2012 Cambridge Business & Economics Conference
ISBN : 9780974211428
Product Growth Patterns of International Consumer Electronic Firms:
An Empirical Determination Using Biological Species Population Models
Dr. Michael J. Harrison
Assistant Professor, Framingham State University
100 State St. Framingham, MA 01701
508-626-4667-Phone
508-626-4030-Fax
mharrison2@framingham.edu
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ISBN : 9780974211428
Product Growth Patterns of International Consumer Electronic Firms:
An Empirical Determination Using Biological Species Population Models
ABSTRACT
This paper utilizes a biological species population model to examine product growth patterns for
home audio products of international consumer electronic firms. Time series analysis of product
unit sales, for six top products in the Home Theater–in-a Box (HTiB) category, employing a
logistic equation, reveals that home audio consumer electronic products behave similarly to a
biological species population. Time series non-linear regression analysis produces a best fit
sigmoid growth curve for each product analyzed. Two population growth parameters, carrying
capacity and growth rate, are determined. The market niche carrying capacity or population
density for the HTiB category is projected to be 203.2 million units. Implications for product
lifecycle planning in consumer electronics are discussed.
INTRODUCTION
The objective of this paper is to determine the market carrying capacity and the growth
rates for consumer electronic products of international firms by incorporating an interdisciplinary
population dynamics approach. This paper utilizes a biological species population model to
examine product growth patterns for home audio products of international consumer electronic
firms. Employing a logistic equation, reveals that home audio consumer electronic products
behave similarly to a biological species population. Manufacturers of Home Theater systems
have achieved a market penetration rate ranging from approximately 21% in 1999 to
approximately 36% in 2011 (Palenchar, 2011). Market penetration rates are significantly lower
than manufacturers anticipated, particularly when benchmarked against the penetration rates of
televisions in the home. Movie theater attendance is decreasing (Germain, 2011) while in-home
movie viewing is increasing (Graham, 2010).
LITERATURE REVIEW
Interdisciplinary studies are increasingly applied to business analysis. Techniques from
the Science disciplines have been used in economics and finance for over a century (Farmer &
Lo, 1999; Peters, 1999;Zovko & Farmer, 2002). More recently biological models have expanded
from economic and finance applications to business strategy and management (Arthur, 1998
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Modis, 1998, Cortright, 2001). Game Theory (Dixit & Nalebuff, 1991) and Chaos Theory
(Williams, 1997; Arthur, 1994) have been applied to business analysis in the context of strategic
decision making. “In the last 5 years, roughly 30 recent physics PhDs have addressed finance
topics with their doctoral research” (Farmer, 1999 p.26). Arthur (1994) uses dynamical systems
modeling to study path dependence in product allocation processes. Kauffman (1993) uses
complexity and chaos theory to explore how systems coordinate complex tasks and exhibit
spontaneous self-organizing order. McKelvey (1999) translates Kauffman’s complexity theory
models in analyzing firm value chain activities (Porter 1985) in order to understand strategic
choices and strategy implementation for firms in coevolutionary pockets. Farmer and Lo (1999)
state that analogies between biology and economics have been discussed for more than a century
and that two of the forefathers of modern economics Thomas Malthus and Adam Smith were
both cited by Darwin. The coupling of biology and finance to create an interdisciplinary
approach for solving business problems is not new. “However, a quantitative foundation for this
approach has been slow to develop” (p.5). Lo (1999) suggests that recent research in finance
indicates that the trend of using quantitative biological methods to study finance will continue to
increase. This paper applies a quantitative biological species population model to determine
product growth patterns for international consumer electronic firms.
Modis (1998) compares the business lifecycle to that of the change of seasons. The
biological species analogy is used. Thus, the growth of a species through the natural change of
seasons is a natural comparison to that of a firm, product or technology’s growth through
business cycles. Like a natural species, the importance of understanding what season it is in
order to function appropriately is critical to a business manager. “If mangers accurately
determine what business season they have entered, how long it will last, and their position in it,
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they can use the seasons metaphor as a guide for setting strategies that favor the rise of another
Spring” (p.74).
Modis (1998) describes the natural lifecycle (birth, growth, maturity decline, death) of
many phenomenon as a natural S-shaped curve. The natural growth cycle of a species, such as a
rabbit population, is used to compare the lifecycle of product sales for a population of computers,
a population of cars, and in a competitive lifecycle between fountain pens and ball point pens.
“Because different business seasons dictate distinctly different behavior, successful strategic
actions become telltale signals for the season a company is in” (p.64).
Cortright (2001) similarly compares the business cycle to a natural biological process.
“The economy is an evolutionary system, not a Newtonian balance that always seeks equilibrium.
Both the micro behavior of economic actors (firms, workers and consumers) and the overall path
of economic development can be pictured by invoking analogies to biological evolution” (p.12).
While neoclassical theory has a difficult time explaining technological change, evolutionary
theory deals with it explicitly (Cortright, 2001). The quantitative analogy that connects
technological change and product growth to biological population growth lies with logistic
equation. “In view of the evolutionary economists, change isn’t the smooth and continuous
adjustment at the margin, but is rather the abrupt and often uneven displacement of the one
technology by another. Economic growth is a dis-equilibrium process, and as the competitive
environment changes, development and improvement of new techniques and changes in markets
cause some firms to grow and others to shrink” (p13-14).
Conventional economic theory is built on the assumption of diminishing returns (Arthur,
1994 p. 1) where there is one equilibrium point, based on supply and demand, for prices and
market share. Technological advances and the advent of technology products themselves allow
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for increasing returns. That is because “not only do the costs of producing high-technology
products fall as a company makes more of them, but the benefits of using them increase.(p4).
The Logistic model was developed by Belgian mathematician Pierre Verhulst (1838) who
suggested that the rate of population increase may be limited. That is, the rate of growth may
depend on population density. The upper limit population growth is the carrying capacity. As
Sharov (1997) notes, the biological species population dynamics is described by the differential
equation:
dN
N
 rN  r 0 N (1  )
dt
K
(1)
N0 K
N 0  ( K  N 0) (  r 0*t )
(2)
With the solution:
Nt 
Where:
Nt = Population at time t
N0= Population at time 0
K = Carrying capacity
r0= Rate of growth
The model has three possible outcomes:
1- N0 < K (Where the population increases and reaches a plateau; this is the sigmoid Scurve or logistic growth curve)
2- N0 > K (Where the population decreases and reaches a plateau)
3- N0 = K or N0 = 0 (Where the population does not change; Note there are two
equilbria)
Sharov (1997) graphically illustrates the possible outcomes in Figure 1 and explains the
relationship of the parameters N and K within the logistic curve.
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Figure 1: Possible outcomes
The logistic model has two equilibria: N = 0 and N = K. The first equilibrium is unstable because
any small deviation from this equilibrium will lead to population growth. The second equilibrium
is stable because after a small disturbance the population returns to this equilibrium state (Sharov,
1997).
Williams (1997) notes that the logistic equation is the most popular equation for studying
chaos and that in species population studies “the population is assumed to change at discrete time
intervals, rather than continuously” (p.161). Similar to a biological species population, product
population growth, measured by unit sales, is accounted for and reported in discrete time
intervals. Applying the logistic model to consumer electronics to determine if product growth
patterns are similar to biological species growth patterns we specifically test for the sigmoid or
logistic pattern and for the carrying capacity or population density. Additionally, we test the
growth rate for each product by determining the slope of the curve at its midpoint.
DATA
Monthly unit sales data for six (6) of the top selling products, as reported by NPD
Intellect, the Consumer Electronics Association and TWICE magazine are analyzed. The six
products represent the top three international firms in the consumer electronic Home Theater-ina-Box category. The top two selling products from each of the three firms are analyzed over a
thirty-seven month period.
METHODOLOGY
Using the first derivative of sales we calculate the cumulative sales per month, in units,
over a 37 month time period to chart the product (species) growth. Next, we test the resulting
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cumulative sales growth pattern to determine the best fit line using three different regression
models. We test the best fit using:
1) an exponential growth model
2) a linear growth model
3) a non-linear, sigmoid or S-curve growth model.
Hypothesis 1
Ho: Non-linear regression does not provide the best fit
Ha: Non-linear regression provides the best fit
Next, using nonlinear regression we test three hypothesis of the biological species
population model to determine the relationship between the parameters of the species population
and the carrying capacity for each product. That is whether:
1) N = K
2) N > K
3) N < K
Where:
N = the population at time t
K = the carrying capacity
Hypothesis 2
Ho: N = K or N < K
Ha: N < K
Next, we test the growth rate of each product (species) to determine if the rate of growth
parameter is same between products (species).
Hypothesis 3
Ho: The growth rate is the same for all products
Ha: The growth rate is different for all products
Finally, we test to determine the overall carrying capacity of the home theater in-a-box category
for all products.
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Hypothesis 4
Ho: The carrying capacity for products is the same
Ha: The carrying capacity for products is different
RESULTS
Correlation coefficient results for the best fit of the product growth patterns for the three
models tested demonstrate that the exponential best fit produces an R2 range between 0.5077
and .8581 for each of the six products. The linear regression best fit produces an R2 between
0.9086 and 0.9956.The non-linear logistic regression results in an R2 between 0.9810 and .9950.
Table 1: Best fit R2 by model type
For each of the four products from both Firm A and Firm B, the non-linear logistic model
(Sigmoid) produces the highest correlation coefficient best fit. For both products from Firm C
the linear regression model produces a slightly higher correlation coefficient best fit than the
logistic model. For FPC-1 the difference between the linear and logistic model is .0058 and for
FPC-2 the difference is .0046.
Figure 2: Product growth patterns
The best fit regression line for Firm A’s two products using a biological species
population model indicates that the relationship between the population parameter and carrying
capacity for Firm Product A1 is N0 < K. That is, the product (species) increases and then reaches
a plateau. At time 37 the relationship is Nt37 = K. That is, the product (species) is at equilibrium.
Firm Product A2 results in logistic growth (N0 < K). However at time = 37, equilibrium (N = K)
is reached at a much lower unit volume. Figure 3 clearly demonstrates the non-linear, S-curve
growth properties of Firm A products 1 and 2.
Figure 3: Firm A’s product growth patterns
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For Firm B the biological species population model indicates that the relationship
between the population parameter and carrying capacity for Firm Product B1 is N = K. That is,
the product (species) is at equilibrium. Firm Product B2 results in N = K at a much lower
carrying capacity level. The graph clearly demonstrates the non-linear, S-curve growth
properties of Firm B products 1 and 2.
Figure 4: Firm B’s product growth patterns
For firm C’s two products the linear regression model produces a slightly better fit than
the non-linear logistic model. The biological species population model indicates that the
relationship between the population parameter and carrying capacity for Firm Product C1
exhibits a pattern where N < K. However, the results are not at strong as the previous 4 products
(species). That is, the product (species) is at equilibrium. Firm Product C2 results in N < K at a
much lower carrying capacity level.
Figure 5: Firm C’s product growth patterns
For Hypothesis 1 we reject the Null Hypothesis for products FPA-1, FPA-2, FPB-1, FPB2, indicating the non-linear logistic growth curve exhibits the best fit. We cannot reject the Null
Hypothesis for products FPC-1 and FPC-2. Although FPC-1 exhibits S-Curve properties the
linear regression model provides a slightly better fit. Clearly for FPC-2 the linear regression
model provides the best fit.
For Hypothesis 2 we reject the Null Hypothesis for FPA-1, FPA-2, FPB-1, FPB-2 and
FPC-1. That is, the population increases and reaches a plateau, exhibiting the logistic curve. We
can not reject the Null Hypothesis for FPC-2. While linear regression produces a slightly better
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fit, FPC-1 exhibits S-Curve characteristics and logistic regression fits the data within acceptable
parameters.
For Hypothesis 3 we reject the Null Hypothesis for FPA-1, FPA-2, FPB-1, and FPB-2.
That is, the growth rate for each product is not the same.
Table 2. Product growth rates
For Hypothesis 4 we do not reject the Null Hypothesis. That is, we determine that the
carrying capacity for FPA-1 and FPC-1 is the same at the .05 significance level. The estimated
carrying capacity for FPA-1 is 191, 928,000 units and for FPC-1 203,209,000 units. Further, we
predict at the 95% confidence level for FPA-1 that the carrying capacity ranges between
183,935,000 to 199,920,000 units and FPC-1 the carrying capacity ranges between 188,769,000
to 217,649,000 units. The R2 for the analysis for each product is .99 and .98 respectively.
Analysis reveals that the environment (market) supports 2 products at the carrying
capacity plateau. A 2nd tier carrying capacity is evident. The 3rd product in each tier attains a
lower level of unit sales. The two product groupings are similar in that the two top products in
each group achieve similar levels while the 3rd product in each group is well below the carrying
capacity.
Analyzing the top 3 products indicate that the tops are not statistically similar. However,
comparing products FPA-1 and FPC-2 indicate the carrying capacity is statistically similar. The
interesting note is that FPC-1, a first mover in the HTiB Category, reaches the carrying capacity
in month 39 while FPA-1, a late mover in the category, reaches the carrying capacity in 19
months. The significant differences in the time taken to reach the carrying capacity is worthy of
note for product planners.
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First understanding the carrying capacity is significant in order to plan for unit sales.
Second, understanding product growth rates will impact the supply chain order process. Sourcing
raw materials, production scheduling and shipping are all impacted. For product brand managers
understanding the carrying capacity of a product brand and its corresponding growth rate
determines the timeframe available for maximizing the current product’s sales while planning for
new product introduction.
LIMTATIONS
The study included six products from the top three firm brands in the Home Theater-in-aBox category over a 37 month period. Data from a broader time-series that includes more
products and firm brands competing within the category will provide further robustness. Data
confidentially limited the breadth and depth of this study.
Study of first movers versus late movers within the product category warrants further
research as does the impact of product pricing on growth rate, carrying capacity and overall
growth patterns. Understanding the reasons home theater market penetration rates are lower than
expected is an area warranting further study.
CONCLUSION
Examining product growth patterns of home audio products produced by international
consumer electronic firms using a biological species population model indicates that the category
carrying capacity is 203.2 million units. Logistic regression provides a best fit that the products
exhibit a logistic growth pattern. Analysis indicates that product growth rates are unique to each
product.
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Figures
Figure 1: Possible Outcomes
Product Growth Patterns
Units in '000
225000
200000
175000
FP A-1
FP A-2
FP B-1
FP B-2
FP C-1
FP C-2
150000
125000
100000
75000
50000
25000
0
0
10
20
30
40
Time (Monthly)
Figure 2: Possible Outcomes
Firm A Products 1&2
200000
FP A-1
FP A-2
Units in '000
175000
150000
125000
100000
75000
50000
25000
0
18 20 22 24 26 28 30 32 34 36 38 40
Time (Monthly)
Figure 3: Firm A’s product growth pattern
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Figures
Firm B Products 1&2
200000
FP B-1
FP B-2
Legend
Units in '000
175000
150000
125000
100000
75000
50000
25000
0
18
23
28
33
38
Time (Monthly)
Figure 4: Firm B’s product growth patterns
Firm C Products 1&2
200000
FP C-1
FP C-2
Units in '000
175000
150000
125000
100000
75000
50000
25000
0
0
10
20
30
40
Time (Monthly)
Figure 5: Firm C’s product growth patterns
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Tables
Product
Model
Exponential
Linear
Logistic
FPA-1
FPA-2
FPB-1
FPB-2
FPC-1
FPC-2
0.5077
0.9344
0.9912
0.5590
0.9086
0.9950
0.7560
0.9550
0.9934
0.6931
0.9633
0.9950
0.8017
0.9868
0.9810
0.8581
0.9956
0.9916
Table 1: Best fit R2 by model type
Product
FPA-1
FPA-2
FPB-1
FPB-2
FPC-1
FPC-2
Growth Rate 0.4570
0.6679
0.3871
0.4259
0.1292
0.1335
Table 2: Product growth rates
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Appendix A
Logistic Growth Patterns by Firm
Firm A Products 1&2
200000
FP A-1
FP A-2
Units in '000
175000
150000
125000
100000
75000
50000
25000
0
18 20 22 24 26 28 30 32 34 36 38 40
Time (Monthly)
Firm B Products 1&2
200000
FP B-1
FP B-2
Legend
Units in '000
175000
150000
125000
100000
75000
50000
25000
0
18
23
28
33
38
Time (Monthly)
Firm C Products 1&2
200000
FP C-1
FP C-2
Units in '000
175000
150000
125000
100000
75000
50000
25000
0
0
10
20
30
40
Time (Monthly)
Growth Patterns Top3 three international firms with the top 6 Home Theater In-A-Box products
(2 from each firm)
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Appendix B
Logistic Growth Patterns by Product Tier
Tier 1
Product 1
200000
Firm A
Firm B
Firm C
Units in '000
175000
150000
125000
100000
75000
50000
25000
0
0
10
20
30
40
Time (Monthly)
Tier 2
Product 2
Units in '000
100000
FP A-2
FP B-2
FP C-2
75000
50000
25000
0
0
10
20
30
40
Time (Monthly)
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Appendix C
Linear and Logistic Regression Best Fit
FPA-1 Linear Fit
FPA-1 Logistic Fit
.90
277
207
138
103
73
.
185
63
690
345
172
90
Units in '000
Units in '000
172
207
0
1.9
0
9.9
93.9
0
47.9
0
138
27
0
7.9
.
731
90
.
185
90
63
103
690
0
9.9
93.9
47
345
0
1.9 0.1
3.6
7.0
10.4
13.9
17.3
0
.90
0
1.9 0.1
20.8
3.6
Time (Monthly)
.
408
39
689
517
344
80
74.0
861
.20
0
69.6
0
34
172
40
103
0
04.4
689
517
344
.80
172
0
0.0 0.1
3.6
7.0
10.4
13.9
17.3
74.0
0
39.2
0
04.4
0
69.6
0
34.8
0
0
0.0 0.1
20.8
3.6
125
.
775
00
.
420
00
0
10.0
0
55.0
313
0
627
13
188
156
65.0
940
125
7.0
10.4
13.9
Time (Monthly)
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13.9
17.3
20.8
17.3
20.8
0
0.0
.
420
940
313
3.6
10.4
.00
775
627
0
0.0 0.1
7.0
FPB-1 Logistic
Units in '000
Units in '000
156
00
20.8
Time (Monthly)
FPB-1 Linear Fit
.
130
17.3
0
8.8
Time (Monthly)
188
13.9
FPA-2 Logistic Fit
Units in '000
Units in '000
861
10.4
Time (Monthly)
FPA-2 Linear Fit
103
7.0
00
65.0
0
10.0
0
55.0
0
0
0.0 0.1
3.6
7.0
10.4
13.9
17.3
20.8
Time (Monthly)
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Appendix C (continued)
Linear and Logistic Regression Best Fit
FPB-2 Linear Fit
Units in '000
794
94.5
95
635
476
317
158
.40
93
953
0
794
.60
96.7
0
97.8
0
98.9
0
Units in '000
93
953
FPB-2 Logistic Fit
635
476
317
94.5
0
95.6
0
96.7
0
97.8
0
98
158
0
0.0 0.0
3.5
7.0
10.4
13.9
17.4
.40
.90
0
0.0 0.0
20.9
3.5
Time (Monthly)
86
140
105
0
2.4
.
067
704
352
27
211
50
176
0
2.6
.
657
70
52.8
0
47.9
0
86
140
13.6
20.4
105
27.1
33.9
.
657
52.8
0
.90
00
43. 0.1
40.6
6.8
421
05
316
210
58
105
0
631
526
0
.10
81.8
0
0
51.7
0
316
6.8
13.6
20.4
27.1
27.1
33.9
40.6
.40
05.1
0
81.8
0
58
105
Time (Monthly)
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75.0
28
421
210
.50
20
35. 0.1
20.4
FPC-2 Logistic Fit
.70
28.4
13.6
Time (Monthly)
Units in '000
Units in '000
51
526
40.6
70
FPC-2 Linear Fit
75.0
33.9
50
Time (Monthly)
631
20.9
0
2.6
47
352
6.8
17.4
0
2.4
.
067
704
00
43. 0.1
13.9
FPC-1 Logistic Fit
Units in '000
Units in '000
176
10.4
Time (Monthly)
FPC-1 Linear Fit
27
211
7.0
.50
20
35. 0.1
6.8
13.6
20.4
27.1
33.9
40.6
Time (Monthly)
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Appendix D
Determination of Carrying Capacity for Home Theater-in-a-Box
Comparison of Fits FPA-1 and FPC-1
Null hypothesis
TOP same for all data sets
Alternative hypothesis
TOP different for each data set
P value
0.1521 0.1521
Conclusion (alpha = 0.05)
Do not reject null hypothesis
Preferred model
TOP same for all data sets
F (DFn, DFd)
2.115 (1,50)
Best-fit values
FPA-1
BOTTOM
0
TOP
191928
V50
25.33
SLOPE
2.188
Std. Error
TOP
3770
V50
0.185
SLOPE
0.1534
95% Confidence Intervals
183935 to
TOP
199920
V50
24.93 to 25.72
SLOPE
1.863 to 2.513
Goodness of Fit
Degrees of Freedom
16
R²
0.9912
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FPC-1
0
203209
17.18
7.742
7101
0.7638
0.5338
188769 to 217649
15.63 to 18.73
6.656 to 8.827
34
0.981
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