The Bass Diffusion Model

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The Bass Diffusion Model

• Model designed to answer the question
• When will customers adopt a new product or
  technology?

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HistoryPublished in Management Science in1969,
A New Product Growth Model For Consumer Durables

• Working Paper 1966

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Empirical Generalization Always (Almost)Looks
Like a Bass Curve
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Color TV Forecast 1966
Peak in 1968
Industry Built Capacity For 14 million units
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Bass Model100s of Applications-An Empirical
GeneralizationWidely CitedNumerous
ExtensionsPublished in Several Languages

• Growing Software Applications

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Assumptions of theBasic Bass Model

• Diffusion process is binary (consumer either
  adopts, or waits to adopt)
• Constant maximum potential number of buyers m
  (N)
• Eventually, all m will buy the product
• No repeat purchase, or replacement purchase
• The impact of the word-of-mouth is independent of
  adoption time
• Innovation is considered independent of
  substitutes
• The marketing strategies supporting the
  innovation are not explicitly included

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Adoption Probability over Time
(a)
1.0
Cumulative Probability of Adoption up to Time t
F(t)
Introduction of product
Time (t)
(b)
f(t) d(F(t)) dt
Density Function Likelihood of Adoption at Time t
Time (t)
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Number of Cellular Subscribers
9,000,000
5,000,000
1,000,000
1983 1 2 3 4
5 6 7 8 9
Years Since Introduction
Source Cellular Telecommunication Industry
Association
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Sales Growth Model for Durables (The Bass
Diffusion Model)

• St p Remaining q Adopters
  Potential Remaining Potential
• Innovation Imitation
  Effect Effect

where St sales at time t p coefficient
of innovation q coefficient of imitation
Adopters S0 S1
St1 Remaining Total Potential
Adopters Potential
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Parameters of the Bass Model in Several Product
Categories
Innovation Imitation Product/ parameter
parameter Technology (p) (q) BW
TV 0.028 0.25 Color TV 0.005 0.84 Air
conditioners 0.010 0.42 Clothes
dryers 0.017 0.36 Water softeners 0.018 0.30 Recor
d players 0.025 0.65 Cellular telephones 0.004 1.7
6 Steam irons 0.029 0.33 Motels 0.007 0.36 McDona
lds fast food 0.018 0.54 Hybrid
corn 0.039 1.01 Electric blankets 0.006 0.24 A
study by Sultan, Farley, and Lehmann in 1990
suggests an average value of 0.03 for p and an
average value of 0.38 for q.
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Technical Specificationof the Bass Model
The Bass Model proposes that the likelihood that
someone in the population will purchase a new
product at a particular time t given that she has
not already purchased the product until then, is
summarized by the following mathematical. Formulat
ion Let L(t) Likelihood of purchase at t,
given that consumer has not purchased until
t f(t) Instantaneous likelihood of purchase at
time t F(t) Cumulative probability that a
consumer would buy the product by time t Once
f(t) is specified, then F(t) is simply the
cumulative distribution of f(t), and from Bayes
Theorem, it follows that L(t)
f(t)/1F(t) (1)
Hazard Rate
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Bass Model Math
Differential Equation
Solution to Differential Equation
or
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Bass Model Math Estimation
Bass Model can also be expressed as
Sales S(t) m f(t) Cum. Sales Y(t) m F(t)
Or, Estimate Directly with this
Run Regression
ap.m, b(q-p) and c -q/m.
p a/m q -c.m
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Peak Time
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Why it Works--Saturation

• S(t)mpqF(t)(1-F(t)

Gets Smaller and Smaller
Gets Bigger and Bigger
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An Example
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Another Example 35 mm Projectors
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Yet Another Example Overhead Projectors
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Factors Affecting theRate of Diffusion

• Product-related
• High relative advantage over existing products
• High degree of compatibility with existing
  approaches
• Low complexity
• Can be tried on a limited basis
• Benefits are observable
• Market-related
• Type of innovation adoption decision (eg, does it
  involve switching from familiar way of doing
  things?)
• Communication channels used
• Nature of links among market participants
• Nature and effect of promotional efforts

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Some Extensions to theBasic Bass Model

• Varying market potential
• As a function of product price, reduction in
  uncertainty in product performance, and growth in
  population, and increases in retail outlets.
• Incorporation of marketing variables
• Coefficient of innovation (p) as a function of
  advertising
• p(t) a b ln A(t).
• Effects of price and detailing.
• Incorporating repeat purchases
• Multi-stage diffusion process
• Awareness è Interest è Adoption è Word of
  mouth

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Some Extensions

• Successive Generations of Technologies
• Generalized Bass Model Includes Decision
  Variables
• Prices, Advertising

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Successive Generations of Technology The Law of
Capture-MigrationGrowth

• The Equations Three Generations
• S1,tF(t1)m11-F(t2)
• S2,tF(t2)m2F(t1)m11-F(t3)
• S3,tF(t3)m3F(t2)m2F(t1)m1
• miincremental market potential for gen.i
• titime since introduction of ith generation and
  F(ti) is Bass Model cumulative function and p and
  q are the same for each generation

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Capture Law- DRAMSNorton and Bass Management
Science (1987)Sloan Management Review (1992)
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Capture Law-Mainframes-Beautiful!
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Generations of PCs
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What About Prices ?

• The Generalized Bass Model
• With Prices, Advertising, and other Marketing
  variables the curve is shifted with different
  policies but the shape stays the same.
• Explain Why Adoption Curves Always Looks The Same
  Even Though Policies Vary Greatly Model Must
  Reduce to Bass Model

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Effects of Different Prices
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Generalized Bass Model Bass, Krishnan, andJain
(1994) Marketing Science

• A Higher Level Theory
• Must Reduce as Special Case to Bass Model
• Prices Fall Exponentially

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The Bass Model (BM) and GBM

• BM f(t)/1-F(t)pqF(t)
• GBM f(t)/1-F(t)x(t)pqF(t)
• where x(t) is a function of percentage change in
  price and other variables (eg advertising)

x(t) 1(dPr(t)/dt)/Pr(t)b1
(dADV(t)/dt)/ADV(t)b2
X(t) t Ln(Pr(t)/Pr(0)) b1 Ln(ADV(t)/ADV(0))
b2
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Impulse Response Comparison GBM and Current
Effects Model Carry-Through Effects for GBM
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Some Applications

• Guessing Without Data
• Satellite Television
• Satellite Telephone (Iridium)
• New LCD Projector
• Wireless Telephone Adoption Around World and
  Pricing Effects
• Projecting Worldwide PC Growth

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Satellite TV Forecast-1993-Guessing By Analogy
and Purchase Intentions

• Use of Adjusting Stated Intention Measures to
  Predict Trial Purchase of New Products A
  Comparison Journal of Marketing Research
  (1989), Jamieson and Bass
• Guessing By Analogy Cable TV vs.Color TV

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1993 Forecast of Satellite TV Penetration in 1999
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Projection of World-Wide PC Demand,
1999-2010-Data From Bill Gates, Newsweek
5-31-99
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Bottom Line and Quotation

• In Forecasting the Time of Peak It is Helpful to
  Know that a Peak Exists
• By Frank Bass