Bass diffusion model

${\displaystyle {\frac {f(t)}{1-F(t)}}=p+qF(t)}$ [2]

Where:

• ${\displaystyle \ F(t)}$ is the installed base fraction
• ${\displaystyle \ f(t)}$ is the rate of change of the installed base fraction, i.e. ${\displaystyle \ f(t)=F'(t)}$
• ${\displaystyle \ p}$ is the coefficient of innovation
• ${\displaystyle \ q}$ is the coefficient of imitation

Expressed as an ordinary differential equation,

${\displaystyle {\frac {dF}{dt}}=p(1-F)+q(1-F)F=(1-F)(p+qF).}$

Sales (or new adopters) ${\displaystyle \ s(t)}$ at time ${\displaystyle \ t}$ is the rate of change of installed base, i.e., ${\displaystyle \ f(t)}$ multiplied by the ultimate market potential ${\displaystyle \ m}$. Under the condition ${\displaystyle \ F(0)=0}$, we have that

${\displaystyle \ s(t)=mf(t)}$

${\displaystyle \ s(t)=m{\frac {(p+q)^{2}}{p}}{\frac {e^{-(p+q)t}}{(1+{\frac {q}{p}}e^{-(p+q)t})^{2}}}}$ [2]

We have the decomposition ${\displaystyle \ s(t)=s_{n}(t)+s_{i}(t)}$ where ${\displaystyle \ s_{n}(t):=mp(1-F(t))}$ is the number of innovators at time ${\displaystyle \ t}$, and ${\displaystyle \ s_{i}(t):=mq(1-F(t))F(t)}$ is the number of imitators at time ${\displaystyle \ t}$.

The time of peak sales ${\displaystyle \ t^{*}}$

${\displaystyle \ t^{*}={\frac {\ln q-\ln p}{p+q}}}$ [2]

The times of the inflection points at the new adopters' curve t**

${\displaystyle \ t^{**}={\frac {\ln(q/p)-\ln(2\pm {\sqrt {3}}))}{p+q}}}$ [4]

or in another form (related to peak sales)

${\displaystyle \ t^{**}=t^{*}\pm {\frac {\ln(2+{\sqrt {3}}))}{p+q}}}$ [4]

The peak time and inflection points' times must be positive. When t* is negative sales have no peak (and decline since introduction). There are cases (depending on p, q values) when the new adopters curve (that begins at 0) has only one or none inflection points.

Derivation

The Bass diffusion model is derived by assuming that the hazard rate ${\displaystyle \lambda (t)}$ for the uptake of a product or service may be defined as:

$${\displaystyle \lambda (t)={f(t) \over {S(t)}}=p+q[1-S(t)]}$$

where ${\displaystyle f(t)}$ is the probability density function and ${\displaystyle S(t)=1-F(t)}$ is the survival function, with ${\displaystyle F(t)}$ being the cumulative distribution function. From these basic definitions in survival analysis, we know that:

$${\displaystyle f(t)=-{dS \over {dt}}\implies \lambda (t)=-{1 \over {S}}{dS \over {dt}}}$$

Therefore, the differential equation for the survival function is equivalent to:

$${\displaystyle {dS \over {S[p+q(1-S)]}}=-dt}$$

Integration and rearrangement of terms gives us that:

$${\displaystyle {S \over {p+q(1-S)}}=Ae^{-(p+q)t}}$$

For any survival function, we must have that ${\displaystyle S(0)=1}$ and this implies that ${\displaystyle A=p^{-1}}$. With this condition, the survival function is:

$${\displaystyle S(t)={e^{-(p+q)t}+{q \over {p}}e^{-(p+q)t} \over {1+{q \over {p}}e^{-(p+q)t}}}}$$

Finally, using the fact that ${\displaystyle F(t)=1-S(t)}$, we find that the Bass diffusion model for product uptake is:

$${\displaystyle F(t)={1-e^{-(p+q)t} \over {1+{q \over {p}}e^{-(p+q)t}}}}$$

Bass (1969)[2] distinguished between a case of p<q wherein periodic sales grow and then decline (a successful product has a periodic sales peak); and a case of p>q wherein periodic sales decline from launch (no peak).

Jain et al. (1995)[11] explored the impact of seeding. When using seeding, diffusion can begin when p + qF(0) > 0 even if p’s value is negative, but a marketer uses seeding strategy with seed size of F(0) > -p/q . The interpretation of a negative p value does not necessarily mean that the product is useless: There can be cases wherein there are price or effort barriers to adoption when very few others have already adopted. When others adopt, the benefits from the product increase, due to externalities or uncertainty reduction, and the product becomes more and more plausible for many potential customers.

Moldovan and Goldenberg (2004)[12] incorporated negative word of mouth (WOM) effect on the diffusion, which implies a possibility of a negative q. Negative q does not necessarily mean that adopters are disappointed and dissatisfied with their purchase. It can fit a case wherein the benefit from a product declines as more people adopt. For example, for a certain demand level for train commuting, reserved tickets may be sold to those who like to guarantee a seat. Those who do not reserve seating may have to commute while standing. As more reserved seating are sold, the crowding in the non-reserved railroad car is reduced, and the likelihood of finding a seat in the non-reserved car increases, thus reducing the incentive to buy reserved seating. While the non-cumulative sales curve with negative q is similar to those with q=0, the cumulative sales curve presents a more interesting situation: When p > -q, the market will reach 100% of its potential, eventually, as for a regular positive value of q. However, if p < -q, at the long-range, the market will saturate at an equilibrium level –p/q of its potential.

Orbach (2022)[13] summarized the diffusion behavior at each portion of the p,q space and maps the extended (p,q) regions beyond the positive right quadrant (where diffusion is spontaneous) to other regions where diffusion faces barriers (negative p), where diffusion requires “stimuli” to start, or resistance of adopters to new members (negative q), which might stabilize the market below full adoption, occur.

A map of the extended Bass p-q space [13]

The model is one of the most cited empirical generalizations in marketing; as of August 2023 the paper "A New Product Growth for Model Consumer Durables" published in Management Science had (approximately) 11352 citations in Google Scholar.[14]

This model has been widely influential in marketing and management science. In 2004 it was selected as one of the ten most frequently cited papers in the 50-year history of Management Science.[3] It was ranked number five, and the only marketing paper in the list. It was subsequently reprinted in the December 2004 issue of Management Science.[3]

The Bass model was developed for consumer durables. However, it has been used also to forecast market acceptance of numerous consumer and industrial products and services, including tangible, non-tangible, medical,[15][16] and financial[17] products. Sultan[18] et al. (1990) applied the Bass model to 213 product categories, mostly consumer durables (in a wide range of prices) but also to services such as motels and industrial/farming products like hybrid corn seeds.

• Diffusion of innovation
• Forecasting
• Lazy user model
• Shifted Gompertz distribution