, adoption rate, and predict the evolution of sales of the new movie “The Chronicles of Narnia: Price Caspian”. A good source of data is available at:

The first part of the assignment focuses on getting information that may help get reasonable values for potential market and diffusion parameters (innovation and imitation) of the movies. Interesting things to focus on are:

  • Are there any movies similar to this movie?
  • What are the sales of similar movies?

Estimation of sales for similar movies is critical since “Prince Caspian” will be released on May 16, 2008 in the United States — which is after the assignment is due.

A good starting place is always Wikipedia since Wikipedia provides an overview and links to primary sources of information:

The movie is a children’s fantasy movie and is a part of Disney’s series “The Chronicles of Narnia.” The first movie in the series was called “The Lion, the Witch, and the Wardrobe.” The movie will be released on June 5 in Australia, and on June 27 in the United Kingdom.

King Miraz is an evil king and he is preventing the rightful heir to the thrown, Prince Caspian, from taking power. The Narnian people are trying to help Prince Caspian and, using a magic horn, have summoned four Pevensie children to help them. Prince Caspian is an adolescent and is in the process of “growing up.” In the movie, unlike the book that its based on, Prince Caspian and Peter Pevensie are somewhat rivals. The Prince is in the process of learning the skills necessary to be a good leader.

The Pevensie children have been in Narnia before. But, while they were gone, one human year (and 1300 Narnaian years) have passed. In the meantime, Narnia has basically been destroyed through the oppression of the Telmarines. The movie is a family movie, but its likely to appeal more to boys than to girls. believes that the movie has a potential audience of 400 million people.

Trailers for the movie can be viewed at:

Looks like there are lots of battle scenes. I’m surprised that its only rated PG.

I collected data from Box Office Mojo for three movies that seem similar to Prince Caspian. These movies are all in the “Fantasy – Live Action” category:

  • The Chronicles of Narnia: The Lion, the Witch and the Wardrobe. Prince Caspian is a sequel to this movie and, so, its obviously similar.
  • Harry Potter and the Order of the Phoenix. This movie is the only movie suggested by Box Office Mojo as a similar movie. To Box Office Mojo, a similar movie is one that is comparable “in terms of audience appeal, genre, tone, timeframe and/or release pattern.” Furthermore, Box Office Mojo suggests that similar movies can be used “to aid in forecasting and to provide suggestions for movie viewing.” This movie contains fantasy/wizardry and battle scenes.
  • Lord of the Rings – The Fellowship of the Ring. This movie was released in 2001. The gross domestic sales were adjusted to 2008 dollars. I’m not sure about the foreign sales. I picked this one because it appears to me that the subject matter is similar. This movie was listed in the “Fantasy – Live Action” category near The Lion, the Witch, and the Wardrobe.