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“Parametric analysis of the Bass model”
AUTHORS Yair Orbach
ARTICLE INFOYair Orbach (2016). Parametric analysis of the Bass model. Innovative Marketing
, 12(1), 29-40. doi:10.21511/im.12(1).2016.03
DOI http://dx.doi.org/10.21511/im.12(1).2016.03
RELEASED ON Wednesday, 27 April 2016
JOURNAL "Innovative Marketing "
FOUNDER LLC “Consulting Publishing Company “Business Perspectives”
NUMBER OF REFERENCES
0
NUMBER OF FIGURES
0
NUMBER OF TABLES
0
© The author(s) 2022. This publication is an open access article.
businessperspectives.org
Innovative Marketing, Volume 12, Issue 1, 2016
29
Yair Orbach (Israel)
Parametric analysis of the Bass model
Abstract
In this research, the authors explore the influence of the Bass model p, q parameters values on diffusion patterns and
map p, q Euclidean space regions accordingly. The boundaries of four different sub-regions are classified and defined,
in the region where both p, q are positive, according to the number of inflection point and peak of the non-cumulative
sales curve. The researchers extend the p, q range beyond the common positive value restriction to regions where either
p or q is negative. The case of negative p, which represents barriers to initial adoption, leads us to redefine the motiva-
tion for seeding, where seeding is essential to start the market rather than just for accelerating the diffusion. The case of
negative q, caused by a declining motivation to adopt as the number of adopters increases, leads us to cases where the
saturation of the market is at partial coverage rather than the usual full coverage at the long run. The authors develop a
solution to the special case of p + q = 0, where the Bass solution cannot be used. Some differences are highlighted be-
tween the discrete time and continuous time flavors of the Bass model and the implication on the mapping. The distor-
tion is presented, caused by the transition between continuous and discrete time forms, as a function of p, q values in
the various regions.
Keywords: Bass model, mapping, diffusion patterns, discrete time, continuous time, seeding.
JEL Classification: M3.
Introduction
The theory of diffusion of innovation, proposed by
Bass (1969), has been explored, implemented and
extended by numerous researches. Some researchers
extend the model to explicitly consider market or
industry characteristics or behavior. Others examine
implementation of the model on various cases and
show that adding more factors improves the capabil-
ity of the model to capture complex behavior in de-
tails. Another direction taken by some researchers
was to explore the factors that influence parameters
values estimation and forecasts accuracy. Some re-
searches explore empirically how the Bass model
parameters vary across products and markets, usual-
ly at the ranges when p << q and 0.1< q < 0.7. Bass
(1969) notes that there are two different categories
of diffusion curve patterns. When q > p, the period-
ic sales grow until they reach a peak and, then, de-
cline asymptotically to zero. When q < p, periodic
sales decline asymptotically to zero starting from
launch. Little attention has been directed since then
to a further comprehensive exploration of the con-
straints and classification of the Bass model parame-
ters and how they affect the adoption curve patterns.
Another issue that has attracted little attention in
diffusion theory is the transition between the conti-
nuous time form and the discrete time form of the
Bass model. While many researchers switch be-
tween these two flavors without further notice, at-
tention needs to be paid to verifying, for each im-
plementation, that this transition has little impact on
parameters’ values and on forecasts’ accuracy.
Yair Orbach, 2016.
Yair Orbach, Ph.D., Dr., Assistant Professor of Marketing, School of
Business Administration, Bar-Ilan University, Israel.
In this paper, we perform a comprehensive explora-
tion and classify and map the different diffusion pat-
terns over the p, q space. The exploration is theoretic
and is not limited to specific empiric cases. We refer
to the different flavors (continuous vs. discrete) and
highlight the differences in mapping, constraints and
classification. We also define the conditions that
need to be checked when switching between them.
1. Literature review
The diffusion of innovation model, known as the
Bass model, has several flavors and numerous ex-
tensions. In this paper we focus on the basic model
of Bass (1969), which has a simple analytic solution
and not to extensions that are usually solved numer-
ically. We brief the different basic models and high-
light the differences between them. We use these
models for mapping the patterns and constraints cat-
egories over the p, q space and show that each fla-
vor has a slightly different map.
The Bass (1969) presents the diffusion of innovation
dynamic equation:
( )( ) ( ) 1 ( ) .
dF tf t p q F t F t
dt (1)
This equation represents the innovation and imita-
tion influence on the remaining potential market.
The periodic sales, or the rate of cumulative sales
change, which is the derivative of the cumulative
adoption, are proportional to the multiplication of
the remaining market by the sum of innovation and
imitation influence. The analytic continuous time
solution that Bass presents to his equation is:
1( ) .
1
p q t
p q t
eF t
qe
p
(2)
Innovative Marketing, Volume 12, Issue 1, 2016
30
The cumulative sales function, as well as its deriva-
tive or periodic sales function, can be outlined as a
graph where time is the X axis and sales, or sales
rate, are the Y axis. The shapes of both curves, the
cumulative and periodic sales, are determined by the
values of the model parameters p and q. Srinivasan
and Mason (1986) note that both p and q must be
non-negative. Acemoglu and Ozdaglar (2009) note
that the levels of p and q scale time, while the ratio
q/p determines the overall shape of the curve.
Bass (1969) refers to the effect of the q/p ratio or q,
p phase and differentiates between two categories.
He notes that when q/p > 1, i.e., p, q phase = 45°,
the product is successful and sales experience
growth and then decline due to saturation. When q/p
< 1, which represents an unsuccessful product, sales
will start at a certain level and keep declining. Bass
(1969) also presents the influence of (p + q) values,
between 0.3 and 0.9, on the growth rate. Sultan et al.
(1990) performed a meta-analysis of 213 applica-
tions of diffusion models from 15 articles published
between 1950 and 1980. They compare several pa-
rameters estimation methods (OLS, MLE, Bayesian
and non-linear least square) and also how the num-
ber of sampling points influences accuracy. They
found that the average p value is 0.03, while the av-
erage q value is 0.38. Van den Bulte (2002) ex-
plored how p and q vary across products and coun-
tries, based on a database containing 1586 sets of p
and q parameters, from 113 papers published be-
tween January 1969 and May 2000. He explains that
the parameters p and q provide information about
the speed of diffusion. A high value for p indicates
that the diffusion has a quick start, but also tapers
off quickly. A high value of q indicates that the dif-
fusion is slow at first, but accelerates after a while.
He also notes that, when q is larger than p, the cu-
mulative number of adopters follows the type of S-
curve often observed for radically innovative prod-
uct categories. When q is smaller than p, the cumu-
lative number of adopters follows an inverse J-curve
often observed for less risky innovations such as
new grocery items, movies, and music CDs.
Lilien et al. (2000) formulate a discrete difference equ-
ation to model diffusion of innovation, used by many
previous and later of the Bass model extensions.
( ) ( ) ( 1) ( )
( ) ( ) 1 ( ) ,
d d d d
d d d d
f n F n F n F n
X n p q F n F n (3)
where Fd(n) is the cumulative adoption and fd(n) is
the periodic sales at period n.
We use the notation pd and qd for the discrete time
model parameters to distinguish them from the p, q
parameters of the continuous time form. The X(n)
function can capture many factors the original Bass
model ignores such as advertisement of price
changes. When X(n) = 1, it converges back to the
original Bass model. For example, Bass et al. (1994)
propose including the influence of advertisement of
price changes by using:
Pr( ) Pr( 1) ( ) ( 1)( ) 1 0, .
Pr( 1) ( 1)
n n A n A nx n Max
n A n (4)
When coefficient captures the percentage increase
in diffusion speed resulting from a 1% decrease in
price, Pr(n) is the price in period n, coefficient
captures the percentage increase in diffusion speed
resulting from a 1% increase in advertising and A(n)
is advertising in period n. The main advantage of the
discrete model is that it can be solved numerically,
in cases where there is no analytic solution. While
Bass (1969) presents an analytic general solution for
the continuous time differential equation (1), neither
he nor others propose an analytic solution for the
discrete time difference equation (3). Bass (1969)
does refer implicitly to the discrete model by pro-
viding an insight that the likelihood of a purchase at
time t, P(n) is calculated by:
)()(1
)()( nFqp
nF
nfnP ddd
d
d. (5)
While many researchers switch between the discrete
model, based on a difference equation (3), and the
continuous time model, based on the differential
equation (1), the transition is not trivial. Van den
Bulte and Lilien (1997) show that OLS or NLS es-
timations of the continuous Bass model parameters
using discrete time data are biased and that they
change systematically as one extends the number of
observations. An analytic discrete time solution for
the Bass continuous time differential equation (1),
developed by Satoh (2001), is:
2
2
1 ( )1
1 ( )( ) .
1 ( )1
1 ( )
n
s s
s s
n
s s s
s s s
p q
p qF n
q p q
p p q
(6)
However, this solution does not solve the difference
equation (3). Satoh (2001) notes that the relation
between his discrete time Bass model parameters ps
and qs and the corresponding continuous time Bass
model parameters p and q is:
)(2
)(2
1
11qp
qp
e
e
qpk ;
qkq
pkp
s
s. (7)
Table 1 compares the Bass solution (2) for typical
values p = 0.01, q = 0.3, Satoh solution (6) for the
corresponding parameters values pS = 0.0097, qS =
Innovative Marketing, Volume 12, Issue 1, 2016
31
0.2907 and a numeric solution for the Lilien equation
(3) that has minimal RMSE with them with pd =
0.0138, qd = 0.2865. While the parameters values and
forecasts of all three flavors are very close, we still
notice that, while Satoh solutions perfectly match
Bass, Lilien solution is very close but not identical.
Table 1. A comparison between Bass, Satoh and Lilien solution
t Bass F(t) Satoh F(n) Lilien Fd(n)
0 0 0 0
1 0.011588 0.011588 0.013808
2 0.026960 0.026960 0.031327
3 0.047166 0.047166 0.053396
4 0.073399 0.073399 0.080948
5 0.106923 0.106923 0.114953
6 0.148907 0.148907 0.156323
7 0.200171 0.200171 0.205758
8 0.260871 0.260871 0.263546
9 0.330179 0.330179 0.329323
10 0.406107 0.406107 0.401864
11 0.485607 0.485607 0.478990
12 0.565000 0.565000 0.557684
13 0.640625 0.640625 0.634465
14 0.709486 0.709486 0.705958
15 0.769662 0.769662 0.769491
16 0.820388 0.820388 0.823493
17 0.861864 0.861864 0.867574
18 0.894941 0.894941 0.902319
19 0.920798 0.920798 0.928920
20 0.940698 0.940698 0.948819
The cumulative sales of Satoh (2001) from (6) per-
fectly match the continuous time Bass sales solution
from (2) for t = 0, 1, 2…n. Satoh (2001) compares se-
veral methods for estimating the discrete time Bass
model parameters for nine data sets, and concludes
that, for most cases, when the time interval is small
enough, the exact solution of the discrete Bass model
provides a very good approximation of the solution of
the conventional Bass model. He also proves that,
when using the transformation (7) to p and q, a solu-
tion of the discrete Bass model (6) provides identical
values to the solution of the continuous model (2). For
two cases (out of nine) where the estimated value of
the p parameter is negative, he notes that the wrong
sign indicates that the data are not appropriate for the
Bass model.
2. Mapping the Bass model parameters
We perform the mapping separately for each fla-
vor of the Bass model. First, we develop a formu-
la of the periodic sales inflection points’ times,
for the continuous time form, and classify the dif-
fusion patterns according to the number of inflec-
tion point and whether there is peak. Second, we
map each category classified to a sub-region of
the positive p and q values of the Euclidean space.
Then, we extend the map to include regions where
either p or q is negative, which add some insight
about seeding and about market saturation. For
the discrete time model, equation (3), which is
more restricted, we remap the p, q space. The map
is similar but has additional constraints about the
absolute values of p, q and their sum. We redo the
mapping for the Satoh model which is a discrete
time solution (6, 7) for the continuous time mode
of equation (1, 3).
2.1. Mapping the Bass model parameters ratio
space. Bass (1969) and Van den Bulte (2002) distin-
guish between cases where q > p, and periodic sales
have a peak after launch, to cases where q < p and pe-
riodic sales keep declining since launch. The periodic
sales peak time, as calculated by Bass (1969), is:
1* ln .
qt
p q p (8)
From (8), we see that when q > p, the peak time is positive. When q < p, peak time is negative, thus, sales keep declining since launch time (t = 0). There are no sales before product launch.
The inflection points’ times (see Appendix A) are calculated as:
qpt
qp
p
q
t)32ln(
*
)32ln(ln
** (9)
Result 1: the time between the inflection points
and between them to peak depends only on (p + q) and not on (p/q).
Innovative Ma
32
Since sales
or peak tim
cumulative,
The parame
constrained
Srinivasan a
p or q (but n
plicitly that,
regression o
der for the
mention that
Chandrasekmentioned bsions of innovalue of theproduct liesthere are difue of the cois 0.001 forveloping cois higher foStates The for a new prand there atries and devbetween indal/medical iimitation thtions The mfor a new pr0.56 for devresearches Chandrasekucts, usuallyal. (2005) rtinguish betage p = 0.3mapping, rewith averagto our mapp
2.2. Can th
or negative
since diffus
(1986) claim
arketing, Volume
begin at t =
mes are neg
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eters p and q
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karan and Teby Boyle (20ovation casese coefficients globally befferences betwoefficient of r developed ountries Theor European cmean value roduct lies glare differenceveloping ecodustrial and innovations
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tion
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gain
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claim
reste
attain
reach
the
meth
imiz
size,
base
effec
and
thod
is us
et al
diffu
e 1 presents tperiodic sale
3 3.73 for
periodic sales
ative. Rafi and
y explain tha
s diffusion m
del reduces t
n p = 0, the B
the Gamma/
when using
usion can sta
egative, but
h seed size o
hematically
p value do
duct is usele
price or effo
others have
rs adopt the
to externali
duct becomes
ntial custom
that there
ufacturers, w
iers are due t
cern regardin
for new or
s popularity
e to adopt it
m that a maj
ed in introduc
n a critical
hed, the initi
network exp
hod for findin
zation) and c
required to
d on the de
ct and cost s
Economides
d on several e
sed not only
l. (1995), bu
usion. Interes
the p, q spaces curves. W
r convenienc
curves
d Akthar (20
at there are t
model. (a) W
to the Expon
Bass model r
/shifted Gom
g seeding, a
art when p
a marketer
of pF )0(
start. The in
oes not nece
ss. There ca
ort barriers
e already ad
benefits fro
ities or unc
s more and m
mers. Katz an
are barriers
where sales a
to reluctance
ng less expe
less popula
y, the barrier
t increases. O
jor problem
cing a good
mass. Onc
ial structural
periences gr
ng the optim
alculating th
o start the m
emand curve
tructure. Far
s (1996) app
empiric cases
for accelerat
ut is a preco
stingly, Jain
ce four regioe use the app
e.
11) allow als
two special c
When q = 0
nential distr
reduces to a
mpertz distr
as in Jain et
0)0(qF
r uses seedi
q , then, di
nterpretation
essarily mea
an be cases
to adoption
dopted. How
m the produ
certainty red
more plausib
nd Shapiro (
to foreign
are often reta
e of custome
erience and t
ar brands. A
rs are lower
Oren and Sm
facing a pro
network is th
ce the critic
inertia is ov
rowth. They
mal price (for
he critical m
market. Their
e, considerin
rrell and Salo
lied and test
s. In such cas
ting diffusion
ondition for
et al. (1995
ns (A to D) proximation
o that p = 0.
cases of the
0, the Bass
ribution, (b)
special case
ribution. In
al. (1995),
. If p value
ing strategy
iffusion can
of a nega-
an that the
when there
when very
wever, when
uct increase,
duction, the
le for many
1985) men-
automobile
arded. These
ers and their
thinner ser-
s the brand
red and the
mith (1981)
oducer inte-
he ability to
cal mass is
vercome and
y propose a
profit max-
ass, or seed
r method is
ng networks
oner (1986)
ted this me-
ses, seeding
n, as in Jain
starting the
) do not re-
fer to such
positive va
can overco
the negativ
condition i
cannot be o
interest rat
justify the
negative p
justification
low and th
will keep t
( qpF0
higher, the
larger.
Cumulative
according
when q va
equal:
tF
1
1
)(
1*t
p q
Without se
duces back
seed size is
we always
since (9) a
for having
0
0
(1 )q F
p qF
Condition
close to its
Figure 2 pr
curves of t
tive p. Wh
gle or two
has two in
when p va
the diffusio
ment that t
the potenti
diffusion c
there is alw
Result 2: T
leased. In
h cases and c
alues of p and
ome the barri
ve value of p
is not fulfill
overcome by
te is low, dif
expenses an
that can be
n for it is in
here is no in
the seeding s
q ). Note th
seed that is
e adoption a
to Jain et a
alue is posit
eqFp
Fq
eqFp
Fp
0
0
0
0
)1(
)1(
0
0
(1 )ln
q F
p qF
eeding, when
k to (2, 8). W
s close to its
have a perio
applies also
two inflectio
2 3.
(11) is alway
minimal lim
resents two c
their correspo
hile the posit
inflection po
nflection poin
alue is negat
on is much
the seed F0
ial market, t
urve belong
ways a period
The assump
such cases,
check seedin
d q. The cas
iers to adopti
, are when q
led, the barr
y seeding. No
ffusion acce
nd sales loss
e overcome w
nherent. Whe
ncentive for
size close to
hat, when th
required to s
and peak tim
al. (1995), a
ive) also fo
tqp
tqp
e
e
)(
)(
;
).
n 00F , e
With negative
s minimal lim
odic sales pe
for negative
on points is:
ys fulfilled w
mit qp for
curves of ne
onding (abso
ive p curves
oints, its cor
nts in both
tive, seeding
slower. Du
qp is le
the correspo
s to either re
dic sales peak
ption of posi
seeding of
ng effect onl
ses where see
ion, expresse
q p . When
riers to ado
ote that, whe
eleration doe
of seeding.
with seeding
en interest ra
acceleration
its minimal
he ratio p
start the mark
me with see
are applied
r negative p
equation (10
e p and whe
mit ( pF0
eak. Furtherm
e p, the cond
when seed si
r any q/p ratio
egative p and
olute value)
s may have a
rresponding c
cases. Note
g is essential
ue to the req
ess than 100
onding positi
egion A or B
k.
itive p can b
at least p
ly for
eding
ed by
n this
option
en the
es not
With
g, the
ate is
n, we
limit
q is
ket is
eding,
(only
p and
(10)
0) re-
en the
qp ),
more,
dition
(11)
ize is
o.
d two
posi-
a sin-
curve
that,
l and
quire-
0% of
ive p
B and
be re-
q is
esse
mar
star
Anois wberg(WOrefeadoptive withneceunpltheadoptrainwhobuy stanloadlikelcarservcourtionWhetheircomthefectsaretheyto onega
tive
the mrang
p
base
that
p
In
ential for st
rket where
rt the diffusi
Fig. 2. Comp
other questiowhether q cang (2004) inOM) effect or to a mix ofption supporteffect. We c
h overall negessarily meanleased with tbenefit frompt. For examn commute, ro would like t
a reserved nding. As mod in the non-lihood to findincreases, thu
ved ticket. Arse that quali
n programmeen there are fr salaries are
mmon, the worequired skis, where mormentioned b
y are usually overuse. Whative q is sim
presents a m
market will rge, as for a re
q , the lon
ed on (2), is
the market
q of its pote
nnovative Mark
arting the m
initial critic
ion.
paring positive
on about then be negativencorporate non diffusion f both positivt and resistanclaim that difgative q valuen that adopttheir purchas
m a product mple, for a creserved tickto guarantee ticked may
ore reserved treserved traind a free seat us, reducing
Another examfies PC techn
ers or a forefew PC techne high. As knorth, and the sill, decline. Nre users makeby Naglerw (caused by co
hile non-cummilar to those
more interestin
reach 100% oegular positiv
g range forec
s tFt
)(
will saturate
ential. Figure
keting, Volume 1
market. It r
cal mass is
e and negative p
e p, q parame. Moldovan negative wo
models. Hove and negatince with an ffusion can pe. A negativers are disape. It can fit adeclines as certain dema
kets may be sa seat. Thosehave to com
tickets are son car is reduat the non-r
the incentivemple is takinnicians or moign languagenicians or oth
knowledge besalaries to thoNegative exte a product l(2011) who ongestion tha
mulative salese with q = 0,
ng feature. W
of its potentiave value of q.
cast for cumu
qp , w
e at an equil
e 3 presents b
12, Issue 1, 2016
33
represents a
required to
p values
meters rangeand Golden-
ord-of-mouthowever, theyive WOM ofoverall posi-
progress evene q does notppointed anda case wheremore people
and level forsold to thosee who do notmmute whileold, the over-uced, and thereserved traine to buy a re-ng a trainingobile applica-e translation.her rare skill,ecomes moreose who haveternalities ef-less valuable,explains thatat occurs dues curve with the cumula-
When qp ,
al at the long. However, if
ulative sales,
which means
librium level
both cumula-
6
a
o
e-hyf-ntdeerete-en-g-.,ee-,teh-
,
gf
,
s
l
-
Innovative Ma
34
tive and nontive and neg
ratio is presing effect tdamping ef
increases an
Result 3: T
leased whe
sales will e
lative ado
equilibrium
the potenti
when p
a portion o
When p + q
though the
this case, th
(( )
dF tf t
dt
The solution
( ) 1F tp
A summary
is presented
It presents t
p – q space
region rathe
Region
A
B
C
D
arketing, Volume
n-cumulative gative q valu
ented too. Ththat contradiffect strength
nd, when p
Non-cum
Fig. 3. Co
The assumpt
en p is posi
xperience st
ption asym
m. The equ
ial market, a
q . The equ
of the potent
q = 0, Bass
discrete time
he Bass equat
)1 (
tp F
n for this cas
1.
1p t
y map of the
d in Figure 4.
the classic (p
e regions, b
er than two)
Table 2. The
Lower bound oq/p
32
1
32
0
e 12, Issue 1, 201
sales curve oues. The effe
he negative qicts the drivhens as cum
qtF )( , t
mulative adop
omparing cum
tion of posit
tive. At bot
teady declin
mptotically
ilibrium ma
as in a “reg
uilibrium is
tial market,
solution (2)
e (3) can stil
tion is:
2( ) .t
se is:
e combinatio
.
positive p an
but in a grea
). The map a
e seven regio
of Upper bounq/p
32
1
32
16
of cases with pfect of the p
q creates a dave to adopt. mulative adop
the drive to a
ption
mulative and no
tive q can be
th, the peri
ne, while, cu
approaches
ay be 100%
gular” diffus
qp / , whic
when qp
is undefined
ll be applied.
ons we discu
nd q) Bass m
ater details (
also presents
ons of q/p, bo
nd of Lower phof q/p
75º
45º
15º
0º
posi-qp /
amp-The
ption
adopt
is b
level
adop
decli
qp /
in Fi
on-cumulative
e re-
iodic
umu-
s its
% of
sion,
ch is
q .
d, al-
. For
(12)
(13)
ussed
model
(four
s re-
gion
dere
We
mine
cumu
Tabl
at ea
flect
satur
oundaries, pe
hasep
Upper phaof q/p
90º
75º
45º
15º
alanced by t
l is a stable
ption starts at
ine (i.e., neg
q . This is de
igure 3.
Cu
adoption with
ns with negat
d to be out-o
Fig. 4
summarize a
e the existe
ulative sales
le 3 presents
ach region a
tion points,
ration level.
eak, inflectio
ase Number inflection po
2
1
1
0
the q dampin
e equilibrium
t a higher lev
gative non-c
emonstrated b
umulative ad
positive and n
tive p or q
of-scope in p
4. Extended p –
all seven reg
ence of thes
in Table 2.
examples fo
and whether
as well as a
on points and
of oints
Peak e
Yes
Yes
No
No
ng effect. Th
m and if the
vel, or even 1
cumulative a
by the broken
doption
negative q
values that w
revious resea
– q space map
gions of q/p
se points on
or the values
there are pe
a need for s
d equilibrium
exists E(
he saturation
cumulative
00%, it will
adoption) to
n line curves
were consi-
arches.
p that deter-
n the non-
s of p and q
eak and in-
seeding and
m
Equilibriumsaturation)
100%
100%
100%
100%
Tab
Region
E
F
G
Table 3. Ex
Re
p – q phase
Peak
Inflection points
Minimal seed
Saturation
p
q
q/p
Note that m
Lilien et al
found that
side in reg
tions, prod
Products in
be adopted
3. Mappin
parameter
When refer
tion (3), fo
the likeliho
(1969), it a
ratio, as pr
solute valu
When refer
non-negativ
sume that a
purchase a
pd. Since P
ly that the
ter a lon
)( nd nF
imply that
range [0:1]
mention th
the range o
the continu
thus, can v
ters of the
bounded in
For region
negative, th
pd and with
ood , see (5
(1) dP p q
ble 2 (cont.).
Lower boundq/p
-1
xamples of di
egion
s
most empiric
., 2000; Cha
most succes
gion A. Still
ducts at reg
n region G c
d only by a po
ng the discre
rs space
rring to the
ormulated by
ood of purch
adds some b
resented abov
ues.
rring to regio
ve, and seed
at launch F(0
at time n = 0
P(n) represen
value of pd i
ng time t
1, thus, P
the value of
]. Indeed, No
hat the values
of [0:1]. Whi
uous Bass m
vary to any p
discrete tim
n the range
s F and G,
he likelihood
h no seeding
5), during the
(1)d d dq F p
. The seven r
of Upper bouq/p
-1
0
-1
ifferent param
A
75°-90°
yes
2
0
100%
c studies (Su
ndrasekaran
ssful product
l, with prop
ion E can
can succeed
ortion of the
ete time Bas
discrete tim
y Lilien et a
hased (5), m
oundaries no
ve, but also t
ons A to D, w
ding in opti
0) = 0, thus,
0, according
nts a probabil
is bounded b
the whole
n pnP )(
f pd+qd is als
oratikah and
s of pd and q
ile the param
model repre
positive num
me Bass mode
of [0:1] and
where p is p
d probability
g Fd(1) = pd,
e following p
(1d d dq p p
regions of q/p
und of Lower pof q/
90º
-45º
-90º
meters sets th
B
5°- °
yes
1
0
100%
ultan et al., 1
and Tellis, 2
t p – q value
per marketing
succeed as
as well, but
potential ma
ss model
e diffusion e
al. (2000), an
entioned by
ot only to th
to parameter
where p and
onal, we ca
the probabili
to (5), is P(
lity, one can
between [0:1]
market ad
dd qp . W
so bounded i
Ismail (201
qd are bound
meters p and
esent a rate
mber, the par
el (pd and qd
d also their
positive, but
y at launch P
purchase lik
period is:
1 ).dq
/p, boundarie
hase/p
Upper phof q/p
º 135º
º 0º
º -45º
hat belong to d
C
5°- °
no
1
0
100%
1990;
2008)
es re-
g ac-
well.
t will
arket.
equa-
nd to
Bass
he p/q
rs ab-
q are
n as-
ity of
(0) =
imp-
]. Af-
dopts
e can
in the
3) do
ded in
q, of
and,
rame-
d) are
sum.
t q is
(0) =
kelih-
(14)
It m
prob
be h
tive
as p
not
valu
abso
thei
Figu
disc
of th
Res
spac
spac
resi
In
es, peak, infle
hasep
Numberinflection p
2
0
0
different regi
D
°- °
no
0
0
100%
means that,
bability, lim
higher than –
, seeding is
dd qp , mu
have further
ues of dp and
olute values
r sum dp
ure 5 presen
crete model
he continuou
Fig. 5.
sult 4. Discr
ce is more c
ce of the co
de in a sub-a
nnovative Mark
ection points
r of points
Peak
Yes
No
No
ons of p – q s
E
°- °
yes
2
|p|/q
100%
-0.1
0.2
-2
since dp is
ited to [0:1]
–1. For regio
essential. Th
ust be positiv
r limitations
d dq . There
of dp and q
dq is betwe
nts the pd – q
(3). Note tha
us model (1)
Discrete time p
rete time mo
onstrained c
ontinuous tim
area of the co
keting, Volume 1
s and equilibr
exists
space and the
F
-45°-0°
no
0
0
100%
-
-
positive an
], the value
on E, where
he values of
ve. For regio
regarding th
e may be case
dq are highe
een 0 and 1.
qd space map
at it include
map (see Fig
pd - qd space m
odel parame
compared w
me map. Its
ontinuous tim
12, Issue 1, 2016
35
rium
Equilibrium(Saturation)
100%
100%
p/|q|
eir properties
G
-90°-45°
no
0
0
p/|q|
0.1
-0.2
-2
nd P(1) is a
of dq must
dp is nega-
dq , as well
on E, we do
he individual
es where the
er than 1, but
pping of the
s a sub-area
gure 4).
map
eters pd – qd
with the p – q
valid areas
me map.
6
5
a
t
-
l
o
l
e
t
e
a
d
q
s
Innovative Ma
36
4. Mapping
solution pa
The Satoh (
continuous
matches per
an integer bu
tions. When
are non-neg
both sp an
Note that t
which appl
0, and also
absolute va
higher than
The p – q sp
arketing, Volume
g the Satoh
arameters sp
(2001) model
diffusion m
rfectly the B
ut it still follo
n referring to r
gative, we can
nd sq are wit
Fig.
there is sing
ies also for
that, for reg
alues of bot
n 1, while th
Table 4
pace map of
e 12, Issue 1, 201
discrete tim
pace
l is a discrete
model (1).
ass solution
ows the prob
regions A to
n directly im
thin the range
. 6. Relation of
gularity wh
p + q = 0 a
gions E to G
th ps and q
heir sum is
. Examples o
Regio
Bass p
Bass q
Satoh k
Satoh ps
Satoh qs
Lillien pd
Lillien qd
the Satoh (2
Fi
16
me analytic
e solution fo
Its solution
(2) when tim
ability (5) res
D, where p a
mply from (7)
e of [0:1]. Fo
f Satoh discrete
en ps + qs
and for pd +
G, the indivi
s may be m
bounded by
of ps, pd and q
on E
-1
20
0
-1
2
-
3
001) is prese
ig. 7. The p – q
or the
(6)
me is
stric-
and q
) that
or re-
gion
may
are h
the
relati
ters t
p –
qs /
e parameters a
= 0,
qd =
idual
much
y –1
and
wher
whil
the p
indiv
qs, qd that ha
E
90 1
00 -2
0.1 0
19
20 -
-2 0
3 -0
ented in Figu
space map of
s E to G, whe
be cases wh
higher than 1
1 ss qp
ion between
to the Bass (1
q phase
qpps / so
and continuous
1. Table 4
re both ps a
le their sum
parameters o
vidual absol
ave absolute v
G
190
200
0.1
19
-20
0.95
0.99 -
ure 7.
the Satoh (200
ere p and q h
here the absol
1. Still, Satoh
1s constra
Satoh (2001)
1969) qp .
of Bass an
o it maintains
s Bass parame
4 presents
and qs have
is 1. Note t
of the discre
lute value hi
values highe
F
200
-190
0.1
20
-19
0.99
-0.95
01)
have opposite
lute values of
h parameters
aint. Figure 6
) k and sp
From (7) we
nd Satoh ar
s the same reg
eters
such three
e high abso
that, for reg
ete model (3
igher than 1.
er than 1
e signs, there
f sp and sq
s still follow
presents the
sq parame-
e see that the
re identical
gion.
examples,
lute values
gion E also,
3) can have
.
Result 5. T
discrete tim
is less cons
5. Switchinand discre
Many rese
between th
time form
many case
cast seem t
time, in som
casts is sig
the properti
tinuous tim
to have eno
how many
Shannon (1
the samplin
quist (1928
in maintain
signal, and
nuous time
little distort
tween suc
In almost a
in the low
between th
seamless a
are some o
in the yello
cases, whe
areas, a tr
crete time
quires care
Result 6: T
time and
most existi
ues reside
where p,
The Satoh pme model p
strained com
ng betweente time form
earchers, inc
he continuou
(3) without
s, the discre
to be very cl
me cases, the
gnificant. The
ies of the cur
me to discrete
ough data po
data points
1948) who p
ng rate boun
8) determines
ning the info
d being able
signal from
tion, we nee
ccessive cur
all empiric ca
distortion (g
he continuou
and causes in
ther cases, w
ow areas wi
ere the value
ransition betw
causes a s
ful adjustme
The transiti
discrete tim
ing empiric
in regions w
q values re
pS, qS space
parameters p
mpared in re
n continuousms
cluding Bass
us form (1)
an explicit
ete time para
ose to those
e difference
e major issu
rve, in the tra
e time form, i
oints per tim
are required
provided a th
ndaries found
s that, when
rmation of th
to restore th
its discrete t
d at least tw
rve peaks.
ases, the val
green) areas w
us and discre
nsignificant
where the val
ith minor im
es of p, q r
ween the con
significant d
ents.
ion between
me form is
c researches,
with low dis
eside in hig
is similar to
pd – qd space
egions F and
s time
s (1969), sw
and the dis
notice. Whi
ameters and
of the contin
between the
e for mainta
ansition from
is the require
me. The theo
d was outline
heoretic basi
d by Nyquist
we are inter
he harmonies
he original c
time samples
wo data point
In the co
Fig. 8. Distor
ues of p, q r
where a trans
ete time form
distortion. T
lues of p, q r
mpact. In the
reside in the
ntinuous and
distortion an
n the contin
justified, a
, where p, q
stortion. In c
gh distortion
o the
e but
d G.
witch
screte
le, in
fore-
nuous
fore-
aining
m con-
ement
ory of
ed by
is for
t. Ny-
rested
s as a
conti-
s with
ts be-
ontext
of t
man
infle
infle
min
(Ro
disc
each
RM
Whe
satu
the
corr
are u
Figu
acro
area
is b
area
0.15
and
tion (RMSE) p
reside
sition
ms is
There
reside
e rare
e red
d dis-
d re-
nuous
as in
q val-
cases
n re-
gion
imp
Con
In thcurvramguisas ireginumanalalsogionprovning
In
he Bass mod
ny data point
ection point
ection point
ned by p, q v
ot Mean Sq
crete time fo
h p, q coordi
0
1 n
i
MSEn
ere n is the
uration and F
continuous
respondingly
used ( pp
ure 8 presen
oss the p, q
as is between
between 0.0
as is more th
54. The map
q values.
p – q space map
ns, such a tr
pact on the f
nclusion
his paper, weve patterns a
meters valuesshing betweein previous ions {A, B, C
mber of the inlytic formula
o extend the ns {E, F, G}vide marketig of the nega
nnovative Mark
del curve, w
ts are betwe
(when we h
s and peak.
values. The d
quare Error)
orecasts diffe
nate as:
1
0
( ) (dF i F
number of
F, Fd are the
(2, 10, 13)
y, when the
dd qqp ; ).
nts how the d
space. The
n 0 and 0.015
15 and 0.07
han 0.07 an
p was genera
p
ransition wo
forecasts.
e explored thand how theys and p/q raen two regionresearch, we
C, D} that arenflection poia for the inflcommon p
} with negatiing intuitionative values
keting, Volume 1
we need to co
en start (t =
have one), a
These time
distortion, o
of the con
erence, is ca
2) .i
periods from
e forecasts a
and discret
same param
distortion (RM
e distortion a
5. At the yel
7. The disto
nd up to a m
ated with 0.0
ould have a
he propertiesy depend on atio. Rather ns, with or we refer to foe categorizedints. We alsoection points– q space toive values o
n or insight and to the m
12, Issue 1, 2016
37
onsider how
0) and first
and between
es are deter-
r the RMSE
ntinuous and
alculated for
(15)
m launch to
according to
te (3) forms
meters values
MSE) varies
at the green
llow areas, it
ortion at red
maximum of
01 steps of p
a significant
of diffusionthe p, q pa-than distin-
without peak,our differentd also by theo develop ans’ times. Weo include re-f p or q andto the mea-
market beha-
6
7
w
t
n
-
E
d
r
o
o
s
s
s
n
t
d
f
p
t
n--,tene-d--
Innovative Marketing, Volume 12, Issue 1, 2016
38
vior. In region E, we define a new motivation for seeding which, unlike previous research, is not used only for accelerating the diffusion, but, in certain conditions is essential for starting the market. Another contribution of this paper is defining the conditions to saturation below 100% (unlike pre-vious concept that any product will finally cover the entire market). We also highlight some differences
between discrete time and continuous time flavors of diffusion models and the map of the regions, where a switch between them is appropriate. Future research may propose an intuition for the regions that are still white in the p – q space maps and ex-plore their properties. Another direction may be de-veloping an analytic solution for the discrete time diffusion difference equation.
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Innovative Marketing, Volume 12, Issue 1, 2016
39
Appendix A: Inflection points calculations
Bass solution for the non-cumulative adoption rate is: 2
2
1
)()(
tqp
tqp
ep
q
e
p
qptf
For finding the inflection points, we need to find where the derivative equals 0.
The derivative is calculated using the formula:
)(
)()(
)()(
)(
)(
)()(
2tg
thdt
tdgtg
dt
tdh
dt
tdf
tg
thtf
4
2
2
1
))((121)()(
)(
tqp
tqptqptqptqptqp
ep
q
eqpp
qe
p
qee
p
qeqp
p
qptf
3
2
1
))((21)()(
tqp
tqptqptqptqp
ep
q
eqpp
qee
p
qeqp
p
qp
3
3
3
3
1
1)(
1
21)(
tqp
tqptqp
tqp
tqptqptqptqp
ep
q
ep
qe
p
qp
ep
q
ep
qee
p
qe
p
qp
When comparing the numerator to 0, since denominator is always positive, as Bass (1969) we can find the periodic
sales peak time t*.
p
q
qpt ln
1*
For finding the inflection points, we need to calculate when the second derivative equals 0.
Changing variables for convenience
tqpetX )( -3
3
)(1
)(1)()(
)(
tXp
q
tXp
qtX
p
qptf
6
2
23
3
)(1
)()()`()(13)(1)`()(2
1)(
)(
tXp
q
tXp
qtXtX
p
qtX
p
qtX
p
qtXtX
p
q
p
qptf
Comparing the numerator to 0
Innovative Marketing, Volume 12, Issue 1, 2016
40
0)()()`()(13)(1)`()(2
1 2
23
tXp
qtXtX
p
qtX
p
qtX
p
qtXtX
p
q
0)(1)(1)(3)(1)(2
1
23
tXp
q
p
qtX
p
qtXtX
p
qtX
p
q
)(1)(3)(1)(2
1 tXp
q
p
qtXtX
p
qtX
p
q
01)(4)(2
2
tXp
qtX
p
q
3232
2
324
2
4164
)(22
22
q
p
p
q
p
q
p
q
p
q
p
q
p
q
p
q
p
q
tX
qp
p
q
tp
qtqp
q
petX
tqp
)32ln(ln
ln)32ln()(32)( )(
Since )32ln()32ln(0)32ln()32ln()32ln( 2
The times of the inflection points are:
qpt
qp
p
q
t)32ln(
*
)32ln(ln
**
Note that the inflection points are at equal distance from the peak.