The Bass Diffusion Model

Model designed to answer the question:

When will customers adopt a new

product or technology?

Assumptions of theBasic Bass Model

• Diffusion process is binary (consumer either adopts, or waits to adopt)

• Constant maximum potential number of buyers (N)

• Eventually, all N 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

Adoption Probability over Time

Time (t)

Cumulative Probability of

Adoption up to Time t

F(t)

Introduction of product

(a)

Time (t)

Density Function: Likelihood of

Adoption at Time t

f(t) = d(F(t))dt

(b)

1.0

Number of Cellular Subscribers

Source: Cellular Telecommunication Industry Association

9,000,000

1983 1 2 3 4 5 6 7 8 9

1,000,000

5,000,000

Years Since Introduction

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 + • • • + St–1

Remaining = Total Potential – # AdoptersPotential

Parameters of the Bass Model in Several Product Categories

Innovation ImitationProduct/ parameter

parameter Technology (p) (q)

B&W TV 0.028 0.25Color TV 0.005 0.84Air conditioners 0.010 0.42Clothes dryers 0.017 0.36Water softeners 0.018 0.30Record players 0.025 0.65Cellular telephones 0.004 1.76Steam irons 0.029 0.33Motels 0.007 0.36McDonalds fast food 0.018 0.54Hybrid corn 0.039 1.01Electric 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.

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.

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 bytime 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)

Technical Specificationof the Bass Model cont’d

The Bass model proposes that L(t) is a linear function:

qL(t) = p + –– N(t) (2)

N

where

p = Coefficient of innovation (or coefficient of external influence)q = Coefficient of imitation (or coefficient of internal influence)

N(t) = Total number of adopters of the product up to time tN = Total number of potential buyers of the new product

Then the number of customers who will purchase the product at time t is equal to Nf(t) . From (1), it then follows that:

qNf(t) = [ p + –– N(t)][1 – N(t)] (3)

N

Nf(t) may be interpreted as the number of buyers of the product at time t [ = (t)]. Likewise, NF(t ) is equal to the cumulative number of buyers of the product up to time t [ = N(t)].

Bass Model cont’d

Noting that [n(t) = Nf(t)] is equal to the number of buyers at time t, and [N(t) = NF(t)] is equal to the cumulative number of buyers until time t, we get from (2):

qNf(t) = [ p + –– N(t)][1 – N(t)] (3)

N

After simplification, this gives the basic diffusion equation for predicting new product sales:

qn (t) = pN + (q – p) [N(t)] – –– [N(t)]2 (4)

N

Estimating the Parameters of the Bass Model Using Non-Linear

Regression

An equivalent way to represent N(t) in the Bass model is the following equation:

qn(t) = p + –– N(t–1) [N – N(t–1)]

N

Given four or more values of N(t) we can estimate the three parameters of the above equation to minimize the sum of squared deviations.

Estimating the Parameters of the Bass Model Using RegressionThe discretized version of the Bass model is obtained from (4):

n(t) = a + bN(t–1) + cN 2(t–1)

a, b, and c may be determined from ordinary least squares regression. The values of the model parameters are then obtained as follows:

–b – b2 – 4acN = ––––––––––––––

2c

ap = ––

N

q = p + b

To be consistent with the model, N > 0, b 0, and c < 0.

Forecasting Using the Bass Model—Room Temperature Control Unit

Cumulative Quarter Sales Sales

Market Size = 16,000(At Start Price) 0 0 0

1 160 160Innovation Rate = 0.01 4 425 1,118

(Parameter p) 8 1,234 4,678 12 1,646 11,166

Imitation Rate = 0.41 16 555 15,106(Parameter q) 20 78 15,890

24 9 15,987Initial Price = $400 28 1 15,999

32 0 16,000Final Price = $400 36 0 16,000

Example computations

n(t) = pN + (q–p) N(t–1) – q N(t–1) 2/N

Sales in Quarter 1 = 0.01 16,000 + (0.41–0.01) 0 – (0.41/16,000) (0)2 = 160Sales in Quarter 2 = 0.01 16,000 + (0.40) 160 – (0.41/16,000) (160)2 =

223.35

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

Some Extensions to the Basic 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 Interest Adoption Word of mouth

Pretest Market Models

• Objective

Forecast sales/share for new product before a real test market or product launch

• Conceptual model

Awareness x Availability x Trial x Repeat

• Commercial pre-test market services

– Yankelovich, Skelly, and White

– Bases

– Assessor

ASSESSOR Model

Objectives

• Predict new product’s long-term market share, and sales volume over time

• Estimate the sources of the new product’s share, which includes “cannibalization” of the firm’s existing products, and the “draw” from competitor brands

• Generate diagnostics to improve the product and its marketing program

• Evaluate impact of alternative marketing mix elements such as price, package, etc.

Overview of ASSESSOR Modeling Procedure

Management Input(Positioning Strategy)

(Marketing Plan)

ReconcileOutputs

Draw &Cannibalization

Estimates DiagnosticsUnit SalesVolume

Preference Model

Trial &Repeat Model

Brand Share Prediction

Consumer Research Input(Laboratory Measures)(Post-Usage Measures)

Overview of ASSESSOR Measurements

Design Procedure Measurement

O1 Respondent screening and Criteria for target-group identification recruitment (personal interview) (eg, product-class usage)

O2 Pre-measurement for established Composition of ‘relevant set’ of brands (self-administrated established brands, attribute weights questionnaire) and ratings, and preferences

X1 Exposure to advertising for established brands and new brands

[O3] Measurement of reactions to the Optional, e.g. likability and advertising materials (self- believability ratings of advertising administered questionnaire) materials

X2 Simulated shopping trip and exposure to display of new and established brands

O4 Purchase opportunity (choice recorded Brand(s) purchased by research personnel)

X3 Home use/consumption of new brand

O5 Post-usage measurement (telephone New-brand usage rate, satisfaction ratings, and repeat-purchase propensity; attribute ratings

and preferences for ‘relevant set’ of established brands plus the new brand

O = Measurement; X = Advertsing or product exposure

Trial/Repeat Model

Market share for new product

Mn = T R W

where:

T =long-run cumulative trial rate (estimated from measurement at O4)

R =long-run repeat rate (estimated from measurements at O5)

W =relative usage rate, with w = 1 being the average market usage rate.

Trial Model

T = FKD + CU – (FKD) (CU)

where:

F =long-run probability of trial given 100% awareness and 100% distribution (from O4)

K =long-run probability of awareness (from managerial judgment)

D =long-run probability of product availability where target segment shops (managerial judgment and experience)

C =probability of consumer receiving sample (Managerial judgment)

U =probability that consumer who receives a product will use it (from managerial judgment and past experience)

Repeat Model

Obtained as long-run equilibrium of the switching matrix estimated from (O2 and O5):

Time (t+1)New Other

New p(nn) p(no)Time t

Other p(on) p(oo)

p(.) are probabilities of switching where

p(nn) + p(no) = 1.0; p(on) + p(oo) = 1.0

Long-run repeat given by:p(on)

r = ––––––––––––––1 + p(on) – p(nn)

Preference Model: Purchase Probabilities Before New Product

Use

where:

Vij=Preference rating from product j by participant i

Lij =Probability that participant i will purchase product j

Ri =Products that participant i will consider for purchase (Relevant set)

b =An index which determines how strongly preference for a product will translate to choice of that product (typical range: 1.5–3.0)

(Vij)b

Lij = ––––––––Ri

(Vik)b

k=1

Preference Model: Purchase Probabilities After New Product

Use

where:

L´it =Choice probability of product j after participant i has had an opportunity to try the new product

b =index obtained earlier

Then, market share for new product:L´in

M´n = En –––I N

n =index for new product

En =proportion of participants who include new product in their relevant sets

N =number of respondents

(Vij)b

L´ij = –––––––––––––––––Ri

(Vin)b + (Vik)b

k=1

Estimating Cannibalizationand Draw

Partition the group of participants into two: those who include new product in their consideration sets, and those who don’t. The weighted pre- and post- market shares are then given by:

Lin Mj = –––

I N

L´in L´in M´j = En ––– + (1 – En) –––

I N I N

Then the market share drawn by the new product from each of the existing products is given by:

Dj = Mj – M´j

Example: Preference Ratings

Vij (Pre-use) V´ij (Post-use)

Customer B1 B2 B3 B4 B1 B2 B3 B4 New Product

1 0.1 0.0 4.9 3.7 0.1 0.0 2.6 1.7 0.2

2 1.5 0.7 3.0 0.0 1.6 0.6 0.6 0.0 3.1

3 2.5 2.9 0.0 0.0 2.3 1.4 0.0 0.0 2.3

4 3.1 3.4 0.0 0.0 3.3 3.4 0.0 0.0 0.7

5 0.0 1.3 0.0 0.0 0.0 1.2 0.0 0.0 0.0

6 4.1 0.0 0.0 0.0 4.3 0.0 0.0 0.0 2.1

7 0.4 2.1 0.0 2.9 0.4 2.1 0.0 1.6 0.1

8 0.6 0.2 0.0 0.0 0.6 0.2 0.0 0.0 5.0

9 4.8 2.4 0.0 0.0 5.0 2.2 0.0 0.0 0.3

10 0.7 0.0 4.9 0.0 0.7 0.0 3.4 0.0 0.9

Choice Probabilities

Lij (Pre-use) L´ij (Post-use)Customer B1 B2 B3 B4 B1 B2 B3 B4 New Product

1 0.00 0.00 0.63 0.37 0.00 0.00 0.69 0.31 0.00

2 0.20 0.05 0.75 0.00 0.21 0.03 0.03 0.00 0.73

3 0.43 0.57 0.00 0.00 0.42 0.16 0.00 0.00 0.42

4 0.46 0.54 0.00 0.00 0.47 0.50 0.00 0.00 0.03

5 0.00 1.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00

6 1.00 0.00 0.00 0.00 0.80 0.00 0.00 0.00 0.20

7 0.01 0.35 0.00 0.64 0.03 0.61 0.00 0.36 0.00

8 0.89 0.11 0.00 0.00 0.02 0.00 0.00 0.00 0.98

9 0.79 0.21 0.00 0.00 0.82 0.18 0.00 0.00 0.00

10 0.02 0.00 0.98 0.00 0.04 0.00 0.89 0.00 0.07Unweighted market share (%) 38.0 28.3 23.6 10.1 28.1 24.8 16.1 6.7 24.3

New product’s draw from each brand (Unweighted %) 9.9 3.5 7.5 3.4

New product’s draw from each brand (Weighted by En in %) 2.0 0.7 1.5 0.7

Assessor Trial & Repeat ModelMarket Share Due to Advertising

•Max trial with unlimited Ad•Ad$ for 50% max. trial•Actual Ad $

•Max awareness with unlimited Ad•Ad $ for 50% max. awareness•Actual Ad $

% buying brand in simulated shopping

Awarenessestimate

Distributionestimate (Agree)

Switchback rate ofnon-purchasers

Repurchase rate of simulation

purchasers

% making first purchaseGIVEN awareness &

availability0.23

Prob. of awareness0.70

Prob. of availability0.85

Prob. of switchingTO brand

0.16

Prob. of repurchaseof brand

0.60

% making first purchase due to

advertising0.137

Retention rateGIVEN trial

for ad purchasers0.286

Response Mode Manual Mode

Long-term market share

from advertising0.39

Source: Thomas Burnham, University of Texas at Austin

Assessor Trial & Repeat ModelMarket Share Due to Sampling

Samplingcoverage (%) 0.503

% Delivered 0.90

% of those deliveredhitting target 0.80

Simulation sampleuse

Switchback rate of non-purchasers

Repurchase rate ofsimulation

non-purchasers

Prob. of switchingTO brand

0.16

Prob. of repurchaseof brand

0.427

Long-term market share

from sampling0.02

% hitting target that get used

0.60

Retention rate GIVEN trial

for sample receivers0.218

Correction for sampling/adoverlap (take out those whotried sampling, but would

have tried due to ad)0.035

Market share tryingsamples0.251

Source: Thomas Burnham, University of Texas at Austin

Assessor Preference Model Summary

Source: Thomas Burnham, University of Texas at Austin

Pre-use constantsum evaluations

Post-use constantsum evaluations

Cumulative trialfrom ad

(T&R model)0.137

Beta (B) forchoice model

Pre-entry market shares

Post-entry marketshares (assuming

consideration0.274

Weighted post entry

market shares0.038

Pre-use preferenceratings

Pre-use choices

Post-use preferenceratings

Proportion of consumers who

consider product 0.137 Draw &

cannibalization calculations

Assessor Market Share to Financial Results Diagrams

Market share0.059

Market size60M

Sales per person$5

JWC factory sales

16.7

Average unit margin

0.541

Ad/samplingexpense4.5/3.5

Net contribution

JWCfactory sales

16.7

Industry averagesales $ for

market share17.7

JWCfactory sales

Frequency of usedifferences

0.9

Unit-dollar adjustment

0.94

Price differences1.04

Returnon sales

Source: Thomas Burnham, University of Texas at Austin

Predicted and Observed Market Shares for ASSESSOR

Deviation Deviation Product Description Initial Adjusted Actual (Initial – (Adjusted – Actual) Actual)

Deodorant 13.3 11.0 10.4 2.9 0.6

Antacid 9.6 10.0 10.5 –0.9 –0.5

Shampoo 3.0 3.0 3.2 –0.2 –0.2

Shampoo 1.8 1.8 1.9 –0.1 –0.1

Cleaner 12.0 12.0 12.5 –0.5 –0.5

Pet Food 17.0 21.0 22.0 –5.0 –1.0

Analgesic 3.0 3.0 2.0 1.0 1.0

Cereal 8.0 4.3 4.2 3.8 0.1

Shampoo 15.6 15.6 15.6 0.0 0.0

Juice Drink 4.9 4.9 5.0 –0.1 –0.1

Frozen Food 2.0 2.0 2.2 –0.2 –0.2

Cereal 9.0 7.9 7.2 1.8 0.7

Etc. ... ... ... ... ...

Average 7.9 7.5 7.3 0.6 0.2

Average Absolute Deviation — — — 1.5 0.6

Standard Deviation of Differences — — — 2.0 1.0

BASES Model

Trial volume estimate

Calibrated DistributionAwarenessPt =

intent score intensitytlevelt

Tt =Pt U0 (1/Sit) (TM) (1/CDI)

where:

Pt = Cumulative penetration up to time t

Tt =Total trial volume until time t in a particular target market

U0 =Average units purchased at trial (t = 0)

Sit = Seasonality index at time = tTM = Size of target marketCDI = Category development index for target market

Repeat volume estimate

Rt = Ni–1,t Yit Ui

i=1

where:

Ni–1,t =Cumulative number of consumers who repeat at least i–1 times by week t (N0,t = initial trial volume)

Yit =Conditional cumulative ith repeat purchase rate at week t given that i–1 repeat purchases were made up to week t

Ui =Average units purchased at repeat level i

Ni–1,t & Yit are estimated based on consumers’ stated “after use intended purchase frequency” and estimate of long-run decay in repeat rate.

Ui is estimated based on consumers’ stated purchase quantities.

BASES Model cont’d

BASES Model cont’d

Total volume estimate

St = Tt Rt + Adjustments for promotional volume

Yankelovich, Skelly and White Model

Forecast market share = S N C R U K

where:

S =Lab store sales (indicator of trial),

N =Novelty factor of being in lab market. Discount sales by 20–40% based on previous experience that relate trial in lab markets to trial in actual markets,

C =Clout factor which retains between 25% and 75% of SN determined, based on proposed marketing effort versus ad and distribution weights of existing brands in relation to their market share,

R =Repurchase rate based on percentage of those trying who repurchase,

U =Usage rate based on usage frequency of new product as compared to the new product category as a whole, and

K =Judgmental factor based on comparison of S N C R U K with Yankelovich norms. The comparison is with respect to factors such as size and growth of category, new product’s share derived from category expansion versus conversion from existing brand.

Some Issues in ValidatingPre-Test Models

• Validation does not include products that were withdrawn as a result of model predictions

• Pre-test and actual launch are separated in time, often by a year or more

• Marketing program as implemented could be different from planned program