Modeling and Maximizing Customer Equity

Electronic copy available at: http://ssrn.com/abstract=2052662

MODELING AND MAXIMIZING CUSTOMER EQUITY

Keywords: customer equity; customer acquisition; customer retention; customer expansion

Jose Carlos Fioriolli

Professor of Marketing

Management School

Federal University of Rio Grande do Sul

Washington Luiz St., 855 – 90010-460, Porto Alegre/RS, BRAZIL

Ph: (55) 51 3308-3536, Fax (55) 51 3308-3991

E-mail: [email protected]

Working Paper - May 2012


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MODELING AND MAXIMIZING CUSTOMER EQUITY

ABSTRACT

One of the most efficient criteria for achieving an optimal balance of resources among

customer acquisition, retention and expansion efforts is the maximum customer equity. In this

article the author proposes a new solution for the customer equity maximization problem. The

proposed solution takes into account two elements not considered in the majority of customer

equity models in the literature: (i) the customer expansion concept and (ii) acquisition,

retention and expansion floor rates. The article presents a double-phase model that employs a

variation of the Lagrange multiplier method to solve this balancing challenge. Its main

contribution consists in providing a more comprehensive framework for optimizing the

balance among customer acquisition, retention and expansion investments. The new approach

contributes also for improving customer relationship management (CRM) decisions.

Additionally, it represents appropriately the complexity of relationships among the customer

equity core variables.


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INTRODUCTION

Marketing theory and practice are changing as a result of the worldwide economy’s

new profile. The impacts generated by market transformations, where services constitute the

predominant activity in the majority of national economies, have required firms to place more

emphasis on activities that build sustainable competitive advantages. In most cases these

efforts materialize in form of marketing investments, carried out in conditions of extreme

uncertainty. This fact has contributed to de-characterizing the cause-effect relations existing

between these investments and the respective returns obtained by the firms, making it

difficult or even impossible to identify and quantify those relations.

In an effort to minimize these difficulties, several studies have been carried out with

the objective of identifying and quantifying the variables that determine the nature and

intensity of the relationship between marketing investments and firm profitability. In recent

years, a variety of work has been done towards this end, using an approach centered on

customer equity (Blattberg and Deighton 1996; Berger and Nasr 1998; Gupta, Lehmann, and

Stuart 2004; Libai, Narayandas, and Humby 2002; Rust, Lemon, and Zeithaml 2004; Seggie,

Cavusgil, and Phelan 2007; Villanueva, Yoo, and Hanssens 2008; Petersen et al. 2009).

For the purposes of this paper, aligned with Blattberg and Deighton (1996) and Berger

and Bechwati (2001), the author defines customer equity (CE) as the present value of the

expected cash flows from the customer to the firm over time. The definition adopted by

Berger et al. (2002), though succinct, succeeds in objectively transmitting the same idea: CE

is the value that customers generate for the firm over the course of life. The basic premise of

CE is that customers are a financial asset that firms should individualize, measure, manage

and maximize just like any other asset it possesses (Blattberg, Getz, and Thomas 2001).

Starting from models that quantify the direct effects produced by marketing actions on

CE, the return on a firm's marketing investments can be estimated on the long term. A model

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is a representation of the most important elements perceived in a real world system. Good

models show more than just the nature of the relationships existing among the variables, but

also the magnitude of their effects (Leeflang et al. 2000).

The increased complexity of a marketing environment can make it worthwhile to

analyze the data using quantitative models instead of the usual treatment in the form of tables

or statistical techniques (Franses and Paap 2001). A quantitative model serves three main

purposes: (i) improve the description of the phenomenon in focus; (ii) increase forecasting

precision and; (iii) support the decision making process. The improvement of the description

generally refers to the investigatory process regarding which explanatory variables have a

statistically significant impact on the dependent variables. Once these variables and their

respective impacts have been identified, the forecasting (elaborated based on the explanatory

variables identified as relevant during the modeling process) tends to be more precise, and

can thereby contribute more productively to increasing confidence in and quality of the

decision making process (Franses and Paap 2001, p. 12).

In this sense, a proposal for a quantitative model that makes it possible to identify in

which conditions the CE can reach its maximum value (providing simple, fast and precise

solutions as to how to optimally balance customer acquisition, retention and expansion

efforts) constitutes an important contribution to understanding and applying the concept of

CE, in both academic and business contexts.

In addition to this introduction, the article contains seven more sections. The next

section contains a review of the main topics related to CE and to balancing customer

acquisition, retention and expansion investments. The third section presents the first part of

the mathematical development of the new CE maximization model (specific to environments

without budget constraints). In the same section, the initial model is validated using the

results of its application without constraints and the respective results encountered via the

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non-linear Generalized Reduced Gradient (GRG) optimization algorithm. The fourth section

presents the second part of the CE maximization model’s development (general model, valid

for all environments, regardless of the degree of budgetary constraints). In the fifth section, it

is applied a numerical example of the proposed model in its final form. The sixth section

analyzes the maximum CE’s sensitivity in the face of possible variations in budget

availability. In the seventh section, the managerial implications of the new model are

presented and at the end, the last section shows this work’s main limitations, conclusions and

several suggestions for future research in the field.

BACKGROUND

This section presents the main drivers that define the scope of CE, as well as the

importance of this for achieving alignment between strategic decisions and the operational

activities developed by firms, and the main research related to integrating CE into the process

of identifying answers to the problem of optimal balance among customer acquisition,

retention and expansion efforts.

Scope and Importance of Customer Equity

Different metrics can be used to support the development of entrepreneurial

guidelines more strongly geared towards the market. According to Ambler (2003), a majority

of firms develop their approach to marketing performance evaluations, especially regarding

their benefits, in five stages: (i) the firm does not place importance on this activity; in this

stage, marketing is not seen formally, as something that requires executives’ special attention;

(ii) the evaluation is seen on financial terms and prioritizes analyses of earnings and losses

and cash flow; (iii) the exclusively financial measures are recognized as inadequate and many

non-financial measures are adopted, generating confusion in these firms; (iv) the firm places

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its focus on the market and combines financial and non-financial metrics, although without

being certain they are using the appropriate metrics; and (v) a scientific method for evaluation

is adopted, allowing the databases and metrics to be analyzed mathematically in such a way

as to generate a reduced list of metrics with a high explanatory and predictive capacity. Firms

which systematically employ the concept of CE in their decision making processes can be

considered to be fifth stage organizations.

In these conditions, CE can be used in ways that present advantages in comparison

with other metrics, providing criteria that aid managers in identifying the main motives

associated with a customer’s decision to buy a product or not. Customers decide to buy from

a firm based on three fundamental factors: (i) the firm’s product is a better value than the

competitor’s; (ii) the firm has the better brand or (iii) the cost of change is very high (Lemon,

Rust, and Zeithaml 2001).

In order to facilitate understanding regarding the scope of CE, it is important to

identify and describe its primary drivers. According to Rust, Zeithaml, and Lemon (2000),

there are three dominant drivers that define CE:

(i) value equity, which corresponds to objectively evaluating a brand's utility,

developed by the customer starting from their perception of what he/she gives in

exchange for what he/she receives (Rust, Zeithaml, and Lemon 2000);

(ii) brand equity, which corresponds to the subjective evaluation developed by the

customer regarding the firm and the goods and services it provides, influenced as

much by the strategies adopted by the firm as by the customer's experiences with

respect to the firm providing them, including the history of brand associations and

memory (Yoo and Hanssens 2005; Rust, Zeithaml, and Lemon 2000); and

(iii) relationship equity, which corresponds to customer’s tendency to stay with a

determined brand, without regard to objective and subjective evaluations about it,

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due to influence from recognition, valorization and socialization actions promoted

by the firm in possession of the brand (Bolton 1998; Rust, Zeithaml, and Lemon

2000).

CE is a concept that is still unknown to many firms and has been studied very little in

the academy. Villanueva and Hanssens (2007) presented a survey of the number of studies

about brand value and about CE conducted in recent years. While studies on brand value

correspond to approximately 50% of the published studies, proposals to quantify CE

correspond to only 5% of the total production presented. This is due to the fact that the

subject is relatively new to the literature.

Gradually recognizing the importance of the customer-centered approach to firm’s

decision making processes, researchers have studied CE and its application in a variety of

situations that require marketing decisions such a pricing strategies, media selection,

customer attraction programs and methods of determining optimal media budgets. Other

authors, like Slater, Mohr, and Sengupta (2009), have adopted a line of complementary

research, seeking to identify which activities are most critical to CE maximization. In their

opinion, identifying the customers with the greatest return potential, development of

customer acquisition strategies and development of integrated customer portfolio

management strategies can be considered prerequisites for CE to reach its maximum values.

A series of articles can be found in the literature that analyzes the alignment between

strategic decisions and operational activities, directly or indirectly using the concept of CE.

Kumar, Lemon, and Parasuraman (2006, p. 89) stated that it is not enough to come up with a

solid customer management strategy, firms need to develop models and metrics that make it

possible to verify if the strategy is effective. Gupta and Lehmann (2003) have suggested that

customers, as assets that are critical to a firms' success, should be constantly measured and

managed. Hanssens, Thorpe, and Finkbeiner (2008) defend the idea that the connections

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between strategic objectives (such as CE maximization) and tactical objectives (such as direct

mailing) don't need to be merely logical: they can be quantified. In accordance with Hogan,

Lemon, and Rust (2002), CE is more than just an indicator; it is an integrated marketing

approach that can contribute to the creation of successful strategies. Bell et al. (2002) are

more emphatic: they state that managers need to implement marketing initiatives that

maximize customer value.

Since the publication of the first articles on quantitative CE models, interest in the

study of its maximization has been growing. Bell et al. (2002) have identified a series of

barriers (characterized as challenges) that can make the full implementation of the CE

concept by academic and entrepreneurial environments difficult. Among the challenges

(seven in all) they referred to, there is one that deals with the CE maximization problem.

Complementarily, it is worth pointing out that there are authors that are developing a

broader line of research, trying to establish relationships between CE and a firm's value.

Although this matter is not dealt with in this article, several of these papers are mentioned

below.

Based on the results of a study involving North American and European

organizations, Bayón and Becker (2004) defend the use of CE as a means of predicting the

future performance of a firm’s shares, characterizing it as an important decision making

criterion for allocation of resources. Bauer, Hammerschmidt, and Braehler (2003)

demonstrate that the possibility of combining and incorporating CE concepts and shareholder

value (SHV) into the process of determining a firm's value is due to the fact that CE

constitutes the central part of the processes that generate cash flow. SHV is defined as the

present value of firm’s future cash flows plus the value of its non-operating assets, minus its

future claims. In the same sense, Bauer and Hammerschmidt (2005) use a non-traditional

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approach (separating cash flow down to the individual customer level) to propose a model

that incorporates the value of future customers into the SHV model.

Gupta, Lehmann, and Stuart (2004) have developed an approach that contributes to

estimating the current and future value of a customer base. These authors propose the use of

this estimate as a proxy of a firm's value. Broadening the scope of this approach, Rust,

Lemon, and Zeithaml (2004) propose a model for measuring return on investments that uses

CE as one of its central concepts; this proposal is considered one of the first attempts towards

financially accounting for marketing actions.

In the same manner as the authors previously cited, Pfeifer, Haskins, and Conroy

(2005) also suggest that CE can be used to establish a firm's value, but the warning presented

by Berger et al. (2006) is also worth pointing out, in the sense that many empirical

replications are necessary before CE can be widely adopted by corporations aiming at

defining their market value.

Balancing Customer Acquisition, Retention and Expansion Investments

A crucial question associated with the dissemination and effective implementation of

the concept of CE is related to the process complexity of identifying solutions to the problem

of optimal balancing customer acquisition, retention and expansion efforts. The main

approaches developed to this end and the results obtained by their respective authors are

presented as follows.

Blattberg and Deighton (1996) propose the use of CE as a central criterion for

determining the optimal balance between acquisition and retention resources (this optimal

point corresponds to the situation in which CE achieves its maximum value). In their model,

the authors use decision calculus (DC) to identify the limiting values related to the response

rates, although they do not consider acquisition and retention together. The DC approach

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“consists of a set of numerical procedures for processing data and judgments to assist

managerial decision making” (Little 1970). This treatment, by separating acquisition and

retention, makes the model fragile and restricts its possibilities for application. Still, the

structure of the solution proposed by these authors is quite consistent and can be used as a

starting point for the development of more precise models. Pfeifer and Carraway (2000)

present a general class of mathematical models that can be used to calculate CE. These

models are based on Markov processes, adapted to represent the dynamic of the customer

relationship with the firms as a function of their flexibility. Berger and Bechwati (2001)

develop a general approach to organizing media budget allocation, where the objective

function is to maximize CE. A key element of this research consists in the use of the DC

approach. The authors suggest that instead of evaluating the direct impacts of decisions on

sales or on profits, managers should ask themselves about the effects of their decisions and

consequential actions on the firm's CE. Thomas (2001) suggests that the customer acquisition

process affects the retention process; this approach contributes to reduce the negative impact

of managerial decisions made without taking their interdependence into account.

Calciu and Salerno (2002) present a variety of examples of CE maximization, though

they are all developed through graphic or interactive processes. Comprehensive solutions are

not presented for the examples. Libai, Narayandas, and Humby (2002) propose a stochastic

model which uses the concept of customer profitability on the industry segment level. The

proposal is an alternative to the individual approach, which is difficult to implement. Pfeifer

(2005) demonstrates that defining the optimal rate of customer acquisition and retention

resource allocation depends on the concepts used (average costs or marginal costs). Reinartz,

Thomas, and Kumar (2005) propose a statistical model (probit) for maximizing customer

profitability based on the optimal balance of resources destined to customer acquisition and

retention. The modeling establishes relationships among customer acquisition, customer-firm

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relationship’s duration and profitability. The authors present the results of a series of

simulations, though they do not succeed in verifying if the investments in acquisition are

more critical than the investments in retention, except for the particular case they studied.

Ayache et al. (2007) present an analytic solution for optimal balancing of customer

acquisition and retention efforts starting from the CE maximization problem proposed by

Blattberg and Deighton (1996). The positive aspect consists in using a simple response

function (retention) and is easier to put into practice than the function used by Blattberg and

Deighton (1996). The solution is deficient in terms of structure, in that it does not

contemplate the investments in customer expansion and their respective impact on CE value.

Dong, Swain, and Berger (2007) present a model for broader CE maximization,

starting from Blattberg and Deighton’s (1996) model. They begin with the premise that the

channels vary in quality and that the customer acquisition and retention response curves

(obtained through the respective functions) can take on different shapes. Although it presents

the decision making problem more realistically, the solution presented is dependent on an

interactive approach (an objective function should be maximized using the Operational

Research tools). Additionally, they do not consider investments in customer expansion and

the respective response rate. Kumar and George (2007) analyze different approaches to

quantifying CE and propose a hybrid model that is able to integrate different levels of

aggregation used in the approaches studied. They deal with CE maximization but do not

present any analytic solution to this problem. Bruhn, Georgi, and Hadwich (2008) focus on

the question of CE maximization; although they present a proposal for identification and

quantification of activities that contribute to this maximization, they don’t formally establish

the level of ideal effort for reaching maximum CE. Calciu (2008) proposes a mathematical

development focused on optimizing customer acquisition and retention costs. The generic

formulas proposed are derivatives of a simplified retention response function, with a structure

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that makes it possible to identify an analytic solution for the problem. However, that paper

does not contain formal elements related to expansion efforts.

Upon analysis of these studies (approaches to balancing efforts and CE

maximization), it can be affirmed that only Blattberg and Deighton (1996), Ayache et al.

(2007), and Calciu (2008) present proposals oriented towards identifying analytic solutions

for the problem of optimal customer acquisition and retention resource balancing. The results

of the research conducted by these authors constitute a relevant contribution to improving

modeling processes in the field of marketing, especially related to CE maximization.

However, the deficiencies identified in these proposals should not be underestimated:

balancing the resources for customer acquisition and retention obtained in an inadequate

manner, through separate treatment of intervening variables (Blattberg and Deighton 1996);

absence of formal elements related to investments in customer expansion (Blattberg and

Deighton 1996; Ayache et al. 2007; Calciu 2008) and the impossibility of dealing with

situations where investments – while null on the short term – do not necessarily imply null

response rates (Blattberg and Deighton 1996).

CUSTOMER EQUITY MAXIMIZATION MODEL (PART ONE)

Similarly to the paper presented by Berger and Bechwati (2001), this article presents

elements suggesting that managers can benefit from adopting a customer-centered marketing

management. In this sense, the objective of maximizing CE is not merely recommendable, it

is necessary to a firm’s success.

This article’s main contribution consists in developing a more comprehensive model

that makes it possible to maximize CE and identify optimal balance for resources destined to

customer acquisition, retention and expansion. Complementarily, the proposed model opens

new perspectives towards the study of the impact that these variables (acquisition, retention

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and expansion) have on CE, especially when it is nearing its maximum value. Usually,

optimization problems like this one are dealt with via Operational Research, in which an

objective function associated with CE is maximized according to the constraints established

by the media budget, imposed by the firm-market environment.

Structure of the Initial Model

This article proposes a new approach for the customer equity measurement problem;

this simplifies the process of identifying maximum CE (and the conditions in which this

value occurs), as well as reducing the time it takes to obtain a response and makes it feasible

to analyze with a sensitivity that has been, until now, practically impossible (high complexity

and excessive processing time, resulting from the innumerous possible combinations, which

is characteristic of problems dealt with via Operational Research).

The structural CE model used in this research is presented in Figure 1. It consists of

an adaptation of Blattberg, Getz, and Thomas (2001, p. 11) by incorporating the broader

concept of customer expansion, as proposed by Gupta et al. (2006, p. 140). With this

adaptation, the customer expansion concept has the same status as the concepts of acquisition

and retention.

< Insert Figure 1 here >

Customer acquisition, from a transactional perspective, includes all the activities that

culminate in the first purchase (Blattberg, Getz, and Thomas 2001). Retention corresponds to

customers' continuing to engage in the process of purchasing products from the firm for a

determined period. Efforts towards retention are accounted for beginning after the first

purchase and continuing over the course of time the relationship persists. Complementarily,

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customer expansion is an activity associated with the sales of any additional products (related

to each other or not) to current customers; in this way, they are efforts directed at retained

customers and constitute what can be called the third movement of CE management, with the

first two being acquisition and retention, respectively (it is common for expansion operation

to be done predominantly, through add-on selling efforts).

In order to calculate CE, the point of departure used is the procedure employed by

Blattberg and Deighton (1996), introducing the following modifications:

i) incorporation of the customer expansion concept;

ii) alteration of the response function structure (acquisition, retention and expansion)

considering the possibility of using floor rates, in such a way as to facilitate the

treatment of situations where investments, while null on the short term, do not

necessarily imply null response rates.

Similarly to what has been proposed by Blattberg, Getz, and Thomas (2001), the

author adopted the premise that the gross margins, investments and customer acquisition,

retention and expansion rates remain constant over time.

Development of the Initial Model

The mathematical model for the CE proposed in this study expands on the model

presented in Blattberg and Deighton (1996), and comes to consider investments and the

respective customer expansion rate as well. Notations used in the partial and final models

presented in this paper were standardized, as detailed in Table 1 (Entry variables), Table 2

(Intermediate and final variables), Table 4 (CE maximization variables) and Table 8 (CE

maximization variables with budget constraints).

< Insert Table 1 here >

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The amplitude of the customer acquisition, retention and expansion rates, expressed in

Equations (1), (2) and (3), correspond to the differences between the respective ceiling and

floor values, as follows:

[c] [f ]acq acq acqi i i∆ = − (1)

[c] [f ]ret ret reti i i∆ = − (2)

[c] [f ]add add addi i i∆ = − (3)

Customer acquisition, retention and expansion rates are calculated using response

functions similar to those used by Blattberg and Deighton (1996). The alterations made in the

structure of the response functions (acquisition, retention and expansion) make it possible to

use floor rates, in such a way as to make it feasible to deal with situations in which

investment return, while null on the short term, does not necessarily imply null rates.

[c] exp( )acq acq acq acqi = i i k ACQ− ∆ − (4)

[c] exp( )ret ret ret reti = i i k RET− ∆ − (5)

[c] exp( )add add add addi = i i k ADD− ∆ − (6)

CE is calculated in accordance with the structure presented in Table 3.


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A firm’s CE is obtained by (i) accounting for the income from each period and

respective costs; and (ii) applying the discount factor for the period contemplated, according

to what is presented next.

( ) ( )( 1)

1 1(1 ) (1 )ret

t tT Tret

acq acq ret add add ret tt td d

iiFCE = N i M M +i M RET +i ADD ACQ

i i

−

= =

+ − − + +

∑ ∑ (7)

The Equation (7) can be rewritten as follows:

( )( ) ( ){ }acq acq ret add add ret retFCE = N i M + M +i M ADD i RET d i ACQ − − − − (8)

From the definition of the CE ( /CE = FCE N ) it follows that:

( )( ) ( )acq acq ret add add ret retCE = i M + M +i M ADD i RET d i ACQ − − − − (9)

Observe that the CE value depends on the total budget for customer acquisition,

retention and expansion, on the respective rates and gross margins and on the discount rate

used. Table 4 presents the notation used to model maximum CE.


In this optimization process, starting with Equation (9), the first partial derivative of

the CE is calculated with respect to customer expansion investments as follow:

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( )exp( ) 1acq ret add add add add

ret

i i i k M k ADDCE=

ADD d i

∆ − −∂

∂ −

(10)

The value of ADD that maximizes CE is obtained by making this derivative equal to

zero, as follows.

[max ] = ( )CE add add add addADD ln i k M k∆ (11)

Substituting in Equation (6), the customer expansion rate that maximizes CE is

calculated.

[max ] [c] 1/( )add CE add add addi = i k M− (12)

Incorporating these values [Equations (11) and (12)] into Equation (9) and calculating

the first partial derivative of the CE with respect to the investment in customer retention:

( ) ( )

( )

[max ] [c]

2

[c]

( ) 1acq ret CE ret ret

ret ret

i i U d RET k d iCE=

RET d i i

∆ Ζ − − − −∂ ∂ − + ∆ Ζ

(13)

where [max ] [max ] [max ]CE ret add CE add CEU = M +i M ADD− and exp( )retk RETΖ = − .

Once again, equaling the derivative to zero, the expression which makes it possible to

calculate the RET in the condition of maximum CE is:

( ) ( )[max ] [c]( ) 1ret CE ret ret retRET ln i U d RET k d i k = ∆ − − − (13a)

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The RET value can be found by applying (separately or together) the following

solution methods:

i) Develop an interactive process, starting with RET = 0 and completing the study

when the RET value converges to [max ]CERET .

ii) Calculate [ ][max ] 1 2 1( )CE retRET = ln V V ln V k−

where 1 [max ] [c]( 1) ( )ret CE ret retV = i U dk d i∆ − − and

( ) ( ) ( )2 [c] [max ] [max ]2 1ret ret CE ret CE retV = i d i U dk U dk ∆ − − −

The solution (i) refers to an Operational Research problem. In these cases, the result

cannot be expressed analytically, since it depends on interactive research, having CE

maximization as its objective function. The second solution (ii), with a closed structure, will

be used in this study because of its analytical nature. This function constitutes an original

contribution to CE maximization modeling; it was obtained by means of a synthetic proof

(using computational simulation). Thus, the expression proposed for determining the

investment that maximizes CE with respect to customer retention is:

[ ][max ] 1 2 1( )CE retRET = ln V V ln V k− (14)

The customer retention rate (in the CE maximization condition) is obtained by

combining Equations (5) and (14).

[ ][max ] [c] 1 2 1( )ret CE ret reti = i i V V ln V− ∆ − (15)

Incorporating Equations (11), (12), (14) and (15) into Equation (9) and deriving

partially with respect to the investment in customer acquisition, the final expression is:

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[max ]

*exp( ) 1acq CEacq acq acq

CE= i k k ACQ M

ACQ

∂∆ − −

∂ (16)

where ( ) ( )[max ]

*

[max ] [max ] [max ] [max ]acq CE acq CE ret CE CE ret CEM = M + U i RET d i− − corresponds to

the new margin associated with customer acquisition and adjusted by way of incorporation of

the values calculated in the Equations (11), (12), (14) and (15).

Equaling this derivative to zero, the value of ACQ that maximizes CE and the

respective customer acquisition rate are:

[max ]

*

[max ] ( )acq CECE acq acq acqACQ ln i k M k= ∆ (17)

[max ]

*

[max ] [c] 1/( )acq CEacq CE acq acqi = i k M− (18)

By substituting the expressions related to Equations (11), (12), (14), (15), (17) and

(18) in Equation (9), the new model for CE maximization (see Table 5) is defined, as follows.

( )( ) ( )[max ] [max ] [max ] [max ] [max ] [max ] [max ]max acq CE acq ret add CE add CE ret CE CE ret CE CECE = i M + M +i M ADD i RET d i ACQ − − − −

(19)

The necessary budget for CE maximization corresponds to the sum of the values

expressed in Equations (11), (14) and (17).

[ ][max ]

*

[max ] 1 2 1( ) ( ) ( )acq CECE acq acq acq ret add add add addB = ln i k M k +ln V V ln V k +ln i k M k∆ − ∆ (20)


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Validation of the Initial Model

In order to validate the CE maximization model, several scenarios were analyzed.

Eight of these scenarios, contemplating several typical marketing situations, are represented

in Table 6.


Table 7 shows the results of CE maximization obtained through the Generalized

Reduced Gradient (GRG, operated in Excel’s Solver function) and the model proposed in

this article.


Note that the performance of the new model is extremely satisfactory (there are no

differences between the methods) for the eight scenarios considered. The results obtained in

the other tests presented the same performance level. In order to detail and obtain greater

comprehension of the GRG algorithm, the author recommends consulting the work of Lasdon

et al. (1978).

CUSTOMER EQUITY MAXIMIZATION MODEL (PART TWO)

Although the model for CE maximization may have presented excellent performance,

the question of balancing budget allocation between customer acquisition, retention and

expansion efforts has yet to be answered. There are cases where the available budget ( [lim]B )

is less than the ideal/necessary budget for optimizing CE ( [max ]CEB ), as defined by Equation

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(20). In these circumstances, might it be possible to establish an optimal balance of these

limited resources by simply distributing them in the same proportion as established for the

ideal budget (according to the model presented in Table 5)? The answer to this question is:

probably no. As the response functions for acquisition, retention and expansion are not linear

(Kumar 2008), the condition of optimality achieved with the ideal budget will most likely not

stay whole. This is where the need arises to develop a complementary model that makes CE

maximization feasible in situations in which the available budget is less than ideal (this

condition [lim] [max ]CEB B< is common in business environments). Table 8 presents the notation

used in this model.

This is a new CE maximization problem, and is subject to a set of budget constraints.

Seeking for an analytic solution, the author uses the Lagrange multiplier method. This

approach is frequently used to solve problems of conditioned optimization (Hughes-Hallett et

al. 1999).

The usage of Lagrange multipliers method involves the calculation of the first partial

derivatives of the CE with respect to investments in customer acquisition, retention and

expansion. These derivatives correspond, respectively, to the following expressions:

*exp( ) 1acqacq acq acq

CE= i k k ACQ M

ACQ

∂∆ − −

∂ (21)

where: ( ) ( )* ( )acq acq ret add add ret retM = M + M +i M ADD i RET d i− − −

( ) ( )

( )

[c]

2

[c]

( ) 1acq ret ret ret

ret ret

i i Ud RET k d iCE=

RET d i i

∆ Ζ − − − −∂ ∂ − + ∆ Ζ

(22)

where: ret add addU = M +i M ADD− and exp( )retk RETΖ = −

22

( )exp( ) 1acq ret add add add add

ret

i i i k M k ADDCE=

ADD d i

∆ − −∂

∂ − (23)


The value of the Lagrange multiplier, λ , is identified by using the first partial

derivatives of the CE and considering B = ACQ+ RET + ADD as follows.

CE B=

ACQ ACQλ

∂ ∂

∂ ∂ (24)

( ) ( )exp( ) ( ) 1

1 exp( )( ) /( )

acq acq acq acq ret add add ret ret

acq acq acq ret ret

i k k ACQ M + M +i M ADD i RET d i=

i k k ACQ RET +i ADD d iλ

∆ − − − − − + ∆ − −

(24a)

CE B=

RET RETλ

∂ ∂

∂ ∂ (25)

( ) ( )[c]exp( ) (( ) ) 1

( ) exp( )

ret ret ret add add ret ret

ret ret ret ret

i k RET M +i M ADD d RET k d i=

d i RET dADD i k k RETλ

∆ − − − − − −

− + + ∆ −

(25a)

CE B=

ADD ADDλ

∂ ∂

∂ ∂ (26)

23

exp( ) 1add add add addi k M k ADDλ = ∆ − − (26a)

Thus, investments in customer acquisition [Equation (27)] and customer expansion

[Equation (28)] can be obtained from Equations (24a) and (26a), respectively. The investment

in customer retention is calculated in a complementary manner, according to the expression

proposed in Equation (29).

( ) ( )( )

1 exp( )( ) /( ) 1

acq acq acq ret add add ret ret

acq

acq acq acq ret ret

i k M + M +i M ADD i RET d iACQ ln k

i k k ACQ RET +i ADD d iλ

∆ − − − = + ∆ − − +

(27)

1add add add

add

i k MADD ln k

λ

∆ = +

(28)

[lim]RET B ACQ ADD= − − (29)

The method used to determine the λ value, as proposed next, constitutes an original

contribution. It consists of four steps:

i) calculation of the initial values [0]ACQ , [0]RET e [0]ADD , using a proportionality

factor, [ ]limFator , equal to the ratio between the available budget (limited) and the

necessary budget (ideal); this factor should be applied to [max ]CEACQ , [max ]CERET

and [max ]CEADD , respectively;

ii) update of the values related to the acquisition, retention and expansion rates,

according to the initial calculated values;

iii) calculation of the initial values of the Lagrange multipliers, [[0]acqλ ,

[0]retλ and

[0]addλ ], according to Equations (24a), (25a) and (26a); and

24

iv) calculation of λ̂ (estimate of the Lagrange multiplier), obtained as a function of

[0]acqλ , [0]retλ and

[0]addλ .

The Equations in the [(30)-(40)] interval formalize the steps of the proposed model.

[ ] [lim] [max ]lim CEFator B B= (30)

[0] [ ] [max ]lim CEACQ Fator ACQ=

(31)

[0] [ ] [max ]lim CERET Fator RET= (32)

[0] [ ] [max ]lim CEADD Fator ADD=

(33)

[0] [c] [0]exp( )acq acq acq acqi = i i k ACQ− ∆ − (34)

[0] [c] [0]exp( )ret ret ret reti = i i k RET− ∆ − (35)

[0] [c] [0]exp( )add add add addi = i i k ADD− ∆ − (36)

Incorporating the initial calculated values [steps (i) and (ii)] into Equations (24a),

(25a) and (26a), the initial values of the Lagrange multipliers are calculated as follows.

( ) ( )[0] [0] [0] [0] [0] [0] [0]exp( ) ( ) 1acq acq acq acq acq ret add add ret ret= i k k ACQ M + M +i M ADD i RET d iλ ∆ − − − − − (37)

( ) ( )

( )

[0] [0] [0] [0] [0] [c]

[0] 2

[c] [0]

exp( ) (( ) ) 1

exp( )

acq ret ret ret add add ret ret

ret

ret ret ret

i i k RET M +i M ADD d RET k d i

d i i k RETλ

∆ − − − − − − =− + ∆ −

(38)

[0] [0]exp( ) 1add add add add addi k M k ADDλ = ∆ − − (39)

25

In order to estimate the value of the multiplier to be used in developing the final

model, the author proposes the use of a simple average [values defined in the Equations (37),

(38) and (39)]. The simple average is a central trend measure that is easy to put into

operation, capable of producing a fast convergence of values in situations like these, in which

the initial values of the Lagrange multipliers [generated in step (iii)] tend to be very close to

each other.

[0] [0] [0]ˆ3

acq ret addλ λ λλ

+ += (40)

Using the λ̂ multiplier [obtained in step (iv)], it is possible to generate a solution for

the system formed by Equations (27), (28) and (29). In this sense, the values [lim]ACQ ,

[lim]RET and [lim]ADD (investments in customer acquisition, retention and expansion subject

to budget constraints) are defined which tend to maximize the firm’s CE, in accordance to the

following mathematical model:

( ) ( )[0] [0] [0] [0] [0]

[lim]

( )

ˆ 1


acq

i k M + M +i M ADD i RET d iACQ = ln k

λ

∆ − − −

+

(41)

[0] [0]

[lim]

[0] [0] [0]ˆ( )

acq ret add add add

add

ret acq ret

i i i k MADD ln k

d i i iλ

∆=

− + (42)

[lim] [lim] [lim] [lim]RET B ACQ ADD= − − (43)

The response rates are calculated as follows:

26

[lim] [c] [lim]exp( )acq acq acq acqi = i i k ACQ− ∆ − (44)

[lim] [c] [lim]exp( )ret ret ret reti = i i k RET− ∆ − (45)

[lim] [c] [lim]exp( )add add add addi = i i k ADD− ∆ − (46)

In this way, replacing the expressions relative to Equations (41), (42), (43), (44), (45)

and (46) in Equation (9), the final model for CE maximization is defined in conditions where

the budget is restricted (see Table 9). Its general form has the following structure:

( )( ) ( )[lim] [lim] [lim] [lim] [lim] [lim] [lim] [lim]max acq acq ret add add ret retCE = i M + M +i M ADD i RET d i ACQ − − − −

(47)


NUMERICAL EXAMPLE

In this example, the environment is characterized by the set of data presented in Table

10. Floor rates, ceiling rates, and the impacts of customer acquisition, customer retention and

customer expansion investments on the respective response rates are obtained through the DC

approach, as described by Blattberg and Deighton (1996). The other values can be extracted

from the data bases maintained by the firm.

This firm invests an average of $9.00 in customer acquisition, $18.00 in retention and

$3.00 in expansion. The available budget of $30.00 generates a CE of $70.39.


27

Is this 9/18/3 distribution appropriate? Is this the greatest value that CE can assume in

this environment? If not, how far is it from the maximum value? How sensitive is CE to

changes of these proportions? How much should be invested in customer acquisition,

retention and expansion in order to maximize CE without disrespecting the budget constraints

($30.00 per customer/period)? In this optimized condition, what would the increase in CE

with respect to the initial value ($70.39)? These are important questions, for which the

majority of firms do not have answers and, for this reason, are creators of serious gaps of

knowledge that compromise the firm’s management. In order to minimize these difficulties,

the first step to be taken includes identifying the maximum CE value, without any budget

constraints. In this example, both the GRG algorithm and the proposed model presented the

same results: a budget of $49.24 increased the CE to its maximum value of $80.20. The

optimal balancing of efforts among customer acquisition, retention and expansion (identified

equally by the two methods) is obtained by investing $17.24 in customer acquisition, $24.68

in retention and $7.32 in expansion. In this manner, it is discovered that investments greater

than $49.24 will probably not produce increases in CE. At the same time, knowing the

differences between the effective values and the maximum values ($30.00 versus $49.24 for

the budget; and $70.39 versus $80.20 for the CE) the decision making processes related to

allocation of investments can be developed with greater speed and confidence.

ANALYSIS OF MAXIMUM CUSTOMER EQUITY SENSITIVITY

In this section, the sensitivity of the maximum CE values is checked against the

variations produced by different levels of budget availability. In this sense, Table 11 shows

the results obtained with the GRG algorithm and the results generated through the

maximization model proposed by this study. The environment (defined in Table 10) was

28

maintained, only altering the levels of budget availability (beginning with $25.00 and ending

with the ideal value of $49.24), in such a way as to facilitate the intended comparison.

<Insert Table 11 here>

It can be seen that even in situations where budgets are very restricted (availability of

resources approximately 50% of the ideal) the performance of the proposed model is highly

satisfactory. The respective pairs of values corresponding to maximum CE, in the worst

situation, present differences less than 0.6% (in the case of a budget of $25.00, equivalent to

50.77% of the ideal value). For availabilities 75% greater than the ideal, the differences

between the values for maximum CE are practically inexistent, being less than 0.1% (budget

equal to or greater than $38.00) according to information presented in Figure 2 (generated

with the maximum CE values that make up Table 11).

The questions presented in the fifth section (Numerical example) can be easily

answered by applying the solution proposed in this article (see Table 9), according to the two

examples presented below.

Q How much should be invested in customer acquisition, retention and expansion in

order to maximize CE without disrespecting the $30.00 per customer/period

budget restriction?

A It can be seen in Table 11 (columns of the Proposed Model) that the values that

maximize CE with a budget of $30.00 are, respectively: $13.08 for customer

acquisition; $14.92 for retention; and $2.00 for expansion. In this case, the

maximum CE is $74.60. The non-analytic solution, maximized with the GRG

algorithm, reached $74.82. Even in unfavorable circumstances, with an available

budget equivalent to 60.93% of the ideal budget ($30.00 versus $49.24), the

29

difference between the two methods is $0.22 (deviation less than 0.3% with

respect to the GRG algorithm solution).

Q In this condition of optimization with budget constraints, how might the CE

increase with respect to its initial value of ($70.39)?

A That would be $4.21 (difference between $74.60 and $70.39), approximately 6%

of the elevation in relation to the initial CE, with the same $30.00 available.

< Insert Figure 2 here >

MANAGERIAL IMPLICATIONS OF THE NEW MODEL

From a managerial perspective, more precise models of CE can serve as references, as

much for planning as for monitoring the administration of firm-customer relations. The

detailed and continuous accompaniment of CE evolution generates information that can assist

in analyzing the performance of a firm’s strategies, contributing to validating the marketing

strategy adopted. Managers, as they implement their marketing actions, need precise data and

objective information that guides them towards maximization of the value of their customer

base (Bell et al. 2002). In practice the use of this modeling reinforces this view and allows:

i) acceleration of the process of adopting the CE concept in firms;

ii) improve decisions related to management of their relationship with customers;

iii) contribute to identifying and quantifying the partial and general impacts

produced by marketing actions on the return obtained by the firm in the market;

iv) reduce the risks inherent to the process of financial resources allocation in the

realm of marketing;

30

v) support planning and execution of media budgets;

vi) facilitate decisions on choice of channel;

vii) subsidize the process of identifying more profitable customers, individually or by

segment;

viii) increase the use of managerial practices that are less sensitive to environmental

pressures, in the short term;

ix) contribute to valorization of the use of metrics in marketing;

x) make the interdependencies that involve customer acquisition, retention and

expansion clearer;

xi) value the firm’s databases, making them more productive;

xii) perfect the process of customer lifecycle management; and

xiii) evaluate the impact of the customer lifecycle in the strategies and in the firm's

marketing mix.

LIMITATIONS, CONCLUSIONS AND SUGGESTIONS FOR FUTURE STUDIES

In the current research, the author adopted the premise that the gross margins,

investments and customer acquisition, retention and expansion rates remain constant over

time. In models of highly complex systems, simplifications like these play a very important

role, since they make it feasible to build initial models, without which it would be very

difficult to move forward in building scientific knowledge to arrive at more potent models

that are more representative of reality. The premises used in this paper are similar to those

used by the authors consulted, although they should be reviewed and redefined in such a way

as to reduce, as much as possible, the distance between the models produced from them and

the real systems they represent.

31

The modeling developed and presented in this article maximizes CE and optimizes the

balance of investments in customer acquisition, retention and expansion. Built starting from

the approaches developed by Blattberg, Getz, and Thomas (2001) and Gupta et al. (2006), it

incorporates the investments in customer expansion into its structure in an innovative

manner. Its operation, inspired by the work of Blattberg and Deighton (1996), is extremely

simple and fast. It is different from the main approaches found in the literature; it makes a

solution feasible for a problem that had, until now, been solved by means of optimization

algorithms, based on interactive research. Its utilization can contribute to improving decision

making processes in the realm of customer relationship management.

The performance of the maximization model presented in this study is extremely

satisfactory. In environments where budget limitations are small or inexistent, the modeling

generates results that are identical to those obtained through non-linear research algorithms

(such as the GRG, used in this research), which confers it high explicative and predictive

capability. In so far as budget constraints increase and the resources available represent

increasingly smaller fractions of the ideal budget (necessary budget to non-conditioned CE

maximization), the model shows a slight drop in performance (differences that oscillated

between zero and 0.6% with respect to the CE value obtained by non-linear optimization

algorithms). In order to reduce possible risks in use of the model, the author suggests that its

use be restricted to situations in which the available budget is equal to or greater than 75% of

the ideal budget.

In so far as they contribute to building more precise metrics for the field of marketing,

the researchers help to establish new criteria and better perspectives for studying the cause-

effect relationships between marketing investments and the returns obtained by firms. To this

end, the author suggests the development of studies that contribute to establishing the

32

operational intervals for reliable use of this new model. In a more general sense, it stimulated

the incorporation of other variables to the processes of modeling CE maximization, such as:

i) variables representative of the competitive environment on CE, particularly on

the conditions in which this reaches its maximum point;

ii) stochastic variables to represent processes of purchase and repurchase;

iii) hybrid response functions (customer acquisition, retention and expansion) in

order to represent the interdependencies in a more adequate manner;

iv) stochastic rates and margins;

v) complementary metrics that contemplate the financial and production areas in

order to broaden the scope of the models and, with that, increase their speed of

dissemination in firms and improve their performance.

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39

Figure 1 – Customer equity structure

Investments

Customer Life Cycle

Returns

Acquisition

Retention

Expansion

40

Table 1 – Entry variables

Symbol Description

[f ]acqi Customer acquisition floor rate

[c]acqi Customer acquisition ceiling rate

[f ]reti Customer retention floor rate

[c]reti Customer retention ceiling rate

[f ]addi Customer expansion floor rate

[c]addi Customer expansion ceiling rate

acqk Impact of acquisition investments on customer acquisition rate

retk Impact of retention investments on customer retention rate

addk Impact of expansion investments on customer expansion rate

acqM Average value of gross margin generated by new customers (1st purchase)

retM Average value of gross margin generated by retained customers

addM Average value of gross margin generated by developed customers

di Discount rate (equivalent to the firm’s cost of capital)

ACQ Average customer acquisition investment

RET Average customer retention investment

ADD Average customer expansion investment

[ ]limB Available budget (limited) per customer

N Number of potential customers T Temporal horizon (number of periods)

41

Table 2 – Intermediate and final variables

Symbol Description

acqi∆ Amplitude of the customer acquisition rate

reti∆ Amplitude of the customer retention rate

addi∆ Amplitude of the customer expansion rate

acqi Customer acquisition rate

reti Customer retention rate

addi Customer expansion rate

d Discount (1 di+ )

B Budget

FCE Firm’s customer equity

CE Customer equity

42

Table 3 – CE calculation structure

t Income Costs Discount factor

0 acq acqi M ACQ 1

1 1 ( )retacq ret add addi i M i M+

0 ( )retacq reti i RET i ADD+

11 (1 )di+



21 (1 )di+



31 (1 )di+

... ... ... ...

T ( )ret

T

acq ret add addi i M i M+ ( 1) ( )ret

T

acq reti i RET i ADD− + 1 (1 )T

di+

43

Table 4 – CE maximization variables

Symbol Description

[max ]CEACQ Average customer acquisition investment that maximizes CE

[max ]acq CEi Customer acquisition rate that maximizes CE

[max ]CERET Average customer retention investment that maximizes CE

[max ]ret CEi Customer retention rate that maximizes CE

[max ]CEADD Average customer expansion investment that maximizes CE

[max ]add CEi Customer expansion rate that maximizes CE

[max ]CEB Budget that maximizes CE: [max ]CEACQ + [max ]CERET + [max ]CEADD

maxCE maximum customer equity

44

Table 5 – CE Maximization

Resources allocation Rate

[max ]

*

[max ] ( )acq CECE acq acq acqACQ ln i k M k= ∆

[max ]

*

[max ] [c] 1/( )acq CEacq CE acq acqi = i k M−

[ ][max ] 1 2 1( )CE retRET = ln V V ln V k− [ ][max ] [c] 1 2 1( )ret CE ret reti = i i V V ln V− ∆ −

[max ] = ( )CE add add add addADD ln i k M k∆ [max ] [c] 1/( )add CE add add addi = i k M−

where:

1 dd = + i

[max ] [max ] [max ]CE ret add CE add CEU = M +i M ADD−

( ) ( )[max ]

*

[max ] [max ] [max ] [max ]acq CE acq CE ret CE CE ret CEM = M + U i RET d i− −

1 [max ] [c]( 1) ( )ret CE ret retV = i U dk d i∆ − −

( ) ( ) ( )2 [c] [max ] [max ]2 1ret ret CE ret CE retV = i d i U dk U dk ∆ − − −

45

Table 6 – Scenarios used in validating the proposed model

Variable Scenario

1 2 3 4 5 6 7 8

[f ]acqi 0% 1% 2% 3% 4% 5% 6% 7%

[c]acqi 20% 20% 18% 18% 16% 16% 14% 14%

[f ]reti 50% 52% 54% 56% 56% 54% 52% 50%

[c]reti 70% 68% 66% 64% 64% 66% 68% 70%

[f ]addi 0% 1% 1% 2% 2% 1% 1% 0%

[c]addi 10% 9% 9% 8% 8% 9% 9% 10%

acqk 0.15 0.10 0.15 0.10 0.15 0.10 0.15 0.10

retk 0.10 0.15 0.10 0.15 0.10 0.15 0.10 0.15

addk 0.30 0.30 0.20 0.20 0.10 0.10 0.05 0.05

acqM $200 $200 $200 $200 $200 $200 $200 $200

retM $250 $250 $250 $250 $250 $250 $250 $250

addM $300 $300 $300 $300 $300 $300 $300 $300

di 18% 20% 22% 24% 24% 22% 20% 18%

46

Table 7 – CE maximization without constraints (GRG versus proposed model)

Scenario GRG (Excel/Solver)a Proposed Modela

[max ]CEACQ

[max ]CERET [max ]CEADD maxCE [max ]CE

ACQ [max ]CERET

[max ]CEADD maxCE

01 18.32 24.68 7.32 79.12 18.32 24.68 7.32 79.12 02 22.57 17.44 6.58 68.01 22.57 17.44 6.58 68.01 03 16.02 18.40 7.84 60.22 16.02 18.40 7.84 60.22 04 19.07 12.10 6.40 51.72 19.07 12.10 6.40 51.72 05 13.73 13.74 5.88 49.32 13.73 13.74 5.88 49.32 06 16.34 14.96 8.75 48.18 16.34 14.96 8.75 48.18 07 11.55 21.27 3.65 47.75 11.55 21.27 3.65 47.75 08 12.81 18.88 8.11 49.17 12.81 18.88 8.11 49.17

a values in $

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Table 8 – CE maximization variables with budget constraints (w.b.c)

Symbol Description

λ Lagrange multiplier

[ ]limFator The proportionality factor, equal to the ratio between the available budget and the ideal budget

[0]ACQ Initial value of the customer acquisition investment

[0]acqi Initial customer acquisition rate

[0]RET Initial value of the customer retention investment

[0]reti Initial customer retention rate

[0]ADD Initial value of the customer expansion investment

[0]addi Initial customer expansion rate

[0]acqλ Lagrange multiplier associated with customer acquisition (initial value)

[0]retλ Lagrange multiplier associated with customer retention (initial value)

[0]addλ Lagrange multiplier associated with customer expansion (initial value)

λ̂ Estimate of the Lagrange multiplier

[lim]ACQ Average customer acquisition investment that maximizes CE w.b.r

[lim]acqi Customer acquisition rate that maximizes CE w.b.r

[lim]RET Average customer retention investment that maximizes CE w.b.r

[lim]reti Customer retention rate that maximizes CE w.b.r

[lim]ADD Average customer expansion investment that maximizes CE w.b.r

[lim]addi Customer expansion rate that maximizes CE w.b.r

[lim]B Available budget (limited) per customer equivalent to [lim]ACQ + [lim]RET + [lim]ADD

[lim]maxCE Maximum customer equity in environments where budget constraints are present

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Table 9 – CE maximization with budget constraints

Item Proposed expressions

Investments

( ) ( )[0] [0] [0] [0] [0]

[lim]

( )

ˆ 1


acq

i k M + M +i M ADD i RET d iACQ = ln k

λ

∆ − − −

+

[0] [0]

[lim]

[0] [0] [0]ˆ( )

acq ret add add add

add

ret acq ret

i i i k MADD ln k

d i i iλ

∆=

− +

[lim] [lim] [lim] [lim]RET B ACQ ADD= − −

Rates

[lim] [c] [lim]exp( )acq acq acq acqi = i i k ACQ− ∆ −

[lim] [c] [lim]exp( )ret ret ret reti = i i k RET− ∆ −

[lim] [c] [lim]exp( )add add add addi = i i k ADD− ∆ −

Model ( )( ) ( )[lim] [lim] [lim] [lim] [lim] [lim] [lim] [lim]max acq acq ret add add ret retCE = i M + M +i M ADD i RET d i ACQ − − − −

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Table 10 – Data for the numerical example

Description Notation Value

Customer acquisition floor rate [f ]acqi 3%

Customer acquisition ceiling rate [c]acqi 20%

Customer retention floor rate [f ]reti 50%

Customer retention ceiling rate [c]reti 70%

Customer expansion floor rate [f ]addi 0%

Customer expansion ceiling rate [c]addi 10%

Impact of acquisition investments on customer acquisition rate acqk 0.15

Impact of retention investments on customer retention rate retk 0.10

Impact of expansion investments on customer expansion rate addk 0.30

Average value of gross margin generated by new customers (1st purchase) acqM $200

Average value of gross margin generated by retained customers retM $250

Average value of gross margin generated by developed customers addM $300

Discount rate (equivalent to the firm’s cost of capital) di 18%

Average customer acquisition investment ACQ $9

Average customer retention investment RET $18

Average customer expansion investment ADD $3

Customer equity CE $70.39

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Table 11 – CE maximization without constraints (GRG versus proposed model)

Budget

($)

GRG (Excel/Solver) Proposed Model

ACQ RET ADD max CE ACQ RET ADD max CE

25.00 12.55 10.69 1.76 71.08 11.86 12.36 0.78 70.67 26.00 12.78 11.26 1.96 71.92 12.11 12.87 1.02 71.56 27.00 13.00 11.84 2.16 72.72 12.36 13.38 1.26 72.40 28.00 13.22 12.41 2.37 73.46 12.60 13.89 1.51 73.18 29.00 13.44 12.99 2.57 74.17 12.84 14.41 1.75 73.91 30.00 13.66 13.57 2.77 74.82 13.08 14.92 2.00 74.60

31.00 13.87 14.15 2.98 75.43 13.32 15.44 2.24 75.24 32.00 14.09 14.72 3.19 76.00 13.56 15.95 2.49 75.83 33.00 14.30 15.30 3.40 76.52 13.80 16.47 2.73 76.38 34.00 14.51 15.87 3.62 77.01 14.03 16.99 2.98 76.89 35.00 14.71 16.45 3.84 77.45 14.26 17.51 3.23 77.35 36.00 14.92 17.02 4.06 77.86 14.49 18.03 3.48 77.77 37.00 15.12 17.60 4.28 78.23 14.72 18.55 3.73 78.16 38.00 15.31 18.18 4.51 78.57 14.94 19.07 3.99 78.50 39.00 15.51 18.75 4.74 78.86 15.16 19.59 4.25 78.81 40.00 15.69 19.34 4.97 79.13 15.38 20.10 4.52 79.09 41.00 15.88 19.92 5.20 79.36 15.60 20.61 4.79 79.33 42.00 16.07 20.49 5.44 79.56 15.81 21.12 5.07 79.54 43.00 16.24 21.07 5.69 79.74 16.02 21.63 5.35 79.72 44.00 16.41 21.65 5.94 79.88 16.22 22.14 5.64 79.86 45.00 16.58 22.23 6.19 79.99 16.42 22.64 5.94 79.98 46.00 16.74 22.81 6.45 80.08 16.62 23.14 6.24 80.08 47.00 16.90 23.39 6.71 80.15 16.82 23.62 6.56 80.14 48.00 17.06 23.96 6.98 80.19 17.01 24.10 6.89 80.19 49.00 17.20 24.54 7.26 80.20 17.19 24.57 7.24 80.20 49.24 17.24 24.68 7.32 80.20 17.24 24.68 7.32 80.20

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Figure 2 – Budgetary availability and its impacts on maximum CE

70

72

74

76

78

80

82

30 40 50 60 70 80 90 100

Available budget (%)

Max

imu

m c

ust

om

er e

qu

ity

($

) .

GRG

Proposed model

Modeling and Maximizing Customer Equity

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