ISSN (Print) 0473-453X

Discussion Paper No. 1024

ISSN (Online) 2435-0982

MULTIPRODUCT FIRMS,
CONSUMER SEARCH,
AND DEMAND HETEROGENEITY

Yuta Kittaka

Revised October 2019
March 2018

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Multiproduct Firms, Consumer Search, and Demand
Heterogeneity
By Yuta KITTAKA∗This study constructs a consumer search model in which someconsumers search for multiple products, whereas others search fora single product. A price diﬀerence arises because of a diﬀerence inthe price elasticity for each group. We show that a positive demand
shock to one of the products decreases the price of another product,
whereas it increases its own price, and a negative correlation between the demands for each product strengthens these tendencies.
Both prices decrease, however, following a positive demand shock
when the demands for each product are positively correlated. We
also show that multiproduct firms set a relatively high price for amore demanded product, as such a product’s price tends to be moreelastic with respect to search costs. A price diﬀerence between products increases as the demand gap between products increases or as
economies of scale in search increase.
Keywords: consumer search, multiproduct pricing, demand heterogeneity
JEL classification: D11, D43, D83, L13

I.

Introduction

Considering consumers’ purchase behavior, multiproduct firms that sell in multiproduct markets must often consider a variety of consumers with diﬀerent
demands. Some consumers will want to purchase a large basket of products,
whereas others will want to purchase only one of the products, i.e., each consumer is heterogeneous in terms of demand. There exists a large literatureinvestigating multiproduct firms’ pricing in the context of diﬀerentiated products and demand-side heterogeneity.1 From another perspective, consumers’∗ Graduate school of economics, Osaka University; mail:kittaka@iser.osaka-u.ac.jp. I would like tothank my supervisor Noriaki Matsushima. I would also like to thank Jidong Zhou for his valuable andhelpful comments, Yu Awaya, Makoto Hanazono, Junichiro Ishida, Shingo Ishiguro, Akira Ishii, Takashi
Shimizu, Akifumi Ishihara, and seminar participants in Kobe, Nagoya, Nanzan, Osaka University andOsaka University of Economics. The article has benefited from comments at the 45th Annual Conferenceof EARIE (Athens), 3rd Conference of APIOC (Melbourne), 2018 JAE spring meeting (Hyogo) and
JAAE spring meeting 2018 (Kyoto). I gratefully acknowledge the financial support from the Japan
Society for the Promotion of Science (JSPS) Grant-in-Aid for JSPS Fellows Grant number 19J10596,
JSPS KAKENHI Grant Number 15H05728, and the Foundation for the Future Young Researchers at
Osaka University.
1 For example, Champsaur and Rochet (1989) and Johnson and Myatt (2003) analyze verticallydiﬀerentiated products with multiple demand segments; Doraszelski and Draganska (2006) investigatesthe segmentation problem in horizontally diﬀerentiated products markets. All of these studies assume
perfect information.

1

2purchase behavior is inherently incurring various costs associated with information gathering, namely search costs. There is a growing literature focusing
on the relationship between consumers’ search behavior and firms’ pricing in asingle-product environment that shows that such search behavior significantly affects firms’ strategy;2 although, only a few studies adopt a multiproduct context(McAfee 1995, Zhou 2014, Rhodes 2014). Furthermore, somewhat surprisingly,
no previous studies focus on the demand-side heterogeneity mentioned above in a
multiproduct environment. The purpose of this study is to reveal how heterogeneity in demand aﬀects multiproduct firms’ pricing, and provide useful implications
in a multiproduct environment.
In our model, we consider a market with two multiproduct firms and two products that are supplied by both firms. There are potentially three groups of consumers: consumers who wish to purchase just one product or the other; andconsumers wishing to purchase both products. Each consumer is initially uninformed about actual prices and her valuation of the product she wants; hence, shemust search to gather product information which incurs a search cost per firm.
Search costs can be diﬀerent between single-product and multiproduct searchers,
and multiproduct searchers do not need to purchase all products in one place.
The firms cannot discriminate prices among consumers. To highlight the demandheterogeneity eﬀect, we mainly assume the two products are symmetric in thesense of the distribution of valuation.
First, we describe the optimal search rules of the three groups and the demand
function for each group, and then derive equilibrium prices for given consumers. A
key factor is that the price elasticity of each group will be diﬀerent because there
exists scale merit of multiproduct search behavior, namely the joint search eﬀect
so termed by Zhou (2014) and economies of scale in search. Hence, the ratio ofconsumers searching for multiple products to consumers searching for just one ofthe products matters the price that firms charge for each product. These eﬀects
provide firms with an extra incentive to lower prices to attract multiproductsearchers. Consequently, a positive demand shock to one of the products (anincrease in the mass of multiproduct searchers) induces firms to lower the price ofthe other product. Not surprisingly, such an increase in demand for one productaﬀects not only the price of the other product but also the own price becausea price reduction for one product induces multiple purchases via the scale meritof multiproduct search. As a result, an increase in demand for one product also
causes an increase in own price resulting in a price diﬀerence between products.
We believe that these results bridge a gap between evidence and theory.3
2 The basic theory of search was developed by Simon (1955), Stigler (1961), Kohn and Shavell (1974),
and Weitzman (1979). The pioneering works of Diamond (1971), Varian (1980), and Stahl (1989) investigate models of price searching. Wolinsky (1986) and Anderson and Renault (1999) construct a modelof search for both actual price and match utility. Recently, in laboratory experiments, Caplin, Dean andMartin (2011) develops a search-theoretic experimental technique and shows that subjects indeed searchsequentially.
3 For example, we often observe shifts in prices that are associated with a change in demand; thatis, countercyclical pricing such as retail prices decreasing during high demand seasons. Indeed, there isa vast body of evidence that shows such demand-changing shifts in prices, as Chevalier, Kashyap andRossi (2003) points out. Nevertheless, in standard price theory and consumer search theory, positive
demand shocks only have either a positive eﬀect or no eﬀect on price. So there was a remaining puzzle.

3

In the latter part of this paper, we consider several extensions that are useful forboth theoretical and empirical analyses: correlation among consumers’ demand,comparison prices and price sensitivity of a popular and a less popular product.
We show that if there is a negative correlation between demands, it strengthens
the impact of a change in demand for one product on the two prices; that is,
the price of one product increases more, while the price of the other productdecreases more. However, a positive correlation negates the eﬀect of an increasing
price associated with an increase in own demand, and such an increase in demandleads to a decrease in both prices. Specifically, more multiple searchers result in
lower prices for both products. We also show that firms charge higher prices forthe popular product, whereas they lower the price for the other product (such as
cross-subsidization pricing). Moreover, the price for a popular product is moresensitive to changes in search costs in many circumstances, which is intuitive andconsistent with standard models of price determination.
The rest of the article proceeds as follows. In the remaining part of this section, we describe both the related theoretical and empirical literature. Section II
presents the basic model of single- and multiproduct searches and gives the optimal pricing rule. Section III analyzes how a change in the proportion of consumers
searching for one of the products (or simply a demand shock to one of the products) aﬀects the prices; introduce correlation between demands; compare prices
and price sensitivity of diﬀerent popularity products. Section IV concludes anddiscusses some important extensions. All proofs and calculations are available in
the appendix.
Related literature

Our model builds on a single-product search model and a multiproduct searchmodel established in the seminal works of Anderson and Renault (1999) and Zhou
(2014), respectively. These papers (and other existing papers on consumer searchtheory) assume that all consumers search for all products in the market. We
combine these models to allow for consumers with diﬀerent demands for productvarieties. More precisely, our idea is similar to that of Zhou (2011), who produced
the working paper version of the work above. A key diﬀerence is that we allow
the demand for each product to be diﬀerent, and investigate how this demandheterogeneity aﬀects firms’ pricing.
There is a growing (but still small) literature that investigates multiproductsearch environments despite most firms selling many products. The pioneering
work of Burdett and Malueg (1981) develops a multiproduct search model inwhich each consumer wants to purchase several products and must discover thevector of prices. They characterize the set of consumers’ acceptable price vectors
as the reservation price frontier. The idea of a reservation frontier in our multiproduct analysis (and that of Zhou (2014)) is similar to their work. McAfee(1995) (and Burdett and Judd (1983)) also studies a multiproduct environment
We show that if the demand for each product are positively correlated, then prices can indeed decrease
in response to demand changes (see Section III-ii (Proposition 2)).

4

in which consumers can discover the vector of prices with a given probability. In
those paper, firms adopt mixed strategies and there are multiple equilibria.
In recent works on multiproduct search, Rhodes (2014) and Rhodes and Zhou
(2019) investigate consumers searching both prices and valuations for multiple
products. Rhodes (2014) studies the relationship between the size of the productrange and prices. He shows that larger stores charge lower prices to attract
consumers having lower product valuations. The methodology of his work is
diﬀerent from that of Zhou (2014), and ours, but his results are similar. Rhodes
and Zhou (2019) examine why small retailers that provide a single product cancoexist with larger multiproduct firms.
Our paper is related to the literature on cross-subsidization strategy; that is,
the strategy of charging a high price for one of the products and charging a low
price for others to compensate for it. Chen and Rey (2012) consider consumers
with homogeneous valuations and both large and small retailers that oﬀer different varieties in a competitive market. They show that a cross-subsidization
strategy will be adopted by the large retailer as an exploitative device. Chen andRey (2019) also investigate cross-subsidization with consumers who are diﬀerent
in terms of search cost. In another related study, Lal and Matutes (1994) examine a multiproduct price-search environment in other ways; that is, using the
Hotelling city model with advertising. They show that all consumers purchaseall products at the same store in equilibrium (one-stop shopping occurs), which
is diﬀerent from our results. They also investigate loss-leader pricing (a strategy in which a very low price is charged (often below marginal cost) to attract
multiple purchases), which is related to our results. We believe that our model
may be complementary to their paper and enhances the explanatory power ofmultiproduct firms’ strategies.
Our paper is also related to the literature on countercyclical pricing as mentioned above (Warner and Barsky 1995, Chevalier, Kashyap and Rossi 2003,
Guler, Misra and Vilcassim 2014). Warner and Barsky (1995) find weekly pricepatterns, such that the mean price decreases on Friday, using retail store data in
Ann Arbor. To examine the patterns, they adopt the idea of economies of scale insearch, which is somewhat similar to our approach, to generate cyclical demand
elasticities. They show that it is optimal for consumers to search during high purchase seasons because of the economies of scale in search, and as a result, lower
prices exist in higher purchase seasons. Chevalier, Kashyap and Rossi (2003) also
examine whether prices fall during high demand seasons using weekly store-level
scanner data of Dominick’s Finer Foods, the second largest supermarket in the
Chicago area. Guler, Misra and Vilcassim (2014) finds that the diﬀerence ofchanges in consumer valuations result in countercyclical pricing. As Zhou (2011)
points out, those paper consider only own-price elasticity. We show that thecross-price elasticity also does matter, and such countercyclical pricing strategies
can be explained well by changes in the number of mass multiproduct searchers(relative to the mass of single-product searchers) in the market. Therefore, weshould take both price elasticities into account in considering multiproduct firms’
pricing.

5

II.

The Model

We base our setting on studies by Anderson and Renault (1999) and Zhou
(2014). There are two firms, I and II, supplying two products, products 1 and 2.
Without loss of generality, we assume that their marginal costs are normalized tozero. We also exclude the possibility of product bundling (both pure and mixed ),so consumers can choose any combination of firms’ products.4 The market size
is normalized to one. Let λi (0 ≤ λi ≤ 1) denote the proportion of consumers
who search for product i, for i = 1, 2.5 Suppose that the consumer demand forthe products is independently distributed; then, λ1 λ2 of consumers search forboth products.6 Firms have perfect information about the mass of consumers.
We assume initially that both λs are independent. We analyses the eﬀects of thecorrelation between λs later.
Initially, each consumer has imperfect information about the actual price andthe utilities of all products, and they sequentially gather information while incurring search cost s. According to Perloﬀ and Salop (1985), and other standard
consumer-search models, we treat both products as diﬀerentiated between firms I
and II, and each consumer has idiosyncratic tastes for each product. We represent
this by assuming that each consumer’s valuation for each product is independently
and randomly drawn from a common cumulative distribution function F, whichis twice diﬀerentiable so it has a common probability density function f and isdefined on the interval [u, ū]. The valuation for product i is denoted by ui for
i = 1, 2 and we call the valuation the match utility. To obtain clear-cut results,
we assume that the two products are neither complements nor substitutes in thesense that each consumer simply obtains the sum of the match utilities when she
searches for both products. We also assume that each consumer has an independent valuation for the two products in the same firm. Additionally, we assume
that each consumer’s valuation is independent across consumers. Therefore, forthe consumers who search for both products, once they go to a firm, they draw
a pair of match values from F (u1 , u2 ) = F (u1 )F (u2 ). The distribution is commonly known. We also impose the regularity condition (the increasing hazardfi (ui )increases in ui for i = 1, 2.
rate condition) such that 1−F
i (ui )We allow search costs to be heterogeneous among groups. Specifically, we allowthat the search cost per product for a multiproduct searcher can be less thanthat for a single-product searcher, which is based on the idea that there will
be economies of scale in search for multiple search behaviors (e.g., saving time,money, etc.). Let sS be the search cost per store for a single-product searcher and
4 For more details, see the discussion in Section IV.
5 We assume that λ is exogenously given to simplify the model and make the analysis more transparent.
This case corresponds to the case such that a consumer who searches only for product i (represented byλi (1 − λj )) will obtain a large basic utility Vi in addition to the realization match utility ui , whereas she
obtains only a small basic utility Vj such that Vj + u¯j − pj < 0; hence, she does not search for information
about product j ab initio. This setting is consistent with many other studies and the assumption of afull covered market.
6 Of course, another market expression can be considered. We adopt this setting in following the previous research, and simplify the notation. In our model, the single-product search model (e.g., Anderson
and Renault (1999)) corresponds to λi = 1 and λj = 0, and the multiproduct search model (e.g., Zhou
(2014)) corresponds to λ1 = λ2 = 1. See the discussion for more details.

6s (= sM ) for a multiproduct searcher. Define δ such that sS = δs (0.5 ≤ δ ≤ 1).7
An interpretation of δ is as follows. If δ = 1, then the search costs per store
are exactly the same among consumers, i.e., a multiproduct searcher can obtain
information for half the cost per product, and therefore, there exists economies
of scale in search for multiproduct searchers. However, for δ = 0.5, the searchcosts per product are the same among consumers. In that sense, δ represents the(relative) measure of economies of scale in search (per product). This is basedon the following ideas. As the main component of a search cost is the trip cost,the search cost per product for multiproduct searchers may be less than it is for
single-product searchers via saving money, time, and so on. In that sense, there
exists economies of scale in search. In this case, δ approaches from 0.5 to 1 as thetrip cost increases relative to the in-store search cost. However, when the main
search cost is the cost for searching inside the store, the search cost per productfor all consumers close to similar. In this case, δ approaches 0.5.
We also include other assumptions, such as no replacement, full covered market,
and free-recall.8 Consumers can memorize information and freely return to firms
they’ve visited to purchase products. In equilibrium, all consumers purchase theproducts they want. These three assumptions are standard in the single-productsearch literature. In the multiproduct search models, however, the free-recall
assumption implies some new features. Once a consumer who is searching for two
products visits a firm, her options are (i) buying both products immediately andending the search activity, or (ii) buying no products and going to another firm.
In this scenario, the other possible option (iii) buying one of the products andthen continuing to search, does not exist because of the free-recall assumption.
We also assume that each consumer’s utility function is linear in that the net
utility from purchasing product i at the price pi is given by (i = 1, 2):ui − pi .
Because this is independent of any search cost, when a consumer purchases bothproducts, her utility is the sum of the individual net utilities.
Each consumer’s optimal search rule follows Kohn and Shavell (1974): in eachperiod of search, a consumer compares the incremental expected benefit of one
more search with the additional search cost. If the former is greater, she goes toanother firm and then compares the observed utilities; otherwise, she immediately
stops and purchases the product (products) at the current firm. We assume thata first observation is inevitable so the search cost for the first observation is sunk,and each consumer visits firms in a random order.
7 This restriction on the parameter range is merely expedient. In general, multiproduct searchers
should search for each product separately if δ < 0.5. But theoretically, δ can take a value less than 0.5
and greater than 1, but such an extension does not aﬀect our conclusions.
8 Of course, in most markets, consumers incur not only a search cost but also a return cost. In addition,
Janssen and Parakhonyak (2014) show that costly recall significantly aﬀects firms’ pricing strategies andconsumers’ behavior. However, considering such a costly recall makes our model more complicated, andmoreover, as we will discuss in Section IV, our main insights are unchanged when we reconsider thisassumption for smaller large costs. Hence, following most of the consumer search literature (including
Zhou (2014)), we adopt this assumption for simplicity and tractability. It is also useful for comparisonof our results with the models based on Wolinsky (1986) and Anderson and Renault (1999).

7

The timing of the game is as follows. In the first stage, firms set prices simultaneously. In the second stage, consumers search according to the optimal search
rule. In this study, we focus on the symmetric equilibrium from the related literature. We adopt the perfect Bayesian equilibrium as a solution concept.
In equilibrium, each firm maximizes its own profit and each consumer maximizes
her own consumer surplus. We assume that each consumer’s belief is consistent in
the equilibrium. We also assume the passive beliefs of consumers: each consumer
does not change her belief, even if she observes a price that diﬀers from her
expectation. To make our analysis interesting, we focus on the case of a relatively
small search cost, which ensures search activity.
Single-product searchFirst, we characterize the single-product search. Consider consumers who
search for only a single product i (i = 1, 2). Recall that λi (1 − λj ) (i ̸= j)represents the proportion of such consumers in the market population. The demand of these consumers is specified as in Anderson and Renault (1999). First,
we define the reservation match utility. Let x represent the first observed value.
Then, the incremental expected benefit from an additional search is given by:∫ u¯i(ui − x)dF (ui ) =

x

∫ u¯i

[1 − F (ui )]dui .

x

For simplicity, following McAfee (1995), we define the following expression.
ζi (x) ≡∫ u¯i[1 − F (ui )]dui , for i = 1, 2.

x

We have the critical value with which an incremental expected benefit and an
additional search cost sS are indiﬀerent for the consumer:
ζi (ûi ) = sS (= δs).
Hereafter, we refer to ûi as the reservation match utility for product i for the
single-product search. Notice that ûi is a decreasing function of both s and δ.
In the following analysis, we focus on the case in which each consumer has arelatively small search cost:(1)sS ≤ s̄S (= δs̄) ≡ ζi (ui ), for i = 1, 2,

i.e., the incremental expected benefit of one more search is at least greater than
the search cost for an additional search even if the consumer drew the lowest
utility. We will state a formal condition in the next subsection. For the moment,
we assume that the inequality above holds.
Now, consider a firm’s incentive to deviate from the equilibrium pricing strategy.
Suppose firms adopt a symmetric equilibrium pricing strategy, but firm I reduces

8

its price by ϵi > 0. Then, the demand for firm I from λi (1 − λj )-type consumers
is given by:(2)

1
DiS (pi ) = [1 − F (ûi − ϵi )][1 + F (ûi )] +2

∫ uˆi −ϵi
F (ui + ϵi )f (ui )dui .
ui

The first term of the RHS represents a stopping probability at firm I. As a consumer searches in a random order and there are two firms, half of the consumers
visit firm I first, and the rest of the consumers who had visited firm II will go
to firm I with probability F (ûi ). If firm I reduces the price by a small value ϵi ,such a deviation induces a purchase with a relatively lower match utility; that is,a purchase by a consumer who observes a lower match utility than her reservation match utility ûi . The second term of the RHS of equation (2) denotes the
expected demand of the remaining (fully informed) consumers who finally decideto buy from firm I.
Multiproduct searchNow, consider λ1 λ2 consumers who want to purchase both products. This partfollows straightforwardly from Zhou (2014). For consistency, we adopt the same
notation as in his model. First, we define the reservation match utilities, which
we denote by the pair of u1 and u2 that satisfy the following equation:
ζ1 (u1 ) + ζ2 (u2 ) = s (= sM ).
To characterize the pair of reservation match utilities, we introduce function
ϕ(u1 ) = u2 , which satisfies:(3)

ζ1 (u1 ) + ζ2 (ϕ(u1 )) = s,and call these pairs of utilities the reservation frontier.
Once the search cost is assumed to be small enough, all consumers searchingfor a single-product will be active. However, for a multiproduct search, such a
discussion is complicated because there exists s′ such that a consumer searchingfor multiproducts will do another search for both products, but will not do so fora single product. To avoid this complexity, we focus on a relatively small searchcost that satisfies the following inequality:(4)s < s̄ ≡ ζi (ui ), for i = 1, 2,

i.e., each consumer searches for only one product. We refer to this inequality(4) as a relatively small search cost condition. Notice that when it holds, it also
ensures search activities for a single-product search because sS = δs(= δsM ) < s̄.
By (4), we can define the minimum reservation match utility ai that satisfies
the following equality.
ζ(ai ) = s, for i = 1, 2,

9and by using the definition of ai , we also obtain two expressions: ϕ(ai ) = u¯j , andϕ(ūi ) = aj . Note that (4) means ai > ui . Figure 1 shows the above expressions
u2

a1

u1 u2

region A(Acceptance set)

f (u1 )=u2

region B(Complement)a2

u1
Figure 1. The optimal stopping rule for the multiproduct searchand definitions (this figure is the same as Zhou (2014)).
We call region A of Figure 1 the Acceptance (Acceptable) set, following Burdett
and Malueg (1981). Region B is the complement of A. Once a consumer observesa pair of utilities, which are an element of Acceptance set A, the consumer stops
and purchases both products immediately because she knows that the incremental
expected benefit from one more search never exceeds the additional search cost.
Suppose now that firm I deviates by ϵ1 and ϵ2 . These deviations move thereservation frontier to the lower left of the figure (in other words, more deviations
lead to more immediate acceptances). There are tradeoﬀs between the benefit
from increasing demand and the loss from the lower prices. Note that unlike thesingle-product model, the deviation of ϵi increases not only demand for producti, but also that for product j, because the reservation frontier moves down.
We use the notations A and B, which denote the region of the acceptance set
and its complement, and give the demand for each product i for firm I as:∫∫
1
M
dF (u) +F (ui + ϵIi )dF (u)Di (p1 , p2 ) =2 AI (ϵ)
B I (ϵ)(5)∫+(1 − F (vi − ϵIi )dF (v),
B II

where u1 and u2 are the match utilities from firm I, v1 and v2 are from firm II, (u) and (v) express (u1 , u2 ), (v1 , v2 ), and A(ϵ) and B(ϵ) are the regions after
deviating (ϵ1 , ϵ2 ), respectively. The first term of the RHS represents the expecteddemand for a consumer who purchases both products immediately. The second

10term represents the expected demand for a consumer who first visits firm I and
decides not to purchase any products, and then goes to firm II, but finally becomesfully informed and returns to firm I, and purchases product i there. The third
term is almost the same as the second term and represents a consumer who visits
firm II the first time.
Equilibrium

We can now derive the first-order condition for firm I when it deviates. We
combine equations (2) and (5) and obtain the profit function for firm I:
2
{
}
1 ∑
(pi − ϵIi ) λi (1 − λj )DiS + λi λj DiM .
π (pi , pj ) =
2
I

(6)

i=1,i̸=j

In the symmetric equilibrium, no firm has an incentive to deviate. Thus, we havethe following Lemma 1.
Lemma 1. In the symmetric equilibrium in which consumers adopt an optimal
search rule and under a relatively small search cost, as in condition (4), theequilibrium prices p1 and p2 are given by the solution to the following equations.
{p1[(1 − λ2 ) f (uˆ1 )(1 − F (uˆ1 )) + 2

]

∫ uˆ1

2

f (u1 ) du1
u1

(7)

[ ∫
]}
∫ u¯2
−1
−1
+λ2 2
f1 (u1 )dF (u) +(1 − F (ϕ (u2 )))f (ϕ (u2 ))f (u2 )du2B

= 1 − λ2 p2

∫ u¯2

a2

(1 − F (u2 ))f (ϕ−1 (u2 ))f (u2 )du2 ,

a2

and,
{
p2

[

(1 − λ1 ) f (uˆ2 )(1 − F (uˆ2 )) + 2

∫ uˆ2

]
2

f (u2 ) du2
u2

(8)

[ ∫
]}
∫ u¯1
+λ1 2
f2 (u2 )dF (u) +(1 − F (ϕ(u1 )))f (u1 )f (ϕ(u1 ))du1B= 1 − λ1 p1

∫ u¯1

a1

(1 − F (u1 ))f (u1 )f (ϕ(u1 ))du1 .

a1where ϕ(u1 ) and ϕ−1 (u2 ) are defined in equation (3).
As Zhou (2014) mentions in his seminal work, it is not easy to show the existenceof an equilibrium. Even if we assume the log-concavity of f (u), we cannot check
whether the profit function (6) is quasi-concave or not. Therefore, we employ

11

a technique developed by Zhou (2014) to check the existence of a symmetricequilibrium. To save space, we show the uniqueness and existence of a symmetricequilibrium in the online appendix, where we show that when δ = 1, and whenthe match utilities are distributed symmetrically and exponentially, there exists
a unique symmetric pure-strategy equilibrium. By following the same procedure,
we may also check the existence of other distributions e.g., uniform, log-normal,
Weibull, Gumbel, logistic, and so on.
III.

Analysis

In the last section, we characterized the symmetric equilibrium. In this section,
we derive the equilibrium prices and analyze how the presence of three groups ofconsumers who are heterogeneous in demand for products aﬀects the equilibrium
prices and profit. First, we begin by analyzing how a change in the proportionof consumers searching for one of the products (or simply a demand shock to
one of the products), say a change in λi , aﬀects the prices of both products. In
that part, we will explain the phenomenon in which both prices fall as demand
increases (countercyclical pricing). Then we introduce correlation (both negativeand positive) between the λs and check how the correlation aﬀects the pricing.
Second, we compare the prices when there is a diﬀerence between the two λs (i.e.,
diﬀerences in popularity), and also compare the price sensitivities after a change
in search cost. Finally, we analyze how a change in λ aﬀects each firm’s profit,
by using some examples with specific distributions of match utilities.
We first note that each product’s price tends to be lower in the multiproduct search environment than in the single-product search environment because of
the presence of the joint search eﬀect (and implicitly, the presence of economies
of scale in search) as mentioned by Zhou (2011). However, the mechanism ofprice adjustment and the characteristics of prices relating to demand in the multiproduct environment are still unclear. That is why we first investigate how ademand shock (a change in λ) aﬀects prices. Hereafter, we refer to a demand
shock to product i simply as a change in λi . There are many examples of positive or negative demand shocks. For example, suppose a new indirect competitor
supplying one of the products appears outside of the market. A fraction of consumers may now purchase that product outside of the market for many reasons,so the demand for the product would fall in the market. Then, how does such a(negative) demand shock aﬀect both prices? In our model, a decrease in λi reflects such a negative demand shock. As a second example, we often observe that
consumers’ shopping patterns change over time and seasons, such as weekends,
holidays, and vacations. Consumers who usually purchase only a single productmay purchase multiple products during weekends, holidays, and the Christmas
and Easter seasons. Furthermore, in those periods, some consumers who usuallypurchase nothing may also purchase single or multiple products. Such behavior
can be expressed as an increased λi or increase in both λs in our model. Then,
how does such a change in λ aﬀect prices? What about the magnitude of the
impacts on each price?

12

i. Equilibrium prices

First, we derive the symmetric equilibrium prices. To simplify the notation, we
first define the following.
αi ≡ f (ûi )(1 − F (ûi )) + 2
βi ≡

∫ u¯j

∫ uˆi
ui

(1 − F (uj ))f (ϕ

aj

∫

−1

(uj ))f (uj )duj , βj ≡∫ u¯jγi ≡ 2fi (ui )dF (u) +
B

∫ u¯i

(1 − F (ui ))f (ui )f (ϕ(ui ))dui ,ai(1 − F (ϕ−1 (uj )))f (ϕ−1 (uj ))f (uj )duj ,aj∫γj ≡ 2(f (ui ))2 dui (for i = 1.2),∫ u¯ifj (uj )dF (u) +B(1 − F (ϕ(ui )))f (ui )f (ϕ(ui ))dui .

aiNote that all terms above are nonnegative. We first explain what each notation
represents. Notice that αi is a partial derivative of the demand for product i byconsumers searching for a single product with respect to pi , i.e., the marginal
eﬀect of a change in pi on the mass of λi (1 − λj ) consumers. Suppose αi increases
while the other parameters are constant, then it induces firms to reduce pi . In thatsense, αi reflects the eﬀect of a single-product searcher on price i. Hereafter, wecall α the (standard) eﬀect of a single-product search. Similarly, γ is a standard
eﬀect of a multiproduct search. We emphasize that αi and γi can take diﬀerent
values, and the latter becomes larger in general. This is because multiproductsearchers enjoy economies of scale in search (there are diseconomies of scale insearch when δ < 0.5), and therefore the impacts of a product’s price adjustment
on each consumer group will be diﬀerent.9 In addition, there exists anotherimportant factor that arises in the multiproduct search environment, namely thejoint search eﬀect, β, which was introduced by Zhou (2014). It highlights a newfeature of multiproduct search pricing in that an undercutting of price pi induces
multiproduct searchers to purchase not only product i but also product j. In thatsense, undercutting one product’s price boosts the demand for both products, andbecause of the presence of this eﬀect, multiproduct firms have extra incentive to
reduce prices. β represents this tendency.
In the following analysis, we assume that the two distributions of match utilities
are symmetric.10 Hence, slightly abusing the notation, we can express α = αi =αj , and so forth. Then, from Lemma 1, and by using the expressions above, we
obtain the equilibrium prices as follows:
(9)

p∗i =

α(1 − λi ) + γλi − βλjfor i = 1, 2, i ̸= j,
X

9 We note that δ aﬀects only α. As we show in the appendix, α is decreasing in δ, as is û.
10 An asymmetric distribution in the match utilities will cause various interesting phenomena; however,this is inconsistent with our objective. Moreover, we have not proved the existence of such an asymmetric
distribution, so we focus only on the symmetric distribution.

13

(
)
where X ≡ α2 (1 − λi )(1 − λj ) + αγ(λi + λj − 2λi λj ) + λi λj γ 2 − β 2 .
Before moving to the analyses, we describe the following search eﬀect condition:(10)

γ + β > α,i.e., the sum of the eﬀect of a multiproduct search on prices is greater thanthat of a single-product search. We emphasize that this search eﬀect condition
predominantly depends on δ, and more importantly, it easily holds in broader
cases because there are two unique eﬀects of multiproduct search: economies ofscale in search and joint search eﬀect.1112
ii. Eﬀect of a change in λ on prices

According to the discussion above, we now investigate how a demand shock (achange in λ) aﬀects prices. We first show that the eﬀect of a change in λ for oneof the products on each price is not the same. From (9), we obtain the following
lemma.
Lemma 2: Suppose the search cost condition (4) holds, and two products are
symmetric. Then, in the symmetric equilibrium, the partial derivatives with respect to equilibrium prices are given as follows.
(11)

βλj (γ + β − α)Aj
∂pi
=
,
∂λi
X2

(12)

∂pj
{(γ + β − α)(α(1 − λj ) + γλj )Aj }
=−
,
∂λi
X2

where Aj = α(1 − λj ) + λj (γ − β) ≥ 0.13
From Lemma 2 we can see that the signs of each partial derivative depend on theterm (γ + β − α): the search eﬀect condition (10). If (10) holds, the sign of (11) is
positive unless β and λj ̸= 0, and the sign of (12) is negative. If not, the opposite
result holds. This yields the following proposition.
Proposition 1: Suppose the search cost condition (4) and the search eﬀect condition (10) hold, and the two products are symmetric. Then, in the symmetricequilibrium,
11 For example, it holds for any δ and s in the exponential case such as F (u) = 1 − e−u . As anotherexample, it is violated for δ < 0.082 for s̄M in the uniform case such as F (u) = u, but such a δ is
implausible. We can easily verify that (10) holds except for an extreme case in other usual distributions.
12 This condition is not a necessary condition. Some of following results are opposite when (10) does
not hold.
13 See the proof of proposition 1 in the appendix.

14(a) the equilibrium price pj always decreases in λi .
(b) pi increases in λi unless both β and λj equal zero. If not, pi does not change
in λi .
If (10) does not hold, the directions of change are opposite.
The interpretation of Proposition 1 is as follows. Suppose both β and λj do notequal zero. This means that there exist consumers searching for multiproducts,
and undercutting one product price aﬀects such consumers’ behavior via the joint
search eﬀect. When each firm cannot price discriminate among consumers, eachfirm must take all three eﬀects above (α, β, and γ) into account in setting prices.
For the moment, suppose that the search eﬀect condition (10) holds. Then, under a given λj , a positive demand shock to product i (an increase in λi ) resultsin changes of three masses: an increase in the mass of consumers searching forboth products (λi λj ) and those for product i (λi (1 − λj )), whereas the mass of
consumers searching only for product j (λj (1 − λi )) decreases. This means thatthe eﬀect of a multiproduct search relative to the eﬀect of a single-product searchon price j becomes large as long as (10) holds. Thus, an increase in λi induces
firms to reduce pj . However, such a reduction in pj boosts multiple purchases
when the joint search eﬀect exists. Hence, it enables firms to increase pi slightly.
Therefore, an increase in λi causes a decrease in pj first, then an increase in pi ;however, the latter eﬀect disappears when β or λj equals zero. We note that sucha price increase eﬀect is not larger than the price reduction for another product,
which is implied by the result of Zhou (2011) which we mentioned above; thatis, firms can make money back slightly by increasing the price, but they cannot
compensate for all the price reductions (we will confirm this in Proposition 3).
Furthermore, a decrease in δ (meaning a decline in the relative eﬃciency of amultiproduct search) implies that it mitigates the eﬀect of a change in λi on pj(and pi ) because it causes an increase in α. It should be noted that Proposition
1-(a) holds even if β = 0; that is, a decrease in λi always aﬀects (and usually
decreases) pj . This is because of the presence of economies of scale in searchwhich is represented by the diﬀerence between γ and δ, and it always intensifies
price competition unless (10) holds. We also note that if (10) does not hold, thenthe eﬀect of a multiproduct search relative to the eﬀect of a single-product searchon price j becomes small, and therefore the opposite results hold. In summary,the directions and sizes of price adjustments hinge on the relative impacts ofsingle-product search and multiproduct search.
Proposition 1 provides answers to the earlier questions; a decrease in demandfor one of the products enables firms to charge higher prices for the other products
because there is a smaller joint search eﬀect and economies of scale in search;14
14 For one example, consider hypermarket stores such as Walmart, and an indirect competitor suchas Amazon. Amazon started in 1995, with strictly increasing revenue from 1995 to 2017. Although one
might think that the emergence of Amazon would hurt the revenues of physical stores, while specialized
stores did suﬀer severely, Walmart’s revenue increased constantly from 1990 to 2014 (it decreased slightly
in 2015, but rebounded in 2017). Proposition 1 can provide (at least partly) a plausible answer to15

a positive demand shock for the product strengthens the joint search eﬀect andeconomies of scale in search, and therefore, it induces firms to reduce other product’s price to induce multiple purchases. This implies that at least one of theequilibrium prices must be lower during high demand seasons, which is consistent
with the literature (we consider more related cases in the next part such as how
simultaneous demand shocks aﬀect prices).
These results suggest that multiproduct search may play an important role inmultiproduct firms’ price adjustments, and therefore, we should take all of thoseeﬀects into account in a multiproduct environment.

Now we introduce the correlation between the two λs, i.e., a change in the
demand for one of the products occurs along with a change in the demand for another product. Such a correlation can be either positive or negative. For example,
when consumers face a constraint of some kind (e.g., budget, time, baggage, etc.),
the λs should be correlated negatively. As in the previous example, consumers
tend to purchase more products in some seasons (e.g. weekends; holidays; vacations) than in other seasons. In this case, demand may be positively correlated.
When there is correlation between the λs, Lemma 2 must be modified as follows.
Lemma 2’: Suppose two products are symmetric and the two λs are correlated.
Then, in the symmetric equilibrium under the search cost condition (4),

(13)

(14)

∂p∗i
∂λi

∂p∗j
∂λi

=

)
(
( )
∂λ
(γ + β − α) βλj Aj − ∂λji (α(1 − λi ) + γλi )Ai

=−

X2

,

(
( )
)
∂λ
(γ + β − α) (α(1 − λj ) + γλj )Aj − ∂λji βλi Ai
X2.

Notice that all the expressions above except for the second terms in the large
parentheses are the same as in Lemma 2. However, (13) no longer equals zero for
β = 0, unlike (11). Note that the second term in the large parenthesis of (13) isalmost the same as the first term of it in (14). This means that when there iscorrelation between the λs, a change in λi now aﬀects pi not only via the joint
search eﬀect but also via a change in λj and vice versa. We can confirm that
this puzzle. In a multiproduct environment, physical stores tend to charge lower prices via the joint
search eﬀect and economies of scale in search than in a single-product environment. Now, a fraction of
consumers purchase a portion of their necessities on Amazon.com, and they purchase few products inphysical stores (they still purchase some products such as food, clothes, and so forth at physical stores
because of various factors, e.g., uncertainty about quality, risk, ability to obtain products immediately).
It reduces the joint search eﬀect, and the equilibrium prices increase. Consequently, multiproduct firms
can maintain constant revenue (while specialized stores suﬀer). Indeed, we confirmed whether or not the
number of stores increased following the emergence of Amazon, which in turn increased Walmart’s profits
by using an AR model and store data from 1978 to 2019 (customer numbers data were not available).
The results showed that the emergence of Amazon had no significant impact on store numbers or on
store growth rates, which implicitly supports our hypothesis.

16both βλA terms above are (generally) smaller than the (α(1 − λ) + γλ)A terms;
that is, the price increase eﬀect is smaller than the price reduction eﬀect for eachequation, as we mentioned in the interpretation of Proposition 1. From Lemma
2’, we obtain the following.
Proposition 2: Suppose the search cost condition (4) and the search eﬀect condition (10) hold, and the two products are symmetric. Suppose also that the two
λs are correlated. Then, in the symmetric equilibrium,(a) a negative correlation strengthens the impacts of a change in λi on both prices,which was shown in Proposition 1.
(b) a positive correlation weakens the price-increasing eﬀect of a change in λi on
pi . If the correlation is not weak, then both prices decrease in λi .
If (10) does not hold, the opposite of (a) holds.
The interpretation of Proposition 2-(a) is as follows. Suppose the λs are negatively correlated and suppose the search eﬀect condition (10) holds. Suppose
also that λi increases. Then, there are two forces that induce firms to reduce
pj . According to Proposition 1-(a), such an increase in λi reduces pj , whereas pi
increases. Moreover, a decrease in λj also aﬀects pj in an opposite way, i.e., it
reduces pj , whereas it increases pi . As a result, firms charge lower pj than thecase with no correlation. Moreover, a decrease in λj induces firms to increase pi
because it mitigates the eﬀect of a multiproduct search (relative to the eﬀect of asingle search) on pi . The latter eﬀect remains when β = 0, and therefore a change
in λi always aﬀects both prices, unlike in Proposition 1. Consequently, a change
in λi induces firms to increase pi and decrease pj , and such price adjustments
are greater than in the case of no correlation. However, Proposition 2-(b) tells
us that it is diﬃcult to obtain clear-cut predictions about price changes when apositive correlation between the λs exists. This is because a change in λi now has
an impact through not only an increase in the mass of consumers searching for
multiproducts, but also through an increase in the mass of consumers searchingfor each product. If the correlation between the λs is suﬃciently weak, then thesigns of the price changes are the same as in Proposition 2-(a), but the priceincrease eﬀect on prices becomes small, because there still exists room for priceincreases via the joint search eﬀect. However, if the correlation is not suﬃciently
weak, firms no longer increase pi as λi increases because the eﬀect of a price reduction relating to multiproduct search dominates. As a result, both products’
prices decrease.15

15 For example, suppose ∂λ /∂λ = 0.1 (weak positive correlation) and s = s̄. Then we can confirmijthat both prices decrease in λi for small λj (λj < 0.55 for any λi ) in the exponential case that we will
specify later. As another example, both prices decrease in λi for λj < 0.18 in the uniform case.

17iii. Popularity

Now consider how the diﬀerence in λ aﬀects the prices. Such a diﬀerence in λcan be interpreted as a diﬀerence in popularity. Thus, we refer to the productwith the higher λ as the more popular product in this subsection. Suppose that
λi > λj (i.e., product i is more popular than product j). Also suppose that thetwo λs are not correlated.16 Without loss of generality, we normalize λi to one,i.e., all consumers want to purchase product i, whereas some of them do not wantproduct j. In that sense, we can interpret λj as representing a measure of theunpopularity of product j relative to product i.
By substituting λi = 1 into (9) for i and j, we obtain the following proposition.
Proposition 3: Suppose the search cost condition (4) and the search eﬀect condition (10) hold, and two products are symmetric. Then, in the symmetric equilibrium,(a) the equilibrium price for the more popular product is relatively high comparedwith that for the less popular product:(15)p∗i > p∗j for λi > λj (i ̸= j),(b) the (relative) price diﬀerence becomes large as λj decreases,(c) the (relative) price diﬀerence becomes large as δ increases.
If (10) does not hold, the opposite result holds.
The interpretation is intuitive. Proposition 3-(a) means that firms tend to charge
a relatively high (low) price for a more (less) popular product in equilibrium. As
we have seen above, each firm must place more focus on consumers searchingfor multiple products when (10) holds. As each firm can adjust two prices tomaximize its own joint profit, it charges a lower price for less popular products
to attract consumers searching for multiple products, whereas it charges higher
prices for more popular products and makes money back from these products.
Proposition 3-(b) shows that firms tend to lower the price of less popular products
as the popularity of such products decreases (i.e., the mass of consumers searchingfor multiple products becomes small). This is because the impact of such a pricereduction (say, reducing pj ) is not large; hence, each firm tends to drive pj down
more to induce multiple purchases, whereas it drives pi up to make money back.
The mechanism of such a price adjustment follows from Proposition 1-(a). Theinterpretation of Proposition 3-(c) is simple, and the mechanism is almost thesame as for Proposition 3-(b). As δ increases, the eﬀect of a multiproduct search
becomes large relative to that of a single-product search. Hence, firms have to
16 Of course, the same arguments of Proposition 2 hold when we introduce the correlation between the

λs.

18

drive pj down to attract multiproduct searchers.
There are various implications for multiproduct firms’ strategies via Proposition
3. For example, we often observe cross-subsidization pricing such that multiproduct firms charge a high price for one product, whereas they charge low prices
for other products. This strategy is often adopted by multiproduct firms e.g.,(fast food) restaurants, supermarkets, and the like. We can partly explain sucha pricing strategy using Proposition 3 and a consumer search-theoretic approach.
Firms can charge marked-up prices because of consumers’ search costs, and in a
multiproduct environment, they tend to charge a relatively high price for popular
products, whereas they charge less for less popular products to stimulate profitable purchases and make money back. The intensity of such pricing depends on
both the size of the mass of multiproduct searchers and the size of the economies
of scale in search. Firms charge a very high price for more popular products andcharge a low price for other products when the mass of multiproduct searchers issmall and when the multiproduct search behavior is suﬃciently beneficial to suchconsumers.17
Now let us consider the sensitivity (the price elasticity) of the two prices to achange in search costs. Search costs may vary for many reasons: psychologicalfactors, technological innovation, physical factors, and so on. As in the case of
prices we described, the eﬀect of such a change in search costs on prices will be
diﬀerent among products.
Before we move on to derive the elasticity, we first discuss the preliminary
results.
(16)∂p∗i⋚ 0 for i = 1.2.
∂sThis is almost the same as Zhou (2014). Recall that our model is basically theconvex combination of his model and Anderson and Renault (1999)’s model, andhence, the same argument about the search costs may hold. We note that ∂α∂s ≤ 0∂γ
18
and ∂s ≤ 0 under a regularity condition. These imply that the standard eﬀectof a single-product search and a multiproduct search on prices will be mitigated ass becomes large, and hence, prices tend to increase as s increases. The intuition is
simple. As each consumer must pay a search cost for gathering information about
products, firms have suﬃciently large market power in terms of pricing. Whensearch costs increase, each consumer becomes more reluctant to search further.
Hence, firms will not lose all consumers, even if they raise prices and firms can
17 In practice, firms sometimes charge lower prices for more popular products in contrast with the result
above. To apply our model to such a counterexample, we should take into account other factors. For
example, introducing product asymmetry is plausible and feasible. As another example, if we considera (positive) correlation between the λs as explained in Proposition 2-(b), then the desired results will
be obtained. We can also consider ex ante information heterogeneity, (e.g., consumers know the match
utility of one of the products because they purchase it frequently), correlation between match utilities(consider complements), and so on. Even with these additional factors, the main insights (and mechanismof price adjustment) of our model remain true.
18 See the proof of Proposition 4 in the appendix.

19

thus charge higher prices as the search cost increases. However, ∂β∂s can change
19
in either direction. This implies that the equilibrium price decreases in s only
20 In
if ∂β
∂s > 0, and such an eﬀect is suﬃciently large relative to the other two.
that case, firms have a greater incentive to reduce prices substantially to attract
multiple purchases.
Let us consider price elasticity further. Let ei be a price elasticity of product i:
(17)

ei =

∂p∗i s· , for i = 1, 2.
∂s p∗iThen, we obtain the following proposition.
Proposition 4: Suppose the search cost condition (4) and the search eﬀect condition (10) hold, and two products are symmetric. Then, in the symmetric equilibrium, the price of the more popular product is more elastic with respect to thesearch cost than the price of the less popular product if and only if,
{
(18)

∂α
−
∂s

}

{
(γ − βλj ) +

∂β
∂s

}

{
(γ + λj (γ − α)) >

∂γ
−
∂s

}
(α − β(1 + λj )).

This shows that the price sensitivity to a change in search costs for each productwill be diﬀerent even if two products are symmetric in the sense of match utility.
Recall that the first and the last sets of curly brackets are nonnegative under aregularity condition, and the first term is nonnegative as proved in Lemma 1.
However, the remaining terms can change in either direction, depending on the
distribution of match utilities and δ. We first present two examples in which (18)
holds.
Uniform example: Suppose the match utilities are uniformly distributed as
F (u) = u on the interval [0, 1]. Also suppose δ = 1. This case corresponds to asituation wherein various consumers with diverse preferences exist, so firms cannot
focus on any specific group, or a situation wherein each consumer evaluates thematch value(s) the same as they do the price(s). We can find many such uniform
examples in practice. Let ∆e denote the diﬀerence between ei and ej . In thiscase, ∆e takes a complicated form. Instead, we present a figure in which ∆e ≥ 0.
The price for a popular product is more elastic with respect to s than the pricefor a less popular product when the match utilities are distributed uniformly.
Exponential example: Suppose the match utilities are exponentially distributed
as F (u) = 1 − e−u on the interval [0, ∞). This case corresponds to the situation
wherein consumers are sensitive to product suitability or there are many lowquality products. In this case, consumers think that it is unlikely they will find abetter product from an additional search in terms of match value; that is, price∫19 ∂β > 0 holds when the density function is decreasing because ∂β = f (a)f (ū) − ū f ′ (u)f (ϕ(u))du.
a∂s∂s
20 Choi, Dai and Kim (2018) present a similar result with advertised prices.

20

Figure 2. Uniform example ∆e is positive for s > 0 when δ = 1.

plays an important role. Then, ∆e is given by:∆e =18(1 − λj )s3> 0, for s > 0, ∀δ.
(6 − s3 )(6 − λj s3 )There are many other types of distributions for which Proposition 4 holds, asin the above examples. Although (18) does not give us clear-cut predictions,
we argue that it may hold in many general cases. From (18), one can see thatthe opposite of (18) holds if: the density function is an increasing function in u( ∂β∂s < 0); the standard eﬀect of a multiproduct search is suﬃciently small relative
to that of a single-product search; and δ is suﬃciently small (α is large enoughrelative to γ). However, as we have mentioned in connection with the search
eﬀect condition, the second and third conditions above are not plausible. Hence,
the opposite result might not be sustained in broader cases; that is, the price ofa popular product tends to be more sensitive to a change in s than it is for a lesspopular product.

21

iv. Profit analysis

In the remaining part, we show in brief how a change in λ aﬀects each firm’s
profit. For simplicity, in the following analysis, we assume that there is no correlation between the λs. Note first that each firm obtains half of the demand in thesymmetric equilibrium. Therefore, each firm’s expected (joint) profit is given by:1∑
pi λi ,
2
2

(19)

π=

i=1

where pi is given by (9).
To calculate the profits, the distributions must be specified. We consider theuniform case and the exponential case. In the following, we assume the search
eﬀect condition (10) holds.
Uniform example: Similar to the last subsection, suppose that the match utilities are distributed uniformly as F (ui ) = ui on the interval [0, 1] for i = 1, 2. In
the uniform case, we can confirm ∂β∂s = 1. By applying this result to the argument∂p∗iabove, it immediately leads to ∂s > 0. We also have:
(20)

∂π
∂π
∂π
> 0,
≥ 0 (= 0 for λi = λj = 1),
> 0.
∂s
∂δ
∂λi

Details are in the appendix. These expressions are quite intuitive, and the intuition is also quite straightforward. As we have mentioned above, an increase in s
makes consumers more reluctant to search further (it mitigates all three eﬀects).
Hence, the incentive for firms to charge higher prices as s increases is always
stronger than the incentive to reduce prices because of the joint search eﬀect. As
a result, prices
and each firm’s profit increase monotonically in s. We note that√α = 2 − 2sδ: that is, an increase in δ reduces the eﬀect of a single-productsearch on price, and therefore such an increase enables firms to drive up prices
as same as s unless all the consumers search both products (λ1 = λ2 = 1). The
third part of (20) arises from a market expansion in λi .
Exponential example: Next, we consider general exponentially distributed
match values. Suppose that the match values are distributed as F (u) = 1 − e−uon the interval [0, ∞). We first note that β = 16 s3 ; hence, ∂β∂s < 0. We also note
that:
(
)
18λj s2 36 + λi λj s6 − 12λi s3
∂p∗i
=−
≤ 0.
∂s
(36 − λi λj s6 )2
Notice that the inequality above holds strictly unless s ̸= 0 and λj ̸= 0. Zhou
(2014) shows that the equilibrium prices decrease in s when λ1 = λ2 = 1 (i.e., thestandard eﬀect of the single-product search is absent), because the marginal joint

22search eﬀect with respect to s on price is greater than the marginal standard eﬀect(of the multiproduct search) on price. Unlike Zhou (2014), our model includes
standard eﬀects that cause firms to charge higher prices (because γ + β > αholds), although the same results hold.21 This example implies that the presenceof the joint search eﬀect plays an important role in multiproduct firms’ pricing.
In particular, this applies for the case in which the density function is decreasingin u.
Now consider the equilibrium profit:22
(21)

(
)
6 3(λ1 + λ2 ) − λ1 λ2 s3.
π(λ1 , λ2 , s) =36 − λ1 λ2 s6

The comparative statics are given as follows.
(22)

∂π
∂π
∂π
< 0,
= 0, and
> 0.
∂s
∂δ
∂λi

The first and second results follow from the argument above. As an increase in sdoes not make consumers reluctant to search further ( ∂α∂s = 0) in the exponential
case, firms cannot increase prices as s increases. However, firms still have an incentive to reduce prices to restrain consumers from leaving and to induce multiple
purchases (an increase in s strengthens this incentive). As a result, each firm’s
profit decreases in s, whereas it is constant in δ. The third one simply arises froma market expansion in λi . The following figure 3 shows how much the joint search
eﬀect aﬀects profit, in the case that λ1 + λ2 = 1 (to fix the market size).23
IV.

Concluding Remarks

In this study, we construct a sequential consumer-search model with diﬀerentiated products where there are three groups of consumers: some consumers search
for multiple products, whereas others search for one of two products. We showthat the diﬀerence in price elasticity between single-product searchers and multiproduct searchers aﬀects firms’ pricing, which arises from two unique eﬀects ofmultiproduct search: joint search eﬀect, and economies of scale in search. Theseeﬀects induce firms to reduce the price more as the mass of multiproduct searchers
increases, and therefore, firms adjust the prices such that they reduce the pricefor one product (in general, a less demanded product) to attract multiple purchases, while they slightly increase the price for another product. If demands are
21 The marginal standard eﬀect of both single-product and multiproduct searches with respect to s are
zero in the exponential case.
22 The reservation match value û can be derived by using,∫ ∞[1 − F (u)]du = e−û = s.
ûNote that to ensure search activity, the search cost must satisfy e0 = 1 ≥ s. We also note that α isindependent of δ, and α = γ = 1. By summarizing these results, we obtain (21).
23 That is, there is negative correlation between the λs. Proposition 2 tells us that the eﬀect of anincrease in λi is strengthened.

23

Figure 3. Profit is decreasing in s, and decreases as the joint search effect increases.

negatively correlated, these tendencies will be strengthened. However, if there ispositive correlation between demands, then both prices will decrease in demand,such as in countercyclical pricing.
Finally, we discuss some important issues.
Another market expression— As we mentioned in the footnote 6, we representa group of consumers by two λs to compare our results with those of previous
studies and to simplify the model. In this setting, a change in one λ representsa change in the mass of each group of single and multiproduct searchers. It is ofcourse possible to treat those searchers as separate groups, and to consider separately the eﬀects on prices of changes in the mass of each group. For example,
we can divide the groups of single and multiple searchers into three independentgroups: λ1 , λ2 , and λ̄. The expression in this paper can be expressed as a specialcase. We emphasize that as we have seen in the main text, the key factor is therelative magnitude of the eﬀects of single-product search (α) and multiproductsearch (γ and β), so our conclusion will be unchanged in the expression.
Costly recall— The free-recall assumption in our model means we did not have toconsider consumers who purchase one of the products at the first store and thencontinue searching for the other product. There are various markets in which
this assumption is justified, but in some cases, it may be necessary to consider
such that a recall is costly. Janssen and Parakhonyak (2014) consider consumers’
behavior and firms’ pricing strategies without the free-recall assumption in the
single-product environment and show that the equilibrium price will be diﬀerent
when each consumer must pay a cost for recalling (in addition, the equilibrium
price is no longer stationary when there are more than two firms). They also show

24

that when both costs are not large, an increase in the returning cost has a similareﬀect to that of increasing the search cost. By applying their results to our model,
it can be anticipated that if both the search cost and returning cost are not large,
and both λs are positive, there exists scope for multiproduct search. Therefore,
our result survives (but the equilibrium prices now depend on both costs). We
emphasize that even though we consider a costly recall, we can treat consumers
who purchase one of the products at the first firm and continue searching for theother one as if they search only for a single product at the second firm. In thatsense, our model is robust, although we adopt a free-recall assumption insteadof costly recall. We predict that when we introduce a costly recall assumption,
it will strengthen the eﬀect of single-product search (α), whereas it weakens thestandard eﬀect of multiproduct search (γ). Hence, the impact of a change indemand on prices will be smaller. However, as we have seen above, our main insights still hold if the eﬀect of the single-product search becomes suﬃciently large
relative to that of the multiproduct search (even if the search eﬀect condition isviolated).
Pure, mixed bundling and market structure— One plausible extension of ourmodel is allowing a bundling strategy. There are two types of bundling strategy: pure bundling strategy such that multiproduct firms sell products only in a
package; and mixed bundling strategy such that firms sell both the package andindividual products. In our environment, mixed bundling is more plausible because a firm adopting a pure bundling strategy loses all the consumers searchingfor a single product, and more importantly, the search intensity of such consumers
is less than that of multiproduct searchers and therefore the opportunity cost will
be large. Despite the plausibility of mixed bundling, such an extension is challenging because it increases the strategy space and makes the problem extremelycomplicated even if we consider a duopoly case. Moreover, we argue that ourmodel can be decomposed into two single-product search parts and a multiproduct search part when we allow firms to adopt mixed bundling, and therefore, Zhou
(2011)’s argument is applicable.
Although pure bundling seems less plausible as we mentioned above, many interesting questions still remain open. If firms cannot discriminate price among
consumer groups and cannot adopt mixed bundling, then we can pose two questions. (i) Do firms have an incentive to adopt pure bundling, and if so, in whatcase? (ii) Is there room for the coexistence of multiproduct firms (generalists)and single-product firms (specialists)? The latter question is also related to theissue of market structure in multiproduct environments.
Advertising and ordered search— There is a growing body of literature that considers advertising and ordered search in search environments, but it mainly focuses
on a single-product environment. In practice, however, we often receive advertisements by multiproduct firms rather than single-product firms. For example,
multiproduct firms often advertise such that some products are sold at very low
prices, and such ads are generally nontargeted. We can modify our model to discuss many aspects of this issue. For example, we show that multiproduct firms’25

pricing hinges on the relative importance of single and multiproduct searchers.
Then, which products the firms should advertise? How diﬀerences in the size of
consumer groups aﬀect advertising performance? Which firm should consumers
visit first? How much product information should they provide? Is there any
possibility of asymmetric equilibria? Can small firms coexist with multiproductfirms as in the example above? What about the relationship between the market
population and coexistence? These questions are interesting from both theoretical and empirical perspectives, and analyses of them would help bridge the gap
between the literature on economics and marketing by providing many insights
into these fields.
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26

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V.

Appendix

Proof of Lemma 1:
The following proof follows Wolinsky (1986) (Anderson and Renault (1999)) andZhou (2014). Let us begin with the former part.
Suppose the search cost condition holds. Let us decompose the first-ordercondition (FOC) of equation (6) into a single-product part and a multiproductpart.24 Equation (8) is obtained by following the same procedure.
24 We can obtain the lemma by diﬀerentiating equation (6) directly.

27

Single-product part: We begin with the former part of equation (7). Let us
focus on product 1. By diﬀerentiating (6) with respect to ϵ1 and evaluating ϵ1 = 0,
we obtain:
{
}
∫ uˆ1
1
∂π
= −λ1 (1 − λ2 )[1 − F (uˆ1 )][1 + F (uˆ1 )] +F (u1 )f (u1 )du1
∂ϵ1
2
u1
{
}
∫ uˆ1
1
f (uˆ1 )[1 + F (uˆ1 )] ++ p1 λ1 (1 − λ2 )f (u1 )2 du1 − F (uˆ1 )f (uˆ1 ) = 0.
2
u1This gives:{p1 λ1 (1 − λ2 )(23)

1
f (uˆ1 )[1 − F (uˆ1 )] +
2

∫ uˆ1

}
2

f (u1 ) du1
u1

1= λ1 (1 − λ2 ) .
2Multiproduct part: Likewise, we can obtain the latter part of (7) by following
the same procedure. Recall that A(ϵ) and B(ϵ) are the regions where firm I
deviates (ϵ1 , ϵ2 ). By using Leibnitz’s rule, the FOC is a half of the following
equation times λ1 λ2 ,∫∫∫−dF (u) + (p1 + p2 )f (u)du2 −F (u1 + ϵ1 )dF (u)
A(ϵ)
C
B(ϵ)
∫
∫
+ p1
f (u1 )dF (u) −[p1 F (u1 + ϵ1 ) + p2 F (u2 + ϵ2 )] f (u)du2
B(ϵ)
C
∫
∫
− [1 − F (v1 − ϵ1 )]dF (v) +p1 f (v1 )dF (v) = 0,
B

B

∫

∫

where C denotes the line integral. We applied Green’s theorem such that∫∂gA ∂x dydx = C gdy. By evaluating ϵ1 = 0, we obtain:

1
(24)
= p1
2

∫

1
f (u1 )dF (u) +2B∫{p1 [1 − F (u1 )] + p2 F [1 − F (u2 )]} f (u)du2 .
C

By multiplying λ1 λ2 and combining equations (V) and (V), we obtain equation(7). We note that the line-integral part can be restated by using the definition of
the reservation frontier as follows:∫∫ u¯1[1 − F (u2 )]f (u)du1 =[1 − F (ϕ(u1 ))]f (u1 )f (ϕ(u2 ))du1 .
C

a1

□

28

Calculation of Lemma 2:
From Lemma 1 and the definitions, and by applying a symmetric assumption, we
can restate the equilibrium price p∗i (i = 1, 2, i ̸= j) as follows.
p∗i =

(25)

α(1 − λi ) + γλi − βλj,X()
where X ≡ α2 (1 − λi )(1 − λj ) + αγ(λi + λj − 2λi λj ) + λi λj γ 2 − β 2 . Notethat a change in λi (or λj ) does not aﬀect both the region of integrals and the
distribution functions directly. By diﬀerentiating pi with respect to λi we obtain:
βλj (γ + β − α)Aj
∂p∗i
=
,
∂λi
X2

(26)

where Aj = α(1 − λj ) + λj (γ − β). By following the same procedure, we obtainthe following partial derivative about p∗j :∂p∗j(γ + β − α)(α(1 − λj ) + γλj )Aj
=−
.
∂λi
X2

(27)

□
Proof of Proposition 1:
Recall that α, β, and γ are all nonnegative. Thus, the sign of ∂p∗i /∂λi depends
on the signs of both (γ + β − α) and Aj . The former one depends on the scale ofδ (it moves in either direction), and hence, we just consider the sign of Aj .
We show that γ > β.25 From the definitions and by applying the symmetry of
distribution, we have:
γ−β >

∫ ū

[1 − F (ϕ(u))]f (ϕ(u))f (u)du −

a

∫ ū
=

∫ ū

[1 − F (u)]f (u)f (ϕ(u))dua[F (u) − F (ϕ(u))]f (ϕ(u))f (u)du.

a

Let u′ solve u = ϕ(u) (hence, it is defined on the interval [a, ū]). Then, the last
expression above can be decomposed into:
∫ ū
(28)
|

u′

∫ u′

[F (u) − F (ϕ(u))]f (ϕ(u))f (u)du .
[F (u) − F (ϕ(u))]f (ϕ(u))f (u)du +
a
{z
} |
{z
}
(i)

(ii)

By changing the integral variable from u to u = ϕ−1 (u) (= ϕ(u)), the second term
25 The following procedure follows straightforwardly from the working paper version of Zhou (2014).

29

can be rewritten as:∫ ūu′(−ϕ′ (u))[F (ϕ(u)) − F (u)]f (u)f (ϕ(u))du.

1−F (t)From the definition, ϕ′ (t) = − 1−F(ϕ(t)) < 0. As ϕ(u) is a decreasing function′in u, and because u solves u = ϕ(u), ϕ(u) < u for u ∈ (u′ , ū). This leads toF (ϕ(u)) < F (u), and −ϕ′ (t) > 0. Hence, we have:∫ ū(i) + (ii) > (i) +u′[F (ϕ(u)) − F (u)]f (u)f (ϕ(u))du = 0.

This result follows from γ > β, and hence, Ai,j > 0 holds. Therefore, the signof (26) is the same as the sign of (γ + β − α), and the sign of (27) is just the
opposite.
□
Proof of Proposition 2:
When the λs are correlated, equation (26) and equation (27) no longer hold.
From (25), we have two new expressions (Lemma 2’):

(29)

(30)

∂p∗i
∂λi

∂p∗j
∂λi

=

(
( )
)
∂λ
(γ + β − α) βλj Aj − ∂λji Ai (α(1 − λi ) + γλi )

=−

X2

.

)
(
( )
∂λ
(γ + β − α) Aj (α(1 − λj ) + γλj ) − ∂λji βλi Ai
X2.

By applying similar arguments that are proved above, the proof of Proposition 2
is completed.
□
Proof of Proposition 3:
Suppose λi = 1. Then, from (25), we have:p∗i =γ − βλj,αγ(1 − λj ) + λj (γ 2 − β 2 )p∗j =α(1 − λj ) − β + λ2 γ.
αγ(1 − λj ) + λj (γ 2 − β 2 )Let ∆p denote the diﬀerence between p∗i and p∗j :(31)∆p ≡ p∗i − p∗j =(1 − λj )(γ + β − α).
αγ(1 − λj ) + λj (γ 2 − β 2 )As γ > β, the sign depends on (γ + β − α). This completes Proposition 3-(a).

30Let us consider Proposition 3-(b). By diﬀerentiating (31) with respect to λj ,we have:∂∆p(γ 2 − β 2 )(γ + β − α)
< 0,
=−
∂λj
(αγ(1 − λj ) + λj (γ 2 − β 2 ))2which means that the (relative) price diﬀerence becomes small in λj .
Now we move on to Proposition 3-(c). Notice that (31) becomes large as αdecreases. As a change in δ aﬀects only α via û, we have:
(
)
∂α
dû ′
2
=
f (û)[1 − F (û)] + f (û) .
∂δdδ
From the definition of the reservation match utility, û = ζ −1 (δss ). By using theformula for the derivative of the inverse function, we obtain:
dδ
d
=
=
dû
dû
dû
dδ
1

( ∫ ū
)1
1[1 − F (u)]du = − [1 − F (û)].
s û
sThus, we have:
(
)
∂α
s
′
2
=−
f (û)[1 − F (û)] + f (û) < 0.
∂δ
1 − F (û)
The second inequality is given by applying the regularity condition. Therefore,
the (relative) price diﬀerence becomes large as δ increases.
□
Proof of Proposition 4:
We first define the diﬀerence between the two elasticities ∆e ≡ ei − ej . Then,from (17), we have:∆e =s(1 − λj )

{(

∂β
∂s

)

(γ + λj (γ − α)) +

(

∂γ
∂s

)

(α − β(1 + λj )) −( ∂α )∂s}(γ − βλj )(γ − βλj ) ((γ − α)λj + α − β)As the denominator is positive, ∆e becomes positive if and only if the expression
in the curly brackets above is positive:{∆e > 0 ⇔

∂α
−
∂s

}

{

∂β
(γ − βλj ) +
∂s

}

{

∂γ
(γ + λj (γ − α)) +∂s}(α − β(1 + λj )) > 0.

We note:∫ ū∂αΛ(ϕ(u))
Λ(û)
∂γ
=−
,
=−
dF (u),∂s
1 − F (û) ∂s1−F (ϕ(u))a∫ ū∂βf ′ (u)f (ϕ(u))du,= f (a)f (ū) −
∂s
a

.

31

where Λ(x) = f (x)2 + f ′ (x)[1 − F (x)]. Under a regularity condition, Λ(·) > 0,∂γ
and hence, both ∂α∂s and ∂s are nonpositive.
□
Calculation of examples in Section III-iv:
Suppose that the match utilities are distributed uniformly as F (ui ) = ui on theinterval [0, 1] for i = 1, 2. The reservation match value û is given by:∫ 1[1 − F (u)]du = δsS ⇔ û = 1 −√2δsS .

ûTo ensure the validity of the search activity, s ≤ 0.5. Then, the profit of eachfirm is given by:
(32)

πS =

α(λ1 + λ2 ) + λ21 (γ − α) + λ22 (γ − α) − 2βλ2.
2 (α2 (1 − λ1 )(1 − λ2 ) + αγ(λ1 + λ2 − 2λ1 λ2 ) + λ1 λ2 (γ 2 − β 2 ))√From the definitions, α = 2 − 2sδ and β = s. Now we consider the region of
acceptance set A and the complement B,√to derive γ. In this case, the acceptancearea A is a quarter-circle with radius 2s. It follows that γ = 2 − 21 πs. By
substituting these results into (32), we obtain each firm’s (joint) profit function.
Unfortunately, such an expression is too complicated to characterize using comparative statics. Instead of mathematical calculation, we just show the numericalresults of the partial derivatives (32) with respect to s and δ for δ = 1, s = 0.5,
and λi in the following figures.

32

Figure 4. The partial derivative w.r.t. s is positive, and reaches its maximum when one λ = 1
and the other λ equals zero.

Figure 5. The partial derivative w.r.t. δ behaves almost the same as that of s (it reaches
zero as both λs approach 1).

33

Figure 6. The partial derivative w.r.t. λ1 is positive but decreases in λ1 when λ2 is large.