ESSAYS ON PRICE DISCRIMINATION AND PRICE DISPERSION
By
Yeonjei Jung

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Economics – Doctor of Philosophy
2014

ABSTRACT
ESSAYS ON PRICE DISCRIMINATION AND PRICE DISPERSION
By
Yeonjei Jung
This dissertation consists of three chapters on price discrimination. The first chapter, “Price discrimination through joint marketing”, studies joint marketing arrangements by
competing firms who engage in price discrimination between consumers who patronize
only one firm (single purchasing) and those who purchase from both competitors (bundle purchasers). Two types of joint marketing are considered. Firms either commit to a
component-price that applies to bundle-purchasers and then firms set stand-alone prices
for single purchasers; or firms commit to a rebate off their stand alone price that will be
applied to bundle-purchasers, and then firms set their stand alone prices. Both methods
allow firms to raise prices and earn higher profits. However, the effect of price discrimination on social welfare depends on how prices are chosen. The rebate joint marketing
scheme increases joint purchasing, whereas bundle pricing diminishes bundle purchases.
So if the marginal social value of a bundle over a single purchase is large, the former increases total welfare. However, welfare can also increase with bundle pricing compared
to non-discriminatory pricing.
In the second chapter, “On the strategic choice of add-on pricing policies”, we use an analytical model to examine the consequences of add-on pricing when firms are both horizontally and vertically differentiated and there is a segment of consumers who are unaware of the add-on fees at the time of initial purchase. We find that consumers who
know about the add-on fees can be penalized by the existence of those who do not know.
Our consideration of quality differences on base good and add-ons leads to several novel
findings regarding firm profits. We then explore the case in which sellers can decide to
provide the complementary good for free. It is shown that there exist an incentive not to

charge separate add-on fees that stems from differences in add-on profitability and that is
dependent on parameter constellations whether a more profitable or a less profitable firm
will want to bundle or not.
In the last chapter, “Consumer search with price competition”, we consider model of consumer search with differentiated products, where consumers can readily observe price
and an observable characteristic, but have to incur search costs to learn about a hidden
characteristic. In this set-up we find that equilibrium prices are decreasing in search costs.
As search costs increase, it becomes more important for firms to attract consumers on their
first visit, and hence they charge a lower price.

To my beloved wife, Yumi,
and my son, Sihyun

iv

ACKNOWLEDGMENTS

I would never have been able to finish my doctoral studies without the guidance of
my committee members, help from friends, and support from family.
First and foremost I am truly indebted to my advisor, Jay Pil Choi, for his invaluable guidance and support throughout the writing of this dissertation. This dissertation
would not have been completed without his help and encouragement in every step. I
also would like to thank, one of my committee member, Thomas D. Jeitschko for his continuous support and inspiration. Hours of conversation with him were both helpful and
memorable. Without their support, this dissertation would have been literally impossible. I also would like to express my special thanks to Carl Davidson and Steven Wildman
for all the detailed and helpful comments in the completion of this dissertation.
I also want to express my special thank to Jaesoo Kim for his valuable comments and
encouragements. Thanks also go out to my colleagues, Se Hoon Bang, Sangjun Lee, Sanghyun Kim, Kyungmin Kim, Kyoungbo Sim and many others for their discussion and
comments on my work.
Last but not least, I owe my deepest gratitude to my family. Especially, I would like to
thank my wife, Yumi for her support, encouragement, and unwavering love. My son, Sihyun, I cannot imagine my life without you and thank you for giving me great happiness
always.

v

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

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1
3
5
6
10
12
13
16
20
20
22
24
28
30
38

CHAPTER 2
ON THE STRATEGIC CHOICE OF ADD-ON PRICE POLICIES .
2.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 RELATED LITERATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 THE MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Consumers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Timing of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Symmetric Firms and Boundedly Rational Consumers . . . . . . . .
2.4.2 Solving the Model When Only One Firm Charges an Add-On Price .
2.5 ASYMMETRIC QUALITIES . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Equilibrium Pricing Strategies . . . . . . . . . . . . . . . . . . . . . . .
2.6 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41
41
44
48
48
49
51
52
54
57
60
64
67
69
80

CHAPTER 1
PRICE DISCRIMINATION THROUGH JOINT MARKETING
1.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 A HOTELLING MODEL WITH JOINT PURCHASES . . . . . . . . . . . .
1.3.1 Joint Purchasing and Uniform Pricing . . . . . . . . . . . . . . . . .
1.3.2 Joint Purchasing and Independent Price Discrimination . . . . . . .
1.4 JOINT MARKETING AND PRICE DISCRIMINATION . . . . . . . . . . .
1.4.1 Bundle Price Marketing . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.2 Rebate Marketing . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 WELFARE ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.1 Profit Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.2 Consumer Surplus . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.3 Total Welfare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER 3
CONSUMER SEARCH WITH PRICE COMPETITION
3.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 THE MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

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85
85
87
89

3.4 THE CASE OF HIDDEN PRICES
3.5 EXTENSION . . . . . . . . . . . .
3.6 CONCLUSION . . . . . . . . . .
REFERENCES . . . . . . . . . . . . . .

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vii

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95
98
99
101

LIST OF TABLES

Table 2.1

Summary of Firm’s payoffs . . . . . . . . . . . . . . . . . . . . . . . . . 59

viii

LIST OF FIGURES

Figure 1.1 The locations of the marginal consumers . . . . . . . . . . . . . . . . .

7

Figure 1.2 Firm i’s inverse demand function . . . . . . . . . . . . . . . . . . . . .

8

Figure 1.3 Best response correspondences (here: v1 > v2 )

9

. . . . . . . . . . . . .

Figure 1.4 Social Welfare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Figure 1.5 Firm i’s best response correspondence . . . . . . . . . . . . . . . . . . 32
Figure 1.6 Equilibrium pattern under uniform pricing . . . . . . . . . . . . . . . 35
Figure 2.1 Sequence of Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

ix

CHAPTER 1
PRICE DISCRIMINATION THROUGH
JOINT MARKETING
1.1

INTRODUCTION

Joint marketing arrangements involving separate firms in which customers are charged
differentially when they patronize multiple firms are not new. Thus, the unilateral cancellation of a joint marketing agreement that involved discounted pricing to skiers who
bought passes to ski resorts run by separate operators gave rise to claims of illegal refusal
to deal in the famed Aspen Skiing antitrust suit.1 In recent years similar joint marketing
arrangements involving separate firms have been on the rise. CityPASS, for example, is
a package that bundles multiple tourist attractions in nine popular destinations in North
America. The package for Chicago allows admission to five attractions: the Shedd Aquarium, the Field Museum, Skydeck Chicago, either of the Adler Planetarium or the Art Institute of Chicago, and either of the John Hancock Observatory or the Museum of Science
and Industry. The attractions compete against each other for time-constrained travelers
who cannot visit more than a few places. At the same time, the participating venues offer
discounted pricing by selling the CityPASS.2,3
1 Aspen

Skiing Co. v. Aspen Highlands Skiing Corp., 472 U.S. 585, (1985).
of April 2014, the total price for visiting five attractions is $187.95, while the discounted price
with CityPASS is $93.95. Also, the CityPASS is valid for 9 days following the first use. See ❤tt♣✿✴✴✇✇✇✳
❝✐t②♣❛ss✳❝♦♠✴❝❤✐❝❛❣♦ for more information.
3 Other examples cover tied promotions and other joint marketing efforts, including entertainment
venues, such as movie theaters, restaurants, or museums, and tour-operators giving reciprocal discounts
involving potential competitors; or newspapers jointly marketed under the Newspaper Preservation Act
of 1970. Also, a very similar arrangement to the one in the Aspen case is currently in place for multiple ski
resorts across four countries that are tied together through joint marketing of the epic-pass.
2 As

1

In this paper we investigate the incentives and welfare implications of instances in
which companies choose pricing strategies that target consumers who make joint purchases across firms. The firms offer horizontally differentiated products and consumers
view the firms’ goods as imperfect substitutes. However, each firm’s product has unique
features and attributes that give a consumer who has already purchased a unit an added
utility from buying the competing product as well, and so consumers are endogenously
divided into two groups. While some consumers purchase a single product from either
firm, others purchase both products.
We show that firms are able to leverage their joint marketing schemes into higher
prices and higher profits compared to both uniform pricing and independent price discrimination. However, the mechanism through which prices and profits are raised depend on the nature of the joint marketing scheme used. When firms market to joint purchasers by each setting a price for their contribution to the bundle (bundle pricing), firms
commit to a high bundle price, which drives consumers into single-purchasing. This enables the firms to capture more surplus from single-purchasing customers. In contrast, if
joint marketing takes the form of each firm setting a rebate offer that applies to the the
stand-alone price when a consumer makes a joint purchase, a generous rebate is offered.
This draws consumers into joint purchasing. The increased demand for the bundle is
reinforced by charging high stand alone prices, which yields higher profits because the
fixed rebate then applies to a high price.
Welfare is affected differently in the two cases. If incremental values associated with
purchasing a second unit are high, then the rebate scheme which induces more joint
purchasing increases welfare; whereas bundle pricing, which reduces joint purchasing,
tends to decrease welfare. However, there are also more subtle welfare implications of
joint marketing. When a firm lowers its price to bundle-purchasers then this has both a
demand-inducing effect in that some new customers are attracted to the firm; as well as a
demand-shifting effect, as existing customers of the firm obtain the second good in order

2

to secure an effective price reduction on the first good. An implication of the latter effect
is that transportation costs under joint purchasing are partly incurred to obtain a better
price on purchases that take place anyway. Because the lower price is welfare neutral (it
is merely a transfer between the firm and the consumer) whereas the transportation costs
are welfare reducing, there are situations in which joint marketing reduces total welfare
regardless of which scheme is used.
The paper is organized as follows. Section 1.2 relates the paper to the literature on
joint bundling, joint purchase and patent pools. In Section 1.3 we introduce a variation
of the Hotelling location model in which we allow for joint purchasing. We solve for the
equilibrium uniform prices for both single and joint purchasing regimes and also study
independent price discrimination. Section 1.4 contains the two forms of joint marketing:
bundle pricing and bundle rebates; and draws some initial comparisons to uniform pricing and joint price discrimination in terms of prices and consumption patterns. Profits,
consumer surplus, and total welfare are presented in Section 1.5, which then also establishes the main findings between the different welfare and consumption implications of
the pricing schemes. The final section concludes. Detailed derivations and proofs are
relegated to the Appendix .

1.2

LITERATURE REVIEW

There is an extensive literature on bundling for the purpose of price discrimination.4
Early papers in this literature, however, restrict attention to the case where bundle discounts are offered by a multi-product monopolist (e.g., Adams and Yellen, 1976; McAfee
et al., 1989; Armstrong, 1996; Rochet and Choné, 1998), rather than independent firms.
Gans and King (2006) are the first to study the situation where bundle discounts are offered by different firms through joint marketing. In contrast to our setting, the two products sold are independent, yet all consumers must purchase both goods so that there are
4 Armstrong

(2006) and Stole (2007) provide excellent surveys.

3

no single purchasers. They show that the unilateral bundling by the pair of firms against
other firms is profitable, whereas bundle rebates by both pairs of firms do not increase
their profits. At the same time, mutual joint marketing diminishes social welfare substantially. In the setting we consider price discrimination through joint marketing is always
profitable and its impact on social welfare depends on the mechanics of the joint marketing.5
The most closely related paper to ours is Armstrong (2013), who studies incentives to
offer bundled discounts by separate sellers in a very general setup. The main argument
of the paper is that competing firms offer bundle discounts to reduce competition by
mitigating the substitutability of products. In the same line of research, our paper looks
more closely at the difference between choosing prices vs. choosing rebates; and examines
the relationship between total consumption and total welfare for the two cases.
Song et al. (2012) discuss the case of pharmaceutical cocktails. Although the focus
of their analysis is on situations in which firms set the same price for a drug, regardless
of whether it is used in isolation or as part of a cocktail, they also consider situations
where firms can set different prices for the two kinds of uses. They document how a
firm selling treatments for HIV/AIDS set different prices for similar chemical depending
on whether the drug was part of a cocktail or not. They estimate a demand system for
colorectal cancer drugs, where there are at least 12 major drug treatments, 6 of which were
cocktails combining drugs from different firms. Gentzkow (2007) estimates the degree
of complementarity between print and online newspapers. In his illustrative model in
Section 1.A, he supposes that the value of the bundle is the sum of the values of the two
individual products plus a constant term (which could be positive or negative).
There is also a nascent literature on the impact of joint purchases to which our paper contributes. Gabszewicz and Wauthy (2003) analyze how joint purchases affect price
5 Brito and Vasconcelos (Forthcoming) build on the investigation of Gans and King (2006) by considering

firms that produce vertically differentiated goods. Their main insight is that bundled discounts may induce
a decrease in consumer surplus and always induce a reduction in total welfare.

4

competition and show that various types of equilibria arise depending on the value of incremental utilities from joint purchasing. Kim and Serfes (2006) and Anderson et al. (2012)
extend Gabszewicz and Wauthy (2003) by investigating joint purchasing in a horizontally
differentiated market. Kim and Serfes (2006) ask under what conditions the “Principle of
Minimum Differentiation” is restored when firms choose their location on the Hotelling
line; and Anderson et al. (2012) find a non-monotonic relationship between equilibrium
prices and qualities under joint purchasing. There, the additional gain by joint purchasing
is valued more by closer consumers so firms have an incentive to sacrifice some sales and
set high prices to prevent joint purchases. In contrast to our work, these papers abstract
from price discrimination as a motivation for joint marketing.6
Somewhat related to joint marketing arrangements are several other recent papers on
joint pricing in the context of patents and patent pooling (e.g., Lerner and Tirole, 2004;
Cheng and Nahm, 2007; Choi, 2010; Rey and Tirole, 2013; Jeitschko and Zhang, 2014).
While this work generally does not consider price discrimination, Lerner and Tirole (2004)
and Rey and Tirole (2013) examine how individually set royalty rates interact with pricing
in patent pools. Our findings shed further light on this issue and provide additional
insights as in our setting we allow for the firms’ contributions to the bundle to asymmetric
and we consider alternative joint marketing structures.

1.3

A HOTELLING MODEL WITH JOINT PURCHASES

We consider an extension of the Hotelling (1929) model. When consumers make a purchase, the intrinsic utility from consuming the product is given by V—regardless of which
firm’s product is purchased. We assume that V is sufficiently large so that all consumers
buy at least one product.7 The model is fairly standard except that each firm’s product
6 Gabszewicz et al. (2001) also consider competition between firms that produce complementary goods.
There, too, multiple equilibria emerge only for intermediate degrees of complementarity.
7 This is not a particularly restrictive assumption; indeed if the added values that each firm provides are
sufficiently high then even with V = 0 the market is covered.

5

has additional idiosyncratic features that are only obtained when purchasing that particular product. Consumers may choose to purchase from both firms—making a joint
purchase—to enjoy the idiosyncratic features of both products. For example, PlayStation
offers free online games, whereas Wii is more family friendly and has more games that
are suitable for children—families who care about both of these attributes may purchase
both systems.
We let vi denote the consumer’s gross added value from using product i’s unique
features. That is, a consumer who purchases product i obtains the base value of having
purchased an item (V) and then the added marginal value of product i’s features (vi ) so
that the total payoff (gross of prices and transportation costs) is V + vi . In contrast, a
consumer who purchases both products receives the base utility and then each product’s
additional marginal value for a total (gross) payoff of V + vi + v j .

1.3.1

Joint Purchasing and Uniform Pricing

When a consumer located at x ∈ [0, 1] purchases product 1 only, her payoff is given by
U1 ( x; p1 ) ≡ V + v1 − tx − p1 , where t denotes the (linear) transportation costs. When she
purchases product 2 only, her payoff is given by U2 ( x; p2 ) ≡ V + v2 − t(1 − x ) − p2 . And
because the common attributes of the product is captured by V, the payoff from a joint
purchase is given by U12 ( p12 ) ≡ V + v1 + v2 − t − p12 , where p12 = p1 + p2 is the price
paid by a consumer who purchases products 1 and 2 together.
Note that as the idiosyncratic value from either of the two products increases, more
consumers undertake a joint purchase—all else equal. This can lead to a corner solution,
in which all consumers purchase both products. Also, if the incremental values of the
idiosyncratic characteristics are too small, then an equilibrium in which some consumers
purchase more than one unit my fail to exist. To rule out these trivial cases we make the
following assumption that holds throughout the paper.
Assumption 1. vi ∈ t − v j , 2t , i, j = 1, 2 and i = j.
6

Letting xi denote a consumer who is indifferent between buying from firm i only and
buying from both firms (see Figure 1.1), the indifference condition Ui ( xi ; pi ) = U12 ( p12 )
provides x1 = 1 − (v2 − p2 )/t and x2 = (v1 − p1 )/t. This implies that the mass of consumers who buy good i only is
ni = 1 −

vj − pj
,
t

i = 1, 2;

and, assuming a positive measure of joint purchasers, the mass of consumers who buy
both goods is given by
n12 = x2 − x1 =

v1 + v2 − p1 − p2
− 1.
t

Firm 1

Firm 2

0

x1

✛

1

x2

✲✛

✲✛

Good 1 only (n1 )

Joint purchase (n12 )

✲

Good 2 only (n2 )

Figure 1.1: The locations of the marginal consumers
This gives each firm’s demand function as

qi ( pi , p j ) =




1 +
2

1
2t ( vi

− v j + p j − pi ) if pi ≥ vi + v j − t − p j ,



 1 ( vi − pi )

(1.1)

if pi ≤ vi + v j − t − p j .

t

Note that demand has an inward kink at vi + v j − t − p j (see Figure 1.2). Above the
kink prices are so high that there are no joint purchasers, yielding the standard Hotelling
model. Below the kink prices are low enough to have joint purchasers. Here consumers
are more price sensitive (demand is flatter), because the value-added of a second unit lies

7

✛

✲✛

Good 1 only

✲✛

Joint purchase

✲

Good 2 only

Figure 1: The location of marginal consumers
pi

vi + v j − t − p j
qi
1
t (t

− vj + pj )

Figure 1.2:
Firm
i’s inverse
demand
Figure
2: Firm
i’s demand
functionfunction
below the value of purchasing the
p2 first unit. Note also that the other firm’s price is irrelevant on this segment and each firm behaves as a monopolist with regard to providing
p2 = v1 + v2 − t − p1

their idiosyncratic value vi —this is because when pricing to the joint-purchasers on the
v +v −t

2
1
margin, there is no business
stealing from the rival.

Given demand, firm i’s profit function is πi ( pi ,Sp j ) = pi qi ( pi , p j ); the Appendix shows
that this yields best responses of
J

φi ( p j ) =




 1 ( t + vi − v j + p j )
2


 1 vi
2

√
p
if pi ≥ pˆ j = ( 2 − 1)1vi + v j − t,
v1 + v2 − t

√
if pi ≤ pˆ j = ( 2 − 1)vi + v j − t.

Figure 3: Partition of the price region according to demand functions

The key implication of the kink in demand is that each firm’s marginal revenue is nonmonotonic, and therefore the first-order condition may be satisfied twice—possibly permitting two different pricing strategies: pricing high to compete for single purchasers
with less elastic demand; or pricing low to attract joint purchasers—whose demand is
more elastic. As a result, firms’ best response
1 correspondences are not continuous. Figure 1.3 depicts the three possible configurations.
The following proposition gives the equilibrium configurations, the proof is in the
Appendix.

√
√
√
√
√
√
Proposition 1. Let ΦS := (v1 , v2 ) | (3 − 2)v1 + 2v2 ≤ 3 2t and (3 − 2)v2 + 2v1 ≤ 3 2t ,
√
√
and Φ J := (v1 , v2 ) | ( 2 + 1)(t − 12 v2 ) ≤ v1 ≤ 2t and ( 2 + 1)(t − 12 v1 ) ≤ v2 ≤ 2t ; then
if
8

p2

p2

BR1
BR2

US
S
( pU
1 , p2 )

BR1

pˆ 2

√

2
2 v2

√

2
2 v2
1
2 v2

1
2 v2

BR2
U

U

( p1 J , p2 J )
pˆ 2

45◦

pˆ 1

45◦

p1
1
2 v1

√

p1
1
2 v1

2
2 v1
S

√

2
2 v1

pˆ 1
J

J

S

(c) when (v1 , v2 ) ∈ Φ \ Φ
(b) Some
Joint Purchasing

(a) when
(v1 , v2Purchasing
) ∈ Φ \Φ
(a) Only
Single

p2

BR1
BR2

US
S
( pU
1 , p2 )

√

2
2 v2

pˆ 2
1
2 v2

U

U

( p1 J , p2 J )

45◦

p1
1
2 v1

√

2
2 v1

S

J

(b) when
1 , v2 ) ∈ Φ ∩ Φ
(c) Both
Eq.(vConfigurations

Figure 1.3: Best response correspondences (here: v1 > v2 )
(i) (v1 , v2 ) ∈ ΦS , there is a single-purchasing regime in which
US

pi
1

US

ni

=

1
= (3t + vi − v j ),
3

3
1
S
(3t + vi − v j ) and nU
=
0,
12
6t
1
U
πi S =
(3t + vi − v j )2 ;
18t

2

9

(ii) (v1 , v2 ) ∈ Φ J , there is a regime with joint purchases in which
U

pi J =
U

ni J =

1
v
2 i

1
(2t − v j )
2t

U

1
(v + v j )
2 i
1
= (v1 + v2 − 2t),
2t

and

p12J =

and

n12J

UJ

πi

U

=

v2i
.
4t

When the additional gain from a second purchase is small, the firms are better off not
attracting joint purchasers and instead charge high prices to single purchasers ((v1 , v2 ) ∈

ΦS ). On the other hand, when a joint purchase adds a large additional gain, secondary

customers are an attractive prospect and the firms are willing to lower their prices to
capture these consumers ((v1 , v2 ) ∈ Φ J \ ΦS ).8 Assuming that both firms can coordinate
to achieve an equilibrium that gives higher profits for both of them, we maintain that the
single purchasing equilibrium configuration prevails whenever (vi , v j ) ∈ ΦS ∩ Φ J .

1.3.2

Joint Purchasing and Independent Price Discrimination

Customers who are located close to a firm purchase from that firm; and those far removed
from this firm will not purchase from the firm. Thus, the purchase decision reveals something about the consumer’s location on the line. Firms can use this information to segment the market and price discriminate even when they do not engage in joint marketing
efforts. This scenario yields the second benchmark.
Specifically, suppose that firms provide a rebate ρi off their price to consumers who
buy the rival’s good—i.e., who purchase both goods. The joint purchaser’s price is p12 =

( p1 − ρ1 ) + ( p2 − ρ2 ), yielding a net payoff of U12 ( p12 ) = V + v1 + v2 − t − ( p1 − ρ1 ) −
( p2 − ρ2 ).
The locations of the marginal consumers follow from Ui ( xi ; pi ) = U12 ( p12 ), implying
that ni = 1 − (v j − p j + ρi + ρ j )/t and n12 = (v1 + v2 − p1 − p2 + 2ρ1 + 2ρ2 )/t − 1.
8 Figure

1.6 in the Appendix illustrates the bounds of the relevant regions.

10

Firm i’s profit maximization problem is given by
max πi = pi ni + ( pi − ρi )n12 ,
pi, ρi

(1.2)

and the first-order condition for this problem yields
Proposition 2. Under independent price discrimination,
piIPD =

1
1
1
IPD
(3t + vi − v j ), ρiIPD = (2t − v j ) and p12
= (2t + v1 + v2 )
3
3
3
1
1
IPD
niIPD = (2t − v j ) and n12
= ( v1 + v2 − t ),
3t
3t
1
πiIPD = (v2i + v2j + 2tvi − 4tv j + 5t2 ).
9t

Note that firm i’s choice of the bundle rebate is independent of vi and is decreasing
in v j , implying that as firm j’s unique features are more valuable, firm i can reduce the
bundle rebate, as the rival product’s attractiveness serves to increase the demand.
Compared to uniform pricing, the stand alone price under independent price discrimination is identical to the equilibrium price in the single purchasing regime. That is, the
price charged to the relatively inelastic (i.e., the ‘captured’) consumers is the same as
when only these are targeted. This follows readily, since the margin on which this price
operates is the same across the two cases.
On the other hand, the price paid by joint-purchasers under uniform pricing is lower
U

than the price paid by joint-purchasers under independent price discrimination, i.e., pi J =
IPD . This is because when the firm price discriminates
vi /2 < (2t + v1 + v2 )/3 = p12

and lowers its price to joint purchasers, it cannibalizes its high-profit sales to inelastic
consumers who otherwise single-purchase—this limits the amount the firm is willing to
lower the price to joint purchasers compared to the case of uniformly low prices (that is,
ni is decreasing in the rebate ρi ).
An implication of consumers being able to obtain rebates upon purchasing the second

11

product is that despite the bundle price being higher than the uniform price under joint
purchasing, the mass of consumers who joint-purchase increases when the firms price
U

IPD = ( v + v − t ) /3t to n J = ( v + v −
discriminate. This can be seen by comparing n12
2
2
1
1
12
U

2t)/2t and n12S = 0. That is, the high stand alone price pushes people into making a joint
purchase in order to obtain the price break.
Thus, independent price discrimination leads to (some) joint purchasing and increased
profits when otherwise there would only be single purchasing under uniform pricing.
Moreover, independent price discrimination also raises profits when under uniform pricing there are some joint purchasers, because both single purchasers and joint-purchasers
now face higher prices, and there are more purchases in total as some consumers shift
from single to joint purchasing to avoid the higher non-discounted price that single purchasers face.
This result is similar to one in Armstrong (2013, Proposition 3). He shows that if there
are separate sellers, then each seller has an unilateral incentive to offer bundle discounts
for consumers who buy goods from both sellers whenever the demand for the bundle
product is more elastic than total demand for one firm’s product.

1.4

JOINT MARKETING AND PRICE DISCRIMINATION

Consider now joint marketing. Firms act non-cooperatively in their pricing decisions
so joint marketing does not take the form of price collusion. Joint marketing is done
in two stages, with the firms first committing to their pricing strategies vis-á-vis joint
purchasers, and then setting their stand alone prices in light of the offer that is made to
joint purchasers.
We consider two types of joint marketing schemes. First we suppose that firms noncooperatively set the price for their product in the bundle, pi , and then choose stand-alone
P ≡ p + p . The second scheme we consider is
prices pi in light of the bundle price of p12
2
1

a rebate scheme in which firms first announce a rebate offer ρi , and then choose the stand
12

alone price in light of the bundle rebate of ρ := ρi + ρ j .9

1.4.1

Bundle Price Marketing

With bundle pricing, each firm first chooses the price pi for its contribution to the bundle.
P ≡ p + p that will be marketed to consumers. The firms’
This yields the bundle price p12
2
1

stand-alone prices pi are then set in light of this bundle price. Given the choice of prices,
consumers make their purchasing decisions, with a firm receiving pi for each bundle that
is sold, and obtaining pi from consumers that purchase only firm i’s product.
P = p + p , a joint purchaser’s net payoff is U ( p P ) ≡
Facing a bundle price of p12
2
1
12 12

P , resulting in n P = 1 − ( v − p P + p ) /t and n P = ( v + v + p +
V + v1 + v2 − t − p12
2
1
1
j
i
12
12
i

P ) /t − 1. We assume that n P ≥ 0, which yields the candidate pricing structure.
p2 − 2p12
12

We then confirm the conditions on the candidate pricing structure that entails a positive
measure of joint purchasers.
P , each firm chooses p to maximize
Given the bundle price p12
i

P
max πiP = pi niP + pi n12
.
pi

From the first-order conditions, each firm’s stand-alone price is:
pi ( pi , p j ) =

t − v j + 2pi + p j
.
2

Note that pi is increasing in pi and p j . Thus, commitment to the component price leads
to raising the stand-alone price. Moreover, pi rises with t and falls with v j , because an
increase in t and a decrease in v j make demand for firm i’s product less elastic.
In anticipation of the stand alone prices as a function of the bundle price, the firms
9 Caminal

and Matutes (1990) make a similar distinction between prices and rebates. In their model,
however, customers are distinguished by purchasing across periods.

13

price their individual contribution to the bundle:
max πiP =
pi

t − v j + 2pi + p j t − v j + p j
vi + v j − pi − p j
·
+ pi ·
.
2
2t
2t

Joint purchases take place provided that the idiosyncratic values of the two firms are
sufficiently high, yielding the following equilibrium.
Proposition 3. When vi + v j ≥ 2t, there exits a joint marketing equilibrium in which firms set
bundle component prices with
piP =

1
1
1
P
(5t + 2vi − v j ), pi = (t + vi ) and p12
= (2t + v1 + v2 ),
4
2
2
1
1
P
niP = (3t − v j ) and n12
= (v1 + v2 − 2t)
4t
4t
1
πiP =
11t2 + 2(2vi − 3v j )t + (2v2i + v2j ) .
16t

Compared to the case of uniform pricing with joint purchasers the price paid by joint
U

P > p J ; and so is the price paid by single purchasers
purchasers is now higher, i.e., p12
12
U

when compared to the case of uniform pricing with only single purchasing, piP > pi S ,
US

(which, of course, is higher than the uniform joint purchasing price, i.e., pi

U

> pi J ).

That is, when jointly marketing the bundle price, both prices (those applied to single purchasers and those applied to joint purchasers) are above the the uniform prices—despite
all prices being set non-cooperatively.
The fact that the bundle price is higher under joint marketing suggests that fewer consumers purchase the bundle. However, the fact that the stand alone price is also high,
makes the bundle attractive in that it leads to a price discount. The former effect dominates the latter so that fewer joint purchases take place when compared to the case of uniU

P = ( v + v − 2t ) /4t < n J = ( v + v − 2t ) /2t,
form pricing with joint purchases, i.e., n12
2
2
1
1
12

given v1 + v2 ≥ 2t; (naturally, more joint purchases take place when compared to uniform
pricing with no joint purchases).

14

Intuitively, by committing to high bundle component prices the firms push demand
towards single purchasing and use this increased demand to charge higher stand alone
prices. Overall then, profits are greater compared to either of the uniform pricing cases,
U

U

i.e., πiP > πi S , πi J .
Consider now the comparison to independent price discrimination. Writing pi = pi −
ρi , the independent price discrimination optimization problem for firm i, Equation (1.2),
is:
max πiIPD ( pi , p j , pi , p j ).
pi ,pi

The first-order condition with respect to the bundle component price is
∂πiIPD
∂n
∂n
= pi i + n12 + piIPD 12 = 0,
∂ pi
∂ pi
∂ pi

(1.3)

where piIPD = piIPD − ρiIPD .
In contrast, with the joint marketing scheme, firm i’s choice of its bundle component
price is implied by
max πiP piP ( pi , p j ), p Pj ( pi , p j ), pi , p j .
pi

From the envelope theorem, the effect of changes in the stand alone price pi on πiP is of
second order (∂πi /∂pi = 0), so the optimal bundle price component of firm i, pi , satisfies
the following first-order condition:
0
P
❃
✚
∂p✚iP ∂πi ∂p j
∂πi ∂πi ✚
dπi
=
+ ✚
+
d pi
∂ pi ✚
∂pi ∂ pi
∂p j ∂ pi
0



= pi



P

∂n
∂n  ∂p j
∂ni
 ∂n✁✕
+ n12 + pi 12 +  pi ✁✁i + pi 12 
= 0.
∂ pi
∂ pi
∂p j
∂p j
∂ pi

(1.4)

✁

=Direct Effect

=Indirect Effect

The first term captures the direct effect that firm i’s component price has on profits.
It mirrors the first order condition that the firm has in independent price discrimination,
15

namely Equation (1.3). The second term is the indirect effect that firm i’s component
price has on profits by affecting firm j’s stand alone price that is set in light of the bundle
prices. The indirect effect is positive, because committing to a bundle component price
leads to the rival raising its stand-alone price. Evaluating the first-order condition (1.4) at
pi = piIPD yields
dπi
dpi

P

pi = piIPD

∂n ∂p j
= pi 12
∂p j ∂ pi

> 0,
pi = piIPD

which implies that piP > piIPD .
Thus, the firms charge a higher bundle price when they commit to the bundle component prices prior to the stand-alone prices. As a consequence, the mass of joint purchaser
is smaller in the bundle price commitment case than in the case of independent price disP < n IPD . This makes single-purchasing relatively more attractive and
crimination, i.e., n12
12

allows the firms to raise the stand alone price as well, piP > piIPD , and so profits on each
unit sold exceed those of independent price discrimination, leading to an overall increase
in profit, πiP > πiIPD .

1.4.2

Rebate Marketing

In the case of non-cooperative joint rebate marketing, each firm determines a discount or
rebate of ρi that will be applied to their stand alone price to any consumer who purchases
both goods. Together the discounts constitute a rebate offer of ρ := ρi + ρ j that will be
jointly marketed to consumers. The firms’ stand alone prices are then set in light of the
R ≡ p + p − ρ so that firm i receives a net
rebate offer. Joint purchasers pay a price of p12
i
j

price of pi − ρi for its own product in the bundle, and obtains pi from single purchasers.

R) ≡
Given the joint marketing rebate of ρ, a joint purchaser’s net payoff is U12 ( p12

R = (v + v −
V + v1 + v2 − t − ( p1 + p2 − ρ) implying niR = 1 − (v j − p j + ρ)/t and n12
2
1

p1 − p2 + 2ρ)/t − 1, with joint purchasing increasing as the rebate offer ρ increases.

16

In light of the joint marketing rebate ρ, firm i solves
R
max πiR = pi niR + ( pi − ρi ) n12
,
pi

where ρi can be thought of as the cost of having a product in the bundle. The first-order
condition yields the individual firm’s stand-alone price
pi ( ρi , ρ j ) =

vi + 2ρi + ρ j
.
2

To compare with the benchmark case of uniform pricing with joint purchasing, the
U

price can be written as pi (ρi , ρ j ) = pi J + ρi + ρ j /2. This shows the advantage of the
joint marketing rebate scheme: each firm raises the stand-alone price to consumers by
its rebate and by half the rival’s rebate as well. That is, since the firms choose the rebate
before choosing their stand-alone prices, their choice of rebate serves as a commitment to
raise the stand-alone prices.
Given the optimal stand-alone prices as a function of the joint rebate, each firm selects
their partial rebate non-cooperatively:
max πiR =
ρi

vi + 2ρi + ρ j 2t − v j − ρi vi + ρ j vi + v j + ρi + ρ j − 2t
·
+
·
.
2
2t
2
2t

The second-order conditions are satisfied, and the problem has the unique solution.
Proposition 4. When firms undertake a joint marketing arrangement in which a rebate is given
to bundle purchasers,
piR =

1
1
1
R
(6t + vi − 2v j ), ρiR = (2t − v j ) and p12
= (4t + v1 + v2 ),
4
2
4
1
1
R
niR = (2t − v j ) and n12
= ( v1 + v2 ),
4t
4t
1
πiR =
12t2 + 4(vi − 2v j )t + (v2i + 2v2j ) .
16t

17

Compared to the uniform pricing regime with joint purchasing, joint marketing again
R =
leads to an increase in the price for joint purchasers. That is, joint purchasers pay p12
U

(4t + v1 + v2 )/4 > (v1 + v2 )/2 = p12J . And, as was also the case for joint marketing
with bundle component pricing, with a rebate being jointly marketed, the stand alone
prices are higher than they are when a uniform price is charged to single purchasers,
U

piR > pi S . In contrast to the case of joint marketing through bundle pricing, in this case
the higher stand alone price drives purchasers to buy the bundle in order to obtain the
U

R > n J . Once
rebate—despite the effective bundle price being higher than before, so n12
12

again, joint marketing leads to increased profits compared to uniform pricing, that is
U

U

πiR > πi S , πi J .
Turning to the comparison with independent price discrimination, without joint marketing of the rebate, firm i solves
max πiIPD ( pi , p j , ρi , ρ j ).
ρi ,pi

The first-order condition with respect to the individual rebate in this case is
∂πiIPD
∂n
∂n
= pi i + ( pi − ρi ) 12 − n12 = 0.
∂ρi
∂ρi
∂ρi

(1.5)

In contrast, when jointly marketing the bundle rebate, the optimal individual rebate is
derived by solving
max πiR piR (ρi , ρ j ), p Rj (ρi , ρ j ), ρi , ρ j ,
ρi

18

with the following first-order-condition
0
R
❃
✚
∂p✚iR ∂πi ∂p j
∂πi ∂πi ✚
dπi
=
+ ✚
+
dρi
∂ρi ✚
∂pi ∂ρi
∂p j ∂ρi
∂n
∂n
∂n
∂n
= pi i − n12 + ( pi − ρi ) 12 + pi i + ( pi − ρi ) 12
∂ρi
∂ρi
∂p j
∂p j
=Direct Effect

∂p Rj
∂ρi

= 0.

(1.6)

=Indirect Effect

Again, the direct effect reflects the condition obtained in independent price discrimination, Equation (1.5); and the indirect effect captures the effect that one’s own rebate has
on the stand-alone pricing decision of the rival. Evaluating the first order condition (1.5)
at ρi = ρiIPD yields
dπi
dρi

ρi =ρiIPD

=

∂n
∂n
pi i + ( pi − ρi ) 12
∂p j
∂p j

∂p Rj
∂ρi

> 0,
ρi =ρiIPD

implying that ρiR > ρiIPD .
That is, firm i increases its rebate when it commits to jointly marketing the bundle
rebate prior to setting the stand-alone price. The effect of this is to increase the attractiveness of the bundle relative to the stand alone price. For this reason, there are more joint
purchasers when rebates are jointly marketed compared to the case of independent price
R = ( v + v ) /4t > ( v + v − t ) /3t = n IPD .
discrimination, i.e., n12
2
2
1
1
12

However, because firms set stand alone prices only after the rebates have been fixed,
this commitment leads to the ability to increase the stand alone prices, dpiR /dρ j > 0. This
commitment is sufficiently strong so that joint purchasers end up paying more with joint
marketing then with independent price discrimination, despite the former getting higher
R = (4t + v + v ) /4 > (2t + v + v ) /3 = p IPD .
rebates, i.e., p12
2
2
1
1
12

In sum, when the firms jointly market the bundle rebate they give generous rebates,
as this allows very high stand alone prices that drive consumers to joint purchasing, who
despite large rebates are very profitable, because the rebate is taken off very high initial
19

(i.e., stand-alone) prices.

1.5

WELFARE ANALYSIS

We now consider the welfare effects of the various pricing schemes derived. We begin
with profits, then move to consumer surplus, and finally compare total welfare for the
cases analyzed.

1.5.1

Profit Comparisons

Profits are derived in the previous section and are given in the propositions covering
the equilibrium configurations. Here, we discuss their relation across the different pricing schemes. Not surprisingly, firms’ profits are increasing in their ability to engage in
and coordinate price discrimination. Thus, profits are lowest under uniform pricing and
firms do better when they independently price discriminate. And regardless of whether
the firms choose a bundle price or a rebate scheme when they launch a joint marketing strategy, their profits increase over and above what independent price discrimination
achieves. This occurs despite the non-cooperative nature of the agreement, because joint
marketing commits the firms to the bundle strategy—a commitment that is leveraged into
higher prices and profits.
Given that profit comparisons are quite intuitive when it comes to the degree of commitment and coordination, the intriguing question is whether joint marketing through
bundle pricing or through rebates is preferred. While it may be the case that for logistical
reasons one method or the other may not be practical or available, whenever firms can
engage in either scheme it is worth knowing which yields the greater profit.
Comparing Proposition 3 and Proposition 4 shows that the answer to which scheme
dominates is not unambiguously clear. In particular, when firms use a joint marketing
scheme in which the bundle is priced, then the effective price for both the bundle and

20

the stand alone products are higher when compared to the case of a rebate scheme, that
P = p P + p P = ( v + v + 2t ) /2 > ( v + v + 4t ) /4 = p R + p R − ρ = p R , and
is, p12
2
2
1
1
2
2
12
1
1

piP = (5t + 2vi − v j )/4 > (6t + vi − 2v j )/4 = piR .

On the flip side, in the case of rebates the mass of joint purchasers is larger when
R = ( v + v ) /4t > ( v + v −
compared to the bundle price marketing arrangement, n12
2
2
1
1

P , so the rebate scheme leads to a greater sales volume when compared to the
2t)/4t = n12

bundle pricing regime.
Which of the two arrangements is more profitable depends on the size of the idiosyncratic values of the two products. Specifically, note from Proposition 3 and Proposition 4
that πiR − πiP = ((v j − t)2 − v2i )/16t, from which it follows that if the idiosyncratic values
are similar to one-another (close in size), then the bundle pricing is preferred whenever
the idiosyncratic values are sufficiently large.
When the idiosyncratic values differ (are asymmetric), constellations can arise in which
the firm with the smaller v would prefer the rebating scheme whereas the other firm
would prefer to institute a bundle price. This suggests that firms that are very different
may find it difficult to decide on a joint marketing scheme. However, even in these cases
it turns out that the overall profit ranking is unambiguous in favor of bundle pricing.
Thus, when idiosyncratic values are large, joint profits are higher under bundle pricing,
even if firms’ individual idiosyncratic values differ substantially. As a consequence, if
side payments are permissable, then even under non-cooperative joint marketing, firms
can agree on which scheme to employ.
In sum, letting Π denote industry profits, we have the following theorem.
Theorem 1 (Profits under Price Discrimination and Joint Marketing). Firms’ profits are
increasing in the extent of price discrimination in that independent price discrimination increases
profits above uniform pricing, and price discrimination through joint marketing raises profits even
more:
ΠUS , ΠU J ≤ Π IPD < Π R , Π P .
21

Moreover, the bundle pricing scheme is preferred to the rebate scheme whenever the idiosyncratic
values are large enough to implement the bundle price; otherwise the rebate scheme is chosen:
Π P > Π R ⇐⇒ v1 + v2 ≥ 2t.

1.5.2

Consumer Surplus

Let CS X be the level of consumer surplus for X ∈ {US , U J , IPD, P, R}, defined by
x1 X

X

CS =

0

= V

U1 ( x;

+ v1 n1X

p1X )dx

+

x2 X
x1 X

1

X
U12 ( x; p12
)dx +

X
+ (v1 + v2 ) n12

+ v2 n2X

−t

x2 X
n1X
0

U2 ( x; p2X )dx

xdx

X
+ n12

+

n2X
0

xdx

X X
− p1X n1X + p12
n12 + p2X n2X .

The first term is the gross utility from consumption, the second term is the transportation
costs, and the final term is the total expenditures.10
It is natural to conjecture that consumer surplus is directly related to total consumption, because consumers only purchase the second product when the added utility exceeds their transportation costs. In that case consumer surplus would be directly proX . Indeed, this logic applies when comparing uniform pricing with single
portional to n12

purchasing to independent price discrimination. The stand alone price is the same for
both cases, but under independent price discrimination some obtain the lower price and
U

IPD > n S and
more consumers are drawn into purchasing each firm’s product. Thus, n12
12

CS IPD > CSUS .
However, comparing uniform pricing with joint purchasing to independent price discrimination reveals that even though there are more purchases under price discriminaU

IPD > n J , consumer surplus is actually lower, CS IPD < CSU J . In this case some
tion, n12
12
10 Note

U

that x1 US = x2 US so n12S = 0.

22

consumers make the second purchase even if the additional marginal value from consuming the second unit is smaller than the added transportation costs, simply because
the second purchase allows them to take advantage of the lower prices for both the first
and the second purchase. And because the stand alone price is higher with price discrimination, consumer surplus is lower, despite more total purchases.
This dynamic, that consumers base their decision on purchasing a second unit in part
on how this determines the effective price paid on the first unit, is important in understanding the impact of joint marketing on consumer surplus. In particular, as was shown
in Section 1.4, firms use their commitment to the joint marketing scheme to increase overall prices. An unambiguous result of this is that consumer surplus is always lower under a joint marketing scheme compared to independent pricing schemes with or without
price discrimination. That is, CSUS , CSU J , CS IPD > CS P , CS R . And yet, because a joint
marketing rebate scheme increases joint purchases, total consumption is larger under this
R = max n X .
scheme compared to any other: n12
X 12

Finally, the positive relationship between consumption volume and consumer surplus
holds again when comparing the two joint marketing schemes. In particular, consumer
surplus is always larger under a rebate scheme compared to the bundle pricing arrangeR > nP .
ment, CS R > CS P , and so is total consumption: n12
12

The main findings concerning consumer surplus are recapped in the following theorem:
Theorem 2 (Consumer Surplus under Price Discrimination and Joint Marketing). All
forms of price discrimination harm consumers, unless firms engage in independent price discrimination and thereby induce joint purchases, when otherwise consumers only purchase a single good
given uniform pricing, CS IPD > CSUS .
Joint marketing raises prices compared to any of the other settings and as a result leads to lower
consumer surplus even when more is purchased, CSUS , CSU J , CS IPD > CS P , CS R .
Finally, consumers are better off under a rebate joint marketing scheme compared to the bundle
23

pricing scheme, CS R > CS P .

1.5.3

Total Welfare

We take the sum of consumer surplus and industry profits as a measure of social welfare
as usual. Because firms’ costs are normalized to zero, consumer expenditures are equal
to industry profits, so it is immediate that social welfare is the difference between gross
utility and transportation costs:
X

TW = V

+ v1 n1X

X
+ (v1 + v2 ) n12

+ v2 n2X

−t

n1X
0

xdx

X
+ n12

+

n2X
0

xdx ,

(1.7)

again with X ∈ {US , U J , IPD, P, R}.
There are two opposing effects of joint purchases on social welfare. First, more joint
purchases increases social welfare because extra surplus is realized from the consumption
of additional features. In the case of the standard Hotelling model with inelastic consumer
demand, there is no demand creation effect with pricing, as long as the market is covered.
Here, however, the demand for a firm’s product increase when it charges a relatively
lower price for joint purchasers. The first term in (1.7) captures the base surplus associated
with consuming either good, as well as the added gain associated with the idiosyncratic
features of the two goods.
The second effect of joint purchasing is that it increases total transportation costs,
which is captured in the second term in (1.7). Recall from the discussion on consumer
surplus that added surplus through additional consumption need not be enough to offset
the added transportation costs, because the consumption decision is based on the price
difference between joint and single purchasing due to price discrimination. As a result,
the net effect on social welfare depends on the relative size of the surplus creation and
transportation cost effects.
While the interests of firms are often directly opposed to those of consumers, assessing

24

overall welfare requires closer scrutiny in all but one case: compared to uniform pricing
in which consumers only purchase one good, independent price discrimination increases
consumer surplus while also raising firm profits. So total welfare is clearly greater under
independent price discrimination compared to uniform pricing when consumers do not
engage in joint purchasing, TW IPD > TW US .
In contrast, independent price discrimination lowers consumer surplus when compared to uniform pricing with joint purchases. The reason for this is that the increased
surplus from added consumption is more than offset by the higher prices paid to firms
by those who only purchased a single good. This latter effect, however, is a welfareneutral price transfer from consumers to firms. Therefore total welfare is also greater under independent price discrimination compared to uniform pricing with joint purchases,
TW IPD > TW U J .
This result is somewhat reminiscent of welfare effects under third degree price discrimination, in which welfare increases when consumption (output) increases (see Varian,
U

U

IPD > n J > n S = 0. However, note that there is a critical
1985), which is the case here: n12
12
12

difference in our analysis compared to third degree price discrimination: in our model
it is the prices that segment the markets into single and joint purchasers, and consumers
self-select into whether they are single or bundle purchasers. Hence the comparison to
pricing across distinct markets without an arbitrage possibility is not apt in our setting.
Nevertheless, the association between greater consumption and higher surplus also
holds across joint marketing schemes. Thus, total consumption under a rebate scheme is
R > n P , and total welfare is also
greater than total consumption under bundle pricing, n12
12

greater in the former case, TW R > TW P .
A positive association between consumption and total welfare holds more generally,
provided that the products’ idiosyncratic values are sufficiently large and not too asymmetric. Thus,
Theorem 3 (Total Consumption and Total Welfare). Whenever the idiosyncratic values are
25

not too small (23v21 + 23v22 − 90tv1 − 90tv2 + 54t2 + 80v1 v2 > 0) and not too asymmetric,
((v1 − 22t/17)2 + (v2 − 22t/17)2 ≤ 288/289), then increases in total equilibrium consumption
imply increases in total welfare:
X
n12
> nY12 ⇒ TW X > TW Y , with

TW R > TW IPD > TW U J > TW P > TW US .
The area identified in the Theorem 3 is given in Figure 1.4.
v2
2t
TW P − TW US > 0

t
10
17 t

TW R − TW IPD < 0
10
17 t

t

v1
2t

Figure 1.4: Social Welfare
The reason for the existence of a threshold that values must exceed is intuitive. Firms
are balancing their pricing decisions across two margins, namely the margin at which
they capture new customers who otherwise only purchase from the rival, and the margin
at which their own customers decide to become joint purchasers or remain single purchasers. As a result, the firms’ pricing decisions induce transportation costs that need not
be offset by marginal increases in consumer surplus tied to consumption of the second
product. However, greater consumption leads to greater surplus provided that the value
from added consumption is sufficiently high to off-set added transportation costs that are
incurred by consumers whose second purchase is partly driven by the desire to lower
26

their expenditure on their first purchase.
Two aspects of the theorem are particularly noteworthy. First, Theorem 3 demonstrates that total welfare is greatest when firms engage in joint marketing that sets a rebate for bundle purchasers compared to any of the other schemes, so joint marketing
can increase welfare above what firms can do independently. While consumer surplus is
lower under this scheme than when compared to independent price discrimination, the
increase in profits more than offsets the reduction in consumer surplus. However, there
is an important caveat to note here. From the second part of Theorem 1 we know that
when idiosyncratic values are high, firms actually prefer the bundle pricing scheme over
the rebate scheme.
Nonetheless, the finding points in the direction of which type of joint marketing arrangements should be viewed as beneficial—especially as the case of bundle pricing leads
to the lowest mass of joint purchasers, and the lowest welfare of any equilibrium configuration in which there is joint purchasing. Because of this, the comparison between
surplus through added consumption and losses through additional transportation is relatively easy for the case of joint marketing: if a joint marketing agreement leads to greater
consumption then it likely also increases total welfare.
The second noteworthy finding in Theorem 3 is that compared to uniform pricing
with (only) single purchasing, (even) joint marketing through bundle pricing can raise
total welfare. And, thus, the welfare implications of joint marketing are not clear ex ante,
but whenever idiosyncratic values are high and not too asymmetric, there’s a chance that
either type of joint marketing may increase total welfare.
There is, of course, an immediate important corollary to Theorem 3. If the idiosyncratic
values of the products are not that large, then equilibrium consumption need not be positively correlated with equilibrium welfare. The intuition for this is straightforward: if the
added value from consuming a second good is not that large, then pricing schemes that
induce the added purchase may generate less additional surplus than the added trans-

27

portation that the second purchase entails, thus reducing total welfare. In fact, when the
idiosyncratic values are very small, then no pricing scheme generates more total welfare
than simple uniform pricing with single purchasing.
Related to this is the case where idiosyncratic values are large yet very asymmetric.
Adverse welfare effects occur on the margin between joint and single purchasing when it
comes to the relatively lower-valued good when switching from independent price discrimination to bundle joint marketing. That is, bundle joint marketing will induce additional purchases of the relatively lower valued good for which the added transportation
costs are not offset by the added consumption value.

1.6

CONCLUSION

We have studied the effect of price discrimination through joint marketing by firms producing imperfectly substitutable goods. The main finding is that the impact on overall
purchasing and welfare depend on wether firms offer a bundle price or offer rebates off
individual prices to consumers who buy from both firms. Firms may want to increase
joint purchasing by setting bundle rebates before determining stand-alone prices. Alternatively, they may want to mitigate price competition by pricing the bundle components
before setting the individual prices.
Loosely speaking, when a second purchase adds little in terms of incremental utility,
firms prefer to offer discounts off their prices. This has the effect of increasing the mass
of customers to both firms, which is beneficial because rent-extraction is limited by the
added value of the second unit. In contrast, if consumers place a high value on each
of the goods and therefore the second unit is relatively valuable, firms prefer to directly
price and market the bundle, as this allows them to increase their stand-alone prices and
extract more rents from their captured customers. However, we show that the firms interests need not be aligned in this respect if their incremental values to the consumers vary
greatly.
28

With respect to welfare we find that if second purchases add a lot of additional utility,
then price discrimination that induces added purchases raises total welfare. However,
as firms find bundle pricing more profitable compared to the rebate scheme when incremental values are high, total purchasing can actually decrease with high additional
values when both joint marketing options are available to the firms. Nevertheless, there
are parameter values for which total welfare increases due to joint marketing compared
to uniform pricing regardless of which joint marketing scheme is used. Indeed, whenever both idiosyncratic values are sufficiently high joint marketing raises total welfare;
otherwise independent price discrimination is better from a total welfare perspective.

29

APPENDIX

30

Appendix for Chapter 1
Derivation of Best Response Correspondences We will derive firm 1’s best response
correspondence first. Firm 2’s problem will be solved in a similar manner. Given the
demand function (1.1), the profit function of firm 1 is given by,

π1 =






1
2

+

v1 − v2 + p2 − p1
2t

p1



 1 ( v1 − p1 ) p1

if p1 ≥ v1 + v2 − t − p2 ,
if p1 ≤ v1 + v2 − t − p2 .

t

Consider first the case in which p1 ≥ v1 + v2 − t − p2 . Using the first order necessary

condition, we identify the candidate candidate best reply b1 ( p2 ) = 21 (t + v1 − v2 + p2 ). It

yields a payoff π1S ( p2 ) = (t + v1 − v2 + p2 )2 /8t. Notice that this case is valid only when
p1 ≥ v1 + v2 − t − p2 , so we need to have the condition:
p1 ≥ v1 + v2 − t − p2 ⇐⇒ p2 ≥

1
(v1 + 3v2 − 3t).
3

For the case in which p1 ≤ v1 + v2 − t − p2 , firm 1 sets the price at p1J = v1 /2, yielding

a payoff π1J = v21 /4t. This payoff is valid only when p1 ≤ v1 + v2 − t − p2 , requiring the
following condition:
p1 ≤ v1 + v2 − t − p2 ⇐⇒ p2 ≤
Thus, for p2 ∈

1
3 ( v1

v1
+ v2 − t.
2

+ 3v2 − 3t), 21 v1 + v2 − t both choices satisfy the requirements.

In order to identify ‘true’ best response of firm 1, it remains to compare the payoffs in
these two cases. From solving π1S ( p2 ) ≥ π1J we have

√
v2
( t + v1 − v2 + p2 )2
≥ 1 ⇐⇒ p2 ≥ p2 = ( 2 − 1)v1 + v2 − t.
8t
4t

31

Thus, firm 1 switches from the joint-purchasing regime to the single-purchasing regime at
p2 . A similar argument applies for firm 2. We therefore summarize firm i’s best response
correspondences φi as follows

φi ( pi , p j ) =




 bi ( p j ) = 1 ( t + v i − v j + p j )
2


 1 vi
2

√
if p j ≥ p j = ( 2 − 1)vi + v j − t,

√
if p j ≤ p j = ( 2 − 1)vi + v j − t.

Figure 1.5 illustrates best response correspondence for firm i.
pi

bi ( p j ) = 12 (t + vi − v j + p j )

√

2
2 vi
1
2 vi

pj
pj
Figure 1.5: Firm i’s best response correspondence

Proof of Proposition 1 We first identify two critical regions in the space prices according
to their resulting demands and denote each set as J and S , respectively, i.e.,

J = {( p1 , p2 ) ∈ R2+ | p1 + p2 ≤ v1 + v2 − t},
S = {( p1 , p2 ) ∈ R2+ | p1 + p2 ≥ v1 + v2 − t}.
There are two candidate Nash equilibria according to the region.
U

U

( p1 J , p2 J ) =
US
S
( pU
1 , p2 ) =

v1 v2
,
2 2
v − v2
v2 − v1
t+ 1
,t+
3
3
32

if ( p1 , p2 ) ∈ J ,
if ( p1 , p2 ) ∈ S

U

U

First, consider ( p1 J , p2 J ). Suppose one firm, say firm 1, deviates to the region S then
its profits becomes
U
π1S ( p1 , p2 J )

U

= p1

J
1 v1 − v2 + p2 − p1
+
2
2t

,

U

given p1 ≥ v1 + v2 − t − p2 J = v1 + 12 v2 − t. Then the optimal deviating price for firm 1
is given by

p1dev =




 1 v1 − 1 v2 + 1 t
2
4
2

if 2v1 + 3v2 ≤ 6t and v1 + v2 ≥ 2t,



 v1 + 1 v2 − t
2

if 2v1 + 3v2 ≥ 6t.

This implies the optimal deviation payoff for firm 1 as follows:
π1dev

=

=

p1dev





U

J
1 v1 − v2 + p2 − p1dev
+
2
2t

1
32t (2t + 2v1

− v2 )2

if 2v1 + 3v2 ≤ 6t and v1 + v2 ≥ 2t,



 1 (2t − v2 )(2v1 + v2 − 2t)
4t

if 2v1 + 3v2 ≥ 6t.
U

In contrast, the profit in the proposed equilibrium in region J is π1 J = v21 /4t. Thus,
U

the condition for first candidate to be a Nash equilibrium is that π1 J ≥ π1dev , and computations show that this is equivalent to (v1 , v2 ) ∈ Φ J .
U

U

We can apply same logic for candidate ( p1 S , p2 S ). Suppose firm 1 deviates to the
region J . Then, its profit becomes
U

π1 J ( p1 ) = p1
U

v1 − p1
t

,

given p1 ≤ v1 + v2 − t − p2 S = 43 v1 + 32 v2 − 2t. Then, the optimal deviating price for firm

33

1 is given by

p1dev =




 4 v1 + 2 v2 − 2t
3
3

if 5v1 + 4v2 ≤ 12t,



 1 v1

if 5v1 + 4v2 ≥ 12t and v1 + v2 ≤ 3t.

2

Thus, the optimal deviation payoff of firm 1 is
π1dev

=

=

v1 − p1dev
t

p1dev



2

2
3 v1

t

+ 31 v2 − t

2t − 31 v1 − 23 v2

if 5v1 + 4v2 ≤ 12t,



 1 v2

if 5v1 + 4v2 ≥ 12t and v1 + v2 ≤ 3t.

4t 1

U

In contrast, the profit in the candidate equilibrium in region S is given by π1 S =

US
US
S
(3t + v1 − v2 )2 /18t, so the condition for ( pU
≥ π1dev . Computations
1 , p2 ) is that π1

show that this is satisfied whenever (v1 , v2 ) ∈ ΦS .

Figure 1.6 illustrates the equilibrium pattern depending on the value of each firm’s
unique features vi and v j . If vi and v j are sufficiently small (ΦS \ Φ J ), (only) single pur-

chasing arises in equilibrium; and if vi and v j are sufficiently large (Φ J \ ΦS ), joint pur-

chasing occurs in equilibrium. For the case of intermediate values of vi and v j (ΦS ∩ Φ J ),
both configurations are possible.

Proof of Theorem 1 Recall that we have derived the firm’s profits under various scenarios (see Proposition 1 – Proposition 4). From the information in each propositions,
we have following industry profits: ΠUS =
Π IPD =
ΠR =

1
2
9t (2v1

1
2
16t (3v1

1
2
9t ( v1

+ 2v22 − 2tv1 − 2tv2 + 10t2 ), Π P =

+ 3v22 − 4tv1 − 4tv2 + 24t2 ).

+ v22 − 2v1 v2 + 9t2 ), ΠU J =
1
2
16t (3v1

1
2
4t ( v1

+ v22 ),

+ 3v22 − 2tv1 − 2tv2 + 22t2 ) and

Independent price discrimination increases profits above uniform pricing, which is

34

v2
2t

√

ΦJ
2t

ΦS ∩ Φ J

√2 t
2 2−1

ΦS

√2 t
2 2−1

√

v1
2t

2t

5: Equilibrium pattern in the benchmark
Figure 1.6:Figure
Equilibrium
pattern under uniform pricing

simply shown as
Π IPD − ΠUS =

Ω1D ∩ Ω2P

1 2
1
(v1 + vv222+ 2v1 v2 − 2tv1 − 2tv2 + t2 ) = (v1 + v2 − t)2 > 0
9t
9t
2t

and
Ω1P ∩ Ω2P

Π IPD − ΠU J = −

1 2
(v + v22 + 8tv1 + 8tv2 − 40t2 ) ≥ 0.
36t 1
Ω1D ∩ Ω2D

Ω2D ∩ Ω1P

The last inequality above comes from the fact that Π IPD − ΠU J is decreasing in vi and it
v1

takes the value of 0 when v1 = v2 = 2t.

2t

The comparison between
joint marketing regime and independent price discriminaFigure 6: Private incentives to form coordination
tion shows
3

1
Π
−Π =
(5v21 + 5v22 + 4tv1 + 4tv2 − 56t2 ) < 0,
144t
1
Π IPD − Π P =
(5v21 + 5v22 − 14tv1 − 14tv2 − 38t2 ) < 0.
144t
IPD

R

Finally, we check whether the bundle pricing scheme is better for the firms when v1 +
v2 ≥ 2t. The difference between the bundle pricing scheme and the rebate scheme is
35

given by
ΠP − ΠR =

1
( v1 + v2 − t ) > 0
8

which holds provided that v1 + v2 ≥ 2t.
Proof of Theorem 2 Computation shows that

( v1 + v2 − t )2
> 0,
36t
1
CS R − CSUS =
(v1 + v2 )2 + 36(v1 + v2 )t + 14v1 v2 − 144t2 < 0,
288t
1
3v21 + 3v22 − 20tv1 − 20tv2 + 56t2 < 0,
CS R − CSU J = −
32t
5
CS R − CS P =
(v1 + v2 − t) > 0.
16

CS IPD − CSUS =

Proof of Theorem 3 From the results in Proposition 1 – Proposition 4, we can obtain the
measure of joint purchasing consumers in each case:

U

n12S = 0
R
n12
=

1
( v1 + v2 )
4t

U

1
(v1 + v2 − 2t)
2t
1
= (v1 + v2 − 2t)
4t

IPD
n12
=

n12J =
P
n12

1
( v1 + v2 − t )
3t

Provided that v1 + v2 > 2t, the rank of measure is
U

U

R
IPD
P
n12
> n12
> n12J > n12
> n12S = 0.

First, consider the case in which (v1 , v2 ) ∈ Φ J . Computation shows the following

36

relationship of total welfare in each regime:
1
(17v21 + 17v22 − 44tv1 − 44tv2 + 40t2 ) > 0,
288t
1
(7v21 + 7v22 − 16tv1 − 16tv2 + 8t2 ) > 0,
TW IPD − TW U J = −
72t
1
TW U J − TW P =
(5v21 + 5v22 − 6tv1 − 6tv2 + 2t2 ) > 0.
32t
TW R − TW IPD = −

Thus, we have
TW R > TW IPD > TW U J > TW P .
Next, consider the case in which (v1 , v2 ) ∈ ΦS . Define the subsets
Φ1S := {(v1 , v2 ) | TW P − TW US ≥ 0},
Φ2S := {(v1 , v2 ) | TW R − TW IPD ≥ 0}.
After some algebra one can see that TW IPD − TW P > 0 for any (v1 , v2 ). So if (v1 , v2 ) ∈
Φ1s ∩ Φ2S , we obtain the following relationship:

TW R > TW IPD > TW P > TW US

37

REFERENCES

38

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40

CHAPTER 2
ON THE STRATEGIC CHOICE OF
ADD-ON PRICE POLICIES
2.1

INTRODUCTION

Consumers are often faced with fees for add-ons not included in the price of a base product (or service). For example, hotel guests, after room rates are already paid, often have
to pay additional charges for services such as Internet access, safe access, parking, a fitness center, or pool use. Banks charge fees for consumers who opt to cancel a check, use
a foreign ATM, or use a debit card. Air travelers frequently find themselves paying for
pets, snacks, checked luggage, and even blankets. Cell phone service providers charge
subscribers to send text messages and download data.
The central focus of this paper is in determining the implications of competing firms
charging separately for the add-ons. Whereas the previous add-on pricing literature assumes quality symmetry between the horizontally differentiated firms, we account for
the fact that competing firms may actually differ in quality. As a consequence, the implications of add-on pricing may be different for a superior firm than for an inferior firm.
In addition to quality asymmetry, this paper uniquely incorporates market realities
regarding consumer segmentation. Specifically, there is often heterogeneity in the willingness to pay for the add-on and heterogeneity in the anticipation of the add-on fee.
To illustrate heterogeneity in willingness to pay for the add-on, suppose that some consumers may need to park a car (or use Internet service), whereas a consumer without a
car (computer) does not value parking (Internet service). With regard to heterogeneity

41

in anticipating the add-on, we allow for some consumers to be knowledgeable about the
add-on price yet still purchase the add-on, whereas other consumers do not take the addon price into consideration at the time of initial purchase. Moreover, they are boundedly
rational in that they do not form rational expectations.
In the following, we enrich the model of add-on pricing by assuming that firms can
decide whether to charge separate prices for add-ons before they set their prices. This
allows us to examine when the add-on price policy is optimal for each firm and why
add-on pricing arises as an equilibrium.
Two reasons to explain the existence of boundedly rational consumers in add-on pricing. First, consumers are sometimes not informed of add-on fees when purchasing a base
product—a recurring theme in the add-on pricing literature that is attributed to the lack
of advertisement of add-on fees and even deliberate information hiding (Verboven, 1999;
Ellison, 2005; Gabaix and Laibson, 2006). Second, although many consumers may know
these add-on fees at the time of their initial purchase decision, there is strong anecdotal evidence that many other consumers are surprised by the charges for services they
thought were included in the posted price. Internet chat rooms are filled with complaints
from hotel consumers who were faced with fees they had not anticipated. Sullivan (2007)
describes consumers who were surprised to learn how much it costs to send each text
message. Our consideration of boundedly rational consumers is also consistent with the
increasing attention on bounded rationality in the broader business research context.
This paper addresses three questions.
1. Who will gain from the advent of add-on pricing?
2. What happens to prices and consumer surplus when more consumers are boundedly rational and do not anticipate the add-on prices?
3. Who will charge for add-ons and when does add-on pricing occur?
The result of the research can be used to determine the implications of a newly realized
42

ability to charge separately for an add-on previously included in the posted price (either
through innovative thinking or technological advances). They also help show the implications of government intervention intended to limit the ability to charge for an add-on
(for instance, President Obama has taken aim at bank fees1 and senator Charles Schumer
has proposed prohibiting certain airline fees2 ). The answers to theses questions are not
straightforward because they each depend critically on the mix of consumer segments
and/or the nature of asymmetry between firms.
In the popular press, add-on pricing is often viewed as a means for firms to boost
profit via locking consumers into decisions that are more costly than anticipated. There
is ample evidence that consumers make a purchase with the inaccurate expectation that
the add-on is included in the posted price. After such a purchase, as one traveler opined,
“It’s like the people are at the mercy of the [companies]”.3 Although the intuition would
suggest that add-on pricing benefits firms, a well-known result in the unadvertised prices
and add-on pricing literature is that charging for add-ons results in equivalent profits as
an all-inclusive price. We aim to resolve this discrepancy by showing when the profit
irrelevance result will hold and when firms will benefit from add-on pricing. We also aim
to identify conditions for when add-on pricing will diminish a firm’s profitability.
Intuitively, the effect on prices of firms serving a greater number of boundedly rational consumers is unclear. On the one hand, the higher add-on revenue from exploiting
boundedly rational consumers incentivizes firms to charge lower posted base prices to
attract consumers. With the add-on price capped by the customer’s valuation of the addon, the influx of boundedly rational consumers increases competitive intensity in base
prices, thereby lowering the total price and thus increasing surplus. This logic is consistent with the predictions of the add-on literature (e.g., Gabaix and Laibson, 2006). On the
other hand, boundedly rational consumers expect a lower total price (i.e., base price plus
1 See

“Obama blasts Bank of America debit card fee,” Washington Post, October 3, 2011.
“Schumer asks airlines to reverse flight change fee,” USA Today, May 20, 2013.
3 See “Fees of flying airlines charge travelers to supplement base fares, raise revenue in tough economic
time,” Pittsburgh Post-Gazette, November 28, 2010.
2 See

43

add-on fee) than observed by consumers who are knowledgeable about add-on fees and
thus may have a higher willingness to pay. The higher willingness to pay may increase
total prices and hurt all consumers. This latter effect has not been found in the current
add-on pricing literature. We aim to resolve these two viewpoints and identify when total
prices will (and will not) increase in the number of boundedly rational consumers.
In the previous literature, firms are symmetric in their add-on prices. In reality, however, firms vary in the add-on prices charged.4 Our model proposes asymmetry in firm
quality as a driver of variance in add-on prices. The question becomes whether the superior firm or inferior firm is more likely to charge add-on prices. Because add-on prices
are a form of price discrimination (Ellison, 2005), applying predictions from another form
of price discrimination, such as rebates, would suggest the inferior firm is more likely to
price discriminate than the superior firm. We examine whether this finding holds true for
add-on pricing.

2.2

RELATED LITERATURE

A well-known result in the unadvertised prices add-on pricing literature is that charging
for add-ons results in equivalent profit as when add-on pricing is prohibited or infeasible.
Any gains from the unadvertised price are competed away through the posted prices.
Lal and Matutes (1994) modeled a retailer’s use of loss-leader products coupled with
unadvertised prices of other products that are not discounted. Verboven (1999) looked
at add-on pricing for products in which the price of the base product (i.e., the car) is
advertised and the firms can choose whether to advertise the prices of premium upgrade
add-ons. Gabaix and Laibson (2006) produced the profit-irrelevancy result of add-on
prices with a segment of boundedly rational consumers who inaccurately believe the addon is included in the posted price. This paper differs from the above papers in that we
4 See

e.g. “Hotels piling on hidden fees,” CNNMoney, October 31, 2011.

44

show add-on pricing can affect firm profits.5
Ellison (2005) identified how add-on pricing can actually lead to improved profit for
the firms if consumers who are more sensitive to interfirm price differences are less likely
to purchase costly add-ons. The correlation between consumer price sensitivity and addon valuation creates an adverse selection problem that softens competition in base prices.
Our paper complements Ellison’s (2005) in that we find alternative venues in which addon pricing can influence firm profits. Furthermore, we show that add-on pricing can have
opposing profit implications for the two competing firms. Ellison and Wolitzky (2012)
extended the sequential search model of Stahl (1989) and showed that concealing prices—
which increases search costs—reduces the fractions of consumers who search and thus
benefits firms.6 Ellison and Ellison (2009) provided empirical support for the explanations
of Ellison (2005) and Ellison and Wolitzky (2012).
Because add-on pricing is a form of price discrimination, our paper relates to a body of
research on targeted pricing. The ability to price discriminate between old and new customers is shown to diminish profits for competing firms (Villas-Boas, 1999) and a durablegood monopolist (Villas-Boas, 2004). Corts (1998) modeled vertically differentiated firms
and showed that price discrimination between customers who value quality and those
who do not value quality will result in lower profit for both the high-quality firm and
the low-quality firm. Chen (2008) modeled firms of differentiated quality and found that
the inferior firm earns less profit when price discrimination based on consumer purchase
history is feasible than when uniform pricing is adopted.
This paper also belongs to the broader literature on competitive price discrimination.7
Much of this literature examines third-degree price discrimination, i.e., models in which
firms can identify whether consumer is “high” or “low” type and charge different prices
5 Chen

and Rey (2012) also found that being able to charge different prices to one-stop shoppers from
multistop shoppers may increase the firm’s profit and result in socially harmful effect.
6 The add-on pricing literature assumes two products: base and add-on products. In contrast, Ellison
and Wolitzky (2012) only consider a single product in a search model.
7 Stole (2007) provides an excellent survey.

45

accordingly. Borenstein (1985) and Holmes (1989) provide some of the most basic results. Borenstein notes that differences across groups in the interfirm price sensitivity are
needed to generate price discrimination. Holmes notes that banning price discrimination will lower prices in one market and raise them in the other; the net effect on profits
is ambiguous. Within this literature few papers analyzes models with both vertical and
horizontal differentiation, no doubt because it is difficult to construct models that are
sufficiently tractable to allow closed-form solutions. (Borenstein (1985) presented his results using numerical simulations.) Two notable exceptions are Armstrong and Vickers
(2001) and Rochet and Stole (2002). Among other contributions, each of these papers derives a non-discrimination theorem. They show that when brand preferences are of the
type generally assumed in discre-choice models and brand preferences are independent
of consumers’ valuations for quality, then the outcome of the competitive second-degree
price discrimination model is that firms do not use quality levels to discriminate.
Establishing conditions that define when equilibrium leads to charging separately for
add-ons is also new to the bundling literature. There are a variety of papers that theoretically and empirically identify various reason to offer bundles rather than price each
element separately (e.g., Adams and Yellen, 1976; Schmalensee, 1984; McAfee et al., 1989;
Fang and Norman, 2006). Specifically, Whinston (1990), Chen (1997), Nalebuff (2004) analyzed bundling in an oligopolistic setup. Whinston (1990) showed that a firm can leverage
its monopolistic power in one market into another oligopolistin market via product tying.
Chen (1997), from a different research angle of ex ante homogeneous firms, showed that
bundling can serve as a tool for product differentiation. Our research also differs from
existing works in that the bundling literature treats all prices as common knowledge,
whereas our paper allows for the possibility that add-on fees are not observable to at least
some consumers. Such unobservability can result in a high magnitude of add-on fees—a
signature dynamic in the add-on literature—that is absent from bundling research.

46

This paper also belongs to the broader literature on switching costs.8 Although the
early switching cost papers stress applications where consumers buy the same product in
multiple periods, many arguments are equally applicable to situations where the product purchased in the second period is different from the product bought in the first period. For example, Klemperer’s (1987a) discussion of situations where profits with infinite
switching costs are identical to profits with no switching costs is essentially the same as
Lal and Matutes’ irrelevance result, and a number of papers have used similar frameworks to discuss market power in aftermarket service, e.g., Shapiro (1995) and Borenstein
et al. (2000). The most basic result in the switching cost literature is that switching costs
can increase or decrease profits because they usually make first-period prices (think base
goods) lower and second-period prices (add-ons) higher. The literature contains several
well-known arguments about why switching costs may tend to raise profits, for example,
Farrell and Shapiro (1988), Klemperer (1987b), and Beggs and Klemperer (1992). The argument in the above papers are inapplicable to add-ons, however, because they require
an assumption that firms cannot differentiate between new and old customers; i.e., that
the firm cannot choose an add-on price different from the price for good L.
This paper is also loosely related to all papers discussing a strategic investment that
softens competition. Chapter 8 of Tirole (1988) reviews a number of such papers. A classic
example is Thisse and Vives (1988), which notes that firms are better off competing in
FOP prices than in delivered prices, because when they choose separate delivered prices
for each location they end up being in Bertrand competition for the consumers at each
location. As in this paper, they also note that FOB pricing in not individually rational in
an extended game in which firms first choose pricing policies, and then compete in prices.
Gabaix and Laibson (2006) were among the first in the add-on literature to allow for
boundedly rational consumers.9 As noted by Spiegler (2006), consumers may find it dif8 Farrell

9 Ellison

sumers.

and Klempere (2007) provide an excellent survey.
(2006) and Spiegler (2011) provide excellent survey on the issue with boundedly rational con-

47

ficult to think and account for all possible dimensions (e.g., all possible services that
may or may not be included in the posted price) and thus resort to simplifying heuristics (e.g., posted price) to save cognitive resources. The concept of bounded rationality
has been around for decades (Simon, 1955), though relatively recently it has gained more
acceptance in economics (Rubinstein, 1998). Conlisk (1996) reviewed a large body of evidence supporting bounds on rationality and provided a discussion of when and why
they should be incorporated in economic models. In our setting, there is a large body of
anecdotal evidence supporting bounded rationality in terms of expectations surrounding
hidden add-on fees. As described by Conlisk (1996), human cognition is a scarce resource,
and thus a consumer’s inability to accurately forecast the price for each possible add-on
is consistent with a process by which cognitive resources are allocated elsewhere.10

2.3
2.3.1

THE MODEL
Firms

The model is one of imperfect competition. There are two competing firms, 1 and 2,
that are horizontally differentiated in that they are located at x1 = 0 and x2 = 1 on a
linear city (Hotelling, 1929). We also allow for asymmetry in the quality of each firm’s
offerings, which we will specify later when we discuss consumer reservation utilities. As
in Liu and Serfes (2005), we assume the cost of developing a certain quality to be a fixed
cost (i.e., the cost of developing a product or establishing a brand name), which has no
effect on the marginal cost of production (see also Motta, 1993; Ikeda and Toshimitsu,
2010).11 Each firm offers a base good with marginal cost c = 0. Each firm also offers
10 See

also Armstrong and Vickers (2012); Grubb (2009, Forthcoming).
asymmetric quality could be modeled as having asymmetric marginal costs of production (e.g., Moorthy, 1988; Kuksov and Lin, 2010; Dahremöller, 2013). Incorporating this assumption would
complicate the model, yet logic suggests the current findings are preserved when the difference in costs is
sufficiently less than the difference in qualities. The basis for this logic is that our results hold when the
difference in costs is zero, and thus hold for local deviations.
11 Alternatively,

48

an add-on, with marginal cost cˆ = 0.12 For example, a base good can be a hotel room,
checking account, wireless plan, or flight seat, and the associated add-on can be pool
access, overdraft protection, data usage, or pillow usage, respectively.
We denote the price charged for the base good by firm i = 1, 2 as pi while the price
for add-on good as pˆ i . Posted prices are easily accessible by consumers at negligible
costs, such as through popular Internet-based price comparison site like Expedia.com,
MyRatePlan.com, and Google Product Search. As a result, consumers have full information over posted prices. In contrast pˆ i is not easily discoverable by consumers. For
example, on popular price comparison sites such as Expedia.com, hotels post their room
rates yet often keep silent on their Internet access, pool usage, or parking fees, and airlines
show their ticket prices but not their charges for check-in baggage, pet fees, or Internet access. Although some consumers may track down this information, others may not know
to do so. Outside the travel industry, marketing letters from credit card companies highlight APRs, yet keep details on how default rates are triggered in fine print.13 Wireless
companies like Verizon and AT&T clearly announce their access fees for voice plans, but
per-unit text messaging rates are in their fine print.

2.3.2

Consumers

A consumer type θ’s utility from buying the base product from firm i is equal to
Ri − t | xi − θ | − pi ,

(2.1)

where transportation cost t is a positive constant, consumer location θ are uniformly distributed along the linear city with a mass normalized to one, and the reservation value for
12 For

example, we can think that the marginal cost of the hotels to provide amenities such as wireless
internet services, telephone calls would be significantly small so that we can ignore that costs. But in some
industries, it could not be the case. Providing meal in flight and overdraft fee in banking industries may be
such examples.
13 One controversial yet popular trigger is called “universal default,” where a credit card’s default rate is
triggered if a consumer defaults on credit cards issued by any other financial institute.

49

the base good from firm i, Ri , depends on the quality of the firm. Without loss of generality, let firm 1 be the qualitatively superior firm with quality normalized to 1 (i.e., R1 = R).
Firm 2 is weakly inferior with quality of the base good (q ≤ 1) (i.e., R2 = qR). Similarly
to Villas-Boas (1999) and Shin and Sudhir (2010), we assume that qR is large enough to
ensure full market coverage.14 A consumer type θ’s utility from buying the add-on from
firm i is equal to
Rˆ i − pˆ i ,

(2.2)

where Rˆ 1 = Rˆ and Rˆ 2 = g Rˆ with firm 1’s quality of the add-on normalized to 1 and firm
2’s quality of the add-on equal to g ≤ 1.
We allow for consumer heterogeneity in willingness to pay for the add-on service as
well as in bounded rationality.15 As such we have three segments that we describe and
provide justification for.
We allow a proportion (1 − α) of consumers do not derive utility from the add-on (i.e.,

Rˆ = 0). Consumers may be in the base segment because they do not value the add-on or
because they have found reasonable substitutes from the outside market. Examples include a wireless customer who does not want to text message, a checking account holder
who balances the books and does not require overdraft protection or cancel-check services, and a flyer who packs light for a trip or carries their own pillow and thus doesn’t
check a bag or pay for use of pillow. In ancillary analyses in Appendix, we confirm our
results when an additional segment exists who will consume the add-on unless it carries
a charge.
Whereas base consumers have zero value for the add-on, all remaining consumers

ˆ i.e.,
have a positive value for the add-on (i.e., Rˆ > 0 for all θ). We assume g Rˆ > c,
the reservation value of the add-on for either firm is higher than the marginal cost of
the add-on. As in Gabaix and Laibson (2006), we allow for consumers to both value the
14 In

Section 2.5 we provide the explicit condition for full market coverage in equilibrium.
Appendix, we examine a model in which consumers know that they will be charged add-on fees,
but do not know what those are before committing to purchase the base product.
15 In

50

add-on and inaccurately believe that the add-on is provided for free. Let β represent
the proportion of nonbase consumers who have this form of bounded rationality (i.e.,
consumers who believe pˆ i = 0). In other words, the proportion of the entire consumer
population who fall in this segment is equal to αβ. As described by Gabaix and Laibson
(2006), these consumers incompletely analyze the future game tree and do not anticipate
the add-on fee at the time of base purchase. There are several reasons why consumers
may be boundedly rational in this sense. It has been documented that even experienced
travelers are surprised to learn that there are charges for services that had at a previous
time been free.16 Alternatively, consumers may not have enough knowledge about the
product to know what services will or will not be included in the price. Consistent with
Conlisk (1996), boundedly rational consumers do not allocate cognitive resources toward
finding the actual add-on fee or toward forming accurate expectations.
The remaining α(1 − β) consumers have Rˆ > 0 and know the add-on fees. Their
knowledge of add-on fees may come from their experience or search. As such, knowledgeable consumers explicitly account for the add-on fee in their base purchase decision.

2.3.3

Timing of the Model

The sequence of game is as follows. In Stage 1, firms firms decide whether to charge separately for add-ons. Given the pricing policy chosen in Stage 1, the firms simultaneously
choose their posted and add-on prices in Stage 2. In Stage 3, consumers choose from
which firm to buy. In Stage 4, all consumers are aware of add-on fee and only nonbase
consumers purchase the add-on if the add-on fee is less than or equal to the utility it provides. Consumers in Stage 4 are locked in to their Stage 3 purchase and cannot switch
firms. Figure 2.1 depicts the sequence of events.
16 See, for example, “As hotels struggle for business, some guests find an upside”, USA Today, February
5, 2009.

51

Stage 1
Stage 2
Firms choose whether Firms 1 and 2 simultaneously
or not to charge add-on
set p1 , p2 , pˆ 1 and pˆ 2

Stage 3
Consumers choose
which product to buy

Stage 4
Consumers choose
whether or not to buy add-on

Figure 2.1: Sequence of Events

2.4

ANALYSIS

We analyze the model using backward induction. We have three different consumer segments to examine: (1 − α) base consumers, αβ myopic consumers, and α(1 − β) sophisticated consumers. For the add-on to be purchased by consumers in Stage 4, it must be
ˆ Otherwise, no consumer will consume the add-on.
that pˆ 1 ≤ Rˆ and pˆ 2 ≤ g R.
To the best of our knowledge this is the first research on add-on pricing that considers
all of the aforementioned three segments of consumers. It turns out, as the results will
demonstrate, that insights new to the literature arise when both firms serve all three segments of consumers in equilibrium. We therefore will focus our attention on the scenario
where all three segments of consumers purchase in equilibrium in the body of the paper,
and we will provide the corresponding regularity conditions.
To derive the quantity of consumers who purchase each base product and each addon in Stage 3 and Stage 4, we identify the marginal consumer in each segment who is
indifferent between purchasing product 1 and purchasing product 2. Given firm prices
(pi , pˆ i ) for i = 1, 2, the marginal consumer in the sophisticated segment satisfies
R + Rˆ − t|0 − θs | − p1 − pˆ 1 = qR + g Rˆ − t|1 − θs | − p2 − pˆ 2 ,
and thus she is located at

θs =

1

p + pˆ 2 − p1 − pˆ 1
1 (1 − q) R + (1 − g) Rˆ
+
+ 2
.
2
2t
2t

52

All consumers in the sophisticated segment with θ ≤ θs will purchase both the base
product and the add-on service from firm 1, whereas those with θ > θs will purchase
from firm 2.
Similarly the locations of the marginal base consumer and the marginal myopic consumer are θb = 1/2 + [(1 − q) R + p2 − p1 ]/2t and θm = 1/2 + [(1 − q) R + (1 − g) Rˆ +

ˆ i , are
p2 − p1 ]/2t, respectively. Demand of each firm’s base product, Di , and add-on, D

then
α(1 − β)( pˆ 1 − pˆ 2 ) + ( p1 − p2 ) − α(1 − g) Rˆ − (1 − q) R
1
− η (i )
,
2
2t
ˆ
ˆ i = α − αη (i ) (1 − β)( pˆ 1 − pˆ 2 ) + ( p1 − p2 ) − (1 − g) R − (1 − q) R ,
D
2
2t

Di =

(2.3)
(2.4)

where η (i ) = 1 if i = 1, and η (i ) = −1 if i = 2. Firm profits are
ˆ i.
πi = ( pi − c) Di + ( pˆ i − cˆ) D

(2.5)

In this model with three consumer segments and two firms that are vertically differentiated along two dimensions as well as horizontally differentiated, the Stage 2 pricing
analysis is relatively involved. Similarly to Shin and Sudhir (2010), we proceed by first
analyzing the symmetric case. Examining the benchmark allows us to isolate the impacts
of having three distinct consumer segments served by the firms and asymmetry in quality
of the base products.
Before analyzing equilibrium prices in the general framework, we focus on the special
case in which two firms are symmetric in that q = 1. Thus, there is no quality differences
between the products of each firm. This case will provide the key to the more general
analysis in the next section. Suppose that each firm charges add-on fees separately. To
ensure that firms choose to serve all three consumer segments we impose the following
assumption regarding the degree of product differentiation relative to add-on value.

53

2.4.1

Symmetric Firms and Boundedly Rational Consumers

The first special case is when two firms are symmetric, i.e., q = 1 and g = 1. We impose
the following assumption regarding the level of horizontal differentiation relative to the
add-on value to ensure that firms choose to serve all three consumer segments.
ˆ
Assumption 2. t > Rˆ − c.
Applying q = 1 and g = 1 to equations (2.3) and (2.4), in this special case we have
ˆ i = α/2 − αη (i )[(1 − β)( pˆ 1 −
Di = 1/2 − η (i )[α(1 − β)( pˆ 1 − pˆ 2 ) + ( p1 − p2 )]/(2t) and D
pˆ 2 ) + ( p1 − p2 )]/(2t). We can derive the equilibrium posted prices and add-on fees from
the first-order conditions of profit expressions in (2.5). It turns out that there are two
mutually exclusive cases.
Lemma 1. Define β¯ ≡ ( Rˆ − cˆ)(1 − α)/ t + ( Rˆ − cˆ)(1 − α) . Given symmetric firms, i.e.,
q = 1 and g = 1:
¯ the unique equilibrium is an interior equilibrium, in which firms charge posted
(i) if β < β,
prices p1 = p2 = c + t − [αβ/ (1 − α)(1 − β) ]t and add-on fees pˆ 1 = pˆ 2 = cˆ +

[ β/ (1 − α)(1 − β) ]t;
¯ the unique equilibrium is a corner equilibrium, in which firms charge posted prices
(ii) if β ≥ β,
ˆ
p1 = p2 = c + t − α( R − c) and add-on fees pˆ 1 = pˆ 2 = R.

See Appendix for all proofs. In contrast to the Gabaix and Laibson’s (2006) prediction
that firms will charge the maximum possible add-on price, Lemma 1 shows that this result
is conditional on the relative magnitude of two ratios β/ (1 − α)(1 − β) and ( Rˆ − cˆ)/t.
Although part (ii) of Lemma 1 is consistent with Gabaix and Laibson’s (2006) shrouded
prices equilibrium, part (i) indicates when the equilibrium add-on price may be less than
the consumer valuation of the add-on.17 Proposition 5 highlights a key insight gained
17 Note that combined with Assumption 2, the interior equilibrium holds for β/ [(1 − α )(1 − β )] < ( R
ˆ−
ˆc)/t < 1. This inequality is true for any Rˆ − cˆ and t that satisfy Assumption 2 if β is small enough and
α < 1.

54

from considering rational consumers who buy the add-on.
Proposition 5. Given symmetric firms, the total price for buying the base product and the add-on
¯ As such, consumer welfare for rational consumers and
is increasing in β if and only if β < β.
myopic consumers is decreasing in the number of myopic consumers (β) if and only if β < β¯ holds.
Proposition 5 identifies when rational consumers who value the add-on are penalized—
and increasingly so—by the existence of myopic consumers. To our knowledge this result
is new to the add-on pricing literature and contrasts with the results of Gabaix and Laibson (2006, p. 517), who found that consumers who account for the add-on during the base
purchase earn greater surplus with more myopic consumers.18 Proposition 5 also shows
that an inattentive consumer is worse off when there are a greater number of peers who
are also inattentive. Gabaix and Laibson (2006) found the opposite result when prices are
hidden.
The intuition behind the contrasting results lies in how the sizes of the consumer segments affect posted prices and add-on fees. When a greater number of consumers will
purchase the add-on, firms engage in increasingly intensified competition over posted
prices to lock in additional profit on the add-on. This effect is present in our model and
that of Gabaix and Laibson (2006). In the Gabaix and Laibson (2006) paper, all consumers
who buy the add-on when its price is hidden are boundedly rational and the add-on
prices are set at their maximum (further increasing them will trigger consumers to stop
buying the add-on). Therefore, the only effect of increasing the number of boundedly rational consumers is to lower posted prices, which in turn means lower total prices. However, when there is a sufficient number of sophisticated consumers and base consumers
¯ add-on prices are not set at their maximum. As such, in this parameter range
(i.e., β < β),
there is a second effect of increasing the number of boundedly rational consumers. In this
18 Gabaix

and Laibson (2006) consider both a shrouded prices equilibrium in which firm hide the add-on
fee from myopic consumers and an unshrouded prices equilibrium in which some myopic consumers know
about the add-on fee. In their shrouded prices equilibrium, the surplus of consumers who know about the
fee increases in the number of myopic consumers; in the unshrouded prices equilibrium, this surplus is a
constant that is lower than the surplus under the shrouded prices equilibrium.

55

scenario, firms soften their competition over the add-on—and thus increase their addon fees—to take advantage of locked-in boundedly rational consumers. The increased
add-on fee dominate the reduced posted prices, resulting in higher total prices that sophisticated and myopic consumers pay.
As the intuition suggests, Proposition 5 is driven by the fact that the add-on price is
set below its maximum level. We refer to this case as the interior equilibrium. The new
result in Proposition 5 regarding total price and consumer welfare requires participation
in the marketplace by each of the three consumer segments. To see this, the condition
from Proposition 5 can be rewritten as follows:
β < β¯ ⇐⇒

⇐⇒

Rˆ − cˆ
αβ
<
(1 − α) · [α(1 − β)]
t

number of boundedly rational consumers
Rˆ − cˆ
<
[number of base consumers] · [number of sophiscated consumers]
t

(2.6)

Equation (2.6) highlights that the equilibrium is the interior equilibrium when there
is a sufficient number of both base consumer and sophisticated consumers. Notice that
the condition holds trivially when there are no boundedly rational consumers, and the
condition never holds when all consumers who buy the add-on are boundedly rational (i.e., there are no sophisticated consumers). Intuitively, the inclusion of enough base
consumers dampens the firms’ incentive to slash posted prices in an attempt to exploit
boundedly rational consumers. Furthermore, when there are enough sophisticated consumers, firms will not set total prices too high, because otherwise they will drive away
sophisticated consumers. the above two dynamics together driver down the add-on price.
Hereafter, we refer to (2.6) as the “interior-equilibrium condition” for ease of exposition.
Its essence that there should be enough numbers of both base and sophisticated consumers to ensure interior add-on fees will hold across the more general case, though the
condition will be relatively more involved.

56

When (2.6) does not hold, Lemma 1 shows that firms will charge the maximum possible add-on fees, under which total price is independent of β. The equilibrium prices are
analogous to the shrouded prices equilibrium in Gabaix and Laibson (2006) and thus do
not add new insight.19 For the rest of this paper, we focus attention on the more novel
case in which equilibrium add-on fees are not corner solutions.
The inclusion of consumers who do not know about the add-on price before the base
purchase but still buy the add-on has a notable impact on the welfare of the consumer
segments. However, this model feature has no effect on the result of profit irrelevancy of
add-on prices. Firm profit in either case of Lemma 1 is equal to t/2. In contrast, when no
firms charge add-on fees (i.e., pˆ 1 = pˆ 2 = 0), the equilibrium prices are
p1N,N = p2N,N = t + cˆ + αcˆ
and profit is also t/2.

2.4.2

Solving the Model When Only One Firm Charges an Add-On
Price

Consider now an asymmetric pricing policy of the following type. Assume that firm 1
charges an add-on fee for its add-on consumer, whereas firm 2 does not charge the addon fee, i.e., pˆ 2 = 0. Intuitively, we would expect that firm 1 charges a higher base price
than firm 2. This intuition is confirmed in the following lemma.
¯ in the pure strategy Nash equilibrium of games in which firm 1 charges an
Lemma 2. For β < β,
19 Clarifying

α(1 − α) as the number of consumers who buy (do not buy) the add-on, the prices in part
(ii) of Lemma 1 are equivalent to the prices in Equations in (3) and (4) of Gabaix and Laibson (2006).

57

add-on price, the equilibrium prices are given as follows:
p1A,N

6ω − 3αβ
t,
=
6ω − αβ2
pˆ 1A,N =

p2A,N

6ω − 2αβ2
=
t,
6ω − αβ2

3β
t,
6ω − αβ2

where ω = (1 − α)(1 − β). Also, in equilibrium, we have p2A,N > p1A,N > 0 and pˆ 1A,N > 0.
Since firm 1 and firm 2 are symmetric, the Nash equilibrium in the case where only
firm 2 uses an add-on pricing policy is analogous to Lemma 2. This lemma implies that
each firm makes strictly positive sales to consumers in each segment in equilibrium. Also,
the fact that the firm who can price discriminate charges a lower base price is not surprising. If a consumer who is ignorant of add-on fees purchases a product from firm 1,
she is locked into its aftermarket. Thus it creates an incentive for the firm to charge the
maximum possible price for the add-on and a lower price for the base to induce more
consumers into its aftermarket.20
Indeed we can observe that firm 1’s add-on fee is increasing in β. Due to the presence of the sophisticated consumers, firm 1 cannot charge the maximum possible add-on
fee, but when there are more myopic consumers the firm will sacrifice some profit gain
from sophisticated consumers and increase the add-on fee. Also increasing the number
of boundedly rational consumers increases the competitive pressure in the base product
market, thus the equilibrium posted price decreases with an increasing of β. However
a price reduction of the base product causes more profit loss for firm 2 compared to the
firm 1, since firm 2 does not have any other way to compensate for this loss whereas firm
1 can make it up from the sales of add-ons. That is | ∂β1 | > | ∂β2 |.
∂p

∂p

¯ we also have another equilibrium in which p1 < 0 < p2 . In order to make it easier to
β > β,
compare the results with the one in Lemma 1, we restrict the parametric space of β.
20 When

58

Lemma 2 allows us to obtain the following equilibrium profits:
π1A,N

9ω (4ω − αβ2 )
t,
=
2(6ω − αβ2 )2

π2A,N

6ω − 2αβ2
=
t.
2(6ω − αβ2 )2

Table 2.1 summarizes the payoffs for the firms.
Table 2.1: Summary of Firm’s payoffs
Firm 1 \Firm 2

A

NA

A

t t
,
2 2

9ω (4ω − αβ2 )
6ω − 2αβ2
t,
t
2(6ω − αβ2 )2 2(6ω − αβ2 )2

NA

9ω (4ω − αβ2 )
6ω − 2αβ2
t,
t
2(6ω − αβ2 )2 2(6ω − αβ2 )2

t t
,
2 2

The analysis so far has compared the equilibrium profits associated with the four different outcomes of firm’s add-on pricing decisions. The equilibrium profits, summarized
in Table 2.1, define a normal-form game in which each firm chooses whether to charge
separate prices for the base product and add-on. It can be verified that firm 1 will charge
an add-on price no matter what pricing policy is adopted by firm 2. Likewise, firm 2
charges an add-on price without regard to firm 1’s price policy. This proves that add-on
pricing is a dominant strategy for each firm. Proposition 6 summarizes the result.
¯ choosing the add-on pricing policy is a dominant strategy for each
Proposition 6. For β < β,
firm and, consequently, ( A, A) with the market prices is the unique subgame-perfect equilibrium.
Proposition 6 allows us to answer the question of whether the firm should charge
a separate price for the add-on or not. We observe that the firms’ optimal strategy is
committing to the add-on pricing policy, which explains why the firms use this pricing
scheme even though they earn the same profits compared to the case where they do not
engage in this form of price discrimination.
Add-on pricing has no impact on total welfare since all consumers have unit demand
for the base good, and they purchase the product from the closer firm with either pricing
59

policy. However, it has different implications on consumer welfare according to the type
of consumers. The total price for add-on users (sophisticated and boundedly rational
consumers) are, from the results in Lemma 1, given by
p A,A + pˆ A,A = t +

β
t > t = p N,N ,
(1 − β )

(2.7)

which implies that when both firms use an add-on pricing policy, the add-on users are
worse off compared to the no add-on equilibrium. On the other hand, those who do
not purchase an add-on can be better off. Also, direct comparison between p A,A and
p N,N shows that add-on pricing policies benefit base consumers as compared to the case
without add-on pricing. Notice that if there are no boundedly rational consumers in the
model (i.e., β = 0), we can see that there are no effects in consumer welfare, as in standard
add-on pricing literature.

2.5

ASYMMETRIC QUALITIES

We now consider the case with quality differentiation on base goods, i.e., q < 1. For
example, hotels can differ in their star ratings. For ease of exposition, we frequently
refer to firm 1 with base good quality 1 as the superior firm and firm 2 with base good
quality q < 1 as the inferior firm. To isolate the consequences of quality differentiation
over base goods, in this special case we consider symmetric quality of the add-on which
is represented by g = 1. Add-ons such as Internet access, check cancellation, and text
messaging likely fall this category because their add-on quality is likely irrespective of
the quality of base offering. We impose the following assumption to ensure that each firm
will serve all three consumer segments:
Assumption 3. t > Rˆ − cˆ + (1 − q) R/3.
After applying g = 1 to Equations (2.3) and (2.4), we can derive the equilibrium posted

60

prices and add-on fees from the first-order conditions of profit expressions in (2.5). Similar to Lemma 1, we again get mutually exclusive cases and present the conditions leading
to the interior equilibrium below.
Lemma 3. Consider symmetric add-on quality, i.e., g = 1. If
1
β
β
< Rˆ − cˆ −
(1 − q ) R ,
ω
t
9ω − 2αβ2

(2.8)

the firms charge the following posted prices and add-on fees:
p1A,A = c + t −

p2A,A = c + t −

αβ
3ω − αβ
t+
(1 − q ) R
ω
9ω − 2αβ2

;

αβ
3ω − αβ
t−
(1 − q ) R
ω
9ω − 2αβ2

;

pˆ 1A,A = cˆ +

β
β
t+
(1 − q ) R
ω
9ω − 2αβ2

(2.9)

pˆ 2A,A = cˆ +

β
β
t−
(1 − q ) R
ω
9ω − 2αβ2

(2.10)

The above parametric condition (2.8) which ensures pˆ i < Rˆ in equilibrium for each
firm i is analogous to the interior equilibrium condition in the symmetric case, yet more
involved because of asymmetric base qualities.21 This condition, combined with Assumption 3, ensures that the indifferent consumers in each consumer segment are located in the
interior. Also note that full market coverage again requires a high enough qR.22 We again
limit our discussion to the more novel case where both firms charge interior add-on fees.
Notice that the interior-equilibrium condition holds trivially as the number of boundedly rational consumers approaches zero (because the left-hand side of the inequality
approaches zero and the right-hand side is strictly positive by fact that R > c). As such,
both the interior-equilibrium condition and Assumption 3 (which does not depend on β)
can hold simultaneously.
21 Note

that firm 1’s (firm 2’s) prices are increasing (decreasing) in R because, as we show in the proof of
Lemma 3, conditions 9ω − 2αβ2 > 0 and 3ω − αβ > 0 hold under the interior equilibrium.
22 We provide the proof, in the appendix, that it is possible to have all regularity conditions for the interior
equilibrium to hold at the same time.

61

Below we discuss the implications of this general model on the firm’s pricing. An
examination of the prices reveals that firm 1’s total price always increases in β when
condition (2.8) holds, i.e., when the equilibrium is an interior equilibrium. Firm 2’s total
price under this condition, however, increases in β if and only if t is above a threshold:
t > t˜ ≡

ω (1 − β) 9 − α(3 − 2β)2
(1 − q) R.
(9ω − 2αβ2 )2

(2.11)

Therefore, when (2.11) holds, the findings of Proposition 5 are replicated in the general
model.23 Nevertheless, when t < t˜, the effect of β on firm 2’s total price is negative.
This is because the inferior firm experiences an additional dynamic that differs from the
symmetric case. Because the superior firm increases its total price to get a higher margin
out of the boundedly rational consumers (while experiencing a greater market share due
to its quality superiority), the inferior firm has an incentive to actually cut its total price
because the marginal increment in market share is now more significant. The second
dynamic dominates the first dynamic when t is small enough: when t is small, consumers
are more price sensitive, and thus the inferior firm can be more effective in acquiring
market share by lowering price.
When both firms use add-on pricing, Lemma 3 leads us to have the following equilibrium profits:
π1A,A =

t
(1 − α ) α
+
(1 − q ) R
2 2t(9ω − 2αβ2 )

2

π2A,A =

t
(1 − α ) α
(1 − q ) R
+
2 2t(9ω − 2αβ2 )

2

+

1
αβ2
1−
(1 − q ) R
3
9ω − 2αβ2

(2.12)

−

1
αβ2
(1 − q) R.
1−
3
9ω − 2αβ2

(2.13)

and

23 Analysis

of the impact of β on total prices is included at the end of the proof of Lemma 3. To see that
the condition in equation (2.11) can be satisfied at the interior equilibrium, observe that the right-hand side
of the inequality is bounded as β approaches zero. Thus, the interior-equilibrium condition and the equality
of equation (2.11) can be satisfied for low value of β.

62

We now solve the model for the game in which neither firm can charge add-on prices
to the consumers. Thus, we solve for the Nash equilibrium in which pˆ 1 = pˆ 2 = 0 is
imposed in expressions (2.3), (2.4), and (2.5). When both firms do not charge separate
add-on fees, we obtain the following Nash equilibrium prices and corresponding profits:
1
p1N,N = t + (1 − q) R,
3
1
1
N,N
π1 =
t + (1 − q ) R
2t
3

2

1
p2N,N = t − (1 − q) R,
3
1
1
N,N
π2 =
t − (1 − q ) R
2t
3

,

(2.14)
2

.

(2.15)

When all consumers know the add-on prices, the add-on prices are set to marginal
cost, the add-on pricing does not affect profit for either firm (i.e., when β = 0, the above
πiA,A is equal to πiN,N for i = 1, 2). This result, which holds even for firms who are
asymmetric in quality, is consistent with the bulk of the add-on pricing research (see Lal
and Matutes, 1994; Verboven, 1999; Gabaix and Laibson, 2006). However, the existence
of boundedly rational consumers in the context of asymmetric base qualities has unique
profit implications.
Proposition 7. Given the presence of myopic consumers and horizontally differentiated firms that
are asymmetric in the quality of the base product but symmetric in the quality of the add-on (i.e.,
β > 0, q < 1, g = 1), add-on pricing by both firms will diminish profits for the superior firm and
increase profits for the inferior firm in the interior equilibrium.
Proposition 7 is the first finding in the add-on pricing literature to identify opposing
effects of add-on pricing on profitability for a superior firm and for an inferior firm. It also
demonstrates how add-on pricing with boundedly rational consumers differ from other
forms of price discrimination. the finding that the superior firm earns less profit when
both firms practice price discrimination via add-on pricing is counter to the finding of
Shaffer and Zhang (2000) that the firm with larger loyal following earns greater profit with
preference-based price discrimination. The finding that the inferior firm can earn greater
profit when both firms practice price discrimination via add-on pricing is counter to Chen
63

(2008), who found that the inferior firm is always worse off with price discrimination
based on purchase histories. It also contrasts with the finding that price discrimination
between consumers who value quality and consumer who do not value quality will lead
to lower profits for both the high- and low-quality firms (Corts, 1998).
To understand this contrasting result, first consider the fact that g = q implies the
reservation value differential between firms is the same for each type of consumer. Specifically, base consumers who do not buy the add-on experience the same reservation value
differential, absent transportation cost, (1 − q) R as both sophisticated and myopic con-

sumers because ( R + Rˆ ) − (qR + Rˆ ) = (1 − q) R. In the absence of the boundedly rational
consumer segment, both firms will earn all profit on the base consumers and give the
add-on at marginal cost.24 However, when the myopic consumer segment exits, both
firms have an incentive to boost revenue by charging higher add-on fees, which does not
affect the base purchase decision by the boundedly rational consumers. As a result, there
is greater competitive intensity in posted prices that will attract consumers who will generate further income via add-on purchases. This diminishes profitability on the nonbase
consumers. However, the superior firm has a larger number of base consumers to which
this diminished profit margin applies. For the superior firm, the loss in profit margin on
base consumers outweighs the profit gain from selling the add-on above cost. The opposite is true for the inferior firm. As a consequence, the superior firm loses profit and the
inferior firm gains profit with the advent of add-on pricing.

2.5.1

Equilibrium Pricing Strategies

We now solve the model in which only one firm charges an add-on price. Because we
assumed asymmetry in qualities, we have two cases according to which firm adopts the
add-on pricing policy.
Lemma 4. Suppose that the interior-equilibrium condition in (2.8) holds.
24 This

follows straightforwardly from simplifying the prices of Lemma 3 with β = 0

64

(i) When only superior firm charges an add-on price, then the equilibrium prices are
p1A,N

2ω − αβ
αβ2
2ω
6ω − 3αβ
A,N
t+
(1 − q) R, p2 = t −
t−
(1 − q) R,
=
2
2
2
6ω − αβ
6ω − αβ
6ω − αβ
6ω − αβ2
3β
β
pˆ 1A,N =
t+
(1 − q) R,
2
6ω − αβ
6ω − αβ2

and we have p2A,N > p1A,N > 0 and pˆ 1A,N > 0.
(ii) When only inferior firm charges an add-on price, the equilibrium prices are
p1N,A = t −

αβ2
6ω − 3αβ
2ω
2ω − αβ
t+
(1 − q) R, p2N,A =
t−
(1 − q) R,
2
2
2
6ω − αβ
6ω − αβ
6ω − αβ
6ω − αβ2
3β
β
pˆ 2N,A =
t−
(1 − q ) R
2
6ω − αβ
6ω − αβ2

and p1N,A > p2N,A > 0 and pˆ 2N,A > 0.
The idea behind the equilibrium prices in Lemma 4 is follows. First, the firm who
price discriminates charges a lower base price and the intuition is same with the one in
Lemma 2. Second, as the quality differences, (1 − q) R, are larger, the base price for superior firm increases while decreases for inferior firm. When the quality difference is large,
firm 1 can attract more consumers in base good market and consequently its incentive
to higher base product price increases as well. Firm 2 on the other hand, need to lower
its base price in order to make up for the quality disadvantage. Third, the superior firm
charges a higher price to the consumer group in which it correctly anticipates the add-on
fees. In doing so, it can generate high margin per consumer in the sophisticated group,
and this marginal benefit exceeds the marginal loss from the decreased sales because the
superior firm also can generate profits from the myopic consumers through the increased
add-on fees.

65

The equilibrium profits for the first case are
π1A,N

2 ω (1 − q) R − (3ω − αβ2 )t
ω (4ω − αβ2 )
A,N
2
((
1
−
q
)
R
+
3t
)
,
π
=
=
2
2t(6ω − αβ2 )2
t(6ω − αβ2 )2

2

(2.16)

and the profits for the second case are

π1N,A

2

2 ω (1 − q) R + (3ω − αβ2 )t
=
,
t(6ω − αβ2 )2

π2N,A =

ω (4ω − αβ2 )
((1 − q) R − 3t)2 (2.17)
2t(6ω − αβ2 )2

Now we can make use of the profit levels to characterize the Nash equilibrium with
respect to the add-on pricing polices with quality differences. First consider the superior
firm. When the rival firm adopts an add-on pricing, the profit for superior firm is larger
when it also charges a separate add-on price (i.e., π1A,A > π1N,A ), provided that the interior
equilibrium condition (2.8) is satisfied). The fact that the superior firm can get higher
profits from add-on pricing holds even when the rival firm does not charge separate addon fees. For the inferior firm, we can observe same results, i.e., the inferior should charge
a separate price for the add-on no matter which pricing policy is used by the superior
firm.
Proposition 8. Given the presence of myopic consumers and asymmetric firms, add-on pricing
policy arises as the unique Nash equilibrium when the interior equilibrium condition (2.8) holds.
This finding is consistent with the well-known results in price discrimination literature
that the use of more instrument enables the industry to better extract consumer surplus,
which causes profits to rise. As a result, the firms wish to use the more ornate pricing
policy even if its rival does not. But as we observed in Proposition 7, the firms might be
worse off when they practice price discrimination at the same time. In other words, the
firm is always better off if it can price discriminate, for give pricing policies offered by
its rivals. However, once account is taken of what rivals too will do, firms in equilibrium
can be worse off when discrimination is used. Firms then find themselves in a classic
66

prisoner’s dilemma.

2.6

CONCLUSION

This paper extends prior work on add-on pricing in two dimensions. First, it allows for
asymmetry in the quality of the base good. Second, it accommodates three segments
of consumers: base consumers who do not derive utility from the add-on, sophisticated
consumers who value the add-on and correctly anticipate the add-on fee, and boundedly
rational consumers who value the add-on yet do not take the add-on price into consideration at the time of initial purchase. Each of these model considerations has substantive
implications and reverses predictions from prior research.
Specifically, previous research in add-on pricing predicts that consumers who account
for the add-on price in their base purchase decision will enjoy greater surplus when there
are more boundedly rational consumers in the marketplace. However, we show that this
is not the case when there are enough base and sophisticated consumers served by the
firms serve. In this case, an increasing number of myopic consumers results in lower base
prices and higher add-on fees. We find the add-on fee increase dominates the base price
reduction for both firms when the quality differential between the firms is not too large.
Consequently, a greater number of boundedly rational consumers can lead to increased
total prices and lower consumer surplus for the boundedly rational consumers and the
sophisticated consumers.
Our consideration of quality asymmetry (along with the inclusion of boundedly rational consumers) leads to new findings regarding firm profits. Two specific features are
discussed: quality asymmetry in the base good, and the presence of boundedly rational
consumers. Strikingly, the standard profit-irrelevancy result of add-on pricing is robust
to any one of these two features, but is challenged in the presence of both features.
We also find when both firms are engaged in price discrimination, they are worse off
compared to the case in which neither firm price discriminates. This is consistent with
67

the price discrimination literature on endogenous price discrimination model.
This research can be extended in several directions. First, an avenue for future research
is to nest the current model in a customer lifetime value framework where customers are
repeat buyers. Such a model could allow boundedly rational consumers to punish firms
that charge add-on prices by switching to competitors. The add-on prices by each firm,
and the feasibility of equilibrium add-on pricing in general, would depend on the firm’s
discount rate on future payoffs, the size of each consumer segment, and the manner in
which consumer beliefs are based on purchase history.

68

APPENDIX

69

Appendix for Chapter 2
¯ i.e., the interior-equilibrium
Proof of Lemma 1 First consider the case where β < β,
condition holds. Plug (1 − q) R = 0 into equations (2.3) and (2.4), then solve the following
first-order conditions simultaneously: ∂πi /∂pi = 0 and ∂πi /∂ pˆ i = 0 for i = 1, 2. We have
base good prices
p1∗ = p2∗ = t −

αβ
t
(1 − α)(1 − β)

(2.18)

and add-on fees

pˆ 1∗ = pˆ 2∗ =

β
t.
(1 − α)(1 − β)

(2.19)

For these prices to constitute an equilibrium, we need all following four conditions to
hold: (i) pˆ i < Rˆ for any i, (ii) second-order conditions hold at the equilibrium prices,
(iii) no firm will deviate by abandoning sophisticated consumers, and (iv) full market
¯ and (iv) holds if Rˆ
coverage. Notice that (i) is apparently true under the condition β < β,
is sufficiently large (as we assumed). We will prove that (ii) and (iii) hold in the proof of
the general case and thus they also hold in this specific case.
¯ The above interior equilibrium (2.18) and (2.19)
Next consider the case where β ≥ β.

ˆ which implies no consumer
cannot hold now because the add-on fee would surpass R,

would buy the add-on. To show that parametric conditions can lead to the corner equilibrium as in part (ii) of Lemma 1, we need the following two conditions to hold at equilibrium prices: (i) ∂πi /∂pi = 0 and (ii) ∂πi /∂ pˆ i > 0 for i = 1, 2. Without loss of generality,
we only consider i = 1. Plugging p2 = t − α Rˆ and pˆ 2 = Rˆ into (2.5), we have
π1 =

1
p1 2t − α Rˆ − p1 + α(1 − β)( Rˆ − pˆ 1 ) + α pˆ 1 2t − α Rˆ − p1 + (1 − β)( Rˆ − pˆ 1 ) .
2t
(2.20)
70

ˆ
It is straightforward to verify that, at p1 = t − α Rˆ and pˆ 1 = R,
∂π1
=0
∂p1
and

α tβ − (1 − α)(1 − β) Rˆ
∂π1
=
>0
∂ pˆ 1
2t

¯
if and only if β ≥ β.
Proof of Lemma 3 The flow of this prof is analogous to that of Lemma 1, albeit more
involved because of quality asymmetry. Recall that we need the following condition from
Assumption 3,
1
t > Rˆ + (1 − q) R.
3

(2.21)

Suppose the interior equilibrium exists. Given profit function in (2.5), we solve firstorder conditions and get equilibrium prices as presented in Lemma 3. For these prices
to constitute an interior equilibrium, we need all of the following conditions to hold: (i)
pˆ i < Rˆ for any i, (ii) second-order conditions hold, (iii) no firm deviates by abandoning
sophisticated consumers, (iv) full market coverage, and (v) marginal consumers between
0 and 1. We will also show that all regularity conditions can hold simultaneously.
First, from (2.9) we get the interior-equilibrium condition that guarantees pˆ 1 < Rˆ in
equilibrium:
1
β
β
(1 − q ) R .
< Rˆ −
ω
t
9ω − 2αβ2

(2.22)

In the subsequent analysis we will frequently use the fact that β < ω = (1 − α)(1 − β),
which directly follows the interior-equilibrium condition and Assumption 3. We will also
frequently use the fact, 9ω − 2αβ2 > 9β − 2αβ2 > 0.

71

For (ii) to be true, we need the determinant of Hessian matrix of profit fuction πi
∂2 π i ∂2 π i
∂2 π i ∂2 π i
−
H=
∂pi ∂ pˆ i ∂ pˆ i ∂pi
∂p2i ∂ pˆ 2i

= 4α(1 − β) − α2 (2 − β)2
= α(4ω − αβ2 ) > 0,
and this inequality always hold because ω > β.
We prove (iii) by contradiction. Suppose that firm 2 prefers to deviate from this equilibrium by abandoning sophisticated consumers (and thus will charge the maximum posˆ firm 2’s profit
sible add-on fee). Given p1 and pˆ 1 as specified in equation (2.9) and pˆ 2 = R,
over base and myopic consumer groups would be
π2 = p1 (1 − θm + 1 − θb ) + Rˆ (1 − θm ),
which implies that optimal deviating posted price for firm 2 as follows:
p2 =

6ω + αβ(1 − 2β)
αβ
2ω − αβ
ˆ
t−
(
1
−
q
)
R
−
R.
2ω
2(1 − α(1 − β))
2(9ω − 2αβ2 )

A contradiction arises if, given the above prices, the marginal sophisticated consumer
is located within the linear city. We have θs < 1 when plugging the above prices into
θs = 1/2 + ((1 − q) R + p2 − p1 + pˆ 2 − pˆ 1 )/(2t) (thus the contradiction) if and only if
t>

ω 6 − 8β − α(2 − β)(3 − 2β)
ω
Rˆ +
(1 − q) R.
1 − α (1 − β )
(2 − α(2 − β)) 9(1 − β) − α(3 − β)(3 − 2β)

(2.23)

Each of the two terms on the right-hand side are decreasing function of β. The first
term is apparently decreasing in β. The second, 6 − 8β − α(2 − β)(3 − 2β) / 9(1 − β) −
α(3 − β)(3 − 2β) , or equivalently, 1 − (3(1 − α) − β + 2αβ)/ 9(1 − β) − α(3 − β)(3 −

72

2β) , is decreasing in β because

∂

3(1 − α) − β + 2αβ
9(1 − β) − α(3 − β)(3 − 2β)

∂β =

9(1 − α)(2 − α) + 2αβ (6 − β)(1 − α) + αβ
9(1 − β) − α(3 − β)(3 − 2β)

2

> 0.

Therefore, if the above inequality (2.23) holds at β = 0, it will hold for any positive
β. At β = 0, (2.23) simplifies to t > Rˆ + (1/3)(1 − q) R, which is our Assumption 3.
Thus the contradiction. For (1 − q) R = 0, the inequality (2.23) at β = 0 simplifies to
Assumption 2. The analysis of firm 1 abandoning sophisticated consumers is analogous
and thus omitted.
For (iv), we now derive the full-market-coverage condition. The market is fully covered in equilibrium if the marginal consumer in each segment who is indifferent between
purchasing product 1 and 2 derives a non-negative utility, i.e., the following three inequalities hold:
R − p1 − tθb > 0,
R + Rˆ − p1 − tθm > 0,
R + Rˆ − p1 − pˆ 1 − tθs > 0.
Plugging equilibrium prices into these inequalities (and afterward it is straightforward
that the second inequality will hold if the first one holds), we can simplify the first and
the third inequalities to the following, respectively:
u θb
u θs

3ω − 2αβ
1
R − (1 − q ) R −
t ≥ 0,
2
2ω
1
3−β
R + Rˆ − (1 − q) R −
t ≥ 0.
2
2(1 − β )

Therefore, full market coverage is true if
min u θb , u θs
73

≥ 0.

(2.24)

We refer to this as the full-market-coverage condition.
For (v), we simplify the location of the marginal consumer who is indifferent between
firm 1 and firm 2 at the equilibrium prices for each segment:
1 (1 − β)(3 − α(3 − 2β))
+
(1 − q) R,
2
2t(9ω − 2αβ2 )
1 (1 − β)(3 − α(3 − 2β))
θm = +
(1 − q) R,
2
2t(9ω − 2αβ2 )
1 3 − 5β − α(1 − β)(3 − 2β)
θs = +
(1 − q) R.
2
2t(9ω − 2αβ2 )
θb =

It can be shown that these values are each between 0 and 1 given Assumption 3 and the
interior-equilibrium condition.
We next show that all regularity conditions for the interior equilibrium are feasible
to hold simultaneously: Assumption 3, the interior-equilibrium condition, and the fullmarket-coverage-condition,. We simplify this feasibility proof using the following fact:
all sides of all the above conditions are continuous functions of (1 − q) R. Therefore, if
all three conditions can hold strictly at (1 − q) R = 0, it must be true that they will also
hold when (1 − q) R is close enough to zero. Applying (1 − q) R = 0 to the above three
regularity conditions, i.e. into equations (2.21), (2.22) and (2.24), they are simplified to
ˆ
t > R,
β
1ˆ
< R,
ω
t
3ω − 2αβ
3−β
min R −
t, R + Rˆ −
t
2ω
2(1 − β )

(2.25)
(2.26)

≥ 0.

(2.27)

When (1 − q) R = 0, the left-hand side of (2.25) goes to zero, thus this first condition

holds if β is low enough. When (1 − q) R = 0, (2.26) holds if Rˆ is sufficiently small relative
to t. Only the third condition (2.27) involves R. When (1 − q) R = 0 and for any given
values of all other model parameters, (2.27) holds if R is large enough.
Given that all sides of all three regularity conditions for this general model are con74

tinuous function of (1 − q) R, we can now conclude that all regularity conditions for the
existence of the interior equilibrium will hold simultaneously when (i) (1 − q) R is small
enough, (ii) β is low enough, (iii) Rˆ is sufficiently small relative to t, and (iv) R is large
enough.
Finally, we show how each firm’s total price changes in β:

(1 − α) 9 − α(3 − 2β)2
t
∂( p1 + pˆ 1 )
=
+
(1 − q) R > 0,
∂β
(1 − β )2
(9ω − 2αβ2 )2
(1 − α) 9 − α(3 − 2β)2
t
∂( p2 + pˆ 2 )
=
−
(1 − q ) R
∂β
(1 − β )2
(9ω − 2αβ2 )2
This last expression is positive if and only if condition (2.11) holds.
Proof of Proposition 7 From the equilibrium profit in (2.12), (2.13), and (2.15), the firm’s
profit differentials between when add-on pricing is adopted and when add-on pricing is
not adopted are
π1A,A − π1N,N = −

αβ2 3t − (1 − q) R
(1 − q ) R < 0
9t(9ω − 2αβ2 )

(2.28)

and
π2A,A

− π2N,N

αβ2 3t + (1 − q) R
=
(1 − q) R > 0.
9t(9ω − 2αβ2 )

(2.29)

It should be noted that, from (2.14), the pricing equilibrium without add-on fees results in p2N,N = t − (1 − q) R/3. Thus, (1 − q) R < 3t is a necessary condition for both
firms to sell to consumers when add-on pricing is not adopted or prohibited.
Add-on Usage Dependent on Price The base consumers never use the add-on regardless of whether it is free or not. In practice, however, there might be scenarios where
consumers who do not need a product or service asks for one if it is offered for free. In

75

this section, we show that our results are robust to such consumer behavior. Consider a
fraction, ρ, of the base consumers who will consume the add-on if it is provided for free
and will not consume the add-on if p1 > 0. Thus, we now have four consumer segments:
sophisticated consumers, boundedly rational consumers, base consumers who never use
add-ons regardless of whether it is free or not, and base consumers who will use add-ons
only if it is free.
Firm i’s profit can now be written as
ˆ i − 1 pˆ =0 · ρ( Di − D
ˆ i )c,
ˆ
πi = ( pi − c) Di + ( pˆ i − cˆ) D
i
where 1 pˆ i =0 is an indicator function that takes the value 1 if pˆ i = 0 and 0 otherwise.
ˆ i )cˆ captures the fact that a fraction ρ of the consumers
Notice the term 1 pˆ i =0 · ρ( Di − D
ˆ i ) will use the add-on if and only
who do not buy the add-on (which is equal to Di − D

if pˆ i = 0. Without add-on pricing, each firm’s optimization problem max pi πi | pˆ i =0 is
concave (∂2 πi /∂p2i = −1/t < 0), and simultaneously solving the first-order conditions
leads to equilibrium prices of
1
p1 = t + c + (α + ρ − αρ)cˆ + [α(1 − g) Rˆ + (1 − q) R],
3
1
p2 = t + c + (α + ρ − αρ)cˆ − [α(1 − g) Rˆ + (1 − q) R].
3
The price effect due to the existence of price-dependent add-on usage is ∂pi /∂ρ = (1 −
α)cˆ ≥ 0, i.e., by increasing and charging a higher price to consumers, the firms try to
compensate the loss from the ‘cherry-picking’ consumers who use add-ons only if it is
provided for free. Notice that for the case of cˆ = 0, the prices are identical with the result
in the benchmark. Intuitively, if the firms do not incur any cost to produce or provide
add-on then consumer’s behavior on add-on good does not affect firm’s profit, so their
optimization problems remain unchanged.

76

When g = 1, equilibrium profits for the firms implied by these prices are
π1N,N =

1
(3t + (1 − q) R)2 ;
18t

π2N,N =

1
(3t − (1 − q) R)2 ,
18t

which give the same profits with the case of α = 0, (see (2.15)). If we assume that α is
common knowledge, each firm is able to set the base good price based on this information
and indeed the firms’ optimal prices are given as the function of α. Thus, the firms can
easily attain the original equilibrium profits by adjusting the equilibrium prices and we
don’t have any other reason to have different level of equilibrium profits. Thus, we can
conclude that Proposition 5 and Proposition 7 are trivially preserved with an introduction
of ‘cherry-picking’ consumers.
Inattention to Add-on Price In the basic model, we assumed that myopic consumers
mistakenly anticipate that they will not be charged for add-ons. However, as more firms
adopt add-on pricing schemes, consumers are now well aware of the extra fees charged
for an add-on. With such learning, one may argue that having boundedly rational consumers who think that any extra goods or services can be utilized for free is not a realistic
assumption. To investigate this issue, the model needs to be modified to allow the possibility of different specifications on boundedly rational consumers. More specifically, let
us assume consumers are rational and know they will be charged for the add-on, but do
not know the add-on price before deciding to purchase the base good. As such, there are
two consumer segments: (1 − α) consumers for whom Rˆ = 0 and α consumers for whom
Rˆ > 0.

Claim 1. Consumers will expect that after they have committed to a firm by buying its base
product, the add-on price will be at the maximum level.
Proof. For any consumer belief, firms optimally charge their maximum possible add-on
fees (i.e., Rˆ for firm 1 and g Rˆ for firm 2). If firms charge above this price, no consumers
will buy the add-on in Stage 4. If firms charge below this price, then the add-on margin
77

will be lower without an effect on sales. Thus, the only rational belief is that the add-on
fees will be at their maximum.
Claim 2. Profit can be expressed as π1 = ( p1 + α Rˆ − αcˆ − c)(t − p1 + p2 + (1 − q) R)/2t

and π2 = ( p2 + αg Rˆ − αcˆ − c)(t − p2 + p1 − (1 − q) R)/2t for superior and inferior firms,
respectively.
Claim 2 follows from the analysis of marginal base consumer and the marginal conˆ
sumer who values the add-on and rationally expects pˆ 1 = Rˆ and pˆ 2 = g R.
The first-order conditions are uniquely satisfied at p1∗ = t − (3α Rˆ − (1 − q) R)/3 and

p2∗ = t − (3α Rˆ + (1 − q) R)/3 leading to equilibrium profits of
1
(3t + (1 − q) R)2 ,
18t
1
(3t − (1 − q) R)2 ,
π2∗ =
18t
π1∗ =

(2.30)
(2.31)

provided differentiation between firms is such that both firms compete for both consumer
segments.
In comparison, when add-on pricing is not used, the superior and inferior firms maximizes
1
p1 ( t − p1 + p2 + (1 − q ) R ) − α ( t − p1 + p2 + (1 − q ) R ) ,
2t
1
π2 =
p2 ( t + p1 − p2 − (1 − q ) R ) − α ( t + p1 − p2 − (1 − q ) R ) .
2t

π1 =

The first-order conditions are uniquely satisfied at p1∗ = t + αcˆ + c − (2α Rˆ + gα Rˆ − (1 −

q) R)/3 and p2∗ = t + αcˆ + c − (α Rˆ + 2gα Rˆ − (1 − q) R)/3, leading to equilibrium profits

of π1∗ = (3t + α(1 − g) Rˆ + (1 − q) R)2 /18t and π2∗ = (3t − α(1 − g) Rˆ − (1 − q) R)2 /18t

provided differentiation between firms is such that both firms compete for both consumer
segments.
The difference in profit from add-on pricing relative to when add-on pricing is not
78

used are π1∗ − π1N = −3α Rˆ (3t + (1 − q) R) < 0 and π2∗ − π2N = −3α Rˆ (3t − (1 − q) R) < 0.
Thus, add-on pricing decreases profits for both the superior firm and the inferior firm,
and we can conclude that the qualitative results derived with the consumers who are
ignorant of add-on fees are robust to the relaxation of this assumption.

79

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84

CHAPTER 3
CONSUMER SEARCH WITH PRICE
COMPETITION
3.1

INTRODUCTION

Models of search costs routinely assume that consumers are completely uninformed about
a firm’s offer before they visited that firm. Only after incurring some search costs and visiting a firm does the consumer learn the prices that the firm charges and, in the case of
differentiated products, how much it likes the product that the firm has on offer. Yet, in
the real world, things are often different. With the advent of the Internet, for example, it
becomes very easy to compare prices. There are many search engines that list all prices
without requiring any effort on behalf of the consumer. Also, it is very easy to find some
characteristics of a product online, without having to make a costly visit to the store. Yet,
some product characteristics are impossible to digitize. A consumer that wants to buy
a car can easily find all technical specifications and prices online, yet still has to make a
costly visit to the dealer to be able to kick the tires and take it for a test drive. Similarly,
the consumer that wants to buy a new jeans may find many details and pictures of this
product online, which allows her to check whether she likes the design, but she still has
to make a costly visit to the store in order to try them on.
In this paper, we try to model these cases. We consider products that feature two
dimensions of horizontal product differentiation. One characteristic is observable for free,
but the other requires a costly visit to the store. We refer to the first characteristic as
observable, and to second as hidden. Prices are readily observable as well. We thus build

85

on models of consumer search with differentiated products by adding a characteristics
that can be observed for free, but by also assuming that prices can be observed at zero
costs.
We find that this has profound implications. Rather than having prices increasing in
search costs, which is true for all consumer search models that we are aware of, we find
that prices decrease in search costs. The intuition is as follows. Different from most other
search models, the order of search in our model is not random. Rational consumers will
first visit the firm where they expect the best deal, that is the firm that offers the best
combinations of price and observable characteristic for this consumer. As search costs
increase, however, consumers are less likely to walk away from a firm that they have
visited. Hence, as search costs increase, it becomes more important for a firm to attract
consumers at their first search. The only way to do so is by charging a lower price.
This intuition suggests that it is crucial for our results that prices are readily observable. We show that this is indeed the case. If consumers have to search for prices as well
as for the hidden characteristic, prices increase in search costs.
Our model builds on the literature on search with differentiated products, pioneered
by Wolinsky (1986) and Anderson and Renault (1999).1 Yet, different from those papers,
we assume that firms are not visited at random. Other papers also drop that assumption.
In Chen and He (2011) and Athey and Ellison (2011) firms pay for placing advertisements
on a search engine. A firm has private information about its quality, that is, the probability
that consumers like its product. In equilibrium, higher placed advertisements represent
higher quality products and consumers rationally search firms in the order in which their
ads are placed. In Arbatskaya (2007), the search order is exogenously given. Prices fall
in the order of search: a consumer that walks away from a firm reveals that she has low
search costs, giving the next firm an incentive to charge a lower price. Zhou (2011) studies
a similar model with differentiated products but finds the opposite effects on price: a con1 Weitzman

(1979) is an earlier paper which studies the general optimal search among options with
stochastic match values. But there is no supply side model.

86

sumer who walks away has fewer options left and is thus price sensitie. In Wilson (2010)
a firm can choose the size of search cost consumers have to incur to visit it. Consumers
are then more likely to first visit firms with low search cost, and prices fall in the order of
search. Armstrong et al. (2009) study a search market with differentiated products where
one firm is always visited first, while the other firms are sampled randomly if a consumer
decides not to buy from the pioneered firm. Haan and Moraga-Gonzalez (2011) study a
model of search with differentiated products where the order of search is affected by the
amount of advertising.
In the literature on search in labor markets, recent papers have analyzed “directed
search”, where employers advertise their wages and workers choose their search order
accordingly. An important additional feature in the labor market relative to a typical
consumer market is that a job vacancy can be filled, so that workers do not necessarily
visit the highest-wage employers first if they anticipate that many other workers will
apply for the same post.2
The remainder of this paper is structured as follows. In the next section, we set up the
model. We solve for the equilibrium in Section 3.3, and show that prices are decreasing
in search costs. Section 3.4 considers the case in which prices are hidden as well. In that
case, prices are increasing in search costs. Section 3.5 considers an extension in which
consumers have to pay a small search costs to observe prices and the unobservable characteristic. We show that, if this cost is low enough, our main result still holds. Section 3.6
concludes.

3.2

THE MODEL

Consider a market where two firms sell horizontally differentiated products. Marginal
costs are constant and normalized to zero. For simplicity, we assume that there is one
2 See

Rogerson et al. (2005, section 5) for discussion of these models.

87

consumer. She derives utility
ui ( pi ) = v +

i

+ ηi − p i ,

if she buys product i at price pi , i = 1, 2. The term v is the gross utility of buying the worst
possible product. We assume that v is high enough, so the market is always covered. The
term

i

+ ηi can be interpreted as the match value between this consumer and the product

of firm i. This match value consists of two components: the observable component ηi
reflects characteristics that can be observed without visiting firm i, the hidden component
i

can only be observed upon visiting firm i. We assume that

i

is the realization of a

random variable that is uniformly distributed on [0, 1], while ηi is the realization of a
random variable that uniformly distributed on [0, h], with h some parameter.
The consumer must incur search costs s if she want to visit a firm to learn the value
of

i.

We assume that the search process is without replacement and there is costless

recall (i.e., the consumer can return to any firm she has visited without extra cost). We
√
assume that h < 1 − 2s. As we will show in our analysis, this assumption assures
that the observable characteristics ηi and η j can never be such that the consumer knows
in advance that she will buy from firm i for sure, regardless of the value of

i

that she

encounters. In other words, we assume that the hidden characteristic can always at least
potentially influence the consumer’s buying behavior. Hence, we require the observable
characteristic to be more noisy than the hidden characteristic. Firms can never observe
either

i

or ηi , so practicing price discrimination is not feasible. Crucially, and different

from most of the search literature, we assume that prices are observable as well. Having
the observable characteristic then becomes crucial to assure pure strategy equilibria.3
For ease of exposition, we define ∆η as the difference in the value of the observable
3 Suppose

that we would not have an observable characteristic. In a symmetric pure strategy equilibrium, both firms would charge the same price and attract half of consumers on their first visit. Yet firms
strictly benefit from being visited first. Slightly undercutting the tentative equilibrium price would then
lead to a discrete increase in first-visits and hence sales, which implies that a firm would benefit from such
a defection, and hence a symmetric pure strategy equilibrium.

88

characteristic for both firms, and ∆ p as the difference in prices:
∆ η ≡ η2 − η1 ,
∆ p ≡ p2 − p1 .
Also, we define ∆ as the net advantage that firm 1 has over firm 2, so
∆ ≡ ∆ p − ∆η .
For the purpose of the analysis, it is also useful to define the probability density of ∆η .
Note that
Γ(z) ≡ Pr(∆η < z) =

max{min{z+η1 ,h},0}
0

h
0

dG (η1 )dG (η2 ).

With a uniform distribution of G on [0, h] this implies

Γ(z) =






1
( h + z )2
2h2



1 −

1
( h − z )2
2h2

if

z≤0

if

z≥0

which in turn implies
γ(z) =




 1 (h + z)

if

z≤0



 12 (h − z)

if

z≥0

h2
h

3.3

ANALYSIS

Consider the best reply of firm 1 to some price p2 . Suppose that firm 1 chooses to set
p1 < p2 . A consumer first goes to firm 1 whenever it expects a better deal there, thus if
v + η1 + E [ 1 ] − p 1 > v + η2 + E [ 2 ] − p 2 ,

89

which simplifies to
∆ > 0.
Total sales of firm 1 will consist of three groups of consumers. First, there will be consumers who have visited firm 1 first, and sufficiently like the hidden characteristic to buy
from 1. Second, there are consumers that visit firm 2 first, but find the hidden characteristic at that firm so disappointing that they are willing to incur additional search costs s to
go to firm 1 and buy there. Third, there are consumers that first visit firm 1, find the hidden characteristic at that firm so disappointing that they are willing to incur additional
search costs s and go to firm 2, only to find out that the hidden characteristic at firm 2 is so
adverse that they prefer to buy from firm 1 anyway. We will consider these three groups
in turn.
Let ω ∈ {1, 2, 12, 21} denote which firms a particular consumer visits, and in what
order. Thus ω = 12 implies that the consumer has first visited firm 1, and then firm 2.
Let qiω denote total demand for firm i from such consumers. Thus q12
1 denotes demand
for firm 1 from consumers that visit firm 1 and 2 in that order, while q11 denotes demand
for firm 1 from consumers that only visit firm 1. Similarly, we define Diω as the set of
consumers that buy from firm i after having visited the firms in ω in the given order.
We first consider consumers that visit firm 1 first and decide to buy there right away.
Given that they prefer to come firm 1 first, these consumers have ∆ > 0. They will immediately buy if the expected gains from search are lower than search costs. The expected
benefits of sampling one more firm equal
1
1 +∆

( −

1

− ∆)d =

[1 − (

1

+ ∆)]2

2

.

(3.1)

A consumer stays at this firm if this is smaller than search costs

[1 − (

1

+ ∆)]2

2

90

<s

(3.2)

or4
1

> 1−

Note that necessarily ∆η > −h > −(1 −

√

√

2s − ∆.

(3.3)

2s), so for small deviations in price we indeed

have that the right-hand side of this inequality is strictly positive, hence that there is
some probability that the consumer will walk away. Also not that ∆ > 0 implies that
the right-hand side is strictly smaller that 1, which implies that the crucial value of

1

is

well-defined.
Let’s define
c≡

√

2s,

so the consumer buys from firm 1 if

1

> 1 − c − ∆.

Define
D11 ≡ {η1 , η2 ,

1, 2

: ∆ > 0 and

1

> 1 − ∆ − c }.

The mass in this set is demand of group 1:
q11 =

D1

dΓ(∆η )dF ( 1 )

(3.4)

so
q11 =

∆p

−h

1
1− c − ∆ p + ∆ η

f ( 1 )γ(∆η )d 1 d∆η

(3.5)

As the probability density function of ∆η has a kink at ∆η = 0, we have to split this
4 Note

that we implicitly√assume that, for the cut-off consumer,
the cut-off value yields 1 − 2s − ∆ < 1 − ∆.

91

1

+ ∆ < 1 so

1

< 1 − ∆. Plugging in

expression to obtain
∆p
1
1
(
h
+
∆
)
d
d∆
+
η
η
1
− h 1− c − ∆ p + ∆ η h 2
0
1− c − ∆ p + ∆ η
1
1
1
1
= c + h + ∆ p − 2 ∆ p (3c∆ p − 3h∆ p + ∆2p − 6ch).
2
6
2
6h

q11 =

0

1

1
(h − ∆η ) d 1 d∆η
h2

Now consider the consumers that go to the other firm first, walk away, then buy from
firm 1. A consumer goes first to firm 2 if ∆ < 0. Similar to the analysis above, she
decided not to buy right away if

2

< 1 + ∆ − c. After having visited firm 1 as well, this

consumer has observed all relevant parameters, and hence decides to buy from firm 1 as
if

2

< ∆+

1.

We thus have

D121 ≡ {η1 , η2 ,

1, 2

: ∆ < 0 and

2

< 1 + ∆ − c and

2

< ∆+

1 }.

(3.6)

For analytic convenience, it is useful to partition this set into two subsets, depending on
whether or not

1

< 1 − c:

21
21
D121 = D1a
∪ D1b

with
21
D1a
≡ { η1 , η2 ,

1, 2

: ∆ < 0 and

1

< 1 − c and

2

< ∆+

21
D1b
≡ { η1 , η2 ,

1, 2

: ∆ < 0 and

1

≥ 1 − c and

2

< 1 + ∆ − c}

1}

21
21
Total demand from these consumers equals q21
1 = q1a + q1b , with

q21
1a =

=
=

h
∆p

1− c

∆+

h

−∆
0
1− c

∆p

−∆ p +∆η

1

dF ( 2 )dF ( 1 )dΓ(∆η )

∆ p −∆η +
0

1

1
(h − ∆η ) d 2 d 1 d∆η
h2

1
(h − ∆ p )2 (6c2 + 4ch − 4c∆ p − 12c + h2 − 2h∆ p − 4h + ∆2p + 4∆ p + 6)
24h2

92

and
1+ ∆ p − ∆ η − c

1

h

q21
1b =

∆p

1
(h − ∆η ) d 2 d 1 d∆η
h2

1− c 0

1 c
( h − ∆ p )2 3(1 − c ) + ∆ p − h .
6 h2

=

Finally, there will be consumers that start out at firm 1, then walk away, but do come
back. For these people, to have them start out at firm 1, we require ∆ > 0. For them to
then walk away requires

1

< 1 − ∆ − c. For them to prefer firm 1 anyway, after having

observed the offer of firm 2, requires
D112 = {η1 , η2 ,

1, 2

2

< ∆+

: ∆ > 0 and

1

1.

Hence

< 1 − ∆ − c and

2

< ∆+

1 }.

We thus have
q12
1 =

∆p

−h

1− ∆ p + ∆ η − c
0

∆ p −∆η +
0

1

dF ( 2 )dF ( 1 )dΓ(∆η )

As the probability density function of ∆η has a kink at ∆η = 0, we have to split this
expression to obtain
0

1− ∆ p + ∆ η − c

∆ p −∆η +

1
(h + ∆η ) d 2 d 1 d∆η
h2
−h 0
0
∆p
1− ∆ p − ∆ η − c
∆ p −∆η + 1
1
+
(h − ∆η ) d 2 d 1 d∆η
h2
0
0
0
1
1
1
1
1
= (1 − c)2 − h2 − h∆ p − ∆2p +
6(1 − c)2 (2h − ∆ p ) + ∆2p (∆ p − 4h) .
2
4
24
6
4
24h

q12
1 =

1

Total demand for firm 1 consists of consumers that visit firm 1 first and buy immediately, those that visit firm 2 first, then walk away and buy from firm 1, and consumers
that visit firm 1 first, then go to firm 2, but end up buying from firm 1 anyway. Thus
12
q1 = q11 + q21
1 + q1 .

93

Adding these expression yields
c2
1
1 3
1
c2 2 1
4
∆ + ∆p − 2 ∆p + .
q1 = ∆ p (3 − h ) − ∆ p +
3
3h
h
2
12h2 p
2h
Profits equal
Π1 = p1 q1 .
Taking the first-order condition and imposing symmetry, we have
1 ∗
c2
1
hp − p∗ − p∗ + = 0
3
h
2
which yields equilibrium price
p∗ =

6c2

3h
.
+ 6h − 2h2

(3.7)

From this, we immediately have
Proposition 9. When prices are observable, equilibrium prices are strictly decreasing in the search
costs of the hidden characteristic. In other words,
∂p∗
< 0.
∂c

(3.8)

The intuition is as follows. Rational consumers will first visit the firm where they
expect the better deal, that is the firm that offers, for this consumer, the best combination
of price and observable characteristic. As search costs increase, however, consumers are
less likely to walk away from a firm that they have visited. Hence, as search costs increase,
it becomes more important for a firm to attract consumers at their first search. The only
way to do so is by charging a lower price.

94

To determine the effect of a change in h, note that
∂p∗
3(3c2 + h2 )
=
> 0.
∂h
2(h2 − 3c2 − 3h)2
Thus, as the hidden characteristic becomes more important, prices are higher.

3.4

THE CASE OF HIDDEN PRICES

Suppose that we are in a world with one observable and one hidden characteristic, as
we had above, but where prices are also hidden. Hence, in this case, to learn the price
consumers also have to fist visit a firm. The decision which firm to visit first now only
depends on the observable characteristic: a consumer will visit firm 1 first if ∆η ≡ η2 −
η1 < 0. We look for the equilibrium price p∗ that is such that, if firm 2 charges p∗ , it is a
best reply for firm 1 to do the same.
As in the standard search literature, we impose that consumers have “passive beliefs”
about the equilibrium price, and the price p a consumer is offered does not alter her
anticipated p∗ . This restriction implies that a consumer’s search and purchase behavior
can be described by an optimal stopping rule: she buys the current product if the utility
she obtains is more than or equal to a threshold value, and continues searching otherwise.
When the consumer faces the decision whether to buy from firm 1 immediately, however, she faces a decision similar to that above: she will walk away if she expects that also
visiting firm 2 is not worth her while, hence if

1

> 1 − ∆ − c,

where again ∆ ≡ ∆ p − ∆η , but now ∆ p ≡ p∗ − p1 . Hence ∆ p is not the true difference
between firm 1 and firm 2, but the price difference that the consumer expects after having
observed p1 and assuming that firm 2 charges p∗ . Using the same notation as in the

95

previous section, we now have
D11 ≡ {η1 , η2 ,

1, 2

: ∆η < 0 and

1

> 1 − ∆ − c }.

This implies
q11 =

=

0

1

− h 1− c − ∆ p + ∆ + η
0
1

dF ( 1 )dΓ(∆η )
1
(h + ∆η ) d 1 d∆η
h2

− h 1− c − ∆ p + ∆ + η

1
1
1
= c + h + ∆p.
2
6
2

Consider consumers that visit firm 2 first. These consumers have ∆η > 0. They decide
not to buy at firm 2 if

2

< 1 − ∆η − c, as these consumers assume that firm 1 also charges

the equilibrium price. They buy from firm 1 if it turns out that
D121 ≡ {η1 , η2 ,

1, 2

: ∆η > 0 and

2

2

< 1 + ∆η − c and

<

2

1

+ ∆. Hence

< ∆+

1 }.

It is convenient to partition this set into two subsets, depending on whether or not
1 − c − ∆p:

21
21
D121 = D1a
∪ D1b

with
21
D1a
≡ { η1 , η2 ,

1, 2

: ∆η > 0 and

1

< 1 − c − ∆ p and

2

< ∆+

21
D1b
≡ { η1 , η2 ,

1, 2

: ∆η > 0 and

1

≥ 1 − c − ∆ p and

2

< 1 − ∆ η − c }.

96

1 },

1

<

21
21
Total demand from these consumers equals q21
1 = q1a + q1b , with
1− c − ∆ p

h

q21
1a =

0

∆+

−∆
h
1− c − ∆ p

0

1

dF ( 2 )dF ( 1 )dΓ(∆η )

∆ p −∆η +

1
(h − ∆η ) d 2 d 1 d∆η
h2
0
−∆ p +∆η 0
1
1
1
1
1
1
= c2 + ch − c + h2 − h +
4
6
2
24
6
4

=

1

and
1− ∆ η − c

1

h

q21
1b =

1− c − ∆ p

0

1
(h − ∆η ) d 2 d 1 d∆η
h2

0

1
= (c + ∆ p )(3(1 − c) − h).
6
Finally, there will be consumers that start out at firm 1, then walk away, but do come
back. For these people, to have them start out at firm 1, we require ∆η < 0. For them to
then walk away requires

1

< 1 − ∆ − c. For them to prefer firm 1 anyway, after having

observed the offer of firm 2, requires
D112 = {η1 , η2 ,

1, 2

2

< ∆+

: ∆η < 0 and

1

1.

Hence

< 1 − ∆ − c and

2

< ∆+

We thus have
q12
1 =

0

1− ∆ p + ∆ η − c

∆ p −∆η +

−h 0
0
0
1− ∆ p + ∆ η − c
∆ p −∆η +

1

dF ( 2 )dF ( 1 )dΓ(∆η )

1
(h + ∆η ) d 2 d 1 d∆η
h2
−h 0
0
1
1
1
1
1
1
= c2 − c − h2 − h∆ p − ∆2p +
4
2
24
6
4
4

=

1

Total demand again equals
12
q1 = q11 + q21
1 + q1

97

1 }.

Adding these expressions yields
1
1
1
1
q1 = ∆ p − c∆ p − h∆ p − ∆2p +
2
3
4
2
Profits equal
Π1 = p1 q1 .
Taking the first-order condition and imposing symmetry yields:
1
1
1
− p∗ 1 − c − h
2
2
3

= 0,

so
p∗ =

3
.
6 − 3c − 2h

(3.9)

Proposition 10. When prices are hidden, equilibrium prices are strictly increasing in search costs.
That is,
∂p∗
> 0.
∂c

(3.10)

Note that this is in line with the intuition we gave for our main result. With observable
prices, an increase in search costs implies that firms are more eager to attract consumers
on their first visit, and hence to lower prices. Once prices are also hidden, as we consider
here, this effect is no longer present and we are back to the standard result that prices
increase in search costs.
The effect of a change in h is qualitatively the same as that in the case of observable
prices: as the hidden characteristic becomes more important, prices increase.

3.5

EXTENSION

So far, we assumed that although hidden characteristics are costly to observe, information about prices and observable characteristics come for free. Of course, this is a simpli98

fication: in the real world there may also be some costs involved from observing these,
although they are relatively small compared to the costs of observing the hidden characteristics. Still, ever since Diamond (1971) it is well known that even infinitesimally small
costs may fundamentally change the nature of equilibrium. In this section we argue that
in our model, that is not the case.
Suppose that it is also costly to observe price and ‘observable’ characteristic, and these
cost equal s per firm, where s

s. An individual consumer now picks one firm at ran-

dom, and observes price and the observable characteristic at a price s . She now has two
options: She either also inspects the hidden characteristic of firm 1, or she first inspects
price and the observable characteristic at firm 2, such that she can decide on the basis of
that which firm’s hidden characteristic to observe. In the latter case, this consumer thus
uses the approach described in this paper.
In the case that s = 0, it is easy to see that this consumer strictly prefers to first do a
check of the observable characteristics at firm 2. By obtaining this information, she will be
able to make a better choice regarding which firm’s hidden characteristic to inspect first,
and hence she will be less likely to have to make additional costly searches, and hence
also be likely to find a better match on average.
But if this is true for the case that s = 0, it will also be true for very small s . Also
if prices and observable characteristics of each firm are costly to observe, it will still be
worth her while for her to do so, rather than to go look for the hidden characteristic of a
firm right away. In turn that implies that also for small s , our analysis still holds true in
equilibrium, and hence we still have that equilibrium price are decreasing in search costs
of the hidden characteristic.

3.6

CONCLUSION

In this paper we considered a model in which consumers can readily observe prices and
some product characteristics, but have to incur search costs to be able to observe other
99

crucial product characteristics. In such an environment, we showed that an increase in
search costs decrease equilibrium prices, as higher search costs imply that consumers are
less likely to walk away from a firm, which implies that firms are more eager to attract
them. We also showed that this result is still valid if there are small search costs involved
to observe both the price and the product characteristics that we assume to be readily
observable.

100

REFERENCES

101

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