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The Bass Diffusion Model. Model designed to answer the question: When will customers adopt a new product or technology?. History Published in Management Science in1969, “A New Product Growth Model For Consumer Durables”. Working Paper 1966.

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The bass diffusion model l.jpg
The Bass Diffusion Model

Model designed to answer the question:

When will customers adopt a new product or technology?


History published in management science in1969 a new product growth model for consumer durables l.jpg

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

Working Paper 1966



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Color TV Forecast 1966

Peak in 1968

Industry Built Capacity For

14 million units


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Bass Model:100’s of Applications-An Empirical GeneralizationWidely CitedNumerous ExtensionsPublished in Several Languages

Growing Software Applications


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Assumptions of the GeneralizationBasic 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 Generalization

(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 Generalization

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) Generalization

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 + • • • + St–1

Remaining = Total Potential – # Adopters Potential


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Parameters of the Bass Model in Several Product Categories Generalization

Innovation Imitation Product/ parameter parameter Technology (p) (q)

B&W 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

Record players 0.025 0.65

Cellular telephones 0.004 1.76

Steam irons 0.029 0.33

Motels 0.007 0.36

McDonalds 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 Specification Generalizationof the Bass Model

The Bass Model proposes that the likelihood that someone in the population will purchase a new product at a particular time tgiven that she has not already purchased the product until then, is summarized by the following mathematical.

Formulation

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)/[1–F(t)] (1)

Hazard Rate


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Bass Model Math Generalization

Differential Equation

Solution to Differential Equation

or


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Bass Model Math: Estimation Generalization

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

a=p.m, b=(q-p) and c = -q/m.

p = a/m

q= -c.m


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Peak Time Generalization


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Why it Works--Saturation Generalization

  • S(t)=m[p+qF(t)][(1-F(t)]

Gets Smaller

and Smaller

Gets Bigger

and Bigger


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An Example Generalization




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Factors Affecting the GeneralizationRate 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 the GeneralizationBasic 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 Generalization

  • Successive Generations of Technologies:

  • Generalized Bass Model: Includes Decision Variables:

  • Prices, Advertising


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Successive Generations of Technology The Law of Capture- GeneralizationMigration&Growth

  • The Equations: Three Generations

  • S1,t=F(t1)m1[1-F(t2)]

  • S2,t=F(t2)[m2+F(t1)m1][1-F(t3)]

  • S3,t=F(t3){m3+F(t2)[m2+F(t1)m1]}

  • mi=incremental market potential for gen.i

  • ti=time 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- DRAMS GeneralizationNorton and Bass: Management Science (1987)Sloan Management Review (1992)



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Generations of PC’s Generalization


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What About Prices ? Generalization

  • 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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Generalized Bass Model: Bass, Krishnan, and GeneralizationJain (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 Generalization

  • BM: f(t)/[1-F(t)]=[p+qF(t)]

  • GBM: f(t)/[1-F(t)]=x(t)[p+qF(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 Model

  • 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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Bottom Line and Quotation Gates,

“In Forecasting the Time of Peak It is Helpful to Know that a Peak Exists”

By Frank Bass


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