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How customisation of pricing and item

availability information can improvee-commerce performance

Hisashi Kurata and Carolyn M. BonifieldReceived (in revised form): 15th February, 2006

Temple University, Japan Campus, Azabu Hall 1F, 2-8-12 Minami Azabu, Minato-ku, Tokyo 106-0047,Japan. Tel: þ 81-3-5441 9864; Fax: þ 81-3-5441 9811;E-mail: [email protected]

Hisashi Kurata received a PhD from the

School of Business Administration, The Uni-

versity of Wisconsin-Milwaukee. He earned an

MS in Management Science from Pennsylvania

State University, and a BA in Economics from

the University of Tokyo. His research interests

include supply chain management, marketing-

manufacturing interface, and retail operations.

Carolyn M. Bonifield is an assistant professor

at the School of Business Administration,

University of Vermont. She earned a PhD

from the University of Iowa. Her research

interests include the role of affect in consumer decision making, post-purchase decision mak-

ing, and framing.

ABSTRACTKEYWORDS: inventory-dependent demand,

pricing, E-business, item availability, air or

hotel industry

Two characteristics of e-commerce, the ability to

micromarket (ie, customising a marketing plan

according to customers’ purchasing patterns) and the ability to selectively offer item availability information

(ie, manipulating whether or not to display the total

number of items available to customers), considerably

increase firms’ potential to improve their performance.

This paper considers e-business in the hotel and

airline industries, which has two customer segments:

one is the leisure segment, which focuses more on

price, and the other group is the business segment,

which focuses heavily on schedule. We propose an

analytical model that determines the optimal pricing

and demonstrates that e-business can improve its

revenue by taking into account customer segmentation

when offering item availability information to

customers. We provide numerical examples that

demonstrate that accuracy in segmenting customers

and the size of each segment will influence the

performance of customised marketing planning. We

also present managerial implications derived from

these analytical findings.

Journal of Revenue and Pricing Management (2007) 5,

305–314. doi:10.1057/palgrave.rpm.5160054

INTRODUCTION

Booking or purchasing services via the internet

is common for various service industries,

including airlines, hotels, rental cars, theatres,

and restaurants. The pricing decision for

services, relative to manufacturing firms, is

particularly challenging for a variety of reasons

including difficulty in determining fixed and

operating costs, the intangible nature of

services, and perishability, which translates intoinability to inventory services. As for types of

pricing decisions, for instance, Talluri and van

Ryzin (2004, Chapter 1) present recommenda-

tions for how to set posted prices, how to price

across product categories, how to price over

time, and how to discount. Clearly, price

significantly influences consumers’ decisions.

www.palgrave-journals.com/rpm

& 2007 Palgrave Macmillan Ltd, 1476-6930 $30.00 Vol. 5, 4 305–314 Journal of Revenue and Pricing Management 305

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In addition to pricing, the firm has the

ability to present the number of available items

to the potential customer. For example,

imagine a common scenario where an indivi-

dual is looking on the internet for a discount air

ticket for his/her business trip. If he/she knows

that only a few seats remain on the desired

departure date, he/she may quickly decide to

make a reservation in order to avoid losing his/

her chance to go on the trip. As this example

implies, the number of remaining items that a

seller offers to customers (we call this item

availability) may influence the customer’s deci-

sion process and change the demand rate. That

is, item availability is one choice factor in

addition to traditionally well-discussed market-

ing activities such as price, promotion, and in-store display.

This paper addresses e-business in the

service industry, such as hotels or airlines.

These areas have the following common

characteristics: the capacity of service

offered is fixed, fixed and operation

costs are exogenously decided or are of lesser

concern compared to revenue, customer

purchasing (or reservation of service) is

often conducted via the internet, a custo-

mised marketing plan is easily offered, and the

date/timing of service is important for

customers.

Our model considers two types of customer

segments: one is a group of customers who pay

more attention to timing of service than to

price. For example, it is common for business

travellers to buy expensive tickets to meet their

schedules. We call this customer segment a

business (traveller) segment . The other segment is

composed primarily of leisure travellers. For

example, the highest priority for a college

student who wants to go on a trip duringsummer break is to search for a highly

discounted air ticket. Usually such a leisure

traveller has high flexibility in schedule but is

very interested in price discounts. Hereafter,

we call this segment the leisure (traveller) segment .

Note that business versus leisure segmentation

is a common approach for revenue manage-

ment (see Talluri and van Ryzin, 2004,

Chapter 10).

The goal of this paper is to determine the

optimal pricing under the item availability

effect for an e-commerce transaction, particu-

larly in the airline or hotel industries, assuming

that e-business can assess customers’ behaviour

(ie, sensitivities to price and item availability)

and manipulate information for both price and

item availability to a specific customer accord-

ing to the segment to which the customer

belongs (hereafter we call this ability to offer

individually customised marketing micromarket-

ing). We will answer the following business

issues:

(1) Optimal pricing under item availability effect :

Using an analytical model, we determine

the optimal pricing under item availability

effect on sales when the e-business can

access the information on the customer

purchasing behaviour. Sensitivity analysis to

the optimal revenue explores how manage-

ment can handle its customers so as to

improve its business performance.

(2) Comparison of segmentation versus non-seg-

mentation strategy. We compare two scenar-

ios for e-business. One is a scenario

where e-business does not utilise segmen-tation information and offers the same

pricing and item availability plan to the

entire market (hereafter called this a

homogeneous scenario). The other is a case

where e-business applies obtained beha-

viour information to develop a customised

pricing and item availability plan to a

specific customer. That is, each customer

receives a different marketing offer from

the e-business (hereafter, we call this a

customised scenario). We compare the per-formance of the two scenarios. We also

investigate how accuracy of the segmenta-

tion decision and market share of each

segment influences the performance of the

customised scenario.

In the remainder of this section, we review

existing papers related to this study. From the

Journal of Revenue and Pricing Management Vol. 5, 4 305–314 &

2007 Palgrave Macmillan Ltd, 1476-6930 $30.00306

Customisation of pricing and item availability information

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operations management side, several papers

have already explored the effects of inventory

level on demand. For example, Baker and

Urban (1988) investigated the optimal ordering

decision for inventory-dependent demand

where deterministic demand rate is a poly-

nomial function of the stocking level. They

developed a nonlinear programming algorithm

to search the solution. Datta and Pal (1990)

analysed the inventory model in which the

demand rate is a linear function of the

inventory level. Using an optimisation ap-

proach, they determined the maximum ex-

pected profit and the optimal order cycle time.

Hwang and Hahn (2000) explored inventory

management for perishable items with an

inventory-dependent demand. Their profitmaximisation model determines the optimal

order-up-to level and the optimal order cycle.

These operations papers focus on the optimum

search, but rarely refer to consumer behaviour

and heterogeneity, which are important to

develop an effective marketing strategy for a

retail business.

Individually customised marketing decisions

have been addressed in the literature by

marketing researchers. Using a game theoretic

framework, for example, Chen et al . (2001)

investigated how a firm’s ability to predict

purchasing behaviour of individual customers

(which they called targetability) influenced the

performance of individually customised mar-

keting. Their finding is that higher targetability

results in higher profit for the firm offering

micromarketing. This is consistent with our

numerical result that examines the effect of

segmentation correctness on revenue.

Note that our assumption that the informa-

tion about the number of remaining items

offered by a retailer influences customers’purchasing decisions is explained by consumer

behaviour and applied psychology research.

Brehm (1966) proposed reactance theory,

which means that when an individual’s beha-

vioural freedom is threatened, the person reacts

to restore his/her endangered freedom. Clee

and Wicklund (1980) discussed psychological

reactance from a marketing perspective. They

provided an example in which consumers

regard the product as more attractive under

conditions of product scarcity. Inman et al .

(1997) empirically studied how the presence of

a restriction (eg, purchase limit) increases the

perceived deal value and encourages consumer

choice.

The remainder of this paper is organised as

follows. The next section explains the funda-

mental concepts of demand-dependent inven-

tory, e-commerce, and response function. We

develop an analytical model to determine the

optimal price over time under the item

availability effect on demand. Then we com-

pare the aforementioned two scenarios and

analyse the difference. In addition, we proposeseveral numerical examples that may offer

additional insights on the effects of a custo-

mised marketing plan on e-business. Finally, we

conclude this paper.

MODEL

This section explains the fundamental concepts

for our model formulation and analysis.

Unique characteristics of e-commerce

Compared to traditional brick-and-mortar

stores, e-commerce has several unique char-

acteristics. For example, Viswanathan (2005)

lists customisation, interactivity, multimedia

abilities, global accessibility, inexpensive acces-

sibility to information, and real-time interac-

tions. This research focuses on the following

two unique characteristics of e-business: (1) an

e-business can perfectly control offers of

inventory level information about the item to

customers, and (2) an e-business can easily trace

the purchasing history and information search

process of a customer.As for the first characteristic, it is common

for e-stores to offer the number of available

items on a website when a customer is

searching product/service information for the

item that he/she is interested in (eg, Amazon.-

com often displays item availability information

such as ‘Only 5 left in stock–order soon’).


Kurata and Bonifield

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Clearly, an e-business can control whether to

display the item availability information to

customers. If an e-business does not display

certain items, it is impossible for customers to

know the actual number of available items,

whereas it is difficult for a traditional store to

perfectly hide inventory information from

customers. As earlier mentioned, item avail-

ability is one of the marketing factors that

might stimulate customers’ purchasing deci-

sions. In particular, manipulating the intensity

of item availability offers to business customers

who put strong emphasis on date and time can

be an effective strategy for e-business.

The second characteristic, e-commerce’s

accessibility to customer’s behaviour, is closely

related to micromarketing, a store’s ability tooffer the customer a specifically customised

marketing action that might be effective with

this particular customer. One good example of

a micromarketing application is the ‘Retail

Service’ offered by Catalina Marketing (Cata-

lina Marketing, 2005), which develops market-

ing actions for their customers according to

their past shopping patterns. Micromarketing is

also frequently applied in e-business: e-com-

merce websites often show recommended

items to the customer based on his/her

purchasing history (eg, Amazon.com recom-

mends a newly released sci-fi movie DVD to

customers who bought a science fiction video

or book in the past). Our model assumes that

e-business can access the customer’s informa-

tion search and purchasing records and deter-

mine which segment a customer who is

currently looking at its website belongs to, a

leisure segment or a business segment.

Demand function

We assume the following relationship betweeninventory level x, retail price p, and demand

rate d

d ¼ m ap bx ð1Þ

where a>0 is a given price sensitivity, b>0 is a

given sensitivity of item availability, and m is a

given market size. We assume that m is large

enough to avoid negative demand. We use a

linear response model. In addition to analytical

tractability, we can present the plausibility of a

linear demand function as follows. A linear

demand function often achieves a satisfactory

fit to the given dataset (see Simon, 1989). Also,

our model follows a tradition of microeco-

nomics analysis (for example, Tirole, 1988;

Wolfstetter, 1999) and marketing research

about brand management and pricing, such as

Raju et al . (1995) and Sayman et al . (2002).

Note that inventory-dependent demand de-

scribed by equation (1) was often applied for

existing inventory-dependent demand papers,

such as Datta and Pal (1990) and Sarker et al .

(1997). Demand in equation (1) behaves as an

increasing function with respect to time, asshown in Figure 1.

Revenue functions

A revenue p( p, x) at a specific time t , where

retail price p ¼ p(t ) and inventory level x ¼ x(t ),is determined as follows.

pð p; xÞjx¼xðt Þ ¼ pd ¼ pðm ap bxÞ

The total revenue P over period from t ¼ 0 tot ¼ T is determined as,

P ¼

Z t ¼T t ¼0

ð pd Þdt ¼

Z t ¼T t ¼0

pðm ap bxÞdt ð2Þ

subject to .

x ¼ d ¼ m þ ap þ bx, and aninitial inventory level is x(t ¼ 0) ¼ x0.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.00.40.81.21.62.0

Remaining Time

D e m a n d

Figure 1: Demand change over time




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Solving the optimal control model above,

we can obtain the following generic solution

for our revenue maximisation.

Proposition 1: Optimal solutions

(a) For the optimal control model in (2), the

optimal multiplier, inventory trajectory, and

price at time t are defined as:

l ¼ ebt þ p

x ¼ ða=bÞðebt 1Þ þ x0

p ¼m þ a bx0

a

ð3Þ

(b) The total revenue at time T is determinedas

P ¼

ðm a bx0Þ

bð1 ebT Þ ð4Þ

Proof: See Appendix.

We know from Proposition 1 that the

optimal inventory level x decreases in t , while

the optimal price p is constant over time.

Figure 2 illustrates the change of x and pover

time.

A constant optimal price is plausible. This is

because, from our observation of pricing

behaviour in e-business in the hotel and air

industries (eg, Expedia.com, Orbitz.com, etc.),

it is rare that e-business changes its retail price

day by day. In contrast, the price is usually kept

constant over time once it has been decided.

Another possibility is that fixing the price at a

specific level over time is a common pricing

strategy in the hotel and air industries. For

example, discount airlines often use price

appeals in their advertising, such as selling a$99 ticket between two major cities. Moreover,

our optimal price equation (4) describes a

realistic pricing situation in which, if many

customers are eager to get accommodations on

the day of a special event, such as Indianapolis

hotels on the day of the Indy 500 race, or New

Orleans hotels on the days of Mardi Gras, then

the price will initially be set very high (ie, high

m and small a will increase p).

The effect of parameter value changes on

the optimal profits is summarised below.

Proposition 2: Sensitivities of the

optimal revenue

(a) P increases in m and T .

(b) P decreases in a, b, and x0.

Proof. See the Appendix.

CUSTOMISED SCENARIO VERSUS

HOMOGENEOUS SCENARIO

This section explores how e-business can

improve its revenue by offering customised

pricing and item availability information to

customers when dividing its customers into

two segments: a business segment and a leisure

segment . Note that, for the homogeneous

scenario, the e-business offers the same price

p and normal item availability information to

all its customers while, for the customised

scenario, management offers high price pB (> p)and intensive information about item avail-

ability to the business segment and low price pL(o p) and weak item availability information to

the leisure segment. Figure 3 visualises the

difference between the two scenarios with

respect to the pricing and item availability

information decision.

10

15

20

0.0 0.5 1.0

t (time)

P r i

c e / I n v e n t o r y l e v e l

p (price) x (inventory)

Figure 2: Trajectories of the inventory level x and

retail price p over time



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Modelling two scenarios

The sensitivities of price and item availability

between business and leisure segments in the

customised scenario are considered to have the

following relationship:

aBoaL and bB4bL ð5Þ

We assume that, for the homogeneous scenario,

the sensitivities are determined as a weighted

average of the two segments in the customised

scenario. That is,

a ¼ ðmBaB þ mLaLÞ=ðmB þ mLÞ

and b ¼ ðmBbB þ mLbLÞ=ðmB þ mLÞ

Note that the sensitivity parameters have the

following relationship:

aBoaoaL and bB4b4bL

The total revenues for homogeneous and

customised scenarios are defined as follows.

Phomtotal ¼P

homB þ P

homL

¼ðmB a bx0Þð1 e

bT Þb

þðmL a bx0Þð1 e

bT Þb

Pcustotal ¼P

cusB þ P

cusL

¼ðmB aB bBx0Þð1 ebBT Þ

bB

þðmL aL bLx0Þð1 e

bLT Þ

bL

To quantify the difference in revenues

between the two scenarios, we determined

e-business

Business segment

Leisure segment

Homogeneous scenario

Normal item

availability offer

Price p

Customers

e-business

Business segment

Leisure segment

Customized scenario

Price p B

Price p L

Intensive item availability

offer for Business segment

Weak item availability offer

for Leisure segment

Customers

Figure 3: Homogeneous scenario versus customised scenario




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DP as

DP ¼ Pcustotal Phomtotal

Note that if DP

>0, then we conclude that itis profitable for e-commerce to utilise con-

sumer behaviour information to determine

optimal profit and item availability customisa-

tion. However, if DPo0, then there is no need

to consider segmentation for the e-business.

Proposition 3 addresses whether customisation

is beneficial or not.

Proposition 3: Effect of a and b on DP

(a) DP¼ 0 when only information of a is

customised.(b) DPX0 when only information of b is

customised.

(c) DP is maximised at the b which satisfies

qðPhomB þ PhomL Þ

qb¼P

cusB P

cusL

bB bL

Proof: See the Appendix.

Proposition 3 suggests that a customised

model outperforms a homogeneous model if

the e-business offers customised item availabil-

ity information to customers. However, seg-

mentation with respect to customer price

sensitivity does not make a customised model

better than a homogeneous model. Note that

there often exists a situation where e-com-

merce is allowed to customise item availability

offers, but retail price is supposed to be the

same throughout the market. One well-cited

pricing mistake is that Amazon.com offered a

higher price to channel loyal customers than

the price for channel switchers, and this kind of

price discrimination was claimed to be unfair (Streitfeld, 2000). That is, one must be aware

that sometimes price discrimination according

to how much a customer is willing to pay for

the item (ie, first degree price discrimination) is

not suitable for e-commerce, even if price

discrimination is not problematic in several

traditionally acceptable situations, such as a

discount for bulk-purchasing (ie, second degree

price discrimination) or discounts for seniors

and students (ie third degree price discrimina-

tion). Thus, our business implication is that e-

commerce can raise its revenue by customising

item availability offers only without risking

price discrimination. We shall note that,

although our analysis shows that e-commerce

can raise revenue as long as management can

categorise customers into either the business or

leisure segment, the remaining question is how

to segment. Possible approaches are usage of

demographic information or application of

specific statistical methods to customers’ panel

data. However, segmentation techniques are

not the objective of this research. We will leave

the details of customer segmentation to Wedeland Kamakura (1999).

NUMERICAL EXAMPLES

The purpose of numerical examples is twofold:

visualisation of our analytical discussion might

support reader’s understanding, and some

business questions are hard to explore via the

analytical approach so that numerical examples

can give us insight on such tough questions.

Particularly, this section tries to answer the

following questions:

(1) Does the difference in total revenue be-

tween the two scenarios (ie, DP) change if a

segmentation share (ie, the ratio of mB and

mL) changes?

(2) How will the profitability of the custo-

mised model be undermined if segmenting

customers to two groups is not accurate

enough?

Arbitrary parameter settings in this section are:

mB0 ¼ mL0¼ 100; aB ¼ 1.4; aL ¼ 1.6; mB ¼ 0.6;

mL ¼ 0.4; x0 ¼ 30; and T ¼ 10.As for the first question, Figure 4 illustrates

the effect of the business segment share on the

difference in revenue between two scenarios.

Note that the business segment share S B is

defined as S B ¼ mB/(mB þ mL).We know from Figure 4 that when the two

segments are of similar size (ie, S BE0.5), the



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customised scenario greatly outperforms the

homogeneous scenario, whereas if one segment

is much larger than the other (ie, S BE0 or S BE1), the advantage of information custo-

misation is dampened. This is intuitive: seg-

mentation has meaning in business strategy

only when more than one segment of sig-

nificant size exists in a market.

As for the second question, we assume that

each customer has his/her own reference price

pR and if the price offered by the e-commerce

is less than or equal to this reference price, he/

she will buy; otherwise, no purchase occurs.

Let pBR and pL

R be the reference prices of a

customer belonging to the business segment

and leisure segment, respectively. We assume

that the reference prices follow a uniform

distribution: pBRBU (mB

RHR,mBR þ HR) and

pLRBU (mL

RHR,mLR þ HR), where HR repre-

sents the half-range of the uniform distribution.

Note that the uniform distribution corresponds

to our linear response function. Assumption (5)

results in pB> pL

. Thus, the distribution of

reference prices belonging to the business

segment is located higher on the price axis

than that of the leisure segment. Under thissetting, a customer’s choice works as follows. If

pL is mistakenly offered to a business traveller

whose reference price is pR, then he/she will

buy if pL

p pR and the retailer receives pL as

revenue. However, compared to the case where

the e-business correctly offers pB to the business

traveller, the retailer loses pB pL

as the cost of

inaccurate segmentation. Also, if pL> pR, the

customer does not buy so that the retailer

receives zero revenue. Note that the same

outcome happens for the case that pB

ismistakenly offered to a leisure customer.

As shown above, the accuracy of the

segmentation decision plays an important role

in the performance of the customised scenario.

If mistakes happen quite often in segmentation,

the e-business may lose large amounts of

revenue. Figure 5 is the outcome of a

simulation analysis of the effect of segmentation

accuracy on performance. In this simulation

example, we change the percentage of wrong

segmentation from 0 to 100 per cent, with

pB ¼ 27.4, pL

¼ 24.0, and HR ¼ 5.0. Note thatthe basic parameter setting in Figure 5 is the

same as that of Figure 4.

What we know from Figure 5 is that the

average revenue for the customised scenario

increases in proportion to the accuracy of the

segmentation decision (ie, percentage of cor-

rect segmentation). In other words, as firms

make more segmentation mistakes, more lost

sales occur to the e-business and we can

reasonably imagine that customers lost due to

segmentation errors might switch to another retailer. This numerical result is consistent with

the analytical result by Chen et al . (2001). As

we have discussed before, the customised

scenario theoretically outperforms the homo-

geneous scenario. However, in a real custo-

mised marketing strategy, perfectly correct

segmentation is hard to achieve: some custo-

250

300

350

400

450

0 10 20 30 40 50 60 70 80 90 100

Share of Business segment (%)

P r o f i t

Customized scenario Homogeneous scenario

Figure 4: Effect of segment share on the difference

between the two scenarios

26

27

27

28

28

29

29

0 10 20 30 40 50 60 70 80 90 100

Percentage of segmentation accuracy

A

v e r a g e p r o f i t

Average profit

Figure 5: Effect of segmentation accuracy on the

customised scenario performance




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mers are mistakenly categorised into the wrong

group. Our numerical example shows that a

reliable segmentation method is critical for the

implementation of a customised marketing

strategy.

CONCLUDING REMARKS

Two characteristics of e-commerce, the ability

to micromarket (ie, to customise a marketing

plan according to customers’ purchasing pat-

terns) and the ability to offer item availability

information, considerably increase the potential

to improve a firm’s performance. We have

analysed revenue management issues for e-

business in the hotel or airline industry, in

which there are two segments: price-conscious

customers (leisure travellers) and schedule-con-scious customers (business travellers). We provide

an analytical model that determines the optimal

pricing under item availability effect over time.

We then show how firms can improve their

total revenue in the e-business area by taking

into account customer segmentation and

offering customised item availability informa-

tion to customers. Our numerical examples

demonstrate that the accuracy of segmenting

customers and the size of each segment will

influence the performance of customised

marketing planning.

Finally, several viable directions for future

research exist. First, as analytical research, we

have formulated a linear response function.

This linear assumption should be confirmed or

adjusted by empirical research that can deter-

mine a specific shape of customers’ response

function to price and item availability. Second,

some e-business firms offer item availability

information, but the timing of purchase is less

important (ie, sales of books or CDs). In our

paper, one of the primary ways that industriessegment is business versus leisure segments. It

would be interesting to analyse another type of

business situation that differs in terms of its

segments. Third, to focus on our main research

goal, our model does not include some other

common occurrences for hotel or airplane

sales, such as overbooking, lost sales due to

limited capacity, and cancellation of booking.

Including these factors may enrich our discus-

sion.

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APPENDIX

Proof of Proposition 1. Set Hamiltonian as

H ¼ pðm ap bxÞ þ lðm þ ap þ bxÞ

where l is a Lagrange multiplier. The condi-tions for optimality are:

H p ¼ ðm ap bxÞ að p lÞ ¼ 0 ðA:1Þ

l0 ¼ H x ¼ bð p lÞ ðA:2Þ

Solving (A.2) with respect to l, we obtain

l ¼ ebt þ p ðA:3Þ

From (A.1) and (A.3),

ðm ap bxÞ þ ae bt ¼ 0

That is .

x ¼ aebt ðA:4Þ

From (A.4) and the initial condition x(0) ¼ x0,we obtain

x ¼ ða=bÞðebt 1Þ þ x0 ðA:5Þ

Plug (A.3) and (A.5) into (A.1),

p ¼ (m þ abx0)/a. Finally,

P

¼Z t ¼T

t ¼0ð pd Þdt ¼

Z t ¼T

t ¼0

ðm a bx0Þ

a

ðaebt Þdt ðm a bx0Þ

bð1 ebT Þ &

Proof of Proposition 2 From the structure of

Equation (4), it is obvious that P increase in m

and T , but linearly decreases in a and x0. As for

the sensitivity of b, set G (b)((ma/b)x0) andH (b)(1ebT). Clearly, both G (b) monoto-nously decreases in b, while H (b) monoto-

nously increases in b. Also, limb-0G (b)

¼N>0, limb-NG (b) ¼ x0o0, limb-0H (b)¼ 0, and limb-NH (b) ¼ 1>0. We are assumingplausibly that m is large enough compared to

H j (b), which is bound between 0 and 1. Hence,

if so, the change of H (b) with respect to the

change of b can be ignored when compared to

qG (b)/qb ¼ (ma)/b2. Thus (qH (b)/qb)/(qG (b)/qb)D0 At this time,

qP=qb ¼ qG ðbÞ

qbH ðbÞ þ G ðbÞ qH ðbÞ

qb

ffiqG ðbÞ

qbH 1ðbÞ¼

ð1ÞðmaÞ

b2 ð1 ebt Þ

Then, ((q2P)/(qb2)D(2(ma))/(b3))(1ebt )>0.As a result, P

is convex in b. &

Proof of Proposition 3

(a) Proposition 2 mentions that P is linear in a

so that, obviously, DP¼ 0.(b) From Proposition 2, P is concave in b so

that, from the definition of a concave

function, DPX0.

(c) The slop of the line connecting the two

points, PBcus and PL

cus, is defined as

PcusB P

cusL

bB bL

It is obvious that when the slop of the

tangent line attaching to P

total

hom

(

b) at

b isequivalent to the slop of the line connect-

ing PBcus and PL

cus, DP is maximised. &




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