ISSN 1 746-7233, England, UK
International Journal of Management Science
and Engineering Management
Vol. 1 (2006) No. 2, pp. 148-160
The dynamic multi-innovation diffusion model with active potential
consumers and its application to the diffusion of local
telephony and mobile telephony in China
Xin Li1∗ , Zhigao Liao2
1
2
CAAC Scientific Research base of Civil Aviation Flight Technology
and Safety Civil Aviation Flight University of China, Guanghan 618307, P. R. China
Management Department, Guangxi Technology University, Liuzhou 545006, P. R. China
(Received August 13 2006, Accepted October 1 2006)
Abstract. New product innovations are emerging in large numbers day by day, and constantly struggling
for market acceptance. Various models have been constructed to explore the principles of innovation diffusion and market share. The relationships between those innovations also draw the attention of scholars, and
many competition models have been established to show today’s competitive social environment in which
increased adoptions of one innovation result in decreased adoptions of other innovations. But that is only
one of the probable relationships between innovations. Moreover, the active colony of consumers is vitally
important for innovation diffusion since the updating rate of products is too frequent in the information age
nowadays. In order to find the principles of the relationships between innovations and the importance of the
affection on the active colony of an innovation, in this research we concentrate on developing and analyzing a
dynamic rate model for multi-innovations with active colony. The innovation interrelationships may include
independence, complementation, substitution, competition and contingency, etc. Furthermore, an empirical
analysis of the diffusion of mobile telephony and local telephony in China has been made. For the diffusion of
those communication innovations , the affection of active colony should not be ignored. The parameters of the
model have been estimated on the basis of the historical data and numerical simulation has been carried out
to show the relationship between the diffusion process of mobile telephony and local telephony in China. The
result shows that the model matches the history data perfectly and better than many empirical estimations in
the past: the increased adoptions of local telephony result in decreased adoptions of mobile telephony, while
the increased adoptions of mobile telephony result in increased adoptions of local telephony.
Keywords: differential equations, diffusion model, active colony, innovation diffusion, mobile telephony,
local telephony
1
Introduction
Since Schumpeter brought forward the concept of “innovation”, people have realized that innovations being brought into product process is one of the determinant factors of economic increasing[32] .
As a result, many scholars have been focusing on the innovation diffusion all over the world, especially
the scholars headed by Bass established dynamic models to research the principles of innovation diffusion and gained abundant achievements in scientific theory[2, 3, 7, 10, 12, 14, 15, 18, 23, 24, 29, 30, 35, 41] and empirical
research[8, 9, 11, 13, 26, 28, 34, 36] . The representative figures mainly include Fourt, Mansfield, Bass and Mahajan.
Fourt and Mansfield who have discussed the diffusion pattern in external influence and internal influence
separately[7, 24] . The sales model for forecasting TV made by Bass in 1969 has paved the way for the following researches on innovation diffusion[2] . The potential for innovation diffusion in these models is enormous.
∗
E-mail address: xinli.li@yahoo.com.cn.
Published by World Academic Press, World Academic Union
International Journal of Management Science and Engineering Management, Vol. 1 (2006) No. 2, pp. 148-160
149
Obviously such an assumption is not tenable with regard to either theoretical study or practice research. Aiming at this shortcoming Mahajan and Peterson proposed a dynamic diffusion model where the potential is
permitted to vary over time[23] , which brought about a new stage in the research on innovation diffusion.
After that, there are also many improved models and empirical researches which are based on such
models mentioned above [3, 8–11, 13–15, 18, 25, 26, 28–30, 34–36, 39–42] . But the objects of most researches are singleinnovation’s diffusion and few researches aim at the multi-innovations’ concomitance diffusion. Because innovations are not isolated since other innovations’ existence will exert an influence on them, either positively
or negatively. So, it’s more realistic and scientific to focus on the multi-innovations’ concomitance diffusion.
In the process of study on the innovation diffusion, many scholars have sufficiently proved the importance
of multi-innovations’ diffusion, such as Peterson and Mahajan’s research on encrypted software’s concomitance diffusion, and Bucklin’s notion that the effect caused by concomitance diffusion would be stronger than
that caused by innovation itself. Moreover, Kalish believed that many relevant products would compete with
it when an innovation enters a market, and he constructed a competition model for the new products with
two Bass models to show the relationship among multi-innovations. Such notions are also cited in the work
of Eliashberg and Jeuland[6] , Rao[4] , Krider and Weinberg[16] , Krishnan and Bass[17] , Rao and Bass[31] . But
these models can’t show the dynamic relations of competition and influence between each other, especially
the influence on the potential adopters who have planned to adopt other innovation is not taken into account in
former competition models. It goes without saying that consumers are all-important for businessmen and market in today’s economic society environment, since they are the active colony in oral communication and they
can be easily affected by other innovations before adopting the innovation. Conversely, after they adopt it, the
influence they exert on their friends and relatives will be far stronger than that exerted by those who adopted
formerly. Based on the above analysis , we construct a new dynamic model for multi-innovation diffusing in
the same time to explore the innovation interrelationships and the influence on the diffusion rate made by the
active colony of an innovation.
We organize this paper as follows. In section 2, an multi-innovations diffusion model taking the influence
of the potential adopters into account is proposed. In section 3, a stability analysis is made on the model and
an empirical analysis is made on the diffusion of the mobile telephony and the local telephony in China in
section 4. Finally, some conclusion remarks are given in section 5.
2
Model constructing
We would introduce the foundation of modelling, with which a conceptional model is given before constructing the mathematical model. After that, a stability analysis is made on this dynamic innovation diffusion
model.
2.1
The foundation of modelling
During the diffusion process of an innovation, new companies or customers adopt it and become it’s
adopters. However, for the influence of the existence of other innovations, the colony who has planed to
adopt this innovation will speed up or slow down their steps, even turn back to adopt other innovations. For
them, the opportunity cost of that will be much lower than that of the consumers who have adopted this
innovation. So they will be easy to change their minds frequently and then the adoption rate is affected as well
as the cumulative number of adoption of an innovation. However, on the contrary, once this colony adopts
this innovation, the affection which they made on their friends and relatives will be much stronger than that
made by those who has adopted this innovation long before since they are the active colony. This made more
potential consumers accelerate their steps to become a adopters or reject it.
2.2
Conceptional model
Based on the foundation above, the concrete consumers flow diagram for such innovation diffusion model
and the diagram illustrating are shown in Fig. 1 and Table 1 respectively, in which PTAITP-Adopters denotes
MSEM email for subscription: info@msem.org.uk
150
X. Li & Z. Liao: Dynamic multi-innovation diffusion model with active potential consumers and its application
the consumers planning to adopt Innovation 1 in this period. The arrowheads just show the direction of consumers flow and not always denote the consumers’ inflow and outflow. More frequently they show that the
existence of an innovation increases or decreases the number of adopters of another. The number is proportional to the numbers of the two end of the arrowhead. For the increased adoptions of one innovation don’t
always result in decreased adoptions of other innovation.
Inno vatio n 2
Inno vatio n 1
Fo rm er Ado pters
Fo rm er Ado pters
A
P TAITP -Ado pters
P TAITP -Ado pters
C
Fo rm er Ado pters
.
P TAITP -Ado pters
F
D
.
Fo rm er Ado pters
E
B
.
inno vatio n n
.
.
.
P TAITP -Ado pters
Inno vatio n i
Fig. 1. Adopters flow diagram for Innovation 1 affected by other innovations
Table 1. The explanation for conversion relation
Conversion name/
Way of Conversion
A/ Oral communication
B/Oral communication
C/Oral communication
D/Oral communication
E/Oral communication
F/Oral communication
2.3
Explanation for conversion
The affection on the consumers of Innovation 2 made by
the consumers planning to adopt Innovation 1 in this period
The affection on the consumers planning to adopt Innovation 1
in this period made by the consumers of Innovation 2
The affection on the consumers of innovation i made by
the consumers planning to adopt Innovation 1 in this period
The affection on the consumers planning to adopt Innovation 1
in this period made by the consumers of innovation i
The affection on the consumers of innovation n made by
the consumers planning to adopt Innovation 1 in this period
The affection on the consumers planning to adopt Innovation 1
in this period made by the consumers of innovation n
Mathematical model
Hypothesis 1: The diffusion rate of every innovations is dependent not only on the number who have
adopted but also on the proportion of the maximum number of adopters not yet realized when they are diffusing
separately.
Hypothesis 2: There does not always exit such relationships between innovations that they compete on
resources or markets. But for other innovations’ exit or being about to enter the market they are affected by
each other. As a result, the diffusion rate of every innovation is affected, hasten, block, or even nothing.
Based on the hypothesizes, and the convert relations above, we build the diffusion equation for Innovation
1. The potential adopters Mi (t)(i = 1, · · · , n) of time t are divided into users and non-users. For the sake of
the convenience for discussion suppose fi (t) = Ni (t)/Mi (t) denote the diffusion ratio of market of innovation
i, so 0≤ fi ≤1, i = 1, · · · , n.
From Hypothesis 1, there are b1 f1 (1 − f1 ) consumers adopting Innovation 1 in time t. From conversion
way C in Fig. 1, because of the flack of new adopters of Innovation 1 to the adopters of innovation i there are
MSEM email for contribution: submit@msem.org.uk
International Journal of Management Science and Engineering Management, Vol. 1 (2006) No. 2, pp. 148-160
151
ui1 b1 f1 (1 − f1 )fi of them turn to adopt Innovation 1. and from conversion way D, there are ui2 b1 f1 (1 − f1 )fi
adopters of Innovation 1 turn to adopt innovation i for they are convinced by the adopters of innovation i.
Then the diffusion equation of Innovation 1 is governed by
n
n
i=2
i=2
X
X
df1
= b1 f1 (1 − f1 ) +
ui1 b1 f1 (1 − f1 )fi −
ui2 b1 f1 (1 − f1 )fi .
dt
Let us define ui = ui2 − ui1 , then the equation above can be written as
n
X
df1
= b1 f1 (1 − f1 )(1 −
ui fi ).
dt
i=2
Similarly, the diffusion equation of innovation j is governed by
n
X
dfj
= bj fj (1 − fj )(1 −
ui fi ), j = 2, · · · , n.
dt
i=1,i,j
Consequently, we could gain the dynamic equation of innovation diffusion which is governed by
n
X
dfj
= bj fj (1 − fj )(1 −
ui fi ), j = 1, · · · , n.
dt
(1)
i=1,i,j
For the sake of the convenience for discussion, we only discuss the case of n = 2, i.e., the diffusion model of
two innovation in the same time:
( df1
dt = b1 f1 (1 − f1 )(1 − u2 f2 )
.
(2)
df2
dt = b2 f2 (1 − f2 )(1 − u1 f1 )
3
The stability analysis on the model
In this section we firstly introduce some related concepts and lemmas to help understand the following
analysis expediently and then give the stability analysis on the innovations diffusion.
3.1
Related concepts
Suppose there is an autonomous system
dx
= f (x), f ∈ C(G ⊆ Rn , Rn )
(3)
dt
where f ∈ C(G ⊆ Rn , Rn ) denotes f (x) is continuous in real set of n dimension. The local behavior of the
nonlinear system (3) near a hyperbolic equilibrium point x0 is qualitatively determined by the behavior of the
linear system
ẋ = Ax
(4)
ẋ =
with the matrix A = Df (x0 ), near the origin. The linear function Ax = Df (x0 )x is called the linear part of
x0 .
Definition 1. [27] A point x0 ∈ Rn is called an equilibrium point or critical point of (3) if f (x0 ) = 0. An
equilibrium point x0 is called a hyperbolic equilibrium point of (3) if none of the eigenvalues of the matrix
Df (x0 ) have zero real part. The linear system (4) with the matrix A = Df (x0 ) is called the linearization of
(3) at x0 .
Lemma 1. [27] Let δ = detA and τ = traceA and consider the linear system (4).
(a) If δ < 0 then (4) has a saddle at the origin.
(b) If δ < 0 and τ 2 − 4δ ≥ 0 then (4) has a node at the origin; it is stable if τ < 0 and unstable if τ > 0.
(c) If δ > 0 and τ 2 − 4δ < 0, and τ , 0 then (4) has a focus at the origin; it is stable if τ < 0 and
unstable if τ > 0.
(d) if δ > 0 and τ = 0 then (4) has a center at the origin.
MSEM email for subscription: info@msem.org.uk
X. Li & Z. Liao: Dynamic multi-innovation diffusion model with active potential consumers and its application
152
3.2
Independent diffusion
If ui =0 (i = 1, 2), each of the equation in systems (2) will degenerate to the Blackman/Fisher-pry
model, which is different from Bass model. There is no affection between innovations and they are diffusing
respectively.
Theorem 1. If ui =0(i = 1, 2), systems (2) have four equilibrium points, which are (0, 0), (1, 0), (0, 1), (1, 1)
respectively. And (0, 0) is a unstable node, (1, 0), (0, 1) are saddles, (1, 1) is a stable node.
Proof. First we will show point (1, 1) is a stable node. Denote M = b1 (1 − f1 )(1 − u2 f2 ), N = b2 (1 −
f2 )(1 − u1 f1 ), the coefficient matrix of the approximately linear equation of systems (2) about point (1, 1) is
Mf0 1 Mf0 2
−b1
0
A=
=
Nf0 1 Nf0 2
0 −b2
Then, T rA = −(b1 + b2 ) < 0, det A = b1 b2 , and (T rA)2 − 4 det A > 0. Accordingly, the equilibrium
point (1, 1) is a stable node from Lemma 1. Similarly we can show (0, 0) is an unstable node, (1, 0), (0, 1) are
saddle. This completes the proof.
Just as shown in Fig.2, there is no affection between each innovation and they are diffusing respectively. At
last systems tend to point (1, 1) and each of them occupies their maximum potential markets.
3.3
Complementary diffusion
If ui < 0(i = 1, 2), the affection between each innovation is promoting, which means that increased
adoptions of one innovation result in increased adoptions of other innovations.
Theorem 2. If ui < 0(i = 1, 2), systems (2) have four equilibrium points, which are (0, 0), (1, 0), (0, 1), (1,
1) respectively. And (0, 0) is an unstable node, (1, 0), (0, 1) are saddle, (1, 1) is a stable node.
Proof. First we will show point (1, 1) is a stable node. Denote M = b1 (1 − f1 )(1 − u2 f2 ), N = b2 (1 −
f2 )(1 − u1 f1 ), the coefficient matrix of the approximately linear equation of systems (2) in point (1, 1) is
Mf0 1 Mf0 2
−b1 (1 − u2 )
0
A=
=
0
−b2 (1 − u1 )
Nf0 1 Nf0 2
then T rA = −[b1 (1 − u2 ) + b2 (1 − u1 )] < 0, det A = b1 b2 (1 − u1 )(1 − u2 ) > 0, and (T rA)2 − 4 det A > 0.
Accordingly, the equilibrium point (1, 1) is a stable node from Lemma 1. Similarly we can show (0, 0) is an
unstable node, (1, 0), (0, 1) are saddles. This completes the proof.
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
0.2
0.4
0.6
0.8
Fig. 2. ui ≤ 0(i = 1, 2)
1
0
0.05
0.1
0.15
0.2
Fig. 3. u1 > 1, u2 < 0
Just as shown in Fig. 2, systems (2) tend to point (1, 1) eventually. That is to say, f1 = f2 = 1, which
means each of the innovations has occupied its maximum potential market. In fact, since u1 , u2 are both
MSEM email for contribution: submit@msem.org.uk
International Journal of Management Science and Engineering Management, Vol. 1 (2006) No. 2, pp. 148-160
153
blow 0, the two innovations accelerate the diffusion rate of each other, until reaching their own maximum
potential markets. For example, more mature software and more efficient hardware will accelerate each other’s
diffusion rate. Similarly, in cases 0 < u1 < 1, u2 < 0 and 0 < u1 < 1, 0 < u2 < 1 , the inequalities
(1 − f2 )(1 − u1 f1 ) ≥0 (the equality is right if and only if f2 =1) and (1 − f1 )(1 − u2 f2 ) ≥0 ( the quality is
right if and only if f1 =1) are exact since 1 − u2 f2 > 0, 1 − u1 f1 > 0. Consequently, it follows f1 = f2 =1
at last. In case 0 < u1 < 1, u2 < 0, Innovation 2 accelerates the diffusion of Innovation 1, while Innovation
1 blocks the diffusion of Innovation 2 too faintly to block off it. So both of them reach their potential. In case
0 < u1 < 1, 0 < u2 < 1, the two innovations both block each other’s diffusion too faintly to hold back each
other , so both reach their maximum potential.
3.4
Increase and decrease’s diffusion
When u1 > 1, u2 < 0, the increased adoptions of Innovation 2 result in increased adoptions of Innovation
1. Conversely, the increased adoptions of Innovation 1 result in deceased adoptions of Innovation 2.
Theorem 3. if u1 > 1, u2 < 0, systems (2) have four equilibrium points, which are (0, 0), (1, 0), (0, 1), (1, 1)
respectively. And (0, 0) is an unstable node, (1, 1), (0, 1) are saddles, (1, 0) is a stable node.
Proof. First we will show equilibrium point (1, 0) is a stable node. The coefficient matrix of the approximately
linear equation of systems (2) at point (1, 0) is
−b1
0
A=
.
0 −b2 (1 − u1 )
Then T rA = −b1 +b2 (1−u1 ) < 0, det A = b1 b2 (1−u1 ) < 0, and (T rA)2 −4 det A > 0. Accordingly,
the equilibrium point (1, 1) is a stable node from lemma 1. Similarly we can show (0, 0) is a unstable node,
(1, 0), (0, 1) are saddles. This completes the proof.
Just as shown in Fig. 3, since u2 < 0, 1 − u2 f2 > 1, the diffusion of Innovation 2 increases that of Innovation
1. However, if u1 > 1, and 1 − u1 f1 > 0, although the existence of market share of Innovation 1 makes great
resistance on the diffusion of Innovation 2, Innovation 2 has its potential to diffuse and enlarge its market
share. But when the market share of Innovation 1 reaches a given point, that is to say when 1 − u1 f1 < 0,
(1−f2 )(1−u1 f1 ) < 0, the market share of Innovation 2 is shrunk gradually till lost entirely with the vigorous
competition of Innovation 1. For example, for those substitute technologies with good compatibility and better
benefit cost, their advantages are specially outstanding since the former generation has no any advantage of
price and the consumers are attracted by the new generation. The bigger market share the former has, the
stronger affection on the new generation, which accelerates the diffusion rate of the new generation. At last,
the new generation occupied the whole market share. This is true in the case of u1 > 1, 0 < u2 < 1.
3.5
Competition or substitutes diffusion
If u1 > 1, u2 > 1, the increased adoptions of one innovation result in decreased adoptions of another.
Theorem 4. If u1 > 1, u2 > 1, systems (2) have five equilibrium points, which are (0, 0), (1, 0), (0, 1), (1,
1), (1/u1 , 1/u2 ) respectively. In which (0, 0), (1, 1) are unstable nodes, (0, 1), (1, 0) are stable nodes, (1/u1 ,
1/u2 ) is a saddle.
Proof. First we shall show the equilibrium point (1/u1 , 1/u2 ) is a saddle. The coefficient matrix of the approximately linear equations of systems (2) at point (1/u1 , 1/u2 ) is


u2
1
0
−b1
1−

u1
u1 

A(1/u1 ,1/u2 ) = 


u1
1
−b2
1−
0
u2
u2
1
1
and T rA = 0, det A = −b1 b2 1 −
1−
< 0, so point A is a saddle. Similarly, we can show (0,
u1
u2
0), (1, 1) are unstable nodes, (0, 1), (1, 0) are stable nodes, just as shown in Fig. 4. This completes the proof.
MSEM email for subscription: info@msem.org.uk
154
X. Li & Z. Liao: Dynamic multi-innovation diffusion model with active potential consumers and its application
Similar to the analysis on Innovation 2 in 3.3, there exists great competition between the two innovations just
as illustrated in Fig. 4. In the area B, although the two innovations have great competition between them,
they also enlarge their market share respectively for the market is large enough or other reasons; in area D,
(1 − f1 )(1 − u2 f2 ) > 0 , (1 − f2 )(1 − u1 f1 ) < 0. That is to say, the diffusion rate of Innovation 1 is positive
and that of Innovation 2 is negative. Systems will tend to point (1, 0). In area A, (1 − f1 )(1 − u2 f2 ) < 0 , and
(1 − f2 )(1 − u1 f1 ) > 0. That is to say, the diffusion rate of Innovation 1 is negative, its market share of it
decrease gradually. While the diffusion rate of Innovation 2 is positive, its market share increases incessantly
and systems will tend to point (0, 1). In area C, (1 − f1 )(1 − u2 f2 ) < 0 , and (1 − f2 )(1 − u1 f1 ) < 0. The
resistance of the two innovations made by each other is too large to decrease the market share of each other
for their diffusion rate is both negative.
f2
1
A
0.8
C
1/u 2
0.6
B
D
0.4
0.2
0
0
0.2
0.4
0.6
1/u 1
0.8
1
f1
Fig. 4. u1 > 1, u2 > 1
30000
25000
20000
Local
Mobile
R_L
R_M
15000
10000
2003
2002
2001
2000
1999
1998
1997
1996
1995
1994
1993
1992
1991
1990
0
1989
5000
Year
Fig. 5. The history curves of the diffusion rate and accumulative total numbers of mobile telephony and local telephony
respectively
Now we take two similar technologies for example. Suppose there is little difference or both have little in
market shares at the time of emerging in a market. That is to say, the competition between the two innovations
is setting out from the origin point in area B. At that time, the market is big enough for them to increase their
diffusion rate and expand their shares. If u1 , u2 , firstly assume 1 < u1 < u2 , that is to say the resistance on
Innovation 1 made by Innovation 2 is bigger than that on Innovation 2 made by Innovation 1. The evolvement
path will reach the boundary between area A and B firstly. From Fig. 4, we can see the path will get across
the boundary, enter area A and reach the node (0, 1), which means Innovation 2 occupies the whole market.
MSEM email for contribution: submit@msem.org.uk
International Journal of Management Science and Engineering Management, Vol. 1 (2006) No. 2, pp. 148-160
155
Similarly, if u1 > u2 , the dominator of the market is Innovation 1. If u1 = u2 , the evolvement path will
run to point (1/u1 , 1/u2 ) and reach state of equilibrium. However, the equilibrium will be broken if even
faintly disturbed. For example, if the company in possession of Innovation 1 increases their expenditure of
advertisement or reduces the price of admittance, the position of point (1/u1 , 1/u2 ) will move leftward, and
the following equilibrium point will be at the boundary between area C and D (Suppose u2 is fixed). Then
Innovation 1 will predominate the market from Fig. 4; similarly, if any negative news of Innovation 1 are
uncovered, the position of point (1/u1 , 1/u2 ) will move rightward, and the following equilibrium point will
be at the boundary between area A and B (Suppose u2 is fixed). Then the path will enter area A, Innovation 2
will predominate the market eventually.
4
The empirical analysis
In 1876, Bell invented telephone which transfers sound by cable and spread all over USA in 10 years and
soon the whole world. However, mobile communication mainly is the mobile communication technology and
product, which emerged in USA, Japanese and other European countries in the beginning of 80s of 20th and
was imported by China in the end of 80s of 20th .
The two technologies discussed above are both communication technologies and in this paper we use the
numbers of the subscribers to denote the extent of diffusion. Since the technology of mobile imported, along
with the increasing of living standard and the deceasing of cost to communication, the market shares vary
drastically and the competition for consumers between the two technologies are also to somewhat unbelieving
extent. Based on this, we try to set up a mathematics model to analyze such relationship between the diffusion
of the two technologies.
Since the consumers can use both mobile telephone and local telephone, even there exits competition
in market between mobile telephone and local telephone, the competition is not absolute. Consequently, it’s
improper to use general competition model before discussing such competition relationship. So we try to use
systems (2) to validate the influence on the diffusion rate caused by the competition between them.
for the convenience of analysis, The potential adopters Mi (t)(i = 1, · · · , n) of time t are supposed
to be constant, which may be in contradiction with our model discussed before. But in this section the key
objective is trying to find the diffusion relationships between mobile telephone and local telephone. Then, for
the convenience of software calculation, systems (2) can be written as a
f1 (k + 1) − f1 (k) = b1 f1 (k)(1 − f1 (k))(1 − u2 f2 (k))
(5)
f2 (k + 1) − f2 (k) = b2 f2 (k)(1 − f2 (k))(1 − u1 f1 (k))
and then
ln(f1 (k + 1) − f1 (k)) = lnb1 + ln(f1 (k)(1 − f1 (k))) + ln(1 − u2 f2 (k))
ln(f2 (k + 1) − f2 (k)) = lnb2 + ln(f2 (k)(1 − f2 (k))) + ln(1 − u1 f1 (k))
(6)
We make linear regression on the systems (6) with the software SPSS11. 0 and obtain the following model with
the hypothesis above and the 15 groups of history data shown in Fig. 5, in which Local T ele, M obile T ele
denote the cumulative number of local telephone subscribers and mobile telephone subscribers respectively
and R Local, R M obile denote the diffusion rate per year of those respectively.
f˙1 = 1.3349018f1 (1 − f1 )(1 − 0.974017969f2 )
(7)
f˙2 = 0.3522845f2 (1 − f2 )(1 + 0.739111255f1 )
In which f1 , f2 denote the accumulative total ratio of diffusion for mobile telephony or local telephony respectively. The R squared of the diffusion rate of mobile telephony is 0.98691 and that of the local telephony is
0.93253. Their market potential are 418.27 million and 373.243 million respectively. Their simulation curves
and the dynamic market shares are shown in Fig. 6, Fig. 7 respectively.
We give approximate answer to the model above and predict as shown in Fig. 8 and Fig. 9 respectively
with Mathematic 4.
MSEM email for subscription: info@msem.org.uk
156
X. Li & Z. Liao: Dynamic multi-innovation diffusion model with active potential consumers and its application
30000
20000
10000
L_TELE
Mean
M_TELE
R_L
0
R_M
1990.00
1992.00
1991.00
1994.00
1993.00
1996.00
1995.00
1998.00
1997.00
2000.00
1999.00
2002.00
2001.00
2003.00
YEAR
Fig. 6. The history and prediction curve of the diffusion rate of mobile telephony and local telephony respectively
Fig. 7. The prediction curve of the shares of mobile telephony and local telephony respectively
Table 2. The history and predicted numbers of local telephone subscribers and mobile telephone subscribers in year 2001 -2003
respectively
year
The cumulative number of users of mobile
The cumulative number of users of telephone
History data
Predicted data
Error rate
History data
Predicted data
Error rate
2001
14522.2
14306.84
-1.48%
18036.8
17961.93
-1.84%
2002
20661.6
21029.47
1.78%
21441.9
21655.89
1.07%
2003
26869.3
26738.00
-0.49%
26330.5
24698.01
-6.2%
The accumulative total ratios of diffusion for mobile telephony and local telephony are 0.955986,
0.962907 in the end of year 2005 respectively and those in year 2010 are 0.965417 and 0.998111 respectively,
which will reach their market limits. After year 2010, although their market shares can still be expanded, the
diffusion potentials left are very little.
We can draw the following conclusions based on the above analysis :
(1) The diffusion rate of mobile telephony is far faster than that of local telephony. Since the intrinsic
diffusion rate of mobile telephony is 1.334901809 and that of local telephony is only 0.3522845. In fact, there
are more than 100 years from the time of telephone invented to now and that of mobile telephony is only less
than 20 years. However, from Fig. 5, we can see in year 2003 the number of mobile telephone subscribers has
exceeded that of local telephone. That may be relevant to the convenience of mobile telephone, the boosting up
MSEM email for contribution: submit@msem.org.uk
157
International Journal of Management Science and Engineering Management, Vol. 1 (2006) No. 2, pp. 148-160
40000
35000
30000
30000
25000
20000
20000
15000
10000
10000
5000
5
10
15
20
25
Fig. 8. The prediction curve of the diffusion of mobile
telephony
10
20
30
40
Fig. 9. The prediction curve of the diffusion of local
telephony
of communication effects and the decreasing of cost. Besides, telephone means a notion of family or company
while that of mobile telephone is individual.
(2) The relationship between the two technologies during the process of diffusion is not mutually competitive or simulative. The block coefficient in model (5) is 0.974017969(u2 ), -0.739111255(u1 ) respectively.
So the exit of local telephony blocks the diffusion rate of mobile telephony and that of mobile telephony results in the speedup of the diffusion rate of local telephony. Apparently, there does exit competition between
their processes of diffusion. But the competition between them is on the cost of communication. In fact, the
aim of price war is to attract more subscribers and to enlarge their market. Moreover, their total numbers of
subscribers do not counteract that of another’s. At the time of using the mobile telephone, many consumers
do not give up using local telephone. Furthermore, the use of mobile telephone make people more reliable on
the communication facility, which result in companies or families to subscribe the local telephone because the
cost is more attractive and communication becomes necessary.
(3) The effect of simulation of this dynamic model is perfect. But from Fig. 8 and Table 2 we can see the
effect of simulation of mobile telephony is more efficient than that of telephone telephony. Here follow the
possible causes: First, this model is based on the internal-influence model. The effect of oral communication
and imitation on the diffusion of mobile telephony is bigger than that of telephone. So the origin model
is fit to describe the diffusion of mobile telephony; Second, this model doesn’t take the economic society
environment into account, and the effect of simulation for telephone is a little worse. From Fig. 5, we can
see the prediction data is bigger than that of history in periods 1989-1991 and 1996-1999. At these periods,
China have experienced internal economic blockage, deflation and economic developing slowly. Those block
the steps of telephone users increasing and make the prediction data below the history data. However, during
1991-1995, the inflation made China developing rapidly and that may be the cause which the history data is
higher than that of prediction. The causes above may lead to the imprecision the prediction for diffusion rate
of telephone.
Furthermore, we also use Bass model to describe and forecast the number of the mobile telephone subscribers or that of the local telephone subscribers without caring about the relationship between them.
Ṅ1 = (0.001651619 + 0.000030570N1 )(30176.876824 − N1 )
(8)
Ṅ2 = (0.001182105 + 7.47469 × 10−6 N2 )(47431.004374 − N2 )
in which the coefficient of external affection of mobile telephone a1 = 0.001651619, the coefficient of the
internal affection b1 = 0.000030570 and those of local telephone respectively are 0.001182105, 7.47469 ×
10−6 . The market potential of mobile telephony is 3.01769 billion and that in country is 4.7431 billion. The
R squared of the first equation of (8) is 0.98132 and that of the second is 0.93163.
In order to distinguish which prediction model is more accurate, we have drawn the plots of the prediction
curves of different models, and compare with those of the historical data, which are shown in Fig. 10, Fig.
11 respectively. Although the effect is not notable, We find that model (7) is more efficient than model (8) to
simulate the diffusion of local telephony and mobile telephony. We also draw the prediction curves of mobile
telephony and local telephony governed by model (7) or model (8), which are shown in Fig. 12 and Fig.
13 respectively. It is shown that the Bass model magnifies the potential of local telephony and shrinks that
MSEM email for subscription: info@msem.org.uk
X. Li & Z. Liao: Dynamic multi-innovation diffusion model with active potential consumers and its application
6000
8000
7000
R_L
4000
6000
P_L_m
e
B_P_L
3000
2000
R_M
5000
P_M_m
e
B_P_M
4000
3000
2000
1000
Fig. 10. The history and prediction curve of the diffusion rate of mobile telephony and telephone respectively
2002
2001
2000
1999
1998
1997
1996
1995
1994
1993
1992
1991
1990
0
2002
2001
2000
1999
1998
1997
1996
1995
1994
1993
1992
1991
1990
1000
1989
0
Subcribers
Subcribers
5000
1989
158
Fig. 11. The history and prediction curve of the diffusion rate of Local telephone respectively
of mobile telephony compared with model (7). From empirical estimation, the potential of mobile telephony
should be bigger than that of local telephony since the subscribers of local telephone are companies or families
and those of mobile telephone are individual persons. However, Bass model is not consistent to such empirical
estimation and may not be suitable to be used to predict such diffusion phenomena of mobile telephone.
40000
G over ned by
model (6)
G ov er n ed b y
40000 m od el (6)
30000
G over ned by model (5)
20000
30000
G ov er n ed b y m od el (5)
20000
10000
10000
5
10
15
20
25
Fig. 12. The prediction curve of the diffusion of mobile telephony governed by model (7) or model (8) respectively
5
10
20
30
40
Fig. 13. The prediction curve of the diffusion of local telephony governed by model (7) or model (8) respectively
Conclusion
This paper is focusing on a dynamic diffusion model in which innovations are having an effect on others
during the process of diffusion. This model is established to show the existing of other innovations is having
effect on the active colony of an innovation. With the stability analysis on such model, we have simulated
all kinds of diffusion process with the help of Mathematic 4. The relationship between innovations in market
could be independent, complementary, substitute or competition, and increase and decrease. We also have
made empirical analysis on the diffusion of mobile telephony and local telephony in China. The parameters
of the model have been estimated on the basis of the history data and numerical simulation has been carried
out out to show the diffusion process for mobile telephony and local telephony in China. We find that the
increased adoptions of local telephony result in decreased adoptions of mobile telephony and such action are
not come to exist for the affection on local telephony made by mobile telephony. It’s shown that this model
is not only efficient to simulate the late diffusion rate but has good guidance for the long-term tendency of
the innovation diffusion since the effect on the active colony should not be ignored for the particularity of the
communication technologies diffusion and such model is also applicable for those similar innovations. But
MSEM email for contribution: submit@msem.org.uk
International Journal of Management Science and Engineering Management, Vol. 1 (2006) No. 2, pp. 148-160
159
we have not take the advertising, price and the economic environment into account, which will be included in
future research.
References
[1] The Committee of Annual. The Annual of China Statistic 2002, Beiking, China Statistic, 2003.
[2] F. M. Bass. A new product growth model for consumer durables, Management Science, 1969, 15 (5): 215-227.
[3] P. T. Chang, C. H. Chang. A stage characteristic-preserving product life cycle modeling, Mathematical and Computer Modeling, 2003 , 37: 1259-1269.
[4] P. K. Chintagunta, V. R. Rao. Pricing strategies in a dynamic duopoly: a differential game model, Management
Science, 1996, 42: 1501-1513.
[5] R. J. Dolan, A. P. Jeuland. Experience curves and dynaminc demand models: implications for optimal pricing
strategies, Journal of Marketing, 1981, 45: 52-73.
[6] J. Eliashberg, A. Jeuland. The impact of competitive entry in a developing market upon dynamic pricing strategies,
Marketing Science, 1986,5: 20-36.
[7] L. A. Fourt, Woodlock. Early prediction of market success for new grocery products, Journal of Marketing, 1960,
25: 31-38.
[8] A. N. Giovangis, C. H. Skiadas. A stochastic logistic innovation diffusion model studying the electricity consumption in greece and the United States, Technological Forecasting and Social Change, 1999, 61: 235-246.
[9] H. Gruber. Competition and innovation: the diffusion of mobile telecommunication in Central and Eastern Europe,
Information Economics and Policy, 2001, 3: 19-34.
[10] D. B. Jun, Y. S. Park. A choice-based diffusion model for multiple generations of products, Technological Forecasting and Social Change, 1999, 61: 45-58.
[11] D. B. Jun, S. K. Kim. Forecasting telecommunication service subscribers in substitutive and competitive environments, International Journal of Forecasting, 2002, 18: 561-581.
[12] S. Kalish. A new product adoption model with price, advertising, and uncertainty, Management Science, 1985, 31
(12): 1569-1585.
[13] Kalish, S. Mahajan, E. Muller. Waterfall and sprinkler new-product strategies in competitive global markets, International Journal of Research in Marketing, 11995, 2: 105-119.
[14] B. Kang, H. Kim. A demand-based model for forecasting innovation diffusion, Computers Industry Engineering ,
1996, 30(3): 487-499.
[15] M. Karshenas, P. Stoneman. Rank, stock, order and epidemic effects in the diffusion fornew process techonologies:
an empirical mode, Rand Journal of Economics, 1996, 24: 503-528.
[16] R. Krider, C. Weinberg. Competitive dynamics and the introduciton of new products: the motion pricture timing
game, Journal of Marketing Research, 1998, 2: 1-15.
[17] T. Krishnan, F. Bass, D. Jain. Optimal pricing strategy for new products, Management Science, 1999, 45 (12):
1650-1663.
[18] N. Kshetriand, M. K. Cheung. What factors are driving china’s mobile diffusion, Electronic Markets, 2002, 12 (1):
22-26.
[19] C. L. Lackman. Gompertz curve forecasting: a new product application, Journal of the Market Research Society,
1978, 20: 45-57.
[20] E. Mansfield. Technical change and rate of imitation, Econometrica, 1961, 29: 741-766.
[21] V. Mahajan, E. Mller, F. Bass. New product diffusion models in marketing: a review and directions for research,
Journal of Marketing, 1990, 54 (1): 1-26.
[22] D. Tigert, B. Farivar. The Bass new product growth model: a sensitivity analysis for a high technology product,
Journal of Marketing, 1981, 45: 81-90.
[23] V. Mahajan, R. A. Peterson. Innovation diffusion in a dynamic potential adopter population, Management Science,
1978, 24: 1589-97.
[24] E. Mansfield. Technical change and the rate of imitation, Econometrica, 1961, 29: 741-766.
[25] A. Mcmeekin, M. Tomlinson. Diffusion of household durables in the UK, Futures, 1998, 30 (9): 873-886.
[26] M. Mohle. The coalescent in population models with time-inhomogeneous environment, Stochastic Processes and
Their Application, 2002, 97: 199-227.
[27] L. Perko. Differential Equations and Dynamical Systems, Springer-Verlag, New York, 1991.
[28] A. Posad, V. Mahajan. How many pirates should a software firm tolerate? an analysis of piracy protection on the
diffusion of software, International Journal of Research in Marketing, 2003, 20: 337-353.
[29] D. Satoh. A discrete bass model and its parameter estimation, Journal of the Operations Research, 2001, 44(1):
1-18.
MSEM email for subscription: info@msem.org.uk
160
X. Li & Z. Liao: Dynamic multi-innovation diffusion model with active potential consumers and its application
[30] E. Montaguti, S. Kuester, s. Thomas. Entry strategy for radical product innovations: a conceptual model and prepositional inventory, Internal Journal Research in Marketing, 2002, 19: 21-42.
[31] R. Rao, F. Bass. Competition, strategy, and price dynamics: a theoretical and empirical investigation, Journal of
Marketing Research,1985, 22: 283-296.
[32] E. M. Rogers. Diffusion of Innovations, New York: Free Press, 1983.
[33] Y. Sheng. Innovation Diffuison and the Marketing of New Product, Beiking, China developement, 2002. (in Chinese)
[34] P. R. Steffens. A model of multiple-unit ownership as a diffusion process, Technological forecasting & Social
Change, 2003, 70: 901-917.
[35] P. Stoneman. The Economics of Technological, Oxford, Blackwell Publishers, 2002.
[36] J. T. C. Teng, V. Gtobrt, W. Guttler. Information technology innovations: general diffusion patterns and its relationships to innovation characteristics, IEEE Transactions on Engineering Management, 2002, 49 (1): 13-27.
[37] S. Wallsten. An econometric analysis of telecom competition, privatization, and rgegulation in Africa and Latin
Amercia, Journal of Industrial Economics, 2001, 49: 1-20.
[38] W. D. Wang , P. Fergola. Innovation diffusion model in patch environment, Applied Mathematics and Computation,
2003, 134: 51-67.
[39] J. P. Xu, X. Z. Chen. The diffusion model of dryland raising seeding and its empirical analysis, Joural of Management Engineering, 2001, 15 (1) : 14-19. (in Chinese)
[40] J. P. Xu, M. A. Hu, Z. N. Hu. Market diffusion analysis of combine Harvesters, the Applied Theory on System
Engineering, 2000, 9 (2): 141-146. In Chinese.
[41] J. P. Xu, Z. G. Liao. Model for innovation diffusion rate, Chinese Journal of Management, 2004, 1(3): 330-340. In
Chinese.
[42] Z. G. Liao, J. P. Xu. A dynamic diffusion model with time-delay for color TV in China, World Journal of Modelling
and Simulation,2006, 2(2): 73-86.
MSEM email for contribution: submit@msem.org.uk