Table of Contents
A.B. Blinov, A.I. Koblov, V.I. Shiryaev
Identification of Carrying Capacity of the Market
And Synthesis of a Cellular Communication C ompany Price Strategy
Abstract
The purpose of the work is to develop the optimum price strategy for the
cdllular communications services company. This is achieved by establishing a new
Piice structure based on the identification of the capacity of the market segment for
the sale of cellular commumication services.
Keywords: system dynamics, growth modd, price strategy, management,
carrying capacity, market, communication services, cumulative sales
Introduction
The task of analyzing existing experimental data and building growth models to
illustrate the cumulative growth of sales volumes of goods and services, has wide
ranging implications to be considered in various areas - fiom the spread of
epidemics to the detail of the procedures used during the compleie life-cycle of the
sale of a product [1,4].
The task of defining the canying capacity of the researched market segment
and synthesis the optimum price strategy for a business fimctioning in a
competitive market during a given period of time, is a mechanism which plays a
Significant role in the information gethered together to make management
decisions [2,3].
The first part of the article describes a model built on the basis of a dynamic
system, relating to the services adoption process of the cdlular commumicetions
company on the competitive market. The model includes information about the
popularity of the services of the celular communications company. Such
popularity provides the growth of the subscriber bese in the initial stages, when the
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number of subscribers is much less than potential. Advertising, dumping etc. can
create such popularity. In the second part of the article, the method of constructing
an growth model with a first-order differential equation is described. The approach
taken is suitable for describing processes of sales where goods are distributed
rapidly during the initial stages (when the number of people who have taken an
interest in the new services and products on offer is small compared with the actual
carrying capacity of the market sector). The result of the first-order differential
equation will be incorrect if it is used to describe the adoption process of services
onto the market with low initial speed of distribution. The third part of the article is
devoted to describing a way of applying a second-order differential equation
resulting in the same type of cuves but taking into account different extemal
circumstances. In both cases (first-order and second-order) methods are described
and illustrated for identifying the capacity of the market segment at which the
services are aimed. The task of establishing the optimal pricing strategy for the
company is also formulated, thus providing a means for maximum income when
1. Model of distribution of cellular communications operator services on
the competitive market
Fig. 1. shows a model of a dynamic system illustrating the distibution of
cellular communication company services on the competitive market. It is based on
John D. Sterman’s approach [4]. The total adoption rate is the sum of adoptions
from “word of mouth’ ae formulated exactly as in the logistic innovation
diffusion model or Bass diffusion model [4]. The probability that a potential users
will adopt as the result of exposure to a given amount of advettising and the
volume of advertising and other extemal influences each period are constant.
When growth process begin, positive feedbacks depending on the installed base
are absent because there are no or only a few users of the company. Initial growth
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is driven by other feedbacks outside the boundary of the simple logistic models.
Rate of penetration
. of the Company .
Potential into theIwlarket -| Subscribers to
Subscribers = the Company
a ) Population
Company <Z of region
—
Word of Mouth «_, Frequency
efticleacy ame contacts
same network”
of oneenes
Efficiency
of contacts
“Cheaper calls
within the
Figure 1. Model of adoption of services of a company into the market
Existing subscribes of the company, having found out about satisfactory
services available, spread the information among unaware people As a result of
such contact there is "word of mouth" and thee is a probability that new
subscriptions will be made. This probability (the efficiency of the distribution of
“word. of mouth”) depends on the important principle of estimating appropriate
costs for the services of the cellular communication operator. For example, the cost
of calls made within the network of the same operator is cheaper than those calls
made to destinations outside of the network (i.e. for calls between relatives, friends
and immediate family it is cheaper to use one communication company).
An important factor influencing the rate at which customers subscribe, is the
informing of potential subscribers of the services available For this purpose,
2. First dynamic growth model
Tt has already been stated that there are two possible scenarios in the realization
of sales, one with a high initial rate of sales, and one with a low initial rate of sales.
From teal data available on sales volumes of services belonging to various price
categories, the cumulative total sales volume for a length of time on the
competitive market (Fig. 1) follows [4] a sales process that can be described by the
first-order differential equation:
TE x=, a, EL, a)
where x(t) - cumulative sales of the services o, - estimation of the potential
market carrying capacity for the services, taking into account issues specific to the
socio-economic factors of the region T - time constant, which depend on
fractional growth rate; L, - interval estimate of the market carrying capacity.
Fig. 2 shows real data curves for sales volumes within various price categories
with a high initial rate of distribution. These can be approximated to a high degree
of accuracy by the result of a differential equation (1).
Sales Volume
WA =a
a8 |
L-—%
‘+ == Product 2 Volume of Sales
so LA eee proauers,approximetion
ia ——— Product 2 Approximation
P
° + —+—_} 4} 44+ ¥ 4
o 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Time
Figure 2. Increasing sales volumes
Suppose that we have point evaluation of parameter a, . In this case x, =x, and
appropriate cumulative sales curve on time interval [0,7] presented on figure 3. In
other cases (x, >x, or x, <x, ) firm will have the smaller received proceeds in an
outcome of non- optimum planning of price policy.
Price strategy of the company [2,3] solved from the following optimization
poblen:
F(a.) = px, tat» mag, Q)
where F(o,) - income from the sale of goods and services by the company; s(a,) -
function of the cost of the goods, based on the attitude of a certain target audience
towards the goods and the affordability of the goods by a given a segment of the
population.
Figure 3. Estimated capacity of the market sector
If the value of «,, set as a citerion for the result of condition (2) is estimated at
its extreme (asymptote) in relation to the capacity of the market segment, a,, the
company’s income is maximized. The function of s(o,) depends on the leva of
competition within the researched market segment and the target consumers;
whether or not the goods are essential (higher or lower purchase priority), social
and economic indices of the region, advertising etc. When the change of parameter
Overtime is omitted from (2) we get:
F(a,) = +1fb—) - > mex. (6))
The use of asymptotic values paroduces the type of function shown in Fig. 4and
hence the function of cost sia,). Therefore, the result of the one-dimensional
optimisation (3) task, chosen specifically for the price bracket on the market to
which the goods and services are to be directed, will allow for the company to
receive maximum income from sales activities.
sp), F@s)
Figure 4. Relationship between price and potential capacity of the market segment
3. Second dynamic growth model
In practice there are various possibilities affecting the growth of volume of
communication services sales with differing initial rates of distibution of the
product on the market. The first model of cumulative sales volumes (Fig. 2.) is not
applicable to the S-shaped curve (Fig. 7) of total volume of sales. In this case we
should use a second-order differential equation:
2 x
T=
dv
where T,€ - constants, relating to the anve of the cumulative increase in total
volume of sales of services (Fig.5, 6). Unlike the first-order differential equation
(1), equation (4) offers more possibilities to look to available expaimental data
sa Bax =a, » GpEL,€— >1, (4)
and thus increase the accuracy of the approximations. This results in a more exact
determination of the course and capacity of the market for the goods.
100
Figure 5. Influence of parameter T on curve of function x(t)
Figure 6. Influence of parameter € on curve of function x{t)
100
90 |-
80h
70}-
60}
BD p-------b--fo-- fo fot oop
40h
sob
20h
10}
10}------- 6-3.
Fig. 7 illustrates the approximation of the real company statistical data for
cumulative volume of services sales at a low initial rate of distribution (i.e when
the number of people who have purchased the goods is low compared with the
capecity of the market). This is the result of the differential equation (4) with zero
values entered for the derivative and function x(t).
Such approximation of experimental data, assuming an S-shaped curve of
distribution of services to potential consumers [4], as well as the content of Fig. 2,
allows us to estimate the capacity, a, of the market sagment at which the goods
are directed. This method also enables us to forecast the demand for the goods
despite the uncertainty of some factors. As new sales take place, the sales data will
correlate to the market research more and more.
300000
250000
200000
2 150000
100000
— - —- Second-order Approxima tion
50000 |_|
VA qq xperime
o
Figure 7. Cumulative sales volumes of the goods on the market
Time
As in the case where the sale of goods is described by a first-order differential
equation, for S-shaped curve distribution of an innovation on the market it is also
analytically possible to utilise function (2)
F(o,,) =0.ps(0t, KT; ~C, —Cy +C, expl\T,) +C, exp(A,T,)) > max. ©)
8
It should be noted that the problem of one-dimensional analytical optimisation
(5) can only be written down and solved under the condition that the analytical
function, s{c,) is the cost of the goods based upon the target audience at which
sales are focussed. A function of type s(a,) can change during the sale of goods
along with changes to the social and economic indices of the region, resulting from
changes to the level of competition from other companies selling the same product,
consumer status and the influence of other factors.
If the price of the product is determined beforehand, or for some reason a
specific value of s(a,) is assumed (Fig.4), this will enable an estimate to be made
of the size of the prospective market, but the result of optimisation exercise (3) or
(5) will be insignificant. If the company is at the stage of creating a price strategy
for the presentation of an innovation on the market, then the function s(a,) is the
key to establishing the optimum pricing policy for the company, which for the
product in question, will result in maximum income from sales.
All data in the paper besed on real volumes of cellular services sales from one
of the Southem- Ural communication company of Russia.
Conclusion
This work has presented a model of a dynamic system, describing the process
of distributing a cellular network operator's services in a competitive market. Two
modes of cumulative volumes of sales of communication services are considered
by way of first-order and second-order differential equations. A method of shaping
an optimum pricing strategy for companies is described, as is the process of
identifying the size of the market Numerical results of the modeling ae
established and comparison is made with real experimental data on the volumes of
sales of communication services in a competitive market.
References
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Golovin Ya, Shiryaev V.I. 2001. Optimal Management of a Firm under
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Shiryaev V.L, Shiryaev E-V., Golovin Ya, Smolin V.V. 2002. Adaptation.
and Optimal Contol of Firm and its State and Parameters Estimation at
Change of a Market Situation. Proceedings of The 20th Intemational
Conference of The System Dynamics Society: 140-141.
Sterman, John D. 2000. Business Dynamics: System thinking and modeling
for complex world. McGraw-Hill Higher Education Co.
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