Do higher search costs make markets less competitive?

Do higher search costs make markets less competitive?∗
José L. Moraga-González†
Zsolt Sándor‡
Matthijs R. Wildenbeest§
March 2014
– P R E L I M I N A R Y & I N C O M P L E T E –
C O M M E N T S W E L C O M E
Abstract
We study price formation under search cost heterogeneity and revisit the question how an
increase in search costs affects the level of prices. In doing so, we dispense with the usual assumption that all consumers search at least once in equilibrium. This allows for an important role of
the price mechanism, namely, that the price ought to affect the number of consumers who make
the decision to search for a product rather than not searching for it at all. Recognising this role
turns out to be critical for our understanding of the effect of higher search costs on prices and
profits. We show that higher search costs may result in more elastic individual demand functions,
and therefore lead to lower prices. This happens because an increase in search costs affects two
margins, the intensive search margin, or search intensity, and the extensive search margin, or
participation. Higher search costs result in less search intensity, making demand more inelastic;
however, higher search costs also lower the participation of consumers who happen to search little,
which makes demand more elastic. We identify one critical condition for higher search costs to
result in lower prices, namely, that the search cost distribution has a decreasing elasticity with
respect to the parameter that shifts the distribution. In that case, the effect of higher search costs
on the extensive search margin is stronger than the effect on the intensive search margin. These
insights hold no matter whether products are differentiated or homogenous and irrespective of
whether consumers search sequentially or non-sequentially.
Keywords: non-sequential search, sequential search, oligopoly, search cost heterogeneity,
homogeneous products, differentiated products
JEL Classification: D43, C72
∗
We would like to thank Remco bvan Eijkel, Knut-Eric Joslin, Dmitry Lubensky, Marielle Non, Vaiva Petrikaitė
and Régis Renault for their comments. Special thanks go to Paulo K. Monteiro for his comments and help since the
beginning of this project. The paper has also benefited from presentations at the CPB Netherlands Bureau for Economic
Policy Analysis, University of East Anglia, VU University Amsterdam, the IV Workshop on Search and Switching Costs,
Moscow 2013 and at the 40th EARIE Meetings (Evora, 2013). Financial support from the Marie Curie Excellence
Grant MEXT-CT-2006-042471, grant ECO2011-29533 of the Spanish Ministry of Economy and Competitiveness and
grant PN-II-ID-PCE-2012-4-0066 of the Romanian Ministry of National Education, CNCS-UEFISCDI, is gratefully
acknowledged.
†
VU University Amsterdam and University of Groningen. E-mail: [email protected]. Moraga is also
affiliated with the Tinbergen Institute, the CEPR, and the Public-Private Sector Research Center (IESE, Barcelona).
‡
Sapientia University Miercurea Ciuc. E-mail: [email protected].
§
Kelley School of Business, Indiana University, E-mail: [email protected].
1
Introduction
Ever since Stigler (1961), search theory has become an important tool for understanding the functioning of real-world markets. Seminal contributions include the homogeneous product market models of
Burdett and Judd (1983), Reinganum (1979) and Stahl (1989), where the important phenomenon of
price dispersion is given a microfoundation based on search theory. When products are differentiated,
the market models of Wolinsky (1986) and Anderson and Renault (1999) show that prices remain
above costs even if firm entry is costless and correspondingly infinitely many firms enter the market.
Another well-known and important result in this literature that is common to homogeneous and differentiated product market models is that higher search costs lead to higher prices, thus benefiting
firms at the expense of consumers.
In this paper we contribute to the search cost literature in two ways. Our first contribution is
to study price formation under consumer search cost heterogeneity. In doing so, we look at models
with differentiated and homogeneous products, as well as with non-sequential and sequential search.
The second contribution is to revisit the question how an increase in search costs affects the level of
prices. As we will see later in more detail, it turns out that the behavior of the equilibrium price
depends on the range of search costs as well as on the properties of the search cost distribution.
When studying search markets with search cost heterogeneity, it is natural to dispense with a
restrictive assumption that the consumer search literature has almost invariably made: most models
assume that the cost consumers have to incur in order to conduct a search is “sufficiently low,” de
facto implying that all consumers search at least once in equilibrium (e.g. Stahl, 1989; Burdett and
Judd, 1983; Wolinsky, 1986). As pointed out by Stiglitz (1989), assuming otherwise that search
costs are large can potentially cause the market to collapse. For example, in the setting of Diamond
(1971), the only price that can be part of a market equilibrium is the monopoly price –this result is
referred to as the “Diamond paradox”. If the search cost is relatively high, the surplus consumers
derive at the monopoly price may be insufficient to cover the cost of the first search, in which case
consumers rather do not search at all and the market fails to exist.
Apart from being simply less general, the standard “low-search-cost” assumption is also restrictive. To understand why, note that in search markets, in addition to the usual extensive and intensive
margins,1 the price mechanism ought to affect the number of consumers who choose to search for a
good deal in the first place, which we refer to as the extensive search margin, as well as the intensity
1
The extensive margin refers to the number of consumers who choose to buy from a given firm while the intensive
margin to the quantity acquired by consumers who choose to buy.
2
with which consumers search, which we call the intensive search margin. The literature, by assuming
that all consumers search at least once, has focused on the effects of the intensive search margin on
price determination and has neglected the role of the extensive search margin.
Failing to recognise this role is critical for our understanding of the functioning of consumer
search markets. In fact, this point is the central tenet in Anderson and Renault (2006), who, by
allowing for arbitrary search costs, are able to reconcile the empirical observation that much of the
advertising we observe in arguably search environments does impart only match information and not
price information. Similarly, Janssen et al. (2005) show that in the standard model of Stahl (1989)
(the distribution of) equilibrium prices will increase as the search cost goes up provided that the
search cost is sufficiently high.
We start the analysis by considering models in which products are differentiated and the price
equilibrium is in pure-strategies, as in the model of Wolinsky (1986). In these models, consumers
do not search for lower prices but for satisfactory products, have unit demands and have correct
expectations about the equilibrium price. We study both the case in which consumers search nonsequentially and the case in which they search sequentially. The chief difference between these two
search protocols is in terms of the ability of an individual firm to affect the different aforementioned
margins. While under sequential search an individual firm can affect the intensive search margin,
this is not the case under non-sequential search. Nonetheless the insights we derive are common to
the two modes of search.
In the non-sequential search model, consumers choose how many products they wish to inspect,
including none, in order to maximise the expected gains from search minus the search costs. When
the search cost of every consumer is sufficiently low, then they all search the entire market in
equilibrium. In this case, the equilibrium price is the same as in Perloff and Salop (1985) and
therefore small changes in the search cost distribution have no bearing on the equilibrium price.2
As search costs become higher (in a first-order stochastic dominance sense), search intensities
become more heterogeneous. Some consumers continue to choose to inspect all the products in
the market while other consumers with higher search costs choose to search fewer products. From
the point of view of an individual firm, consumers who sample many firms are more elastic than
consumers who visit just a few. Optimal pricing must then trade-off the incentives to extract profits
from less elastic consumers and the incentives to compete for the more elastic ones. Eventually as
2
As far as we know, the only study of a non-sequential search model with differentiated products is in Anderson
et al. (1992); in their model, however, all consumers have the same cost of search and this makes their analysis very
different from ours.
3
search costs continue to rise, some consumers choose to opt out of the market altogether.
We demonstrate that the effect of an increase in search costs on the equilibrium price in a situation
where all consumers have relatively low search costs and correspondingly they all search at least once
differs fundamentally from the case in which search costs are more dispersed and some consumers
opt out of the search market altogether. The reason is that, everything else constant, in the first
case an increase in search costs has a bearing only on the intensive search margin while in the second
case both the intensive and the extensive search margins are affected.
In the first case, we show that higher search costs unambiguously lead to higher prices, in line with
the literature results. Intuitively, what happens is that when search costs are low and all consumers
participate, an increase in search costs results in a clockwise rotation of the demand, making it
more inelastic. This occurs because consumers from more elastic groups choose to search less after
search costs go up thereby enlarging the size of the more inelastic groups. Firms, anticipating a more
inelastic demand, raise their prices to maximise profits.
However, when the search costs of some consumers are that high that they choose not to search in
the first place, a shock that makes search costs even higher results in a further decrease in consumer
participation. If participation were not affected by the increase in search costs, only the intensive
search margin would play a role and demand would again become more inelastic. However, since
consumer participation falls, the extensive search margin plays a fundamental role. We show that,
in contrast to the case of low search costs, under a simple and intuitive necessary and sufficient
condition, higher search costs result in a counterclockwise rotation of the demand function that
makes it more elastic, instead of more inelastic. In such cases, we prove that higher search costs
result in lower prices.
The condition has to do with how the different groups of consumers that form the demand are
affected by a shock that raises search costs. When an increase in search costs is more felt at the
higher percentiles of the search cost distribution, then the impact on the extensive search margin
has a dominating influence over the effect on the intensive search margin and then prices decrease
as search costs increase. In this model, we identify a sufficient condition in terms of the elasticity
of the search cost distribution with respect to a parameter that shifts the distribution downward.
When this elasticity increases in search costs, then prices fall as search costs rise.
The insight that higher search frictions can result in lower prices and profits depends neither on
our assumption of consumers searching for differentiated products nor on the search protocol. In
fact, we generalise the standard sequential search model with differentiated products of Wolinsky
4
(1986) by allowing for search cost heterogeneity. Under the additional requirement that the search
cost density is log-concave, we prove existence and uniqueness of an equilibrium in pure strategies.
As mentioned above, altering the search protocol has a bearing on price determination because
under sequential search the price of an individual firm has an effect on search intensity, while this
is not the case under non-sequential search. This makes it somewhat more difficult to interpret the
necessary and sufficient condition under which the equilibrium price decreases as search costs go
up. Nevertheless, we compute the equilibrium for a family of search cost distributions and show
that the equilibrium price rises with search costs when the elasticity of the search cost distribution
family with respect to the shift-parameter is increasing in search costs. This suggests that prices
respond to higher search costs in a similar fashion no matter whether consumers search sequentially
or non-sequentially.
We finally move to examine the case of search in homogeneous products markets. For this, we
study a model in the spirit of Burdett and Judd (1983) and allow for search cost heterogeneity. In
such a model, the equilibrium is in mixed pricing strategies. Nevertheless, in line with the results
for the differentiated product models, we show that the entire equilibrium price distribution can
shift to the left when consumer search costs increase, thereby making lower prices more frequent.
Interestingly, the necessary and sufficient condition we derive for this to occur is exactly the same as
that in the case of differentiated products. This is somewhat surprising because in the model with
homogeneous products the distribution of prices is endogenous while in the model with differentiated
products the distribution of match values is exogenous.
An important of this paper is to derive conditions on search cost heterogeneity under which higher
search costs may result in lower prices and lower profits for the firms. This result is important for
the recent literature on obfuscation, which points out that firms have incentives to obfuscate their
products (Ellison and Wolitzky, 2012; Wilson, 2010) by raising search costs. Our result tells that
under some conditions firms may benefit from doing exactly the opposite, that is, to lower search
costs instead.
This is not the only paper showing that higher search costs need not result in higher prices.
As mentioned above, Janssen et al. (2005) study the effects of higher search costs in the Stahl’s
(1989) homogeneous products sequential search setting and show that prices will surely fall provided
the search cost is sufficiently high. As shown in this paper, this outcome is the result of a special
assumption about consumer heterogeneity. Chen and Zhang (2011), who enrich Stahl’s (1989) setting
by adding loyal consumers, show that a reduction in the search cost sometimes leads to higher
5
equilibrium prices. In a different framework where search is price-directed, Armstrong and Zhou
(2011) show that higher search costs lead to lower prices.
The structure of the paper is as follows. In Section 2 we study our model of non-sequential search
for differentiated products. In Section 3 we explore the case of sequential search for differentiated
products. In Section 4 we study the case of non-sequential search for homogeneous goods. The paper
closes with a Conclusions Section.
2
Non-sequential consumer search for differentiated products
We next present a duopoly model of firms selling horizontally differentiated products to consumers
who search the market for satisfactory goods.3 The model is in the spirit of Wolinsky (1986) but our
consumers have heterogeneous search costs and search non-sequentially.4
On the supply side of the market, two firms produce horizontally differentiated products using
the same constant returns to scale technology of production; the marginal cost is equal to r. The
two firms choose their prices simultaneously, aiming at maximizing their expected profits. On the
demand side of the market, there is a unit mass of consumers. A consumer m has tastes for a product
i described by the following indirect utility function: uim = εim − pi , if she buys product i at price
pi ; and zero if she does not buy it. The parameter εim is a match value between consumer m and
product i. We assume that the match value εim is the realization of a random variable distributed
on the interval [0, ε] according to a differentiable cumulative distribution function denoted by F.
Match values εim are independently distributed across consumers and products. Moreover, they
are private information of consumers so personalized pricing is not possible. Let f be the density
function of F . We assume that f is log-concave. For later use, we define the monopoly price as
pm = arg maxp (p − r)(1 − F (p)).
Consumers differ in their costs of search. A buyer’s search cost is drawn independently from a
differentiable cumulative distribution function G with support (0, c) and positive density g everywhere. A consumer with search cost c sampling k firms incurs a total search cost kc, k = 0, 1, 2. As
mentioned above, we assume that consumers search non-sequentially; this means that they choose
the number of firms to visit, including none, in order to maximize expected utility. Once they have
3
The N -firm model is examined in section 2.2. Later in Section 4 we study the case of homogeneous products, as
in Burdett and Judd (1983).
4
A classic paper on non-sequential search is Burdett and Judd (1983), where firms sell homogeneous products. To
the best of our knowledge, only Anderson et al. (1992, ch. 7) has considered a model of consumer non-sequential
search with differentiated products. In their model, search costs are assumed to be small and all consumers search the
same number of firms in equilibrium. The equilibrium price (weakly) increases in the search cost. For an empirical
application of our model, see Moraga-González et al. (2012).
6
visited the desired number of firms, they buy from the store offering them the best deal. While
making such a decision, they have correct beliefs about the equilibrium price.5
In what follows we characterize a symmetric pure-strategy Nash equilibrium. Let us start examining the problem of the consumers. Assume both firms charge a price p∗ ∈ [r, pm ]. Because
consumers have correct expectations about the equilibrium price, a consumer with search cost c that
chooses to sample one firm only expects to obtain a utility equal to
E[ε − p∗ |ε ≥ p∗ ] =
Z
ε
(ε − p∗ )f (ε)dε − c.
(1)
p∗
For a consumer to conduct at least one search, such an expected utility has to be positive. If an
individual with search cost c ∈ [0, c] exists such that (1) is equal to zero, then this means that for
some consumers it is not worthwhile to conduct a first search. Correspondingly, we define the critical
search cost value:
Z ε
∗
c0 (p ) ≡ min c,
(ε − p )f (ε)dε .
∗
(2)
p∗
If c0 (p∗ ) is strictly lower than the upper bound of the search cost distribution c, a fraction of the
consumer population will abstain from searching. We refer to the effect of the equilibrium price on
the decision to search as the extensive search margin. If consumers expect a higher equilibrium price,
then fewer consumers will search. The standard assumption in the search cost literature has been
that c0 (p∗ ) ≡ c, which implies that all consumers search at least once. Because c0 (p∗ ) might depend
on the equilibrium price, this assumption is clearly restrictive: it basically boils down to assuming
little consumer heterogeneity, or relatively low search costs for all consumers, which implies that the
extensive search margin will be unresponsive to the equilibrium price.6
Consider now a consumer with search cost c for whom it is worth to conduct at least one search.
This consumer has to choose between searching one firm only or searching the two firms. Let
z2 ≡ max{ε1 , ε2 } and note that the distribution of z2 is F (ε)2 . Then, the utility a consumer expects
to get when sampling the two firms is equal to
∗
∗
E[z2 − p |z2 ≥ p ] =
Z
ε
(ε − p∗ )2F (ε)f (ε)dε − 2c.
p∗
5
The choice of search protocol is based on theoretical tractability. Later in Section 3 we examine a model where
consumers search sequentially, as in Wolinsky (1986), and show that the main result of this paper does not qualitatively
depend on whether consumers search non-sequentially or sequentially.
6
The assumption that all consumers search is akin to a fully-covered market assumption where all consumers are
required to buy. While such an assumption is often adopted for convenience, it is known to be restrictive.
7
Comparing this utility with that derived from searching only one firm, she will prefer to visit the
two firms provided that
Z
ε
Z
∗
ε
(ε − p )2F (ε)f (ε)dε − 2c ≥
p∗
(ε − p∗ )f (ε)dε − c.
p∗
Correspondingly, we define the critical search cost value c1 (p∗ ) above which and below c0 (p∗ ) consumers prefer to search one time only:
Z ε
∗
c1 (p ) ≡ min c,
(ε − p )[2F (ε) − 1]f (ε)dε .
∗
(3)
p∗
It is straightforward to check that c1 (p∗ ) ≤ c0 (p∗ ). Individuals with search cost below c1 (p∗ ) prefer
to search twice. Hence, the population of consumers can be split into three groups of consumers.
These three groups comprise consumers not searching at all, searching one time and searching two
times. Denoting the group of consumers searching k times by µk (p∗ ), we have:
µ0 (p∗ ) = 1 − G(c0 (p∗ )); µ1 (p∗ ) = G(c0 (p∗ )) − G(c1 (p∗ )), and µ2 (p∗ ) = G(c1 (p∗ ))
(4)
Figure 1 illustrates. In the graph of Figure 1(a) we represent a case where all consumers search; in
particular the vertical (blue) line denoted µ1 (p∗ ) depicts the share of consumers who search once,
while the vertical (blue) line denoted µ2 (p∗ ) shows the fraction of consumers who search twice. Note
that when c is very low, all consumers will search twice. By contrast, in the graph of Figure 1(b)
the fraction of consumers µ0 (p∗ ) chooses not to search at all.
G(c)
G(c)
G(c)
1.0
1.0
0.8
0.8
µ0(p*)
µ1(p*)
0.6
0.6
0.4
0.4
µ1(p*)
G(c)
0.2
0.0
0.0
0.2
µ2(p*)
0.1
c1(p*)
0.2
0.3
c
c0(p*)
0.0
0.0
0.4
c
µ2(p*)
0.1
0.2
c1(p*)
(a) c0 sufficiently large
c0(p*)
0.3
0.4
c
c
(b) c0 sufficiently small
Figure 1: Equilibrium search intensities and search costs
We now move to the problem of the firms. To characterize the symmetric pure-strategy equilibrium we start by deriving the payoff of a firm i that deviates from equilibrium pricing by charging
8
a price pi 6= p∗ , given that the rival firm charges p∗ and given consumer search behaviour. The
expected payoff to the deviant firm i is:
µ1 (p∗ )
∗
∗
∗
πi (pi ; p ) = (pi − r)
Pr[εi ≥ pi ] + µ2 (p ) Pr [εi − pi ≥ max{εj − p , 0}] ,
2
(5)
where the symbol Pr stands for probability. This payoff formula is easily understood. The perconsumer profit is pi − r. Consumers who search only once happen to visit firm i with probability
1/2; these consumers buy firm i’s product when the match values they obtain there are higher than
the price pi . Consumers who search twice only buy from firm i when firm i’s deal is better than the
rival’s and the outside option of 0.
When firm i deviates by charging a higher price than the rival, pi > p∗ , the payoff in (5) can be
written as follows:7
πi (pi > p∗ ; p∗ ) = (pi − r)
µ1 (p∗ )
(1 − F (pi )) + µ2 (p∗ )
2
Z
ε
F (ε − (pi − p∗ ))f (ε)dε .
(6)
pi
The first order condition (FOC) in this case is:
Z ε
µ1 (p∗ )
dπi (pi )
∗
=
(1 − F (pi )) + µ2 (p )
F (ε − pi + p∗ ) f (ε) dε
dpi
2
pi
Z ε
µ1 (p∗ )
∗
∗
∗
− (pi − r)
f (pi ) + µ2 (p )
f (ε − pi + p ) f (ε) dε + F (p )f (pi )
=0
2
pi
(7)
Setting pi = p∗ in (7), replacing µ1 (p∗ ) and µ2 (p∗ ) by their corresponding values in terms of the search
cost distribution and rearranging, we obtain the necessary condition for a symmetric equilibrium price
p∗ . Let us define the function
H (p) ≡ N (p)G (c1 (p)) − D(p)G (c0 (p)) .
(8)
where the functions D(p) and N (p) are given by
D(p) ≡ − [1 − F (p) − (p − r) f (p)]
Z
N (p) ≡ F (p)(1 − F (p)) − 2 (p − r)
p
ε
1
f (ε) dε + F (p)f (p) − f (p) .
2
2
The necessary condition for a symmetric equilibrium price p∗ is
H (p∗ ) = 0
(9)
7
When firm i deviates by charging a lower price, the payoff formula is different:
"
#!
Z ε+pi −p∗
µ1 (p∗ )
∗
∗
∗
∗
∗
πi (pi < p ; p ) = (pi − r)
(1 − F (pi )) + µ2 (p ) 1 − F (ε + pi − p ) +
F (ε − (pi − p ))f (ε)dε
.
2
pi
However, the condition that a symmetric price equilibrium must satisfy is the same as the one we derive below in
equation (9).
9
Equation (9) cannot be solved for an explicit solution in p∗ . However, we now note that a candidate
equilibrium price p∗ ∈ [r, pm ] exists. We observe first that when we set p = r we obtain
H (r) = (1 − F (r)) [F (r)G (c1 (r)) + G (c0 (r))] > 0.
Second, if we set p = pm then we get that
H (pm ) = N (pm )G (c1 (pm ))
(10)
just because the price pm satisfies the first order condition for the monopoly problem: 1 − F (pm ) −
(pm − r) f (pm ) = 0. The sign of H(pm ) depends on the sign of N (pm ), for which we can write:
Z ε
1
2
m
m
m
m
m
m
m
f (ε) dε + F (p )f (p ) − f (p )
N (p ) = F (p )(1 − F (p )) − 2 (p − r)
2
pm
Z ε
2
m
m
m
m
m
f (ε) dε + F (p )f (p )
= [1 + F (p )] [1 − F (p )] − 2 (p − r)
pm
m
m
m
Z
ε
= (p − r) f (p ) [1 − F (p )] − 2
f (ε) dε ,
2
(11)
pm
where we have used again the relation 1 − F (pm ) − (pm − r) f (pm ) = 0. Upon observing (11) it
follows that the sign of H(pm ) depends on the sign of the expression inside the squared brackets.
Let us define
Z
M (p) ≡ f (p) [1 − F (p)] − 2
ε
f (ε)2 dε.
p
Taking the derivative of M with respect to p gives f 0 (p)(1 − F (p)) + f (p)2 , which is greater than
zero by logconcavity of f (see Corollary 2 in Bagnoli and Bergstrom (2005)). Since M is increasing
in p and is equal to zero when we set p = ε, we conclude that M (pm ) < 0. Hence H (pm ) < 0.
Since H is a continuous function with H (r) > 0 and H (pm ) < 0, we conclude that for any
log-concave density f, there exists a candidate price equilibrium p∗ ∈ [r, pm ]. Note also that at the
candidate equilibrium price p∗ we must have dH(p∗ )/dp < 0.8 Further, we can prove that:
Proposition 1 Depending on the magnitude of the upper bound of the search cost distribution c,
there may exist three types of symmetric Nash equilibrium (SNE).
(A) A SNE where all consumers search twice and firms charge a price given by the solution to
Z ε
1
2
2 ∗
∗
∗
∗
(1 − F (p )) − (p − r)
f (ε) dε + F (p )f (p ) = 0.
(12)
2
p∗
8
In case there are multiple equilibria, because the number of equilibria will generically be odd, this will also be true
for the highest price equilibrium.
10
This equilibrium is unique and exists provided that
ε
Z
c≤
(ε − p∗ )[2F (ε) − 1]f (ε)dε.
(13)
p∗
(B) A SNE where a fraction G
R
ε
p∗ (ε
− p∗ )[2F (ε) − 1]f (ε)dε of consumers searches the two
firms and the rest just one, in which case the equilibrium price p∗ is given by the solution to (9). For
this equilibrium to exist c must satisfy the inequality
Z
ε
Z
∗
ε
(ε − p )f (ε)dε ≥ c >
p∗
(ε − p∗ )[2F (ε) − 1]f (ε)dε.
(14)
p∗
When F is the uniform distribution, an equilibrium exists.
R
ε
(C) Finally, a SNE where a fraction G p∗ (ε − p∗ )[2F (ε) − 1]f (ε)dε of consumers searches
R
R
ε
ε
the two firms, a fraction G p∗ (ε − p∗ )f (ε)dε − G p∗ (ε − p∗ )[2F (ε) − 1]f (ε)dε of consumers
searches one firm only and the rest do not search at all, in which case the equilibrium price p∗ is
given by the solution to (9). For this equilibrium to exist c must satisfy the inequality
Z
ε
c>
(ε − p∗ )f (ε)dε.
(15)
p∗
When F is the uniform distribution, an equilibrium exists.
Proof. (A) If all consumers search both firms, the payoff function in (6) coincides with that in
Perloff and Salop (1985):
∗
∗
Z
ε
πi (pi > p , p ) = (pi − r)
F (ε − pi + p∗ )f (ε)dε
pi
From Caplin and Nalebuff (1991) we know that under log-concavity of f , this payoff function is
quasi-concave in pi and therefore p∗ is the unique symmetric equilibrium price. In order for all
consumers to search twice, we need that c1 (p∗ ) = c, which is guaranteed under condition (13).
(B and C) When c is relatively large some consumers search once and some search twice. In such
a case, the candidate equilibrium price is given by the solution to equation (9). We now note that
the payoff (6) involves the sum of two log-concave functions. Unfortunately, such a sum need not
be quasi-concave, which implies that we need to impose additional restrictions on the primitives of
the model in order to guarantee the existence of a pure-strategy equilibrium.9 We now show that
9
This problem is quite common in search models where demand stems from various consumer types. For example,
in the sequential search model of Anderson and Renault (1999) demand stems from consumers who happen to visit a
firm for the first time, and from consumers who happen to walk away from a firm and later return to it to conduct a
purchase.
11
when F is the uniform distribution, the payoff function in (6) is strictly concave. The second order
derivative of (6) is:
Z ε
d2 πi (pi > p∗ )
µ1 (p∗ )
∗
∗
∗
= −2
f (pi ) − 2µ2 (p )
f (ε − pi + p ) f (ε) dε + F (p )f (pi )
2
dp2i
pi
Z ε
µ1 (p∗ ) 0
∗
0
∗
∗
∗ 0
− (pi − r)
f (pi ) − µ2 (p )
f (ε − pi + p ) f (ε) dε + f (p )f (pi ) − F (p )f (pi )
2
pi
(16)
where f 0 denotes the derivative of f .
For the uniform distribution, we have F (ε) = ε/ε, f (ε) = 1/ε and f 0 (ε) = 0. Plugging these
values in (16) and simplifying gives:
d2 πi (pi > p∗ )
µ1 (p∗ )
ε − pi p∗
1
∗
=−
− 2µ2 (p )
+ 2 + (pi − r) µ2 (p∗ ) 2
2
2
ε
ε
ε
ε
dpi
εµ1 (p∗ ) + µ2 (p∗ )(2ε − 3pi + 2p∗ + r)
=−
ε2
which is clearly negative because pi cannot be greater than the monopoly price, which in this case
of the uniform distribution is given by pm = (ε + r)/2. In a similar way, we can compute the second
order condition for prices pi < p∗ , which gives
1
d2 πi (pi < p∗ )
= − (µ1 (p∗ ) + 2µ2 (p∗ )) < 0.
ε
dp2i
Because of strict concavity of the payoff, the equilibrium exists.
In order for consumers to search as prescribed in part B, we need that c1 (p∗ ) < c < c0 (p∗ ), which
is guaranteed under condition (14). Finally, for consumers to search as prescribed in part C, we need
that c0 (p∗ ) < c, which gives condition (15). 2.1
The effect of higher search costs on the equilibrium price
We now study how the equilibrium price derived in Proposition 1 depends on the magnitude of search
costs. In order to address this question, we parametrize the search cost distribution G by a positive
parameter β and use the notation G(c; β). Specifically, we assume that an increase in β implies an
increase in search costs in the sense of first-order stochastic dominance (FOSD), i.e. ∂G(c; β)/∂β < 0
for all c. For later use, we define the elasticity of the search cost distribution with respect to β as
G,β (c; β) ≡ −
∂G(c; β) β
> 0.
∂β G(c; β)
We shall denote the equilibrium price corresponding to a given search cost distribution G(c; β) by
p∗ (β) and we will examine how p∗ (β) changes with β.
12
The first observation we make is that the price in part (A) of Proposition 1, given by the solution
to the FOC (12), is completely independent of a small change in the search cost distribution. As
mentioned above, this is because search costs are so low in this case that they do not restrict
consumers’ search behaviour at all and, as a result, all consumers search the two firms in equilibrium.
The other cases, namely (B) and (C), are the most interesting ones. In the cases of (B) and
(C), if an equilibrium exists, the price is given by the solution to the FOC (9). Because we have
parametrized G by β, let us denote by H(p∗ ; β) the corresponding parametrized function defined
by the FOC (9). By the implicit function theorem, the comparative statics effect of an increase in
search costs is then given by
∂H
dp∗ (β)
∂β
= − ∂H .
dβ
∂p∗
(17)
We have already noted above that the denominator of (17), ∂H/∂p∗ , is negative. We now study the
sign of the numerator of (17). For this we now distinguish between cases (B) and (C) in Proposition
1. Consider first the situation in (B). In this case, the upper bound of the search cost distribution is
neither too high nor too low, which implies that all consumers search at least once, i.e., G (c0 (p∗ ), β) =
1, and some consumers do search twice, i.e. µ2 (p∗ ) = 1 − µ1 (p∗ ) = G(c1 (p∗ ), β). In such a case, the
numerator of (17) is
∂H
∂G (c1 (p∗ ), β)
= N (p∗ )
> 0,
∂β
∂β
because D(p∗ ) < 0 and existence of a candidate equilibrium implies that N (p∗ ) < 0. As a result, since
∂H/∂p∗ < 0 and ∂H/∂β > 0, we have demonstrated that dp∗ (β)/dβ > 0. That is, an increase in
search costs results in higher prices, which is the standard result in the search cost literature. In the
present case where all consumers search, an increase of search costs has only a bearing on consumers’
search intensity, the intensive search margin, and not at all on consumers’ participation, the extensive
search margin. When search costs increase, consumers search less and prices go up. That consumers
search less is reflected here in G(c1 (p∗ ), β) falling in β, which, by definition, means that the fraction
of consumers searching twice decreases and, by implication, the fraction of consumers searching once
increases. Facing fewer consumers who compare the products of the two firms after search costs
increase, the producers safely increase their prices.
Consider now the situation in case (C). In this situation G (c0 (p∗ ), β) < 1 and therefore for the
numerator of (17) we have
∂G (c1 (p∗ ), β)
∂G (c0 (p∗ ), β)
∂H
= N (p∗ )
− D(p∗ )
∂β
∂β
∂β
13
Using the equilibrium condition (9), we can rewrite this as follows
∗
∂H
G (c0 (p∗ ), β) ∂G (c1 (p∗ ), β)
∗ ∂G (c0 (p ), β)
= D(p∗ )
−
D(p
)
∂β
G (c1 (p∗ ), β)
∂β
∂β
∗
∗
∗
G (c0 (p ), β) ∂G (c1 (p ), β) ∂G (c0 (p ), β)
∗
= D(p )
−
G (c1 (p∗ ), β)
∂β
∂β
∗
∂G (c1 (p ), β)
∂G (c0 (p∗ ), β)
1
1
∗
∗
= D(p )G (c0 (p ), β)
−
G (c1 (p∗ ), β)
∂β
G (c0 (p∗ ), β)
∂β
∗
∗
∗
D(p )G (c0 (p ), β)
β
∂G (c1 (p ), β)
∂G (c0 (p∗ ), β)
β
=
−
β
G (c1 (p∗ ), β)
∂β
G (c0 (p∗ ), β)
∂β
∗
∗
D(p )G (c0 (p ), β)
=
[G,β (c0 (p∗ )) − G,β (c1 (p∗ ))] .
β
(18)
(19)
(20)
(21)
(22)
The sign of ∂H/∂β is therefore ambiguous; it depends on the values that the elasticity of the search
cost distribution with respect to the shift-parameter β takes at the cutoff points c0 (p∗ ) and c1 (p∗ ).
The interesting issue is that this derivative can be negative,10 in which case the equilibrium price
will decrease when search costs increase. The next proposition summarizes our findings and provides
a sufficient condition for the equilibrium price to decrease in search costs. We explain the intuition
behind this surprising result after stating it precisely.
Proposition 2 Let G (c; β) be a parametrized search cost cdf with positive density on [0, c] and with
derivative ∂G(·)/∂β < 0. Then the comparative statics of the symmetric equilibrium price described
in Proposition 1A,B,C with respect to β is as follows:
(A) The equilibrium price given by Proposition 1A is independent of β. Therefore, higher search
costs do not have a bearing on the equilibrium price.
(B) The equilibrium price given by Proposition 1B unambiguously increases in β. Therefore,
higher search costs always result in higher prices.
(C) The equilibrium price given by Proposition 1C decreases in β if and only if G,β (c0 (p∗ )) −
G,β (c1 (p∗ )) > 0. Moreover, if G,β (c) increases (decreases) in c, then the equilibrium price decreases
(increases) in β. Therefore, for search cost distributions with increasing (decreasing) elasticity with
respect to β, an increase in search costs results in lower (higher) equilibrium prices.
The contrast between the results in Proposition 2B,C is important in that it demonstrates that
the standard result about the relationship between search costs and prices is based on a restriction
on the magnitude of search costs. When search costs are initially low, an increase in search costs has
effects only on the intensive search margin. Confronting more difficulties to try products, consumers
10
Since D(p∗ ) < 0, this occurs when G,β (c0 (p∗ )) < G,β (c1 (p∗ )).
14
engage in less product-comparison. Buyers who stop comparing products enlarge the group of buyers
who do not, and this results in a lower elasticity of the demand of an individual firm. Correspondingly,
firms adjust their prices upwards.
However, when search costs are not restricted to be initially low, increases in search costs have a
bearing on both the intensive and the extensive search margins. At the intensive search margin, the
same effect happens. The share of consumers who used to inspect the two products goes down and
this tends to decrease the elasticity of demand of an individual firm. However, at the extensive search
margin, more consumers drop from the market altogether when search costs go up and this tends to
increase rather than decrease the elasticity of demand. This tradeoff is resolved in favour of lowering
prices when the elasticity of the search cost distribution with respect to the shifter parameter is
increasing in search costs. The reason for this is that in such a case, an increase in search costs is
more noticeable at higher percentiles of the search cost distribution than at lower, which implies that
the effect on the extensive search margin is stronger than the effect on the intensive search margin.
In order to illustrate these arguments, we refer the reader to Figure 2. In this figure we represent
the effect of an increase in search costs on the intensive and extensive search margins. Initially
consumer search costs are given by the blue search cost distribution. This search cost distribution
has the property that its elasticity with respect to parameter β is increasing in c.11 The increase in
search costs is represented by the shift from the blue distribution to the red one. As the graph shows,
the increase in search costs is much more felt at the higher percentiles of the search cost distribution.
In the graph of Figure 2(a) we represent the case discussed in Proposition 2B. Before the increase
in search costs, the blue fractions of consumers µ1 (p∗ ) and µ2 (p∗ ) represent the equilibrium fractions
of consumers searching once and twice, respectively. Because here search costs are small for all
consumers (c0 (p∗ ) = c), they all search at least once. Keeping prices constant, an increase in search
costs results in a fall in the number of consumers who search twice and, correspondingly, in an
increase in the number of consumers who search once. This lowers demand elasticity and firms raise
their prices.
The graph of Figure 2(b) shows the case discussed in Proposition 2C. In this case search costs
are sufficiently large (c0 (p∗ ) < c) and the fraction of consumers µ0 (p∗ ) does not find it worth to
search thereby opting out of the market altogether. When search costs increase, keeping prices fixed,
the share of consumers who do not even start searching increases a lot. This causes the share of
11
To be sure, we plot here the Kumaraswamy’s (1980) distribution with parameters a = 1, b = 1/2 and upper bound
β = 0.3; in section 2.1.1 we introduce such a distribution and demonstrate that it has an increasing elasticity for those
parameter values.
15
inelastic consumers to fall more than the share of elastic consumers, which increases overall elasticity
of demand and leads to lower prices.
G(c)
G(c)
G(c)
G(c)
1.0
1.0
0.8
0.8
0.6
µ1(p*)
µ1(p*)
G(c)
µ0(p*)
µ0(p*)
0.6
0.4
0.4
0.2
0.2
G(c)
µ1(p*)
µ1(p*)
µ2(p*) µ (p*)
2
µ2(p*) µ2(p*)
0.0
0.1
0.2
c1(p*)
0.3
0.4
c
c
0.0
0.1
0.2
c1(p*)
c
(a) c0 sufficiently large
c0(p*)
0.3
0.4
c
c
c
(b) c0 sufficiently small
Figure 2: The effect of an increase in search costs
2.1.1
An example: the Kumaraswamy’s (1980) distribution
Definition: The Kumaraswamy distribution has cdf G and pdf g given by
a b
c
, c ∈ [0, β] , a, b > 0
G (c) = 1 − 1 −
β
a b−1
c
ab c a−1
1−
.
g (c) =
β β
β
(23)
The Kumaraswamy’s (1980) distribution is often used in lieu of the beta-distribution (see e.g. Ding
and Wolfstetter (2011)). This distribution turns out to be quite useful in our setting because it can
have increasing, decreasing and constant elasticity with respect to the shifter parameter β, which
is exactly the sufficient condition stated in Proposition 2.12 Note that parameter β multiplies the
search cost c and scales the support of the distribution. An increase in β therefore shifts the search
cost distribution rightward, which signifies that search costs are higher for all consumers.
Note that
ab
dG (c; β)
=−
dβ
β
a a b−1
c
c
1−
< 0;
β
β
correspondingly, the elasticity of the search cost distribution with respect to β is then
ab
εG,β (c; β) =
12
a b−1
1 − βc
.
h
a ib
1 − 1 − βc
a c
β
Observe that the uniform case obtains by setting a = b = 1 in the Kumaraswamy distribution above.
16
(24)
We now let
a
c
.
t≡1−
β
Note that t ∈ (0, 1) and that t is monotonically decreasing in c. We can rewrite (24) as
εG,β (t) =
ab(1 − t)tb−1
,
1 − tb
and then take the derivative of εG,β (t) with respect to t. This gives
dεG,β (t)
abtb−2 (b − 1 − bt + tb )
=
.
dt
(1 − tb )2
We now argue that this derivative is positive for all b > 1 and negative for all 0 1 case. Let h (t) ≡ b − 1 − bt + tb . Then h (0) = b − 1 > 0, h (1) = 0, and
h0 (t) = −b 1 − tb−1 < 0. So h is monotonically decreasing and hence h (t) > 0 for any t ∈ (0, 1).
As a result, εG,β (t) increases in t and decreases in c. By Proposition 2, this implies that when
condition (15) holds, for the Kumaraswamy family of search cost distributions with parameter b > 1,
the equilibrium price increases as search costs rise.
Second, assume 0 0. Hence h (t) < 0 for any t ∈ (0, 1) . As a result, εG,β (t) decreases in t and increases
in c. By Proposition 2, this implies that when condition (15) holds, for the Kumaraswamy family of
search cost distributions with parameter 0 < b < 1, the equilibrium price decreases as search costs
go up.
For completeness, let b = 1. Plugging b = 1 in (24) gives εG,β (c; β) = a so the elasticity is constant
and therefore the equilibrium price when condition (15) holds does not vary with β.
The following result summarizes these findings.
Corollary to Proposition 2 Assume that search costs are distributed on the interval [0, β] according
to the Kumaraswamy distribution. Then:
(A) The equilibrium price in Proposition 1A is independent of β.
(B) The equilibrium price in Proposition 1B unambiguously increases in β.
(C) For all a, the equilibrium price in Proposition 1C decreases in β if 0 1.
2.2
The N -firm model
The previous non-sequential search model with differentiated products can be generalized to the case
of N > 2 firms. The problem of a consumer with search cost c is to choose a number k of firms to
17
be sampled in order to maximize her expected utilitity:
Z ε
∗
k−1
max
(ε − p )kF (ε) f (ε)dε − kc .
k
p∗
It can easily be checked that this problem is well-behaved and that a unique solution exists. Such
a solution defines a partition of the consumer population into groups of buyers µk (p∗ ) that search
P
∗
k = 0, 1, 2, ..., N firms, with N
k=0 µk (p ) = 1; as above, some of these groups may have zero mass
as the upper bound of the search cost distribution decreases.
In order to determine the size of these groups, let us define the critical search cost parameters
Z ε
∗
∗
c0 (p ) = min c,
(ε − p )f (ε)dε
p∗
Z ε
∗
∗
k−1
ck (p ) = min c,
(ε − p ) [(k + 1)F (ε) − k] F (ε) f (ε)dε , k = 1, 2, ..., N − 1.
p∗
The fractions of consumers searching k times are then given by the expressions:
µ0 = 1 − G(c0 (p∗ ))
µk = G(ck−1 (p∗ )) − G(ck (p∗ )), k = 1, 2, ..., N − 1
(25)
µN = G(cN −1 (p∗ )) − G(cN (p∗ )) = G(cN −1 (p∗ )) since cN = 0.
If cN −1 (p∗ ) = c for example, then all consumers will search the N firms in equilibrium and the
situation will again resemble the perfect information model of Perloff and Salop (1985). When
cN −1 (p∗ ) < c < cN −2 (p∗ ), a fraction µN = G(cN −1 (p∗ )) of consumers will visit the N firms and the
remaining consumers will each visit N − 1 randomly selected firms; and so on and so forth.
Let zk ≡ max {ε1 , ε2 , ..., εk }. In general, the expected payoff of a firm i that deviates from the
symmetric equilibrium price by charging a price pi 6= p∗ is
∗
πi (pi ; p ) = (pi − r)
!
N
X
kµk (p∗ )
µ1 (p∗ )
∗
Pr[εi ≥ pi ] +
Pr [εi − pi ≥ max{zk−1 − p , 0}]
2
N
(26)
k=2
As before, the demand of the deviant firm i stems from the various consumer groups and a consumer
who searches k times compares the offer of firm i with the offers of k − 1 other firms.
For the case where the deviant firm charges a higher price than the rest of the firms, the expression
in (26) becomes
"
N
X kµk (p∗ )
µ1 (p∗ )
πi (pi > p∗ ; p∗ ) = (pi − r)
(1 − F (pi )) +
N
N
k=2
18
Z
ε
pi
#
F (ε − (pi − p∗ ))k−1 f (ε)dε . (27)
Taking the FOC gives:
∗
µ1 (p )(1 − F (pi )) +
N
X
Z
∗
− (pi − r)
∗
F (ε − pi + p∗ ))k−1 f (ε)dε − (pi − r)µ1 (p∗ )f (pi )
pi
k=2
N
X
ε
kµk (p )
Z
ε
∗ k−2
(k − 1)F (ε − pi + p )
kµk (p )
∗
∗ k−1
f (ε − pi + p )f (ε)dε + F (p )
f (pi )
= 0.
pi
k=2
(28)
After imposing symmetry, simplifying and rearranging we obtain:
µ1 (p∗ ) [1 − F (p∗ ) − (p∗ − r)f (p∗ )] +
N
X
kµk (p∗ )
k=2
∗
− (p − r)
N
X
∗
Z
kµk (p )
k=2
ε
k−2
(k − 1)F (ε)
2
Z
ε
F (ε)k−1 f (ε)dε
p∗
∗ k−1
f (ε) dε + F (p )
f (p ) = 0.
∗
(29)
p∗
In the appendix we show that a candidate equilibrium p∗ ∈ [r, pm ] exists.
Depending on the magnitude of the upper bound of the search cost distribution c, there may
exist N + 1 types of equilibria:
1. When
Z
ε
c≤
(ε − p∗ ) [(N + 1)F (ε) − N ] F (ε)N −1 f (ε)dε
p∗
then all consumers search the N -firms in the market and the equilibrium price is given by the solution
to the FOC (29) after setting µi (p∗ ) = 0 for all i = 1, 2, ..., N − 1 and µN (p∗ ) = 1. This equilibrium
exists an is unique, as it is the same as that in Perloff and Salop (1985).
2. When
ε
Z
Z
(ε − p∗ ) [(N + 1)F (ε) − N ] F (ε)N −1 f (ε)dε < c ≤
p∗
ε
(ε − p∗ ) [N F (ε) − (N − 1)] F (ε)N −2 f (ε)dε
p∗
then a fraction of consumers µN (p∗ ) = G(cN −1 (p∗ )) searches N firms and the rest of the consumers
search N − 1 firms, and the equilibrium price is given by the solution to the FOC (29) after setting
µi (p∗ ) = 0 for all i = 1, 2, ..., N − 2 and replacing µN −1 (p∗ ) and µN (p∗ ) by their corresponding values
in (25).
3. When
Z
ε
∗
(ε−p ) [N F (ε) − (N − 1)] F (ε)
N −2
Z
ε
f (ε)dε < c ≤
p∗
(ε−p∗ ) [(N − 1)F (ε) − (N − 2)] F (ε)N −3 f (ε)dε
p∗
then a fraction of consumers µN (p∗ ) = G(cN −1 (p∗ )) searches N firms, a fraction of consumers
µN −1 (p∗ ) = G(cN −2 (p∗ )) − G(cN −1 (p∗ )) searches N − 1 firms and the rest of the consumers search
N −2 firms, and the equilibrium price is given by the solution to the FOC (29) after setting µi (p∗ ) = 0
19
for all i = 1, 2, ..., N − 3 and replacing µN −2 (p∗ ), µN −1 (p∗ ) and µN (p∗ ) by their corresponding values
in (25).
4,5,...,N-1. So and so forth.
N. When
Z
ε
Z
∗
ε
(ε − p ) [2F (ε) − 1] F (ε)f (ε)dε < c ≤
p∗
(ε − p∗ )f (ε)dε
p∗
then a fraction of consumers µN (p∗ ) = G(cN −1 (p∗ )) searches N firms, a fraction of consumers
µk (p∗ ) = G(ck−1 (p∗ )) − G(ck (p∗ )) searches k = 2, 3, ..., N − 1 firms and the rest of the consumers
search just one firm. In this case the equilibrium price is given by the solution to the FOC (29) after
replacing µ1 (p∗ ), µ2 (p∗ ), ..., µN (p∗ ) by their corresponding values in (25).
N+1. Finally, when
Z
ε
c>
(ε − p∗ )f (ε)dε
p∗
then a fraction of consumers µN (p∗ ) = G(cN −1 (p∗ )) searches N firms, a fraction of consumers
µk (p∗ ) = G(ck−1 (p∗ )) − G(ck (p∗ )) searches k = 1, 2, 3, ..., N − 1 firms and the rest of the consumers
do not search at all. In this case the equilibrium price is given by the solution to the FOC (29) after
replacing µ1 (p∗ ), µ2 (p∗ ), ..., µN (p∗ ) by their corresponding values in (25).
Except in case 1 above, the payoff functions involve sums of quasiconcave functions and because
of this feature the payoff function need not be quasiconcave. This makes it very hard to provide
existence results. Nevertheless, we can prove the following existence result:
Proposition 3 Let N = 3 and assume that F is the uniform distribution. Then, depending on
the magnitude of the upper bound of the price distribution c there exist 4 types of equilibria. These
equilibria can be obtained from equation (29) after replacing µk (p∗ ), k = 1, 2, 3 by their respective
values given in (25).
Proof. See the appendix.
To study how an increase in search costs affects the equilibrium price we proceed by solving the
model numerically. We focus on the most interesting case where search costs are sufficiently large so
that not all consumers search in equilibrium. Assuming that search costs follow the Kumaraswamy
distribution with upper bound β, we set a = 1, pick β sufficiently high so that all fractions of
consumers defined above in (25) are strictly positive and compute the price equilibrium and search
intensities for various levels of the parameter b. For the case N = 2, our Proposition 2 shows
mathematically that, after an increase in search costs, prices go down when the parameter b of the
Kumaraswamy search cost distribution is less than 1; for b = 1, prices do not change; while for b > 1,
20
prices increase. Table 1 shows that the same results obtain in a market where N = 5, r = 0 and
match values are uniformly distributed on the set [0, 10].
β=4
b = 0.5
β=5
β=6
β=5
b = 1.00
β=6
β=7
β=5
b = 1.25
β=6
µ0
µ1
µ2
µ3
µ4
µ5
0.7518
0.0118
0.1259
0.0456
0.0220
0.0425
0.8063
0.010
0.096
0.0358
0.0174
0.0339
0.8410
0.0087
0.0781
0.0294
0.0143
0.0281
0.5540
0.0292
0.2083
0.0833
0.0416
0.0833
0.64322
0.0233
0.1666
0.0666
0.0333
0.0666
0.7027
0.0194
0.1388
0.0555
0.0277
0.0555
0.4715
0.03824
0.2369
0.0995
0.0506
0.1030
0.5720
0.0303
0.1938
0.0804
0.0407
0.0826
0.6406
0.0251
0.1637
0.0674
0.03410
0.0689
p∗
π
CS
CS/(1 − µ0 )
W elf are
3.2612
0.1492
0.7028
2.8326
1.4492
3.2507
0.1161
0.5525
2.8655
1.1333
3.2442
0.0951
0.4554
0.4919
0.9310
3.2163
0.2646
1.3015
2.9188
2.6249
3.2163
0.2117
1.0412
2.9188
2.0999
3.2163
0.1764
0.8676
0.4779
1.7499
3.1922
0.3115
1.5664
2.9643
3.1242
3.1981
0.2527
1.2638
2.9530
2.5274
3.2017
0.2124
1.0588
2.9463
2.1209
β=7
Table 1: Non-sequential search for differentiated products: price equilibrium and search intensities
(Kumaraswamy distribution, a = 1)
The table also illustrates the impact of higher search costs on profits, consumer surplus and
welfare. What we see is that, even if higher search costs result in lower prices, consumer surplus goes
down in search costs. This is clearly due to the impact higher search costs have on the extensive
search margin, which is of first order. In fact notice that conditional on searching, consumers benefit
from higher search costs just because prices fall.
Another interesting result is that firm profits always decrease when search costs increase, even if
prices increase. Once again, this is due to the impact of higher search costs on the extensive search
margin.
3
Sequential search for differentiated products
The standard model of search for differentiated products is by Wolinsky (1986). In his model,
consumers search sequentially, instead of non-sequentially as we have assumed in the previous section.
The critical distinction between these two search protocols is that with sequential search consumers
do not commit ex-ante to a number of searches but, instead, at every point in time they decide
whether to continue searching or not based on the match values they have observed so far. In terms
of price formation, since an individual firm can influence the intensity of search in the sequential
search model and cannot in the non-sequential search model, prices are typically lower in the first. In
what follows, we allow for heterogeneous search costs in Wolinsky’s (1986) model. For the rest of the
model features, the model we study next is exactly the same as the model in Section 2 except that
21
the market has infinitely many firms.13 To the best of our knowledge we are the first considering this
generalisation of Wolinsky’s seminal contribution. We first provide a new existence of equilibrium
result; then, we examine whether higher search costs can also result in lower prices in this model.
Since the analysis is more involved, sometimes we specialise the model in order to get further results.
We start by computing the symmetric Nash equilibrium. Let p∗ denote the equilibrium price.
Consider the (expected) payoff to a firm i that deviates by charging a price pi 6= p∗ . In order to
compute firm i’s demand, we first need to characterize consumer search behavior. Since consumers
do not observe deviations before searching, we can rely on Kohn and Shavell (1974), who study the
search problem of a consumer who faces a set of independently and identically distributed options
with known distribution. Kohn and Shavell show that the optimal search rule is static in nature and
has the stationary reservation utility property. Accordingly, consider a consumer with search cost c
and denote the solution to
Z
h(x) ≡
ε
(ε − x)f (ε)dε = c
(30)
x
in x by x̂(c). The left-hand-side (LHS) of (30) is the expected benefit in symmetric equilibrium from
searching one more time for a consumer whose best option so far is x. Its right-hand-side (RHS) is
her cost of search. Hence x̂(c) represents the threshold match value above which a consumer with
search cost c will optimally decide not to continue searching for another product. The function h
is monotonically decreasing. Moreover, h(0) = E[ε] and h(ε) = 0. It is readily seen that for any
c ∈ [0, min{c, E[ε]}], there exists a unique x̂(c) that solves (30); note that x̂(E[ε]) = 0. Later we
show that x̂(c) is a decreasing and convex function of c on [0, min{c, E[ε]}].
In order to compute a firm i’s demand, consider a consumer with search cost c who shows up at
firm i to inspect its product after possibly having inspected other poducts. Let εi − pi denote the
utility the consumer derives from the product of firm i. Obviously, if alternative i is not the best
one so far, the consumer will discard it and search again. Therefore, only when the deal offered by
firm i happens to be the best so far will the consumer consider stopping searching and buying it
right away. For this decision, the consumer compares the gains from an additional search with the
costs of search. In this comparison, the consumer holds correct expectations about the equilibrium
price so she expects the other firms to charge the equilibrium price p∗ . The expected gains from
Rε
searching one more firm, say j, are equal to εi −pi +p∗ [εj − (εi − pi + p∗ )]f (εj )dεj . Comparing this
13
The finite number of firms case is more complicated because of the so-called “comeback consumers.” These are
consumers who happen to visit all the firms in the market and return to a given firm to conduct a purchase. The
presence of “comeback consumers” complicates the proof of existence of equilibrium, as well as the analysis in general
(for details, see Anderson and Renault, 1999). We have also studied a 2-firm model numerically and the insights
obtained are the same. Details are available from the authors upon request.
22
to (30), it follows that the probability that buyer c stops searching at firm i is equal to Pr[εi − pi >
x̂(c) − p∗ ] = 1 − F (x̂(c) + pi − p∗ ). With the remaining probability, the consumer finds the product
of firm i not good enough and continues searching; with infinitely many firms, such a consumer will
buy at another firm.
To obtain the payoff of firm i we need to integrate over the consumers who decide to participate
in the market, that is, those who derive expected positive surplus from participation. To compute
the surplus a consumer with search cost c obtains from participation, we note that the consumer
will stop and buy after the first search when ε > x̂(c); otherwise she will drop the first option and
continue searching, in which case she will encounter herself exactly in the same situation as before
because, conditional on participating, the consumer will continue searching until she finds a match
value for which it is worth to stop searching. Recursively, denoting by CS(c) her consumer surplus,
we must have:
Rε
CS(c) = −c + (1 − F (x̂(c)))
x̂(c) (ε
− p∗ )f (ε)dε
1 − F (x̂(c))
+ F (x̂(c))CS(c).
Solving for CS(c) we obtain the surplus from participation:
Rε
∗
x̂(c) (ε − p )f (ε)dε − c
CS(c) =
.
1 − F (x̂(c))
Setting this surplus equal to zero we get that the critical search cost value above which consumers
will refrain from participating in the market is given by the solution to
Z ε
(ε − p∗ )f (ε)dε − c = 0
x̂(c)
Let c̃(p∗ ) denote such a solution. Note that c̃(·) is the inverse function of x̂(·) on [0, E[ε]]. Therefore,
Z ε
c̃(p) =
(ε − p)f (ε)dε
(31)
p
Correspondingly we have
c0 (p∗ ) = min{c, c̃(p∗ )}.
Therefore the payoff to the deviant firm i is:
Z c0 (p∗ )
∗
π(pi ; p ) = (pi − r)
[1 − F (x̂(c) + pi − p∗ )]g(c)dc.
0
The FOC is given by:
Z c0 (p∗ )
Z
[1 − F (x̂(c) + pi − p∗ )]g(c)dc − (pi − r)
0
0
23
c0 (p∗ )
f (x̂(c) + pi − p∗ )]g(c)dc = 0.
(32)
Applying symmetry, pi = p∗ , we can rewrite the FOC as:
p∗ = r +
R c0 (p∗ )
0
[1 − F (b
x(c))]g(c)dc
.
R c0 (p∗ )
f (b
x(c))g(c)dc
0
(33)
We now show that a candidate symmetric equilibrium price exists. First consider the case in
which search costs are low, i.e. c < c̃(p∗ ). Under this parameter constraint, c0 (p∗ ) = c and therefore
expression (33) gives the equilibrium price explicitly. In this case, there exists a unique candidate
equilibrium price.14
When search costs are not restricted to be small then c0 (p∗ ) = c̃(p∗ ). In this case the equilibrium
price is given implicitly by the solution to (33). We now show that (33) has a solution also in this
case. For this we define the function
Z
c̃(p)
Z
c̃(p)
[1 − F (b
x(c))]g (c) dc − (p − r)
L(p) ≡
f (b
x(c))g (c) dc
0
0
for p ∈ [r, pm ], where pm denotes the monopoly price. Note that
Z
c̃(r)
(1 − F (b
x(c)))dG(c) > 0.
L (r) =
0
Observe also that
m
Z
L (p ) =
c̃(pm )
[1 − F (b
x(c)) − (pm − r)f (b
x(c))]g (c) dc < 0.
(34)
0
This inequality follows from the fact that the integrand 1 − F (b
x(c)) − (pm − r)f (b
x(c)) ≤ 0 for all
c[0, c̃(pm )]. To see this, note that by logconcavity of f , because x
b(c) decreases in c, it follows that
f (b
x(c))/ [1 − F (b
x(c))] decreases in c, which implies that 1 − F (b
x(c)) − (pm − r)f (b
x(c)) increases
in c. Because x
b(c(p)) = p, if we set c = c̃(pm ) in the integrand we get the monopoly pricing rule
1 − F (pm ) − (pm − r)f (pm ) = 0. Since the integrand in (34) is increasing in c and takes on value zero
when we compute it at the upper bound of the integral, we conclude that L (pm ) < 0.
We conclude then that, because L (r) > 0 and L (pm ) < 0, a candidate equilibrium price p∗ ∈
[r, pm ] exists. Note also that
dL (p)
dc̃ (p)
= [1 − F (p) − (p − r) f (p)] g (c̃ (p))
−
dp
dp
Z
c̃(p)
f (b
x(c))g (c) dc
0
is negative for any p ∈ [r, pm ] because [1 − F (p) − (p − r) f (p)] ≥ 0 (since it is the first order
derivative of the monopoly payoff (p − r) (1 − F (p)), which is log-concave) and dc̃ (p) /dp < 0 (since
14
To be more precise, the condition we need for this is c <
24
Rε
x̂(c̃(p∗ ))
(ε − p∗ )f (ε)dε, where p∗ = r +
Rc
x(c))]g(c)dc
0R[1−F (b
c
x(c))g(c)dc
0 f (b
.
from (31) c̃ is decreasing in p). In particular, at the candidate equilibrium price p∗ we must have
dL(p∗ )/dp < 0. More importantly, we conclude that there is a unique candidate equilibrium price,
which implies that if a symmetric equilibrium exists then that is unique.
Our next result provides conditions for existence of a symmetric price equilibrium.
Proposition 4 In the model of search for differentiated products, assume that consumers search
sequentially and that there are infinitely many firms. Then:
(A) If a SNE exists where all consumers conduct at least a first search, then c < c̃(p∗ ) and the
price is given by the expression (33) after setting c0 (p∗ ) = c, where x
b(c) solves (30). Moreover, if
the search cost density g is log-concave, then the equilibrium exists and is unique.
(B) Otherwise, if a SNE exists where not all consumers search, then c > c̃(p∗ ) and consumers
Rε
with search cost c ≤ p∗ (ε − p∗ )f (ε)dε conduct at least a first search while the rest of the consumers
do not search at all. In this case the price is given by the expression (33) after setting c0 (p∗ ) =
Rε
∗
p∗ (ε − p )f (ε)dε. Moreover, if the search cost density g is log-concave, then the unique equilibrium
exists. Alternatively, if the search cost density g is the Kumaraswamy density (23) on [0, 1] with
a = 1/2, b ∈ (0, 2) and ε is distributed uniformly on [0, 1], the unique equilibrium exists.15
Proof. It remains to prove that the equilibrium exists when the search cost density g is logconcave. For this, we prove that the demand function of a firm i in (32)
∗
Z
D(pi , p ) ≡
c0 (p∗ )
[1 − F (x̂(c) + pi − p∗ )]g(c)dc
(35)
0
is a log-concave function of its own price pi . Given this, the inverse of the demand function of a
firm i is convex and by Proposition 3 in Caplin and Nalebuff (1991), the firm profit function (32) is
quasi-concave in own price.
Under the assumption that g is log-concave, the integrand in (35) is log-concave in c and in pi
provided that 1 − F (x̂(c) + pi − p∗ ) is log-concave in c and in pi . Let us now show that this is indeed
the case. For this we need to prove that the function m(c, pi ) ≡ ln[1 − F (x̂(c) + pi − p∗ )] is concave
in c and in pi , where the symbol ln denotes the natural logarithm. Taking derivatives we have:
∂m
f (x̂(c) + pi − p∗ )
=−
x̂0 (c)
∂c
1 − F (x̂(c) + pi − p∗ )
∂m
f (x̂(c) + pi − p∗ )
=
∂pi
1 − F (x̂(c) + pi − p∗ )
15
In this case the Kumaraswamy density is not log-concave.
25
To construct the Hessian matrix, we now compute the necessary second order derivatives:
nh
i
1
∂2m
2
0
∗
0
∗ 00
f
(x̂(c)
+
p
−
p
)
[x̂
(c)]
+
f
(x̂(c)
+
p
−
p
)x̂
(c)
[1 − F (x̂(c) + pi − p∗ )]
=
−
i
i
2
∂c2
[1 − F (x̂(c) + pi − p∗ )]
o
2
+ [f (x̂(c) + pi − p∗ )x̂0 (c)]
n
h
i
1
2
0
0
∗
∗
∗ 2
=−
[x̂
(c)]
f
(x̂(c)
+
p
−
p
)
[1
−
F
(x̂(c)
+
p
−
p
)]
+
[f
(x̂(c)
+
p
−
p
)]
i
i
i
2
[1 − F (x̂(c) + pi − p∗ )]
∗ 00
+ f (x̂(c) + pi − p )x̂ (c) [1 − F (x̂(c) + pi − p∗ )]}
The sign of this expression depends on the sign of the part in curly brackets. We note that, because
f is log-concave, the first summand of the expression in curly brackets is positive. To determine the
sign of the second summand, we need to study the sign of x̂00 (c). Differentiating successively (30), we
obtain
x̂0 (c) = −
x̂00 (c) =
1
< 0.
1 − F (x̂(c))
f (x̂(c)) [x̂0 (c)]2
>0
1 − F (x̂(c))
Because x̂00 (c) > 0, we conclude that ∂ 2 m/∂c2 < 0.
We now observe that
n
o
∂2m
1
0
∗
∗
∗ 2
f
(x̂(c)
+
p
−
p
)
[1
−
F
(x̂(c)
+
p
−
p
)]
+
[f
(x̂(c)
+
p
−
p
)]
=
−
i
i
i
∂p2i
[1 − F (x̂(c) + pi − p∗ )]2
which is negative again by the log-concavity of f.
Finally we derive
n
o
x̂0 (c)
∂2m
0
∗
∗
∗ 2
=−
f
(x̂(c)
+
p
−
p
)
[1
−
F
(x̂(c)
+
p
−
p
)]
+
[f
(x̂(c)
+
p
−
p
)]
i
i
i
∂pi ∂c
[1 − F (x̂(c) + pi − p∗ )]2
Defining
ψ(c, pi ) ≡
f 0 (x̂(c) + pi − p∗ ) [1 − F (x̂(c) + pi − p∗ )] + [f (x̂(c) + pi − p∗ )]2
[1 − F (x̂(c) + pi − p∗ )]2
the Hessian matrix is
∗
H=
00
(x̂(c)+pi −p )x̂ (c)
− [x̂0 (c)]2 ψ(c, pi ) − f1−F
(x̂(c)+pi −p∗ )
−x̂0 (c)ψ(c, pi )
−x̂0 (c)ψ(c, pi )
−ψ(c, pi )
It is straightforward to check that the determinant of H is equal to
!
f (x̂(c)+pi −p∗ )x̂00 (c)
1−F (x̂(c)+pi −p∗ ) ψ(c, pi ),
which is
strictly positive. As a result the function m(c, pi ) is strictly concave and by implication the integrand
of (35) is log-concave in c and in pi .
26
We now invoke Theorem 6 in Prékopa (1973) showing that the integral over a convex subset of
the real line of a log-concave function is also log-concave, which implies that the demand function
(35) is log-concave in pi . Therefore, an equilibrium exists and is unique.
In the second case assume that the search cost density g is the Kumaraswamy density (23) on
(0, 1) with a = 1/2, b ∈ (0, 2) and ε is distributed uniformly on [0, 1]. The basic idea of the proof is the
following. We show below that π (pi ; p∗ ) is increasing for pi ∈ (r, p∗ ), is decreasing just above p∗ and
is quasi-concave for pi ∈ (p∗ , 1), which will imply that it is decreasing for pi ∈ (p∗ , 1). Consequently,
p∗ will be the global maximum of the payoff.
We start by taking pi ∈ (p∗ , 1). For pi > p∗ the expression x̂ (c) + pi − p∗ involved in D (pi , p∗ )
can exceed ε = 1. Denote by ĉ (pi ) the solution in c of the equation x̂ (c) + pi − p∗ = 1, and since x̂ (c)
is strictly decreasing in c, x̂ (c) + pi − p∗ < 1 for all c > ĉ (pi ). In our case ε is distributed uniformly
√
on [0, 1], so x̂ (c) = 1 − 2c, c0 (p∗ ) = (1 − p∗ )2 /2 and ĉ (pi ) = (pi − p∗ )2 /2. We have
Z c0 (p∗ )
∗
π (pi ; p ) = (pi − r)
[1 − (x̂ (c) + pi − p∗ )] g (c) dc
ĉ(pi )
Z
= (pi − r)
(1−p∗ )2 /2 h√
(pi −p∗ )2 /2
i
2c − (pi − p∗ ) g (c) dc.
The derivative of the payoff for pi > p∗ is
"
Z (1−p∗ )2 /2 √
dπ (pi ; p∗ )
∗
=
2cg (c) dc − (2pi − p − r) G
dpi
(pi −p∗ )2 /2
(1 − p∗ )2
2
!
(pi − p∗ )2
2
−G
!#
,
and since
Z
(1−p∗ )2 /2 √
(pi −p∗ )2 /2
Z
2cg (c) dc ≤
(1−p∗ )2 /2 √
∗
2cg (c) dc = (p − r) G
0
(1 − p∗ )2
2
!
,
where the equality comes from the candidate equilibrium condition (33), we have that
!
!
∗ )2
∗ )2
dπ (pi ; p∗ )
(p
−
p
(1
−
p
i
≤ (2pi − p∗ − r) G
− (2pi − 2p∗ ) G
.
dpi
2
2
We prove that there is ω > 0 such that dπ (pi ; p∗ ) /dpi < 0 for all pi ∈ (p∗ , p∗ + ω). Note that the
right hand side of the above inequality can be written as

!
∗ )2 /2
2
G
(p
−
p
∗
i
(pi − p )
(pi − p∗ ) 2G
+ (p∗ − r)
− 2G
2
pi − p∗
and the limit of the expression in the brackets as pi & p∗ is
g (pi − p∗ )2 /2 (pi − p∗ )
0 + (p∗ − r) lim ∗
− 2G
pi &p
1
27
p ∗ )2
(1 −
2
(1 − p∗ )2
2
!
,
!
,
(36)
where for computing the limit of the second term we use the l’Hospital rule. Now using the Kumaraswamy distribution for a = 1/2 this expression becomes
(p∗ − r)
1 − p∗ b
√
√
b
−2+2 1−
.
2
2
Since p∗ is less than the monopoly price, which is (r + 1) /2, we have that p∗ − r < 1 − p∗ . So the
above expression is less than
(1 − p∗ )
1 − p∗ b
b √
−2+2 1− √
.
2
2
√
Introduce now the notation α = (1 − p∗ ) / 2 < 1 and define the function k (b) = αb − 2 + 2 (1 − α)b ,
which is in fact the above expression. We observe that k (0) = 0 and k (2) = 2 (1 − α) (−α) <
0.Further, the function k is convex in b because it is the sum of a linear and a strictly convex
function. We can conclude that k (b) < 0 for all 0 0. Therefore, the payoff π (pi ; p∗ ) is
decreasing for pi ∈ (p∗ , p∗ + ω).
Next we prove that π (pi ; p∗ ) is quasi-concave for pi ∈ (p∗ , 1). This will follow from showing that
1/D (pi ; p∗ ) is convex (Caplin and Nalebuff 1990, Proposition 3), which can be proved by establishing
that for pi ∈ (p∗ , 1)
d2 D (pi ; p∗ )
D (pi ; p∗ ) − 2
dp2i
dD (pi ; p∗ )
dpi
≤ 0.
We have that
D (pi ; p∗ ) =
Z
(1−p∗ )2 /2 √
(pi −p∗ )2 /2
"
2cg (c) dc − (pi − p∗ ) G
(1 − p∗ )2
2
!
−G
(pi − p∗ )2
2
!#
!
!
(pi − p∗ )2
(1 − p∗ )2
−G
2
2
!
(pi − p∗ )2
(pi − p∗ ) .
2
dD (pi ; p∗ )
=G
dpi
d2 D (pi ; p∗ )
=g
dp2i
Note that
∗
Z
D (pi ; p ) ≤
v
u
t2
(1−p∗ )2 /2 u
(pi −p∗ )2 /2
"
= (1 − pi ) G
!
"
(1 − p∗ )2
∗
g (c) dc − (pi − p ) G
2
!
!#
(1 − p∗ )2
(pi − p∗ )2
−G
.
2
2
28
(1 − p∗ )2
2
!
−G
(pi − p∗ )2
2
!#
Since d2 D (pi ; p∗ ) /dp2i ≥ 0
dD (pi ; p∗ ) 2
d2 D (pi ; p∗ )
∗
D (pi ; p ) − 2
dpi
dp2i
!
!#
"
(pi − p∗ )2
(1 − p∗ )2
−G
≤ G
2
2
(
!
"
(pi − p∗ )2
× g
(pi − p∗ ) (1 − pi ) − 2 G
2
(1 − p∗ )2
2
!
(pi − p∗ )2
2
−G
!#)
.
We will show that the expression in the curly brackets is negative. In order to simplify the notation
√
√
√
let x = (pi − p∗ ) / 2, δ = (1 − p∗ ) / 2, so 1 − pi = 2 (δ − x) and 0 < x < δ < 1. The expression
in the curly brackets becomes
k (x) = 2x (δ − x) g x2 − 2 G δ 2 − G x2 .
Using the Kumaraswamy distribution with a = 1/2, we have
k (x) = b (δ − x) (1 − x)b−1 + 2 (1 − δ)b − 2 (1 − x)b .
At x = 0 and x = δ we have k (0) = bδ + 2 (1 − δ)b − 2 and k (δ) = 0. If we regard bδ + 2 (1 − δ)b − 2
as a function k1 (b) then we observe that k1 (0) = 0 and k1 (2) = 2 (1 − δ) (−δ) < 0 and in addition k1
is convex in b because it is the sum of a linear and a strictly convex function, so k1 (b) ≤ 0; therefore,
k (0) = bδ + 2 (1 − δ)b − 2 < 0. Since
b<2
dk (x)
= b (1 − x)b−2 (1 − x − (b − 1) (δ − x)) > b (1 − x)b−2 (1 − δ) > 0,
dx
we conclude that k is increasing in x, so k (x) < 0 for all x such that 0 < x < δ. This implies that
π (pi ; p∗ ) is quasi-concave for pi ∈ (p∗ , 1).
This together with the fact that π (pi ; p∗ ) is decreasing for pi ∈ (p∗ , p∗ + ω) implies that π (pi ; p∗ )
is decreasing for pi ∈ (p∗ , 1). It remains to prove that π (pi ; p∗ ) is increasing for pi ∈ (r, p∗ ). In this
case the derivative of the payoff is
dπ (pi ; p∗ )
=
dpi
Z
(1−p∗ )2 /2 √
∗
2cg (c) dc + (p + r − 2pi ) G
0
(1 − p∗ )2
2
!
,
which from the candidate equilibrium condition (33) is equal to
dπ (pi ; p∗ )
= 2 (p∗ − pi ) G
dpi
(1 − p∗ )2
2
!
.
This is positive for pi ∈ (r, p∗ ), so the payoff π (pi ; p∗ ) is increasing for pi ∈ (r, p∗ ). 29
3.1
The effect of higher search costs
We now study the impact of higher search costs on the equilibrium prices of Proposition 4. We
proceed as before: we parametrize the search cost density by a scalar β and study how the equilibrium
price responds to a change in β. Consider first the case of Proposition (4)A. When the upper bound
of the search cost distribution is sufficiently low, the price is
R c(β)
[1 − F (x̂(c))]g(c, β)dc
∗
p = r + 0 R c(β)
.
f (x̂(c))g(c, β)dc
0
(37)
Notice that we allow the upper bound of the search cost distribution to increase in β.
The effect of an increase in search costs on the equilibrium price follows from taking the derivative
of (37) with respect to β. We provide a number of results below in Proposition 5.
We now move to the the case of Proposition (4)B. In this case the equilibrium price is given by
the solution to the implicit equation L(p∗ ) = 0. By the implicit function theorem, the effect of an
increase in β is given by the sign of
∂L
dp∗
∂β
= − ∂L .
dβ
∂p∗
We have already argued above that in a neighborhood of the equilibrium ∂L(p∗ ; β)/∂p < 0. In regard
to ∂L(p∗ ; β)/∂β, using the notation gβ (c; β) = dg (c; β) /dβ, we note that
Z c0 (p∗ )
Z c0 (p∗ )
∂L
=
(1 − F (b
x(c)))gβ (c; β)dc − (p∗ − r)
f (b
x(c))gβ (c; β) dc
∂β
0
0
R c0 (p∗ )
Z
Z c0 (p∗ )
∗
(1 − F (b
x(c)))g (c; β) dc c0 (p )
0
f (b
x(c))gβ (c; β) dc
(1 − F (b
x(c)))gβ (c; β) dc −
=
R c0 (p∗ )
0
0
f (b
x(c))g (c; β) dc
0
" R c (p∗ )
#
R c (p∗ )
Z
∗
x(c)))gβ (c; β) dc β 0 0
f (b
x(c))gβ (c; β) dc
β 0 0 (1 − F (b
1 c0 (p )
=
(1 − F (b
x(c)))g (c; β) dc R c (p∗ )
− R c (p∗ )
0
0
β 0
(1
−
F
(b
x
(c)))g
(c;
β)
dc
f (b
x(c))g (c; β) dc
0
0
(38)
where we have used the equilibrium condition (33).
Proposition 5 Let G(c; β) be a parametrized search cost cdf with positive density on [0, c] and with
derivative ∂G(·)/∂β < 0. Then the comparative statics of equilibrium prices in Proposition 4A,B are
as follows: (A) The equilibrium price given by Proposition 4A unambiguously increases in β if either
of the following conditions hold: (i) f is non-decreasing (ii) g is uniform.
(B) The equilibrium price given by Proposition 4B increases in beta if and only if
R c (p∗ )
R c (p∗ )
f (b
x(c))gβ (c; β) dc
β 0 0 (1 − F (b
x(c)))gβ (c; β) dc β 0 0
− R c (p∗ )
>0
R c0 (p∗ )
0
(1
−
F
(b
x
(c)))g
(c;
β)
dc
f
(b
x
(c))g
(c;
β)
dc
0
0
Moreover, if G is the uniform distribution, then the equilibrium price does not depend on β.
30
(39)
Condition (59) is quite complicated in general. In order to make sense of it, let us specialise the
model now by assuming that match values are distributed uniformly and search costs are distributed
according to the Kumaraswamy distribution.
For the uniform distribution, c0 (p∗ ) = (1 − p∗2 )/2 ∈ (0, 1/2), so x
b (c) ∈ (0, 1) so the condition in
(39) is:
#
R (1−p∗2 )/2 √
cgβ (c; β)dc
βGβ ((1 − p∗2 )/2; β) β 0
−
.
− R (1−p∗2 )/2 √
G((1 − p∗2 )/2; β)
cg(c; β)dc
"
0
The sign of (3.1) depends on the sign of the expression in the square brackets. Plugging the density
of the Kumaraswamy distribution and integrating, for the expression in square brackets we can write:
R (1−p∗2 )/2 √
cgβ (c; β)dc
βGβ ((1 − p∗2 )/2; β) β 0
− R (1−p∗2 )/2 √
∗2
G((1 − p )/2; β)
cg(c; β)dc
0
=
=
abδ a [1
−
δ a ]b−1
1 − [1 − δ a ]b
abδ a [1 − δ a ]b−1
1 − [1 − δ a ]b
1 − δa −
−
2aδ a+1/2 [1−δ a ]b
1
,b)
B (δ a ;1+ 2a
2 [1 − δ a ]
1 aδ a+1/2 [1 − δ a ]b−1
+
1
2
B δ a ; 1 + 2a
,b
−
where δ = (1 − p∗2 )/2β and the function B (y; c, d) =
Ry
0
(40)
tc−1 (1 − t)d−1 dt is the incomplete beta-
function. We now let a = 1/2 and b arbitrary. Then the expression in (40) becomes
b−1
b−1
1 bδ 1/2 1 − δ 1/2
1 1 δ 1/2+1/2 1 − δ 1/2
−
− +
2 1 − 1 − δ 1/2 b
2 2
B δ 1/2 ; 2, b
(
b−1
b−1 )
bδ 1/2 1 − δ 1/2
δ 1/2·2 1 − δ 1/2
1
=−
1+
b −
2
B δ 1/2 ; 2, b
1 − 1 − δ 1/2
Let x ≡ δ 1/2 and let
1
h (x) ≡ −
2
(
bx (1 − x)b−1
x2 (1 − x)b−1
1+
−
B (x; 2, b)
1 − (1 − x)b
)
;
We now show that h (x) < 0 if b > 1 and h (x) > 0 if b < 1 for any x ∈ (0, 1). For this, first define
J : (0, 1) → R such that
1
h (x) = − J (1 − x) ;
2
then
J (x) = −2h (1 − x) = 1 +
b (1 − x) xb−1
(1 − x)2 xb−1
−
,
B (1 − x; 2, b)
1 − xb
where
B (1 − x; 2, b) =
1 − (b + 1) xb + bxb+1
.
b (b + 1)
31
We study the sign of J (x) now. We have
1 + bxb−1 − (b + 1) xb B (1 − x; 2, b) − (1 − x)2 xb−1 − x2b−1
J (x) =
(1 − xb ) B (1 − x; 2, b)
1 + bxb−1 − (b + 1) xb 1 − (b + 1) xb + bxb+1 − b (b + 1) (1 − x)2 xb−1 − x2b−1
=
b (b + 1) (1 − xb ) B (1 − x; 2, b)
1 − b2 xb−1 + 2 b2 − 1 xb − b2 xb+1 + x2b
=
.
b (b + 1) (1 − xb ) B (1 − x; 2, b)
Since the denominator is positive, we only need to study the sign of the numerator. Define
k (x) ≡ 1 − b2 xb−1 + 2 b2 − 1 xb − b2 xb+1 + x2b .
Note that k (1) = 1 − b2 + 2 b2 − 1 − b2 + 1 = 0 and k (0) = 1 if b > 1 and k (0) = −∞ if b < 1.
The derivative is
k 0 (x) = −b2 (b − 1) xb−2 + 2 b2 − 1 bxb−1 − b2 (b + 1) xb + 2bx2b−1
h
i
= bxb−2 −b (b − 1) + 2 b2 − 1 x − b (b + 1) x2 + 2xb+1 ,
where since bxb−2 > 0 the sign is determined by the sign of the expression in squared brackets; let
` (x) = −b (b − 1) + 2 b2 − 1 x − b (b + 1) x2 + 2xb+1 .
We have ` (1) = −b (b − 1) + 2 b2 − 1 − b (b + 1) + 2 = 0, ` (0) = −b (b − 1), which is negative if
b > 1 and positive if b < 1. The derivate is
`0 (x) = 2 b2 − 1 − 2b (b + 1) x + 2 (b + 1) xb = 2 (b + 1) b − 1 − bx + xb .
The sign is determined by the sign of
m (x) ≡ b − 1 − bx + xb ,
where m (1) = 0, m (0) = b − 1, which is positive if b > 1 and < 0 if b < 1. Since the derivative
is m0 (x) = −b 1 − xb−1 , which is negative if b > 1 and positive if b < 1, we get that m (x) > 0 if
b > 1 and m (x) < 0 if b < 1.
Consequently, if b > 1 then m (x) > 0, so `0 (x) > 0, which implies that ` (x) < 0; therefore, k 0 (x) < 0, so k (x) > 0 and hence J (x) > 0. This implies that h (x) < 0 and, hence,
∂L(p∗ ; β)/∂β > 0. The case b < 1 can be done similarly, and we obtain that h (x) > 0 and, by
implication, ∂L(p∗ ; β)/∂β < 0.
32
Proposition 6 Assume that search costs are distributed on the interval [0, β] according to the Kumaraswamy distribution with parameter a = 1/2 and that match values are uniformly distributed.
Then, in the model of search for differentiated products where consumers search sequentially:
(A) The equilibrium price in Proposition 4A increases in β; as a result, higher search costs
unambiguously result in higher prices.
(B) The equilibrium price in Proposition 4B decreases in β if 0 1; as a result, higher search costs result in lower (higher) prices if 0 1).
4
Homogeneous products
In this Section we study the effects of higher search costs in consumer search models for homogeneous
products. The main difference with the case of differentiated products is that in homogeneous product
models consumers search for low prices and the symmetric equilibrium is characterized by mixedstrategies. We will show that our result above in Proposition 2 that higher search costs can result
in lower prices also arises with homogeneous products. This is interesting because the main insight
of this paper has nothing to do with the mixed- or pure-strategy nature of equilibria.
We start by examining a duopoly model where consumers search non-sequentially as in Burdett
and Judd (1983).16 Except in that products are homogeneous, the model is similar to the model in
Section 2. Firms produce a homogeneous good at constant unit costs r ≥ 0. There is a unit mass of
buyers. Each buyer inelastically demands one unit of the good and is willing to pay for it a maximum
of v > r. Let θ ≡ v − r. Consumers search for prices non-sequentially and buy from the cheapest
store they know. Search costs differ across consumers. A buyer’s search cost is drawn independently
from G(c) with support (0, c) and positive density g(c) everywhere.17 Searching k times costs the
consumer kc, k = 0, 1, 2.
Firms and buyers play a simultaneous moves game. An individual firm chooses its price taking rivals’
prices as well as consumers’ search behavior as given. A firm i’s strategy is denoted by a distribution
of prices Fi (p). Let F−i (p) denote the vector of prices charged by firms other than i. The (expected)
profit to firm i from charging price pi given rivals’ strategies is denoted by Π(pi , F−i (p)). Likewise,
16
For a dynamic version, see Fershtman and Fishman (1992) and for an oligopoly version see Janssen and MoragaGonzález (2004). These models do not allow for search costs heterogeneity.
17
As before, the critical issue will be the relationship between c and v. We will see that when c > v, some consumers
will opt out of the market and will not search at all. By contrast, when c < v every consumer will make at least one
search.
33
an individual buyer takes as given firm pricing and decides on his/her optimal search strategy to
maximize his/her expected utility. The strategy of a consumer with search cost c is then a number
k of prices to sample, k = 0, 1, 2. Let the fraction of consumers sampling k firms be denoted by µk .
We shall concentrate on symmetric Nash equilibria. A symmetric equilibrium is a distribution of
prices F (p) and a collection {µ0 , µ1 , µ2 } such that (a) Πi (p, F−i (p)) is equal to a constant Π for all
p in the support of F (p), ∀i; (b) Πi (p, F−i (p)) ≤ Π for all p, ∀i; (c) a consumer with search cost c
i
h
Rv
chooses to sample k(c) firms such that k(c) = arg mink∈{0,1,2} kc + p pk(1 − F (p))k−1 f (p)dp ; and
P
(d) 2k=0 µk = 1. Let us denote the equilibrium density of prices by f (p), with maximum price p
and minimum price p.
The following 2 lemmas follow directly from Burdett and Judd (1983). The first indicates that,
for an equilibrium to exist, there must be some consumers who search just once and others who
search twice. The second shows that prices must be dispersed in equilibrium.
Lemma 1 If a symmetric equilibrium exists, then 1 > µk > 0, k = 1, 2, and µ0 ≥ 0.
The intuition behind this result is simple. Suppose all searching consumers did search twice (µ0 +µ2 =
1); then pricing would be competitive. This however is contradictory because then consumers would
not be willing to search that much in the first place. Suppose now that no consumer did compare
prices (µ0 + µ1 = 1); then firms would charge the monopoly price. This is also contradictory because
in that case consumers would not be willing to search at all.18
Lemma 2 If a symmetric equilibrium exists, F (p) must be atomless with upper bound equal to v.
This is easily understood. If a particular price is chosen with strictly positive probability then a
deviant can gain by undercutting such a price and attracting all price-comparing consumers. This
competition for the price-comparing consumers cannot drive the price down to zero since then a
deviant would prefer to raise its price and sell to the consumers who do not compare prices.
We now turn to consumers’ search behavior. Expenditure minimization requires a consumer with
search cost c to continue to draw prices from the price distribution F (p) till the expected gains of
drawing one more price fall below her search cost. The expected net gains from searching once rather
18
In the original model of Burdett and Judd (1983), it is assumed that the search cost is lower than the surplus
consumers get at the monopoly price. As a result, all consumers buy no matter the equilibrium price distribution and
therefore there always exists an equilibrium where all firms charge the monopoly price (cf. Diamond, 1971). Since we
have arbitrary search cost heterogeneity, this assumption is relaxed. A by-product is that a Diamond-type of result
cannot be an equilibrium any longer.
34
than not searching at all are given by v − E[p] − c, while the expected net gains from searching twice
rather than once are given by E[p] − E[min{p1 , p2 }] − c, where E denotes the expectation operator.
Since the search cost distribution has support on [0, c̄], we can define the critical consumers c0 and
c1 satisfying the following equalities:
c0 = min {c̄, v − E[p]} ,
(41)
c1 = E[min{p1 , p2 }] − E[p].
(42)
From Lemma 1, it must be the case that c1 > 0 and c0 > c1 . c0 is the minimum of the search cost of
the consumer who is indifferent between searching and not searching at all and of the upper bound
of the search cost distribution. When the upper bound of the search cost distribution c̄ is sufficiently
high c0 = v − E[p] and all consumers with search cost above c0 will not search at all. When c̄ is
small enough, all consumers will search at least once. In particular, consumers for whom c1 < c ≤ c0
will indeed search once and consumers for whom c ≤ c1 will search twice.
Lemma 3 Given any atomless price distribution F (p), optimal consumer search behavior is uniquely
characterized as follows: the fractions of consumers searching once and twice are given by
Z
c0
µ1 =
Z
c1
dG(c) > 0; µ2 =
c1
dG(c) > 0
(43)
0
while the fraction of consumers not searching at all is
Z
c̄
dG(c) ≥ 0,
µ0 =
(44)
c0
where c0 and c1 are given by (41)-(42)
We now examine firm pricing behavior taking consumer search strategies as given. Following Burdett
and Judd (1983), a firm i charging a price pi sells to a consumer who searches one time provided the
consumer samples firm i, which happens with probability 1/2, and sells to a consumer who searches
twice provided the rival firm charges a price higher than pi , which happens with probability 1−F (pi ).
Therefore the expected profit to firm i from charging price pi when its rivals draw a price from the
cdf F (p) is
Πi (pi ; F (p)) = (pi − r)
1
µ1 + µ2 [1 − F (pi )] .
2
In equilibrium, a firm must be indifferent between charging any price in the support of F (p) and
charging the upper bound p. Thus, any price in the support of F (p) must satisfy Πi (pi ; F (p)) =
35
Πi (p; F (p)). Since Πi (p; F (p)) is monotonically increasing in p, it must be the case that p = v. As a
result, equilibrium pricing requires
(pi − r) {µ1 + 2µ2 [1 − F (pi )]} = µ1 (v − r).
(45)
Solving this equation for F (pi ) leads to the following result, also in Burdett and Judd (1983):
Lemma 4 Given µ1 and µ2 , there exists a unique symmetric equilibrium price distribution F (p). In
i
h
1
+
r,
v
according to the price
equilibrium firms charge prices randomly chosen from the set (v−r)µ
µ1 +2µ2
distribution
F (p) = 1 −
µ1 v − p
.
2µ2 p − r
(46)
Notice that F (p) depends on the search cost distribution via its effect on µ1 and µ2 ; moreover,
notice that F (p) is increasing in µ2 and decreasing in µ1 . Hence, if an increase in search costs
results in a higher (lower) ratio of “price-comparing to non-price-comparing” consumers, then the
price distribution shifts up (down) and prices decrease (increase).
For the price distribution (46) to be an equilibrium of the game, the conjectured groupings of
consumers has to be the outcome of optimal consumer search. This requires that
 v

Zv
 Z

c0 = min c, F (p)dp and c1 = F (p)(1 − F (p))dp


0
(47)
0
Since the price distribution F (p) in (46) is strictly increasing in p, we can find its inverse:
p(z) =
v−r
+ r.
1 + 2 µµ21 (1 − z)
(48)
Using this inverse function, integration by parts and the change of variables z = F (p), we can state
that:
Proposition 7 If a symmetric equilibrium exists then consumers search according to Lemma 3, firms
set prices according to Lemma 4, and c0 and c1 are given by the solution to the following system of
equations:
Z 1
G(c0 ) − G(c1 )
c0 = min c, (v − r) 1 −
du ,
0 G(c0 ) − G(c1 )(1 − 2u)
Z 1
[G(c0 ) − G(c1 )] (1 − 2u)
c1 = (v − r)
du
G(c0 ) − G(c1 )(1 − 2u)
0
36
(49)
(50)
Relative to Burdett and Judd (1983), this Proposition is our first contribution. It is useful because of
two reasons. First, it provides a straightforward way to compute the market equilibrium. For fixed v,
r, c and G(c), the system of equations (49)–(50) can be solved numerically. If a solution exists, then
the consumer equilibrium is given by equations (43)–(44) and the price distribution follows readily
from equation (46). Secondly, this result enables us to address the issues of existence and uniqueness
of equilibrium, which are the subject of our second contribution.
Proposition 8 (A) For any consumer valuation v and firm marginal cost r such that v > r ≥ 0 and
for any search cost distribution function G(c) with support (0, c) such that either g(0) > 0 or g(0) = 0
and g 0 (0) > 0, a symmetric Nash equilibrium exists. (B) For the family of polynomial distribution
functions G(c) = (c/c)a , a > 0, the equilibrium is unique.
The proof of this result is in the Appendix.19 Proposition 8 establishes uniqueness of equilibrium
when the search cost distribution has the described polynomial form. General results on uniqueness
prove to be very difficult because we cannot compute the equilibrium explicitly and the system of
equations (49)–(50) is non-linear.
4.1
The effect of higher search costs on prices
The next step in the analysis is to study how an increase in search costs affects prices. As mentioned above, for this it suffices to study how the ratio of “price-comparing to non-price comparing”
consumers λ is affected by an increase in search costs. To do so, let us parametrize the search cost
distribution G by a positive parameter β and use the notation G(c; β). Specifically, assume that
an increase in β implies an increase in search costs in the sense of first-order stochastic dominance
(FOSD), i.e. G(c; β) > G(c; β 0 ) for all c, for all β 0 > β. We shall denote the equilibrium price
distribution corresponding to a given β by F (p; β) and we will examine how F changes with β.
To understand the effect of an increase in β on the equilibrium price distribution, we study how
the solution to the system of equations that determines c0 , c1 and c2 depends on β; this, in turn,
determines how µ1 and µ2 , and therefore λ, depend on β. We start with the (most interesting) case
where the upper bound of the search cost distribution is sufficiently high so that c0 < c. This means
that some consumers have search costs so high that they opt out of the market altogether. Using
19
Elsewhere, we have extended this existence result to the case of an arbitrary number of firms N (see MoragaGonzález et al., 2010).
37
the change of variables xk ≡ G(ck ; β) in (49)–(50) gives
Z 1
x0 − x1
x0 = G θ − θ
du; β ,
0 x0 − x1 + 2x1 u
Z 1
x0 − x1
x1 = G θ
(1 − 2u) du; β .
0 x0 − x1 + 2x1 u
Let y ≡ x1 /x0 ∈ [0, 1] . Then the previous system of equations is equivalent to
yG (c0 (y) ; β) − G (c1 (y) ; β) = 0.
(51)
where
1
1−y
(1 − y)
1−y
c0 (y) = θ 1 −
and
du = θ 1 +
ln
2y
1+y
0 1 − y(1 − 2u)
Z 1
(1 − y) (1 − 2u)
θ (1 − y)
1−y
c1 (y) = θ
du = −
2y + ln
.
1 − y(1 − 2u)
2y 2
1+y
0
Z
(52)
(53)
Note that 0 ≤ c1 (y) ≤ c0 (y) ≤ θ for any y ∈ [0, 1]. For later use, notice that 0 = c1 (0) = c0 (0) and
that c01 (0) > 0.
Rewriting (51) gives:
1−y
1−y
1−y
1−y
H(y; β) ≡ yG θ 1 +
2y
+
ln
;
β
−
G
θ
1
+
ln
; β = 0.
2y 2
1+y
2y
1+y
(54)
An equilibrium of the model is given as a solution to equation H(y; β) = 0. Let y(β) denote such a
solution. If we obtain y(β), then using (52) and (53) we can immediately derive the corresponding
c0 (β) and c1 (β) and hence µ1 (β), µ2 (β), λ(β) and the equilibrium price distribution F (p; β). To be
sure, for a given β, we notice again the relationship between the variables we have introduced
y = x1 /x0 , x0 = G(c0 ), x1 = G(c1 ), µ2 = x1 , µ1 = x0 − x1 .
(55)
Since
µ2
=
µ1
1
y
1
,
−1
a decrease in y results in an decrease in µ2 /µ1 and, correspondingly, in an increase in prices. We now
study how y(β) depends on the shifter β of the search cost distribution G(c; β)
Let y(β) be the solution to equation (54). The Implicit Function Theorem implies
∂H(y; β)
dy (β)
∂β
=−
.
∂H(y; β)
dβ
∂y
38
(56)
In order to sign this derivative, we consider first its numerator.
∂H(y; β)
= yG0β (c0 (y) ; β) − G0β (c1 (y) ; β)
∂β
G (c1 (y) ; β) 0
=
G (c0 (y) ; β) − G0β (c1 (y) ; β)
G (c0 (y) ; β) β
"
#
0
0
G (c1 (y) ; β) βGβ (c0 (y) ; β) βGβ (c1 (y) ; β)
=
−
β
G (c0 (y) ; β)
G (c1 (y) ; β)
where the second equality follows from the equilibrium condition (54). Therefore, we conclude that
∂H(y; β)
> 0 if and only if εG,β (c0 (y); β) − εG,β (c1 (y); β) > 0,
∂β
where
εG,β (c; β) ≡
(57)
βG0β (c; β)
G (c; β)
denotes the elasticity of the search cost distribution with respect to the shift-parameter β. Since
c0 (y(β)) > c1 (y(β)), a necessary and sufficient condition for ∂H(y; β)/∂β > 0 to hold is that the
elasticity of the search cost distribution with respect to β increases in c. If the elasticity is decreasing
then ∂H(y; β)/∂β < 0.
Consider now the denominator of (56). For a given β, ∂H(y; β)/∂y is the derivative of H at the
solution y. We note that for y = 0 and y = 1 we have
H(0; β) = 0 · G (c0 (0) ; β) − G (c1 (0) ; β) = −G (0; β) = 0,
H(1; β) = G (c0 (1) ; β) − G (c1 (1) ; β) = G (1; β) − G (0; β) = G (1; β) > 0.
(58)
Consider now the value of ∂H (y; β) /∂y at y = 0. Since 0 = c1 (0) = c0 (0) and c01 (0) > 0 we have
∂H(0; β)
= G (0; β) − G0 (0; β) c01 (0) = −G0 (0; β) c01 (0) < 0.
∂y
Given these three observations (i.e. H (0, β) = 0, H (1, β) > 0 and ∂H (0, β) /∂y < 0), we conclude
that there exists at least one equilibrium at which H is increasing in y.20 We then obtain the following
result:
Proposition 9 Let G (c; β) be a parametrized search cost cdf with positive density on [0, c] and with
derivative dG(c; β)/dβ < 0 for all c. Assume that c is sufficiently large so that c0 defined in (49)
20
We ignore ill-behaved situations where at the solutions of (54) H (·; β) is tangent to the horizontal axes, that is, we
assume that ∂H (y, β) /∂y 6= 0 at any solution y. Moreover, if there are multiple equilibria, the number of equilibria is
odd. In such situation each odd-numbered solution y (β) satisfies ∂H (y, β) /∂y > 0, while each even-numbered solution
y (β) satisfies ∂H (y, β) /∂y < 0.
39
satisfies c0 < c. Assume also that ∂H (y, β) /∂y 6= 0 at any y for which (54) holds. Then, if there
exists a unique equilibrium and the elasticity of the search cost distribution with respect to β increases
(decreases) in c, we have F (p; β̂) < (>)F (p; β) for all p.21
This result shows that prices can increase or decrease after search costs go up for all consumers.
When search costs increase, holding constant the prices of the firms, two effects take place. On
the one hand, at the intensive search margin, fewer consumers price-compare and as a result firms
have a tendency to raise their prices. On the other hand, at the extensive search margin, more
consumers leave the market without searching at all, which gives firms an incentive to lower their
prices. Our theorem shows that whether the impact at the extensive search margin dominates that
at the intensive search margin depends on whether the search cost distribution has an elasticity with
respect to the parameter β that increases or decreases in c.
The case in which the parameter β enters multiplicatively is easier to interpret. In such a case, it
is straightforward to verify that the search cost distribution has an increasing (decreasing) elasticity
with respect to the parameter β provided that it has a decreasing (increasing) search cost density.
When the search cost density is decreasing, the impact of an increase in search costs on the extensive
search margin is weaker than on the intensive search margin and hence prices increase as search costs
go up. By contrast, when the search cost density is increasing, the effect on the extensive search
margin is stronger and has a dominating influence. Hence prices decrease when search costs go up.
We now continue with the case where the upper bound of the search cost distribution is sufficiently
low so that c0 = c. This implies that µ0 = 0. In this case higher search costs only have an effect
at the intensive search margin and thereby we should obtain the standard result that higher search
costs lead to higher prices. For this case,
µ2
1
=
− 1.
µ1
µ1
As a result, the equilibrium price distribution is uniquely determined by µ1 , which in turn depends
on
Z
c1 = θ
0
1
[1 − G(c1 ; β)] (1 − 2u)
du
1 − G(c1 ; β)(1 − 2u)
(59)
Using the change of variables y ≡ G(c1 ) we can write (59) as y − G (c1 (y)) = 0 where c1 (y) is given
21
If there exist multiple equilibria, this result also holds for the odd-numbered equilibria. For the even-numbered
equilibria, we have the opposite, that is, if the elasticity of the search cost distribution with respect to β increases
(decreases) in c, we have F (p; β̂) > (<)F (p; β) for all p.
40
in (53). Rewriting gives:
1−y
θ (1 − y)
2y + ln
;β = 0
H(y; β) ≡ y − G −
2y 2
1+y
(60)
As above, an equilibrium of the model is given as a solution to equation (60). Note that since G is
monotone and 0 = c1 (0) = c1 (1), the equilibrium is unique in this case.
If we consider the parametrized search cost distribution above G (c; β) and compute the derivative
of H with respect to β we get ∂H (y; β) /∂β = −G0β (c1 (y) ; β) > 0. This implies that the sign of (56)
is negative. As a result:
Proposition 10 Let G (c; β) be a parametrized search cost cdf with positive density on [0, c], and
with derivative dG(c; β)/dβ < 0 for all c. Assume that c is sufficiently low so that c0 defined in (49)
is equal to c. Assume also that ∂H (y, β) /∂y 6= 0 at y for which (60) holds. Then, if β̂ > β we have
F (p; β̂) < F (p; β) for all p.
4.1.1
The Kumaraswamy distribution
Proposition 9 is illustrated in Figure 3. In this Figure, we show the mean equilibrium price as a
function of parameter β for various values of the parameter b, keeping a fixed to 1. In all cases the
mean price is first increasing in β, up to the point where c0 = c. Thereafter, the mean price increases
when b = 0.5 (red curve), is constant for b = 1 (green curve) and decreases when b = 1.25 (blue
curve).
Figure 3: Expected price for increasing (red), constant (green) and decreasing (blue) search cost
densities
41
4.1.2
The general N -firms case.
The non-sequential search model we have presented above can easily be generalized to the case of
N firms.22 In such a case, the payoff to a firm i charging price pi given the N − 1 rivals use the
equilibrium price distribution F (p) is
"
Πi (pi ; F (p)) = (pi − r)
N
X
kµk
k=1
N
#
k−1
(1 − F (pi ))
.
The fractions of consumers searching k times are given by
Z c
dG(c)
µ0 =
c0
Z ck−1
dG(c), for all k = 1, 2, . . . , N ;
µk =
(61)
(62)
ck
where




 Zv

c0 = min c, F (p)dp ;




(63)
p
Zv
ck =
F (p)(1 − F (p))k dp, k = 1, 2, . . . , N − 1; cN = 0
(64)
p
The equilibrium distribution function follows from the constancy-of-profits condition
N
X
kµk (1 − F (pi ))k−1 =
k=1
µ1 θ
,
(pi − r)
(65)
from which we can calculate the inverse of the equilibrium price distribution
p(z) = PN
µ1 θ
k=1 kµk (1
− z)k−1
+ r.
Using (66, we can rewrite the critical points {ck }N
k=0 as:
(
!)
Z 1
G(c0 ) − G(c1 )
c0 = min c, θ 1 −
du
;
PN
k−1
0
k=1 k[G(ck−1 ) − G(ck )]u
Z1
[G(c0 ) − G(c1 )] kuk−1 − (k + 1) uk
ck = θ
du, k = 1, 2, ..., N − 1; cN = 0.
PN
k−1
k=1 k[G(ck−1 ) − G(ck )]u
(66)
(67)
(68)
0
As mentioned above, we have proven elsewhere that an equilibrium always exists (see MoragaGonzález et al., 2010).
22
For empirical applications of the N -firm model see Moraga-González and Wildenbeest (2008) and Moraga-González,
Sándor and Wildenbeest (forthcoming). Hong and Shum (2006) estimates search costs using a model with infinitely
many firms.
42
The impact of higher search costs on the equilibrium price distribution (66) is however very
difficult to analyze in the general N -firms case because the system of equations (67)–(68) is nonlinear and therefore it is hard to say something about how its solution depends on β. Nevertheless,
it is straightforward to check numerically that the spirit of the result in Theorem 8 remains. For
this we take a market with N = 10 firms, set v = 10 and r = 0 and use the family of Kumaraswamy
search cost distributions presented above. We choose β high enough so we are sure c0 < c. Table 2
shows how market equilibria evolve as we increase the parameter β from 8 to 9 and to 10. We do
this for a = 1 and let b take on values that cover the regions in Proposition 2.1.1A, in particular
b = {0.5, 1, 1.25}.
Table 2 clearly shows that the results in Proposition 9 hold true more in general. In particular,
when b = 0.5 an increase in search costs leads to lower prices. We can see that higher search costs
lead to overall less search, i.e., as search costs increase a given consumer searches (weakly) less (all µ’s
decrease except µ0 ). The effect is more noticed at the higher quantiles of the search cost distribution.
This is due to the fact that for b = 0.5, the search cost density is increasing and thereby there is
more mass of consumers at higher search costs. Relative to the non-price-comparing consumers,
the number of price-comparing consumers increases (all fractions muk /µ1 increase), which makes
the market more competitive. As a result prices decrease. Though aggregate social welfare falls as
search costs increase, some consumers benefit. This can be seen in the row CS/(1 − µ0 ), which is
the consumer surplus conditional on searching at least one time.
When search costs follow the uniform distribution (b = 1), prices are constant. What happens
is that the numbers of price-comparing and non-price comparing consumers fall exactly in the same
proportion. Consumer surplus conditional on searching also goes up in this case.
Finally, when the search cost density is decreasing (b = 1.25), an increase in search costs results
in higher prices, lower consumers surplus (conditional and unconditional) and lower welfare.
5
Conclusions
This paper has studied models of price competition under search cost heterogeneity and has revisited
the question how an increase in search costs affects the level of prices. Traditional consumer search
models have typically assumed that all consumers search at least once in equilibrium. By doing
so, the standard literature has neglected an important role of the price mechanism, namely, that
the price ought to affect the number of consumers who choose to search for a product in the first
place. This assumption cannot easily be reconciled with the idea that search costs, to the extent
43
β=8
b = 0.50
β=9
µ0
µ1
µ2
µ3
µ4
µ5
µ6
µ7
µ8
µ9
µ10
0.800
0.131
0.032
0.013
0.007
0.004
0.003
0.002
0.001
0.001
0.005
µ2 /µ1
µ3 /µ1
µ4 /µ1
µ5 /µ1
µ6 /µ1
µ7 /µ1
µ8 /µ1
µ9 /µ1
µ10 /µ1
E[p]
p
PS
CS
CS/(1 − µ0 )
Total Welfare
b = 1.00
β=9
β = 10
β=8
b = 1.25
β=9
β = 10
β = 10
β=8
0.823
0.116
0.029
0.012
0.006
0.004
0.002
0.002
0.001
0.001
0.004
0.842
0.103
0.026
0.011
0.006
0.003
0.002
0.001
0.001
0.001
0.004
0.622
0.241
0.064
0.027
0.014
0.008
0.005
0.004
0.003
0.002
0.010
0.664
0.214
0.057
0.024
0.013
0.007
0.005
0.003
0.002
0.002
0.009
0.697
0.193
0.051
0.022
0.011
0.007
0.004
0.003
0.002
0.002
0.008
0.541
0.287
0.080
0.034
0.018
0.011
0.007
0.005
0.003
0.002
0.012
0.591
0.257
0.071
0.030
0.016
0.009
0.006
0.004
0.003
0.002
0.011
0.630
0.232
0.064
0.027
0.014
0.008
0.005
0.004
0.003
0.002
0.010
0.247
0.103
0.054
0.032
0.021
0.014
0.010
0.008
0.037
0.249
0.104
0.054
0.032
0.021
0.014
0.010
0.008
0.037
0.251
0.105
0.055
0.032
0.021
0.014
0.010
0.008
0.037
0.267
0.113
0.059
0.035
0.023
0.015
0.011
0.008
0.040
0.267
0.113
0.059
0.035
0.023
0.015
0.011
0.008
0.040
0.267
0.113
0.059
0.035
0.023
0.015
0.011
0.008
0.040
0.278
0.118
0.062
0.037
0.024
0.016
0.012
0.009
0.042
0.277
0.118
0.062
0.037
0.024
0.016
0.012
0.009
0.041
0.275
0.117
0.061
0.036
0.023
0.016
0.012
0.009
0.041
7.118
3.433
7.100
3.409
7.086
3.390
6.973
3.240
6.973
3.240
6.973
3.240
6.894
3.139
6.905
3.152
6.913
3.163
1.313
0.683
3.4132
1.996
1.155
0.608
3.4401
1.763
1.032
0.548
3.4612
1.580
2.409
1.355
3.583
3.764
2.141
1.207
3.588
3.348
1.927
1.087
3.593
3.015
2.873
1.687
3.676
4.560
2.569
1.502
3.669
4.072
2.323
1.354
3.664
3.677
Table 2: Equilibrium search intensities for Kumaraswamy distribution (a = 1)
44
that they are related to consumer demographics such as income, age, marital status etc., might be
very heterogeneous. In this paper we have shown that recognising this role of the price turns out to
be critical for our understanding of the effect of higher search costs on prices and profits. The main
results of the paper have been on characterising conditions of search cost distributions under which
higher search costs result in lower prices.
Allowing for search cost heterogeneity, besides being more realistic, allows for prices to increase or
decrease when search costs go up. We have identified a critical property of search cost distributions
that play a decisive role, namely, the elasticity of the search cost distribution with respect to the
parameter that shifts it. When this elasticity is increasing in search costs, an increase in search
frictions affects consumers with higher search costs more strongly than it affects consumers with low
search costs. This makes demand more elastic and correspondingly prices decrease. We have shown
that these insights are quite robust since they hold no matter whether products are differentiated or
homogenous and irrespective of whether consumers search sequentially or non-sequentially.
45
Appendix
Proof that a candidate equilibrium price p∗ ∈ [r, pm ] exists in the non-sequential search
N -firm model with differentiated products.
Recall that the FOC (29) is given by
∗
∗
∗
∗
µ1 (p ) [1 − F (p ) − (p − r)f (p )] +
N
X
Z
∗
kµk (p )
k=2
∗
− (p − r)
Note that when we set
N
X
∗
Z
kµk (p )
k=2
p∗ =
ε
F (ε)k−1 f (ε)dε
p∗
ε
k−2
(k − 1)F (ε)
∗ k−1
2
f (ε) dε + F (p )
f (p ) = 0.
∗
p∗
r, the LHS of this equation is strictly positive. We now show that when
we set p∗ = pm , then it is strictly negative, which implies that there exists a candidate equilibrium
price p∗ ∈ [r, pm ].
Since the monopoly price pm satisfies 1 − F (pm ) − (pm − r)f (pm ) = 0, when we evaluate the LHS
of the FOC at pm we obtain:
N
X
Z
N
h
i
X
m k
m
∗
µk (p ) 1 − F (p ) − (p − r)
kµk (p )
∗
k=2
ε
(k − 1)F (ε)
k−2
2
m k−1
f (ε) dε + F (p )
f (p ) ,
m
pm
k=2
(69)
where we have used the fact that
Rε
pm
F (ε)k−1 f (ε)dε = 1 − F (pm )k .
We now claim (69) is negative. To show it, we first observe that
m k
m
1 − F (p ) = (1 − F (p ))
k−1
X
m j
m
m
F (p ) = (p − r)f (p )
j=0
k−1
X
F (pm )j ,
j=0
where we have used again the monopoly pricing rule, and write (69) as follows:
(
)
Z ε
N
N
m )k
X
X
1
−
F
(p
(pm −r) f (pm )
µk (p∗ )
−
kµk (p∗ )
(k − 1)F (ε)k−2 f (ε)2 dε + F (pm )k−1 f (pm )
.
1 − F (pm )
pm
k=2
k=2
Putting terms together, this simplifies to
Z ε
N
X
1 − F (pm )k
m k−1
k−2
2
(pm − r)
µk (p∗ ) f (pm )
−
kF
(p
)
−
k
(k
−
1)F
(ε)
f
(ε)
dε
.
1 − F (pm )
pm
k=2
We now note that the expression within curly brackets is increasing in pm . In fact, its derivative is
equal to
1 − F (pm )k
1 − F (pm )k − kF (pm )k−1
m k−1
2 m
f (p )
− kF (p )
+ f (p )
1 − F (pm )
(1 − F (pm ))2
1 − F (pm )k − kF (pm )k−1 (1 − F (pm )) 0 m
=
f (p )(1 − F (pm ) + f 2 (pm ) > 0
m
2
(1 − F (p ))
0
m
46
where the sign follows by log-concavity of f. Since it is increasing in pm and it is equal to zero when
we set pm = ε, we conclude it is always negative. This shows that a candidate equilibrium price
p∗ ∈ [r, pm ] exists in the non-sequential search N -firm model with differentiated products.
Proof of Proposition 3. In order to prove the result we compute the second order derivative of
the payoff function and show that for the case N = 3 it is strictly negative so the payoff is strictly
concave. The proof also serves to illustrate the difficulties to prove strict concavity when N ≥ 4.
Let us denote the demand of a firm i charging price pi as
Z
N
X
µ1 (p∗ )
kµk (p∗ ) ε
∗
di (pi , p ) =
(1 − F (pi )) +
F (ε − (pi − p∗ ))k−1 f (ε)dε
N
N
pi
k=2
The first order derivative of demand with respect to own price is:
Z ε
N
X
µ1 (p∗ )
∂di (pi , p∗ )
kµk (p∗ )
=−
(k − 1)F (ε − pi + p∗ )k−2 f (ε − pi + p∗ )f (ε)dε + F (p∗ )k−1 f (pi )
f (pi )−
∂pi
N
N
pi
k=2
and the second order derivative is:
N
i
X
µ1 (p∗ ) 0
kµk (p∗ ) h
∂ 2 di (pi , p∗ )
∗ k−1 0
∗ k−2
∗
=
−
f
(p
)
−
F
(p
)
f
(p
)
−
(k
−
1)F
(p
)
f
(p
)f
(p
)
i
i
i
N
N
∂p2i
k=2
Z ε
N
h
X
kµk (p∗ )
(k − 1) (k − 2)F (ε − pi + p∗ )k−3 f (ε − pi + p∗ )2 + F (ε − pi + p∗ )k−2 f 0 (ε − pi +
+
N
pi
k=2
For the case of the uniform distribution, these derivatives simplify to:
Z ε
N
∂di (pi , p∗ )
µ1 (p∗ ) 1 X kµk (p∗ )
(ε − pi + p∗ )k−2
p∗k−1
=−
−
(k − 1)
dε +
∂pi
N ε
N
εk
εk
pi
k=2
N
µ1 (p∗ ) 1 X kµk (p∗ ) (ε − pi + p∗ )k−1
=−
−
N ε
N
εk
k=2
and
N
N
k=2
k=2
∂ 2 di (pi , p∗ ) X kµk (p∗ ) (k − 1)p∗k−2 X kµk (p∗ )
=
+
N
N
∂p2i
εk
=
(ε − pi + p∗ )k−3
(k − 1)(k − 2)
dε
εk
pi
Z
ε
N
X
k(k − 1)µk (p∗ ) (ε − pi + p∗ )k−2
k=2
N
εk
The second order condition of the maximization problem is then:
∂ 2 πi (pi , p∗ )
∂di (pi , p∗ )
∂ 2 di (pi , p∗ )
=2
+ (pi − r)
2
∂pi
∂pi
∂p2i
=−
=−
N
N
k=2
k=2
X k(k − 1)µk (p∗ ) (ε − pi + p∗ )k−2
2µ1 (p∗ ) X 2kµk (p∗ ) (ε − pi + p∗ )k−1
−
+
(p
−
r)
i
Nε
N
N
εk
εk
(p∗ )
2µ1
Nε
N
1 X kµk (p∗ ) (ε − pi + p∗ )k−2
+
[(pi − r)(k − 1) − 2(ε − pi + p∗ )]
k
N
N
ε
k=2
47
(70)
For the N = 3 case we obtain
2µ1 2µ2
∂ 2 πi (pi , p∗ )
=−
+ 2 [(pi − r) − 2(ε − pi + p∗ )]
3ε
9ε
∂p2i
µ3 (ε − pi + p∗ )
[2(pi − r) − 2(ε − pi + p∗ )]
+
3
ε3
The sign of this second order derivative depends on the sign of the expressions in squared brackets.
Take first the expression (pi − r) − 2(ε − pi + p∗ ) and notice that it increases in pi . If we set pi equal
to the monopoly price (ε + r)/2, which is the maximum price, we get −(ε − r) − 2p∗ ) < 0. Since it
is increasing in pi and at teh maximum price it is negative, we conclude (pi − r) − 2(ε − pi + p∗ ) is
negative for all pi > p∗ . Take now the second expression in squared brackets 2(pi − r) − 2(ε − pi + p∗ )
and notice that it is also increasing in pi . Setting pi equal to the monopoly price we get −2p∗ < 0
so, by the same argument, (pi − r) − 2(ε − pi + p∗ ) is negative for all pi > p∗ . We conclude that for
N = 3, the payoff function is strictly concave so there exists a unique equilibrium that is symmetric.
For N ≥ 4, the summation in (70) involves additional terms and some of these terms are not
always negative; because of this, we can no longer prove that the payoff function is stricly concave.
However, using numerical methods we have been able to check that the payoff function is quasiconcave so we are quite confident that the equilibrium exists and is unique.
Proof of Proposition 8. Since x0 = G(c0 ) and x1 = G(c1 ), we have
Z 1
x0 − x1
x0 = G θ − θ
du ;
0 x0 − x1 + 2x1 u
Z 1
(x0 − x1 ) (1 − 2u)
x1 = G θ
du .
x0 − x1 + 2x1 u
0
An equilibrium of the model is given by a solution to
H (y) ≡ yG (θ − θ (1 − y) I(y)) − G (θ (1 − y) J(y)) = 0,
where
Z
1
I(y) =
0
Z
J(y) =
0
1
1
log (1 + y) − log (1 − y)
du =
;
1 − y + 2yu
2y
1 − 2u
log (1 + y) − log (1 − y) − 2y
du =
.
1 − y + 2yu
2y 2
We note that for y = 0 and y = 1 we have
H(0) = 0 · G (c0 (0)) − G (c1 (0)) = −G (0) = 0,
H(1) = G (c0 (1)) − G (c1 (1)) = G (1) − G (0) = G (1) > 0.
48
Consider now the value of ∂H (y) /∂y at y = 0. Since 0 = c1 (0) = c0 (0) and c01 (0) > 0 we have
∂H(0)
= G (0) − G0 (0) c01 (0) = −G0 (0) c01 (0) < 0.
∂y
Given these three observations (i.e. H (0) = 0, H (1) > 0 and ∂H (0) /∂y < 0), we conclude that
there exists at least one equilibrium.
We now prove the part on uniqueness of equilibrium. Let G (c) = (c/β)a for some a > 0 with
support [0, β]. From equation (51), since the case y = 0 is not interesting and G (c0 (y)) > 0 for
y > 0, it is sufficient to prove that the equation
y=
G (c1 (y))
G (c0 (y))
(71)
has a unique solution. Since the LHS of (71) is increasing in y, it suffices to show that the RHS
decreases in y. Let h (y) denote the RHS of (71):
a
h (y) = c1 (y)
β
c0 (y)
β
a =
c1 (y)a
c0 (y)a
The derivative of h (y) is
0 (y) a−1
1 (y) a−1
c1 (y) ca0 (y) − aca1 (y) dcdy
c0 (y)
a dcdy
dh (y)
=
2a
dy
c0 (y)
a−1
a−1
ac1 (y) c0 (y) dc1 (y)
dc0 (y)
=
c0 (y) − c1 (y)
.
dy
dy
c2a
0 (y)
Since
2y (2 + y) − (1 + y) (2 − y) ln 1+y
dc1 (y)
1−y
=
,
3
dy
2y (1 + y)
1+y
−2y + (1 + y) ln 1−y
dc0 (y)
=
,
dy
2y 2 (1 + y)
we obtain that
dc1 (y)
dc0 (y)
c0 (y) − c1 (y)
=
dy
dy
= 4y 2 (1 + 2y) + 2y (1 + y) (2 − y) ln
1−y
1−y
+ 1 − y 2 (1 − y) ln2
.
1+y
1+y
This expression is negative for 0 < y < 1, so dh (y) /dy < 0, and therefore, the equilibrium is unique.
49
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