The Big Picture…

Bass’s diffusion framework delivers hands-on insights into how and when innovations spread. It defines “the probability of adoption at time t, provided adoption hasn’t yet happened” using two parameters. The first, p, denotes the rate of adoption driven by external sources. The second, q, the imitation coefficient, represents adoption spurred by internal forces (e.g., word-of-mouth).

By calibrating p and q, the model frequently mirrors real-world sales curves with impressive precision. This enables us to tackle forecasting challenges such as:

• determining the right moment to introduce a new product generation;

• projecting production and supply-chain requirements;

• sizing sampling programs;

• estimating losses from piracy or patent breaches;

• valuing companies; and

• diagnosing market saturation and identifying growth avenues.

In this article, we’ll go deeper into the mathematics and then experiment with the model ourselves.

(You can jump to this article for a quick refresher on diffusion fundamentals.)

Ready? Let’s dive into the math!

Bass’s Diffusion Model – The Background

Frank Bass’s seminal paper “A New Product Growth Model for Consumer Durables” first laid out the formula behind this diffusion theory. For more context, explore the Bass Basement Research Institute.

Before unveiling the equation, reflect on whether this shape looks familiar:

(Figure 1)

Chances are it resembles the S-curve Rogers depicted in Diffusion of Innovations. If you need to revisit, see here:

(Figure)

It’s reassuring that a single equation can recreate Rogers’s empirically observed curve. Although Rogers’s shape approximates a normal distribution, Bass’s model isn’t simply a Gaussian—it only mimics that form for certain p and q values. The fit becomes clear when we compare to real data.

Subtitulo – Bass’s Model and Real Life

Beyond matching Rogers, Bass’s S-curve can be fitted to actual sales figures. Observe these examples:

(figure 3)

The three lower charts—originating from Bass’s 1969 work—plot sales of clothes dryers, black-and-white TVs, and power lawnmowers as solid lines, overlaid with Bass model fits in dots. The alignment, while not flawless, is remarkably close.

Above them, two modern charts show iPhone and Samsung Galaxy sales (dots) against the standard Bass model (solid line). The iPhone curve aligns almost exactly; the Galaxy plot is more scattered due to Samsung’s frequent price shifts. Even so, the underlying Bass shape persists.

Now that we’ve seen the model in action, let’s examine its formula.

The Formula

At its core, Bass’s approach is a differential equation summarized by Bass as:

The fraction of the potential market adopting at time t, given they haven’t adopted yet, is a linear function of prior adopters.

Majahan, Muller and Bass restate:

Probability of adoption at time t (conditional on no prior adoption) = p + q × (F(t)),
where F(t) is the cumulative adoption fraction up to t.

Here, p and q are the innovation and imitation coefficients, respectively.

We can visualize their separate effects here:

(figure 4)

Bass Model Formula

At last, the big reveal—here is the equation for f(t), the share adopting at time t:

( figure 5 )

We’ve introduced p, q, f(t), and F(t). To convert into actual sales S(t) for a market size M, multiply M by f(t). Rearranged, it appears as follows:

(figure 6)

Time to Get Hands-On

Download the Excel workbook from the Bass Basement Research Institute to play with the model yourself. You’ll encounter editable inputs for M (market size), p (innovation rate), and q (imitation rate), alongside dynamic charts:

(figure 7)

Tweak M, p, and q, and observe how the diffusion curve shifts.

Predicting Diffusion

With estimates for M, p, and q, you can simulate adoption over time. Ask questions such as:

• How many staff are needed in year 1?

• What stock levels must the supply chain hold?

• When will adoption peak?

To obtain these parameters, use market surveys for M, fit the model to historical sales for p and q, or rely on industry benchmarks and refine as real data arrives.

Lessons from Predicting Colour TV Sales

Bass recalled forecasting U.S. color-TV sales to peak at 6.7 million units in 1968—an estimate validated by reality, whereas industry forecasts overshot and led to excess capacity.

Predicting for Services (to be developed)

Although less documented, services likely diffuse similarly to products. However, given repeat usage, tracking cumulative adoption and customer churn may prove more relevant than single-purchase curves.

Typical Sizes of Bass’s Coefficients

According to “Diffusion of New Products: Empirical Generalizations and Managerial Uses”:

• p + q typically ranges from 0.3 to 0.7

• average p ≈ 0.03 (often ≤ 0.01)

• average q ≈ 0.38 (rarely < 0.3 or > 0.5)

Refining Adopter Type Sizes

Rogers assigned adopter categories to normal-distribution segments (e.g., 2.5 % innovators). Empirical studies by Bass et al. reveal broader bands:

• Innovators: 0.2–2.8 %

• Early Adopters: 9.5–20 %

• Early/Late Majorities: 29.1–32.1 % (vs. Rogers’s 34 %)

• Laggards: 21.4–23.5 % (vs. 16 %)

These ranges shift where the transition from external to internal influence occurs and challenge fixed-percentage heuristics.

(figure 9)

Further empirical insights appear here:

(figure 10)

What About Generations of Innovations?

Bass extended his model to successive “generations” of products—from 1970s kit computers to early-2000s internet PCs—each exhibiting the characteristic S-curve:

(figure 11)

This guides optimal timing for launching incremental innovations.

Do Price and Other Attributes Affect the Model?

The classic Bass model omits decision variables (price, advertising). A Generalized Bass Model incorporates these factors, defaulting back to the standard form when decision variables are static.

Uses of Diffusion Models

Beyond forecasting and logistics, diffusion models support:

• Pre-launch & launch: supply planning, sample distribution strategies

• Post-launch: next-generation timing, undersupply/oversupply impact

• Piracy estimation: by comparing actual vs. projected sales

• Company valuation: from projected market penetration

• Saturation analysis: spotting when and where to expand

Wrapping Up

Bass’s diffusion model is a rigorous counterpart to Rogers’s qualitative framework. With well-estimated M, p, and q, it accurately reproduces historical adoption patterns and underpins reliable forecasts. Empirical research offers realistic parameter ranges and adjustable adopter-type proportions—equipping strategists to ground marketing and product decisions in solid mathematics.