Optimal pricing and delayed incentives in a heterogeneous consumer market

Optimal pricing and delayed incentivesin a heterogeneous consumer market

Moutaz Khouja�, Stephanie S. Robbins and Hari K. RajagopalanReceived (in revised form): 10th July, 2007

�Business Information Systems and Operations Management Department, The Belk College of BusinessAdministration, The University of North Carolina — Charlotte, Charlotte, NC 28223, USATel: þ 1 704 687 3242; Fax: þ 1 704 687 6330; E-mail: [email protected]

Moutaz Khouja received a BS in Mechanical

Engineering, an MBA from the University of

Toledo and a PhD in Operations Management

from Kent State University. Currently, he is a

Professor of Operations Management in the

Belk College of Business Administration at the

University of North Carolina at Charlotte. His

research interests are in the areas of inventory

management, production planning and control,

pricing and forecasting. His publications have

appeared in many leading journals including

Computers and Operations Research, Deci-

sion Sciences, IIE Transactions, European

Journal of Operational Research, International

Journal of Production Research, International

Journal of Production Economics, Journal

of the Operational Research Society, and

Omega.

Stephanie S. Robbins received a PhD in

Management from Louisiana State University.

She is currently a Professor of MIS/OM at

The University of North Carolina at Charlotte.

Dr Robbins does research in the areas of

management information systems, marketing

management and strategy development for

non-profit organisations. Her publications have

appeared in journals such as International

Journal of Electronic Commerce, European

Journal of Operations Research, International

Journal of Production Economics, Information

and Management, The Journal of Computer

Information Systems, Behavioral Science and

the Journal of the Academy of Marketing

Science. She has also presented numerous

papers at international, national and regional

professional meetings.

Hari K. Rajagopalan earned his PhD in

Information Technology from the University of

North Carolina at Charlotte in 2006. Apart from

his PhD, he also has an MBA in Finance and

an MS in Computer Science. His research

interests include locating emergency response

systems, pricing of digital products and ob-

solescence in the high-technology industry.

His research has published in the European

Journal of Operational Research, Computers

and Operations Research and other journals.

He is also an active participant at INFORMS,

Decision Sciences, and European Working

Group in Transportation Meeting and Mini

EURO Conferences. He is currently the

Assistant Professor in Management at Francis

Marion University.

ABSTRACT

KEYWORDS: pricing, delayed incentives,

mail-in cash rebates

Delayed incentives in the form of cash mail-in rebates

have become very popular. We develop and solve a

model for jointly determining optimal price and rebate

value for a heterogeneous consumer market. Con-

sumers are divided into three segments: rebate

independent, fully rebate dependent, and partially

rebate dependent. Partially rebate-dependent consu-

mers’ redemption probability depends on the value of

www.palgrave-journals.com/rpm

& 2008 Palgrave Macmillan Ltd, 1476-6930 $30.00 Vol. 7, 1 85–105 Journal of Revenue and Pricing Management 85

the rebate relative to a reference value. Data show

that the probability of redemption increases linearly in

rebate value for small rebate values and at a

decreasing rate as the rebate value becomes large.

The model shows that two consumer attributes

are critical in determining the effectiveness of rebates.

The first is the reference value. The larger the

reference value, the larger the optimal rebate

value and the higher the profit. The second is the

distribution of consumers among the three segments.

Profit decreases as the proportion of consumers in the

probabilistic redeemers segment decreases. There is,

however, a threshold value where profit increases

as the proportion of consumers in the partially

rebate-dependent segment decreases. This occurs

because as the rebate-independent and fully rebate-

dependent consumers are priced out of the market, a

seller can increase both price and rebate value

substantially without having to be burdened by

the heavy cost of rebates for the fully rebate-dependent

segment. If the reference value for the partially

rebate-dependent segment is high, such a strategy

may prove profitable.

Journal of Revenue and Pricing Management (2008) 7,

85–105. doi:10.1057/palgrave.rpm.5160111

INTRODUCTION

Some researchers in marketing contend that

mail-in rebates present a great opportunity for

firms to increase their profit (Mouland, 2004).

Two main advantages of rebates are that few

consumers redeem them and that they do not

lower consumers’ future price expectations of

the product. Manufacturers mail-in rebates are

the most common type of rebates. Lately many

retailers are, however, offering store rebates.

The main focus of research has been on

consumer attitudes and behaviour with respect

to mail-in cash rebates and there has been

little research on determining the optimal

rebate strategy for sellers. Furthermore, there

is less research on joint optimal pricing

and rebate strategy. Like pricing, rebates are

used to manipulate demand. Therefore, deter-

mining optimal rebate strategy cannot be

accomplished without simultaneously deter-

mining product price.

In this paper, we develop a profit-max-

imisation model for jointly determining the

optimal price and rebate value. Based on survey

data, consumers are divided into three seg-

ments: rebate independent, fully rebate depen-

dent, and partially rebate dependent. The data

also indicate that the redemption rate is

increasing and is linear or slightly convex for

low rebate value and becomes concave at high

rebate values. Therefore, the model is solved

for linear and convex redemption rates that

apply for small ticket items for which the rebate

is less than $20. Results of the model indicate

that three consumer attributes are critical in

determining the profitability of rebates. First is

the reference value, which can be thought of as

the value consumers place on the time required

to redeem the rebate. The larger the reference

value, the larger the optimal rebate value and

the higher the profit. Second is the distribution

of consumers among the three segments. Profit

decreases as the proportion of consumers in the

partially rebate-dependent segment decreases.

There is, however, a threshold value where

profit increases as the proportion of consumers

in the partially rebate-dependent segment

decreases. This occurs because as the rebate-

independent and fully rebate-dependent con-

sumers are priced out of the market, a seller can

increase both price and rebate value substan-

tially without having to be burdened by the

heavy cost of rebates for the fully rebate-

dependent consumers. If the reference value for

the partially rebate-dependent segment is high,

rebates may prove profitable. Third is the ratio

of the increase in demand due to a $1 increase

in rebate value to the increase in demand due

to $1 decrease in price. The closer the amount

is to a $1, the more profitable the rebate

programme. This ratio, termed rebate attrac-

tiveness, is in part determined by the distribu-

tion of consumers among the three segments

and approaches zero as the proportion of

consumers in the rebate-independent segment

approaches 100 per cent.

In the next section, the literature on mail-in

rebates is reviewed. In a further section, results

Journal of Revenue and Pricing Management Vol. 7, 1 85–105 & 2008 Palgrave Macmillan Ltd, 1476-6930 $30.0086

Optimal pricing and delayed incentives

from the data are discussed and the linear

redemption model is formulated and solved

and then the convex redemption model is

formulated and solved followed by numerical

examples and sensitivity analysis. We close with

conclusions and suggestions for future research

in the last section.

LITERATURE REVIEW

One marketing tool that has received little

attention in the production-marketing inter-

face literature is cash mail-in rebates. The use

rebates has become very popular among

manufacturers and retailers (Bulkeley, 1998;

McGinn, 2003). Most rebates are offered by

manufacturers, but rebates are also becoming

popular among retailers. While many of the

studies on delayed incentives focus on con-

sumer perceptions and behaviour, little has

been done to determine their optimal value, or

to examine their impact on inventory policy

and profit.

Several studies have examined consumer

perception and response to delayed incentives

(Soman, 1998; Folkes and Wheat, 1995; Tat et

al., 1988; Jolson et al., 1987). The literature

suggests a few reasons for the popularity of

mail-in rebates (Jolson et al., 1987) which

include negligible accumulation requirements

by the consumer, ease of scheduling by

manufacturers and retailers, minimal risk by

consumers, and slippage. Slippage refers to the

proportion of consumers who purchase the

product because of the rebate but never redeem

it. Preliminary evidence indicates that most

consumers never redeem the rebate offers. For

example, Jolson et al. (1987) reports that 70 per

cent of consumers whose buying decisions are

influenced by rebate offers never redeem them.

Some estimates of the total proportions of

consumers who redeem their rebates are in the

range of 5–10 per cent (Bulkeley, 1998). Even

for rebates as high as $100, some experts

suggest that 50 per cent of consumers do not

make an effort to collect them (McGinn,

2003). A fifth reason suggested for the popular-

ity of mail-in rebates among manufacturers is

that they allow them to offer price discounts

directly to the consumers whereas with tradi-

tional price cuts, retailers may not fully pass the

price reductions to the consumer (Bulkeley,

1998). A sixth reason which makes rebates

preferable to using coupons or sales is that

consumers’ price expectations are higher for

products with rebates than the same products

with sales or coupons (Folkes and Wheat,

1995). In other words, when the promotions

are offered using coupons or sales, empirical

evidence indicate that the savings are likely to

be integrated into the regular price, which

means consumers will have future price

expectations closer to the promoted price

(Folkes and Wheat, 1995). This integration

and decrease in future price expectations are

much less likely to occur when the promotion

is delivered using mail-in rebates.

Based on an empirical experiment, Soman

(1998) tested and found support for two

hypotheses. The first hypothesis is that con-

sumer choice behaviour is more sensitive to the

rebate face value at the time of purchase than it

is to the level of effort required for redemption.

The second hypothesis is that the redemption

rate after purchase is more sensitive to the level

of effort than the rebate face value. In other

words, consumers underestimate the effort

required for redemption at the time of

purchase. Therefore, mail-in rebates provide

retailers with an excellent opportunity for

increasing profit, especially when the optimal

rebate face value is determined jointly with

price and order quantity. A retailer can increase

the per unit price but at the same time can

increase the mail-in rebate face value to offset

the impact of the price increase on demand. If

the redemption rate is low enough, then the

net impact will be an increase in profit. Silk

(2005) conducted a series of experiments that

tracked the purchase and redemption of a real

rebate offer. The experiments showed that

increasing the rebate value or the length of the

period for which it is valid make consumers

more confident that they will redeem the

rebate and increase the likelihood of purchasing


Khouja, Robbins and Rajagopalan

the product. This increased confidence did

not result in significant increase in actual

redemption later on. Therefore, increasing

rebate value has much less impact on redemp-

tion behaviour than purchase behaviour.

Furthermore, contrary to expectations, increas-

ing the length of valid redemption period

decreases redemptions.

Tat (1994) investigated three consumer

motives for rebate redemption: price con-

sciousness, perceived time and efforts associated

with rebate redemption, and perceived satisfac-

tion from using rebates to obtain the savings.

All three factors were found to be significant

predictors of consumers’ decisions to redeem

rebates. Perceived satisfaction was the most

significant, followed by price consciousness,

and then perceived time and efforts. The study

confirmed earlier results reported by Tat et al.

(1988) on the negative relationship between

the perceived effort and the difficulty of

redemption and redemption rates. Tat and

Schwepker (1998) empirically investigated the

relationships between rebate redemption mo-

tives. The authors found that perceived price

consciousness was positively related to rebate

redemption; however, the relationship was not

statistically significant, which implies that more

price conscious consumers do not redeem

rebates more frequently than consumers

who are not as price conscious. Similarly,

the authors found a negative, but statistically

insignificant, relationship between time and

effort required for redemption and rebate

redemption. In other words, the time and

effort needed to redeem a rebate offer does not

seem to directly decrease its probability of

being redeemed.

Ali et al. (1994) developed a model for

determining the optimal refund rate of rebates

as a proportion of the retail price. Their model

considered four ways by which rebates con-

tribute to incremental sales: brand switching,

repeat purchases, purchase acceleration, and

category expansion. The authors developed a

myopic model that considers incremental profit

due to sales during the rebate period only. The

authors also developed a long-term model that

also considers the repeat purchases in subse-

quent cycles due to brand switching and

showed that an optimal long-term rebate value

exists. Soman (1998) took another step toward

establishing optimal rebate value by formulat-

ing a profit maximisation model in which

demand and redemptions are linear functions

of rebate value. The model is developed for a

pre-determined price and under the assump-

tion that total redemptions depend only on

rebate face value and are independent of the

quantity sold. Gilpatric (2003), using the

present-biased preference model, identified

market conditions under which a rebate

programme is profitable. In this model, some

consumers assume that their preferences will be

unchanged in the future and buy the product

thinking they will redeem the rebate. Later on,

because their preferences change, they do not

redeem it. Chen et al. (2005) introduced an

explanation of rebate usage based on the idea of

‘utility arbitrage’. In this model, rebates are

viewed as state-dependent discounts. As re-

demption occurs after purchase, the rebate

utility to the consumer will depend on their

income utility at that time, which can be

low or high. The authors compared a rebate

policy with no rebate policy and presented

a set of necessary conditions for an optimal

rebate policy for the performance of utility

arbitrage.

Mail-in cash rebates have not received as

much research attention as other sales incentives

such as coupons (see Anderson and Song (2004)

for a brief review). Dhar et al. (1996) analysed

the profit impact of package coupons on profit.

They compared three types of package coupons:

Peel-off coupons that are redeemed at time of

purchase of the item, on-pack coupons that can

be redeemed at future purchases of the item, and

in-pack coupons that are similar to on-pack

coupons except consumers are unaware of their

presence at purchase. Other researchers dealt

with the impact of other types of coupons.

A distinguishing characteristic of mail-in rebates

is that they require an additional effort for



redemption over coupons, which may reduce

their redemption rates.

Our review of the literature did not identify

any models that analyse the pricing and rebate

value decisions while taking into account the

diverse behaviour of consumers regarding

rebates. In this paper, we develop a model for

determining the optimal price and rebate value

of a manufacturer operating in a market with

three consumer segments. Based on empirical

evidence, consumers are divided into three

segments with different demand sensitivity to

price and rebate value and redemption behaviour.

MODEL 1: LINEAR REDEMPTION

FUNCTION

Data on actual rebate redemptions are closely

guarded and firms are reluctant to release it. We

relied on exploratory data gathered from 204

graduate students using a questionnaire.

Summary of the data is shown in Appendix

A. In designing the questionnaire, we relied on

literature from the present biased preferences

model (O’Donoghue and Rabin, 1999, 2001).

In this model, consumers are aware that they

may have self-control problems in the future. In

the case of rebates, this self-control problem is

whether they will have the discipline to expend

the time and effort required to redeem the

rebate offer. Individuals are classified into one

of two groups: sophisticated individuals who

have full knowledge of their future selves and

preferences, albeit different from their current

selves and preferences, and naı̈ve individuals

who assume that their preferences will not

change over time. Naı̈ve consumers may find

that after purchase of the product, their

preferences have changed and they do not

redeem the rebate. We therefore divide

consumers into three segments with the first

two making up the sophisticated consumers

and the third making up the naı̈ve consumers.

1. Rebate-independent (RI) consumers. These con-

sumers do not redeem rebates and their

demand is unaffected by rebate value. They

may have been adversely affected by negative

past redemption experience or realise

from past redemption failures that they do

not redeem rebates. The existence of this

consumer segment is supported by earlier

research (Oldenburg, 2005).

2. Fully rebate-dependent (FRD) consumers. These

are consumers who are very sensitive to the

net cost of the product and always redeem

the rebate offer.

3. Partially rebate-dependent (PRD) consumers.

These are consumers who are sensitive to

rebates and intend to redeem them at the

time of purchase, but may not act on their

intentions.

The data from the questionnaire were used to

provide three characteristics of the consumers:

(1) The distribution among the three segments:

RI, FRD, and PRD, (2) the propensity of PRD

to redeem rebates, and (3) how do the PRD

and FRD trade-off price decrease versus

rebate increase. The data are used to identify

reasonable values for the model parameters.

Obviously, values of these parameters will

depend on the type of product as it determines

the consumer base, the ease of the redemption

process, the time allowed for redemption, etc.

The following notation is defined:

t, f, p denote rebate-independent consumers,

fully rebate-dependent consumers, and

partially rebate-dependent consumers

P price per unit, a decision variable

R value of the rebate, which does not

include its processing cost if it is

redeemed, a decision variable

r processing cost per rebate redeemed,

which is incurred in addition to

rebate value

h cost per unit of the product

xi demand for the product by consumer

segment i, (i¼ t, f, p)

X total demand for the product and

p profit, an effectiveness measure.

RI consumers are sensitive to only price and,

therefore, their demand can be written as

xt ¼ at � btP ð1Þ



FRD consumers are sensitive to both price and

rebate value and, therefore, their demand can

be written as

xf ¼ af � bf P þ cf R ð2ÞPRD consumers are also sensitive to both price

and rebate value but in different degrees than

FRD consumers. Therefore, their demand can

be written as

xp ¼ ap � bpP þ cpR ð3ÞEquations (1)–(3) imply that demand decreases

(btþ bfþ bp) units for each $1 increase in price

and increases (cfþ cp) units for each $1 increase

in the value of the rebate. The ratio

L ¼ ðcf þ cpÞðbt þ bf þ bpÞ

ð4Þ

is a measure of the effect of a $1 increase in

rebate value relative to a $1 decrease in price on

demand and will be referred to as ‘rebate

attractiveness’. An L¼ 1 indicates that there

are no RI consumers and as many consumers

buy the product because of a $1 increase in

rebate value as those who will buy it because of a

$1 decrease in price. An L value close to zero

indicates that most consumers are RI or

insensitive to rebates and that increasing rebate

value is much less effective in increasing demand

than decreasing price by the same amount.

Total rebate redemptions depend on the total

number of FRD and PRD consumers who

purchase the product. All FRD consumers will

redeem the rebate. For PRD consumers, the

proportion of rebates redeemed depends on

rebate value relative to a reference value, denoted

by V, resulting in xpR/V redemptions. The use

of the linear function for the proportion of PRD

who redeem the rebate results in a reasonably

good fit for the self-reported data for values of up

to $20 with adjusted R2 value of 0.93. Therefore,

the total number of rebate offers redeemed is

g ¼ xf þ xp

R

Vð5Þ

A reference value is used instead of unit price

because as observed by Soman (1998), redemp-

tion occurs after purchase and at that time the

redemption probability is no longer influenced

by the price paid for the product.

To simplify the analysis, we introduce the

following assumption with respect to the

parameters of the demand functions:

For all non-negative P and R such that

0pRoP; xtoxf oxp ð6Þ

which implies that the demand of the RI

segment vanishes first, followed by the FRD

segment, and finally the PRD segment. Let

S1 ¼ ðP;RÞ : RX0 and hoPoat

bt

;

� �ð7Þ

S2 ¼ ðP;RÞ : RX0 andat

bt

oPoaf þ cf R

bf

� �ð8Þ

S3 ¼(ðP;RÞ : RX0 and

af þ cf R

bf

oPoap þ cpR

bp

) ð9Þ

All three consumer segments are served

If the demand for all three consumer segments

are positive, the profit, which is revenue minus

cost, is given by

Z ¼ XðP � hÞ � gðR þ rÞ ð10Þ

Using the expressions for xi and g from equations

(1–3) and (5) in equation (10) and simplifying

gives the following expression for profit:

Z ¼ ðP � hÞðA � BP þ CRÞ � ðr þ RÞ

� af � bf P þ cf R þ Rðap � bpP þ cpRÞV

� �ð11Þ

where A¼P

iai, B¼P

ibi, and C¼P

ici.

Appendix B shows the conditions under which

the Hessian matrix is negative semi-definite.



Under those conditions, the sufficient condi-

tions for optimality are given by

P� ¼

ðA þ Bh þ bf rÞV þ ½ðbf þ CÞVþbprR þ bpR2

2BVð12Þ

and

R� ¼

bpP � aþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða� bpPÞ2 � 3cpV ½af þ Ch

�ðbf þ CÞP � 3cprðap � bpP þ cf V Þ

s

3cp

ð13Þwhere

a ¼ ap þ cpr þ cf V ð14ÞAt convergence, the values of P� and R� should

be used to compute xp, xf, and xt. If a consumer

segment’s demand is negative, then as shown

later, the optimal price and rebate value may

result in no sales to that consumer segment.

Not all consumer segments are served

Let Zfp be the profit if only the FRD and PRD

consumers are served with a maximum at Pfp�

and Rfp� (the subscript fp denotes only the FRD

and PRD segments). Zfp is obtained from

equation (11) with bt¼ at¼ 0, which should

also be used in computing A and B. Equations

(12) and (13) can be used to compute Pfp� and

Rfp� . The conditions under which the profit

function without the RI segment is concave are

shown in Appendix B.

Let Zp be the profit if only PRD consumers

are served with a maximum at Pp� and Rp

� (the

subscript p denotes only the PRD segment). Zp

is obtained from equation (11) with

bt¼ at¼ bf¼ af¼ cf¼ 0, which should also be

used in computing A, B, and C. Lemma 1

shows the optimal rebate value.

Lemma 1. A solution is optimal if and only if

R�p ¼ cpV � bpr

2bp

ð15Þ

Proof. The proof is shown in Appendix B.

Substituting for Rp� from equation (15) into

equation (12) gives

P�p ¼

½4bpðap þ bphÞ þ 3c2p V V � bprð2cpV � bprÞ8b2

pV

ð16ÞThe sufficiency of equation (16) is proven in

Appendix B. The global optimal solution may

occur when all three consumer segments are

served, only two segments are served, or only

one segment is served.

Identifying the global optimal solution

is based on the following properties of the

profit function which derive from assumption

in (6).

Property 1. For all P and R in S1,

Z >Zpf >Zp.


Property 2. For all P and R in S2 and

P>hþ rþR, ZoZfp and Zfp>Zp.


Property 3. For all P and R in S3, ZfpoZp

and Zp >Z.


Lemmas 2–4 are used to develop an algorithm

for identifying the global optimal solution.

Lemma 2 If equations (12) and (13) with all

three consumer segments converge to P� and

R� in S2 or S3 then it is optimal not to serve the

RI consumer segment.


Lemma 3. If equations (12) and (13) with only

the FRD and PRD segments converge to Pfp�

and Rfp� in S1 or S3 then it is not optimal to serve

only the FRD and PRD consumer segments.




Lemma 4. If equations (12) and (13) with

only the PRD segment converge to Pp� and Rp

�

in S1 or S2 then it is not optimal to serve only

the PRD consumer segment.


Boundary solutions

If the optimal solution occurs at the boundary

between S1 and S2 where P¼ at/bt (ie the

demand of the RI segment is zero), then

substituting P¼ at/bt into equation (11) and

differentiating with respect to (w.r.t.) R, yields

the following necessary condition, which is

sufficient under the condition shown in

Appendix B, for R to be optimal:

R� ¼ �b1 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2

1 � 4b0b2

q2b2

ð17Þ

where

b0 ¼ atbf � af V þ Cðat � bthÞþ rðap þ atbp=bt � cf V Þ ð18Þ

b1 ¼ 2atbp=bt � 2ðap þ cpr þ Vcf Þ ð19Þ

b2 ¼ �3cp ð20Þ

If the solution occurs at the boundary between

S2 and S3 where P¼ (afþ cfR)/bf (ie the demand

of the FRD segment is zero), then substituting

P¼ (afþ cfR)/bf and at¼ bt¼ 0 into equation

(11) and differentiating w.r.t. R, yields (17) with

b0 ¼ bf ½V ðapcf � bf cph þ bpcf hÞ � apbf rþ af ½V ðbf cp � 2bpcf Þ þ bpbf r

ð21Þb1 ¼ �2½ðbf cp � bpcf Þðbf r � cf V Þ

� af bpbf þ apb2f ð22Þ

b2 ¼ 3bf ðbpcf � bf cpÞ ð23Þ

as the necessary condition for optimality, which

is also sufficient under the condition shown in

Appendix B.

Algorithm to identify the optimal solution

Step 1. Using all three consumer segments

in equations (12) and (13) compute

P� and R�.If P� and R� are in S1 then P� and R�

are a local maximum.

If P� and R� are in S2 or S3 then R�

is given by equations (17)–(20) and

P� ¼ at/bt.

Compute the maximum profit in S1

denoted Z1� using equation (11).

Step 2. Using only the FRD and PRD

segments in equations (12) and (13)

compute Pfp� and Rfp

� .

If Pfp� and Rfp

� are in S2 Pfp� and Rfp

� are

a local maximum. Compute Zfp�

If Pfp� and Rfp

� are in S3 then use

equations (17) and (21–23) to com-

pute Rfp1� which should be used to

compute Pfp1� ¼ (afþ cfRfp1

� )/bf and

then Zfp1�

Step 3. Using only the PRD consumer

segment in equations (15) and (16)

compute Pp� and Rp

�.If Pp

� and Rp� are in S3 then Pp

� and Rp�

are a local maximum. Compute Zp�.

If Pp� and Rp

� are in S2 then use

equations (17) and (21)–(23) to

compute Rp1� which should be used

to compute Pp1� ¼ (af þ cfRp1

� )/bf and

then Zp1�

Step 4. Compare all local maximum profits

and select the largest.

MODEL 2: REDEMPTIONS INCREASE

AT AN INCREASING RATE IN REBATE

VALUE

As described by Soman (1998), one of the

problems with the linear redemption function

is that it does not reflect actual consumer

behaviour at extreme values of the rebate

where very few or very many consumers

redeem them. This observation is supported

by the collected data especially at large values of

the rebate exceeding $25 where the redemp-

tion function clearly increases at an increasing



rate. As we are restricting our attention to small

ticket items, we assume proportion of rebate

offers that are redeemed by the PRD is an

increasing convex function in the value of the

rebate and is given by:

g ¼ R

u � Rð24Þ

where u is an empirically determined constant.

Equation (24) implies that 100 per cent of

consumers redeem the rebate (ie g¼ 1) when

its value is R¼ u/2. In other words, u/2 has the

same meaning as the reference value in the

linear redemption function. A graph of the

linear redemption function versus the redemp-

tion function given by equation (24) for V¼u/

2¼ $50 is shown in Figure 1.

Using the expressions for xi and g from

equations (1)–(3) and (24) in equation (10)

and simplifying gives the following expression

for profit:

Z ¼ðP � hÞðA � BP þ CRÞ � ðr þ RÞ

� af � bf P þ cf R þ Rðap � bpP þ cpRÞu � R

� �ð25Þ

Let

o0 ¼ u½rðbpP � apÞ � ðaf � CðP � hÞ� bf P þ cf rÞu ð26Þ

o1 ¼ 2u½af � ap � CðP � hÞ þ Pðbp � bf Þþ rðcf � cpÞ � cf u ð27Þ

o2 ¼ af � ap þ Pðbf � bpÞ þ CðP � hÞþ rðcp � cf Þ þ uð4cf � 3cpÞ

ð28Þ

o3 ¼ 2ðcp � cf Þ ð29Þ

Appendix B shows the conditions under which

the Hessian matrix is negative semidefinite.

Under those conditions, the sufficient condi-

tions for optimality are given by

P� ¼

½A þ Bh þ bf r þ ðbp þ CÞRðu � RÞþbpRðr þ RÞ

2Bðu � RÞð30Þ

and

o3R3 þ o2R

2 þ o1R þ o0 ¼ 0 ð31Þ

At convergence, the values of P� and R� should

be used to compute xp, xf, and xt. If a segment’s

demand becomes negative, then as shown later,

the optimal price and rebate values may result

in no sales to that consumer segment.

1

0.8

0.6

0.4

Pro

prtio

n of

reb

ates

red

eem

ed

0.2

10 20 30R ($)

40 50

Figure 1: Linear versus nonlinear redemption functions for the PRD consumer segment



NOT ALL CONSUMER SEGMENTS

ARE SERVED

Let Zfp be the profit if only the FRD and PRD

consumers are served (ie its demand is greater

than zero) with a maximum at Pfp� and Rfp

� . Zfp is

obtained from equation (25) with bt¼ at¼ 0,

which should also be used in computing A and

B. Equations (30) and (31) can be used to

compute Pfp� and Rfp

� . The conditions under

which the profit function without the RI

segment is concave are shown in Appendix B.

Let Zp be the profit if only PRD consumers

are served with a maximum at Pp� and Rp

�. Zp is

obtained from equation (25) with bt¼ at¼ bf¼af¼ cf¼ 0, which should also be used in

computing A, B, and C. Lemma 5 shows the

optimal rebate value.

Lemma 5. A solution is optimal if and only if

R�p ¼ u �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibpuðbp þ cpÞðr þ uÞ

pbp þ cp

ð32Þ


Substituting for Rp� from equation (32) into

equations (26) gives

P�p ¼ 1

2

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibpuðbp þ cpÞðr þ uÞ

pbp þ cp

þbp þ cpu

b2

þ h � 2u � r

�ð33Þ

The sufficiency of equation (33) for optim-

ality is proven in Appendix B. The global

optimal solution may occur when all three

consumer segments are served, only two

segments are served, or when only one segment

is served.

Properties 1–3 and Lemmas 2–4 also hold

for this redemption function. Therefore, the

same algorithm can be used to find the optimal

solution after replacing equations (12, 13), (15,

16), (17) (18–20), and (21–23) with equations

(30, 31), (32, 33) (31), (34–37) (38–41),

respectively.

Boundary solutions

Similar to the linear redemption function, if

the optimal solution occurs at the boundary

between S1 and S2, then substituting P¼ at/bt

into equation (15) and differentiating w.r.t. R,

yields equation (31) with

o0 ¼ u½apbtr þ btðaf þ cph þ cf ðh þ rÞÞu� apðbpr þ ðbf þ CÞuÞ

ð34Þ

o1 ¼� 2u½apbt þ atðbf � bp þ CÞ� btðaf þ cpðh � rÞ þ cf ðh þ r � uÞÞ

ð35Þo2 ¼ apbt þ atðbf � bp þ CÞ

� bt½af þ cf ðh þ r � 4uÞ þ cpðh � r þ 3uÞð36Þ

o3 ¼ 2btðcp � cf Þ ð37Þ

as the necessary condition for R to be optimal,

which is also sufficient under the condition

shown in Appendix B. If the solution occurs

at the boundary between S2 and S3 where

P¼ (afþ cfR)/bf, then substituting P¼ (afþ cfR)/

bf and at¼ bt¼ 0 into equation (25) and

differentiating w.r.t. R yields equation (31)

with

o0 ¼ u½af ðbpbf r þ bf cpu � 2bpcf uÞþ bf ðbpcf hu � apbf r � cpbf hu þ apcf uÞ

ð38Þ

o1 ¼� 2u½apbf ðbf þ cf Þ þ cf ðbf cp � bpðbf þ 2cf ÞÞ� ðbf cp � bpcf Þðbf h � bf r þ cf uÞ ð39Þ

o2 ¼ apbf ðbf þ cf Þ þ af ½bf cp � bpðbf þ 2cf Þ� ðbf cp � bpcf Þ½bf ðh � r þ 3uÞ þ 4cf u

ð40Þ

o3 ¼ 2ðbf þ cf Þðbf cp � bpcf Þ ð41Þ

as the necessary condition for optimality, which

is also sufficient under the condition shown in

Appendix B.



NUMERICAL EXAMPLES AND

SENSITIVITY ANALYSIS

Numerical Example 1

Based on the collected data, we use the

consumer distribution among segments shown

in Table A1. We assume a total market of

50,000 consumers. In addition, we assume a

loss of the consumer base of 5.5 per cent, 5.0

per cent, and 4.5 per cent for each $1 increase

in price for the RI, FRD, and PRD segments,

respectively. Based on the data, PRD con-

sumers are assumed to be indifferent between

$1.72 increase in rebate and $1.00 decrease in

price whereas FRD consumers are assumed to

be indifferent between $1.69 increase in rebate

and $1.00 decrease in price. The resulting

demand functions are: xt¼ 3,450�189.75P,

xf¼ 20,350�1017.50Pþ 602.78R, xp¼ 26,250�1181.25Pþ 686.37R, and regression analysis of

the data for rebate value up to $20 indicate that

V¼ $24.64. The curve has a better fit at the

lower values of R while the fit becomes poor as

R becomes larger. At unit cost of h¼ $5, the

product is sold to all three segments (ie the

solution is in S1) with P� ¼ $14.55, R� ¼ $2.62,

and Z� ¼ $148,373. Also, 10.6 per cent of the

PRD consumers redeem the rebate offer and

44.3 per cent of all consumers redeem the

rebate offer. At unit cost of h¼ $10, the

product is sold to all three segments (ie the

solution is in S1) with P� ¼ $17.09, R� ¼ $2.69,

and Z� ¼ $69,936. Also, 10.9 per cent of the

PRD consumers redeem the rebate offer and

42.9 per cent of all consumers redeem the

rebate offer. At unit cost of h¼ $15, the

product is sold to only the FRD and PRD

segments (ie the solution is in S2) with

P� ¼ $21.10, R� ¼ $4.72, and Z� ¼ $23,502.

Also, 19.2 per cent of the PRD consumers

redeem the rebate offer and 41.3 per cent of all

consumers redeem the rebate offer. At unit cost

of h¼ $20, the product is sold to only the PRD

segment (ie the solution is in S3) with

P� ¼ $24.08, R� ¼ $6.66, and Z� ¼ $4,776.

Also, 27 per cent of the PRD consumers

redeem the rebate.

Numerical Example 2

To illustrate the impact of different parameters

on pricing and rebates, we consider another

example of a product with demand functions

xt¼ 40,000�3,000P, xf¼ 40,000�3,000Pþ2,500R, and xp¼ 100,000�6,000Pþ 2,000R

for the RI, FRD, and PRD consumer

segments, respectively. The cost of the product

is h¼ $4 per unit, r¼ $1 per redemption, and

the proportion of consumers who redeem

the rebate offer is linear in its value with a

reference value of V¼ $50. Equation (B.9)

shows that the profit function is concave for all

prices below $21.00 and rebates up to 100 per

cent of price. The algorithm results in an

optimal price of P� ¼ $10.45 per unit and

optimal rebate value of R� ¼ $2.49, which is in

S1. Equation (11) yields an optimal profit of

Z� ¼ $365,180. The demand for the three

consumer segments are xp¼ 42,300 units,

xf¼ 14,883 units, and xt¼ 8,662 units. The

rebates redeemed by the FRD and PRD

consumer segments are 14,883 and 2,105,

respectively.

If bt is changed from 3,000 to 4,000, then it

is optimal not to serve the RI segment

Pfp� ¼ $11.75 and Rfp

� ¼ $4.01, which is in S2,

and the optimal profit is Zfp� ¼ $316,188. If, in

addition to the change in bt, bf is changed from

3,000 to 5,000, then it is optimal to serve only

the PRD segment in S3 with Pp� ¼ $12.33 and

Rp� ¼ $7.33 and Zp

� ¼ $289,560.

For the case in which the proportion of

consumers who redeem the rebate offers is an

increasing nonlinear convex function given by

equation (24), we use u¼ $100, which results

in 100 per cent redemptions at R¼ $50 (the

same as the value of V in the linear redemption

function). Equation (B.25) shows that the profit

function is concave for all prices below $16.41

and rebates up to 60 per cent of price. The

algorithm results in an optimal price of

P� ¼ $10.65 per unit and an optimal rebate

value of R� ¼ $3.16, which is in S1. Equation

(25) yields an optimal profit of Z� ¼ $369,583.

The demand for the three consumer segments

is xp¼ 42,441 units, xf¼ 15,964 units, and



xt¼ 8,058 units. The rebates redeemed by the

FRD and PRD consumer segments are 15,964

and 1,386, respectively. Even though both the

linear and nonlinear redemption functions

reach 100 per cent redemption rate at $50,

the nonlinear case has an optimal price, rebate

value, and profit which are $0.21, $0.71, and

$4,404 larger, respectively.

If bt is changed from 3,000 to 4,000, then

the optimal solution is in S2 where the RI

consumers are not served with Pfp� ¼ $12.21,

Rfp� ¼ $5.17, and Zfp

� ¼ $325,196. Unlike the

linear redemption case, if, in addition to the

change in bt, bf is changed from 3,000 to 5,000,

then it is optimal to only serve the PRD

segment with Pp� ¼ $13.87 and Rp

� ¼ $11.75,

which are given by equation (31) with

equations (38)–(41), and Zp� ¼ 329,141. The

results of the last case are counter-intuitive as

the demand of the FRD consumers becomes

more sensitive to price, profit increases. This is

explained by the fact that if it is not profitable

to serve the FRD segment (because the sum of

the optimal rebate value, rebate processing

cost, and unit cost exceeds the optimal

price, any sales to this segment result in a loss),

then an increased sensitivity to price of the

FRD consumers will allow a firm to drive

that segment’s demand to zero at smaller

values of price. Therefore, the solution space

of focusing on the PRD consumers alone, that

is S3, is enlarged and the optimal profit

increases.

Numerical sensitivity analysis

Numerical analysis indicate that there are three

key consumer/market attributes that determine

the effectiveness of a rebate programme. First is

the reference value, V. Figures 2 and 3 show the

optimal profit, rebate value, and price and for

values of V up to $100. Figures 2–5 are

constructed based on the distribution of

consumers shown in Table A1 with unit cost

of h¼ $10. As the figures shows, optimal profit,

price, and rebate value increase with V. The

rate of increase in the optimal profit, price, and

rebate value changes at a value of V between

$20 and $30 and becomes larger. This is due to

the optimal solution moving from region S1,

where all three consumer segments are served

to region S2, where only the FRD and

PRD segments are served. As the RI segment

is only sensitive to price, when it is priced out,

the optimal price increases and so does the

optimal rebate value. As price can be increased

without losing as many consumers, the rate of

increase in the optimal profit with V becomes

even larger.

The second key consumer/market attribute

is the rebate attractiveness defined in equation

(4). As Figure 4 shows, as rebate attractiveness

increases, the optimal profit increases at an

increasing rate. Figure 5 shows the optimal

price and rebate value, which are both

increasing in rebate attractiveness. As the figure

shows, there is a jump in both optimal rebate

value and price at L of about 0.52 as the RI

$67,000

$69,000

$71,000

$73,000

$75,000

$77,000

$79,000

$81,000

$83,000

$85,000

$- $20 $40 $60 $80 $100 $120

Reference value (V)

Opt

imal

Pro

fit (

Z*)

Figure 2: Optimal profit as a function of reference value



consumer segment is priced out of the market.

An important conclusion from Figure 4 is that

having a large RI segment can have a negative

impact on profit. The reason is that even if

consumers in the PRD and FRD segments are

indifferent between $1 increase in rebate value

and $1 decrease in price but the RI segment is

large, the rebate attractiveness will be small.

$(3.00)

$2.00

$7.00

$12.00

$17.00

$22.00

$27.00

$- $20 $40 $60 $80 $100 $120

Reference value V

Uni

t pric

e an

d re

bate

val

ue

P*

R*

Figure 3: Optimal price and rebate value as a function of reference value

Opt

imal

pro

fit (

Z*)

$-

$50,000

$100,000

$150,000

$200,000

$250,000

$300,000

$350,000

0.49 0.54 0.59 0.64 0.69 0.74 0.79 0.84 0.89 0.94Rebate attractiveness (L)

Figure 4: Optimal profit as a function of rebate attractiveness, V¼ $100

$-

$10.00

$20.00

$30.00

$40.00

$50.00

$60.00

0.49 0.54 0.59 0.64 0.69 0.74 0.79 0.84 0.89 0.94Rebate attractiveness (L)

Pric

e an

d re

bate

val

ue

P*

R*

Figure 5: Optimal price and rebate value as a function of rebate attractiveness



This is an important motive for firms to

simplify the redemption process and ensure

prompt refunds upon redemption to avoid

causing consumers becoming RI due to

negative redemption experience.

The third key consumer/market attribute is

the distribution of consumers among the three

consumer segments. To analyse the impact of

different consumer demographics that may

exist for different products on rebate profit-

ability, we again use a market of 500,000

consumers and a product with a unit cost of

h¼ $10.00. For each consumer segment, 5 per

cent of the demand is lost for each $1 increase

in price. PRD consumers are assumed to be

indifferent between $1.72 increase in rebate

value and $1.00 decrease in price, whereas

FRD consumers are assumed to be indifferent

between $1.25 increase in rebate value and

$1.00 decrease in price. Figures 6 and 7 show

the optimal profit, optimal rebate value, and

optimal price and for V values of $75 and $100.

Figure 6 shows that as the relative size of the RI

and FRD increase, profit tends to decrease but

not over all ranges. For V¼ $75 there is a region

around (RI¼ 3 per cent, FRD¼ 38 per cent,

16.00

18.00

20.00

22.00

24.00

26.00

28.00

30.00

1,36,63 2,37,61 3,38,59 4,39,57 5,40,55 6,41,53 7,42,51 8,43,49 9,44,47 10,45,45Market Segments (%RI, %FRD, %PRD)

Pric

es a

nd r

ebat

e va

lue

R*, V = $75

R*, V = $100P*, V = $100

P*, V = $75

Figure 7: Optimal price and rebate value as a function of consumer distribution among segments

Pro

fit (

Z*)

$68,000

$73,000

$78,000

$83,000

$88,000

$93,000

$98,000

$103,000

$108,000

$113,000

$118,000

1,36,63 2,37,61 3,38,59 4,39,57 5,40,55 6,41,53 7,42,51 8,43,49 9,44,47 10,45,45

Market segments (%RI, %FRD, %PRD)

V=$100

V=$75

Figure 6: Optimal profit as a function of consumer distribution among segments



PRD¼ 59 per cent) where the profit shows

an increase. A more significant increase occurs

for V¼ $100 around (RI¼ 5 per cent,

FRD¼ 40 per cent, PRD¼ 55 per cent). Both

of these increases in profit occur when the

optimal solution moves from the boundary of

regions S2 and S3 into region S3. Such a

phenomena is counter-intuitive as it implies

that as the proportion of consumers who are RI

and FRD increase, there is a threshold where

the optimal profit increases. The explanation is

that at a certain threshold in terms of the

distribution of consumers among the three

segments, it becomes optimal to price the FRD

out of the market. As Figure 7 shows, at this

threshold distribution of consumers among

segments, optimal prices show a sudden

increase to price the FRD out of the market.

As FRD are 100 per cent redeemers, having

them priced out of the market causes the

optimal rebate value to also increase. In some

cases, such as for V¼ $100, optimal rebate

value becomes even larger than the optimal

price.

Figures 6 and 7 have an important implica-

tion in terms of the time dimension. As

consumers gain better understanding of their

behaviour with respect to rebates over time,

they may move from the PRD segment to

the RI or the FRD segments. A consumer

who finds that she/he did not redeem any

rebates, in spite of intending to at the time of

purchase, may decide that they will no longer

take rebates into account in making the

purchase decision and becomes part of the RI

segment. Similarly, a consumer who finds

that she/he have successfully redeemed all

rebate offers over time may place more

emphasis on rebates in making a purchase

decision and becomes part of the FRD

segment. As Figure 6 shows, this change in

the relative size of the consumer segments in

favor of the RI and FRD segments will cause

rebate effectiveness to decrease over time,

which may be a contributing factor to some

retailers eliminating cash mail-in-rebates alto-

gether (Albright, 2005).

CONCLUSIONS AND SUGGESTIONS

FOR FUTURE RESEARCH

In this paper, we have developed a profit-

maximisation model for a firm using mail-in

cash rebates. The model takes into account the

presence of three heterogeneous consumer

groups: a RI segment whose demand is

unaffected by the rebate and whose consumers

do not redeem rebates, an FRD segment whose

consumers always redeem rebates, and a PRD

segment whose consumers intend to redeem

rebate offers at the time of purchase but may

not do so later. The proposed model is solved

for a demand that is linearly increasing in rebate

value and decreasing in price and for redemp-

tion functions that are increasing (linear or

convex) in rebate value. These redemption

functions are valid for rebate values up to about

$20 beyond which the redemption function

becomes concave.

Our analysis indicates that there are three

consumer/market characteristics that deter-

mine the profitability of a rebate programme:

The first is the reference value of the consu-

mers in the PRD segment. The reference

value can be thought of as the value at which

consumers believe that the rebate is fully

sufficiently large to compensate for the time

and effort they will spend on redeeming it. The

larger the reference price, the larger the

increase in profit a rebate programme brings

about. This suggests that rebate programmes

may be more profitable for products targeted

toward consumers with relatively large dispo-

sable income since these consumers may

place a larger value on their time. The second

characteristic is the rebate attractiveness

which is given by the ratio of the increase in

demand due to a $1.00 increase in rebate

value to the increase in demand due to a

$1.00 decrease in price. The larger the rebate

attractiveness, the larger the increase in profit.

The third characteristic, which partially deter-

mines the second characteristic, is the distribu-

tion of consumers among the three segments.

The larger the proportion of consumers in the

PRD consumers, the larger the increase in



profit from using rebates. This observation,

however, is not true in certain regions of the

solution space where the optimal solution

changes in terms of the composition of the

served market.

There are several important research ques-

tions that remain. A very important question in

light of the increased negative perception of

rebates among consumers (Oldenburg, 2005;

Chuang, 2003, Spencer, 2002) is the effect

of simplifying the rebate redemption pro-

cess and requirements on redemption rates.

The negative perception of rebates has even

caused some retailers such as Best Buy to

phase-out mail-in rebates altogether (Smith,

2005) and causing many consumers to become

RI. If simplifying the redemption process

has only a small effect on the redemption

rates, then manufacturers and retailers may be

able to avoid the negative perceptions that is

overtaking the market place and may be

causing many consumers to become RI while

maintaining profitable rebate programmes.

Another area or research involves determining

reference values of customer groups and its

relationship to disposable income. Obviously,

the reference value of consumers may be

positively correlated with their disposable income.

While some products are targeted to a broad

consumer market, some products are targeted to

only higher disposable income consumers. Dif-

ferent pricing and rebate strategy may greatly

enhance the profitability of these products.

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APPENDIX A

A pilot study was conducted in order to

develop a profile of consumer behaviour

relative to redeeming manufacturers’ mail-in

cash rebates. A brief four-part questionnaire

was designed and distributed to graduate

students over a three-week period during the

spring semester of 2005. Of the 204 ques-

tionnaires collected, consumers fell into the

three categories as shown in Table A1.

Respondents indicating they always or

sometimes redeem mail-in cash rebates were

asked the following question: ‘If you were

willing to purchase a product priced at $40 that

has a $10 mail-in cash rebate, how much would

the rebate have to be if the price were raised to

$50?’ The average responses were $17.21,

$16.88, and $17.07 for the PRD, FRD, and

the PRD and FRD combined, respectively.

PRD consumers were asked to indicate their

future probability of redemption based on nine

dollar amounts ranging from $1.00 to $100.

After collecting 147 questionnaires, it became

clear that the rebate increments were too large

to estimate redemption rates for rebates that fall

below $20. Therefore, in the next 57 ques-

tionnaires, the increment in mail-in cash rebate

values were reduced to seven, ranging from

$1.00 to $20. Figures A1 and A2 represent the

data from the two parts of the survey. Both

graphs indicate that for rebate value up to $20,

the rebate redemption function is well approxi-

mated by a linear function. Actually, regression

analysis for the second group of respondents

using a linear regression model results in an

adjusted R2 of 93 per cent and a reference value

of $24.64. While this value may be a good

estimate for rebate values up to about $20,

beyond this value, Figure A1 shows that the

redemption function becomes concave. Further-

more, Figure A2 shows that at smaller rebate

values, of up about $10, there may be slight

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 20 40 60 80 100Rebate value (R)

Pro

babi

lity

of r

edem

ptio

n

Figure A1: Self-reported redemption probability as a function of rebate values up to $100

Table A1: Manufacturer’s mail-in cash rebate

survey responses

Consumer category Number of

responses

Percentage

Rebate independent: RI 14 6.9

Partially rebate dependent:

PRD (sometimes redeem)

107 52.5

Fully rebate dependent:

FRD (always redeem)

83 40.7



convexity in the rebate redemption function. It

is worthwhile to stress that many firms who have

used rebates in the past have data on the

amounts of each rebate they offered and the

percentage of rebates redeemed, which may

enable them to better estimate the redemption

function and not rely on self-reported data.

APPENDIX B

DERIVATIVES AND HESSIAN MATRIX OF

PROFIT FOR LINEAR REDEMPTION

FUNCTION

Demand for all three segments

The first partial derivatives of Z with respect to

P and R are

qZ

qP¼

V ½A þ Bðh � 2PÞ þ bf r þ ½ðbf þ CÞVþbprR þ bpR2

V

ðB:1Þand

qZ

qR¼ bf P þ CðP � hÞ � af � cf R

�

ðap � bpP þ cpRÞR þ ðR þ rÞ�ðap � bpP þ cf V þ 2cpRÞ

V ðB:2Þ

The solution to setting the partial derivatives in

equations (B.1) and (B.2) to zero are given by

equations (8) and (9). The Hessian matrix of Z is

H ¼�2B bf þ cp þ cf Vþbprþ2bpR

V

bf þ cp þ cf Vþbprþ2bpR

V

2ðap�bpPþcf Vþcprþ3cpRÞV

" #

ðB:3ÞThe determinant of the first principal minor of

H is

D1 ¼ �2B ðB:4ÞThe determinant of the second principal minor

of H is

D2 ¼ �

½ðbf þ CÞV þ bpðr þ 2RÞ2 þ 4BV

�ðbpP � ap � cf V � cpr � 3cpRÞV 2

ðB:5ÞAs D1o0, Z, is concave if D2>0. Let

R ¼ rV ðB:6ÞSubstituting from equation (B.6) into equation

(B.5) and simplifying gives

D2 ¼ 4cf B � ðbf þ CÞ2

þ 4Bðcp � bpP þ cprÞ � 2bprðbp þ CÞV

�b2

pr2

V 2� 4b2

pr2

�4r½bpV ðbf � 2cp þ cf Þ þ b2

pr � 3cpV ðbt þ bf ÞV

ðB:7Þ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20Rebate value (R)

Pro

babi

lity

of r

edem

ptio

n

Figure A2: Self-reported redemption probability as a function of rebate values up to $20



equation (B.7) has a single root at

Ps ¼� V

"ðbf þ CÞ2 � 4cf B

� 4Bðap þ cprÞ � 2bprðbf þ CÞV

þb2

p r2

V 2þ 4b2

pr2

þ4r½bpV ðbf � 2cp þ cf Þ þ b2

p r � 3cpV ðbt þ bf ÞV

#

=ð4bpBÞ ðB:8ÞD2 switches signs at Ps. Let e be a small positive

number. For P¼Ps�e, D2¼ 4Bbpe/V>0.

Therefore, Z is concave for PoPs. A plot of

Ps for 0prp1 shows the values of P for which

Z is concave for all positive rebate values up to

price.

Demand from only the FRD and PRD

segments

Following the same analysis for the three-

segment case, Zfp is concave for PoPs, where

Ps is given by equation (B.8) with at¼ bt¼ 0. A

plot of Ps for 0prp1 shows the values of P for

which Zfp is concave for all rebate values up to

price.

Demand from only the PRD segment

Substituting from equation (15) into the profit

function and taking the first and second

derivative w.r.t. P gives d2Zp/dP2¼�2bpo0.

Therefore, Zp is concave and equation (16) is a

sufficient condition for optimality.

Proof of Lemma 1. 1. By contradiction, suppose

R1¼ (cpV�bpr)/2bpþD is optimal. Then de-

creasing the rebate by D and decreasing the

price by Dcp/bp will leave demand unchanged.

The change in revenue is:

xp P � Dcp

bp

� �� P

� �¼ � xpDcp

bp

ðB:9Þ

The change in total rebates paid is

xp

R1 � DV

ðR1 � Dþ rÞ

� xp

R1

VðR1 þ rÞ ¼ � xpDðbpDþ cpV Þ

bpV

ðB:10Þ

As the revenue decreases and rebate cost also

decreases, the net change in total profit is

� xpDcp

bp

� � xtDðbpDþ cpV ÞbpV

� �

¼ xpD2

V40 ðB:11Þ

Therefore, R1 cannot be optimal.

2. By contradiction, suppose R2¼ (cpV�bpr)/

(2bp)�D is optimal. Then increasing the rebate

by D and increasing the price by Dcp/bp will

leave demand unchanged. The change in

revenue is

xp P þ Dcp

bp

� �� P

� �¼ þ xpDcp

bp

ðB:12Þ

The change in total rebates paid is

xp

R2 þ DV

ðR2 þ Dþ rÞ

� xp

R2

VðR2 þ rÞ ¼ xpDðcpV � bpDÞ

bpV

ðB:13ÞThe net change in total profit is

xpDcp

bp

� xpDðc2V � bpDÞbpV

¼ xpD2

V40 ðB:14Þ

Therefore, R2 cannot be optimal.

Proof of Property 1. Subtracting Zfp from Z gives

Z � Zfp ¼ ðP � hÞðat � btPÞ40 for all P and R in S1

ðB:15Þ

Subtracting Zp from Zfp gives

Zfp � Zp ¼ ½P � ðh þ r þ RÞðaf � bf P þ cf RÞ40

for all P and R in S1 ðB:16ÞFrom equations (B.15) and (B.16), Z>Zfp and

Zfp>Zp.

Proof of Property 2. From equation (B.15),

Z�Zfpo0 for all P and R in S2. Therefore,

ZoZfp. From equation (B.16), Zfp�Zp>0 for

all P and R in S2 and P>hþ rþR. Therefore,

Zfp>Zp.



Proof of Property 3. From equation (B.16),

Zfp�Zpo0 for all P and R in S3 and

P>hþ rþR. Therefore, ZfpoZp. Subtracting

Zp from Z gives

Z � Zp ¼ ðP � hÞðat � btPÞþ ½P � ðh þ r þ RÞðaf � bf P þ cf RÞo0 for all P and R in S3 ðB:17Þ

From equation (B.17), ZoZp.

Proof of Lemma 2. By Property 1, if P� and R�

are in S2, then Z(P�, R�)oZfp(P�, R�). There-

fore, there is a P2 and R2 in S2 such thatZ(P�, R�)oZfp(P

�, R�)pZfp(P2, R2) and P� andR� cannot be optimal. By Property 2, if P� andR� are in S3, then Z(P�, R�)oZp(P

�, R�).Therefore, there is a P3 and R3 in S3 such thatZ(P�, R�)oZp(P

�, R�)pZp(P3, R3) and P� andR� cannot be optimal.

Proof of Lemma 3. By Property 3, if Pfp� and Rfp

�

are in S3, then Zfp(Pfp� , Rfp

� )oZp(Pfp� , Rfp

� ).Therefore, there is a P3 and R3 in S3 such thatZfp(Pfp

� , Rfp� )oZp(Pfp

� , Rfp� )pZp(P3, R3) and Pfp

�

and Rfp� cannot be optimal. By Property 1, if

Pfp� and Rfp

� are in S1, then Z(Pfp� , Rfp

� )>Zfp(Pfp

� , Rfp� ). Therefore, there is a P1 and R1

in S1 such that Z(P1, R1)>Z(Pfp� , Rfp

� )>Zfp(Pfp

� , Rfp� ) and Pfp

� and Rfp� cannot be optimal.

Proof of Lemma 4. By Property 1, if Pp� and Rp

�

are in S1, then Zp(Pp�, Rp

�)oZ(Pp�, Rp

�). There-fore, there is a P1 and R1 in S1 such thatZp(Pp

�, Rp�)oZ(Pp

�, Rp�)pZ(P1, R1) and Pp

� andRp� cannot be optimal. By Property 1, if Pp

� andRp� are in S2, then Zp(Pp

�, Rp�)oZfp(Pp

�, Rp�).

Therefore, there is a P2 and R2 in S2 such thatZfp(P2, R2)XZfp(Pp

�, Rp�)>Zp(Pp

�, Rp�) and Pp

� andRp� cannot be optimal.

DERIVATIVES AND HESSIAN MATRIX OF

PROFIT FOR NONLINEAR REDEMPTION

FUNCTION

The first partial derivatives of Z with respect to

P and R are

qZ

qP¼

½A þ Bðh � 2PÞ þ bf r þ ðbf þ CÞR�ðu � RÞ þ bpRðr þ RÞ

ðu � RÞðB:18Þ

and

qZ

qR¼ �

ðap � bpP þ cpRÞRðu � RÞ þ ðR þ rÞ½uðap � bpPÞ þ cf ðu � RÞ2 þ cpRð2u � RÞ

ðu � RÞ2

þ bf P þ CðP � hÞ � af � cf R ðB:19Þ

The solution to setting the partial derivatives in

equations (B.18) and (B.19) to zero are given by

equations (30) and (31). The elements of the

Hessian matrix of Z are

q11 ¼ �2B ðB:20Þ

q12 ¼ q12 ¼ bf þ C þ bp½uð2R þ rÞ � R2ðu � RÞ2

ðB:21Þ

and

q22 ¼2½ðap � bpPÞðr þ uÞu þ cpðR2ðR � 3uÞ

þu2ð3R þ rÞÞðR � uÞ3

ðB:22Þ

The determinant of the second principal minor of

H is

D2 ¼1

ðu � RÞ4½½ðbf þ CÞðu � RÞ2

þ bpðRð2u � RÞ þ ruÞ2

þ 4BðR � uÞ½cf ðR � uÞ3

þ uðbpP � apÞðr þ uÞ � cp½R2ðR � 3uÞþ u2ðr þ 3RÞ ðB:23Þ

As q11o0, Z is concave if D2>0. As (u�R)4>0, it is

sufficient that the term inside the brackets be

positive. Let

R ¼ km ðB:24Þ

Substituting from (B.24) into (B.23) and simplifying

gives

D2 ¼u2½�½uðbf þ CÞðk� 1Þ2

þ bpðr � uðk� 2ÞkÞ2

þ 4Bðk� 1Þ½cf u2ðk� 1Þ3

þ ðbpP � apÞðr þ uÞ� cpu½r þ ukð3 þ kðk� 3ÞÞ ðB:25Þ



equation (B.25) has a single root at

Pc ¼1

4bpBðr þ uÞð1 � kÞ ½4apBðr þ uÞð1 � kÞ

þ 4Bcf u2ðk� 1Þ4 þ 4Bð1 � kÞcpu

�½r þ ukð3 þ ðk� 3ÞkÞ� ½ðbf þ CÞuð1 � kÞ2

þ bpðr � uðk� 2ÞkÞ2 ðB:26ÞD2 switches signs at Pc. Let e be a small positive

number. For P¼ Pc�e, D2¼�4bpBu2(rþ u)

(1�k)e. Therefore, as long as ko1 (ie rebate values

less than price), D2o0 for PoPc and Z is concave.

A plot of Pc for 0pko1 shows the values of P for

which Z is concave for all rebate values up to price.

Demand for the FRD and PRD only

Following the same analysis for the three-

segment case, Zfp is concave for PoPc, where

Pc is given by equation (B.26) with at¼ 0 and

bt¼ 0. A plot of Pc for 0prp1 shows the

values of P for which Zfp is concave for all

rebate values up to price.

Demand for the PRD only

Substituting from equation (32) into the profit

function and taking the first and second

derivative w.r.t. P gives d2Zp/dP2¼�2bpo0.

Therefore, Zp is concave and equation (33) is a

sufficient condition for optimality.

Proof of Lemma 5. The proof of Lemma 5 can

be established in the same way as Lemma 1

using and

R1 ¼ u �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibpu ðbp þ cpÞð r þ uÞ

pbp þ cp

þ D

and

R2 ¼ u �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibpu ðbp þ cpÞð r þ uÞ

pbp þ cp

� D

ðB:27Þ

Solution at boundary — Linear

redemption

At P¼ at/bt, d2Z/dR2o0 for R>[atbp�bt(apþcfVþ cpr)]/(3btcp) and Z is concave. At P¼

(afþ cf R)/bf, d2Z/dR2o0 for Ro(bpaf�apbf)/

[3(bc cp�bpcf )]þ (cfV–bfr)/(3bf) and Z is concave.

Solution at boundary — Nonlinear

redemption

At P¼ at/bt,

d2Z=dR2 ¼ k0 þ 6k1u2R � 6k1uR2

þ 2k1R3 ðB:28Þ

where

k0 ¼� 2u½ðapbt � atbpÞðr þ uÞþ btuðcpr þ cf uÞ

ðB:29Þ

k1 ¼ btðcp � cf Þ ðB:30Þ

For k0o0 (which holds for realistic problems)

and R>0, d2Z/dR2 switches sign at

Rc ¼ mþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�2k3

1u3 � k2

1k0Þ2k3

1

3

sðB:31Þ

If k1o0 then d 2Z/dR2o0 for all R>Rc and Z

is concave. If k1>0, then d2Z/dR2o0 for all

RoRc and Z is concave.

At P¼ (afþ cfR)/bf, d2Z/dR2 is given by

(B.28) where

k0 ¼� 2u½ðcf bp � cpbf Þðbf r � cf uÞu

þ bf ðaf bp � apbf Þðr þ uÞ ðB:32Þ

k1 ¼ ðbf þ cf Þðbf cp � bpcf Þ ðB:33Þ

If k0>0 and k1o0, k0o0 and k1>0,

or k0o0 and k1o0, then d2Z/dR2 switches

sign at Rc given by equation (B.31) and

for all RoRc, d2Z/dR2o0 and Z is

concave. If k0>0 and k1>0 then d2Z/dR2

switches sign at Rc given by equation (B.31)

and for all R>Rc, d2Z/dR2o0 and Z is

concave.



Optimal pricing and delayed incentives in a heterogeneous consumer market

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