Relationship between the Bass and the logistic market adoption models

The simplified market adoption model I described on previous postings (1,2) is an Excel implementation of a kind of logistic function. The Bass model is one of the most popular models used in marketing, and management of technology to think about product introductions. (See Wikipedia article). From a mathematical perspective, when the parameter p is 0, the Bass model reduces to the logistic function.

What is most interesting, from a business perspective, is how you arrive to each of those functions by modeling real-world interactions. On both models, you can conceptualize the world as two different pools of people (or stocks, in the system dynamics terminology). One is the pool of potential adopters, and the other is the pool of adopters. The flow between these two pools is controlled by the adoption rate, a variable that models how probable is that a potential adopter becomes “infected” by a current adopter. On the logistic model, it depends solely on how much they interact, how big the total population is, and how “contagious” the product is. On the Bass model, an additional parameter accounts for external factors, the most common being advertising. The Bass model overcomes what is called the startup problem of the logistic model: how a initial base of zero adopters can spread “infection” of the product.

There are more refinements that can be done to the Bass model: accounting for changes in the total population over time, learning and experience curves, etc. For projects where the sensitivity of the model to these factors is high, I definitely recommend to spend more time calibrating your model, understanding which of the different available curves fits better any data you may have, and most critical of all, whether the chosen coefficients for any of the functions have strong impacts on the critical business issues you want to model — capacity planning, pricing, profitability, etc.

For many projects like business plans, revenue projections, etc. I’m willing to sacrifice the ability to fine tune parameters in a model like the BDM for the clarity provided by a model like the Excel logistic function I described. I can generate more tangible conversations with executives by discussing what they believe will be the takeover time, when they believe it will be the start of the fast growth, how much share they believe will be reached in steady state, etc.

Market partition – Mekko chart in Excel, no add-ins

Mekko charts are two dimensional graphs that analyze how data is partitioned against two variables, the X and Y axes. The width of the columns is proportional to data represented by the columns. Individual segment height is a percentage of the respective bar total value.

Yet another in-cell Excel bar chart technique

Two improvements over the technique described by Juice Analytics and posted in Lifehacker: better resolution and solid-looking bars that show better at different font sizes.

If your values are integers in a range 0-9 or so, you can use the REPT formula as presented there, and perhaps you like the dashed type bars, so the formula as shown would work perfectly for you. If not, keep reading.
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Math on the simplified market adoption s-curve for Excel

I’ve got a number of questions on the simplified Excel s-curve formula I published some time ago, so here are more details for those interested in the math behind it. The previous posting focused on how business analysts sometimes need to model market adoption, and provided a simple and easy to maintain formula to do so in Excel.

The formula =saturation/(1 + 81^((hypergrowth + takeover/2 – year)/takeover)) suggested for Excel is a simplification of the formula for a sigmoid function (See the Wikipedia article)

The graphic below shows the shape of both functions is identical. The saturation parameter just scales the function to a desired value, instead of going from 0 to 1. The factor 81 on the Excel formula determines how “sharp” the curve is, in this particular case, reaching 0.1 at the period hypergrowth and 0.9 at hypergrowth + takeover. Note that 81^x can be re-written as e^(ln(81)*x), so whatever factor is used there is simply going to affect the shape by compressing or expanding it horizontally.

This is how the scaling factor can be computed. Let’s say we want the penetration to be 5% at the period specified by hypergrowth. We can work out the solution off the second function. We need to solve for 1/(1+e^(-x) == 0.05, which gives x=-2.94444. Since the function is symmetrical, we also know for x=2.94444 P(x) == 0.95.

Since factor^((hypergrowth + takeover/2 – year)/takeover)) can be re-written as e^(ln(factor)*(hypergrowth + takeover/2 – year)/takeover)), we can solve ln(factor)*(hypergrowth + takeover/2 – (hypergrowth + takeover))/takeover == 2.94444. Reducing all the math, we arrive to
1/(1 + e^(-0.5*ln(factor))) == 0.95, and factor would be 361. If the desired penetration at hypergrowth is 20%, then we solve 1/(1 + e^(-0.5*ln(factor))) == 0.80, leading to factor == 16