Ordered Search, Product Differentiation, and
Price Competition∗ (Job Market Paper)
Jian Shen†
November 16, 2013
Abstract
As the Internet has significantly reduced price search costs, I propose a sequential search
model in which price information is transparent and consumers search for best matches given
their ex-ante “brand” preferences. Unlike in a random search model, firms’ pricing strategies
effectively influence consumers’ search order decisions, while such decisions vary greatly among
consumers with different ex-ante preferences. I show that the relationship between price and
search cost is non-monotonic, as it depends on both the size of the search cost and the shape of
the ex-ante preference distribution. When the search cost decreases, firms usually gain while
both the consumer and total surplus may decline. The model is extended to show that firms
always have incentives to disclose prices. This framework can also be used to study the impact of
the Internet in differentiated products markets.
Keywords: Product search, ordered search, search costs, brand preferences, the Internet.
JEL: D43, D83, L13, L86
∗ I am grateful to Dan Levin, Lixin Ye, and Huanxing Yang for their advice, guidance and comments. I would also like to
thank Yaron Azrieli, Katie Coffman, Lucas Coffman, Matt Lewis, Jingfeng Lu, Jim Peck, Andrzej Skrzypacz, Wing Suen,
Matthijs Wildenbeest, Jidong Zhou, and participants of the NASM (USC, 2013), Midwest Economic Theory Conference
(MSU, 2013), MEA Annual Conference (Columbus, 2013), Oligo workshop (Corvinus University of Budapest, 2013), and
the micro theory seminar (2013) and micro lunch (2012) at OSU for their helpful comments and suggestions. All remaining
errors are mine.
† Department of Economics, The Ohio State University, 410 Arps Hall, 1945 North High Street, Columbus, OH 43210.
Email address: [email protected]
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1 I NTRODUCTION
The Internet is changing consumers’ search behavior. Before the birth of the Web, consumers spent
most of their time searching for price information or the presence of a particular product. With
the development of powerful web search engines, such information becomes a lot easier to find, and
today what consumers are mainly searching for is the details of a product. As Brynjolfsson et al.
(2010) finds in an online book market, consumers face significant search costs when searching on a
shopbot and they put relatively more weight on non-price factors. Considering books are a nearly
homogeneous good, their findings are quite surprising. Likewise, in a differentiated products market
(e.g., automobiles or cosmetics), we should expect similar patterns.
However, such a significant change in consumers’ search behavior is not properly reflected in
the existing search literature, in which consumers are always assumed to learn both the price and
how much they like the product at the same time. It is therefore necessary to incorporate this
trend into a standard search model and study the impacts of this change. To this end, I set up a
simple sequential search model in which price information is costless to observe1 and consumers are
endowed with ex-ante “brand” preferences before search. Consumers search for how well a product
fits their tastes or needs: the “match value.”2 In my model, price becomes a determining factor in
the order of consumers’ search for a particular product. Firms therefore set prices strategically to
influence consumers’ search order decisions. This extra incentive, which I call the “order” effect, is
absent in the current literature but plays an important role in my ordered search model.
Because of the heterogeneity in ex-ante preferences, search thresholds in my model are different
across consumers and firms, and some consumers may leave the market early without buying before
running out of options. A consumer may choose to participate in hopes of buying at her favorite firm.
However, when she finds a bad match at that firm, the consumer will choose to leave the market if
other firms’ products are not attractive enough for her to risk paying another search cost. Hence,
discussions in this paper are generally divided into two scenarios based on the size of the search
cost. In the partial-participation scenario, the search cost is relatively high so that some consumers
leave the market too early. In the full-participation scenario, the search cost is relatively small and
everyone can afford to search multiple times.
The first major result of this paper concerns the impact of search costs on the equilibrium price.
Standard economic intuition (see, e.g., Wolinsky (1986) and Stahl (1989)) suggests that prices are
1 My model is not affected as long as the cost of price information is small enough so that all consumers are willing to
“click their mouses.”
2 It is worth noting that match values are idiosyncratic, meaning that they are independent across consumers and firms.
Product quality is therefore not part of the match value. If quality was included in the model and was observable to the
consumers, the analysis and main results in this paper will still hold. If quality was not observable, the price could be
used as a signaling device, which would interfere with the effect of search costs on the equilibrium price. For this reason,
quality has been intentionally excluded from my model so that it better serves the purposes of this study.
2
higher when it becomes more difficult for consumers to search, which boils down to the well-known
“lock-in” effect: consumers search less when search costs are increased. However, the order effect
in my model provides an opposite prediction: when search costs are higher, consumers tend to stay
with their initial picks and it becomes more important for firms to attract consumers to visit their
stores first, which would lead to lower prices. The direction in which the equilibrium price responds
to a decrease in search cost depends on whether the lock-in effect or the order effect dominates.
Specifically, consumers with different brand preferences can vary a lot in their search order decisions.
Hence, which effect dominates is greatly influenced by the shape of the brand preference distribution.
In the partial-participation scenario, if most consumers are brand neutral, the population as a
whole becomes price sensitive because these brand-neutral consumers’ search order decisions are
mostly affected by price changes. The order effect thus always dominates, leading to a lower equilibrium price if the search cost increases. In contrast, if most consumers have strong brand preferences,
the lock-in effect can become more pronounced. As the search cost increases gradually, the equilibrium price exhibits an inverted U pattern: the price first increases and then decreases. Suppose
initially the search cost is so small that even a brand-loyal consumer visits firms other than her
favorite one. When the search cost increases, many of these brand-loyal consumers will exit the
re-search market, giving market power back to firms. The lock-in effect hence dominates and the
equilibrium price increases. As the search cost becomes larger, more consumers leave the market
early. At some point, only the brand-neutral consumers will search multiple times, and firms will
focus on competing for them. Since these consumers are price sensitive, the order effect dominates
again and firms instead want to lower prices.
In the full-participation scenario, the lock-in effect is weakened as consumers’ search decisions
are not constrained by the search cost. As long as there are some consumers who are completely
unconcerned about brands (i.e., having no brand preference), the order effect dominates and both the
equilibrium price and profit decrease with search cost. Compared to the full-information case (i.e., no
search costs), firms earn strictly lower profits when the search cost is positive. Competition therefore
becomes more intense and harmful to firms when consumers have imperfect information.
I then continue to investigate the welfare impact from a reduction in search cost. While firms
usually gain from a search cost reduction, whether consumers and the whole society benefit becomes
ambiguous since the prices are higher in equilibrium. My analysis shows that the welfare impact of
the search cost also heavily depends on the shape of the brand preference distribution. In particular,
in the full-participation scenario, the welfare level monotonically increases with search cost if there
are moderately many brand-neutral consumers. In my model, the search cost therefore is more than
just the “friction” of the market. First, a higher search cost intensifies price competition through
the order effect, and the resulting lower price increases the market coverage. Second, it reduces
3
the negative externality among consumers. After the initial search, two types of consumers emerge:
those who are lucky to have received positive utilities and those who are unlucky to have received
negative utilities. Since the unlucky consumers will continue to search irrespective of the size of the
search cost, a change in the search cost affects only the lucky consumers. When the search cost is
reduced, the lucky consumers search more as they expect to get higher gains, which supports higher
market prices and imposes negative externalities on the unlucky ones. The society as a whole is then
worse off. A higher search cost thus reduces such negative externalities and improves social welfare.
With all these results inconsistent with findings from most studies on consumer search, this paper highlights the significance of the order effect. As better online search technologies are being
developed, this effect will become more pronounced and should not be overlooked in the search literature. Studies that investigate markets having strong Internet presence should be more careful in
selecting the proper search model and interpreting their results.
This paper will add to the vast literature on consumers searching for variety of goods. Since the
seminal work of Wolinsky (1986), there has been a number of important studies working in this area.
Bakos (1997) looks at several different models to study the implications for electronic marketplaces.
Anderson and Renault (1999) investigates the effect of product diversity on prices, and later they
consider situations in which a fraction of consumers are aware of the match values but not the prices
in Anderson and Renault (2000). In an alternative scenario of his sequential procurement model,
Wolinsky (2005) considers the possibility that a price quote has no cost for the consumers. He finds
that the equilibrium outcome is not efficient; a parametric example shows that the equilibrium price
either increases with search costs or is zero. More recently, Zhou (2012) extends the standard search
model to study multiproduct search. Bar-Isaac et al. (2012) and Larson (2013) incorporate product
design while Yang (2013) focuses on explaining the “long-tail” phenomenon. None of these papers
consider the trend that the Internet is separating product search from price search, and they do not
take ex-ante product differentiation into account either. Hence, my paper studies a setting that is in
some sense more realistic and up-to-date.
My paper is also closely related to the growing literature on ordered search. Arbatskaya (2007) is
the first in this strand, who studies a sequential search model in which the search order is predetermined. Zhou (2011) adds product fitness and finds that prices rise in the order of search. Armstrong
et al. (2009) explores the effect of prominence in a market where all consumers will visit the prominent firm first. In Wilson (2010), the order of search is endogenously determined by the size of each
firm’s search cost, while Haan and Moraga-González (2011) considers the effects of advertising in
which consumers first visit the firm that advertises most. In this research, I model the search order
as a response to both the prices and ex-ante preferences. Armstrong and Zhou (2011) studies a pricedirected search model that is similar to my full-participation scenario. However, they do not consider
4
ex-ante preference and the match values in their model are correlated, meaning that a consumer
knows her match value attached to either product as long as she has visited one firm. They do not
investigate the welfare impact either, which has some interesting implications in my model.
There are two papers predicting prices can decrease with search costs. Janssen et al. (2005)
studies homogeneous products. They modify Stahl (1989)’s model to allow for partial consumer participation and concludes that expected price decreases in search cost. Moraga-González et al. (2013)
considers the dispersion of search costs and also looks at differentiated products. They find that
higher search costs can result in lower prices when the search cost distribution has a decreasing
elasticity with respect to the parameter that shifts the distribution. In essence, both results are driven by the possibility that higher search costs may deter consumer participation. I show that, even
when all consumers participate in the market, the equilibrium price may still decrease with search
cost due to the presence of the order effect.
The next section presents the model. Consumers’ behaviors are derived in Section 3. Section 4
studies firms’ pricing strategies and comparative statics. Section 5 concludes, and discusses firms’
price disclosure decisions and an application on the impact of the Internet. All proofs are in the
Appendix.
2 T HE M ODEL
Consider a duopoly model in a differentiated products market. Two firms are respectively located
at the two endpoints of the classic linear city á la Hotelling, which is interpreted as a product space
instead of a physical space. Each firm sells only one product. A continuum of consumers with
measure one is distributed along the line. Therefore, a consumer’s location, which she knows before
search, represents her “brand” preference.
Besides the ex-ante brand preference, a consumer also attaches a match value to the product
sold by each firm. The match is learned only through costly search. Every consumer has the same
per-search cost c ≥ 0, which includes the travel cost as well as the time and energy associated with
testing the product. The match values are idiosyncratic, meaning that they are independent across
consumers and firms. A consumer is then characterized by a triple (x; ²1 , ²2 ). Here x ∈ [0, 1] denotes
the distance to firm 1, and ² i represents the match value attached to firm i’s product. ²1 and ²2 are
assumed to be identically and independently distributed according to the uniform distribution on
[0, 1], and they are independent of x. The distribution of x is drawn from a family of distributions
H s (x) that are twice differentiable and with symmetric densities h s (x): h s (x) = h s (1 − x). The parameter s characterizes the shape of the distribution and a change in it induces a rotation of the density
5
function3 . In what follows, I focus on the region where x ≤ 1/2 as the other half is a mirror image.
For ∀ s ∈ [s, s], there is a family of rotation points x†s ∈ [0, 1/2] such that
dh s (x)
≶ 0 when x ≶ x†s .
ds
Intuitively, an increase in s is equivalent to a counter-clockwise rotation of the density function.
Without loss of generality, let s = 1 represent the uniform distribution. As a result, when s > 1 the
density h s (x) is a bell-shaped function meaning that more consumers are around the center, while
s < 1 describes a U-shaped one meaning that more mass are around the two firms. By symmetry and
differentiability, H s (1/2) = 1/2 and h0s (1/2) = 0 for ∀ s ∈ [s, s].
Each consumer demands up to one unit of the product. Given her type (x; ²1 , ²2 ), a consumer’s
utility (before search cost) is given by
u i (x) = ² i − p i − k| x − i + 1|
i = 1, 2
if she purchases from firm i at price p i . Here the last term captures the disutility a consumer suffers
when not consuming her ideal brand.4 The parameter k > 0 is referred to as the degree of “brand”
preference, which measures how ex ante differentiated the market is. When k is large, consumers
suffer a lot for buying a product that they don’t like, suggesting that the two products are quite
distinct. A consumer’s type is her private information and not observable to the firms, while the
search cost c is common knowledge. The outside option is normalized to zero.
The marginal cost of production is constant and the same across the two firms, which is normalized to zero. After production, each firm simultaneously posts its price p i , which will be fixed for the
rest of the game. Observing the prices, consumers search sequentially. First, they decide whether to
participate in the market and, if they do, which firm to start with. After having searched one firm
at cost c, a consumer can buy at that firm, visit the second firm, or simply exit the market without
making a purchase. If a consumer visits both firms, she can buy either one’s product. Search is with
perfect recall, meaning that consumers can always go back to the first firm without paying c again.
Following the literature (e.g. Wolinsky (1986) and Robert and Stahl (1993)), I assume consumers
must search to buy.
3 I borrow this concept from Johnson and Myatt (2006) and modify it to serve my purposes here.
4 The disutility is sometimes referred to as the “transport” cost, which should not be confused with the search cost. The
disutility is incurred if and only if the consumer makes the purchase, while the search cost arises whenever the consumer
conducts a search at a new firm.
6
3 M ARKET D EMAND
In a standard sequential search model without ex-ante preference, a consumer uses a static stopping
rule, and she always buys at the last firm she visited unless she has visited all firms. However,
when consumers have ex-ante preferences as in my model, they may leave the market early without
buying since the stopping rule for each firm is no longer the same. Next, I start the analysis with
consumers’ search and purchase behaviors and then derive the demand function.
A rational consumer considers not only the first search outcome but also the possibility for finding
a better match from the other firm when making her participation decision. If a consumer indeed
participates, she will choose to start with the firm providing the higher total expected utility. Since
match values are identically and independently distributed in my model, consumers will first visit
the firm closer to their ideal spots. Formally define
1 p2 − p1
+
,
2
2k
x̂ =
and consumers will visit firm 1 first if x ≤ x̂.5 A consumer’s search order thus depends on both her
brand preference and the prices.
In what follows I focus on consumers with x ≤ x̂. By symmetry, consumers on the right half
behave similarly. After paying a visit to firm 1, a consumer learns her match value ²1 , upon which
she decides whether to continue the search.
If ²1 − p 1 − kx < 0, the consumer gets negative utility from firm 1. She then will visit firm 2 if and
only if the continuation value is greater than c:
Z
1
p 2 + k(1− x)
(²2 − p 2 − k(1 − x)) d ²2 ≥ c.
(1)
If ²1 − p 1 − kx ≥ 0, the opportunity cost of a second search increases and the consumer will continue
if and only if
Z
1
²1 + p 2 − p 1 + k(1−2 x)
{²2 − [²1 + p 2 − p 1 + k(1 − 2x)]} d ²2 ≥ c.
(2)
For convenience, define a to be the unique solution to
1
Z
a
² − ad ² = c.
(3)
Conditions (1) and (2) then can be rewritten in terms of a:
x ≥ x̃1 ≡ 1 −
a − p2
k
if
²1 − p 1 − kx < 0;
5 After deriving the expected utilities, it is not difficult to see that x̂ is the right threshold dividing the consumers.
7
(4)
²1 ≤ a − p 2 + p 1 − k(1 − 2x)
if
²1 − p 1 − kx ≥ 0.
(5)
According to (3), a monotonically decreases with c on the unit interval a ∈ [0, 1]. For the rest of
the paper, I will mainly use a to (inversely) measure the size of the search cost and term it the
“reservation value.”
Notice that Condition (5) implicitly requires x ≥ x̃1 ; otherwise ²1 − p 1 − kx would become negative.
Therefore, only consumers with x ≥ x̃1 will ever visit firm 2. A similar cutoff x̃2 ≡ (a − p 1 )/k can be
found on the other half of the city. Interestingly, even though the two firms’ prices may be different,
x̃1 and x̃2 are always symmetrically located around x̂, as long as they are within the unit interval.
Consumers’ behaviors are best illustrated in Figure 1.
Figure 1: An Illustration of Consumers’ Behaviors
This figure illustrates the case when both firms post the same price. The horizontal axis represents the Hotelling city, while the match values from the two firms are denoted on the two vertical
axes, respectively. The vertical dashed line x = x̂ separates the consumers visiting firm 1 first from
those visiting firm 2 first. The solid line, ²1 = p 1 + kx, represents the zero-utility condition, and the
dashed line, ²1 = a − p 2 + p 1 − k(1 − 2x), determines whether consumers want to visit firm 2 conditional
on them having received nonnegative utilities. Note that the dashed line always hits ²1 = a when
evaluated at x̂ regardless of p 1 and p 2 . These two lines combined with x = x̃1 divide the left half
space into four regions. In region I, the consumers get good match values and buy directly at firm
1. In region II, though having received positive utilities, these consumers still want to try their luck
8
at firm 2. In region III, the consumers receive negative utilities from firm 1, but they will continue
the search because firm 2 is not too far away. In region IV, visiting firm 2 is not worthwhile and the
consumers leave the market without buying.
As the search cost c decreases so that the reservation value a increases, the dashed line shifts
upwards and x̃1 moves towards 0. Therefore, more consumers will search twice when the search
cost becomes smaller. As long as x̃1 ≥ 0, region IV exists and there are always some consumers
who search only once and leave the market too early. This case is termed the “partial-participation”
scenario. When c gets small enough so that x̃1 ≤ 0, region IV disappears and everyone can afford
to search twice. I therefore call this situation the “full-participation” scenario. I require a ≥ k for
the rest of the paper so that both scenarios will arise in my model.6 Note that if c becomes so large
that x̃1 crosses over x̂, both regions II and III disappear and no one in the market will ever search
twice. In fact, a gap will emerge between the two firms’ coverage and consumers located in the gap
will not participate. Each seller then effectively becomes a local monopolist and pricing decisions
become trivial.7 As a result, I only look at the situation in which x̃1 < x̂ in equilibrium, that is
a ≥ k/2 + (p 1 + p 2 )/2.
Given their search behaviors, I show that all consumers will visit at least once firm as long as
there exists direct competition in the market. Note that the minimum expected utility a consumer
at x can get is
Z
1
p 1 + kx
(²1 − p 1 − kx)d ²1 − c.
Given that x ≤ x̂ and the “no-local-monopoly” requirement a ≥ k/2 + (p 1 + p 2 )/2,
p 1 + kx ≤ p 1 + k x̂ =
k p1 + p2
+
≤ a,
2
2
which implies that
Z
1
p 1 + kx
(²1 − p 1 − kx)d ²1 − c ≥ 0.
Therefore, participating in the market is always beneficial to every consumer, as long as the search
cost is not too large so that there exists direct competition between the two firms.
I next derive the demand function D 1 for firm 1. Firm 2’s demand is simply a mirror image.
Figure 1 suggests that a firm’s demand is composed of three different groups. First, there are consumers from region I who directly purchase at firm 1 without visiting firm 2. I call them the “loyal”
customers. Second, for some of the consumers from region II, they continue to search firm 2 but
will come back to purchase. They are called the “returning” customers. Finally, there are consumers
6 If a < k, only the partial-participation case will arise and very little will be changed. All the results will still hold.
7 Anderson and Renault (2006) studies the monopoly case in detail with a focus on search and advertising.
9
whose first choice is not firm 1. However, some of them will search twice and may eventually purchase at firm 1. These consumers are called the “switching” customers. The total demand is the sum
of the demands from these three groups.
Not like in a random search model, here p 1 and p 2 enter the calculation of demand through x̂
and x̃1(2) . Firms therefore set prices strategically to influence consumers’ search order and stopping
threshold decisions.
I first write down the demand function for the partial-participation scenario. Recall that the
cutoff x̂ = 1/2 + (p 2 − p 1 ) /2k, and x̃1 = 1 − (a − p 2 ) /k ≥ 0 and x̃2 = (a − p 1 ) /k ≤ 1 in this case.8 The
demand from the loyal consumers is given by
x̃1
Z
S1 =
0
Z
(1 − p 1 − kx)h s (x)dx +
x̂
x̃1
(1 − 2kx + k − a + p 2 − p 1 )h s (x)dx.
The first integral comes from the consumers who are very close to firm 1. They will purchase as long
as they get positive utilities from firm 1. In the second term, the consumers are further away from
firm 1 and have higher opportunity costs to buy directly from firm 1, meaning that they need higher
match values to be “loyal”.
If the consumers are not satisfied with the match values they get from firm 1, they will visit firm
2. Some of them will become returning customers, from whom we derive the following demand
Z
S2 =
x̂ Z a− p 2 + p 1 − k(1−2 x)
x̃1
p 1 + kx
(²1 − p 1 + p 2 + k(1 − 2x))d ²1 h s (x)dx.
For the switching consumers, their demand can be calculated as follows
Z
S3 =
x̃2 ·Z p 2 + k(1− x)
+
p 2 + k(1− x)
Z x̂ Z p1 +kt
t=2 x̂− x
=====
Z
+
(1 − p 1 − kx) d ²2
x̂
0
Z a− p1 + p2 +k(1−2 x)
x̂ Z
x̃1
¸
[1 − ²2 + p 2 − p 1 + k(1 − 2x)] d ²2 h s (x)dx
[1 − p 2 − k(1 − t)] d ²2 h s (2 x̂ − t)dt
x̃1 0
a− p 2 + p 1 − k(1−2 t)
p 1 + kt
[1 − ²2 + p 1 − p 2 − k(1 − 2t)] d ²2 h s (2 x̂ − t)dt.
S 3 contains two parts. The first term comes from the consumers who receive negative utilities from
firm 2. They will purchase at firm 1 as long as the firm provides positive utilities. In the second
term, these consumers get positive utilities from firm 2 and need higher match values from firm 1 to
make their switches. I change the variable here in S 3 so that the domain of integration is consistent
8 Taking p as given, there might be situations in which firm 1 prices “crazily” so that x̃ ≤ x̂ or x̃ ≥ 1. This crazy pricing
2
2
2
problem is taken care of in Proposition 1
10
with S 1 and S 2 , which helps later derivations of the profit function and the pricing strategy. The
total demand D 1 = S 1 + S 2 + S 3 . In the full-participation scenario, we need to only substitute x̃1 with
0 and x̃2 with 1 in the above expressions.
4 S YMMETRIC M ARKET E QUILIBRIUM
The equilibrium concept used here is subgame perfect. Given the symmetric game setting, I focus on
symmetric equilibria in which the two firms adopt the same pricing strategy. As mentioned in the
preceding section, both the partial- and full-participation scenarios will be discussed.
In the partial-participation scenario, 0 ≤ x̃1(2) ≤ 1. In equilibrium, p 1 = p 2 = p∗ and firm 1’s
demand is given by the following simple form:
1/2
Z
D 1 = S1 + S2 + S3 =
(1 − p − kx)h s (x)dx +
0
1/2 £
Z
∗
¤
1 − p∗ − k(1 − x) (p∗ + kx)h s (x)dx.
x̃1
(6)
Meanwhile, dD 1 /d p 1 is calculated as follows:
¯
dD 1 ¯¯
d p 1 ¯ p 1 = p 2 = p∗
µ
=
¶¯
dS 1 dS 2 dS 3 ¯¯
+
+
d p 1 d p 1 d p 1 ¯ p 1 = p 2 = p∗
(7)
1 h s (1/2)
h s ( x̃1 )
= − −
(1 − a)2 −
(1 − a)(2p∗ + k − a) −
2
2k
k
Z
1/2
x̃1
(1 − k(1 − 2x)) h s (x)dx.
Let G 1 (p) ≡ dD 1 /d p 1 | p1 = p2 = p∗ . For convenience, I drop the superscript ∗. The equilibrium price p is
then derived from
D 1 (p) + pG 1 (p) = 0.
(8)
In the full-participation scenario, a ≥ p + k so that x̃1 ≤ 0 and x̃2 ≥ 1, the equilibrium demand D 1
and dD 1 /d p 1 are given by:
1/2
Z
D1
G 1 (p) ≡
¯
dD 1 ¯¯
d p 1 ¯ p1 = p2 = p
=
0
1/2
Z
(1 − p − kx)h s (x)dx +
1 h s (1/2)
= − −
(1 − a)2 −
2
2k
0
1/2
Z
0
(1 − p − k(1 − x)) (p + kx)h s (x)dx
(1 − k(1 − 2x)) h s (x)dx.
(9)
(10)
Plugging the above two expressions in condition (8), we will obtain the equilibrium price. Note that
both G 1 (p)’s are less than zero as seen from (7) and (10).
As the reservation value a becomes relatively larger, we move gradually from the partial-participation
scenario to the full-participation scenario. However, there exists a transition area between the two
scenarios in which the equilibrium price turns out to be a corner solution.
Suppose in the partial-participation scenario there exists an aL (k) such that the interior equi-
11
librium price p(aL ) derived from condition (8) satisfies p(aL ) = aL − k. Similarly, we see an a H (k) in
the full-participation scenario such that the equilibrium price p(a H ) = a H − k when a = a H (k). Since
aL (k) never equals a H (k), a gap must appear between the two cases. Indeed, if we denote the LHS of
condition (8) by LHS L(H ) (p), in which the subscript L(H) represents the partial (full)-participation
case, then aL(H ) is the solution to LHS L(H ) (p = a − k) = 0. Since
LHS L (p = a − k) = LHS H (p = a − k) − p
h s (0)
(1 − a) (2p + k − a) ,
k
we must have LHS L (p = a H − k) < 0 and LHS H (p = aL − k) > 0. Therefore, a H (k) falls outside the
range of a ≤ p + K, and vice versa. The corner solution p = a − k thus arises when a falls between
aL (k) and a H (k).
Whether aL < a H or aL > a H depends on the distribution H s (x) and k. Because LHS L(H ) (p = a − k)
is a cubic function of a, it is possible that there is more than one solution to LHS L(H ) (p = a − k) = 0.
However, if there is indeed only one solution on a ∈ [k, 1], then aL must be strictly smaller than a H
since LHS L(H ) (p = a − k) > 0 at a = k and that LHS L (p = a − k) lies below LHS H (p = a − k). If there
are multiple solutions, then as a moves from k to 1 we pass the corner solution area more than once,
and aL might be greater than a H .9
The next proposition presents the existence and uniqueness result of the equilibrium when H s (x)
is uniform.
Proposition 1. When s = 1, there exists a unique equilibrium, which is symmetric, for ∀a ∈ [k, 1]10 .
¡
¤
Specifically, there exist aL (k) ∈ (k, 1] and a H (k) ∈ aL (k), 1 such that: when a ≤ aL , the partialparticipation scenario arises and the equilibrium price p ≥ a − k; when a ≥ a H , the full-participation
scenario arises and p ≤ a − k; when a ∈ (aL , a H ), p = a − k. If k > 0.653, then only the partialparticipation scenario occurs.
The detailed proof is involved and can be found in the Appendix. The sketch of proof has two
steps. First I show that there exists only one solution to LHS L(H ) (p = a − k) = 0, meaning that
aL < a H . Then I move on to show the existence and uniqueness in the interior solution cases: a ≤ aL
and a ≥ a H . In the second step, the existence result is derived from Kakutani’s fixed-point theorem
by showing the log-concavity of the demand function and the uniqueness comes from Proposition 6
in Caplin and Nalebuff (1991). I also take care of the “crazy” pricing issue in the proof. For example,
9 Besides these two cutoffs a L and a H , I also require a to be large enough so that the local monopoly case would not
occur. Similar to the above analysis, there exists an a which separates the model from the local monopoly case. Note that,
R (a− p)/k
in the local monopoly case, D 1 = 0
(1 − p − kx)h s (x)dx and dD 1 /d p 1 = −1/2 − (1 − a)h s (1/2)/k when p 1 = a − k/2. a is
thus derived from simple comparison with (7). I therefore require a ≥ a. However, I keep k as the lower bound of a since
imposing a does not change my analysis or results. Indeed, when a > k, what hold for a ≥ k continue to hold for a ≥ a.
When k ≥ a, the local monopoly case will not arise as long as we impose a ≥ k.
10 It can be easily verified that the local monopoly case never occurs in the s = 1 case as long as a ≥ k.
12
in the partial-participation case, if firm 1 prices too differently from the equilibrium price p∗ it is
possible that x̃2 exceeds the interval [ x̂, 1]. The demand function then needs to be modified, and it
might not be log-concave globally. Nevertheless, I managed to show that it is still in firm 1’s interest
to charge p∗ even if such crazy pricing behaviors are allowed.11
The partial-participation scenario thus arises when a ≤ aL , while the full-participation scenario
occurs when a ≥ a H . In what follows, I use the more economic terms and the more mathematical
terms interchangeably. An interesting finding is that in the transition area p = a − k, meaning that
the equilibrium price decreases as the search cost gets larger, and this result holds for any horizontal
distribution H s (x). Most previous studies suggest that a larger search cost helps firms to build up
their market power because consumers search less. However, in my model the two firms face more
intense competition due to the Internet and here the conventional wisdom no longer holds. The
impact actually extends to areas other than the transition region, and in the next subsection I will
focus on the two interior-solution regions.
4.1 The Equilibrium Price and Profit
In this subsection, I show that a decrease in the search cost can lead to an increase or decrease of
the equilibrium price. How firms’ profits are affected is also studied.
The direction in which the equilibrium price responds to a decrease in the search cost depends on
whether the lock-in effect or the order effect dominates. Before diving into the details, it is instructive
to look more closely at these two effects. When deciding which firm to visit first, consumers with different brand preferences behave very differently. The brand-loyal consumers are mostly influenced
by the lock-in effect since their search order decisions are not easily affected by a moderate price cut
of the other firm. In contrast, the brand-neutral consumers are very sensitive to price changes when
making their ranking decisions and the order effect becomes the dominating force among these consumers. Who and how many are affected turn out to be important questions for firms when making
their pricing decisions. Hence, both the size of the search cost and the shape of the brand preference
distribution will influence the result greatly.
In the partial-participation scenario, a decrease in search cost encourages more consumers to
search twice. In particular, region IV in Figure 1 shrinks, meaning that more consumers re-enter
the search market. As x̃1 moves from 1/2 to 0, the mass of consumers affected by a change in search
cost largely depends on the shape of h s (x). Firms’ pricing decisions hence crucially hinge on h s (x)
as well. Next I formalize the analysis and, for simplicity, I impose the following restrictions on the
distribution: for x ∈ [0, 1/2], h0s (x) ≥ 0 when s ≥ 1 (bell-shaped) and h0s (x) ≤ 0 when s ≤ 1 (U-shaped). I
11 Equilibrium existence is not the main concern of this paper. Nevertheless, I show that, for any H (x), a unique
s
p
symmetric equilibrium exists in the full-participation scenario when a ≥ k − 1 + 2 + k2 /4.
13
also drop the subscript s for now.
Given the equilibrium condition (8), d p/da is derived from:
µ
∂D 1
∂p
+ G1 + p
¶
∂G 1 d p
∂p
da
+
∂D 1
∂a
+p
∂G 1
∂a
= 0,
(11)
in which we have
∂D 1
∂p
+ G1 + p
∂D 1
∂a
+p
∂G 1
∂p
∂G 1
∂a
1 h(1/2)
h0 ( x̃1 )
p(2p + k − a)(1 − a)
= − −
(1 − a)2 −
2
2k
k2
h( x̃1 )
−
[2(1 − a)(2p + k − a) + p(1 − 2p − k)]
k
µ
¶
Z 1/2
1
− H( x̃1 ) −
(1 − k(1 − 2x))h(x)dx − (2p + k) − H( x̃1 ) .
2
x̃1
·
¸
h( x̃1 )
h0 ( x̃1 )
h(1/2)
= (1 − a)
(2p + k − a) +
p(2p + k − a) +
p ;
k
k
k2
(12)
(13)
The absence of a closed-form solution for the equilibrium price makes the analysis difficult. I hence
concentrate on deriving sufficient conditions to illustrate the point.
First the following lemma ensures that expression (12) is negative; hence, to determine the sign
of d p/da, I need to focus on expression (13) only. A negative (12) suggests that ∂ p 1 /∂ p 2 < 1 at the
equilibrium point, which is a reasonable requirement. Indeed, expression (12) is equivalent to
¶¯
µ
∂ p 1 ¯¯
.12
SOC · 1 −
∂ p 2 ¯ p1 = p2 = p
(13)
Here SOC stands for the second order condition, which must be negative in equilibrium. The sign
of (12) is therefore determined by ∂ p 1 /∂ p 2 that is derived from the best-response function. I want to
focus on the case expression (12) being negative because it seems natural that ∂ p 1 /∂ p 2 < 1 so that a
firm would not increase his price faster than the opponent.
12 It is derived as follows:
∂G 1
+ G1 + p
∂p
∂p
"
#¯
∂D 1 ∂D 1 ∂D 1
∂2 D 1
∂2 D 1 ¯¯
+
+
+ p1
+ p1
¯
∂ p1
∂ p2
∂ p1
∂ p1 ∂ p2 ¯
∂ p21
p1 = p2 = p

¯
¯
¯

¯
 ∂D

2
2
∂ D 1 ∂D 1
∂ D 1 ¯¯

1
+ p1
+
+ p1
2
¯
2
 ∂ p1
∂ p2
∂ p 1 ∂ p 2 ¯¯
∂ p1


|
{z
}
|
{z
}
¯
¯
−SOC·∂ p 1 /∂ p 2
SOC
p1 = p2 = p
µ
¶¯
∂ p 1 ¯¯
.
SOC · 1 −
∂ p2 ¯ p1 = p2 = p
∂D 1
=
=
=
14
Lemma 1. In the partial-participation scenario (i.e. a ≤ aL ),
∂D 1
∂p
+ G 1 + p ∂∂Gp1 < 0 if the following set
of conditions holds for ∀ x ∈ [0, 1/2]:


p
a ≤
 h0 (x)a(1 − a)(2a − k) ≥
(3+ k)2 −8 k
;
8
¡ 2
¢
1
− k + h( 2 )k(1 − a2 ) .
3+ k+
(14)
Condition (14) suggests a should be relatively small and that h0 (x) should not be “too negative.”
Should the second inequality in condition (14) fail, a firm might behave too “aggressively.” Suppose
there is a big change in h(x) over some range along the interval x ∈ [0, 1/2] and that x̃1 falls into this
range in equilibrium. A decrease in p 2 then might lead to a big drop in D 1 , which drives firm 1 to
cut its price even lower to compete. Such an equilibrium point does not seem to be “stable”, and we
thus impose condition (14). In what follows, I continue the analysis taking condition (14) as given.
When s ≥ 1, there are more consumers around the center and we must have ∂D 1 /∂a + p∂G 1 /∂a > 0.
To see this, note that 2p + k − a > 0 since x̃1 ≥ 0 in equilibrium. Plus, by assumption, h0 (x) ≥ 0 for
∀ x ∈ [0, 1/2]. We therefore have d p/da > 0, meaning that the equilibrium price decreases with search
cost.13 Intuitively, more brand-neutral consumers, who are price-sensitive, means more intense price
competition between firms and that the order effect is very strong. As the search cost increases, it
becomes very important to capture these marginal consumers at their first search as they less likely
to search again, which leads to a lower equilibrium price.
The result is less clear-cut for s < 1 in which more consumers possess strong brand preferences.
When x̃1 is close to 1/2, a change in the search cost affects mostly the brand-neutral consumers’
search order decisions. Firms therefore focus on competing for these consumers and the order effect
again dominates. The equilibrium price hence increases when the search cost is reduced. Note that
this intuition works for any h(x) since firms only focus on the consumers around the center when
x̃1 → 1/2. On the other hand, when x̃1 is close to 0, the brand-loyal consumers start to reconsider their
search decisions and the lock-in effect becomes more important. As c decreases, many brand-loyal
consumers re-enter the market, which is a big loss to a firm since s < 1. However, given that these
consumers have strong brand preferences, the cost to make them loyal again is not very expensive.
All the firms need to do is to lower their prices a little bit and these consumers will exit the re-search
market. When between 0 and 1/2, according to continuity, there must exist a point at which the
above two effects cancel each other completely. The equilibrium price therefore increases and then
decreases as the search cost becomes smaller. The next lemma presents the conditions under which
the above intuition holds.
13 For the case when s = 1, the sufficient condition (14) is redundant and the equilibrium price always decreases with the
search cost. The proof is available upon request.
15
Lemma 2. When s < 1, the equilibrium price first increases and then decreases with the search cost
over the interval a ∈ [k, aL ] if the following conditions hold:
1
a−k 0
h (0) + h( ) < − h(0);
k
2
a − k 00
k(a − k/2) 1
h (x) ≥
h( ) − h0 (0)
k
2
(a − k)2
(15)
for
µ
¶
1
∀ x ∈ 0, .
2
(16)
Proof is again moved to the Appendix. Condition (15) ensures that expression (13) is negative
when x̃1 → 0. It suggests that we need | h0 (0)| to be large enough so that a small change in the search
cost will lead to a large change in the demand when near the boundary. In addition, we need a
small h(1/2) so that there are not too many brand-neutral consumers on the market and that the
competition in the market is not very intense. Note that condition (16) implies that h00 (x) must be
positive, and therefore h0 (x) increases gradually with x. Should this condition fail, the locus of p(a)
might become multi-peaked. I skip the discussion of this unlikely event as the main point of this
section has be clearly made: the shape of h(x) matters.
The following proposition summarizes the above results.
Proposition 2. In the partial-participation scenario (i.e. a ≤ aL ), when condition (14) holds, the
equilibrium price responds to a change in the search cost in the following way.
1. If s ≥ 1, the equilibrium price decreases with search cost.
2. If s < 1, with conditions (15) and (16) satisfied, the equilibrium price first increases and then
decreases with search cost.
This proposition highlights the important impacts of brand preference distribution on firms’ pricing strategies. Both s ≥ 1 and s < 1 can happen in real markets, and my results may shed some light
on the mixed empirical findings. In particular, Moraga-González et al. (2012) studies consumers’
search behaviors in the Dutch automobile market and they look at specifically the product search
costs. Their simulation result in Table 7 suggests that, for at least eight car brands ranging from
Volkswagon Golf to Mercedes S/CL, the prices would go up if there ware no search costs. In addition,
their survey data shows that about 90% of the households search less than four car dealers14 . More
work needs to be done to fully explain their finding. However, the intuitions that drive my theoretical
predictions provide a promising explanation. If search costs were zero, consumers would search more
anyway and lower prices would become less appealing to them. Price competition then becomes less
intense and firms could charge higher prices.
14 Half of these households search only one dealer.
16
Since in my model the equilibrium price may increase when the search cost is reduced, such a
change can actually benefit the firms. Formally,
·
µ
¶
¸¯
∂D 1
∂D 1 d p 1
∂D 1 d p 2 ¯¯
d π1 (p∗ )
= p1
+ D 1 + p1
+ p1
.
da
∂a
∂ p 1 da
∂ p 2 da ¯ p1 = p2 = p∗
(17)
Given that D 1 + p 1 ∂D 1 /∂ p 1 = 0 in equilibrium, all we need to know is the signs of ∂D 1 /∂a and ∂D 1 /∂ p 2 .
If the demand loss from the increased price is not too large, firms will earn higher profits.
Corollary 1. In the partial-participation scenario, the equilibrium profit decreases with an increasing search cost for ∀ s ≥ 1 as long as the reservation value a is not too large.
The upper bound is derived in the proof. For the uniform distribution, the equilibrium price
always decreases with the search cost, and this profit result holds for ∀a ≤ aL . This corollary suggests
that firms jointly prefer lower search costs, which is in contrast with the conventional wisdom that
firms benefit from higher search costs. When s < 1, the profit can go up or down with search cost,
which depends on the distribution and other parameters and is not the focus of this paper.
I next look at the full-participation scenario.15 In general, the intuitions derived from the partialparticipation scenario continue to hold here. The additional assumptions imposed on h s (x) are
dropped and I re-install the subscript s.
D 1 and G 1 are given by (9) and (10), in which D 1 is not a function of a and that G 1 is not a
function of the equilibrium price p. The first order effect of a on p, which is derived in condition (11),
therefore takes the following simple form:
µ
∂D 1
∂p
¶
+ G1
∂G 1
dp
+p
= 0.
da
∂a
The following derivatives can be easily calculated:
∂D 1
∂p
∂G 1
∂a
Z 1/2
1
k
= − +
(1 − 2p − k)h s (x)dx = − p − < 0;
2
2
0
h s (1/2)
=
(1 − a) ≥ 0.
k
Since G 1 < 0, d p/da is greater than zero and the equilibrium price decreases with search cost. Note
that this result does not depend on the shape of H s (x). Since the search cost is small, all the consumers in the market can afford to search twice. When the search cost is changed, the total demand
stays the same because no one is prevented from searching again by the search cost. Though the
15 The corner solution case a ∈ [a L , a H ] can be treated as a special case of a ≥ a H as they share the same demand function
with the only difference being that p = a − k in the corner solution case while the solution is interior when a ≥ a H . All the
results derived in the case of a ≥ a H thus also hold for the corner solution case.
17
proportion of loyal consumers varies with a change in the search cost, it does not vary with a change
in the shape of H s (x). The lock-in effect thus stays at the same level for any H s (x). The order effect,
however, depends on the brand preference distribution, and as I show above it dominates as long as
h s (1/2) > 0. When h s (1/2) = 0, the order effect offsets the lock-in effect completely. I summarize the
results in the following proposition.
Proposition 3. In the full-participation scenario (i.e. a ≥ a H ), for any H s (x), both the equilibrium
price and profit decrease with search cost.
The proof of the profit result can be found in the Appendix. This proposition shows that the
profit is higher when c = 0, suggesting that adding information opacity on the consumer side may
eventually hurt firms. When c > 0, allocation is not efficient because some consumers only visit
one firm, which increases the potential gain by locking consumers in. Hence, the price competition
becomes even more intense than the case when all information is transparent. Note that this result
is not due to the presence of outside option. Simple calculation shows that same results hold even
when consumers are assumed to always buy from either one of the two firms. Hence, Proposition 3
is a result of more intense competition directly between the firms.
Both Corollary 1 and Proposition 3 imply that firms may have the incentive to lower consumers’
costs of finding out their favorite products. In industries where products are largely differentiated,
for example cameras and cosmetics, consumers are always faced with plenty of choices. Many business strategies in these industries can be associated with the practice of reducing search costs. For
instance, firms often provide free test samples, and better product web pages are being designed and
used. By making it easier for consumers to search, price competition is softened. Lower search costs
hence may arise as an equilibrium outcome in markets, which is in contrast to the common belief
that firms prefer higher search costs.
Besides search cost, consumers’ brand preferences also have a big influence on the equilibrium
price. Note that we have two concepts related to brand preference: the shape of the brand preference
distribution, which is captured by s, and the degree of brand preference: k. These two concepts are
similar in the sense that both reflect the competitiveness of the market to some degree. However, as
suggested in the following proposition, the two parameters influence the equilibrium price in very
different ways.
Proposition 4. In the full-participation scenario, the equilibrium price always decreases as there
are more brand-neutral consumers (bigger s). Meanwhile, the equilibrium price may increase as
products become less differentiated (smaller k) if the following condition holds:
µ
1−a
2k
R 1/2
¶2
<
0
xh s (x)dx
h s (1/2)
18
.
(18)
The proof is again moved to the Appendix. A change in s affects only the preference distribution,
which has a direct influence on the competition between the two firms. Hence, an increase in s
uniformly brings down the equilibrium price as firms compete more fiercely. In contrast, k reflects a
consumer’s taste for variety, from which the impact on competition is indirect. As k becomes smaller,
consumers view the two products more alike, which intensifies competition. Meanwhile, a consumer
also suffers less from not buying her ideal brand, which increases her willingness to pay. Condition
(18) says that the second effect is more likely to dominate when a is larger. Indeed, a bigger a implies
higher prices, leaving consumers less surplus. A decrease in k then relatively benefits consumers a
lot, which brings down the elasticity of demand and thus supports higher prices. When initially k is
large, the same intuition applies and the result holds.
4.2 Welfare Analysis
It is now necessary to turn our attention to the welfare analysis. Due to the strong order effect,
firms generally gain higher profits from a search cost decrease. It implies, however, that consumers
may be worse off with a smaller search cost since they have to pay higher prices. If indeed the
consumer surplus declines following a decrease in the search cost, how the society is affected as
a whole becomes even more ambiguous. I therefore take up this task and study welfare in this
subsection. In particular, I would like to see under what circumstances a smaller search cost will
reduce the welfare. It turns out that the brand preference distribution again plays an important role
here.
I first look at the impact on consumer surplus (CS). As an example, the case when x is uniformly
0.08
0.079
0.075
0.078
0.07
0.077
0.065
0.076
consumer surplus
consumer suplus
distributed is presented in the following two figures.
0.06
0.055
0.05
0.075
0.074
0.073
0.045
0.072
0.04
0.071
0.5
0.55
0.6
0.65
0.7
a
0.75
0.8
0.85
0.07
0.86
0.9
0.88
0.9
0.92
0.94
0.96
0.98
a
Figure 2: CS against a: Partial-participation
Figure 3: CS against a: Full-participation
19
1
The numerical results in Figures 2 and 3 are derived for k = 0.5.16 The horizontal axis represents
the reservation value a, and the vertical axis is the consumer surplus. In this example the cutoffs
are aL = 0.832 and a H = 0.862. In the interior solution regions, consumers are better off from a
reduction of the search cost. According to my computations, the same pattern applies to all k ∈ (0, 1).
Therefore, the total surplus also increases if the search cost is reduced as the equilibrium profit
always decreases with c for the uniform distribution.
However, between aL and a H , we see a drop in consumer surplus from 0.75 to 0.71. This negative impact of a reduced search cost is not restricted to this transition area. In fact, when there are
enough, but not too many, brand-neutral consumers in the market, both the consumer and total surplus may decrease as the search cost is reduced. In what follows, my focus is on the full-participation
scenario in which prices always increase with a. The partial-participation scenario will be briefly
discussed at the end of this section.17
To calculate consumer surplus, I first write down the expected utility derived for a consumer at
x ∈ [0, 1/2]:
Z
EU(x) =
1
a− k(1−2 x)
(²1 − p − kx − c)d ²1 +
Z
a− k(1−2 x)
(1)
EU
p+ kx
d ²1 +
p+ kx
Z
0
EU(2) d ²1 ,
in which


EU(1)


=



=
(2)
EU
R1
²1 + k(1−2 x) (²2 − p − k(1 − x))d ²2
R ² +k(1−2 x)
+ 01
(²1 − p − kx)d ²2 − 2c
R1
p+ k(1− x) (²2 − p − k(1 − x))d ²2 − 2c
when
²1 ≥ p + kx;
when
²1 < p + kx.
The first term in EU(x) is for the case when the consumer receives a very good match, and the second
and third terms reflect situations when she receives a moderate and a bad match, respectively. By
R 1/2
symmetry, the aggregate consumer surplus is given by: CS/2 = 0 EU(x)dH(x). Taking derivative
with respect to a,
dCS/2
=
da
1/2
Z
0
dEU(x)
dH(x),
da
in which it can be calculated that
dEU(x)
dp
= (1 − a)[1 + a − k(1 − 2x)] + [−1 + (p + kx)(p + k(1 − x))]
.
da
da
Note that in equilibrium the demand D 1 =
R 1/2
0
[1 − (p + kx)(p + k(1 − x))]dH(x) according to (9). The
16 Matlab codes are available upon request.
17 In the transition region, it is easy to show that the consumers surplus decreases with a for any h (x). Moreover, the
s
total surplus also decreases with a if a ≥ 2k +
q
1
k2
3 − 12 .
20
impact on the consumer surplus is therefore given by
dCS/2
= (1 − a)
da
1/2
µZ
0
¶
[1 + a − k(1 − 2x)] dH(x) −
dp
D1.
da
(19)
In what follows, I derive an upper bound for dCS/da, which can be shown to be less than zero
R 1/2
under some sufficient conditions. Given that 0 xdH(x) < 1/4 and
1
D1 ≥ −
2
1/2
Z
0
(a − kx)[a − k(1 − x)]dH(x) >
1 a(a − k) k2
−
−
≡ D,
2
2
8
it can be derived that
Ã
!
µ
¶
a 1
dCS/2
a 1
dp
p · h(1/2)/k
< (1 − a)
+
−D
< (1 − a)
+ −D·
,
2
da
2 2
da
2 2
p + k/2 + 1 + h(1/2)
2 k (1 − a)
in which d p/da is replaced and bounded by
p · ∂G 1 /∂a
(1 − a)p · h(1/2)/k
dp
=−
>
.
da
∂D 1 /∂ p + G 1 p + k/2 + 1 + h(1/2) (1 − a)2
2k
Since the above lower bound is increasing with p and h(1/2), if I can bound the price from below,
dCS/da will be less than zero when h(1/2) is large enough. I next derive the lower bound for p.
The equilibrium price p∗ is derived from condition (8), which can be re-written as follows:
⇐⇒
D1 + p · G1 = 0
¶
µ
Z 1/2
Z 1/2
1
p2
h(1/2)
2
2
xdH(x) = − k
x(1 − x)dH(x).
+ p 1+
(1 − a) + 2k
2
2k
2
0
0
Let p be the solution to the following auxiliary equation:
Ã
!
k
1 k2
M
p2
2
= − ,
+ p 1+
(1 − a) +
2
2k
2
2 8
where M is the carefully chosen upper bound of h(1/2).18 By contradiction, it is not difficult to show
that p∗ > p.19
With this result, I continue to relax the upper bound of dCS/da and define M as the solution of
18 Indeed, h(1/2) cannot be too large. Otherwise, both p and d p/da becomes too small and a decrease in the search cost
definitely benefits the consumers. The detail will be discussed later in Proposition 5.
19 To see this, note that both the LHS’s of the above two equations are increasing with p. Suppose p∗ ≤ p, then
¡ ∗ ¢2
p
2
µ
¶
Z 1/2
h(1/2)
+ p∗ 1 +
(1 − a)2 + 2k
xdH(x)
2k
0
21
h(1/2) to the following equation:
p · h(1/2)/k
a 1
= 0.
+ −D·
2
2 2
p + k/2 + 1 + h(1/2)
2 k (1 − a)
Note that M is a function of a and k only. Considering that neither D nor p depends on h(1/2),
when h(1/2) ≥ M, the LHS of the above equation becomes less than zero and we can conclude that
dCS/da < 0. Therefore, the sufficient condition of the consumer surplus decreasing with a is that
there are enough, but not too many, brand-neutral consumers.20 Indeed, the order effect is intense
when h(1/2) is large. Following an increase in the search cost, firms are willing to cut their prices a
lot to compete, which benefits the consumers. A lower price also increases market coverage, which
enhances consumer surplus as well.
The total surplus (TS) is the sum of firms’ profits and the consumer surplus. In a symmetric equilibrium, the two firms make the same profit, and the total surplus is given by TS/2 = π1 + CS/2, where
π1 = pD 1 is firm 1’s equilibrium profit earning, and consumer surplus is derived from expression (19).
Taking derivative with respective to a, we derive
dTS/2 d π1 dCS/2
=
+
da
da
da
Z 1/2
dp
dp
dD 1 d p
=
[1 + a − k(1 − 2x)]dH(x) −
D1 + p
+ (1 − a)
D1
da
d p da
da
0
Z 1/2
dD 1 d p
[1 + a − k(1 − 2x)]dH(x).
+ (1 − a)
=p
d p da
0
(20)
As can be seen from above, what consumers pay goes into firms’ pockets. Therefore, the condition
under which total surplus decreases with a reduction in c is more stringent. However, there indeed
exists an interval of h(1/2) in which this counter-intuitive result holds. The following proposition
summarizes the welfare analysis.
Proposition 5. In the full-participation scenario, when h(1/2) is moderately large so that M ≤
h(1/2) ≤ M, the consumer surplus will decrease over the interval a ∈ [a H , 1] if the search cost is
reduced. When h(1/2) → ∞, the consumer surplus will instead increase if the search cost is reduced.
p2
≤
2
p2
<
2
µ
¶
Z 1/2
h(1/2)
+ p 1+
(1 − a)2 + 2k
xdH(x)
2k
0
Ã
!
Z 1/2
M
k
1 k2 1
+ p 1+
(1 − a)2 +
= −
< − k2
x(1 − x)dH(x),
2k
2
2
8
2
0
which contradicts the first order condition.
20 A numerical example is given here to show that the existence is not a problem. When a = 0.9 and k = 0.4, M can be
written a function of M, which must be less than M when M falls in the interval [7.3, 1000]. For instance, when M = 10,
dCS/da < 0 as long as h(1/2) ≥ M ≈ 7.5.
22
Consumers are also better off following a reduction in the search cost if h(1/2) → 0.
A set of similar results holds for total surplus.
I leave the proof as well as the conditions for total surplus to the Appendix. When h(1/2) → ∞ so
that there are too many brand-neutral consumers, the consumer surplus decreases with a because
the order effect is too strong. Recall that in Proposition 4 I show that the equilibrium price uniformly
decreases with s. When h(1/2) becomes too large, the equilibrium price is driven too low, leaving
very little room for firms to further lower their prices following a search cost increase. Consumers
are thus worse off when the search cost is bigger. The same result holds when there are very few
brand-neutral consumers, but for a different reason. The resulting order effect when h(1/2) → 0 is
too weak. Therefore, an increase in search cost is not going to bring down the price a lot, which in
the end hurt consumers.
The total surplus decreases with a mainly because less consumers are covered when the search
cost decreases. When there are sufficiently many brand-neutral consumers in the market, the equilibrium price becomes very “sensitive” to a change in search cost because of the strong order effect. A
small decrease in search cost will push the price much higher, leading to a slump in market coverage
and a lower level of total surplus.
One implication from Proposition 5 is that the presence of search costs, as part of transaction
costs, can enhance social welfare in a market. It is not difficult to see that in the first best of this
economy, p = 0 for any H(x) and costless search is always desirable for society. The situation however,
is very different in a market. When the search cost is reduced, consumers who have already received
good match values become better off because they now expect to get even better matches. Meanwhile,
they impose a negative externality on those unfortunate consumers who received poor match values
at their first visits. Since these consumers have to search again regardless of the size of search
cost, a reduction in the search cost does not help to improve their expected utility. However, they
need to pay the higher prices pushed up by the fortunate consumers. As a whole, consumers are
worse off following a search cost reduction. By maintaining the search cost at a relatively high level,
the negative externality is alleviated and the unfortunate consumers can receive higher utilities.
The search cost thus works like a “redistributing” tool reducing the “matching inequalities” among
consumers, which improves social welfare.
A final comment is made for the partial-participation scenario. When the search cost is relatively
large, analytical results are missing because the analysis becomes a lot more complicated given
that x̃1(2) now enters the demand calculation. Intuitively, we expect a more positive influence from
a reduction in search cost. As the search cost falls, though we still experience an increase in the
equilibrium price, the market coverage can expand as it becomes cheaper for consumers to search
again. This additional impact on the market demand is beneficial to both the demand and supply
23
sides. The social welfare is therefore more likely to increase following a reduction in search cost.
5 C ONCLUDING D ISCUSSION
This paper highlights the important changes that the Internet brings to a search market. Since
price information becomes much transparent, firms can effectively influence consumers’ search order
decisions. The resulting order effect generates a set of new findings. First, the relationship between
price and search cost heavily depends on both the size of the search cost and the shape of the brand
preference distribution. Second, higher search costs are generally no longer preferred by firms. They
sacrifice profits that could have been earned from higher search costs to compete with each other.
Finally, when there are moderately many brand-neutral consumers, consumers as a whole strictly
prefer larger search costs, which causes a serious conflict between the demand side and the supply
side. One potential way to resolve this conflict is to improve search “accuracy,” which is made possible
by personalized search. When search becomes more personalized, every consumer can make a better
decision regarding which firm to search first, making all consumers “luckier” on their first search.
The externality problem discussed in Section 4.2 could then be alleviated, and a decrease in the
product search cost may benefit both the consumers and firms.
In what follows, I discuss the possibility that firms intentionally hide prices on the Internet, and
I also discuss one application of the model that compares prices before and after the Internet. More
formalized analysis can be found in a previous version of the paper.
The Random Search Model When firms hide their prices on the Internet, we are back to the more
classic random search model. Compared to the main ordered search model, the price information and
the match value are “bundled” and therefore have to be discovered at the same time. Following the
search literature, I assume consumers correctly anticipate the prices on the equilibrium path and the
equilibrium concept used in this model is the symmetric oligopolist equilibrium defined in Wolinsky
(1986).
Since consumers now form beliefs about prices before search, firms have little control over their
search decisions. Specifically, x̂ does not not depend on the actual prices and always equals 1/2 if
consumers expect same prices from both firms. In addition, since consumers are assumed to hold the
same belief as before even if they observe an off-equilibrium price at their first search, x̃1(2) depends
on the initial expected price only. The order effect is therefore removed and firms enjoy higher
market power as the search cost goes up. In particular, in the full-participation scenario, it holds
for any H s (x) that both the equilibrium price and profit increase with search cost, which is in sharp
contrast to Proposition 3. This model is useful in showing that the inclusion of ex-ante preference
24
does not alter the prediction from a standard random search model.
Price Disclosure It is therefore interesting to see if firms will hide their prices in equilibrium so
that they can earn higher profits. Suppose that, before consumers search, each firms simultaneously
decide whether to disclose the price as well as the price to be charged. The equilibrium concept used
is perfect Bayesian equilibrium (PBE). In such a setting, it can be shown that 1) the equilibrium
derived in the random search model is no longer an equilibrium, and 2) the equilibrium derived in
the ordered search model can be indeed supported as an equilibrium.
To achieve the first goal, it is enough to show that firms have the incentive to deviate from
the equilibrium derived in the random search model. Let p 1R and p 2R be the equilibrium prices in a
random search equilibrium. Given firm 2’s strategy unchanged, if firm 1 discloses p 1R on the Internet,
the profit will stay the same since consumers already correctly anticipated its price. However, firm 1
can do better by charging a lower price. Indeed, since both x̂ and x̃2 are now affected by p 1 , ∂D 1 /∂ p 1
becomes smaller, meaning that firm 1 should charge a price lower than p 1R . Deviating is therefore
beneficial for firm 1 and the random search equilibrium is destroyed.
To support the ordered search equilibrium as a PBE, we need a proper belief system on offequilibrium paths. Suppose consumers correctly anticipate the price when a firm deviates from
disclosing. Since off-equilibrium paths are not reached, all belief systems are consistent with Bayes
rule. Next I show that deviating is not beneficial under the proposed consumer belief.
O
Let pO
1 and p 2 be the equilibrium prices in an ordered search equilibrium. Firm 1’s profit is
O
O
given by pO
1 D 1 (p 1 ; p 1 ) where the second argument in D 1 represents consumers’ belief of p 1 . Since
prices are observable in the ordered search model, the expected price p 1e always equals pO
1 . When
firm 1 deviates, p 1 will definitely go up since ∂D 1 (p 1 ; p 1e )/∂ p 1 becomes larger. However, as consumers
can correctly guess the hidden price, say p 1h , firm 1 will end up with earning a profit of p 1h D 1 (p 1h ; p 1h ).
O
O
h
O
Taking firm 2’s strategy as given, it must hold that p 1h D 1 (p 1h ; p 1h ) < pO
1 D 1 (p 1 ; p 1 ) since p 1 6= p 1 . Firm
1 is therefore strictly worse off by hiding its price. As a result, price disclosure is not an assumption
but a valid equilibrium outcome in my static model.
The Impact of the Internet
Another contribution of this paper is that it provides a framework
that can be applied to study the impact of the Internet in search markets. Unlike homogeneous
products, the Internet affects differentiated products markets in many aspects. To this end, we
first identify what factors might be changed by the Internet in such a market. Besides disclosing
price information to consumers at negligible costs, the Internet may also reduce the product search
costs. After the Internet, people do not have to call or make a trip to a local store to find out some
specifications of a product, suggesting that the search costs for match values are reduced in general.
25
Meanwhile, the Internet also affects consumers’ preferences prior to search. Compared to traditional
advertising campaigns, Internet advertising is more direct and personalized, which may become
more effective. Therefore, the shape of h s (x) might also be altered by the Internet.
By comparing my ordered search model with a random search model, I attempt to draw some
implications for the impact of the Internet in search markets.
Figure 4: The Impact of the Internet
The possible impact from the Internet is illustrated in the above figure in which price is plotted
against search cost. p R (c) and pO (c) denote the equilibrium prices in the random search and ordered
search models, respectively. When c = 0, the two models are equivalent and generate the same
equilibrium price. While p R (c) increases with c, pO (c) slopes downward. Suppose we are interested
in a market which requires a search cost of c 0 before the Internet. If the Internet reduces the search
cost to c 1 and it also alters h s (x) so that there are more brand-loyal consumers and pO (c) shifts
upwards to pO A (c), the resulting equilibrium price will be changed from p 0 to p 1 and whether the
Internet increases or decreases product prices becomes ambiguous.
This simple example has the following implications for real markets. First, the impact of the
Internet on prices differs across markets. Specifically, firms producing more sophisticated products
that require higher search costs are hurt more by the Internet, compared with firms whose products
are less sophisticated. Second, in markets requiring smaller search costs, we should expect to obtain
mixed observations more often regarding the impact of the Internet on prices. Exsiting empirical
work mostly focus on nearly “homogeneous” products (e.g., books and CD’s). Their findings on price
levels are somewhat mixed, which should not be surprising given the type of industries they look at.
Third, given the large profit drop, firms in markets with higher search costs have stronger incentives
26
to collude and hide their prices from the Internet. For example, most local grocery stores do not post
prices online. Since people have to physically visit the stores to buy groceries, the product search
costs are not lowered by the Internet. Had they posted all prices online, their profits would plunge.
Hence, the common practice in this industry is to hide prices from the Internet.
R EFERENCES
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Arbatskaya, Maria. “Ordered Search.” The RAND Journal of Economics 38, 1: (2007) 119–126.
Armstrong, Mark, John Vickers, and Jidong Zhou. “Prominence and Consumer Search.” The RAND
Journal of Economics 40, 2: (2009) 209–233.
Armstrong, Mark, and Jidong Zhou. “Paying for Prominence.” The Economic Journal 121, 556: (2011)
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Johnson, Justin P, and David P Myatt. “On the Simple Economics of Advertising, Marketing, and
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A PPENDIX
Proof of Proposition 1. First we show that there exists only one solution to LHS L (p = a − k) = 0.
When s = 1, this condition can be rewritten as follows
1
LHS L (a) = −
2
1/2
Z
0
¶
µ
k 1 − a2
(a − k + kx)(a − kx)dx + (a − k) −a + −
.
4
2k
28
Since p is replaced by a − k, we write LHS L as a function of a. It is not difficult to check that
LHS L (a = k) > 0 and

 ≤0
LHS L (a = 1) =
 >0
if
0 < k ≤ 0.653;
if
0.653 < k ≤ 1.
As a result, when k ≤ 0.653, by continuity, there must exist a aL (k) such that LHS L (aL ) = 0. When
k > 0.653, it can be shown that LHS L (a) > 0 for ∀a ∈ [k, 1]. Indeed, differentiating LHS L (a) with
respect to a gives us
LHS 0L (a) = −4a +
1
3a2 7k
+
−
.
2k
4
2k
When evaluated at a = 1, LHS 0L (a) ≤ 0 if and only if k ≥ 2/7. Since 0.653 > 2/7, when k > 0.653 it
must be the case that LHS L (a) is monotonically decreasing with a and LHS L (a) ≥ LHS L (a = 1) > 0.
Therefore, the case a > p + k never arises if k > 0.653 and we focus on k ≤ 0.653 for the remaining
analysis.
The uniqueness of aL can be derived by checking the shape of LHS L (a). When k ≥ 2/7, LHS 0L (a) ≤
0 for ∀a ∈ [k, 1] and thus aL is unique. When k < 2/7, there exists a ã(k) such that LHS 0L (a) ≤ 0 for
a ≤ ã(k) and LHS 0L (a) > 0 for a > ã(k). Considering that LHS L (a = 1) < 0, aL must be unique as well.
Further, it must be the case that LHS 0L (aL ) is less than zero. As a result, when a < aL , LHS L (a) > 0
and the equilibrium price p∗ must be interior and greater than a − k. When a > aL , LHS L (a) < 0 and
interior solution is not feasible and in equilibrium the corner solution p∗ = a − k arises. A similar
practice can be done for LHS H (p = a − k) = 0 and the solution can be shown to be unique as well.
Next we study the existence and uniqueness of interior solutions for the cases a ≤ aL and a ≥ a H .
We will start with the case a ≤ aL 21 and then move on to a ≥ a H .
The Case of a ≤ aL :
The proof of the first case contains two steps. First we prove that “crazy” pricing is not optimal,
which is shown in the following lemma. Let p∗ be the “equilibrium” price derived from condition 8.
p∗ ∈ [a − k, a − k/2] given that a ≤ aL . Suppose firm 2 charges p∗ , then pricing outside [a − k, a − k/2]
is never optimal for firm 1.
Lemma 3. Given that the opponent firm charges p∗ , it is better off for the firm to price inside
[a − k, a − k/2].
Proof. If firm 1 lowers the price such that p 1 < a − k, her demand can be written as follows:
D 1l
Z
=
D1 −
( p∗ − p 1 )/ k Z p 1 + kx £
x̃1
0
¤
1 − p∗ − k(1 − x) d ²2 dx
21 Technically we need a lower bound on a so that the local monopoly case won’t happen. However when s = 1, a ≥ k
already ensures the direct competition between the two firms.
29
Z
−
≡
( p∗ − p 1 )/ k Z 2 kx− k+a− p∗ + p 1 £
x̃1
D 1 − ∆1 − ∆2 ,
¤
1 − ²2 + p 1 − p∗ − k(1 − 2x) d ²2 dx
p 1 + kx
where D 1 is calculated from (6). The profit function π1l = p 1 D 1l = p 1 D 1 + p 1 (−∆1 ) + p 1 (−∆2 ). If we
can show d π1l /d p 1 ≥ 0 for p 1 ≤ a − k then it is not profitable to lower the price below a − k. As is
discussed in Footnote 23, D 1 is log-concave for p 1 ≤ a − k and therefore π1 = p 1 D 1 is quasi-concave
in this range. Given that p∗ is an interior equilibrium price, we must have d π1 /d p 1 ≥ 0 at p 1 = a − k
and for ∀ p 1 < a − k. Let π∆1 ≡ p 1 (−∆1 ) and π∆2 ≡ p 1 (−∆2 ). Then what we need to show is that both
d π∆1 /d p 1 ≥ 0 and d π∆2 /d p 1 ≥ 0 hold. Since both π∆1 and π∆2 are quasi-concave and we only need
to check the signs of the above two derivatives at p 1 = a − k. Simple algebra shows that both are
nonnegative and therefore it is not optimal for store 1 to lower the price below a − k.22
On the other hand, if firm 1 picks a price such that p 1 ≥ 2a − k − p∗ , her demand becomes
D 1h =
x̂
Z
0
(1 − p 1 − kx)dx.
If she further increases the price such that p 1 ≥ 2 − k − p∗ , her demand is given by
D 1hh
(1− p 1 )/ k
Z
=
(1 − p 1 − kx)dx.
0
First we show that pricing above 2 − k − p∗ is not profitable for store 1. The first order effect of p 1
on the profit is given by
d π1hh
d p1
= D 1hh + p 1
dD 1hh
d p1
(1− p 1 )/ k
Z
=
(1 − 2p 1 − kx) dx <
0
(1− p 1 )/ k
Z
0
[1 − 2(2 − a − k/2)] dx.
Since a + k/2 < 3/2, we conclude that d π1hh /d p 1 < 0 for ∀ p 1 ≥ 2 − k − p 2 .
Now assume 2a − k − p∗ ≤ p 1 < 2 − k − p∗ . Since D 1h is log-concave, π1h = p 1 · D 1h is quasi-concave.
If d π1h /d p 1 < 0 when evaluated at p 1 = 2a − k − p∗ , it is less than zero on the whole range:
¯
d π1h ¯¯
¯
d p1 ¯
1−(a− p∗ )/ k £
Z
=
0
p 1 =2a− k− p∗
µ
<
¤
1
1 − 2(2a − p∗ − k) − kx dx −
(1 − a)
2k
¶
¤ 1
a − p∗ £
1−
1 − 2(2a − p∗ − k) −
(1 − a).
k
2k
If the expression in the squared brackets is non-positive, we are done. If it is not, the first whole
22 If firm 1 lowers the price further so that x̂ ≥ 1, in a similar fashion we can show that it is not profitable to do so.
30
term is then increasing with p 2 and therefore
¯
d π1h ¯¯
¯
d p1 ¯
<
p 1 =2a− k− p 2
1
1
(1 − a),
[1 − 2(a − k/2)] −
2
2k
which is less than zero.
Lemma 3 shows that, if there exists an equilibrium price when p ∈ [a − k, a − k/2], it must be also
“globally” optimal. Next we prove the existence result within this closed interval and all we need
to show is the log-concavity of the demand function D 1 on p ∈ [a − k, a − k/2]23 . The profit function
must be quasi-concave. The existence result then follows from Kakutani’s fixed-point theorem. The
uniqueness comes from Proposition 6 in Caplin and Nalebuff (1991). In addition, since the optimal
pricing function for each store is symmetric, the unique equilibrium must be symmetric as well.
We first fix firm 2’s price p 2 ∈ [a − k, a − k/2]. Denote by d 1 the first order derivative of D 1 with
respect to p 1 and by d 10 the second order derivative. If we can show d 12 − D 1 d 10 ≥ 0, then D 1 is logconcave in p 1 and the task is completed. We follow this strategy and let g(p 1 , p 2 , a; k) denote this
expression. Next we show that g(p 1 , p 2 , a; k) ≥ 0 for p 1 ∈ [a − k, a − k/2]. First let’s write down the
functions d 1 and d 10 :
·
µ
¶ ¸
k p1 + p2 2
1
1−
;
+
= − x̂ +
(1 − p 2 − k(1 − x)) dx −
2k
2
2
x̃1
1 p1 + p2
=
+
> 0.
2
2k
Z
d1
d 10
x̂
Clearly, d 1 < 0 because p 2 + k(1 − x) ≥ 0 and thus the sum of the first two integrals is negative.
To show the result, we first establish that g(p 1 , p 2 , a; k) is decreasing with a and p 1 , respectively.
Since d 1 < 0,
∂g
∂a
∂d1
∂a
=
1−a
≥0
k
= 2d 1
and
∂d1
∂a
−
∂D 1
∂a
∂D 1
∂a
=
d 10 , and
1−a
(p 1 + p 2 + k − a) ≥ 0.
k
And differentiating g with respect to p 1 gives us
∂g
∂ p1
23 D
= d 1 d 10 −
D1
< 0.
2k
1 is in fact log-concave on [0, a − k/2] as the lower bound is free in the proof of the result.
31
As a result, g(p 1 , p 2 , a; k) ≥ g(1 − k/2, p 2 , 1; k) and we have the following result for the latter term
1 − p2 − k
g(1 − k/2, p 2 , 1; k) ≥ g(2 − k − p 2 , p 2 , 1; k) =
k
µ
¶2
1
−
k
1−(1− p 2 )/ k
Z
0
(p 2 + k(1 − x) − 1)dx
since p 2 ≤ a − k/2. It is easy to show that g(2 − k − p 2 , p 2 , 1; k) increases with p 2 and thus g(p 2 ; k) ≥
g(1 − k; k) = 0. Therefore, g(p 1 , p 2 , a; k) ≥ 0 and D 1 is log-concave in p 1 . Combined with Lemma 3, we
finally establish that the equilibrium price p∗ ∈ [a − k, a − k/2] is globally optimal for both stores.
The Case of a ≥ a H :
Recall that in this case x̃1 ≤ 0 and x̃2 ≥ 1. Similar to the previous case, we want to show the
log-concavity of D 1 on p ∈ [0, a − k]. The crazy pricing now involves the following two cases: x̃2 < 1
when p 1 is too high and x̂1 > 1 when p 1 is too low. This time we will start with the log-concavity part
and then discuss the crazy pricing.
Fix firm 2’s price p 2 ∈ [0, a − k]. We next show that D 1 , which is derived from 9, is log-concave in
p 1 ∈ [0, a − k]. Note that, in order for x̂ to be in the unit interval, p 1 ’s domain may vary with different
values of a, k, or p 2 . However [0, a − k] is the superset of those changing domains and our goal is
achieved as long as we can show that D 1 is log-concave in p 1 ∈ [0, a − k].
First we show that d 1 < 0. From simple algebra, we have
d 10 =
1 p1 − p2
+
> 0,
2
2k
and therefore d 1 increases with p 1 . Differentiating d 1 with respect to a, we get
∂d1
∂a
1−a
> 0,
k
=
so d 1 increases with a as well. As a result,
p2 − 1
d 1 ≤ d 1 (a = 1, p 1 = 1 − k) = −1 −
+
2k
Z
1+( p 2 −1)/2 k
1+( p 2 −1)/ k
·
µ
¶ ¸
1 + p2 2
1
1−
.
1 − p 2 − k(1 − x)dx −
2k
2
It can be shown that d 1 (a = 1, p 1 = 1 − k) decreases with p 2 and therefore when evaluated at p 2 = 0
d 1 (a = 1, p 1 = 1 − k, p 2 = 0) =
1 − 4k
< 0.
4k
The inequality holds because k ≥ 1/2 in this case.
To show g ≥ 0, we prove that g is monotonic in p 1 , a, and p 2 so that we only need to check the
boundary values. Differentiating g with respect to p 1 gives us
∂g
∂ p1
= d 1 d 10 − D 1 d 100 < 0,
32
where d 100 = ∂ d 10 /∂ p 1 . So g is decreasing with p 1 .
Now let’s look at the first order effect of the reservation value a on g:
∂g
∂a
= 2d 1
∂d1
∂a
−
∂D 1
∂a
d 10 .
It is calculated that
∂d1
∂a
∂D 1
∂a
=
1−a
≥ 0;
k
=

 ≥0
1−a
(p 1 − p 2 )
 <0
k
if
p1 ≥ p2
if
p1 < p2
.
When p 1 ≥ p 2 , it is clear that ∂ g/∂a ≥ 0. When p 1 < p 2 ,
∂2 g
∂ a∂ p 1
=
¤
1−a £ 0
d 1 − (p 1 − p 2 )d 100 > 0
k
and thus ∂ g/∂a increases with p 1 . Therefore the first order effect of a on g is no greater than that
when evaluated p 1 = p 2 , which is less than zero. We thus derive that g decreases with a.
As a result, we only need to check the sign of the following function:
g(a = 1, p 1 = p 2 + k)
when
g(a = 1, p 1 = 1 − k)
when
1
and p 2 ≤ 1 − 2k;
2
1
1
k ≥ or k < and p 2 > 1 − 2k.
2
2
k<
The second case applies here since k ≥ 1/2.
∂ g(a = 1, p 1 = 1 − k)
∂ p2
= 2d 1
∂d1
∂ p2
−
∂D 1
∂ p2
d 10 − D 1
∂ d 10
µ
¶
1 − p2
∂D 1
D1
=
−2d 1 −
+
∂ p2
2k
∂ p2
2k
since ∂ d 1 /∂ p 2 = (p 2 − 1)/2k and d 10 = (1 − p 2 )/2k. Therefore we only need to show that −2d 1 − ∂D 1 /∂ p 2 ≥
0. In doing so, we calculate the following
d1 +
∂D 1
∂ p2
µ
¶ Z 1+( p2 −1)/2k
·
µ
¶ ¸
p2 − 1
1
1 + p2 2
= − 1+
+
1 − p 2 − k(1 − x)dx −
1−
2k
2k
2
1+( p 2 −1)/ k
·
µ
¶2 ¸ Z 0
Z 1+( p2 −1)/2k
1 + p2
1
−
1−
+
1 − k(1 − x)dx +
p 2 − 1 + k(1 − x)dx
2k
2
1+( p 2 −1)/ k
1+( p 2 −1)/ k
¶ Z 1+( p2 −1)/2k
µ
Z 0
p2 − 1
−
=
p 2 − 1 + k(1 − x)dx − 1 +
p 2 dx
2k
1+( p 2 −1)/ k
1+( p 2 −1)/ k
µ
¶ Z 1+( p2 −1)/2k
Z 0
p2 − 1
<
p 2 − 1dx − 1 +
−
p 2 dx < 0.
k
1+( p 2 −1)/ k
1+( p 2 −1)/ k
33
As a result, −2d 1 − ∂D 1 /∂ p 2 > 0 and g(a = 1, p 1 = 1 − k) increases with p 2 . Checking the boundary
conditions, we conclude that g > 0. Finally we have shown that D 1 is log-concave and thus π1 =
p 1 D 1 is quasi-concave in p 1 on the reasonable support. Because of symmetry, there must exists a
symmetric equilibrium in which the equilibrium price p∗ ≤ a − k.
Following the same logic as before, given that there exists an interior equilibrium, raising (decreasing) the price too high (low) unilaterally would never be optimal. For example, when p 1 ≥ a − k
so that x̃2 < 1, the demand function can be written as
D 1h
Z
=
x̃1
Z
p 1 + kx
D1 −
Z
+
(1 − p 2 − k(1 − x)) d ²2 dx
2 x̂−1 0
x̃1 Z 2 kx− k+a− p 2 + p 1
2 x̂−1 p 1 + kx
(1 − ²2 + p 1 − p 2 − k(1 − 2x)) d ²2 dx.
It can be verified that D 1 is log-concave for p 1 ≥ a − k as well. Plus the fact the the last two terms
of D 1h are both log-concave, we follow the strategy taken in Lemma 3 and show that d π1h /d p 1 ≤ 0 for
∀ p 1 ≥ a − k. Therefore increasing the price is not profitable. The intuition is that when x̃2 < 1, store
1 loses more switching consumers from store 2 than when x̃2 ≥ 1 and thus increasing p 1 above the
upper bound is not optimal. Similarly, when p 1 is too low such that x̂ ≥ 1, she cannot gain more loyal
and returning consumers compared to the case when x̂ ≤ 1 and therefore undercutting is not optimal
as well.
Proof of Lemma 1. We copy the expression of (12) to here:
∂D 1
∂p
+ G1 + p
∂G 1
∂p
1 h(1/2)
h0 ( x̃1 )
= − −
(1 − a)2 −
p(2p + k − a)(1 − a)
2
2k
k2
h( x̃1 )
−
[2(1 − a)(2p + k − a) + p(1 − 2p − k)]
k
µ
¶
Z 1/2
1
− H( x̃1 ) −
(1 − k(1 − 2x))h(x)dx − (2p + k) − H( x̃1 ) .
2
x̃1
The only uncertainty comes from h0 (x) and the second line. We next show that the expression in the
squared brackets from the second line must be greater than zero in equilibrium and then discuss the
size of h0 (x). For expositional ease, we denote the squared brackets by SQ.
If 1 − 2p − k ≥ 0 in equilibrium, provided that a − k ≤ p ≤ a − k/2, SQ > 0. If 1 − 2p − k < 0,
k
k
k
SQ ≥ 2(1 − a)(2p + k − a) + (a − )(1 − 2p − k) = (2p + k)(2 − 3a + ) + 2a2 − a − .
2
2
2
34
When 2 − 3a + k/2 ≥ 0, that is a ≤ (4 + k)/6, we derive the following:
k
k2 5k2
k
−
.
SQ ≥ (2a − k)(2 − 3a + ) + 2a2 − a − = −4a2 + (3 + 4k)a −
2
2
2
2
In order to show SQ > 0, all we need to check is then the sign at the two end points: k and (4 + k)/6.
Simple algebra shows that at both points SQ > 0 and we are done in this case.
When s − 3a + k/2 < 0, that is a > (4 + k)/6, we can instead bound SQ from below as follows:
SQ > 2a(2 − 3a + k/2) + 2a2 − a −
k
k
= −4a2 + (3 + k)a − ,
2
2
where the last line obtains its maximum at a∗ = (3 + k)/8. Note that the lower bound of a here
may vary depending on the value of k. Specifically, when k ≤ 4/5, (4 + k)/6 ≥ k and thus a > (4 + k)/6.
Otherwise a > k. It turns out that under both cases a∗ is smaller than the lower bound of a. Therefore
the expression in the last line is monotonically declining as a increases and it is greater than zero if
a≤
3+k+
p
(3 + k)2 − 8k
,
8
which is part one of condition (14).
If the second part of condition (14) holds, the first line in the expression of (12) will be less than
zero. To see this, when h0 (x) ≥ 0, the result automatically follows. When h0 (x) < 0, we use the fact
that p < a − k/2 in equilibrium and the condition is derived.
Proof of Lemma 2. I first look at the two extreme cases. x̃1 → 1/2 means a → a. Here a is the lower
bound of a that separates the game from the local monopoly case (See discussion in footnote 9). This
additional requirement does not conflict with a ≥ k. When a > k, a ≥ a is a finer requirement than
a ≥ k. When k ≥ a, the pattern I derived for a ≥ a continues to hold for a ≥ k. It is easy to see
d p/da > 0 since h0 (1/2) = 0 by the differentiability of h(x).
When a → aL , x̃1 → 0. Condition (15) makes sure that expression (13) is negative. To see this,
note that here (13) can be rewritten as
∂D 1
∂G 1
µ
¶
(1 − a)(a − k)
a−k 0
1
+p
=
h(0) +
h (0) + h( ) < 0.
∂a
∂a
k
k
2
Now I derive the condition such that, when a moves from a to aL , there is only one unique peak on
the p(a) locus. First, it is necessary to show that aL > a so that we have an idea of whether expression
(13) is increasing or decreasing with a. Recall that, as discussed at the beginning of Section 4, aL is
the solution to the first order condition (8): D 1 + pG 1 = 0 when evaluated at p = a − k. Since there
may exist multiple solutions, the relationship between aL and a becomes ambiguous. Fortunately,
35
it turns out that the solution to the above equation is unique given conditions (14) and (15). To see
this, we take derivative of the first order condition, evaluated at p = a − k, with respect to a:
µ
∂D 1
∂p
+ G1 + p
∂G 1
∂p
¶
·
∂p
∂a
+
∂D 1
∂a
+p
∂G 1
∂a
µ
=
∂D 1
∂p
+ G1 + p
∂G 1
∂p
¶
·1+
∂D 1
∂a
+p
∂G 1
∂a
< 0.
As a result, D 1 + pG 1 monotonically decreases with a and aL is the unique solution, which means
that for ∀a such that x̃1 > 0 we have a < aL . Therefore a is indeed smaller than aL .
Now what we need to show is that
µ
¶
d ∂D 1
∂G 1
<0
+p
da ∂a
∂a
whenever ∂D 1 /∂a + p∂G 1 /∂a crosses 0. Since a < aL and that expression (13) is positive when evaluated at a and negative when at aL , if the above claim is true then our task is done.
Given that d p/da = 0 when ∂D 1 /∂a + p∂G 1 /∂a crosses 0, we have
µ
¶
·
¸
d ∂D 1
∂G 1
d
h0 ( x̃1 )
1
+p
=
h( x̃1 )(2p + k − a) +
p(2p + k − a) + h( )p
da ∂a
∂a
da
k
2
³
´
³
´
(2p + k − a) 0
p
p
=−
h ( x̃1 ) + h00 ( x̃1 ) − h( x̃1 ) + h0 ( x̃1 )
k
k
k
´
p 00
h(1/2)p
(2p + k − a) ³ 0
h ( x̃1 ) + h ( x̃1 ) +
.
=−
k
k
2p + k − a
Since a − k ≤ p ≤ a − k/2, the above expression is less than zero if condition (16) holds.
Proof of Corollary 1. As in the proof of Proposition 3, all we need to show is ∂D 1 /∂ p 2 ≥ 0. From
straightforward algebra we derive the following:
∂S 1
∂ p2
∂S 2
∂ p2
∂S 3
∂ p2
= H( x̂) +
Z
=−
x̂
x̃1
x̂ ·
Z
(p 2 + k(1 − x))h(x)dx +
a
( k+ p 1 + p 2 )/2
p 1 + kx
²1 d ²1
h( x̂)
;
2k
¸
h0 (2 x̂ − x)
dx
−(p 1 + kx)h(2 x̂ − x) +
(1 − p 2 − k(1 − x))d ²2
k
x̃1
0
µ
¶µ
¶
k p1 + p2
k p 1 + p 2 h( x̂)
h(2 x̂ − x̃1 )
+
− (p 1 + p 2 + k − a)(1 − a)
+
1− −
2
2
2
2
2k
k
¸
Z x̂ ·
Z
h0 (2 x̂ − x)
a− p 2 + p 1 − k(1−2 x)
+
−(1 − p 2 − k(1 − x))h(2 x̂ − x) + p 1 + kx
(1 − ²2 + p 1 − p 2 − k(1 − 2x))d ²2
dx
k
x̃
Z 1a
h( x̂)
+
(1 − ²2 )d ²2
.
2k
( p 1 + p 2 + k)/2
Z
=
1−a
h( x̂) − H( x̃1 );
2k
Z
36
In equilibrium, p 1 = p 2 and x̂ = 1/2. ∂D 1 /∂ p 2 thus can be derived as follows:
∂D 1
1
h(1/2)
= − H( x̃1 ) +
(1 − a)2 +
∂ p2 2
2k
Z
1/2
x̃1
Z 1/2
(1 − 2p − 2k(1 − x))h(x)dx
h(1/2)
1
(1 − a)2 +
> − H( x̃1 ) +
(1 − 2p − 2k(1 − x̃1 ))h(x)dx
2
2k
x̃1
µ
¶
h(1/2)
1
1
(1 − a)2 + (1 − 2a) h( ) − h( x̃1 ) .
= − H( x̃1 ) +
2
2k
2
If a ≤ 1/2, we are done. If a > 1/2, the above expression can be bounded from the below as follows:
∂D 1
∂ p2
>
µ
¶
h(1/2)
1
(1 − a)2 + (1 − 2a) h( ) − h( x̃1 ) .
2k
2
Therefore, as long as the above expression is non-negative, we have ∂D 1 /∂ p 2 ≥ 0. Since the above
quadratic function reaches the minimum at 1 + 2k(1 − h(0)/h(1/2)), which is greater than 1/2. We only
need to make sure that a does not exceed the smaller root of the function for ∂D 1 /∂ p 2 ≥ 0 to hold. In
the case of s = 1, this condition does not bind.
Proof of Proposition 3. We only need to prove the profit result. As before, the impact of the search
cost on the profit is given by (17), namely the following expression:
·
µ
¶
¸¯
d π1 (p∗ )
∂D 1
∂D 1 d p 1
∂D 1 d p 2 ¯¯
.
= p1
+ D 1 + p1
+ p1
da
∂a
∂ p 1 da
∂ p 2 da ¯ p1 = p2 = p∗
Since in equilibrium ∂D 1 /∂a = D 1 + p 1 ∂∂Dp11 = 024 and d p 1 /da = d p 2 /da ≥ 0, all we need to show is
∂D 1 /∂ p 2 ≥ 0 in equilibrium, which indeed holds for any H s (x).
The subscript s in H s (x) is suppressed here. The derivatives we need are given by:
∂S 1
∂ p2
∂S 2
∂ p2
∂S 3
∂ p2
1 h(1/2)
+
(1 − a);
2
2k
Z
Z 1/2
h(1/2) a
=
²1 d ² 1 −
(p + k(1 − x))h(x)dx;
2k
p+ k/2
0
¸
Z 1/2 ·
Z p+kx
h0 (1 − x)
=
−(p + kx)h(1 − x) +
(1 − p − k(1 − x))d ²2
dx
k
0
0
µ
¶µ
¶
k
h(1/2)
k
h(0)
+
1− − p
+p −
p(1 − p − k)
2k
2
2
k
¸
Z 1/2 ·
Z a−k(1−2 x)
h0 (1 − x)
+
−(1 − p − k(1 − x))h(1 − x) +
(1 − ²2 − k(1 − 2x))d ²2
dx
k
0
p+ kx
=
24 For the corner solution case, we still have ∂D /∂a = 0, while D + p ∂D 1 > 0 when evaluated at p = a − k. The following
1
1
1 ∂ p1
1
analysis does not change.
37
h(1/2)
+
2k
Z
a
h(0)
(1 − ²2 )d ²2 −
k
p+ k/2
a− k
Z
p
(1 − ²2 − k)d ²2 .
∂D 1 ∂ p 2 is the sum of the above three and it can be simplified as follows:
∂D 1
∂ p2
=
∂D 1
Z
1/2
(2p + k)h(x)dx
∂ p1
0
Z 1/2
Z 1/2
1
h(1/2)
2
−
(p + k(1 − x))h(x)dx +
(1 − a) +
[1 − 2(p + k(1 − x))]h(x)dx
2
2k
0
0
Z 1/2
+
[1 − (p + k(1 − x))]h(x)dx.
= −
−
0
Only the second term is negative here, and it is less than 1/2 because p + k(1 − x) < p + k < a ≤ 1. We
thus have shown that ∂D 1 /∂ p 2 > 0 in equilibrium.
Proof of Proposition 4. I first look at the impact from a change in s and then k. d p/ds is derived from
the equilibrium condition (8) as follows:
µ
∂D 1
∂p
¶
+ G1
d p ∂D 1
∂G 1
+
+p
= 0.
ds
∂s
∂s
Since ∂D 1 /∂ p + G 1 < 0, all we need to know is the sign of ∂D 1 /∂ s + p∂G 1 /∂ s. From the demand function
D 1 derived in (9), I calculate the derivative as below:
∂D 1
∂s
= − k2
1/2
Z
0
x(1 − x)
dh s (x)
dx.
ds
Note that
1/2
Z
0
dh s (x)
x(1 − x)
dx =
ds
x†s
Z
0
x†s
Z
>
=
0
dh s (x)
x(1 − x)
dx +
ds
1/2
Z
dh s (x)
x†s (1 − x†s )
dx +
x†s (1 − x†s )
ds
1/2
Z
0
x(1 − x)
x†s
Z
1/2
x†s
dh s (x)
dx
ds
x†s (1 − x†s )
dh s (x)
dx
ds
dh s (x)
dx = 0,
ds
where the inequality comes from the definition of the rotation and the monotonicity of x(1 − x) on
R 1/2
[0, 1/2]. The last line holds because 0 h s (x)dx = H s (1/2) = 1/2 for ∀ s ∈ [s, s]. As a result, ∂D 1 /∂ s < 0.
Similarly, I derive ∂G 1 /∂ s as follows.
(1 − a)2 dh s (1/2)
=−
−
∂s
2k
ds
∂G 1
1/2
Z
0
38
(1 − k(1 − 2x))
dh s (x)
dx,
ds
which is less than zero as well. To see this, we know dh s (1/2)/ds ≥ 0 and
1/2
Z
0
(1 − k(1 − 2x))
dh s (x)
dx >
ds
=
x†s
Z
0
(1 − k(1 − 2x†s ))
(1 − k(1 − 2x†s ))
dh s (x)
dx +
ds
1/2
Z
0
1/2
Z
x†s
(1 − k(1 − 2x†s ))
dh s (x)
dx
ds
dh s (x)
dx = 0.
ds
Therefore, both ∂D 1 /∂ s and ∂G 1 /∂ s are negative, meaning that d p/ds < 0 in equilibrium.
As before, d p/dk is derived from the following condition:
µ
∂D 1
∂p
¶
+ G1
d p ∂D 1
∂G 1
+
+p
= 0.
dk
∂k
∂k
The sign of d p/dk depends on ∂D 1 /∂ k and ∂G 1 /∂ k that are given by
∂D 1
∂k
∂G 1
∂k
Z 1/2
p
x(1 − x)h s (x)dx < 0;
= − − 2k
2
0
Z 1/2
h s (1/2)
2
(1 − 2x)h s (x)dx > 0.
(1 − a) +
=
2k2
0
Therefore,
∂D 1
∂G 1
¶
Z 1/2
1 ∂G 1
− 2k
x(1 − x)h s (x)dx.
+p
= −p
−
∂k
∂k
2
∂k
0
µ
One sufficient condition for ∂D 1 /∂ k + p∂G 1 /∂ k < 0 to hold is thus 1/2 − ∂G 1 /∂ k > 0, which is equivalent
to condition (18). We therefore conclude that d p/dk < 0 in equilibrium.
Proof of Proposition 5. We only need to prove the second part of the proposition. We first look at the
impact on the consumer surplus when h(1/2) → ∞(0) and then move on to the discussion of the total
surplus.
From the first order condition, the equilibrium price can be explicitly written out as follows:
p
p∗ = − b + b2 + 2c, where
Z 1/2
h(1/2)
(1 − a)2 + 2k
xdH(x),
2k
0
Z 1/2
1
2
c= −k
x(1 − x)dH(x).
2
0
b = 1+
Therefore, p∗ converges to zero as h(1/2) → ∞. We re-write d p/da as follows:
dp
da
=
=
1 − a p∗ · h(1/2)
·
k
p∗ + b
p
1 − a h(1/2)(− b + b2 + 2c)
·
p
k
b2 + 2c
39
=
´
p
1−a ³
· − b + b2 + 2c ·
k
·µ
b
h(1/2)
¶2
+
2c
2
h (1/2)
¸−1/2
,
p
which converges to zero as well because b/h(1/2) converges to some constant and − b + b2 + 2c → 0.
As a result, d p/da → 0, meaning that dCS/da > 0 when h(1/2) → ∞. When h(1/2) → 0, p∗ converges
to a constant and thus d p/da → 0, which implies that dCS/da > 0 as well.
Next, we follow the analysis in the text and discuss the impact on the total surplus. Derived from
expression (20), the total surplus is given by
dTS/2
da
1/2
¶
¸
µ
k dp
[1 + a − k(1 − 2x)]dH(x) − p p +
2 da
!
à 0
µ
¶
a 1
k
p · h(1/2)/k
.
< (1 − a)
+ − p p+
·
2 2
2 p + k/2 + 1 + h(1/2) (1 − a)2
2k
·Z
= (1 − a)
As before, we define M 0 as the solution of h(1/2) to
¶
µ
p · h(1/2)/k
k
a 1
+ − p p+
·
= 0.
2 2
2 p + k/2 + 1 + h(1/2) (1 − a)2
2k
When h(1/2) ≥ M 0 , dTS/da will become less than zero. The case when h(1/2) → ∞(0) follows the same
analysis as above.
40