Optimal Decision of Exploding Offer based on Consumer Search
Model
Lei Wang
School of Economics Shandong University, Jinan, Shandong, China
Keywords: Exploding Offer, Free Recall, Consumer Search.
Abstract: Exploding offer becomes more and more popular business management strategy among firms. This paper
studies firm’s choices of price and strategy (whether to choose exploding offer) as well as the welfare
implications in duopoly competition based on consumer search model. Through backwards induction method,
we find that both firms choose free recall (an exploding offer) with a small (large) search cost; and with a
moderate search cost, one firm chooses free recall while the other chooses an exploding offer. Consumer
surplus reaches its maximum if the search cost is low (high) and therefore both firms choose an exploding
offer (free recall). In addition, this paper extends the basic model with two extensions, considering the
existence of consumer’s observational learning and limited comparability of price. Our conclusions may offer
practical suggestions about business management.
1 INTRODUCTION
Exploding offer is commonly observed in many
business cases. For example, in door-to-door selling,
a salesman often claims that he wouldn’t visit again
and customers will never get his products otherwise
they buy now; many e-commerce platforms, such as
Taobao and Jingdong in China and Gilt, Rue Lala,
HauteLook and Vinfolio in America, often conduct
the strategy of exploding offer about a variety of
goods, which is also called as flash sales. Although,
there are still many firms just conducting the strategy
of free recall only or in most instances, which allows
consumers to return to their products freely.
In this paper, we explore under what conditions a
firm prefers an exploding offer to free recall based on
a duopoly model with consumer search and
investigate the logic behind the choice of firms as
well as the welfare implications. In consideration of
worries about prices of exploding offer in people’s
mind due to their quick decisions, we discuss whether
products are cheaper indeed when offered without
another chance than when people are allowed to
reconsider products freely. Our analysis shows that in
equilibrium firms’ choice depends crucially on the
value of the search cost. Specifically, with a small
(large) search cost, both firms choose free recall (an
exploding offer); and with a moderate search cost,
one firm chooses free recall while the other chooses
an exploding offer. Moreover, the price is higher
when both firms choose an exploding offer than that
when both firms choose free recall; however, when
the two firms choose different strategies, the price of
a firm with an exploding offer is lower than that with
free recall. In addition, consumer surplus reaches its
maximum if the search cost is low (high) and
therefore both firms choose an exploding offer (free
recall).
The literature about exploding offer in consumer
search model is scarce. With respect to consumer
search, there are many works assuming products are
homogeneous or heterogeneous, analyzing firm’s
pricing strategy (Ellison, Wolitzky, 2012),
advertising management (Moraga-Gonzalez, 2011),
and so on. While they don’t consider firm’s strategies
of exploding offer and free recall. Durmus et al.
(Durmus, et al, 2015) shows that exploding offer can
promote sales of luxury goods, however they don’t
analyze consumer’s search behaviors and firm’s
strategic choice of exploding offer. In contrast, our
model studies under what conditions a firm prefers an
exploding offer to free recall in consideration of
consumer’s search and compare prices under
different strategies.
Wang, L.
Optimal Decision of Exploding Offer based on Consumer Search Model.
DOI: 10.5220/0011160600003440
In Proceedings of the International Conference on Big Data Economy and Digital Management (BDEDM 2022), pages 87-91
ISBN: 978-989-758-593-7
Copyright
c
2022 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
87
2 MODEL AND METHODS
The introductions of model are as follows:
Firms. There are two firms, firm 1 and firm 2,
producing horizontally differentiated goods at zero
marginal cost which are labelled as product 1 and
product 2 respectively. The two firms need to choose
one of two strategies from exploding offer and free
recall, as well as pricing their product with
i
p
,
1, 2i =
. If the firm chooses exploding offer,
consumers can buy the product only in their first
search of this firm and have no chance to return. If the
firm chooses free recall, consumers can buy the
product whenever they want.
Consumers. Consumers search products
sequentially and know their valuation
i
u
,
1, 2i =
about products in the search process.
i
u
is an i.i.d
draw from the distribution function
()
F
u on the
support
max
[0, ]u
, and its density function is
()
f
u .
The probability of consumers search firm 1 or firm 2
first is equally. When consumers reach a firm claiming
exploding offer, they can buy this product at once, or
continue to search another product without no chance
to return the first firm. When consumers reach a firm
claiming free recall, they can buy this product at once,
or continue to search another product with the chance
to return the first firm freely. Without loss of
generality, we assume that the cost of the first search
is zero and the cost of the second search is
s
.
The timing of the game is as follows. In the first
stage, two firms choose exploding offer or free recall
simultaneously and compete in prices. In the second
stage, consumers begin to search and decide whether
to buy and when to buy.
According to Armstrong and Zhou (2016), we
define that
max
() [max{0, }] ()
i
u
iu ii i
p
Vp E u p Qudu≡−=
()
i
Vp
is the consumers’ utility when consumers
reach the product
i
, which is a decreasing function
of
i
p
. If
()Va s=
, ()Va is the utility of the
product
i
when consumers are indifferent to
whether to continue searching or not.
When two firms choose the same strategy, the
outcome is given as Armstrong and Zhou (2016).
Specifically speaking, given the uniform distribution
of
()
F
u , when two firms claim free recall together,
the equilibrium price
f
p
satisfies
2
1(1)
ff
pap−=+ , and when two firms claim
exploding offer together, the equilibrium price
ex
p
satisfies the following equation:
2
(2 2 ) 1
ex ex
psp−+ =
.
Due to our analysis allowing for free choice for
firms instead of prior strategy, we explore the
asymmetry situation. When firm 1 chooses free recall
and firm 2 chooses exploding offer, the equilibrium
price
1d
p
and
2d
p
is decided by
2
212
12122
1(1)(2)
(1 )(1 3 ) 2 0
ddd
ddddd
papp
ppppps
−=+ −
+−+ +=
.
As depicted in Figure 1,
f
p
,
ex
p
,
1d
p
and
2d
p
are represented separately by thick dash line,
dot dash line, fine dash line and full line.
Proposition 1. Compared to the price under free
recall, the price under exploding offer is not cheaper
all the time:
(a) The price when both firms choose exploding
offer is higher than that when both firms choose free
recall.
(b) In the asymmetric situation, the price under
exploding offer is lower than that under free recall.
To understand the first point, we need to analyze
the difference of consumers between free recall and
exploding offer. When claiming exploding offer, firm
i
owns two groups of consumers: (i) who buy
product
i
at once when reaching product
i
at the
first time, (ii) who continue to search and buy product
i
after searching product
j
first. However, when
claiming free recall, firm
i
owns three groups of
consumers. Besides the two groups of consumers
aforementioned, the third group is consumers who
return to buy product
i
after searching product
i
and product
j
. Compared to the situation where
both firms choose exploding offer, each firm has
incentives to reduce price to attract the third group of
consumers when both firms choose free recall so that
the equilibrium price is lower. This point identifies
with Armstrong and Zhou (Armstrong, Zhou, 2016).
Nevertheless, Armstrong and Zhou (Armstrong,
Zhou, 2016) assume two firms choose the same
strategy ex-ante and explore whether a firm has
incentives to deviate from the symmetry. In contrast,
we allow for free choice for firms instead of prior
strategy, discussing how firms choose strategies
according to consumer search. As a consequence, we
get the second point in the proposition 1.
The intuition of the second point is as follows. In
the situation where firm
i
chooses exploding offer
and firm
j
chooses free recall, firm
i
always has
incentives to conduct price-off promotions because
its consumers would never return back once they
BDEDM 2022 - The International Conference on Big Data Economy and Digital Management
88
decide to continue searching. Theoretically, firm
j
can also price lower than firm
i
.If firm
j
prices
products in this way, it will face more demand but too
lower price, which reduces its revenue and can’t lead
to an equilibrium outcome.
Figure 1: Price in different situation.
Figure 2: Demand in different situation.
Figure 3: Profit in different situation.
Proposition 2. For horizontally differentiated
firms, their choice of free recall or exploding offer
relies crucially on the value of search cost. Concretely
speaking,
(a) When
1
[0, ]
s
s∈
, both firms choose free
recall;
(b) When
12
[, ]
s
ss∈
, one firm chooses free
recall and another one chooses exploding offer;
(c) When
2max
[, ]sss∈
, both firms choose
exploding offer.
To understand the first point in the proposition 2,
we need to explain why firm
i
’s best response is
always free recall no matter what firm
j
chooses
when
1
[0, ]
s
s∈
. When the search cost is very low,
consumers incline to search more. If firm
j
claims
free recall, firm
i
’s exploding offer will refuse these
consumers who reach firm
i
first but continue
searching, which is called the strategic effect of
exploding offer. In order to attract more consumer to
buy at once after their first search of product
i
, firm
i
must reduce its price so as to increase its demand.
However, the promotion of demand can’t compensate
the loss caused by low price, which leads firm
i
not
to choose exploding offer. If firm
j
claims
exploding offer, the demand of firm
i
under
exploding offer will be low because its high price (as
depicted in figure 1) as well as the strategic effect of
exploding offer aforementioned. The loss caused by
low demand overweighs the promotion due to high
price, which leads firm
i
not to choose exploding
offer. In a word, firm
i
’s best response is always
free recall no matter what firm
j
chooses when
1
[0, ]
s
s∈
.
When
12
[, ]
s
ss∈
which is moderate, the
consumers’ motivation of searching is not
too intense,
which means that the strategic effect of exploding
offer won’t be too obvious. If firm
j
claims free
recall, the price of firm
i
under exploding offer is
lower compared to free recall, which will attract more
consumers who buy at once as well as who continue
searching firm
i
after visiting firm
j
. The
promotion caused by high demand compensates the
loss due to the strategic effect of exploding offer. If
firm
j
claims exploding offer, the price of firm
i
under free recall is lower compared to
exploding offer,
which increases demand from the three groups of
consumers aforementioned. The promotion of
demand can’t compensate the loss caused by low
price. In a word, when
12
[, ]
s
ss∈ , one firm chooses
Optimal Decision of Exploding Offer based on Consumer Search Model
89
free recall and another one chooses exploding offer.
When
2max
[, ]sss∈
, firm
i
’s best response is
always exploding offer no matter what firm
j
chooses. If firm
j
claims free recall, the price of
firm
i
under exploding offer is lower compared to
free recall, which will attract more consumers who
buy at once as well as who continue searching firm
i
after visiting firm
j
. The promotion caused by
high demand compensates the loss due to the strategic
effect of exploding offer. If firm
j
claims
exploding offer, the demand of firm
i
under
exploding offer doesn’t decrease much because the
high search cost deters consumers to continue to
search. Considering the higher price under exploding
offer compared to free recall, the promotion caused
by high price compensates the loss of demand. As a
result, firm
i
’s best response is always exploding
offer no matter what firm
j
chooses.
Proposition 3. When search cost is low, the
condition where both firms choose exploding offer is
best for consumers; when search cost is high, the
condition where both firms choose free recall is best
for consumers.
The intuition of proposition 3 is as follows. When
search cost is low, consumers feel more incentives to
continue to search after the first visit. However, when
both firms choose exploding offer, consumers will
search less which avoids the decrease of consumer
surplus. Although the price under both firms’
exploding offer is high, the decrease of consumer
surplus caused by high price is overweighed by the
positive effect due to the search deterrence effect of
exploding offer. When search cost is high, consumers
will search little, which avoids the decrease of
consumer surplus. What’s more, the price
competition is intense when both firms choose free
recall together, which reduces the payment of
consumers. The positive effect of low price
compensates the weakly negative effect of consumer
search.
3 TWO EXTENSIONS OF MODEL
3.1 The Existence of Observational
Learning
In general, consumers’ decisions are not always
independent. For example, a consumer who plans to
buy a new laptop, may begin his search with Dell if
he observes his friend’s purchase of a Dell laptop. In
consideration of consumer’s observational learning,
how a firm and its competitor choose from free recall
and exploding offer and how they price
their products?
Observational learning has received much attention
in the study of firm’s pricing strategy (Galeotti, 2010,
Campbell, 2013, Kovac, Schmidt, 2014), while there
are few studies investigating firm’s choice from
exploding offer and free recall under observational
learning. In this part, we explore under what
conditions a firm prefers an exploding offer to free
recall based on a duopoly model with consumer
search allowing for consumers’ observational
learning.
Firms. There are still two firms, firm 1 and firm
2, producing horizontally differentiated goods at zero
marginal cost which are labelled as product 1 and
product 2 respectively. The two firms need to choose
one of two strategies from exploding offer and free
recall, as well as pricing their product with
i
p
,
1, 2i =
. with the goal of maximize their discount
revenue with discount factor
δ
( [0,1)
δ
∈ ).
According to Daniel and Sandro (2018), nature
chooses a state
Ω from three possible states
012
{,, }Ω= Ω Ω Ω
.The state
0
Ω=Ω
is realized
with probability
1
ρ
−
, in which the utility of
product 1 and product 2 both draws from
()
Gu(with
associated density
()
g
u
) on the support
[, ]uu
. The
state
1
Ω=Ω
(
2
Ω=Ω
)is realized with probability
2
ρ
, in which the utility of product 1 (product 2) draws
from
()
Gu on the support
[, ]uu
, while the utility
of product 2 (product 1) is
()
bb
uu u<
. That is to say,
there are two states where one of the two products is
worse than another.
Consumers. Consumer
( 1, 2,3...)ii= arrives at
the market sequentially, making his purchasing
decision and leaving the market after observing his
predecessor. Consumer
( 1, 2,3...)ii= can only
observe the predecessor’s final decision without
knowing his process of search and payment.
Meanwhile, Consumer
( 1, 2,3...)ii= can’t know
whether firms claim exploding offer or free recall
unless he begins his search. The probability of the
initial consumer search firm 1 (firm 2) first is
1
2
,
while the probability of the rest consumers search
firm 1 (firm 2) first is related with what they observe.
Proposition 4. Considering the existence of
consumer’s observational learning,
(i) compared to the price under free recall, the
BDEDM 2022 - The International Conference on Big Data Economy and Digital Management
90
price under exploding offer is not cheaper all the time,
which is similar to proposition 1.
(ii) both firms choose free recall when the search
cost is low; duopoly firms choose asymmetric
strategies when the search cost is high.
The intuition of part (i) of proposition 4 is similar
to proposition 1. As for part (ii), we show that both
firms won’t choose exploding offer together when the
search cost is high, which is different from the
conclusion of proposition 1. If firms both choose
exploding offer, they will cut down the price to
increase their demand of prior consumers so that
following consumers will increase because of less
search after observational learning, while the loss
caused by low price can’t be compensated by the
promotion due to increasing demand.
3.2 The Existence of Limited
Comparability
It’s common that consumers often face limited
comparability of price in their search process, and a
number of papers study the relationship between this
phenomenon with firm’s competition using different
models (Dow, 1991, Chen, et al, 2010, Piccione,
Spiegler, 2012, Kutlu, 2015). However, our
contribution is to explore under what conditions a
firm prefers an exploding offer to free recall based on
a duopoly model with consumer search in
consideration of the existence of consumer’s limited
comparability of price, which current works haven’t
discussed about.
In this part, when both firms claim free recall,
customers can return to firms freely to compare prices
accurately; when both firms claim exploding offer,
customers will be confused when they search the
second product and forget the first product’s price
due to no chance of return, and they will purchase at
random; when duopoly firms choose asymmetric
strategies, customers also face limited comparability
of prices. There are two conditions that consumers
will choose to continue searching: (i) when the utility
of the first product they search is lower than its price,
customers will be unsatisfied with the first product;
(ii) when the utility of the first product they search is
higher than the payment of price and search cost,
customers will be prone to search another product.
Proposition 5. Considering the existence of
consumer’s limited comparability, duopoly firms
choose asymmetric strategies when the search cost is
low; both firms choose free recall when the search
cost is high.
In this extension, duopoly firms won’t choose
exploding offer together. Because if consumers aren’t
satisfied with the second product they search, they
can’t return to the first firm so that they will leave the
market without purchase, which decreases the total
demand of the markets. The loss caused by demand
surpasses the gain caused by price, which explains
why duopoly firms won’t choose exploding offer
together.
4 CONCLUSIONS
In this paper, we explore under what conditions a firm
prefers an exploding offer to free recall based on a
duopoly model with consumer search. Our analysis
shows that in equilibrium firms’ choice depends
crucially on the value of the search cost.
Specifically, with a small (large) search cost, both
firms choose free recall (an exploding offer); and
with a moderate search cost, one firm chooses free
recall while the other chooses an exploding offer.
Moreover, the price is higher when both firms choose
an exploding offer than that when both firms choose
free recall; however, when the two firms choose
different strategies, the price of a firm with an
exploding offer is lower than that with free recall. In
addition, consumer surplus reaches its maximum if
the search cost is low (high) and therefore both firms
choose an exploding offer (free recall).
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