MBA New Product Diffusion Concept Card Template

New Product Diffusion Concept Card: [Product, market or course]

A concept card on forecasting the adoption of a new product with the Bass (1969) diffusion model, for MBA marketing analytics and new-product electives. It builds on the core treatment in mba/mba-marketing-guide (section 4, "New products and diffusion": Rogers's adopter categories, the Bass hazard, $F(t)$, $N(t)$ and the peak time $t^{*}$) and does not repeat it. This card goes further: the peak size, the split between innovators and imitators, estimating $m$, $p$ and $q$ from early sales with regression, why early estimates are unstable, pre-launch forecasting by analogy, and the main extensions (marketing-mix effects, successive generations, repeat purchases, takeoff). It is for second-year MBA students and for product, marketing and finance managers who must put a launch forecast in a business case. Source basis: Bass (1969), Mahajan, Muller and Bass (1990), Srinivasan and Mason (1986), Sultan, Farley and Lehmann (1990), Bass, Krishnan and Jain (1994), Norton and Bass (1987), Golder and Tellis (1997), written up in original words with fictional Corvane Robotics, Northfield Outdoor Co. and Tallow Creek Medical Center data. Replace every [bracketed] value and every row marked Example with your own course or company data.

Document control

Contents

1. When to use it
2. The idea in plain language
3. Formulas and notation
4. Assumptions and limits
5. How managers use it
6. Worked example A (fully solved)
7. Worked example B (half solved): estimation from early sales
8. Exercise C (unsolved)
9. Extensions beyond the basic model
10. Connections to other cards
11. Common mistakes and misconceptions
12. Self-test, notes, sources, answer key

When to use it

Use the Bass model when a decision depends on when first purchases of a new product will happen, not only on how many there will eventually be:

• Business cases for a launch. The NPV of a new product depends heavily on how fast adoption builds. A five-year window can cut a project off just before the peak, as the Fastpack case in mba/mba-marketing-guide shows.
• Capacity and supply planning. Production, installation crews, service staff and inventory must be in place for the peak, then scaled back. A diffusion curve tells operations when.
• Funding a start-up. Investors in a venture ask when revenue takes off; a diffusion forecast with ranges is a disciplined answer (see mba/mba-venture-capital-guide).
• Launch strategy. Comparing the external-influence and internal-influence parts of adoption tells you whether to spend on mass awareness or on seeding reference customers.
• Successive generations. When a second generation replaces the first, the model helps to time the switch.

The idea in plain language

Every period, some of the people who have not yet adopted do so. Two forces push them. External influence (the coefficient $p$) acts on everyone who has not adopted, whether or not anyone they know has: advertising, sales calls, trade press, the product itself on the shelf. Internal influence (the coefficient $q$) grows with the share of the market that has already adopted: word of mouth, seeing the product in use, reference customers, lower perceived risk. Early on, few have adopted, so external influence dominates and sales are low. As adopters accumulate, internal influence takes over and sales accelerate. Eventually the pool of remaining non-adopters shrinks and sales fall. The result is the familiar bell-shaped curve of first purchases and the S-shaped curve of cumulative adoption.

Bass (1969) wrote this as a hazard model and showed that three numbers, the market potential $m$ and the two coefficients, fit the sales histories of many consumer durables well and could forecast the timing and size of the peak. Mahajan, Muller and Bass (1990) reviewed two decades of extensions. In practice the model is a structured way to think about three questions a launch team must answer anyway: how big can this get, how fast does it start, and how much does it feed on itself?

The ratio $q/p$ is the most useful single summary. A high ratio (imitation strongly dominates) means a slow start, a late peak in a nearly symmetric bell, and a strong case for seeding early adopters. A low ratio means a fast start, an early peak and a curve skewed toward launch. In the meta-analysis of Sultan, Farley and Lehmann (1990), the average coefficients across published applications were about $p = 0.03$ and $q = 0.38$ per year, a ratio of about 13, with wide variation across products and countries.

Formulas and notation

Hazard (Bass 1969). The probability of adopting at time $t$, given no adoption yet:

$\frac{f(t)}{1 - F(t)} = p + q\,F(t)$

Closed form (continuous time, $F(0) = 0$), cumulative adopters and adoption rate:

$F(t) = \frac{1 - e^{-(p+q)t}}{1 + \frac{q}{p}\, e^{-(p+q)t}}, \qquad N(t) = m\,F(t)$

$f(t) = \frac{(p+q)^2}{p}\cdot\frac{e^{-(p+q)t}}{\left(1 + \frac{q}{p}\,e^{-(p+q)t}\right)^2}, \qquad n(t) = m\,f(t)$

Peak. The adoption rate peaks at $t^{*}$, when the cumulative share and the peak rate are:

$t^{*} = \frac{\ln(q/p)}{p+q}, \qquad F(t^{*}) = \frac{1}{2} - \frac{p}{2q}, \qquad n(t^{*}) = \frac{m\,(p+q)^2}{4q}$

An interior peak exists only if $q > p$. If $q \le p$, adoption is highest at launch and declines from there.

Innovators and imitators. Adoption in a period splits into the part due to external influence and the part due to internal influence:

$n(t) = \underbrace{p\,[m - N(t)]}_{\text{innovators}} + \underbrace{q\,\frac{N(t)}{m}\,[m - N(t)]}_{\text{imitators}}$

Discrete form for estimation (Bass 1969). With $n_t$ the adopters in period $t$ and $N_{t-1}$ the cumulative adopters at the end of the previous period:

$n_t = a + b\,N_{t-1} + c\,N_{t-1}^2, \qquad a = p\,m, \quad b = q - p, \quad c = -\frac{q}{m}$

Recover the parameters from the regression coefficients (take the positive root, with $c < 0$):

$m = \frac{-b - \sqrt{b^2 - 4ac}}{2c}, \qquad p = \frac{a}{m}, \qquad q = -c\,m$

Excel. With $p$, $q$, $m$ in named cells and $t$ in A2:

• Cumulative adopters: =m*(1-EXP(-(p+q)*A2))/(1+(q/p)*EXP(-(p+q)*A2))
• Adopters in period $t$: the difference of two consecutive cumulative values, =B3-B2
• Peak time: =LN(q/p)/(p+q); peak rate: =m*(p+q)^2/(4*q)
• Regression: =LINEST(C2:C8, A2:B8) with $N_{t-1}$ in column A and $N_{t-1}^2$ in column B. LINEST returns the coefficients in reverse column order: $c$, then $b$, then $a$.
• Market potential from the coefficients: =(-b-SQRT(b^2-4*a*c))/(2*c)

Assumptions and limits

The model forecasts first purchases. It does not forecast revenue until you add price, usage, churn and repeat behaviour. It also says nothing about why a market takes off; that needs customer research, experiments and the marketing plan.

How managers use it

• Before launch: forecasting by analogy. Without sales data, choose two to five analogous products (similar benefit, price relative to income, buying unit, need for complementary infrastructure), take their published or internal $p$ and $q$, and weight them by similarity. Estimate $m$ separately from market sizing (mba/mba-market-sizing-drills) and purchase-intent research, adjusted downward because stated intent overstates purchase. Present a range, not a point.
• After four to eight periods: re-estimation. Fit the discrete regression or non-linear least squares (Srinivasan and Mason 1986) to actual sales. Expect the estimates to move a lot until the inflection point has passed (Worked example B).
• Capacity and staffing. Plan installation crews, service engineers or production around the peak rate $n(t^{*})$ and the time to peak, with a buffer for forecast error.
• Seeding. When $q/p$ is high, the first adopters are worth more than their own purchases because they feed later imitation. Firms give pilot discounts, fund reference sites and target well-connected customers first.
• Launch NPV. Feed the adopter forecast into a cash-flow model and test the horizon: a project with a late peak needs a long enough window (mba/mba-npv-irr-card). The option to stop after a pilot adds value when uncertainty is high (mba/mba-options-real-options-card).
• Crossing the chasm. Moore (1991) argued that technology products often stall between early adopters and the early majority, who want proven, complete solutions and references from people like them. A Bass curve that assumes smooth imitation will miss a stall; watch the early-majority pipeline separately.

Worked example A (fully solved): Corvane robots-as-a-service for mid-market warehouses

Situation (fictional). Corvane Robotics, a venture-backed warehouse-robotics company, plans to offer its autonomous mobile robots as a subscription (robots-as-a-service) to mid-market warehouse operators that cannot fund a large capital purchase. The strategy team defines the adopting unit as a warehouse and estimates:

Step 1: the ratio. $q/p = 0.45/0.02 = 22.5$, well above the meta-analytic average of about 13: a slow start, then strong contagion through reference customers.

Step 2: peak time. $t^{*} = \ln(22.5)/0.47 = 6.625$ years. Excel: =LN(0.45/0.02)/(0.02+0.45).

Step 3: share adopted at the peak and peak size. $F(t^{*}) = 0.5 - 0.02/0.90 = 0.4778$, so 1,146.7 warehouses have adopted by the peak. The peak adoption rate is $2{,}400 \times 0.47^2/(4 \times 0.45) = 294.5$ warehouses per year. Excel: =0.5-0.02/(2*0.45) and =2400*(0.02+0.45)^2/(4*0.45).

Step 4: the forecast, year by year. Cumulative adopters from the closed form, new adopters as the difference, and the split into innovators and imitators (each part integrated over the year).

Excel, with the year in A2 and named cells m, p, q: cumulative =m*(1-EXP(-(p+q)*A2))/(1+(q/p)*EXP(-(p+q)*A2)); new adopters in year 1 =B2, then =B3-B2; contract value =new*0.25. The innovator and imitator split integrates $p\,[m - N(s)]$ and $q\,N(s)[m-N(s)]/m$ over each year; in a spreadsheet, approximate it with monthly steps, or use the discrete split $p\,(m - N_{t-1})$ and $q\,(N_{t-1}/m)(m - N_{t-1})$ for a quick check.

Step 5: interpretation.

• The highest year is year 7 (292.9 new warehouses), consistent with $t^{*} = 6.6$. Cumulative adoption passes 16% (the innovators and early adopters of Rogers's scheme) at 3.6 years and half of $m$ at 6.8 years. Excel Goal Seek on the cumulative formula finds these times.
• Over twelve years 14.8% of adopters are driven by external influence (328.5 of 2,214.5) and 85.2% by imitation. In year 1, 79% of adopters are innovators; by year 7, 91% are imitators. Advertising and direct sales start the curve; reference customers carry it.
• Contract value grows from USD 14.94m to a peak of USD 73.23m in year 7. A venture investor looking at years 1 to 3 sees slow growth; the model says this is the normal shape for a high $q/p$ product, provided early adopters are happy enough to be referenced.

Step 6: sensitivity. Doubling $p$ to 0.04 (a larger direct-sales force) brings the peak forward to 4.94 years and raises cumulative adoption at year 5 from 28.8% to 46.4%. Cutting $q$ by a quarter to 0.3375 (poor early experiences, fewer referrals) delays the peak to 7.90 years, lowers the peak rate to 227.2 a year and leaves only 21.8% adopted at year 5. The second risk is the one to manage: early customer success matters more than launch advertising for this product.

The chart rounds each part to whole warehouses, so a bar can differ from the table total by one.

Worked example B (half solved): estimating the model from early sales

Situation (fictional). Northfield Outdoor Co. launched a battery-heated hiking jacket seven years ago. First purchases (one per buyer, replacement sales excluded) were:

1. Put $N_{t-1}$ in A2:A8, $N_{t-1}^2$ in B2:B8 (=A2^2) and $n_t$ in C2:C8. Run =LINEST(C2:C8, A2:B8). The result is $c = -5.4600 \times 10^{-6}$, $b = 0.299683$, $a = 1{,}275.8971$ ($R^2 = 0.9994$).
2. Market potential: $m = \dfrac{-b - \sqrt{b^2 - 4ac}}{2c} =$ [ ] buyers. Excel: =(-0.299683-SQRT(0.299683^2-4*1275.8971*(-5.46E-06)))/(2*(-5.46E-06)).
3. $p = a/m =$ [ ] and $q = -c\,m =$ [ ] (four decimals). Check: $q - p$ should equal $b$.
4. Peak time $t^{*} = \ln(q/p)/(p+q) =$ [ ] years, and the share adopted at the peak $F(t^{*}) =$ [ ].
5. Using the discrete form $n_t = (p + q N_{t-1}/m)(m - N_{t-1})$ with your estimates, forecast new buyers in year 8: [ ]. Then years 9 and 10 are 5,336 and 5,360.
6. Re-run the regression on years 1 to 4 only. The coefficients are $a = 1{,}253.313$, $b = 0.33396$, $c = -1.183 \times 10^{-5}$. Compute $m$ from them: [ ]. In two sentences, explain why the two estimates of $m$ differ so much and what that means for a forecast made in year 4.

Exercise C (unsolved): remote patient monitoring at Tallow Creek

Tallow Creek Medical Center, a regional hospital, launches a remote monitoring service for patients with chronic heart failure. About 6,000 eligible patients could enrol. Planning assumptions per quarter: $p = 0.03$ (clinician referral and mailed invitations) and $q = 0.38$ (patients hearing from other patients and from their family doctors).

1. Compute the peak quarter $t^{*}$, the number of patients enrolled by the peak and the peak enrolment rate. Write the Excel formulas.
2. How many patients are enrolled after four and after eight quarters?
3. The nurse coordinator team can onboard at most 600 new patients a quarter. In which quarters, if any, does forecast demand exceed that? What happens to the curve if onboarding is capped, and how would that bias a later estimate of $q$?
4. A proactive outreach programme would double $p$ to 0.06. Recompute $t^{*}$ and enrolment after four and eight quarters. Is outreach worth considering if the clinical benefit comes mainly from early enrolment?
5. Name two reasons why a single $m$ of 6,000 may be wrong for this service.

Extensions beyond the basic model

Marketing-mix effects: the Generalized Bass Model. Bass, Krishnan and Jain (1994) multiplied the hazard by a function of current marketing effort, $x(t)$, built from percentage changes in price and advertising (discrete form, with $\Delta Pr(t) = Pr(t) - Pr(t-1)$; in the original, only increases in advertising enter):

$\frac{f(t)}{1-F(t)} = \left[p + q\,F(t)\right] x(t), \qquad x(t) = 1 + \beta_1 \frac{\Delta Pr(t)}{Pr(t-1)} + \beta_2 \max\left\{0, \frac{\Delta Adv(t)}{Adv(t-1)}\right\}$

They showed why the basic model fits so well without decision variables: when price and advertising change at roughly constant percentage rates, $x(t)$ is close to a constant and is absorbed into $p$ and $q$. The practical lesson is that a sudden price cut or campaign burst shifts the curve, while a steady price decline is already "inside" the fitted coefficients.

Successive generations. Norton and Bass (1987) modelled a new generation that attracts both new adopters and earlier-generation users who upgrade. Sales of the old generation fall as the new one diffuses. Use it to time a second-generation launch and to avoid double-counting $m$ across generations.

Repeat and replacement purchases. Total unit sales are first purchases plus replacements (durables) or repeat purchases (consumables, subscriptions). For a subscription, treat the Bass forecast as gross new sign-ups and apply a retention model to get the active base, the same way CLV handles churn (mba/mba-customer-lifetime-value-card).

Takeoff. Golder and Tellis (1997) studied when sales of really new consumer durables first jump sharply, and found that takeoff tended to come several years after introduction and to be associated with falling prices. A Bass curve fitted before takeoff is close to guesswork; that is when analogies and scenarios matter most.

Social contagion versus heterogeneity. An S-shaped curve does not prove that people copy each other. If buyers differ in their willingness to pay and prices fall, adoption also follows an S-shape. Van den Bulte and Stremersch (2004) found that the $q/p$ ratio across countries is associated with income inequality as well as with cultural measures, so part of what the model labels "imitation" may reflect heterogeneity. The distinction matters for strategy: seeding works only if contagion is real.

Estimation methods. The discrete regression is simple but has known problems: the three coefficients are estimated from collinear regressors, and the discrete form approximates a continuous process. Srinivasan and Mason (1986) proposed non-linear least squares on the closed form, which also yields standard errors. In Excel, set up the closed-form forecast, a column of squared errors and use Solver to minimise their sum by changing $m$, $p$ and $q$, with constraints $m \ge N_T$, $p > 0$, $q \ge 0$.

Connections to other cards

• Core marketing (mba/mba-marketing-guide): adopter categories, the Bass formula and the Fastpack launch case.
• Customer lifetime value (mba/mba-customer-lifetime-value-card): diffusion gives the number of new customers per period; CLV gives what each is worth. Together they give the value of a launch.
• NPV and real options (mba/mba-npv-irr-card, mba/mba-options-real-options-card): the horizon and the option to stop or scale after a pilot.
• Conjoint analysis and pricing (mba/mba-conjoint-analysis-card, mba/mba-pricing-card): estimate the share of the market that would buy at a given price, an input to $m$.
• Marketing experiments (mba/mba-marketing-experiments-card): test seeding or referral incentives before assuming they raise $q$.
• Innovation portfolio and technology strategy (mba/mba-innovation-portfolio-card, mba/mba-technology-strategy-guide): diffusion speed is one input to how a firm balances core, adjacent and transformational bets.
• Platform strategy (mba/mba-platform-strategy-card): with network effects, value to each adopter rises with the installed base, a stronger form of internal influence.
• Regression (mba/mba-regression-for-managers-card, academic/stats-linear-regression): the estimation step.

Common mistakes and misconceptions

• Mixing time units. An annual $q$ of 0.40 is not a quarterly $q$ of 0.40. Re-estimate or convert, and state the unit in every table.
• Forecasting revenue with first purchases only. The Bass curve stops at first adoption. Add replacement, repeat, usage and churn before you put revenue in a business case.
• Treating $m$ as the total population. $m$ is the number of eventual adopters at the expected price and product quality, usually much smaller than everyone who could in theory buy.
• Trusting estimates from data before the inflection point. Worked example B shows the market potential estimate changing by almost a factor of two between four and seven years of data.
• Reading a supply constraint as weak demand. If stock-outs or onboarding capacity capped sales, the fitted $q$ is biased down and $m$ may be too.
• Forgetting the order of LINEST output. Excel returns coefficients in reverse column order; swapping $a$ and $c$ produces nonsense parameters.
• Assuming the S-curve proves word of mouth. Falling prices and heterogeneous buyers create the same shape. Test contagion before you fund a seeding programme.
• Cutting the evaluation window before the peak. A five-year NPV for a product that peaks in year 7 is biased against the launch; show the horizon sensitivity.
• Using one point forecast before launch. Analogies give a range. Show at least a low, base and high case for $p$, $q$ and $m$.

Self-test questions

1. In one sentence each, what do $p$, $q$ and $m$ represent?
2. What happens to the timing and shape of adoption when $q/p$ rises, holding $p + q$ fixed?
3. With $p = 0.03$ and $q = 0.38$ (the approximate meta-analytic averages, per year), when does adoption peak, and what share of $m$ has adopted by then?
4. In the discrete model with $m = 10{,}000$, $p = 0.02$ and $q = 0.5$, how many adopt in period 1 and in period 2?
5. Why does the Bass model need $q > p$ for an interior peak?
6. A regression of $n_t$ on $N_{t-1}$ and $N_{t-1}^2$ gives a positive coefficient on $N_{t-1}^2$. What does that mean?
7. Why can the basic Bass model fit well even though it has no price or advertising variables?
8. Name two ways to forecast $p$ and $q$ before any sales exist.
9. A start-up's sales grew slowly for three years. Give one reason this could be consistent with a strong product, and one test you would run to tell the difference.
10. How would you turn a Bass forecast of first purchases into a revenue forecast for a subscription product?

My notes and next steps

Sources and further reading

• Bass, F. M. (1969). A new product growth for model consumer durables. Management Science, 15(5), 215-227.
• Bass, F. M., Krishnan, T. V., and Jain, D. C. (1994). Why the Bass model fits without decision variables. Marketing Science, 13(3), 203-223.
• Golder, P. N., and Tellis, G. J. (1997). Will it ever fly? Modeling the takeoff of really new consumer durables. Marketing Science, 16(3), 256-270.
• Mahajan, V., Muller, E., and Bass, F. M. (1990). New product diffusion models in marketing: a review and directions for research. Journal of Marketing, 54(1), 1-26.
• Moore, G. A. (1991). Crossing the Chasm. HarperBusiness.
• Norton, J. A., and Bass, F. M. (1987). A diffusion theory model of adoption and substitution for successive generations of high-technology products. Management Science, 33(9), 1069-1086.
• Rogers, E. M. (1962). Diffusion of Innovations. Free Press.
• Srinivasan, V., and Mason, C. H. (1986). Nonlinear least squares estimation of new product diffusion models. Marketing Science, 5(2), 169-178.
• Sultan, F., Farley, J. U., and Lehmann, D. R. (1990). A meta-analysis of applications of diffusion models. Journal of Marketing Research, 27(1), 70-77.
• Van den Bulte, C., and Stremersch, S. (2004). Social contagion and income heterogeneity in new product diffusion: a meta-analytic test. Marketing Science, 23(4), 530-544.
• Further reading (by name only): Lilien, Rangaswamy and De Bruyn, Principles of Marketing Engineering and Analytics; Urban and Hauser, Design and Marketing of New Products; Kotler and Keller, Marketing Management; OpenStax, Principles of Marketing (free, CC BY-NC-SA 4.0, openstax.org).

Answer key

Worked example B. (2) $m = 58{,}857$ buyers (58,857.4). (3) $p = 1{,}275.8971/58{,}857.4 = 0.0217$; $q = 5.46 \times 10^{-6} \times 58{,}857.4 = 0.3214$; check $0.3214 - 0.0217 = 0.2997 = b$. (4) $t^{*} = \ln(0.3214/0.0217)/0.3431 = 7.86$ years; $F(t^{*}) = 0.5 - 0.0217/0.6428 = 0.4663$, so the peak is still ahead (about a year away by the continuous formula; the discrete recursion in step 5 keeps rising to year 10, a reminder that the two forms time the peak differently). (5) Year 8: $(0.0217 + 0.3214 \times 19{,}334/58{,}857)(58{,}857 - 19{,}334) = 5{,}029$ new buyers, cumulative 24,363 (with unrounded parameters). (6) Years 1 to 4 give $m = 31{,}574$, $p = 0.0397$, $q = 0.3737$. Before the inflection point the sales series is still convex and almost any $m$ fits it; the curvature that pins down $m$ appears only as growth slows. A year-4 forecast would have understated the market by about 46% relative to the seven-year estimate, so in year 4 the team should have combined the fit with analogies and reported a wide range.

Exercise C. (1) $t^{*} = \ln(0.38/0.03)/0.41 = 6.19$ quarters (=LN(0.38/0.03)/(0.03+0.38)); enrolled by the peak $6{,}000 \times (0.5 - 0.03/0.76) = 2{,}763$ patients; peak rate $6{,}000 \times 0.41^2/(4 \times 0.38) = 663.6$ per quarter (=6000*(0.03+0.38)^2/(4*0.38)). (2) After four quarters 1,399 (=6000*(1-EXP(-0.41*4))/(1+(0.38/0.03)*EXP(-0.41*4))); after eight quarters 3,910 (65.2%). (3) Forecast new enrolments by quarter: 215, 296, 393, 496, 588, 648, 659, 616, 534, 432, 332, 244. Quarters 6, 7 and 8 exceed 600. A cap flattens and lengthens the peak and pushes enrolments later; if the capped data were later used to estimate the model, $q$ (and possibly $m$) would be biased down. (4) With $p = 0.06$: $t^{*} = 4.20$ quarters, 2,377 enrolled after four quarters and 4,903 after eight. Enrolment comes about two quarters earlier, which matters if the clinical benefit depends on early enrolment; check that onboarding capacity can absorb the earlier and higher peak. (5) Eligibility changes as patients are diagnosed, move or die, so $m$ is not fixed; some patients lack connectivity or devices, so the practical $m$ is lower than the eligible count; referrals depend on clinicians, so adoption is segmented by practice.

Self-test.

1. $p$: the rate at which non-adopters adopt because of influences outside the adopter population (advertising, sales effort, the product itself). $q$: the additional rate per unit of adopted share, from previous adopters (word of mouth, visibility). $m$: the number of eventual adopters.
2. The start is slower and the peak comes later; the curve becomes a more symmetric bell (with $p + q$ fixed, the peak rate per unit of $m$, $(p+q)^2/(4q)$, is slightly lower), and a larger share of adopters is driven by imitation.
3. $t^{*} = \ln(0.38/0.03)/0.41 = 6.19$ years; $F(t^{*}) = 0.5 - 0.03/0.76 = 0.4605$, about 46%.
4. Period 1: $p\,m = 200$. Period 2: $(0.02 + 0.5 \times 200/10{,}000)(10{,}000 - 200) = 0.03 \times 9{,}800 = 294$.
5. The rate of adoption rises only if the boost from new adopters ($q$ times the change in $F$) outweighs the shrinking pool of non-adopters; that needs $q > p$, and $t^{*} = \ln(q/p)/(p+q)$ is positive only then.
6. $c = -q/m$ must be negative. A positive coefficient means the data have not yet shown any slowing (no inflection), so the model cannot identify $m$: use analogies or wait for more data.
7. Bass, Krishnan and Jain (1994): when price and advertising change at roughly constant percentage rates, their effect is a near-constant multiplier on the hazard and is absorbed into $p$ and $q$.
8. Analogous products (internal history or published estimates, weighted by similarity); meta-analytic averages as a starting point; managerial judgement elicited for the time to peak and the peak share, from which $p$ and $q$ can be backed out.
9. A high $q/p$ product has a slow start by construction, so slow early growth can coexist with strong eventual adoption. Test it by tracking referral-driven sales, reference-customer satisfaction and the share of new customers who name an existing user as the reason, or by fitting the model and checking whether imitation is rising.
10. Treat Bass adopters as gross new subscribers per period, apply retention to get the active base (the survivors of each cohort), multiply by price or margin per period, and discount. That is the cohort logic of the CLV card applied to the diffusion forecast.

Revision history

Standard and licence note

Based on the published new-product diffusion literature (Bass 1969; Mahajan, Muller and Bass 1990; Srinivasan and Mason 1986; Sultan, Farley and Lehmann 1990; Bass, Krishnan and Jain 1994; Norton and Bass 1987) as taught in MBA marketing analytics electives; no single standard governs diffusion forecasting. The explanations, worked examples, data and layout are original and fictional; no textbook text, published case or exam question is reproduced. The template is not affiliated with or endorsed by any business school or publisher. Check your own course materials for the notation and conventions they require. Template licence: MIT.