Demonstration 1 of 4
The S-curve and the year new adoption peaks
How do the innovation and imitation coefficients decide when a new product's adoption is busiest?
The left panel is cumulative adoption bending toward the ceiling; the right panel is its slope, new adopters per year. When q is larger than p, imitation pressure builds as adopters accumulate, so the slope rises, peaks and falls. When q is not larger than p, there is no steep middle: the rate is highest at launch.
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N is the cumulative number of adopters at time t, m the ceiling (100,000 here), p the coefficient of innovation and q the coefficient of imitation, both rates per year. t star is the year new adoption peaks; ln is the natural logarithm.
Predict first. With p 0.025 and q 0.35, will new adoption peak before or after year 5?
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Constructed data: the chapter notebook's Bass curve (p 0.025, q 0.35 per year, ceiling 100,000), with p and q varied.
Calculated values
- Peak year of new adopters
- 7.04
- Adopters at the peak
- 46429
- New adopters per year at the peak
- 10045
- Adopters after 10 years
- 73461
Here q is larger than p, so new adoption rises before it falls. q / p = 0.350 / 0.025 = 14.00, ln(14.00) = 2.639, and the peak comes at 2.639 / 0.375 = 7.04 years, when 100000 x (0.350 - 0.025) / (2 x 0.350) = 46429 have adopted. The ceiling scales every count but does not move the peak year.
Worked steps
- Ratio of imitation to innovation: 0.350 / 0.025 = 14.00.
- Peak year: ln(14.00) / (0.025 + 0.350) = 2.639 / 0.375 = 7.04.
- Adopters at the peak: 100000 x (0.350 - 0.025) / (2 x 0.350) = 46429.
Use the idea
Before reserving capacity for a launch, read the incidence curve for the busiest year and the cumulative curve for the eventual scale; if your rates have q not above p, do not plan for a later peak.
Where the conclusion applies
The rates and ceiling are the notebook's constructed values, not estimates for any product. The basic model has no price, advertising or repeat purchase in it, and the ceiling is fixed.
Check your understanding: With p 0.02 and q 0.3 per year, in which year does new adoption peak?
Chapter 19 source: section "Section One: Three Weeks with a Graph".
Demonstration 2 of 4
Five early years cannot choose the ceiling
If several Bass curves fit the first five years about equally well, do the early data tell you how big the market is?
The dots are five constructed early observations. Each line is the Bass curve fitted to them with a different ceiling held fixed; the shaded band is the data. Inside it the lines agree; beyond it they diverge, because p, q and m trade off against each other when the inflection has not been seen.
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m is the assumed ceiling, held fixed while p and q are fitted to the five early points by least squares. RMSE is the root mean square gap between the fitted curve and those points.
Predict first. The ceilings 60k and 180k differ by a factor of three. Will their fits to the early years differ by a similar factor?
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Constructed data: the chapter notebook's five-year calibration (Bass curve p 0.025, q 0.35, ceiling 100,000 plus seeded noise, seed 20260937) and its three assumed ceilings.
Calculated values
- Fitted p
- 0.0251
- Fitted q
- 0.3465
- Early fit error (RMSE, adopters)
- 62
- Projected adopters in year 15
- 94667
- Projected peak year
- 7.06
Assuming a ceiling of 100000, the fit misses the five early points by 62 adopters on average (the three ceilings range from 62 to 160), yet year 15 lands anywhere from 59.5 to 149.6 thousand: 149.6 - 59.5 = 90.1 thousand apart. The early years constrain the start of the curve, not where it stops; the ceiling has to come from outside the fit.
Worked steps
- Ceiling held at 100000; p and q fitted to five points: p 0.0251, q 0.3465.
- Root mean square miss on those points: 62 adopters.
- Spread of the year 15 projections across the three ceilings: 149.6 - 59.5 = 90.1 thousand.
Use the idea
Fit with several defended ceilings from market sizing, show all of them, and treat close early fits as weak identification, not as a reason to prefer the ceiling the sponsor likes.
Where the conclusion applies
The three ceilings are sensitivity scenarios, not a calibrated prediction band, and carry no probabilities. The data were generated by a known curve (ceiling 100,000) plus seeded noise, which the reveal control shows; real launches give no such answer key.
Check your understanding: A second analyst fits the same five years with a ceiling of 300,000 and gets a small error too. What does that add to the case for 300,000?
Chapter 19 source: section "Section Four: The Full Methodology".
Demonstration 3 of 4
Trial times repeat: diagnosing a missed share
If a frequently bought product misses its share forecast, how do you tell a trial problem from a repeat problem?
The first bar is trial; the second keeps only the triers' purchases that stay with the brand; the third adjusts for how heavily those triers buy the category. The dashed line is the plan built from trial 0.18, repeat 0.42 and B 1.1.
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T is the share of category buyers who ever try the brand, R the share of triers' category purchases that go to the brand at equilibrium, and B the buying-rate index (1.1 here: triers buy the category slightly more than average).
Predict first. Trial comes in at 0.24 instead of 0.18, but repeat falls to 0.336. Will the share beat the plan of 8.32 percent?
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Constructed data: the companion's example Parfitt-Collins inputs for this chapter (T 0.18, R 0.42, B 1.1) with T and R varied, computed with the companion's own tool.
Calculated values
- Trial T
- 0.18
- Repeat R
- 0.420
- Buying-rate index B
- 1.1
- Long-run share
- 8.32 percent
- Share with repeat 20 percent lower or higher
- 6.65 to 9.98 percent
- Gap to planned share
- 0.00 points
Share = 0.18 x 0.420 x 1.1 = 0.0832, so 8.32 percent against a planned 8.32. This launch matches the plan. Because the factors multiply, the same share can come from wide trial with weak repeat or narrow trial with strong repeat, and the remedy differs.
Worked steps
- Trial times repeat: 0.18 x 0.420 = 0.0756.
- Times the buying-rate index: 0.0756 x 1.1 = 0.0832.
- Gap to plan: 8.32 - 8.32 = 0.00 points.
Use the idea
When a launch misses, split the miss into T, R and B before choosing a remedy: awareness and distribution for trial, the product itself for repeat.
Where the conclusion applies
T, R and B are the companion's example inputs, not panel measurements. The band of 20 percent either side of R is the companion tool's sensitivity range, not a confidence interval.
Check your understanding: Trial reaches 0.30, repeat is 0.20 and B is 1.0. What long-run share does Parfitt-Collins predict?
Chapter 19 source: section "Section Four: The Full Methodology".
Demonstration 4 of 4
The contagion you don't see, year by year
Is q, the imitation coefficient, the share of people who adopt because others did?
The gold band is adoption from external influence, p times the remaining pool; the teal band is adoption from imitation, q times N/m times the pool. Early on the gold band dominates; once enough people own the product, the teal band does. Neither coefficient is a population share: their contributions depend on how far adoption has gone.
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The hazard p + q N/m splits into an external part p, constant, and an imitation part q times N/m, the adopted fraction. Each part times the remaining pool m minus N gives that part's new adopters per year.
Predict first. With p 0.025 and q 0.35, q is 14 times p. In year 1, will imitation supply most new adopters?
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Constructed data: the chapter notebook's Bass rates (p 0.025, q 0.35 per year, ceiling 100,000), with q varied; the check is the notebook's workshop arithmetic.
Calculated values
- Share of the ceiling adopted, N/m
- 0.188
- External hazard p
- 0.0250
- Imitation hazard q x N/m
- 0.0658
- Share of new adopters from imitation
- 72.5 percent
In year 4, 0.188 of the ceiling has adopted, so the imitation hazard is 0.35 x 0.188 = 0.0658 against p = 0.0250: 0.0658 / (0.0250 + 0.0658) = 0.725, and imitation now supplies most new adopters. The split changes year by year, which is why q / p = 14.0 is a ratio of rates, not the share of buyers who imitate.
Worked steps
- Adopted so far: N/m = 0.188.
- Imitation hazard: 0.35 x 0.188 = 0.0658.
- Share of the total hazard from imitation: 0.0658 / (0.0250 + 0.0658) = 0.725.
Use the idea
Do not report q / p as the fraction of buyers driven by word of mouth; report the imitation share for the period you mean.
Where the conclusion applies
Constructed rates from the notebook, not estimates for any product. The split is a property of the model's hazard, not an observation of why any individual bought.
Check your understanding: A market has ceiling 10,000, 2,000 adopters so far, p 0.02 and q 0.3 per year. How many new adopters a year does Bass predict now?
Chapter 19 source: section "Section Five: The Contagion You Don't See".