International Game Theory Review, Vol. 7, No.2 (2005) 137-150
@ World Scientific Publishing Company
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DYNAMIC
ONLINE AND OFFLINE CHANNEL
PRICING FOR HETEROGENEOUS
CUSTOMERS
IN VIRTUAL ACCEPTANCE
GILA E. FRUCHTER.
Graduate School of Business Administraction
Bar-Ran University
Ramat-Gan 52900, Israel
fruchtg@mail.biu.ac.il
CHARLES S. TAPIERO
De1Xlrtmentof Mathematicsand Interdisciplinary Studies
Bar-Ran University
Ramat-Gan 52900,Ismel
We consider a manufacturer's dual distributions channels consisting on the one hand of
a virtual (online) channel operated directly by a manufacturer and on the other hand
of a real (offline) channel operated by an intermediate retailer. Customers are assumed
heterogeneous in their virtual acceptance, deriving a surplus according to the channel
they shop at. Assuming that customers' derived benefits are random with a known
probability distribution, we obtain a probabilistic model, which is used to construct an
inter-temporal model for shopping online. In addition, we suppose that the retailer usesa
markup pricing strategy and has a strategic role. This results in a Stackleberg differential
game where the manufacturer is leader and the retailer is a follower. The optimal policy
shows that the manufacturer charges the same price across both channels. This finding
is consistent with classical results in economics. However, our research goes beyond this
observation and indicates that the online price, the retailer's markup and the probability
to buy are affected by consumers' heterogeneity in a specific manner. Moreover, we show
that while the retailer sets a price equal to the product value, the online price is lower
and is equal to the product value less the guarantee provided by the manufacturer for
the risk the customer take to buy online. This guarantee is not discriminating and is
set to the risk of the customer with the lowest virtual acceptance. Finally, we show that
the introduction of the online store is a win-win strategy; both the customers and the
manufacturer are better off.
Keywords: Online marketing; dynamic pricing; channels of distribution;
e-commerce;
competitive strategy; game theory.
1. Introduction
The advent of the Internet has made it easier for many manufacturers who traditionally distribute their products through retailers to engage in online sales.
.Corresponding author.
137
138 G. E. Pr1J.chter
& C. S. Tapireo
According to a survey reported in the New York Times (Tedeschi, 2000), about
42% of top suppliers (e.g., IBM, Pioneer Electronics, Cisco System, Estee Lauder,
and Nike) in a variety of industries have begun to sell directly to consumersover
the Internet. In the computer industry, in addition to IBM, major manufacturers such as Compaq and HP now sell their products directly over the Internet.
Even in a country such as Japan where traditional distribution channelshave been
complex and close-knit, global electronics giant Sony has recently launched online
selling directly to Japaneseconsumers(Rahman, 2000). While more and more manufacturers are engagedin direct sales (see also, McWilliams, 1997), retailers voice
the belief that orders placed through a manufacturer's direct channel are orders
that should have been placed through them. However, direct sales typically are
modest. In 1999, the conventional bricks-and-mortal stores rang up 93% of United
States retail salesrevenue; e-commerce,by contrast, accounted for about 1% and
catalog sales the other 6% (Collet, 1999). The importance of electronic commerce
and purchasing has also been recognizedin academic research,e.g., Benjamin and
Wigand (1995) Smiee (1998) Javalgi and'Ramsey (2001), Burroughs and Sabherval
(2002).
Why does a manufacturer add an online store to distribute directly his product
rather than letting an independent retailer distribute it? To answer this question
we develop a simple model consisting of a manufacturer distributing a product to
a market of potential heterogeneouscustomers. The manufacturer may distribute
the product through either or both of two shopping choices:a network of physical
stores operated by an independent retailer and an online store operated directly
by the manufacturer. For each individual customer, the choice is between the closest physical store and the online store. We assumethat customers differ in their
acceptanceof the online channel and let consumers'heterogeneitybe expressedby
a known probability distribution. Moreover, we assumethat the probability to buy
online changesover time, reflecting technological trends and improvements in online shopping aswell as consumers'increasedsophistication in using online channels.
This results in a theory for optimal dynamic pricing of the dual channelsimplied
by the solution of a dynamic hierarchical game-theoretic problem, where the manufacturer is the price setting leader and the retailer, is a follower, setting a markup
pricing competitive strategy. A number of results are then obtained. For example,
we find that under risk neutral pricing, the manufacturer chargesthe sameprice to
the retailer and to the online customer. Our results also shed light on the way the
customer heterogeneousacceptanceof online shopping affects the pricing strategy.
Finally, we show that the introduction of an online store is a win-win strategy where
both the manufacturer and customers are better off. We are also pointing out that
with the technology improvements when more and more customers will switch to
online purchase,virtual stores will be predominant.
Previous work on dual channels (online manufacturer + independent retailer)
has focused on customers that are homogeneousin their acceptanceof online buying (Chiang et at., 2003). Dual channels with heterogeneousmarket in a static
Dynamic Online and Offline Channel Pricing
139
environment has been investigated by many researchers;among them, Moriarty
and Moran (1990), show that dual channels may reach potential buyer segments
that could not be reached by a single channel. Other streams of related literature deal with the channel coordination problem between a manufacturer and a
retailer. Among an extensive literature in a static environment we refer to Jeuland
and Shugan (1983), Ingene and Parry (1995) and over time to Jorgensenand Zaccour (2002). Finally, the related literature on supply-chain managementhas also
contributed broadly to this problem. For example, we note contributions by Abad
(1994), Parlar and Wang (1994), Weng (1994), Tersine and Barman (1995), Gilo
(1999), Netessine and Rudi (2001) and Ertek and Griffin (2002).
The rest of the paper is organized as follows. The model is discussedin the next
section. The optimal policy for a risk-neutral manufacturer and a strategic retailer
is developedin the third section. The last section reviews the results and discusses
directions for future research.
2. The Model and Notation
We consider a manufacturer who distributes a product to a market of N potential
customers.The manufacturer may distribute the product to either or both of two
shopping choices:a network of physical stores operated by an independent retailer
and an online store operated directly by the manufacturer. To each individual customer, the choice is betweenthe closestphysical store and the online store, so from
now on our discussion in the paper will be framed as if there is only one physical
store. Such choicesare broadly available to customerstoday. For example, a potential traveler may buy travel tickets through a travel agent or through the Internet
-
accessingdirectly to the company.
Buying through an online store versus an offline store (operated by a retailer)
inducesan economic-addedvalue, which is a function of the buying processparticularities. For example, buying a good in a retail store allows the customer to sensethe
product prior to its acquisition in addition to an implied guaranteeassociatedto the
retailer responsibilities. In contrast, buying through an online store provides a virtual inspection of the product. For theseand many other and related reasons,buying
through one channel or the other induces different economic-addedvalues. Manufacturers are aware of such specificities and thus they design policies that recognize
both customers' propensity to buy from one or the other channel and of course,the
selling terms exercisedat each channel. For instance, it is broadly expected that
buying travel tickets through an online site may have lower prices, usually compensated by a lack of personal contact and service, which a customer may require.
To better understand a customer acceptanceof the online channel, we restrict
this model to a homogeneousproduct while customers value on-line shopping
heterogeneouslywith a known distribution. Let V be the value of a homogenous
product in real inspection, i.e., when it is bought through a retailer. When the product is bought online through virtual inspection, we assumethat it has a smaller
140
G. E. Fr1Jchter
fj C. S. Tapireo
value (}V, where 0 ~ () ~ 1, due to the lack of contact between the consumer
and the product at its acquisition. The parameter () thus representsthe acceptance
of the virtual inspection by the consumer. We label this term as a consumer virtual acceptance.To expressconsumers'heterogeneity in the virtual acceptancewe
assume that () is a random variable with cumulative density function Fe('). The
determination of Fe(.) is an empirical question revealed by customer distribution
observed along channels acquisition. However, in the sequel, Fe(') is consideredas
an arbitrary general distribution function.
The economic-addedvalue provided to a consumerby the product can be measured by the concept of consumersurplus, which is equal in our model to the consumer value from the product, lessits price. Therefore, if the product is sold through
a retailer at a price Pr, then the resulting consumersurplus is V - Pro If the product
is sold online at the price Pan, then the resulting consumersurplus is (}V - Pan.
The decision to buy online based on the customer surplus is thus a random
event and occurs if:
-
tJT1'
v' -pan---'
~
T1' -
.
-Pr
~'>
n..
v..
v~
1
- !?:-
v-
~
Pan
(21)
\-.~,
Given a pair of prices Pan and Pr, the probability that a customer buys online is
thus:
ij(Pon,Pr)=
F9
1-
Pr - Pan
V
Dynamics: Over time, the propensity to buy online as well as the pricing policies
the manufacturer can exercise,may change.As a result, customersmay switch from
one channel to the other according the higher customer surplus they may derive
at a specific time by buying in one or the other channels. For example, buying
through a retailer may be induced by a changein the online price policy, reducing
consumer's surplus. As a result the probability that a customer will buy online or
through a retailer will also changeover the time.
Let q(t) be the probability that a customer will buy the product through an
online site at time t. We assumethe following specification for the rate of growth
of the probability q(t):
q(t)
= -a(q(t) - ij(Pon(t),Pr(t))),
q(O)= qQ, 0 ~ q ~ 1,
(2.3)
where qQis the initial condition and a > O. Note that from (2.3), if 0 ~ ij ~ 1,
we also have 0 ~ q ~ 1. In Eq. (2.3) we assume that the rate of change is a
monotonically nonincreasing (nondecreasing)function of the distance between the
current probability of buying online and the limit-probability of buying onlinel if
q(t) > ij(q(t) < ij).
Revenues:The manufacturer's revenue comes from both online and offline stores
sales.We assumethat the manufacturer supplies the retailer of the physical store at
1That is, the probability at steady-state, i.e., when q = O.
~
Dynamic Online and Offline Channel Pricing
141
a wholesaleprice, w. For simplicity, we consider that the retailer will use a markup
b for pricing, i.e.,
Pr
= (1+ b)w.
Thus, (2.2) becomes
ij(Pon,Pr:
- Fe( 1-
(1 + b)w - Pan
V
To avoid arbitrage it is necessarythat
w ~ Pono
Using the offline channel,the manufacturer incurs a cost per unit Cr that includes
the cost of manufacturing and logistics, while using the online channel, a cost per
unit Con.Normally we would expect that Con< Cr. Assuming that the manufacturer is risk-neutral, then the manufacturer's expected revenueat time t, denoted
by R(t), is given by
R(t)
.(Pon
Con)q(t)+ (w
Cr)(
q(t))]N,
where Pan = Pon(t) and w = w(t). Note that in this model, we assumethat the
manufacturer has no influence on the total number of customers,thus it is specified
exogenously.Therefore, the problem associatedto efforts that may contribute to a
growth of customers and increasethe number customers is not considered in this
model.
The decision problem: The manufacturer's decision problem, acting as a leader, is
to choosethe pricing strategy, Won,w) to maximize its own discounted profits over
a given time horizon, given that the retailer acts as a strategic follower in choosing
the markup b to maximize its own profits. Next we find the optimal pricing policy.
3. Optimal
Pricing Policy
Assumea risk-neutral manufacturer and a strategic retailer. This implies a dynamic
hierarchical game-theoretic problem. More exactly, we consider a Stackelberg differential game, with the manufacturer acting as the leader and the competitive
retailer as the follower.
Assuming that firms maximize the present value of their profit stream over an
infinite horizon, the manufacturer maximizes llm with respect to Pan and w, where
llm = l°O[(Pon - con)q+ (w - Cr)(1 - q)]Ne-rt dt.
(3.1)
The competitive retailer maximizes fir with respect to b, where
fir
= l°O[(Pr
- w)(1 -
q)N]e-rtdt.
(3.2)
142 G. E. Fruchter & C. S. Tapireo
Considering (2.4), we obtain
IIr
= hoo
bw(l
-
q)N e-rt dt.
(3.3)
Overall, the manufacturer and retailer's optimization problems can be summarized
by the following differential game:
Max IIm
=
roo {(Pon - con) q + (w
10
=
roo bw(l
10
Pon,W
Max
b
..
IIr
- q)N e-rt
s.t. Q=-a[Q-1+F91,
- Cr)(l
- q)} Ne-6t dt
dt
1-
(3.4)
(1
+ b)w
- Pan
.
.
v
--
q(O) = QQ.
To solvethe problem (3.4), given (2.6), we usedynamic optimization techniques,
see for example, Tapiero (1977, 1986, 1998) and Kamien and Schwartz (1991). We
start by solving the follower's problem, which is an optimal control probleQl. We
compute the markup of the competitive retailer as the rational reaction to the prices
used by the manufacturer. Subsequently,we solve an optimal control problem of
the manufacturer, taking into account the constraints on the manufacturer and the
reaction function of the competitive retailer. We obtain the following result with
the proof relegated to the appendix.
Theorem 1 (Optimal Dual Channel Policy). Consider a risk-neutral manufacturer who distributes a product directly by an online store and indirectly by a
retailer to a fixed number of customers that are heterogeneouswith respect to the
virtual acceptance.The manufacturer acts as a leader and the retailer as a follower.
Assume that the retailer is using a markup pricing, b, and there is no arbitrage, i.e.,
the wholesaleprice, w, is not exceedingthe online price, Pon. Then, optimal pricing
stmtegies (P~n'w*) and b*, are given by,
NV(l - q(t))
P~n = w*
= Vu
and
b* = ~ u
1
with u = u(t;0) = IiI
a Irp[q(t)]I
(3.5)
and representa Stackelbergequilibrium for the leader and follower. The corresponding retail price is p; = V. In (3.5), V is the customerproductvaluationthrougha
real inspection, q(t) is the probability to buy online and is as in (3.4), N is the total
number of customers, () is the virtual acceptance,fe is the corresponding density
function; the function cp(q) = cp[q(t)] is the solution of the following differential
equation,
-a[q -1 + Fe(u)]cp'(q)= (r
+ a)cp(q)+ (1- u)VN,
Jim cp(q(t))e-rt o. (3.6)
t-+oo
where Fe is the correspondingcumulative density function.
Proof.
SeeAppendix.
0
Dynamic Online and Offline Channel Pricing
143
ConsideringTheorem 1, the manufacturer price policy is the samefor both channels,
expressinga uniform marginal pricing procedure. This is consistent with classical
results in economics. However, both the retailer's markup and the online price
changeover time and are affected by consumers'heterogeneity in a specific manner
"
,
through u(t; fJ) = IiI I'~~
1)
,
.
/
Online and offline purchase.Considering Theorem 1 we can also characterizeonline
and offline purchase. Considering (3.5) the online customer surplus is (0 - u)V
and the offline customer surplus is always zero. Thus the offline customer gets
exactly what he pays. However, as the online purchase occurs only if the online
customer surplus exceedsthe offline customer surplus, (see (2.1)), we obtain that
the online customer surplus (0 - u)V is always positive and is zero only for the
marginal customer, that is the customer with virtual acceptance0 = u. Thus a
customer is better off with an online purchase. Moreover, all the customers with
virtual acceptancein the interval [u,1] buy online purchase, otherwise they buy
through the retailer. Thus, customers'with higher virtual acceptance (closer to 1)
have higher benefit from the online purchase.
Let
r(8) = (1 - 8)V.
The increment r( 9) can be thought as the risk a customer takes to buy online.
As 9 increasesthe customer's risk decreases.The increment
g(t; (J)= (1 - u)V
can be thought as an online guaranteeprovided by the manufacturer for the risk a
customer takes to buy online or shortly risk guarantee.For a customer with virtual
acceptance0 < u(t), this guarantee is lower than his risk thus he buys offline. For
a customer with virtual acceptance0 > u(t), the guaranteeexceedshis risk thus he
buys online. In our model, if the customer is more sophisticated and experienced
(i.e., his virtual acceptanceis closer to 1) the benefit from the guarantee provided
is higher. This happens because in our model the manufacturer's policy is nondiscriminating.
Note that, given an arbitrary distribution of customers with respect to their
virtual acceptance,online purchase market share will be equal to 1 - F9(U) (see
Fig. 1) which is exactly the limit-probability to buy online ij. Following Fig. 1, for
a given distribution, as u = u(t; 0) decreases,the market share of online purchase
increasesand so the probability to buy online; this is because the online price
decreases.Moreover, as the risk guarantee increases, the limit-probability to buy
online increases.
Figure 1 illustrates also that the increment u determines the propensity of consumers to buy online or offline. When the distribution f9(.) shifts to the left due
to increased sophistication of consumersin shopping online or through a gradual
144 G. E. Fruchter & C. S. Tapireo
Market Share
Fig. 1. Distribution of virtual acceptance.
integration of online shopping in consumershabit, then obviously the propensity
to buy online will grow.
A special case,that assumesthat () is uniform is also revealing.
Special Case:Let the consumervirtual acceptance() be uniformly distributed in the
interval [a, 1]. In this case, Fe(x) = (x - d)/(1 - a) and fe(x) = Fe(x) = 1/(1- a)
and therefore u = (1 - a) is constant and 1 ?: () ?: 1 - a. Thisimplies:
p~ = V(l-a)
From (3.4), the optimal
probability
trajectory
n
and b* = -.::.--
for probability
to buy and the equilibrium-
becomes,
Properties and Implications: For a risk-neutml manufacturer and a stmtegic
retailer, the pricing policy is non-discriminating:
(i) The manufacturer charyes the same price to the retailer and to the online
customer,i.e., P~n = w" (seeTheorem1).
(ii) The retailer sets a price equal to the product real value, V (see Theorem1).
(iii) The manufacturer sets a price equal to the product real value minus the risk
guarantee, Vu = V - g (see(3.5) and (3.8)).
(iv) The retailer's markup and the online price change over time and are affected
,-
by consumers' heterogeneity in a specific manner I through u(tj ())
=
f9-1
Dynamic Online and Offline Channel Pricing
145
(v) A customer is better off in buying online. Specifically, the online customer's
surplus is always positive and equal to (lJ - u)V, and is higher as his virtual
acceptanceis closer to 1. The offiine customer's surplus is always zero.
(vi) The market share to buy online is exactly the limit-probability to buy online.
Moreover, since the risk guaranteeincreasesthe market share and the limitprobability increaseas well.
(vii) Particularly, if the customer virtual acceptanceis uniformly distributed in the
interval [a , 1], with density 1/ (1- a), the online price, the wholesalesprice, the
retailer's markup and the equilibrium probability take the values,P~n= w* =
V (1- a), b* = ~, and ij = ~, respectively.The risk guaranteeis a V. Thus,
as the parameter "a" increases,the online price decreases.However, the risk
guarantee, the market share as well as the limit-probability of buying online
increase.
In these circumstances, why would a manufacturer add an online store? To answer
this question we compare the optimal pr?fit of the manufacturer for q
= 0 (without
the online store) and 0 < q < 1 (with the online store). We conclude with the
following result.
Corollary 1. Assuming that Con< Cr (the online distribution's cost is less than
the offline distribution's cost), then the introduction of the online store increases
the profit of the manufacturer and decreasesthe profit of the retailer.
Proof.
SeeAppendix.
0
Considering Corollary 1 and Property (v) of "Properties and Implications" we
conclude with the following result.
Corollary 2. Assuming that Con< Cr (the online distribution's cost is less than
the offline distribution's cost), then the introduction of an online store is a win-win
strategy. Both the manufacturer and customers are better off.
An interesting question that raises up now is the following: Does the manufacturer is better off with a mi.'1:ed
(dua~ channel composedfrom both the online and
the offline store or with a pure online channel? To answer this question we compare the optimal profit of the manufacturer for q = 1 (pure online channel) and for
0 < q < 1 (mixed channel). We conclude with the following result.
Corollary
3. Assuming that Con< Cr (the online distribution's cost is less than
the offline distribution'scost),then the mixed(dual) channelthat consistsfrom an
online and offline storeis lessprofitablethan a pure online channel.
Proof.
SeeAppendix.
0
Obviously, as long as the online shopping technology is not sufficiently friendly
for some customers. there will still be a need for an intermediate retailer to make
146 G. E. Fruchter & C. S. Tapireo
additional profits on that part of consumers reluctant to buy online. However,
technology improvements may contribute to customers switching from offline to
online purchase.Accordingly, Corollary 3, points out that eventually, virtual stores
will be predominant.
The results developed in this paper lead us to a number of research questions however which we discussbelow as well as provide somedirections for future
research.
4. Conclusions
and Future Directions
We have considered in this paper a manufacturer's dual channel of distribution
that focuseson a virtual channel operated directly by the manufacturer and a real
channel operated by an intermediate retailer. The virtual channel's customers are
different in their acceptanceof the online channel and randomly distributed. This
leads to a probabilistic model, where the random distribution determines the probability to buy online. We have also assumedthat the retailer used a markup prjcing
strategy and has a strategic role. Assuming that the probability to buy online
changesover time we developed the optimal pricing policy for the dual channel
implied from a dynamic hierarchical game-theoretic problem. We found that the
manufacturer adopts a non-discriminating pricing policy. Specifically, the optimal
policy shows that the manufacturer charges the same price across both channels.
This finding is consistent with classicalresults in economics.However,our research
goesbeyond this and indicates that the online price, the retailer's markup and the
probability to buy are affected by consumers' heterogeneity in a specific manner.
Moreover, we show that while the retailer sets a price equal to the product value,
the online price is lower and is equal to the product value lessa value that worth the
guaranteeprovided by the manufacturer for the risk a customer take to buy online.
This guarantee is not discriminating and is set to the risk of the customer with the
lowest virtual acceptance.Finally, we show that the introduction of the online store
is a win-win strategy; both the customers and the manufacturer are better off. We
are also pointing out that with the technology improvements when more and more
customerswill switch to online purchase, virtual stores will be predominant.
A number of extensions might be considered however. First, the marketing
mix policy (for example, including advertising, services provided by the retailer,
guaranteesprovided by the manufacturer) can be consideredand their effects on
consumers' channel selection assessed.Second,while we have used a risk neutral
framework, it is evident that risk aversion if considered would favor a discriminating price policy and assumethe survival of a channels portfolio. In this framework, we expect that the distribution of consumers' acceptancelevel will play an
important role. Finally, while the model consideredhere is of a theoretical nature,
empirical data regarding channels selection by consumers (reveling thereby the
consumers' acceptance level) can provide a strong empirical verification to out
analvtical results.
Dynamic Online and Offiine ChannelPricing 147
Appendix
Proof of Theorem 1.
1. The retailer (follower) optimization problem
The current-value Hamiltonian of the competitive retailer is given by
Hr(q,Pon(q),w(q),b(q),Ar:
= bw(l - q)N + Ar '-a
[q
-
1 +
F9
1-
(1 + b)w - Pan
'
"
v
,
,
,.I'
where Ar is the adjoint (or co-state) variable. The necessarycondition for this
problem is aHr/{}b = 0, or
I
(l+b)w-(1 - q)N + ArbaFg
.
Pan
1-
v
=0.
SinceF8 > 0, (AI) impliesthat
>'r'< O.
From (AI) we have
(1 - q)NV
v
-+
w
ArQ
The adjoint equation,2
-\.r
= TAr
-w
Pan
- 8Hr/8q, is
-\.r = TAr + bwN + QA,
and the terminal condition is
lim
t-+oo
Ar(t)e-rt
=
o.
(A5)
In what follows we use (A3) as the characterization of the competitive retailer's
reaction. The equation,
q= -a
{q-1
+ F9 rF~-l (
then becomesa constraint of the manufacturer's optimization problem.
Next we analyze the optimal control problem of the manufacturer.
2. The manufacturer (leader) optimization problem
The current-value Hamiltonian of the manufacturer is,
where Am is the adjoint (or co-state) variable.
2In case of an open loop solution.
(A6)
148 G. E. Fruchter & C. S. Tapireo
It is easy to seethat the necessaryconditions for this problem, oHm/oPon = 0
and oHm/ow = 0, cannot be satisfied. Thus, there is no interior solution for this
problem. Moreover, since ~
= qN > 0 and Pon .$:w is necessarythat P~n = w*.
tJPon
In addition, since ~
= (1 - q)N > 0 and Pr = (1 + b)w .$:V is necessarythat
w* = m. Thus we conclude with
and p; = v:
b= [l-F~-l(_i!~:!::)]
(- (l-q)NV
~'-l
9
ArQ )
where
.).r
= (r+a)Ar + [l-F~-J
Let
>'r = cp(q).
Considering (All)
(A9) becomes
(All)
and (A2) and since F8 = f(), where f() is the density function,
b* = .!. - 1
u
(A12)
where
(1 - q)NV
u = fil(
,
Icp(q)la
(A13)
,
Considering the boundary condition, we obtain that cp(q)satisfies,
-a[q
-1
= (r +a)cp(q) + (1- u)VN,
+F8(U)]cp'(q)
Finally substituting
lim cp(q(t))e-rt =0.
t-+oo
(A12) in (A8) we conclude with
P~n = w* = V u.
0
Proof of Corollary 1.
To provethis we needto comparethe optimal profit of the manufacturerfor q = 0
(without the online store) and 0 < q < 1 (with the online store). Considering
Theorem1 the optimal profit of the manufacturerbecomes
n:n =
Loo[(P~n
- con)q+ (w* - Cr)(l - q)]Ne-rtdt
= 100 {w* -lconq + Cr(l-
q)]}Ne-rtdt.
(A15)
Let
7rm(q)
= w* -
rCanq
+ c..(l - q)l.
0 <
Q <
I.
(A16)
Dynamic Online and Offline Channel Pricing
149
As we assumeCon< Cr, obtain for'v'q E (0,1),
7rm(q)= w*
On the other
- [conq+ Cr(l- q)] = w* hand, given that
q >
[(con - Cr)q + Cr] > w*
0 the retailer's
Jooob*w*(l- q)Ne-rtdt is diminished by (1- q).
-
optimal
Cr
= 7rm(O).
profit
n;
=
0
Proof of Corollary 3.
To prove this we compare the optimal profit of the manufacturer for q = 1 (pure
online channel) and for 0 < q < 1 (mixed channel). Using the notation in the proof
of Corollary 1, and again since Con< Cr, we obtain for'v'q E (0,1),
7rm(q) = w*
- [conq+ Cr(l- q)] = w*
- [(Cr-
con)(l- q)] - Con< w* - Con= 7rm(l).
0
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