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Essays on consumer product evaluation and online shopping intermediaries
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Content
Essays on Consumer Product Evaluation and Online Shopping Intermediaries
by
Lin Liu
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
May 2014
Copyright 2014 Lin Liu
1
DEDICATION
I would like to dedicate my dissertation to
my parents, Chunhua Ye and Xingyuan Liu, for their unreservedly support;
my wife, Luxin Wang, for all her love and encouragement;
and my sweet little girl, Aijia Liu, for the endless happiness she brings to us.
2
ACKNOWLEDGEMENTS
I would like to thank my dissertation advisors Dr. Anthony Dukes and Dr. Sha Yang for their
constant guidance, support, and inspiration throughout my fiveyear study towards my Doctoral
Degree. Without their advice and encouragement, it would not be possible for me to reach where
I am now. I am also indebted to my committee members Dr. Shantanu Dutta, Dr. Matthew
Selove, and Dr. Guofu Tan for their insightful suggestions and discussions.
3
TABLE OF CONTENTS
DEDICATION……………………………………………………………………………………………..1
ACKNOWLEDGEMENTS……………………………………………………………………..................2
LIST OF TABLES…………………………………………………………………………………………5
LIST OF FIGURES………………………………………………………………………………………..6
ABSTRACT……………………………………………………………………………………..................7
CHAPTER ONE: CONSIDERATION SET FORMATION: THE CASE OF WITHINFIRM &
ACROSSFIRM EVALUATION COSTS…………………………………………………………………9
1. Introduction……………………………………………………………………………………9
2. Model………………………………………………………………………………...............15
2.1 Consumer Choice and Consideration Set Formation ………………………………........15
2.2 Firms’ Game in Prices and Product Line Length ………………………………………..21
2.3 The Equilibrium and Its Key Properties……………………………………………........24
2.4 Additional Comparative Statics………………………………………………………….29
3. Equilibrium versus the Social Optimum……………………………………………...............32
4. Conclusion…………………………………………………………………………………….35
CHAPTER TWO: CONSUMER SEARCH WITH LIMITED PRODUCT EVALUATION…................37
1. Introduction…………………………………………………………………………………..37
2. A Model of Partial Evaluation…………………………………………………………….…41
3. MultiProduct Firms…………………………………………………………………………55
4. Conclusion…………………………………………………………………………...............62
CHAPTER THREE: ONLINE SHOPPING INTERMEDIARIES: THE STRATEGIC DESIGN OF
SEARCH ENVIRONMENTS……………………………………………………………………………64
1. Introduction……………………………………………………………………………..…...64
2. Model……………………………………………………………………………..…….…....68
4
3. Heterogeneous Consumers…………………………………………………………………77
4. Intermediary Competition………………………………………………………….............79
5. Conclusion………………………………………………………………………….............83
REFERENCES…………………………………………………………………………………………..85
APPENDICES…………………………………………………………………………………...............89
APPENDIX 1……………………………………………………………….…………………..89
APPENDIX 2……………………………………………………………….…………………..98
APPENDIX A………………………………………………………………………….98
APPENDIX B………………………………………………………………………….103
APPENDIX 3……………………………………………………………….………………….108
5
LIST OF TABLES
TABLE 1. Notation…………………………………………………. ……………………………….17
TABLE 2. Optimal Evaluation Plans for Both Consumer Types ……………………………………78
6
LIST OF FIGURES
FIGURE 1. Consumer’s Optimal Sample Plan with Corner Solution and Interior Solution……………..20
FIGURE 2. Equilibrium Prices with Two Restricted Evaluation Modes…………………………………48
FIGURE 3. Equilibrium Prices with Unrestricted Evaluation Depth……………………………………..53
FIGURE 4. Consumer Surplus and Social Welfare…………………………………….………………...54
7
ABSTRACT
My dissertation aims to advance our understanding of consumer product evaluation and its managerial
implications in ecommerce. The conventional search models normally focus on one dimension (e.g.
across firms). However, consumer search is often multidimensional (e.g. attribute and product line) and
consumers are free to explore all these possibilities. In this dissertation, I focus on multidimensional
search: within a product (e.g. attributes), within a firm’s product line (e.g. how many products from a
firm). I show that search among these dimensions bring interesting marketing insights in various e
commerce settings.
My dissertation first considers a theoretical setting in which firms carry multiple products and
consumers incur evaluation costs, not only across firms, but also within firms. Consumers judiciously
decide the number of firms to include in their consideration sets as well as how many products from those
firms. This decision depends on the relative tradeoffs of evaluating an additional product and whether it
is from a firm already included in the consideration set or from an entirely new firm. The composition of
the consumer’s consideration set affects how firms compete in prices and in the number of products to
offer. Contrary to previous literature, firm differentiation can reduce firms’ product lines and withinfirm
evaluation costs have either a positive or a negative effect on firms’ prices. Interestingly, we show that
withinfirm evaluation costs and acrossfirm evaluation costs are different constructs. The number of
products firms offer in equilibrium can exceed the socially optimal level if withinfirm evaluation costs
are significant.
In the second chapter, I study the situation where consumers need not evaluate all available
product information before making a purchase. This may arise because shopping environments prevent a
full evaluation (e.g. online). We study a model of simultaneous search in which consumers have limited
ability in product evaluation. We find that consumers may evaluate more firms, enjoy lower prices and
higher surpluses despite this limited ability. In fact, we show that if consumers are endowed with the
ability to choose how much information to acquire from a searched product, they may choose limited
product evaluation. This implies a nonmonotonic relationship between prices and search costs. We then
8
extend our setting to the case of multiproduct firms and find similar effects due to changes in withinfirm
search costs.
My final chapter studies the strategic design of online shopping intermediaries’ search
environments. An online shopping intermediary is an internet platform for consumers and thirdparty
sellers to transact. Examples are Taobao Mall, YahooShopping and Amazon.com. Shopping
intermediaries provide a search environment (e.g. search aids) to lower consumers’ search costs in
finding and evaluating sellers’ products. In a theoretical model, we study strategic incentives of an
intermediary in the design of its search environment as a means to ease search costs. An important aspect
of our analysis is that consumers optimally decide how many sellers to evaluate and how deeply (e.g.
number of attributes) to evaluate each of them. We find that the equilibrium search environment embeds
enough search costs to prevent consumers from evaluating too many sellers, but not too much to prevent
them from evaluating sellers’ products partially. When facing consumers of heterogeneous search
abilities, the search environment has all consumers evaluating products at full depth and consumers with
higher evaluation abilities evaluating more sellers. We also show that intermediaries embed weakly less
search costs with competition.
9
CHAPTER ONE: CONSIDERATION SET FORMATION: THE CASE OF WITHIN
FIRM & ACROSSFIRM EVALUATION COSTS
1. Introduction
A consumer making a specialty purchase will often face a large number of alternatives and determining
which of these alternatives best satisfies her tastes may be nontrivial. Not only may there be many firms
to consider, but each firm can carry a long line of products. If the consumer incurs a cost to inspect and
evaluate each product, she will find it sensible to limit the number of options to consider (Hauser &
Wernerfelt, 1990). How many and which products to include in her consideration set depends on the
relative tradeoffs of evaluating an additional product, which may further depend on whether the
additional product is from a firm that will already be included in the consideration set or from an entirely
new firm. Two products from the same firm, for example, will contain similar features and be easier to
evaluate than two products from differentiated firms. By the same token, the consumer improves her
chances of finding a better fitting product if she expands her consideration set to include products from
more firms. Therefore, when facing several competing firms, each offering multiple products, a consumer
must decide how many firms to consider and the number of products from each firm to evaluate. In this
way, the consumer’s consideration set formation problem has two dimensions. Specifically, the consumer
must examine the withinfirm and acrossfirm tradeoffs of adding another product to her consideration
set.
To illustrate the consideration set formation in twodimensions, consider a simultaneous search
situation in which a consumer contacts several firms to request product brochures. She first decides on
the firms to contact and on the product brochures to request from each contacted firm. She then spends
time contacting the intended firms to request brochures. After receiving all brochures from all contacted
firms, she spends time evaluating the products. Finally, she makes her purchase decision. This example
serves simply to illustrate the consumer’s consideration set formation process as we model it and may not
10
realistically capture the way consumers typically collect information. In addition, firms may be able to
strategically control the informational content of the brochures, in particular the products that firms
choose to offer.
In general, the composition of the consumer’s consideration set will depend on product
evaluation costs and product differentiation (Roberts & Lattin 1991), both of which have a withinfirm
and an acrossfirm dimension. As noted above, evaluation costs will be lower for a second, third or fourth
product of a firm already included in the consideration set relative to the first product from another firm.
Product differentiation governs the consumer’s benefit from product evaluation. Since products from the
same firm tend to be less differentiated than products from a different firm, the expected marginal benefit
of withinfirm evaluation is lower than acrossfirm. Consequently, the consumer has an incentive to
sample more firms and fewer products from each firm when firms are more differentiated. And, if firms
are strategic, they will compete with knowledge of the makeup of the consumer’s consideration set.
The composition of consumers’ consideration sets also has implications for a firm’s product line
decision. Since maintaining a product line involves costs, a firm’s product line decision will optimally
consider the extent of evaluation within the firm. For instance, if consumers consider only a few products
of a given firm, then the firm will find it optimal to maintain a short product line. Conversely, a firm’s
product line decision can have an impact on the consumer’s consideration set. If, for example, product
line costs are significant, then a firm may find it optimal to pare its product line to such an extent that the
consumer would evaluate additional products were they offered. Thus, in this case, a firm’s product line
decision can put an external constraint on the consumer’s consideration set. The interaction of consumers’
twodimensional consideration set formation problem and firms’ strategic price and product line
decisions is at the heart of this research. Our objective is to study the implications of firm competition
and product variety when consumers incur withinfirm product evaluation costs in addition to acrossfirm
evaluation costs. Extending the consumer’s evaluation problem to include withinfirm factors allows us
to delineate two dimensions of the consideration set formation, offering new insights about how firms
compete in product lines and how product evaluation costs affect firms’ prices.
11
The first new insight regards the relationship between firm differentiation and product variety.
Conventional intuition suggests that as firms become less differentiated, their incentives to invest in
additional products are diminished by the corresponding decrease in margins (e.g. Cachon et al. 2008).
But this intuition ignores a consumer’s withinfirm evaluation costs and her judicious selection of
products to include in her consideration set. To illustrate this distinction, consider the following scenario.
Suppose that several electronics manufacturers carry product lines of televisions, which vary by complex
feature combinations (e.g. screen sizes, picture resolutions, and internet connectivity). Yet, the styles of
each product line offered by manufacturers are relatively similar (e.g. all have the latest flat screen
technology with a sleek appearance). With nonzero evaluation costs, the net expected benefit of
evaluating a few products of many firms is outweighed by the net expected benefit of evaluating
additional products from only a few firms. Strategic firms, aware of consumers’ desire for deeper within
firm evaluation, have a competitive incentive to expand their product lines to provide a better fit for the
consumer. That is, less firm differentiation can induce more product variety.
The second new insight concerns the impact of withinfirm evaluation costs on firms’ prices and
product lines. Intuitively, raising the consumer’s cost of identifying the best product among a firm’s
product line implies lower potential surplus from shopping at that firm and, therefore lower prices. But
this intuition ignores the interplay between the two different dimensions of the product evaluation process
when consumers face competitive multiproduct firms. We show that prices and profits can increase from
larger withinfirm evaluation costs. When small, withinfirm evaluation costs play a role similar to
conventional acrossfirm evaluation costs. In these cases, an increase in withinfirm evaluation implies
that it is more costly to evaluate an additional product at any inspected firm, which serves to raise the
costs of evaluating an additional firm, implying softened price competition. In addition, similar to across
firm evaluation costs, an increase in withinfirm evaluation costs shortens product line length. Thus, firms’
price and profits increase. This is not the case when withinfirm evaluation costs are large. In these
situations, as we show, withinfirm evaluation costs play a role opposite to conventional acrossfirm
evaluation costs. For instance, an increase in withinfirm evaluation costs induces consumers to evaluate
12
more firms. As a result, firms compete more aggressively in price. In addition, in contrast to acrossfirm
evaluation costs, an increase in withinfirm evaluation costs shortens product line length. These findings
show that withinfirm evaluation costs and acrossfirm evaluation costs can have either similar or
opposite impacts on firms’ price and product line competition.
These insights arise from our analysis of an equilibrium model in which consumers optimally
form their consideration sets with two dimensions of evaluation costs, while firms chose their product
line lengths and prices. The equilibrium exhibits the following fundamental property: any firm’s product
line length always matches the number of products from that firm in consumers’ consideration set.
Essentially, firms’ have no incentive to provide more products than consumers are willing to consider.
Similarly, consumers cannot evaluate more products than a firm provides in its product lines. This
property turns out to be crucial to understanding our results because it demonstrates how the composition
of a consumer’s consideration set intrinsically depends on firms’ incentives and viseversa. If withinfirm
evaluation costs are large, then the optimal number of firms and products at each evaluated firm is the
interior optimum of the consumers consideration set formation problem. In this case, firms’ product line
length decision in equilibrium is determined solely by the depth of consumers’ withinfirm evaluation. In
contrast, when withinfirm evaluation costs are small, consumers wish to evaluate more products at a
considered firm than the firm provides in equilibrium. That is, because of product line costs, the firm
finds it optimal to impose a binding constraint on the consumer’s consideration set formation problem.
Consequently, an increase in withinfirm evaluation costs does not have a direct effect on consumer’s
withinfirm evaluation, but rather indirectly induces firms to cut their product line because of competitive
incentives.
As suggested above, consumer’s withinfirm evaluation costs can affect the amount of product
variety in a competitive market. But does the competitive market provide too many or too few products
relative to the social optimum? This is the final question we address in this paper. The literature on
multiproduct firms has suggested that firms offer too few products because they do not fully account for
the social benefit of providing a better fit for consumers (Anderson & de Palma 1992). But when there
13
are costs for inspecting additional products within a firm, it is natural to ask whether the consumer’s
judicious consideration set process mitigates this market inefficiency. By evaluating the equilibrium
number of products offered by firms when consumers optimally collect product and firm information, we
are able to assess the factors that affect the market level of product variety and how that compares to the
socially optimal level. Our analysis shows, in fact, that small withinfirm inspection costs have a
correcting influence on the market failure identified in Anderson & de Palma (1992). But, when these
costs are large, the previous result is overturned and firms provide too many products in equilibrium
relative to the social optimum. This finding further highlights the distinctive implications of the within
firm dimension of product evaluation.
Notions of consideration set formation are typically based on either a consumer’s limited
information processing ability or limited information gathering ability (Manrai & Andrews 1998 and
Mehta et al. 2003). Our research focuses on the latter notion, which recognizes that acquiring product
information can be costly and, therefore, implies that consumers optimally stop short of inspecting all
available options (Hauser & Wernerfelt 1990). In particular, we assume that consumers have a limited
ability in gathering productmatch information due to the costs associated with examining a product and
evaluating its net benefit (Ratchford 1982). We extend this literature by distinguishing how these costs
may differ depending on whether the consumer chooses to consider an additional firm or inspect an
additional product with an evaluated firm. This is important because products from the same firm share
similar traits with each other, implying that the cost of inspecting another product within a firm already
evaluated has lower costs than evaluating a whole new firm.
The marketing and economics literature has also recognized that the cost associated with
acquiring product information has implications for price competition (e.g. Stigler 1961, Morgan &
Manning 1985, and Anderson & Renault 1999). This literature assumes that firms carry a single product
and therefore does not account for within firm product evaluation. This distinction is important with
multiproduct firms because price competition is internalized for product comparisons at the same firm
14
implying that each dimension of evaluation costs affects competitive incentives differently. As we show,
this has implications for the results on price competition.
Our paper also contributes to the growing literature on multiproduct firms and the incentives for
providing product variety. Anderson & de Palma (1992) establish a useful consumer choice setting to
model multiproduct firms that compete in prices and the number of products to offer. Their model
generates a number of insights about competition among multiproduct firms and how well the market
level of product variety compares to the social optimum. We utilize Anderson & de Palma’s (1992)
consumer choice framework to model firm competition and incorporate the notion of costly product
evaluation. As we show here, the inclusion of costly product evaluation suggests new insights regarding
product variety. In a competitive model with multiproduct firms, Cachon et al. (2008) demonstrates how
lower evaluation (search) costs can lead to more product variety and higher prices. Evaluation costs in
Cachon et al. (2008), however, apply only across firms. In contrast, we study the interplay of withinfirm
evaluation on the composition of consideration set and on firm competition. This allows us to identify
that withinfirm evaluation cost can have an anticompetitive effect in price and product line.
Like our paper, Baumol & Ide (1956), VillasBoas (2009) and Kuksov & VillasBoas (2010)
consider withinfirm evaluation costs and identify reasons that a firm may strategically limit product
variety. Baumol & Ide (1956) study the tradeoff faced by a retailer in stocking enough variety of
products to make shopping worthwhile while being aware of a consumer’s burden from too many
products to evaluate. VillasBoas (2009) shows that a monopolist may curtail product variety to avoid
holdingup the consumer when she incurs an evaluation cost before the monopolist commits to a price.
Kuksov & VillasBoas (2010) illustrate that too much product variety by a monopolist can force a
shopper to engage in too much evaluation, causing the consumer to avoid shopping altogether. Unlike
these works, we study a competitive setting in which firms choose prices and product line length when
consumers judiciously choose the number of firms to consider and how many of each firms’ products to
evaluate. In addition, we consider the tradeoff of the interactive roles between consumers and firms by
considering firms’ product line cost. Doing so, we identify another incentive for firms to strategically
15
limit product variety: if consumers limit the number of sampled products because of large withinfirm
evaluation costs, then firms cut costs by keeping product lines short.
1
In the next section, we develop a model of the consumer’s consideration set formation and firm
competition and derive the equilibrium. In section 3, we find the socially optimal outcome and compare
that to the equilibrium outcome. Section 4 concludes.
2. Model
There are 2 firms, each selling a set of differentiated products. We denote
as the product variety,
the number of products, or product line length, offered by firm . There is no systematic quality
difference across firms or products within each firm. The mass of consumers is normalized to one and
each consumer has demand for one product. The timing of the model is as follows: first, firms decide the
number of products to carry for their product lines. Second, after observing the number of products
chosen by other firms, firms choose prices of their products. Firms observe each other’s choices but
consumers do not. Third, consumers form their consideration sets by choosing the number of firms to
evaluate, and the number of products to inspect from each firm. Finally, consumers purchase a product
from their consideration set.
2.1 Consumer Choice and Consideration Set Formation
We assume the consumer initially has imperfect information about the attributes of the differentiated
products. Specifically, product match is idiosyncratic to the consumer and she must evaluate a product to
determine the utility from consuming it. Because evaluation is costly, a consumer judiciously forms a
consideration set that does not include all products offered. We model the formation of her consideration
set as a simultaneous search process (Morgan & Manning, 1985) in which the consumer commits to
evaluating a set of products before making a purchase. Such a process may not be very appealing in many
important situations in which sequential search is possible and this remains a potential issue for future
1
Despite the fact that firms in our model may cut their product lines to match consumers’ limited withinfirm
inspection, our results indicate that product lines may be longer than the socially optimal length.
16
research. Our setting implies that the consumer cannot explicitly observe firms’ choices regarding prices
and number of products offered. Despite the fact that consumers do not observe firms’ decisions, we
assume they are able to deduce the equilibrium strategies of all firms. Nevertheless, the consumer must
engage in costly product evaluation in order to determine her match utility (e.g. its color, styling, or fit).
A product’s match utility has a firmlevel component and a product level component, each of which
requires a cost to learn. Specifically, a consumer incurs a cost for each firm from which she inspects at
least one product and a cost for each product additional inspected at that firm.
A consumer’s consideration set is determined before she evaluates products.
2
And, by assumption,
products are a priori identical before evaluation. At this point, firms are also otherwise identical except
perhaps in terms of the number of products they offer,
and their corresponding prices. Recall that
consumers cannot directly observe firms’ decisions, but can rationally deduce them. We can now
describe the construction of the consumer’s consideration set by the decision to evaluate
0 products
from firm 1, … , . We assume that the first evaluated product of a considered firm does not cause any
evaluation costs to the consumer. From this set, the consumer picks the best alternative from the ∑
products.
The utility of product from firm is given by
,
where is a base level of utility
3
,
is the product’s price, and
and
are random utility terms from
the firm and product, respectively. We assume these terms are random variables with extreme value
distributions and independent from each other and across products and firms. The coefficients
and
capture the across firm and within firm levels of heterogeneity, respectively. A large value of
reflects a
high degree of differentiation between firms and a large value of
means that any given product is not
very substitutable with the other products from the same firm. We assume that
, which ensures
2
We assume that firms have no direct means to influence the makeup a consumer’s consideration set. In many
settings, however, a firm can attempt to be included in their consideration sets. For example, Amaldoss & He (2010)
study a firm’s use of informative advertising to manipulate the content of consumers’ consideration sets.
3
We assume that is sufficiently large to induce consumer search.
17
that our MNL approach to consumer choice (described below) is consistent with utility maximization
(McFadden, 1978 and BenAkiva & François, 1983).
Suppose the consumer contacts firms and inspects 1
products from each
contacted firm 1, … .
4
We only focus on the interior equilibria where not all firms are contacted by
the consumer. This assumption, however, may not hold when there are only a few competing firms. She
incurs a cost 0 for each firm contacted and the first product of each contacted firm and a cost 0
for each additional product inspected. Due to this definition, it is natural to have . Thus, for a
consideration set
!
"
#
, her total evaluation cost is $ % ∑
#
"
. The number of firms and the
number of products from each firm she has in her consideration set depends not only on these costs, but
also on the expected benefit of the choice resulting from the inspection of
!
"
#
products. We model
this choice as a multinomial nested logit model with nests 1, … , and choices 1, … ,
within
nest . The multinomial logit (MNL) approach allows us to derive a closed form expression for consumer
demand and provides us a mathematically tractable choice model from a set of multiproduct firms
(Anderson & de Palma, 1992). Table 1 summarizes all notations.
Symbol Definition
Acrossfirm heterogeneity
Withinfirm heterogeneity
Acrossfirm evaluation cost
Withinfirm evaluation cost
Product line length
& Product line cost
Consumer withinfirm evaluation depth
Consumer acrossfirm evaluation
Product price
Table 1. Notation
4
To simplify index notation, we relabel the firm indices as 1, … , for the firms in the consumer’s
consideration set.
18
For every product in the consumer’s consideration set
!
#
, the consumer knows its utility.
From this set, she chooses the product with the highest utility. The demand for product from firm is its
choice probability, which is derived as follows. Let '
be the (unconditional) probability that the
consumer buys from firm and let (

be the (conditional) probability she chooses product given that
she’s buying from . Specifically, the probability of product being chosen conditional that a consumer
has chosen firm and has inspected
of its products is
(

*
$+,
./
%/1
2
∑ *
$+,
.3
%/1
2
4
.
356
; 1, … ,
, (1)
where
is the number of products inspected at firm . The probability for firm to be chosen is
'
*
7
.
/1
6
∑ *
7
8
/1
6
9
856
; 1, … , (2)
where :
;$max
?@
.
$
%% as the attractiveness of firm given that she will inspect
of
its products. We can then write the demand of product of firm as (
'
A (

.
We focus on symmetric equilibria of the form $, % B C
D
, where represents the product line
length of all firms and is the price of all products at each firm. Specifically, we assume that consumers
believe that symmetric firms will play symmetric strategies. In light of this symmetry, firms are a priori
undifferentiated and the consumer’s optimal sampling decision can be expressed simply as selecting, at
random, a number of firms, , and the number of products
!
"
#
to inspect from each considered firm.
The consumer benefits from considering additional products because it improves the possibility of a
better match to her idiosyncratic tastes. Given the utility formulation above, this benefit is expressed by
ln ∑ expI$
/
% ln
J
!. Since all products have the same price across all firms, we can characterize
the consumer’s optimal sampling plan $ K, ̂
!%, as the maximization of the following objective:
ln M∑ exp NO
P
2
P
6
Q ln
R
S $ % ∑
, (3)
subject to ̂
for all . Two points about (3) are worth emphasizing. First, because $, % represents
all firms’ equilibrium strategies, consumers assume that no firm has an incentive to deviate in prices or
product variety. Consequently, the consumer can take $, % as fixed when determining her optimal
19
sampling plan. Second, from the perspective of the optimization of (3), firms are completely identical.
Therefore, consumers are concerned only about the number of firms to consider and the number of
products
! from those firms to evaluate. Hence, the consumer’s consideration set formation problem
5
is
reduced to the optimization of (3) by the simultaneous choice of $,
!%. The following lemma
characterizes this optimization.
Lemma 1 Given firms’ symmetric product line lengths x and equal prices , a consumer’s optimal
sampling plan is symmetric across firms (̂
̂ for all ) and therefore can be characterized by the
pair $ K, ̂%, where
̂$% min M, O
UVW
P
6
VP
2
Q
P
2
W
S and K$% max M
P
6
VP
2
UVW
,
P
6
UVWDWX
S.
An important implication of Lemma 1 is that the consumer’s optimal sampling plan involves
inspecting the same number of products ̂ at each firm. Consequently, the consumer’s evaluation decision
reflects the simple tradeoff between considering fewer products at more firms or considering more
products at fewer firms. In addition, even though a consumer never considers all firms, she may evaluate
all products within a firm.
The lemma describes two situations. First when the consumer is unconstrained by the number of
products offered by the firm. In this case, her optimal sampling plan is the interior solution to (3), ̂$%
O
UVW
P
6
VP
2
Q
P
2
W
and K $%
P
6
VP
2
UVW
. In this unconstrained case, the marginal cost and benefit of evaluating
an additional product within a firm is and
/, respectively. Thus, the first order condition of at
the symmetric interior optimum implies that the total number of inspected products depends only on the
ratio of withinfirm evaluation parameters: K̂
/. As this ratio increases, the consumer benefits
more from evaluating more products at each firm and more products in total. An increase in $
5
The consumer’s consideration set formation process can be equally interpreted as a simultaneous search process
(Morgan and Manning, 1985).
20
%/$ %, however, implies greater relative benefits from considering more firms and evaluating
fewer products within each firm.
Figure 1: Consumer’s Optimal Sample Plan with Corner Solution (̂ ) and Interior Solution (̂ )
The second situation occurs when the consumer would optimally evaluate more products if firms
offered more. In this case, the maximization of (3) is the boundary solution with ̂$% O
UVW
P
6
VP
2
Q
P
2
W
and K $%
P
6
UVWDWX
. Because the consumer cannot evaluate more products within the firms’ product
lines, it expands its consideration across firms beyond the interior optimal number of firms. Here the
number of products evaluated is K̂
P
6
X
UVWDWX
. Similar to the unconstrained case, the consumer samples
fewer products as withinfirm evaluation costs increases. Unlike in the unconstrained case, however, the
number of products evaluated decreases in acrossfirm evaluation costs, , and increases in product line
length, and in firmlevel differentiation,
.
Figure 1 provides a graphical illustration of the consumer’s optimal sample plan and helps
highlight the important distinction between the interior and boundary solutions to the consumer’s
Constant Evaluation costs: $ %
K$̂ %
K$̂ %
̂ ̂
Constant Evaluation Benefit:
ln
ln
21
sampling problem. A typical isocost curve is drawn as the flatter curve. Its downward slope represents
the positive costs of each dimension of evaluation. Its slight curvature indicates that evaluation costs are
not a linear combination of and . Specifically, recall that evaluating firms requires the cost
$ %. A pair of isobenefit curves are also drawn on the figure. At the interior optimum, the
consumer considers K firms and evaluates ̂ products at each firm such that the isocost and iso
benefit curves are tangent. At the corner solution, the isobenefit curve cuts the isocost curve “from the
topleft”, which implies that the marginal benefit of withinfirm evaluation exceeds the marginal cost.
This is an important feature of the corner solution. As withinfirm evaluation costs increase, the isocost
curve shifts downward. Holding ̂ fixed, the optimal number of firms considered decreases. Even
though increases the cost of evaluating an additional product within the firm, the consumer does not
reduce . However, because the cost of evaluating another firm, ̂ also increases, the consumer
reduces the number of firms considered. This observation plays a crucial role in the results of section 3.
2.2 Firms’ Game in Prices and Product Line Length
We now consider the game played by the firms. Each of the firms chooses its product line length,
,
a price
for each of its 1, … ,
products. We assume each firm has zero marginal cost but incurs an
operational cost &$% & , & 0 to maintain a product line of length products. Firms 1, … ,
simultaneously choose
.Then , firms 1, … , observe the product line lengths of their rivals and then
choose prices Y
Z, 1, … ,
. We do not consider the realistic possibility that firms have the ability to
control the positioning of their products.
We next determine the equilibrium of this game given that consumers acquire product
information as described above. In deriving the equilibrium, several points should be noted. First, given
that there are no systematic differences in quality among a firm’s products, each firm sets the same price
for all of its products in equilibrium, a fact established in Lemma 2:
for all 1, … ,
.
This further implies that the demand for each product at firm is the same for each and [:
22
(
(
\
for all , [ 1, … ,
and ] [.
Second, consumers’ sampling plans are based on rational expectations at the symmetric equilibrium and,
therefore, do not react to deviations from any symmetric equilibrium. This permits us to consider
deviations from a potential equilibrium without considering corresponding changes to consumer sampling
behavior. Consumers’ purchases, on the other hand, may depend on such deviations because purchases
occur after they have inspected the products.
Given Lemma 1, firms expect consumers to employ a symmetric sampling plan. Based on this
observation, we analyze firms’ equilibrium behavior for all possible symmetric sampling plans $, %. To
do so, we utilize the result of Anderson & de Palma (1992), who analyze a game without consumer
evaluation costs in which symmetric firms first choose the number of products to offer and then set prices.
There are three key differences to keep in mind when applying the result. The first key difference is that a
consumer cannot observe firms’ actions when forming her consideration set. We must, therefore, specify
the consumer’s beliefs about how a firm chooses its product line length. (Beliefs about how firms choose
prices need not be specified since a consumer’s sampling plan does not depend on prices under our
symmetry condition.) . We assume that consumers believe that firms would not choose product line
lengths such that a slightly longer product line would simultaneously benefit consumers and firms.
6
Second, in our model, consumers consider only a subset of existing firms and products. However,
because firm deviations from equilibrium do not alter sampling decisions, we can, under the belief
structure above, apply the Anderson & de Palma (1992) result to a game of firms taking into account
that there is a / 1 probability that a firm will be considered by a consumer. The last key difference
is that a firm may constrain its product line length to be shorter than that dictated by the interior optimum
length whenever consumers constrain their evaluation within a firm. The following definitions help
clarify the arguments.
6
This assumption rules out equilibria in which firms’ choices are constrained by arbitrary consumers’ beliefs about
the number of products carried by each firm. For instance, an equilibrium in which firms carry no products can be
supported by a consumer belief that firms carry no products.
23
Definition An $, %equilibrium is a symmetric equilibrium in firms’ strategies $ ^$, %, ̂$, %%
when consumers employ the sampling plan $, %.
As we argue below, there is a unique $, %equilibrium for each sampling plan $, %. Furthermore,
firms’ strategies in any equilibrium of the overall game must be an $, %equilibrium. Note that only
symmetric equilibria of the overall game are considered.
To determine an $, %equilibrium, we focus on firm , assuming all other firms employ the
strategy $, % and consumers employ the sampling plan $, %. Under this condition, we can directly
compute the conditional demand of firm ’s products as a function of strategic choices
and
using (1)
and (2):
(
$
,
, , , %
_
`
a
`
b
*
1
2
1
6
8cde
.
f
$#V%*
g
1
2
1
6
8c$e %,
,
.
1
6
h
D*
1
2
1
6
8cde
.
f
i
,
$#V%*
,
.
1
6 D
i
,
j , (4)
where the probability of firm being evaluated is /. Firm ’s expected profit is then expressed:
k
$
,
, , , %
O
#
l
Q (
$
,
, , , % &
. (5)
Firm chooses
to maximize (5) given its prior choice of
, other firms’ choice $, %, and consumer’s
sampling plan $, %. Accounting for reactions in
, firm first chooses
. For any sampling plan
$, %, we define firms’ symmetric equilibrium product line length as ^$, % and price as ̂$, %.
We report in Lemma 2 the necessary conditions for the firms’ pricing and product line
strategies $ ^, ̂% in equilibrium.
Lemma 2 For any sampling plan $, %, there exists a unique $, %equilibrium, which is
characterized as follows:
(i) ^$, % min M,
P
2
ml
#
2
V#
#
2
V#D
S; and
24
(ii) ̂$%
#
#V
.
Part (i) of this lemma ensures us that, in equilibrium, firms never extend their product line
beyond what consumers are willing to evaluate: ^$, % . This implies that, if consumers significantly
limit their evaluation within firms, the optimal productline length will be a corner solution to the
optimization of (5). Part (ii) states that the equilibrium price is a function only of firm heterogeneity,
,
and the number of firms considered, . In particular, the ability of firms to set prices depends on the
extent to which consumers make price comparisons, as reflected by , but not on the number of products
evaluated within the firm, . (Hence, we abbreviate notation to ̂$%.) Because firms’ pricing strategies
in equilibrium do not depend on , ̂$% does not depend directly on the constraint ^ .
2.3 The Equilibrium and Its Key Properties
The previous sections illustrate (i) how consumers form their consideration set and their final product
choice given a fixed set of prices and product lines and (ii) how firms compete in prices and product line
lengths given a fixed sampling plan of consumers. In this section, we bring these two pieces together,
deriving the equilibrium of the overall model and then discuss its key properties.
To define the equilibrium of the overall model, we use Lemmas 1 and 2.
Definition A symmetric equilibrium is a quadruple $
A
,
A
,
A
,
A
% such that
A
and
A
constitute an
$
A
,
A
%equilibrium where $
A
,
A
% is the optimal sampling plan given firm strategies:
A
and
A
.
Equivalently:
A
K $
A
%,
A
̂$
A
%,
A
̂$
A
%, and
A
^$
A
,
A
%.
It is immediate from the previous results that a unique symmetric equilibrium exists. Also evident
from these two lemmas is the fact that in this equilibrium, either firms or consumers will be at the corner
solution of their respective optimization problems. Furthermore, the properties of this equilibrium depend
on whether
A
A
is binding for the firm or
A
A
is binding for the consumers. Understanding which
25
agent has the binding condition is crucial for understanding the comparative statics properties of the
equilibrium. Lemma 3 demonstrates how these two different regimes depend on the size of withinfirm
evaluation costs.
Lemma 3 There exists a unique threshold n B $0, % with the following properties.
(i) For large withinfirm evaluation costs, n, consumers’ equilibrium sampling plan
$
A
,
A
% is characterized by the interior solution of (3), with
A
A
a corner solution to the
maximization of (5).
(ii) Otherwise, for small withinfirm evaluation costs, n, the consumers’ equilibrium sampling
plan $
A
,
A
% is a corner solution of (3), with
A
A
satisfying the first order condition of an
interior maximizer of (5).
The threshold n is defined by the parameter constellation that equates the interior solutions of the
firms and consumers. Specifically, n uniquely solves
P
2
ml
o
#
A
2
V#
A
#
A
2
V#
A
D
p O
UVW q
P
6
VP
2
Q O
P
2
W q
Q,
where
A
$
%/$ n% . Part (i) of Lemma 3 implies that for large values of withinfirm
evaluation costs, n, the consumers limit their evaluation within firms to such an extent that it
constrains firms’ product line. Conversely, part (ii) means that for small withinfirm evaluation costs,
n, firms’ costs dictate the extent of their product line even though consumers have an incentive to
evaluate more products within firms. Finally, note that Lemma 3 also implies that in the knifeedge case
of n , the symmetric equilibrium is an interior solution to both the firm’s and consumer’s respective
optimization. Therefore, the symmetric equilibrium outcome $
A
,
A
,
A
,
A
% is continuous in at n. This
continuity property will be useful when comparing the equilibrium with the social optimum.
The threshold n is a function of market parameters. This threshold is increasing in the firms’ cost
of extending the product line, &. As this cost increases, firms’ interior optimal product line length
26
shortens and tends to constrain consumers’ withinfirm evaluation. On the other hand, as the consumer’s
relative benefit of across firms, $
%/$ % increases, consumers substitute more acrossfirm
evaluation for withinfirm inspection, tending to constrain firms’ product line length decision.
Lemma 3 can also be used to show the distinct impacts of each dimension of evaluation costs. To
do so, first consider the extreme case when acrossfirm evaluation cost and withinfirm evaluation cost
are equal ( ). Since n , consumers’ equilibrium sampling plan is characterized by the interior
solution of sampling plan. That is, consumers’ evaluation decision dictates firms’ product line length. In
the alternative extreme, when the withinfirm evaluation cost is zero ( 0 n ), firms’ equilibrium
product line length,
A
, is characterized by the interior solution and consumers’ equilibrium sampling
plan is constrained. In this case, firms’ incentives alone determine product line length. Hence, as these
two extreme cases demonstrate, product line length is dictated by different agents reacting to different
incentives depending on which dimension of the two evaluation costs is present. Later we illustrate the
differential impacts of these two types of evaluations costs when both are present.
To understand the equilibrium properties further, we respectively examine the equilibrium
outcomes when consumers’ withinfirm evaluation costs are relatively large: n and when it is
relatively small n. Our objective is to characterize the equilibrium outcomes in this setting and
discuss their key comparative statics properties.
Proposition 1 characterizes the equilibrium choices of consumers and firms categorized based on
the magnitude of consumers’ withinfirm evaluation costs.
Proposition 1
(i) With large withinfirm evaluation cost n, the symmetric equilibrium has firms choosing
price
A
and product line length
A
, where
A
#
A
#
A
V
A
A
,
and consumers considering
A
firms and evaluating
A
products at each firm, where
27
A
O
UVW
P
6
VP
2
Q O
P
2
W
Q
A
P
6
VP
2
UVW
.
(ii) With smaller withinfirm evaluation cost n , the symmetric equilibrium has firms choosing
price
A
and product line length
A
, where
A
#
A
#
A
V
A
P
2
lm
o
#
A
2
V#
A
#
A
2
V#
A
D
p,
and consumers considering
A
firms and evaluating
A
products at each firm, where
A
A
A
P
6
UVWDWX
A
.
With large withinfirm evaluation cost ( n), the impacts of parameters
,
, and on
consumer evaluation are seen directly from the expressions in part (i) of the proposition. Specifically,
from Lemma 3 we know that consumers’ sampling plan $
A
,
A
% is an interior solution to their optimal
consideration set formation problem. Therefore, in equilibrium, changes in evaluation costs and product
heterogeneity parameters follow the same logic as discussed after Lemma 1. With small yet positive
withinfirm evaluation costs ( n), the equilibrium productline length and price $
A
,
A
% correspond to
the interior maximum of firms’ profits while the consumers’ optimal sampling plan $
A
,
A
% is a corner
solution to (3). Consequently, as exhibited by part (ii) of the proposition, the productline length,
A
depends directly only on the product heterogeneity parameter
and indirectly on the parameters , ,
,
which occurs through the equilibrium number of considered firms,
A
. An increase in withinfirm
product differentiation induces the firm to expand its product line to better match consumer tastes.
However, for all other parameters, the impact on productline length is a competitive reaction to the
amount of firms the consumer has considered. If the consumer considers more firms, then a firm
competes more aggressively on product line length to encourage purchase from that firm.
7
Proposition 1 implies several interesting comparative statics results. The following two
corollaries present the key comparative statics properties of the equilibrium with respect to evaluation
7
In fact, the term r $% s
#
2
V#
#
2
V#D
1 can be interpreted as an inverse measure of the degree of market power each
firm has with respect to its product line as a function of . lim
#tu
r$% 1.
28
cost parameters (Corollary 1) and firm differentiation (Corollary 2). We provide the complete set of
comparative statics results in section 2.4.
Corollary 1 (Evaluation Costs)
(i) If n, then an increase in withinfirm evaluation cost lowers prices and shortens firms’
product line length while an increase in acrossfirm evaluation cost has the opposite effect.
(ii) If n, then an increase in withinfirm evaluation cost raises prices and shortens firms’
product line length while an increase acrossfirm evaluation cost has the same effect.
The insights coming from Corollary 1 relate to how firms’ prices and product line lengths
respond to changes in evaluation costs. Part (i) of Corollary 1 demonstrates that when withinfirm
evaluation cost is large, an increase in induces consumers to evaluate few products at each firm,
inducing firms to pare their product lines. In addition, they consider more firms, implying lowered prices.
8
Note that these effects are opposite to the conventional effects of acrossfirm evaluation costs.
This is not the case, when withinfirm evaluation costs is small: n , as indicated in part (ii) of
the corollary. When withinfirm evaluation costs are small, consumer’s sampling plan is at boundary
solution. Thus, larger withinfirm evaluation costs increase the cost of evaluating another firm and thus
consumers react as if there are larger acrossfirm evaluation costs. This results in higher prices and
shortened product line lengths.
We turn to the impact of firm differentiation on the equilibrium in the following corollary.
Corollary 2 (Firm Differentiation)
(i) If n, then greater firm differentiation (
) decreases product line length. If n, then
greater firm differentiation increases product line length.
8
It is possible to connect this result to that of Kuksov (2004), who shows that lower (acrossfirm) search costs
induce competitive firms to invest in product differentiation. Our model suggests that lower (withinfirm) evaluation
costs lead to more investment in product line length.
29
(ii) If n, then prices and profits are nonmonotonic in firm differentiation. Furthermore,
profits may decrease in
even if prices are increasing. If n, then prices are increasing in
firm differentiation, while the impact on profits is ambiguous.
Part (i) of the corollary focuses on the impact of firm differentiation on product line length. With
large withinfirm evaluation costs, we know from Lemma 2 that the firm provides exactly the number of
products in its product line that the consumer optimally evaluates. Thus, increases in the relative benefit
of considering an additional firm by expanding firm differentiation
induces the consumer to consider
more firms and evaluate fewer products within each firm. As a result, firms reduce the length of the
product line. With small withinfirm evaluation costs, the equilibrium productline length corresponds to
the interior maximum of firms’ profits. As a result, the productline length,
A
depends indirectly on firm
differentiation
, which occurs through the competitive reaction to the amount of firms the consumer has
considered. With small withinfirm evaluation costs, the consumer considers more firms as firms become
more differentiated. Thus, a firm competes more aggressively on product line length to encourage
purchase from that firm.
9
As part (ii) of Corollary 2 indicates, firm differentiation has nonmonotonic effect on prices and
profits for large withinfirm evaluation costs ( n). Obviously, greater firm differentiation grants
firms more pricing power. But additional firm differentiation induces consumers to expand their search to
include additional firms, which puts competitive pressure on prices. This latter effect can dominate and
imply that prices decrease in firm differentiation when
/$ % is small. For larger values of
/$
% , the former effect dominates and prices react to firm differentiation in the expected way. This
relationship between prices and firm differentiation has been established in earlier work on single product
firms in Anderson & Renault (1999). There is a noteworthy distinction, however, with multiproduct firms.
9
Another interpretation of this result is possible. Similar to Anderson and Renault (1999), under certain conditions,
more firm differentiation implies more firm evaluation, which intensifies competition. In this case, more
competition can lead to less product variety.
30
Namely, equilibrium profits may decrease in firm differentiation despite increasing prices. The
distinction has to do with how firms modify their product line length to changes in firm differentiation.
Specifically, the fact that consumers consider more firms as a result of larger firm differentiation implies
that they compete more aggressively on product lines, whose extra costs can outweigh the extra revenue
from higher prices. This result holds for intermediate values of
/$ %.
10
For small withinfirm
evaluation costs ( n ), then prices react to firm differentiation in the expected way. The complexity of
the equilibrium expression for profits prevents us from deriving the comparative statics result with
respect to
.
Numerical examples further demonstrate the two main results from our equilibrium model. The
table below illustrates the first main result: As the bolded numbers indicate, for n, withinfirm
evaluation costs induce consumers to evaluate more firms, which implies lowered prices. Acrossfirm
evaluation costs have the opposite effects. For n, withinfirm evaluation costs induce consumers to
evaluate fewer firms, which implies higher prices. In addition, withinfirm evaluation costs always reduce
product line length.
A
A
A
A
n
& 1
5 20 13.68 0.11 0.21 13 11
10 10 14.44 0.11 0.31 13 11
n
& 0.4
20 5 13.75 0.1 0.3 11 10
5 10 12.22 0.2 0.3 11 10
n
& 47
8.1 5 625 0 100 500 400
5.97 2.11 1000 27.5 100 500 400
The table below illustrates the second main result: As the bolded numbers indicate, product line
length decreases in
when n and increases in
when n .
A
A
A
A
n
& 1
10 10 13.33 0.11 0.21 12 11
5 20 13.68 0.11 0.21 13 11
n
& 200
2.2 1.95 1560 0 400 780 700
3 3 1800 0 400 1200 700
10
The exact conditions on
/$ % are given in Corollary 3.
31
2.4 Additional Comparative Statics
In this section we provide the complete set of comparative statics results.
Corollary 3 The equilibrium has the following comparative statics properties:
y z {

{
}
~
A
A
– + – +
z y y q €
A
– + –
�
A
+ +/–
a
+
‚
A
? +/–
c
+/–
b
+/–
d
A
~
A
– – + +
y y q €
A
– – + –
�
A
+ + + +
‚
A
+ + ? +/–
d
Positive iff (a)
ƒ
$ %
$— %
; (b)
P
6
U
A
$
A
1%
mlP
2
UW
O
#
A
V
#
A
Q
11
; (c) $
%$1
ƒ
c…
†
O
1
2
1
6
Q
%; (d)
l
O
#
A
#
A
V
Q
m
W
Evident from Corollary 3 is that when n an increase in acrossfirm evaluation costs
raises firms’ profits for the usual reasons so long as product line cost & is not too large. (See condition (c).)
Recall that an increase in induces firms to offer longer product lines. If & is large, firms compete too
aggressively on product lines and actually see profits decrease despite reduced number of firms evaluated.
Interestingly, with large withinfirm evaluation costs, withinfirm product differentiation
has
similar comparative statics properties as . In our model of two dimensional consumer evaluation,
changes in
and have similar consequences for consumer’s sampling incentives. But, unlike firm
evaluation costs,
has a slightly larger effect on number of products evaluated than . Consequently, the
condition (d) for firm profits to increase in
is stronger than condition (c). Finally, Corollary 3 also
provides the exact conditions required for the nonmonotonic relationship between profits and firm
differentiation discussed after Corollary 2. Note that condition (a) is stronger than condition (b), so that
prices increasing in firm differentiation are sufficient, but not necessary, for profits to increase in firm
11
Note that
A
is a function of
. Thus, the necessary condition for (b) to hold requires
to be large. We do not
give the exact range to avoid messy algebra.
32
differentiation. However, firm differentiation
is quite different in comparative statics properties from
.
In addition, an increase in withinfirm product differentiation (
) always raises prices but may
decrease profits when &/ is large. Greater benefit from withinfirm evaluation, as reflected by an
increase in
, encourages the consumer to consider fewer firms in favor of evaluating more products
within each considered firm. Firms react by competing less aggressively on price (raise
A
) and more
aggressively on productline (expand
A
). Similar to the discussion after Proposition 1, more aggressive
competition in productline raises firms’ costs and, can actually lower profits if the cost of product line
expansion is severe (large &).
3. Equilibrium versus the Social Optimum
We now turn to the question of whether firms’ market incentives induce them to offer product lines
which are excessively long or short relative to the social welfare maximizing length. A firm’s choice of
extending the product line by one product does not fully account for the social benefit of a consumer’s
improved fit from the additional product nor the consumers’ costs to evaluate products. On the consumer
side, the decision to evaluate an additional product is based on the relative costs and benefits of product
evaluation and not on the firm’s cost to extend the product line. So, while it is perhaps intuitive that the
product line length chosen by the market does not coincide with the socially optimal length, it is not clear
on which side it lies.
A second and related question of market performance is whether the consumer evaluates more or
fewer products than socially optimal. This question is not obvious since the consumer makes a tradeoff
between withinfirm and acrossfirm evaluations. Thus, the consumer may substitute too few or too many
firms in its overall sampling plan relative to the social optimum. The fact that either the firm or consumer
is constrained in its choice of products, produced or evaluated, respectively, further complicates the
question. We explore these two questions in this section.
33
We start by defining social welfare. Because firms are symmetric, we assume that all firms carry
the same length of product line, . And, by virtue of the argument in Lemma 1, we can restrict ourselves
to the symmetric case in which consumers evaluate the same number of products, , at each of the
firms considered. Under this symmetric setting, social welfare is defined as
‡ˆ
ln
ln $ 1% &, (6)
and the social optimum as the maximization of ‡ˆ over the choices , , and , which we denote by
$
‰Š
,
‰Š
,
‰Š
%. Note that we ignore price since it is simply a transfer of surplus between firms and
consumers and, therefore, does not affect social welfare. Observe that a benevolent social planner has no
reason to set
‰Š
]
‰Š
since otherwise there would be excessive production or excessive product
evaluation. Therefore,
‰Š
‰Š
. The first order conditions for the maximization of ‡ˆ in (6) with
respect to and subject to gives the socially optimal number of evaluated products:
‰Š
‰Š
P
2
W
ml
W
A
‰Š
. (7)
This expression reflects the social tradeoff with respect to both dimensions of evaluation as well as the
cost of maintaining a product line. We now compare the optimal number of product evaluations given in
(7) with the equilibrium number of products inspected,
A
A
. For large withinfirm evaluation costs,
n, the consumer’s optimal sample plan corresponds to the interior optimum of (3). Note as well,
that the only difference between (3) and (6) is the term &, which is the total social cost of a symmetric
product line of length . In the case of n , we have
A
A
/, which is the first term on the right
hand side of (7). That is,
A
A
‰Š
‰Š
ml
W
‰Š
.
The difference arises because the consumer ignores the firm’s cost of expanding the product line
and evaluates too many of each firm’s products relative to the social optimum. In particular, even though
this cost is internal to the firm, the firm has no incentive to cut the product line since its marginal profit
exceeds & when n. As for the number of firms considered, recall from Lemma 1 that this
number is independent of withinfirm evaluation costs. The social optimal
‰Š
, however, depends on
and exceeds
A
, when withinfirm evaluation is costly.
34
Proposition 2 When n , the symmetric equilibrium always leads to socially excessive
product lines and insufficient number firms considered:
‰Š
A
and
‰Š
A
. The total number
of products evaluated is excessive in equilibrium
‰Š
‰Š
A
A
.
The comparison of the equilibrium and social optimum is not as direct for the case of small
withinfirm evaluation costs ( n). Because the consumer’s sample plan constitutes a corner solution
to (3), we are not able to make a comparison of $
‰Š
,
‰Š
% and $
A
,
A
% for all values of n. We can,
however, utilize the analysis above and the continuity of the equilibrium at n to make the
comparison in the neighborhoods of 0
D
and n
V
.
From Lemma 1, we know that when n , the equilibrium expressions in the two regimes (as
described in Propositions 1) coincide. Therefore, the equilibrium is continuous at n . Furthermore,
Proposition 3 indicates excessive product evaluation in equilibrium for n. Thus, we have
excessive product evaluation for values of close to, but smaller than n. In contrast to the intuition from
Proposition 2, however, the economic inefficiency arises because firms do not internalize the social cost
of additional consumer evaluation when expanding their product lines. And, while the consumer does
internalize this cost, the benefit of additional withinfirm evaluation exceeds the nontrivial social cost
0.
But what about very small withinfirm evaluation costs? We examine the case 0 as a
benchmark. It is straightforward to show that the socially optimal amount of products evaluated at each
firm reflects the withinfirm evaluation benefit and product line costs:
‰Š
/&. The firm’s choice of
product line length, however, depends on its internal benefit, which is its margin, rather than the entire
surplus. Therefore, in equilibrium
A
P
2
lm
#
A
2
V#
A
#
A
2
V#
A
D
‰Š
. Hence, there is a socially insufficient amount
of products evaluated in equilibrium when withinfirm evaluation costs are absent. By continuity, this
argument applied to positive withinfirm evaluation costs in the neighborhood of 0
D
. To compare the
35
number of firms evaluated, note from the solution to the maximization of (6) and Proposition 1,
‰Š
A
P
6
UVWDW@
‹Œ
P
6
UVWDWX
A
0 since
‰Š
A
. Thus, the socially insufficient amount of product variety
at a given firm drives the consumer to evaluate socially excessive number of firms. Proposition 3
formalizes the above results.
Proposition 3 When n, there exists values �
, �
B $0, n% with the following properties.
(i) If 0 �
then competitive firms provide socially insufficient product lines
‰Š
A
and the
number of firms considered is excessive
‰Š
A
in equilibrium with insufficient product
evaluation
‰Š
‰Š
A
A
.
(ii) If �
n then the symmetric equilibrium always has socially excessive product lines and an
insufficient number of firms considered:
‰Š
A
and
‰Š
A
. The total number of products
evaluated is excessive in equilibrium
‰Š
‰Š
A
A
.
As Propositions 2 and 3 (ii) demonstrate, significant withinfirm evaluation costs can lead to
excessively long product lines. Either the firm does not internalize the consumers’ withinfirm evaluation
costs when dictating product line length (Proposition 3 (ii)) or the consumer does not internalize the
firm’s cost of expanding the product line (Proposition 2). This is in contrast to the result with little or no
withinfirm evaluation costs (part (i) of Proposition 3 and Anderson & de Palma, 1992), which states that
firms keep product lines too short, relative to the social optimum since firms do not fully account for the
consumers’ benefit from a better fit.
4. Conclusion
It is natural to assume that in many specialty purchase situations, consumers incur a positive cost to
compare products. Generally, the existing literature supposes that consumers incur an evaluation cost
when comparing firms when each firm carries exactly one product. But, because firms often offer
multiple products, it is reasonable to suppose additional evaluation costs within the firm as well. This
36
paper has proposed a modeling framework for analyzing a setting of consumer evaluation in two
dimensions: within and across firms. And, as our model shows, the addition of a second dimension of
product evaluation leads to new results, not obtained in the classic literature on one dimensional sampling.
For example, higher evaluation costs within a firm can actually either increase or decrease firms’ prices.
Also, firm differentiation decreases product line length. Withinfirm evaluation costs and acrossfirm
evaluation costs can have either similar effects or opposite effects in firms’ product lines or prices.
Finally, we compared the equilibrium outcome with the social optimum. We found that firms can
provide too many products relative to the level that maximizes social welfare when consumers face
significant within firm evaluation costs. This result counters the extant literature on multiproduct firms.
When withinfirm evaluation costs are small, however, firms’ product lines are too short relative to the
social optimum.
37
CHAPTER TWO: CONSUMER SEARCH WITH LIMITED PRODUCT EVALUATION
1. Introduction
In many purchase situations, a consumer faces a large number of products, each of which may
have a complex set of attributes and descriptive product information. To assess whether a
product meets a consumer’s needs, she will invest effort to acquire the information. The classic
literature on search models assumes this is a discrete process in which the consumer puts in
effort and acquires all information available or puts in no effort and acquires no information.
Furthermore, it is common to assume that a necessary condition for the purchase of a product is
for the consumer to have acquired all relevant information about that product. In this paper, we
relax these assumptions and allow consumers to partially evaluate any given product and
purchase a product without the entire amount of information.
The settings we study are quite familiar. Consider, for instance, a shopper wishing to
purchase a shirt from a number of sellers who sell both through conventional stores and online.
Conventional stores provide an environment where she can physically evaluate the shirts for
quality and fit before making a purchase. For instance she can feel the materials and try the shirt
on. Shopping online saves her time and effort, but her ability to fully evaluate the shirts is
limited. In particular, by shopping online, she can inspect the colors and styles of available shirts,
but not physically inspect the material and may have to take the chance that the purchased shirt
will not fit properly.
In this paper we investigate the implications of such limited (or partial) evaluation on
consumers’ search behavior, on firms’ prices, and on consumer welfare. We address these
questions using a model of simultaneous search when consumers partially evaluate products.
Products are horizontally differentiated by the idiosyncratic fit with any given consumer. We
38
find that environments which limit a consumer’s ability to evaluate product information can lead
to higher consumer surplus as compared to environments with full evaluation. Furthermore, we
show that prices and consumer welfare are nonmonotonic in search costs.
We start with the case in which consumers can choose their restricted shopping
environment (e.g. conventional stores versus online shopping) and ask: Under what conditions
does a consumer willingly choose a partial evaluation environment instead of a full evaluation?
We find that when search costs are high consumers may forego full evaluation in order to save
on evaluation costs. The saved evaluation costs may permit consumers to expand their search to
more firms, which induces additional competition. Furthermore, limited evaluation means that
firms must discount even further to compensate consumers for the residual uncertainty.
Next, we study the case in which, consumers have unrestricted flexibility in deciding
how much information they wish to acquire about each evaluated product. That is, consumers
may decide for themselves the depth of evaluation when shopping, and the depth can be different
on each evaluated product. In this setting, we ask: What is the consumer’s optimal level of depth
for partial evaluation and how is it affected by market conditions? We find that, in equilibrium,
consumers engage in full product evaluation, but only if search costs are low. As search costs
pass a threshold, consumers reduce evaluation depth. Again, consumers are not necessarily
worse off as a result of larger search costs.
Finally, we extend our setting to consider the case of multiproduct firms. All products
sold from the same firm share some degree of fitness with the consumer – such as brand level
attributes. Furthermore, a firm does not compete for the sale of its own products. These
distinctions imply that the question of how partial product evaluation affects competition with
multiproduct firms may not directly follow from the single product case. Like with single
39
product firms, the equilibrium prices of multiproduct firms are lower when a consumer reduces
evaluation depth, but the mechanism is different. Unlike the single product firm case, in which
consumers perceive lower levels of product differentiation, lower prices stem from added firm
search. Specifically, when the costs to evaluate products within a firm increase, consumers
reduce the evaluation depth of evaluated products and devote more effort learning more firm
level information (which we assume is full evaluation). In other words, as it becomes more
costly to evaluate products, consumers look at more firms at the brand level. Thus, the
multiproduct firm analysis provides another mechanism by which partial product evaluation
reduces firm market power in search markets.
This paper contributes to the literature on product search, which examines how
consumers acquire product information in the presence of search costs (McCall 1970, Morgan &
Manning 1985, Wolinksy 1986) and the corresponding implication for price competition
(Scitovsky 1950, Stigler 1961, Diamond 1971, Anderson & Renault 1999). This body of work
assumes that product information acquisition is all or nothing. But there is a good bit of evidence
that consumers adapt their decision making strategies to the specific purchasing environment
(Payne et al. 1993) and limit their acquisition of available information if evaluation of product
attributes is costly for the consumer (Hauser 2011). Our paper extends the literature on product
search to account for such partial product evaluation. And, as we show, this distinctive feature
has implications for the relationship between search costs and firm market power.
Our paper also contributes to the literature on product differentiation and the incentives
for search. Diamond (1971), Wolinsky (1986), Anderson & Renault (1999) and others have
pointed out the importance of firm differentiation on the incentives for search and its
implications on competitive prices. Again, the underlying assumption is that information is “all
40
or nothing” and that firm differentiation is fully assessed by consumers. With partial product
evaluation, however, actual ex post firm differentiation can be different than ex ante
differentiation at the time of the purchase decision. In other words, consumers who partially
evaluate a product have residual uncertainty about product match. Evaluated products, therefore,
are perceived to be less differentiated. Consequently, firm competition is intensified as result of
partial product evaluation, a finding that is new to this literature.
Like our paper, Hagerty & Aaker (1984), Bang & Kim (2013), BarIsaac et al. (2012)
and Branco, Sun, & VillasBoas (2012) consider partial product evaluation. Hagerty & Aaker
(1984) is the first study to provide normative model of consumer partial product information
search and its implication on brand choice but only examines the issue from the consumer’s
perspective and without attention to an equilibrium with firm decisions. Bang & Kim (2013),
BarIsaac et al. (2012) and Branco et al. (2012) provide an equilibrium model of partial
evaluation under a monopoly setting and show that a monopolist strategically reacts to the
evaluation depth in changing its price or quality investment. Unlike these works, we study a
competitive setting, which allows us to determine the strategic interactions of firms under partial
product evaluation. Strategic interactions among firms are important to understand in light of the
growing use of online shopping intermediaries (e.g. Amazon.com), which facilitate consumers’
search among sellers and allow for partial evaluation.
We study a consumer’s endogenous choice of evaluation depth from the perspective of
simultaneous search. A good bit of the theoretical search literature has pointed out that a
simultaneous search process can be inferior to a process involving sequential search (McCall
1970) or a combination of simultaneous and sequential (Morgan & Manning 1985). However,
modeling sequential search with partial product evaluation is analytically less tractable than the
41
simultaneous approach. Our focus on simultaneous search, therefore, allows us to gain an initial
understanding of partial product evaluation and its impact on equilibrium outcomes. In addition,
the focus of simultaneous search has been growing in recent theory literature (Kircher 2009,
Chade and Smith 2006) and is empirically demonstrated to be easier (Honka 2011) and more
accurate (De los Santos et al. 2012) in depicting consumer search behaviors compared to the
sequential search models.
In the next section, we develop a model of the consumer’s endogenous partial product
evaluation and firm competition and derive the equilibrium. Section 2 begins with the case in
which consumers choose between two restricted shopping environments: a full depth mode or a
partial depth mode. The remainder of this section studies a more generalized situation in which
consumers are endowed with unrestricted flexibility in deciding how much information they
wish to acquire about evaluated products. Section 3 examines the case of multiproduct firms and
we concluded in Section 4.
2. A Model of Partial Evaluation
There are 2 firms, each selling a single product. The products are heterogeneous and there is
no systematic quality difference across products. The mass of the consumers is normalized to
one and each consumer has a unit demand. We consider a three stage game played by the n firms
and consumers. In the first stage, each firm chooses a price for its product. Then, in the second
stage, consumers choose the evaluation depth. Finally, consumers select the number of products
to sample, evaluate the sampled products and purchase the best alternative based on their
evaluation.
In what follows, we introduce the model and its basic insights by restricting consumers in
the second stage to deciding their evaluation depth by choice of a shopping environment (a full
depth mode or a partial depth mode). For instance, consumers choose between evaluating
42
products in physical stores at full depth and evaluating products at a restricted partial depth
dictated by online shopping environment. We then generalize the setting by giving consumers
unrestricted flexibility in their choice of depth across all products that are chosen for evaluation.
We assume that the consumer initially has imperfect information about the differentiated
products and must, therefore, evaluate a product to determine its idiosyncratic match.
Specifically, the consumer incurs an evaluation cost to learn the match utility (e.g. its color,
styling, or fit). In addition, we assume that the consumer cannot explicitly observe firms’ price
choices but can deduce the equilibrium strategies of all firms before determining her evaluation
depth and evaluation breadth (or the number of evaluated products).
A consumer’s sampling plan (including evaluation depth and evaluation breadth) is
determined before she evaluates products. So, by assumption, all the products are a priori
identical before evaluation. In addition, we assume that firms have no direct means to influence
a consumer’s sampling plan. Thus, her sample plan is a function of the expected product utility
rather than on products’ realized utility. However, because products differ by their ex post fit
realizations after evaluation, a consumer’s purchase depends on the realization of product
random utility. The ex post utility of product is given by
, (1)
where is a base level of utility,
12
is the product’s price, and
is the random utility term.
The parameter captures the degree of differentiation between the firms’ products. We assume
that
!
"
l
are i.i.d. and are drawn from a standard extreme value distribution (with cumulative
distribution function Ž
V*
,�
) and independent across products. We use the extreme value
12
We assume that is sufficiently large to induce consumer search.
43
(Gumbel) distribution to represent consumer’s idiosyncratic utility for two reasons. First, this
discrete choice approach permits closed form expression for consumer demand when using the
logit model (Anderson et al. 1992). Second, this random variable has a selfdecomposable
property, described below, which allows us to model the consumer’s partial product evaluation.
To see how the selfdecomposability property is used to model partial product evaluation,
let �‘$0,1J denote the evaluation depth of a product where � 1 means full product evaluation,
and �‘$0,1% means partial evaluation.
We assume that the consumer cannot buy a product unless
she evaluates it by some positive amount (� � 0). For instance, a consumer cannot purchase
unless she learns the name (brand) and location of the seller.
13
The assumed distribution of
implies that it has a selfdecomposability property that we use to capture the partial acquisition
of product information.
14
This property implies that for any given � B $0,1J, there exist random
variables ’
and “
$�%, which are independent of
, such that
�’
“
$�%
"
”
, where ’
follows standard extreme value distribution.
15
Thus, for a consumer evaluating a product at the
depth �, she learns the realization of the random variable �’
for each product 1, … , ,
which has extreme value distributions (with cumulative distribution function Ž
V*
,�/13
).
16
She
does not observe the realized value for the remaining portion “
$�% but she knows its expected
value ; I“
$�%J $1 �%•.
17
Independent of the evaluation depth, � � 0, we assume that
the consumer learns the price of seller i’s product,
.
13
The parameter � plays no important role until later in the generalized situation, when we allow consumers to
choose different depths across products with unrestricted flexibility.
14
See Corollary 3 of Shanbhag and Sreehari (1977). Steutel and van Harn (1979) was the first to define self
decomposability in terms of a random variable.
15
The random variable “$�% has Le ́ vy density function
š
$% 0 for 0 and
š
$%
*
e/1
V
*
e/31
V
for 0 (Steutel & Van Harn, 2003). Note that “$1% has a degenerate distribution.
16
Continuous evaluation depth k can be interpreted as coming from an evaluation process of discrete product
attributes. Details can be found in the Appendix B.1.
17
• is EulerMascheroni constant (› 0.5772).
44
We assume that a consumer begins her evaluation of a product with simple fit attributes.
For example, when evaluating cameras, a consumer may first look at the brands or sellers’
names, followed by sizes and colors, after which she may want to inspect more difficult
attributes like viewfinder types and resolutions. This suggests that the evaluation cost of a
product is convex in evaluation depth (�). For simplicity, we assume the evaluation cost is
quadratic in depth. We denote $ 0% as the evaluation cost for the consumer to evaluate a
product’s fit completely, or equivalently at full depth (� 1). Thus, the evaluation cost for the
consumer to evaluate a product’s fit at depth � is given by �
.
18
We first examine a simple setting in which the consumer chooses one of two restricted
evaluation modes (or equivalently, a partial depth or a full depth shopping environment).
Correspondingly, there are two restricted depths for the consumer to choose: (P)artial with
�
�
B I�, 1% or (F)ull, with �
ž
1. As suggested above, we can envision these two options
corresponding to shopping situations in which the consumer chooses to shop online or visit
physical stores, respectively. Note that the consumer only considers products in the selected
shopping environment. Later we give the consumer the unrestricted ability to choose any
�
B I�, 1J for each of the 1, … , products that are selected for evaluation.
To derive demand under partial evaluation in the restricted situation, suppose a consumer
partially evaluates firms’ products, where 1 , each at the depth of � �
�
Ÿ 1.
Because the random utility terms, �’
!
"
#
, are i.i.d. random variables with extreme value
distribution, we can utilize a logit choice model which allows us to derive a closed form
expression for consumer demand of a partially evaluated product. Among the partially
18
In addition to the evaluation costs for a product’s fit, the consumer may also have to incur fixed costs (e.g. travel
cost) in the evaluation process. We provide a model with the fixed costs in the Appendix B.2.
45
evaluated products, the consumer chooses the one with the highest expected utility associated
given observations �’
!
"
#
:
A
argmax
??#
�’
;I“
$�%J.
Because the expected value of the unobserved portion, ; I“
$�%J, is independent of , the demand
for product can be expressed as the probability the consumer purchases this product given a
search of firms’ products:
'
*
$+,
.
%/31
∑ *
$+,
/
%/31
9
/56
; 1, … , . (2)
Under this demand system, the consumer obtains an expected benefit of evaluating
products at depth � equal to ; $
A%
A � ln$% • at a cost of �
. Our
formulation of partial evaluation implies that increasing evaluation depth � increases the
expected utility of the chosen product,
A
, purely through the benefit of improved information
about available choices, rather than through any systematic changes in product utility.
Specifically, the mean of the chosen product (neglecting price and basic level of utility) is the
sum of the expected value of the maximum observed utility ;Imax �’
!
"
#
J � ln$%
�• and the expected value of the unobserved utility ; I“
$�%J $1 �%•. Thus, ;$
A%
increases with � only through the expected benefit of a better product choice, � ln$%. Note, in
fact, that the marginal benefit of depth, ln$% is increasing in , starting at zero when 1.
Thus, our formulation of partial evaluation allows us to focus on the influence of evaluation
depth on the benefit of improved product information only, without being affected by the
systematic changes in product utility. Finally, because we consider symmetric firms, we consider
only equilibria in which firms charge identical prices. Furthermore, when consumers determine
how many firms to evaluate, they believe prices are the same for all firms:
for all
46
1, … , . Under this symmetry condition, the consumer’s expected utility of evaluating
firms at a depth � is given by
£
š
$% � ln$% • �
. (3)
Consequently, the number of firms a consumer chooses to evaluate depends only on the relative
benefit � ln$% and cost �
of product evaluation, independent of prices.
The consumer determines the number of firms to evaluate by maximizing (3). Even
though refers to a discrete number of firms, we shall henceforth allow the consumer to treat
as a continuous variable in order to ease analysis. For each evaluation mode, the optimal number
of firms evaluated is:
�
A
P
š
¤
U
,
ž
A
P
U
(4)
To ensure the consumer evaluates at least one product, we assume the evaluation cost is always
less than a critical value ( ¥ s
P
š
¤
). Given constant prices and the optimal number of firms
evaluated, a consumer’s expected utility from the two evaluation modes is given by
£
�
�
�
ln O
P
š
¤
U
Q • �
�
,
£
ž
ln O
P
U
Q • .
The consumer chooses to engage in full evaluation if £
ž
£
�
and engage in partial evaluation
otherwise. The following proposition summarizes the consumer’s equilibrium choice of
restricted evaluation mode.
Proposition 1: Consider the equilibrium when consumers choose between restricted
evaluation modes (P)atrial, with �
�
B I�, 1%, � 0, and (F)ull, with �
ž
1. Let
47
∆s �
�
3
¤
6,3
¤
Ž
V
. Then consumers engage in Full evaluation if 0 ∆ and engage in
Partial evaluation if ∆ ¥.
The main message of this proposition is that whether the consumer chooses to partially
evaluate products depends on the unit cost of evaluating product fit. When costly (∆ ¥),
the consumer chooses to partially evaluate the products. Otherwise, she chooses to fully evaluate
the products.
As the previous argument indicates, market parameters determine the extent to which
consumers sample firms and how deeply they evaluate products. Firms, with the knowledge of
consumers’ sampling plans (including the selected restricted depth and the number of evaluated
firms), will compete in prices accordingly. Suppose firm sets a price
and all other firms play
a symmetric price . Then firm earns profits:
19
k
O
#
A
l
Q §
*
,

.
31
*
,

.
31
DO
1
3¨
VQ*
,

31
© (5)
for � �
�
, 1. The term in the first set of parentheses in (5) reflects the probability that a firm is
one of the
A
firms chosen for evaluation and the bracketed term is the probability that firm ’s
product is the best one conditional on being evaluated. For both evaluation modes firms’ payoffs
in (5) imply the equilibrium prices given in Corollary 1.
Corollary 1: The competitive equilibrium prices (
A
šP
V
6
9
A
) in each evaluation mode
are:
�
A
š
¤
P
V
3
¤
¨
1
ž
A
P
V
¨
1
.
19
We make the assumption of full market coverage in the model to keep the analysis simple. Allowing a nobuy
option does not change the main results.
48
This corollary illustrates that firms’ prices increase in evaluation depth (�) but decrease in
evaluation breadth (
A
). Specifically, � has a direct effect in increasing firms’ prices. Evaluation
depth affects consumers’ perception of firm differentiation at the time of the purchase decision.
In other words, when consumers partially evaluate products they are perceived to be less
differentiated. Consequently, firm competition is intensified, as result. In contrast, when
evaluating products at greater depth, consumers perceive the evaluated products to be more
differentiated.
20
In addition, as suggested by the conventional results in search models, we also
show that wider evaluation breadth (larger
A
) decreases firms’ prices. Consumers evaluate
more competitive products and this puts downward pressure on firms’ prices.
Corollary 1 also implies a nonmonotonic relation between equilibrium prices relative
search costs . Prices with full evaluation and partial evaluation both always increase in search
20
In addition to affect the perceived product differentiation (or the spread of the random utility distribution),
changing k has another effectshifting the center of the distribution of the observed random utility. However, this
later effect does not affect firms’ pricing power because firms’ competitive advantage is only determined by the
relative difference between their products, rather than the systematic shift in their utility which is competed away
across products.
¥ ∆
A
Figure 2: Equilibrium Prices with Two Restricted Evaluation Modes: �
�
�
ž
1
Partial
Evaluation
Full
Evaluation
49
costs. The regions for the prices are depicted in Figure 2. The higher dotted line depicts how
equilibrium price increases when consumers are forced to evaluate products fully when partial
evaluation is preferred. With partial evaluation, however, consumers have the option to cut their
search costs by evaluating each product partially. As increases beyond ∆, consumers switch
from full evaluation to partial evaluation at the depth of �
�
1, which forces firms to cut prices.
Interestingly, one can extend the above framework to show price dispersion among
online and conventional offline stores.
21
For instance, suppose each firm sells both online and
offline, but the online store does not facilitate full product evaluation (�
�
1%. Further, suppose
there are two segments of consumers with different search costs,
and
, respectively. Under
the condition that
∆
, it can be shown that consumers with low search costs (
) shop
offline, with full evaluation, while those in the high search cost segment (
) shop online with
partial evaluation. Furthermore, those with higher search cost may pay a lower price:
�
A
ž
A
.
From a market performance perspective, partial product evaluation provides more
degrees of freedom for consumers and can therefore only enhance social welfare relative to
models in which evaluation is restricted to full depth. What is not so obvious is whether some
level of � is optimal for consumers and how that optimum compares to the social optimal depth
of evaluation. In an equilibrium with partial evaluation, consumer surplus (the consumers’ utility
in (3) evaluated at
A
P
šU
: ª‡
š
£
š
$
A
%, for � �
�
, 1) and total social welfare (consumer
surplus plus firm profits, k
A
A
) are expressed:
ª‡
š
� ln O
P
šU
Q �
šP
V
3¨
1
•, (6)
21
We thank an anonymous referee for suggesting this implication.
50
‡ˆ
š
� ln O
P
šU
Q � •, (7)
where these two values differ only by the equilibrium price: ª‡
š
‡ˆ
š
A
. Social welfare is
independent of prices because, with full market coverage, price is a simple transfer of surplus.
Inspection of ª‡
š
and ‡ˆ
š
shows
22
that they are both single peaked with argmax
š
ª‡
š
argmax
š
‡ˆ
š
, equality holding if and only if � 1. Recall from Corollary 1 that prices are
increasing in �. The consumer benefits from shallower evaluation because of lower prices. This
benefit does not apply to ‡ˆ
š
.
We now turn to the more general case in which evaluation depth is unrestricted and
completely determined by the consumer and can vary across products. Such a setting
corresponds to shopping situations in which a consumer can spend as much or as little time
reading product information as she wants.
23
We assume that a consumer chooses how many
products, m, to evaluate and, for each of those products 1, … , , an evaluation depth,
�
B I�, 1J. Specifically, a consumer chooses a sampling plan $, �
!
"
#
% and incurs the cost
∑ �
#
"
. As before, the consumer cannot purchase any product for which she has not evaluated
by some minimum amount, � 0. Firms choose their prices
, 1, … , , before knowing
consumers’ sampling plan. We focus on symmetric strategies played by firms and assume
consumers (correctly) believe firms’ equilibrium prices are identical,
A
, all , before
deciding their sampling plan. Therefore, for a given sampling plan, the consumer’s the expected
utility is
£$, �
!
"
#
% ; Imax
??#
�
; I“
$�
%J!J ∑ �
. (8)
22
Second derivatives of ª‡ and ‡ˆ with respect to k are negative on (0,1) and therefore singlepeaked. Since
«ª‡/«�< «‡ˆ/«� we conclude the ordering of the corresponding maximizers.
23
It is also conceivable to consider a hybrid environment in which consumers can choose among two shopping
environments, one of which is full product evaluation (e.g. visit a physical store) and the other which facilitates
flexible, yet limited evaluation with any � �
¬
1 (e.g. online store). We discuss this situation later in this section.
51
The consumer chooses $, �
!
"
#
% to maximize (8), subject to the constraint that �
� 0.
We focus on the symmetric equilibrium and denote the symmetric optimizer of (8) by (
A
, �
A
),
in which �
A
�
A
, for all 1, … ,
A
.
Proposition 2: Suppose that consumers choose a sampling plan $, �
!
"
#
% to maximize
(8) and let the minimum required evaluation depth of any product to be �
$/%Ž
V
ƒln O
*
2
D
*
2
Q. Then the symmetric equilibrium is characterized by $
A
, �
A
% and
A
as follows.
(i) If 0 Ž
V
then consumers engage in full product evaluation with
�
A
1,
A
P
U
, and
A
P
VU/P
.
(ii) If Ž
V
then consumers engage in partial product evaluation with
�
A
P
U
Ž
V
�,
A
Ž
, and
A
P
2
$*
2
V%U
.
Like the former simpler case of restricted k, the consumer will engage in full evaluation as long
as search costs are not too large (condition (i)). Otherwise, she will evaluate products only
partially. The key novelty of the case with unrestricted evaluation depth, which can be seen from
part (ii) of the proposition, is that equilibrium prices can decrease in search costs, . Once
exceeds a threshold, consumers curtail their evaluation depth and consider firms’ products as
less differentiated. Recall from the previous discussion, when consumers evaluate products at a
depth �
A
1, the ex post variation in product “fit” �
A
!
"
#
decreases. In this way, consumers
perceive products as less differentiated and firms compete more aggressively on price. This
effect is more pronounced as increases because �
A
is decreasing in . This relationship between
52
equilibrium prices and market parameters is graphically presented in Figure 3.
24
Furthermore,
Proposition 2 implies that equilibrium prices in the partial evaluation regime decrease in faster
than they increase in the full evaluation regime:
ž
A
/« «
�
A
/« in the neighborhood of
Ž
V
. (In Figure 3, the curve to the right of Ž
V
is steeper than to the left.) The
difference in magnitude and sign underscore the importance of considering partial evaluation in
search models.
Without the requirement that �
exceeds the threshold �, the consumer may want to add a
“cheap” default option (evaluate a product with small �) in her sampling plan. In other words, in
our simultaneous search model, the consumer may want to retain the option to pick from a less
certain outcome in the event that all realized values of
!
"
#
A
turn out to be too small or even
negative. This is ruled out with for the threshold � specified in the proposition.
The above results allow us to consider a hybrid environment in which consumers can
choose among two shopping environments, one of which is full product evaluation (e.g. visit a
physical store) and the other which facilitates flexible, yet limited evaluation with any � �
¬
1 (e.g. online store), for some fixed �
¬
, which is determined by the shopping environment. The
results of Propositions 1 and 2 readily allow us to deduce the equilibrium of such a model. For
instance, when �
¬
Ž
V
/ , it is optimal for the consumer to increase evaluation depth up to �
¬
.
Thus, we can apply the results of Proposition 1 by setting �
�
�
¬
and determine the evaluation
regime (and the corresponding prices from Corollary 1) based on the ordering of and ∆.
Alternatively, if �
¬
Ž
V
/, then Proposition 2 assures that the consumer can always select the
24
Another thing to note from Proposition 2 is that under partial evaluation, consumers’ optimal number of sampled
firms is fixed and independent of market parameters. This is a feature of our logit demand framework. As we see in
the next section, with multiproduct firms,
A
depends on market parameters and .
53
optimal partial evaluation depth of �
A
�
¬
. The equilibrium in this case will be determined
precisely by the conditions given in (i) and (ii) of Proposition 2.
25
Finally, we ask whether the price reduction resulting from higher search costs benefits
consumers to a greater extent than the decreased ability in finding a good match among
evaluated products. Consumer surplus, ª‡
š
A £
š
A$
A
%, is fully expressed in equilibrium as
ª‡
š
A
ln O
P
U
Q •
P
VU/P
, 0 Ž
V
P
2
$*
2
V%*
2
U
•, Ž
V
j (9)
Indeed, as one can show from equation (9), consumer surplus is increasing in search costs
when consumers partially evaluate products. While an increases in has a direct negative effect
on consumer surplus through reduced evaluation ability, the consumer reduces her evaluation
depth and the consequential reduction in prices reverses the direct effect, implying that consumer
surplus increases in .
25
We thank an anonymous referee for suggesting this extension.
1
Full
Evaluation
A
Ž
V
Ž
1
Figure 3: Equilibrium Prices with Unrestricted Evaluation Depth
Partial
Evaluation
54
Because the market is fully covered, these price effects do not affect overall social
welfare as search costs increase. However, unrestricted flexibility in depth choice gives
consumers’ more control over the amount of search costs they incur, which mitigates the impact
of an increase in . Formally, social welfare is computed ‡ˆ
š
A ª‡
š
A k
A
, where total
industry profits are simply equilibrium prices,
A
:
‡ˆ
š
A
ln O
P
U
Q • , 0 Ž
V
O
P
2
U
Q Ž
V
•, Ž
V
j . (10)
As reflected in (10), overall social welfare is decreasing in search costs. However, in the regime
of high search costs: B $Ž
V
, %, consumers evaluate products only partially in equilibrium
and the marginal decrease in ‡ˆ
š
A is less than with full evaluation. This can be seen in Figure 4,
where the ‡ˆ
š
A curve slopes downward in partial evaluation, but at a shallower slope than when
consumers are forced to fully evaluate products in equilibrium (represented with the dashed line).
3. MultiProduct Firms
Many firms carry multiple products, for example in a product line, so as to provide a better
chance of matching consumers’ idiosyncratic fit (e.g. various sizes and colors). This
Figure 4: Consumer Surplus and Social Welfare
Partial
Evaluation
Full
Evaluation
Ž
V
ª‡
š
A
‡ˆ
š
A
55
consideration adds another dimension to the consumers’ evaluation decision. Specifically, when
considering single product firms, consumers allocate their evaluation resources in two
dimensions: across firms and the depth within a single product. In this section, we consider a
third dimension, namely within the set of products of a multiproduct firm. That is, a consumer
can decide how many of a firm’s products to evaluate.
One key distinction of the multiproduct case is the fact that products within a firm’s
product line do not compete with each other (Liu & Dukes 2013). Furthermore, unlike the single
product case for which there is a product level fit, with multiproduct firms there also can be firm
level aspects, such as brand attributes, that apply to all of the firm’s products. As such, a
consumer can invest in learning product level attributes (as before), but also learn firm level
attributes. For example, each of Apple’s laptops may have different color or size, but all Apple
branded laptops can share the same operating system and keyboard design. The interaction of
these two evaluation dimensions and the internalized price competition within a firm’s products
suggest that the results of the single product case need not directly apply to the multiproduct firm
case. Specifically, it is no longer clear whether the impact of evaluation costs in the multi
product firm has the same impact on prices and welfare as the singleproduct firm case.
Our model setup borrows from Liu & Dukes’ (2013) multiproduct firm search model by
extending to the case of partial product evaluation. We consider a three stage game played by n
multiproduct firms, each carrying a large set of ¥ Ž
products, and consumers of mass 1. In
the first stage, each firm chooses prices for its products. Then in the second stage, consumers
endogenously determine evaluation depth. Finally, consumers simultaneously make the within
firm and acrossfirm sampling plan, evaluate the sampled products and purchase the best
alternative based on their evaluation. Consumers’ knowledge of firms’ behavior follows the
56
same set up in the singleproduct firm case. Specifically, consumers do not know product fit
information without evaluation and they do not observe firms’ pricing decision. They can,
however, deduce the correct prices in equilibrium. This timing and information structure implies
that firms’ choices in price do not affect consumer’s decision about how many firms and
products to sample nor the decision of how deep to evaluate.
Starting from the last stage, a consumer simultaneously chooses the number of firms and
the number of products from each firm to evaluate. We use to denote a specific firm. The
realized utility of product j from firm i is given by
, (11)
where is a base level of utility,
is the product’s price, and
and
(standard extreme
value random variable) are random utility terms from the firm and its th product, respectively,
with cumulative distribution function e
V®
,�
(
or
). Note the inclusion of a firmlevel fit
applies to any choice j within firm . This can be interpreted, for instance, as a consumer’s fit
with the overall firm or brand of the firm’s product line. Furthermore, relative to the single
product case, consumers learn some information about the fit of a firm’s product (e.g. firmlevel
fit) regardless of the evaluation depth of a specific product. This implies that acrossfirm
evaluation (i.e. learning
) provides some degree of substitutability with evaluation depth.
Following the setting in the previous section, we assume the utility terms are independent
from each other and across products and firms. The parameter
captures the across firm level
heterogeneity and a large value of
reflects a larger firm differentiation. The within firm level
57
of heterogeneity is
. We assume
, which implies that products at a different firm are
more different from each other than to products within a given firm.
26
A consumer must decide how many firms, , and how many products ¥ from
each firm to evaluate. Liu and Dukes (2013) show that there is a unique symmetric equilibrium
for firms’ prices as well as consumers’ optimal sampling plan $, % of the two dimensional
evaluation. Before deciding on a sampling plan, consumers select the depth � B I0,1J of their
evaluation of products. We make two simplifying assumptions. First, unlike the single product
case above, we restrict the consumer to choose a symmetric evaluation depth � � 0. Second,
we assume that partial evaluation applies to the random product term
and not the firm term
.
Allowing for partial evaluation on both dimensions complicates the analysis to such an extent
that we are unable to deduce meaningful results. Because the single product firm case in section
2 relied only on a firm level term, we choose here to focus on the productlevel terms and
assume the firmlevel term
is fully known if one product from that firm is evaluated.
Given that there are no a priori differences in products at a given firm, it can be shown
that firms will set identical prices for all its products in equilibrium:
for all and all .
Furthermore, we focus on symmetric equilibria in prices, so that
for all . Hence, in
forming her optimal sampling plan, a consumer need not account for differences in price across
firms and products. Let
be the costs of evaluating a firm (or acrossfirm evaluation cost) and
the costs of evaluating a product (or withinfirm evaluation cost) at full depth. Following the
development in Liu & Dukes (2013), the utility formulation in (11) implies the expected utility
from a sampling plan $, % with evaluation depth � is given by
£
š
$, %
ln �
ln
•
�
, (12)
26
This assumption also ensures that our MNL demand system is consistent with utility maximization (McFadden,
1978 and BenAkiva & François, 1983).
58
which the consumer maximizes sequentially by first choosing � � and then choosing $, % as
a function of �.
27
To obtain a sense of the role of partial product evaluation on the consumer’s
sampling plan (
š
,
š
), we present in the following lemma the case of fixed �.
Lemma 1: The optimal sampling plan for a fixed evaluation depth � B I� , 1J, � 0 is
given by
š
P
6
VšP
2
U
6
,
š
U
6
P
6
VšP
2
P
2
šU
2
, and
š
š
P
2
šU
2
.
The lemma elaborates how consumers allocate their evaluation resources in two
dimensions: across firms and within firms, given the fixed evaluation depth �. This lemma
indicates that, like the single product case, consumers evaluate fewer firms as evaluation depth
increases (
š
is decreasing in �). As consumers save resources on product evaluation within
firms, they will search more firms.
This lemma also shows that the number of products evaluated within a firm, however,
may increase or decreases in �. Evaluation depth has two opposing effects (a direct effect and a
substitution effect) on the number of products evaluated within a firm. First, larger evaluation
depth implies larger benefit of withinfirm evaluation �
, however, this also implies that larger
cost of withinfirm evaluation �
such that the relative benefitcost
/�
decreases in �.
Thus, larger � directly induces the consumer to evaluate fewer products within a firm. In
addition, larger � increases the attractiveness of withinfirm evaluation relative to acrossfirm
evaluation, implying that the consumer substitutes acrossfirm evaluation for additional within
firm evaluation. This substitution effect works in the opposite direction to the direct effect on the
27
We consider here only the case that a consumer evaluates the same number of products for each firm sampled.
Liu & Dukes (2013) formally show that it is never optimal for the consumer to do otherwise.
59
number of products evaluated within a selected firm. Specifically, when firm heterogeneity is
large
$1 �%
, the relative increased benefit of withinfirm evaluation relative to across
firm evaluation is small when evaluation depth � is larger. Thus, the substitution effect is
dominated by the direct effect and the product line evaluation with partial evaluation exceeds
that with full evaluation:
š¯
š"
. However, when
$1 �%
, product line evaluation
is shorter with � 1 than with full evaluation because substitution effect plays a major role.
Finally, note that despite the nonmonotonic effect of � on product line evaluation, the total
number of products evaluated
š
š
P
2
šU
2
is always decreasing in evaluation depth.
Given the results of the lemma above, consumer’s equilibrium sampling plan will be
fully determined by using the values $
š
,
š
% in (12) and maximizing with respect to �. Under
this sampling plan, suppose firm sets its price to
while all other firms’ prices are set to .
Then the demand for firm is given by
28
(
$
, �%
*
31
2
1
6
°± 4
3
$#
3
V%*
g
31
2
1
6
°± 4
3
,
,
.
1
6
h
D*
31
2
1
6
°± 4
3
, (13)
Since consumers search only
š
of the firms, the probability of firm is chosen is
š
/.
Firm ’s expected profit is then expressed as
k
$
, �%
O
#
3
l
Q (
$
, �%. (14)
Maximizing (14) with respect to
and invoking symmetry implies a unique equilibrium in
prices. The equilibrium price, as described below, depends on the evaluation mode of consumers.
Proposition 3: Consider the symmetric equilibrium with multiproduct firms and endogenous
product evaluation. Firms’ equilibrium prices are given by
A
P
6
V/#
A
.
28
One can derive this demand by following Anderson & de Palma (1992).
60
(i) If 0
/
U
6
*
2
$P
6
VP
2
%
then consumers engage in full product evaluation:
�
A
1,
A
P
6
VP
2
U
6
,
A
U
6
P
6
VP
2
P
2
U
2
and
ž
A
P
6
V
6
1
6
,1
2
¨
6
.
(ii) If
U
6
*
2
$P
6
VP
2
%
/
1
29
then consumers engage in partial product evaluation:
�
A
P
6
2P
2
ƒ
P
6
2
4P
2
2
τ
6
U
2
*
2
,
A
1
6
2
³
1
6
2
4
1
2
2
2
Ž
2
U
6
,
A
Ž
and
�
A
P
6
V
¨
6
1
6
2
³
1
6
2
4
¨
6
1
2
2
τ
2
´
2
.
Proposition 3 characterizes the parametric conditions under which the consumer will engage in
full or partial evaluation. The key dimension of product evaluations in this multiproduct firm
model is the withinfirm dimension. Therefore, we focus the remaining discussion on the impact
of withinfirm search cost variable
. Mimicking the impact of acrossfirm search costs in the
single product firm case, when withinfirm evaluation costs are low, the consumer engages in
full evaluation. This implies that prices also do not depend on
because the number of firms
searched does not. However, when withinfirm search costs are larger, as described in part (ii) of
the proposition, the consumer engages in partial evaluation. In contrast to the full evaluation
situation of part (i), acrossfirm search depends on
in a partial evaluation equilibrium. In
particular, even though the number of products evaluated at a given firm
A
does not depend on
, the number for firms evaluated
A
does.
30
Therefore, withinfirm search costs have an
influence on equilibrium prices. With larger
, a consumer evaluates the same amount products
at a given firm, but each of them less deeply. The saved evaluation costs in the withinfirm
dimension encourage her to search more firms. Consequently, the consumer learns more firm
29
Extra conditions are needed to guarantee that product evaluation generates positive expected net benefit.
Specifically, it requires that
and
are small enough to satisfy,
$
P
6
ƒ
P
6
2
µ
U
6
P
2
2
U
2
*
2
%/ exp ¶ƒ1
µP
2
2
U
6
P
6
2
U
2
*
2
·.
30
This echoes the case of Proposition 2 (ii) and, again, is a consequence of the MNL specification of demand.
61
level, or brand, information inducing firms to compete more aggressively.
31
That is, she can
make acrossfirm evaluation a (less than perfect) substitute for evaluation depth. To understand
this result, consider the following example. If evaluating laptops’ product level features becomes
difficult (larger
), a consumer scales back the depth of individual attributes (e.g. size, speed, or
memory), and expands her evaluation across more firms, relying more on brand level attributes
(e.g. brand image or styling) to make her choice.
The above distinction from the singleproduct firm case is noteworthy because partial
evaluation only applies to withinfirm product evaluation. That is, the impact of partial within
firm product evaluation has a spillover effect to acrossfirm search. In the singleproduct case of
section 2, lower prices resulted from less perceived product differentiation. By assumption that
effect is not present in the multiproduct firm case because consumers always fully evaluate firm
level fit
among searched firms. The lower prices, instead, result from the fact that consumers
search additional firms in response to larger within firm search costs. Thus, the multiproduct
firm case points to an alternative and indirect effect of partial product evaluation on firm
competition.
The impact of search costs on social welfare and consumer surplus in the multiproduct
firm case is different than in the singleproduct firm case. To see this, note that consumer surplus
is defined as the utility expressed in (12) evaluated at the equilibrium: ª‡ £
š
A$
A
,
A
% and
social welfare as the sum of consumer surplus and firm profits: ‡ˆ ª‡
A
.
ªˆ ¸
P
6
V
¨
6
¹
ln
º
U
6
2�
A
• if
U
6
*
2
$P
6
VP
2
%
U
2
P
2
1
P
6
V
6
1
6
,1
2
¨
6
ln
P
6
VP
2
U
6
ln
P
2
P
6
VP
2
U
6
U
2
• if 0
U
2
P
2
U
*
2
$P
6
VP
2
%
j
31
This finding provides another interpretation that larger withinfirm evaluation costs lower firms’ prices in
addition to that of Liu and Dukes (2013).
62
‡ˆ
ln
º
U
6
2�
A
• if
U
6
*
2
$P
6
VP
2
%
U
2
P
2
1
ln
P
6
VP
2
U
6
ln
P
2
P
6
VP
2
U
6
U
2
• if 0
U
2
P
2
U
6
*
2
$P
6
VP
2
%
j ,
where ¼
1
2
ƒ
1
2
4
2
2
Ž
2
1
2
. When there is full evaluation in equilibrium, social welfare and
consumer surplus decline in withinfirm search costs due to the usual reasons. However, because
prices do not depend on
, ª‡ and ‡ˆ decline at the same rate. This is not the case with large
withinfirm search costs. As
increases beyond the threshold that induces partial evaluation,
consumer benefit from the added price competition noted above. And, unlike the case of single
product firms, consumers do not need to tradeoff as much losses in product uncertainty when
reducing evaluation depth because they have an additional dimension for search: acrossfirm.
This means that, as
increases, the impact on ‡ˆ with multiproduct firms is less severe than
with singleproduct firms. Relating to Figure 4 of the single product case with partial evaluation,
the ª‡ curve would be upward sloping more steeply and the ‡ˆ curve would be downward
sloping less steeply.
4. Conclusion
We introduced the notion of partial product evaluation in the context of consumer search and
developed a modeling framework to assess its implications for oligopoly pricing. The ability of
consumers to engage in partial product evaluation arises in many practical settings (e.g. online)
and challenges the notion that market frictions associated with search costs necessarily enhances
firms’ market power. Prior research on search costs focused on product evaluation that is
discrete in nature. Consumers can access either all of none of the available product information.
In a familiar model of simultaneous search, we considered a setting in which consumers can
63
continuously tradeoff accuracy about idiosyncratic product fit with search effort. As we showed,
consumers fully evaluate products in equilibrium, but only if search costs are sufficiently small.
Otherwise, they engage in partial evaluation. When consumers partially evaluate products they
perceive firms as less differentiated. As such, equilibrium prices can decrease in search costs. In
fact, our model indicates that prices decrease to such an extent, that lower prices more than
offset the added costs of search. We extended this setting to the case of multiproduct firms,
which shows that when consumers face more withinfirm search costs, consumers reduce the
depth of evaluation of products within a firm and expand their search to include more firms
(which we is assume is full evaluation). With more firms considered, equilibrium prices decrease.
Our modeling utilized the discrete choice logit framework (Anderson et al. 1992), which
facilitated analytical tractability with closed form expressions for demand and sample plans.
Furthermore, the extreme value distributions associated with idiosyncratic match utility
permitted us to model partial product evaluation using a convenient selfdecomposability
property. As a result, our model is a highly stylized version of reality with the usual caveats. The
same can be said for our use of simultaneous search, which has been theoretically shown to be a
suboptimal search mechanism. Additional research is needed to determine how robust our
findings are to alternative demand and search environments.
64
CHAPTER THREE: ONLINE SHOPPING INTERMEDIARIES: THE STRATEGIC
DESIGN OF SEARCH ENVIRONMENTS
1. Introduction
Consumers increasingly use the internet to evaluate product information and make purchases. Following
this trend, online intermediaries, such as Taobao Mall, YahooShopping and Amazon, offer platforms for
consumers and thirdparty sellers to interact. And, to help consumers navigate and evaluate the huge
number of sellers’ products, they provide interactive search tools in their search environments. According
to a manager from one of the above online platforms, serving both thirdparty sellers and consumers of
these sellers creates two goals for any online intermediary: help consumers to find what they want easily,
and keep sellers’ competition in check. The problem, however, is that often times these are conflicting
tasks and this raises the following managerial question: How should the intermediary design its search
environment to help consumers find a desirable product while ensuring the profitability of their hosted
sellers?
Unlike conventional retailers, online shopping intermediaries host thirdparty sellers, give them
freedom to choose sale prices, and then typically charge a percentage of the final price as a referral fee.
This revenue sharing scheme between intermediaries and sellers incentivizes intermediaries to protect
sellers from fierce competition in order to benefit from higher prices. This research studies the strategic
considerations of an online shopping intermediary in designing its search environment to balance the
needs of its consumers and the benefits of its sellers. We focus, in particular, on the strategic aspects of
search environment design, which is operationalized as a means to ease search costs,
32
as it affects
consumers’ evaluation incentives.
The conventional wisdom in the search literature suggests that consumers scale back the number
of sellers they consider when faced with higher search costs and this softens sellers’ price competition
32
We use search costs and evaluation costs interchangeably.
65
(c.f. Anderson & Renault 1999).
33
However, in reality, consumers always have the flexibility to decide
the amount of information they acquire about any given product. The recent search literature suggests
that consumers can react to high search costs by endogenously deciding how much to invest in
information acquisition of any given product (Branco, Sun and VillasBoas 2012). Incorporating this
withinproduct search dimension (or evaluation depth in the following text) makes consumers’ evaluation
incentives not transparent. In this paper, we study how the twodimensional search – breadth (across
sellers) and depth (within products) – is affected by the design of the search environment and what an
intermediary’s strategic considerations are in this design.
To illustrate the tradeoff of a consumer in her depth and breadth choice, consider a consumer
who wants to buy a camera. She knows her tastes about cameras (e.g. style, color, size, viewfinder
type, …) and her price range, but does not have a particular camera in mind. At the intermediary’s
website, she will evaluate a set of models offered by the thirdparty sellers. Since evaluating cameras is
costly, it is up to the consumer to decide how many sellers to evaluate (evaluation breadth) and how
much information to acquire for each model (evaluation depth). She may decide, for instance, to look into
every technical detail available (e.g. megapixels, screen size, memory, aspect ratio etc.). But doing so,
she might not be able to consider many camera brands. Alternatively, she may consider many brands but
she might have to forego some technical details. A shopping intermediary (e.g. YahooShopping) can
create an interactive search environment that affects this tradeoff.
The depth and breadth of the consumer’s evaluation plan affects her knowledge of the evaluated
products. Specifically, evaluation depth affects how much uncertainty she has about her most preferred
product and evaluation breadth affects how many sellers are competing for her purchase. In classic search
models, a larger search cost implies that a consumer will evaluate fewer sellers (lower evaluation
breadth). In our setting, however, a consumer can react to higher search costs by instead decreasing the
depth of evaluation rather than by only cutting back on the number of evaluated sellers. Because of the
33
In the context of online shopping, Lal and Sarvary (1999) shows that the presence of the internet can increase the
costs for consumers to evaluate another product at a physical store implying softened price competition.
66
endogenous interplay between evaluation depth and breadth, we can model the intermediary’s strategic
design of the search environment by its choice of conventional search costs.
Our model generates several new insights. First, we provide a necessary and sufficient condition
for the intermediary’s profit maximizing search environment. Specifically, the intermediary’s optimal
search environment maximizes consumer search cost subject to the condition that the consumer does not
partially evaluate products. If search costs exceed this point, the consumer will evaluate products partially
(e.g. not evaluating some product attributes), being unable to fully appreciate the value of her most
preferred product. This puts downward pressure on sellers’ prices. But, by making search less costly, the
consumer broadens the set of sellers evaluated, inducing sellers to price more competitively. The
intermediary’s optimal search environment reflects, therefore, a balance between lowering search costs
sufficiently so that the consumer fully evaluates the products she considers and fully knows what she is
buying, but not too much that she searches a lot of sellers.
Second, we demonstrate that deepening evaluation depth plays a larger role in sustaining higher
prices than limiting evaluation breadth. This is because deeper evaluation depth allows consumers to
better appreciate the value of their most preferred product, which plays a direct role in increasing sellers’
prices. In contrast, limiting evaluation breadth plays a weaker role in sustaining prices because it acts
indirectly through sellers’ competitive interactions. This result reinforces the importance of full
evaluation in the optimality condition discussed above. This finding provides some guidance for the
intermediary facing consumers with heterogeneous evaluation abilities. It suggests, in particular, that the
major consideration of intermediary’s design objective of search environment should be to err on the side
of sufficiently lower search costs such that the consumers with lower evaluation abilities can evaluate
products at full depth, although doing so allows the consumers with higher evaluation abilities to broaden
their evaluation breadth.
The third new insight concerns the impact of product differentiation on the intermediary’s
optimal design. Less differentiation implies a lower benefit of product evaluation, to which consumers
tend to react by scaling back evaluation depth – evaluating products only partially. To ensure full
67
evaluation, the intermediary should provide a more helpful search environment to reduce search costs.
However, both the intermediary and sellers are hurt by the reduced prices stemming from lower product
differentiation.
Our final insight regards how competition among intermediaries affects the design of search
environment. We find that the optimal search environment for a competitive intermediary is identical to
the monopolist’s, but only when intermediaries are relatively differentiated. If intermediaries are less
differentiated, intermediaries provide a perfectly helpful search environment, with very low search costs.
In this case, intermediaries are drawn into a prisoner’s dilemma. That is, while they would jointly be
better off providing more costly search environments, the intense competition means that each can
acquire a substantial increase in consumer traffic at their sites by making their search environment more
helpful.
Our work builds off the literature about accuracy and effort tradeoffs of a decision maker’s
decision rule selection. Specifically, people select a particular decision rule in a specific environment by
weighing the costs and benefits of a set of rules, and adapt their decision making strategies to the specific
environment (Payne 1982). Using this framework, Häubl & Trifts (2000) showed that the online
shopping environment lowers consumers’ efforts, allowing them to evaluate selected products at a greater
depth and making better purchase decisions. However, their study focuses on how the search
environment helps consumers’ product evaluations, without regard to the pricing incentives of thirdparty
sellers. In contrast, we propose an equilibrium model with a representative consumer, intermediaries, and
competing sellers and study their strategic interactions in order to identify the intermediary’s optimal
level of search costs in its shopping environment.
Our methodology is based on the literature of partial product evaluation that has been growing in
recent theory research (BarIsaac et al. 2012, Branco et al. 2012). Most of these studies focus on a
monopolistic seller’s strategic reactions to consumers’ partial evaluation. However, because consumers
often consider and evaluate several alternatives when shopping at an online intermediary, the study of an
68
intermediary’s incentives requires a model to capture sellers’ competitive reactions to consumers’ partial
evaluation. Thus, in this paper, we employ a competitive framework.
The marketing literature has recognized two basic roles of online shopping intermediaries to
serve consumers – providing product information of price and nonprice attributes (Häubl & Trifts 2000,
Iyer & Padmanabhan 2006). Chen et al. (2002) and Iyer & Pazgal (2003) consider the intermediaries’
role as disseminators of price information.
34
Chen et al. (2002) study the role of an intermediary as a price
discrimination mechanism, while Iyer & Pazgal (2003) focus on the motivation of internet retailers to
join an intermediary’s service. Like our paper, these studies consider an intermediary that hosts third
party sellers. However, both of these studies focus on situations in which a consumer knows the specific
product she wants to buy and uses the intermediary to find the best price. Our paper, in contrast,
examines the case in which the consumer uses the intermediary to help her find a good fitting product
based on nonprice attributes. In addition, our paper considers the consumer’s evaluation process as a
function of the intermediary’s designed search environment. Doing so, this research is among the first to
incorporate consumers’ optimal search behavior into the intermediary’s design of the search environment.
Finally, our work is also related to the literature on common agency. Bernheim & Whinston
(1985, 1986) show that a common agency may allow sellers to collude and achieve maximal cooperative
profit if they delegate the pricing or output decisions to the agency. In contrast, in our paper sellers are
assumed to delegate the controls of shopping environment to the intermediary. As we show, although the
intermediary cannot help sellers to implement monopoly pricing, it can design the search environment to
protect them from fierce competition.
2. Model
There are thirdparty sellers, each selling a single product. The products are horizontally differentiated
and have no systematic quality difference. The mass of the consumers is normalized to one and each
34
Jiang et al. (2011), which also studies the price informative role of intermediary, focuses on the potential threat
from the intermediary to compete against a thirdparty seller that incentivizes the seller to mask the popularity of its
product from an intermediary.
69
consumer has demand for one product. We assume that the consumer initially has imperfect information
about the fit of products’ attributes (e.g. color or styling). She must therefore evaluate a product to
determine its idiosyncratic utility. Product evaluation is costly to the consumer. But an intermediary
provides a search environment to reduce her evaluation costs. Observing such a search environment, but
not products’ fit, the consumer chooses how many products to evaluate (evaluation breadth) and how
much information to acquire about each considered product (evaluation depth).
These assumptions imply that all the products are a priori identical before evaluation, but may
differ after evaluation by their ex post fit realizations and prices, which affect the consumer’s product
choice. The timing of the model is as follows. First, the intermediary chooses its search environment
(operationalized by choosing search costs). Second, each seller chooses a price for its product. The
consumer then selects the evaluation depth (how much information to acquire about a product) and
breadth (how many sellers to evaluate). At last, she evaluates products at the selected depth and
purchases the best one based on her evaluation.
Because products’ ex post realized utilities affect the consumer’s product choice, we first define
the overall ex post realized utility of a product (seller ’s product) which is observed after full evaluation
,
where is a base level of utility,
35
is price, and
is a random utility term. We assume that
! are
i.i.d. and are drawn from a standard extreme value distribution (with cumulative distribution function
Ž
V*
,�
). The coefficient captures the degree of differentiation between products. The random utility of
seller ’s product
is a random variable with extreme value distribution (with cumulative distribution
function Ž
V*
,�/1
).
The conventional search literature assumes that by incurring evaluation costs a consumer
uncovers the realization of
. However, she cannot observe this overall ex post realized utility of a
product if she chooses to evaluate it partially. To define the utility of a partially evaluated product, we
35
We assume that is sufficiently large to induce all consumers to participate in search.
70
adopt the following framework. Denote ½‘I0,1J as the evaluation depth of a given product where
½ 0, 1 corresponds to no or full evaluation, respectively, and ½‘$0,1% means partial evaluation. With
the assumed distribution,
has a selfdecomposability property,
36
which enables us to define the ex post
realized utility of a product that is partially evaluated.
This property states that for any ½‘I0,1J,
can be
written (equal in distribution) as
½̂
¾
$½%
"
¿
where ̂
is a random utility term that is drawn from a
standard extreme value distribution, independent from
and ¾
$½%.
37
Thus, if the consumer evaluates
product at depth ½, she takes a draw from the random utility ½̂
but does not observe the realized value
for the remaining portion ¾
$½%. The random variables {½̂
} are i.i.d. with extreme value distribution
(cumulative distribution function Ž
V*
,�/À
). Independent of the evaluation depth, we assume that the
consumer learns the price of seller i’s product,
.
We assume that a consumer begins her evaluation of a product with simple fit attributes. For
example, when evaluating cameras, a consumer may first look at the brands or sellers’ names, followed
by sizes and colors, after which she may want to inspect more difficult ones like resolutions and
viewfinder types. This suggests that the evaluation cost of a product is convex in evaluation depth (½),
which, we assume for simplicity to be quadratic. We denote $ 0, referred as to the baseline evaluation
cost% as the evaluation cost for the consumer to evaluate a product’s fit completely, or equivalently, at
full depth (½ 1). To ensure that the intermediary serves consumers when faced with nontrivial search
costs in absence of search environment, we assume that Ž
V
. Thus, the evaluation cost for the
consumer to evaluate a product’s fit at depth ½‘$0,1J is given by ½
. It is noteworthy that this evaluation
cost is proportional to the resolved uncertainty (measured by the variance of the explained random
utility).We assume that a consumer’s search process is simultaneous.
38
Before evaluating products, a
consumer decides Á sellers at an evaluation depth of ½‘I0,1J for each of the Á sellers. Let this choice
36
See Steutel & van Harn (1979).
37
The random variable ¾$½% has Le ́ vy density function
”
$% 0 for 0 and
”
$%
*
e
V
*
e/À
V
for 0 (Steutel & Van Harn, 2003). Note that ¾$1% has a degenerate distribution.
38
Alternatively, one might model consumer search as a sequential process (Wolinksy 1986). However, a
simultaneous search model is sufficient for capturing the basic search incentives without unduly complicating the
analysis.
71
$Á, ½%, which we specify later, be called the evaluation plan. With this evaluation plan, the consumer
uncovers the realized utility of the evaluated portions ½̂
!
"
Â
but does not observe the remaining utility
¾
$½%!
"
Â
. And, among these Á partially evaluated products, she chooses the one with the highest
expected utility:
A
argmax
",…,Â
½̂
; I¾
$½%J!,
where ;I¾
$½%J $1 ½%• denotes the expected value of the unobserved random utility of a product
that is evaluated at depth ½.
39
Note that this value is independent of , implying that products differ by
their ex post fit realizations and prices ½̂
!
"
Â
. Thus, the best product can be expressed by
A
argmax
",…,Â
½̂
. Since ½̂
!
"
Â
are i.i.d. random variables with extreme value distribution,
we can utilize logit model to derive a closed form expression for the choice probability of product :
40
'
*
$+,
.
%/À1
∑ *
$+,
/
%/À1
Ã
/56
; 1, … , Á.
The consumer’s evaluation breadth and depth are determined before evaluation. Thus, they
depend on the expected evaluation benefits rather than on products’ realized utility. By assumption, all
products are a priori identical before evaluation. Furthermore, the consumer believes (correctly) that all
sellers charge the same price in equilibrium:
for all 1, … , . Under this symmetric condition,
the expected benefit of evaluation plan (½, Á) is given by
; $
A% ; $½̂
A% ; I¾
A$½%J!,
which simplifies to
½ ln$Á% •.
We now consider the role of the online shopping intermediary. The intermediary chooses its
search environment to control consumers’ evaluation costs. Let Ä‘I0,1J denote the extent the search
39
;I¾
$½%J ; $
% ; $½
% $1 ½%•, where • is EulerMascheroni constant (› 0.5772).
40
We make the assumption of full market coverage in the model to keep the analysis simple. Allowing a nobuy
option does not change the main results. See also section 4 in which consumers have the option to shop at a
competing intermediary.
72
environment lowers evaluation costs. Then the expected utility from evaluation plan $½, Á% in an search
environment (Ä) is given by
£$½, Á; Ä% ½ ln$Á% • $1 Ä%Á$½
%. (1)
When Ä 0, the search environment does not make product evaluation any easier and thus the
consumer’s evaluation costs are not reduced compared to the earlier situation without considering the
intermediary. Intermediate cases of Ä‘$0,1% means that the search environment partially lowers her
evaluation costs. If Ä 1, then the search environment makes search so easy that the consumer can
costlessly evaluate all sellers’ products. The following lemma characterizes the consumer’s optimal
evaluation depth and breadth as a function of the search environment $Ä%.
Lemma 1: Let Ž
. For any given Ä, there exists a unique pair Æ½
Ç
$Ä%, Á
¬
$Ä%È maximizing
£$½, Á; Ä% s.t. ½
Ç
$Ä% B I0,1J and 1 Á
¬
$Ä% .
½
Ç
$Ä% É
P
$V‰%U
Ž
V
0 Ä 1
P
U
Ž
V
1 1
P
U
Ž
V
Ä 1
j
Á
¬
$Ä%
_
`
a
`
b
Ž
0 Ä 1
P
U
Ž
V
P
$V‰%U
1
P
U
Ž
V
Ä 1
P
Ul
1
P
Ul
Ä 1
j .
Furthermore, ½
Ç
Ê
$Ä% 0 Á
¬
Ë$Ä% for Ä B N0, 1
P
U
Ž
V
Q, ½
Ç
Ê
$Ä% 0 Á
¬
Ë$Ä% for
Ä B N1
P
U
Ž
V
, 1
P
Ul
R, and ½
Ç
Ê
$Ä% Á
¬
Ê
$Ä% 0 for Ä B O1
P
Ul
, 1R.
Lemma 1 characterizes how the consumer chooses her evaluation depth and breadth as a function
of the search environment. Specifically, when the search environment modestly lowers search costs
(0 Ä 1
P
U
Ž
V
), the consumer partially evaluates a fixed number of products. And, in this interval,
as an environment that makes search easier is offered, her evaluation costs are reduced. This allows her to
73
evaluate these products at a greater depth. As the search environment reaches a threshold (Ä 1
P
U
Ž
V
),
the consumer engages in full evaluation (½
Ç
$Ä% 1). And, when Ä exceeds this threshold, her evaluation
costs are further reduced and this allows her to evaluate more products, at a full depth. As even more
helpful search environment are provided (1
P
Ul
Ä 1), the consumer fully evaluates all the products
available at the intermediary (Á
¬
$Ä% ). In addition, product differentiation () also influences
evaluation depth and breadth. Specifically, larger implies larger benefit of evaluation and this implies
that the consumer may evaluate more products at a greater depth. It is noteworthy that most online
shopping intermediaries provide different search environment (e.g. different sets of search aids) for
different products.
A key property of the optimal search plan given in Lemma 1 is the derivative Á
¬
Ê
$Ä% 0 for small
Ä B I0,1 /Ž
J. In this range, as a more helpful search environment is provided, the consumer
increases the depth of her evaluation (½
Ç
Ê
$Ä% 0) without expanding the breadth. This is due to the fact
that for ½ 1, the consumer’s net marginal benefit of depth always exceeds that of breadth. This can
formally be seen in how the partials of search objective of (1) with respect to ½ and Á can be influenced
by Ä. When Ä is small and consumers evaluate partially, a more helpful search environment lowers the
marginal cost of depth to a greater extent than that of breadth. Thus, an increase in Ä induces deeper
evaluation without expanding the number of considered sellers. Only when the search environment is
sufficiently helpful, Ä 1 /Ž
, does the further search costs reduction induce the consumer to
expand her search breadth: Á
¬
Ê
$Ä% 0. In this case, the consumer is evaluating products at full depth. This
property plays an important role in assessing the intermediary’s optimal choice of Ä in stage 1.
We now consider the game played by the sellers. The consumer’s evaluation plan is made
before evaluation and thus depends on her rational expectations of the sellers’ symmetric equilibrium
price. That is, her evaluation plan is not affected by any deviation by a firm from the symmetric
equilibrium price. The consumer’s product choice, however, depends on such deviations because her
purchase decision is determined after she observes products’ prices. To determine the equilibrium price,
74
we focus on seller by assuming that it charges price at
while all other sellers charge price at . Under
this condition, for any evaluation plan (½,Á), we can write the conditional demand for seller ’s product
(given that it is evaluated) as the choice probability of product :
(
*
$+,
.
%/À1
*
$+,
.
%/À1
D$ÂV%*
$+,
/
%/À1
; 1, … , Á
The consumer selects a subset of the available products on the intermediary to evaluate. Because all
the products are a priori identical before evaluation, she randomly selects Á products and there is a Á/
probability for product being selected for evaluation.
41
Thus, the unconditional demand for seller ’s
product is
Â
l
(
. Seller ’s expected profit is given by:
k
$1 Ì%
Â
l
(
. (2)
where Ì‘$0,1% denotes the referral fee paid to the intermediary. Seller chooses
to maximize its
expected profit k
. The following lemma characterizes sellers’ symmetric equilibrium prices given
optimal evaluation plan (½
Ç
$Ä%, Á
¬
$Ä%).
Lemma 2: For any Ä, with the corresponding optimal evaluation plan Æ½
Ç
$Ä%, Á
¬
$Ä%È, sellers’
equilibrium prices are given by
̂$Ä%
”
¬
$‰%P
V
6
Ã
Í
$‹%
_
`
a
`
b
*
2
V
P
2
$V‰%U
0 Ä 1
P
U
Ž
V
P
2
PV$V‰%U
1
P
U
Ž
V
Ä 1
P
Ul
Pl
lV
1
P
Ul
Ä 1.
j
This lemma illustrates two main points. First, it shows how evaluation depth and breadth affect
prices. Specifically, as the consumer evaluates products at a greater depth (larger ½), she can better
appreciate the value of her most preferred product. This effect, which we call the evaluation depth effect,
41
Products can be made more prominent than others on an online shopping intermediary so that consumers may
first evaluate the prominent ones. This remains a potential issue for further research.
75
increases sellers’ prices in equilibrium. In contrast, as she considers more products (larger Á), the
competition between sellers pushes prices downwards – an effect we call the evaluation breadth effect.
Second, this lemma illustrates how the search environment affects prices through the evaluation
depth and breadth. Specifically, when the search environment only modestly lowers search costs (0
Ä 1
P
U
Ž
V
), the consumer evaluates products at a greater depth as more helpful environment are
provided (see Lemma 1). Thus, the evaluation depth effect applies and prices increase. However, when
the search environment passes the threshold (1
P
U
Ž
V
Ä 1
P
Ul
), the consumer fully evaluates more
products as more search costs are lowered. Evaluation breadth effect, therefore, applies and prices
decrease. When an even more helpful environment is provided, the consumer fully evaluates all products
available at the intermediary. Therefore, larger s affects neither consumers’ evaluation plan nor firms’
prices.
We now consider the intermediary’s design of its search environment. As argued earlier, the
intermediary chooses its search environment (Ä B I0,1J) to control the consumer’s evaluation costs.
Assume that the intermediary incurs zero cost in providing search environment. The intermediary’s
expected profit is given by
k
Î
Ì̂$Ä% (3)
where ̂$Ä% is the symmetric equilibrium price as given in Lemma 2.
The intermediary’s objective in (3) is simply to maximize sellers’ equilibrium prices. From
Lemma 2, we see ̂$Ä% is single peaked at Ä 1
P
U
Ž
V
. Any search environment with Ä lower than this
identified point (or equivalently, any environment with higher evaluation costs) induces the consumer to
partially evaluate products, making her unable to fully appreciate the value of her most preferred product.
With partial evaluation, sellers cut prices in equilibrium. Alternatively, any search environment with Ä
larger than 1
P
U
Ž
V
(or equivalently, any environment with lower evaluation costs) encourages the
consumer to broaden the set of sellers evaluated, inducing sellers to price more competitively. The
intermediary’s optimal search environment reflects, therefore, a balance between sufficiently lowering
76
search costs so that the consumer fully evaluates the considered products and fully knows what she is
buying, but not too much that she searches a lot of sellers.
Proposition 1: In equilibrium, the intermediary has a search environment defined by Ä
A
1
P
U
Ž
V
and sellers’ prices are given by ̂$Ä
A
%
P
V*
,2
.
Proposition 1 demonstrates that even though the intermediary designs the search environment
with positive evaluation costs, the consumer evaluates sellers’ products at full depth. Furthermore, the
intermediary sets the search costs in such a way that consumers’ “residual” evaluation depth that the
consumer evaluates without any help from the search environment is free: Ä
A
1 ½
Ç
$0%. In other words,
suppose the consumer, in the absence of such an environment, would evaluate only a few attributes. The
intermediary steps in with search environment to lower search costs so that the consumer evaluates the
remaining attributes costlessly. This proposition also illustrates the impact of product differentiation on
the intermediary’s optimal search environment. Less differentiation implies a lower benefit of product
evaluation, which induces the consumer to scale back evaluation depth. To maintain full evaluation, the
intermediary reduces search costs by offering a more helpful search environment. However, both the
intermediary and sellers hurt by the reduced prices stemming from lower product differentiation.
To gain additional intuition and help understand the insight of the next section, consider an
intermediary choosing between two extreme cases: extremely helpful search environment (s = 1) and a
search environment that offers no help (s = 0). The intermediary always prefers to provide extremely
helpful search environment and this makes search so easy that consumers evaluate all sellers’ products at
full depth. This earns the intermediary (and its sellers) more profits than if it made no effort to help
consumers’ search. With a search environment that offers no help, consumers scale back their evaluation
depth to such an extent, that products appear to them as less differentiated. Knowing that the consumer
sees all products relatively undifferentiated, firms compete more aggressively on prices. This result
77
suggests, therefore, that evaluation depth is perhaps more critical to intermediary profits than evaluation
breadth. And, as we shall see in the next section, this insight can help the intermediary when facing
heterogeneous consumers with different baseline evaluation costs.
3. Heterogeneous Consumers
The main argument from the above setting suggests that the intermediary uses search environment to
choose search costs in order to influence the evaluation depth and breadth of the products consumers
consider in order to maintain sellers’ prices. The previous insight, however, is built off the assumption
that all consumers have the same baseline evaluation costs which reflects that they are homogenous in the
product evaluation abilities. In reality, it is likely that some consumers have higher abilities than others
due to their superior knowledge in a category. And importantly, intermediaries (e.g. YahooShopping or
Taobao Mall) neither customize their design of search environment based on consumers’ evaluation
abilities nor make it feasible for their sellers to customize their prices. This requires the intermediary to
carefully consider the overall effect of its search environment on consumers with different evaluation
abilities, which in turn affects sellers’ prices. To understand this consideration, we extend the previous
setting to the case of consumer heterogeneity in evaluation abilities.
We assume that there are two types of consumers, different in their baseline evaluation costs.
Denote and Ï (0 Ï 1) respectively as the baseline evaluation costs for highcost consumers and
lowcost consumers (henceforth Hconsumers and Lconsumers respectively). Smaller Ï implies larger
difference in evaluation abilities between the two consumer types. We assume that the sizes of both types
of consumers are equal.
42
In addition, suppose that sellers and the intermediary do not discriminate
consumers based on their types. Denote by (½
Ð
, Á
Ð
) and (½
Ñ
, Á
Ñ
) as the optimal evaluation plan of H
consumers and Lconsumers, respectively. For a given Ä, one can apply Lemma 1 to derive (½
Ð
, Á
Ð
) and
(½
Ñ
, Á
Ñ
) and verify that Lconsumers will evaluate no fewer products at a no shallower depth (Á
Ñ
Á
Ð
,
42
Equal mass assumption allows us to focus on the effect of consumer heterogeneity on intermediary’s search
environment design without being influenced by size of the segmentation.
78
½
Ñ
½
Ð
) due to their superior evaluation abilities. The following table illustrates optimal evaluation
plans for both consumer types based on the search environment.
Search
Environment
Ä B g 0, 1
Ž
Ï
p Ä B g 1
Ž
Ï
, 1
Ž
h
Ä
B O1
Ž
, 1R
Evaluation Depth
½
Ð
½
Ñ
1 ½
Ð
½
Ñ
1 ½
Ð
½
Ñ
1
Evaluation
Breadth
Á
Ð
Á
Ñ
Ž
Á
Ð
Ž
Á
Ñ
Ž
Á
Ð
Á
Ñ
Consumer
Evaluation
Behavior
Both consumer
types partially
evaluate products
Hconsumers partially
evaluate products; L
consumers fully evaluate
products
Both consumer
types fully
evaluate
products
Table 2: Optimal Evaluation Plans for Both Consumer Types
As indicated in Table 2, when the search environment modestly reduces search costs, Ä B
N0, 1
P
*
2
ÒU
Q, both consumer types partially evaluate a constant number of products. Alternatively, when
the search environment greatly lowers search costs (Ä B O1
P
*
2
U
, 1R), both consumer types fully evaluate
more products. Clearly from Lemma 2, neither situation is optimal to the intermediary, implying that the
intermediary’s profit maximizing search environment falls into the moderate interval Ä B N1
P
*
2
ÒU
, 1
P
*
2
U
R
. Recall from the previous setting, with homogenous consumers, the optimal search environment is
specified simply as the point that maximizes search costs conditional on full evaluation. However, such a
condition is impossible to specify in the case of heterogeneous consumers. In this moderate interval, an
easier search environment helps Hconsumers to evaluate products at a greater depth, but at the same time
it allows Lconsumers to fully evaluate more products. That is, greater evaluation depth by Hconsumers
is associated with wider evaluation breadth by Lconsumers. How the intermediary optimally handles this
tradeoff is given in the following proposition.
79
Proposition 2: If the sizes of consumer types are equal, then the intermediary’s profit is maximized
at Ä
A
1
P
U
Ž
V
so that both L and Hconsumers evaluate products at full depth. In addition, L
consumers evaluate Á
Ñ
A
*
2
Ò
sellers when Ï B N
*
2
l
, 1Q and evaluate Á
Ñ
A
sellers when Ï B O0,
*
2
l
Q.
This proposition echoes the previous message that greater evaluation depth is more important for
the intermediary’s profit than narrower evaluation breadth. In equilibrium, the intermediary would rather
help Hconsumers to evaluate products at full depth than limit Lconsumers from evaluating more sellers.
Thus, the intermediary’s design objective should be to err on the side of embedding sufficiently low
search costs such that the consumers with lower evaluation abilities can evaluate products at full depth.
This result reinforces the importance of full evaluation in the optimality condition identified in
Proposition 1. It is important to note that the main design objective to ensure full evaluation depth is so
strong that even doing so can allow Lconsumers to fully evaluate all the sellers available on the
intermediary when their evaluation abilities are very high (Ï B $0, Ž
/% ).
4. Intermediary Competition
The previous section studies how one intermediary optimally designs its search environment. Because a
more helpful search environment reduces evaluation costs, an intermediary can use them as a means to
attract consumers from its rivals. Therefore, the design of search environment requires an intermediary to
consider consumer demand, in addition to protecting sellers from fierce competition. In this section, we
study how intermediary competition affects the strategic design of the search environment. We ask, in
particular, how the optimal search environment condition found in Proposition 1 is affected by
competition among intermediaries.
Consider a market with two competing intermediaries (denoted by 1, 2), each with sellers.
We assume that the consumer has individual preferences over both intermediaries (e.g. horizontal
features that distinguish both intermediaries) and only chooses one intermediary at which to shop. All
80
other assumptions about consumers are the same as in section 2. This set up implies that the consumer’s
intermediary choice depends on her expected surplus of evaluating and buying at each intermediary,
which includes two factors: expected net benefit of product evaluation and her individual preference for
an intermediary.
The timing of the model is as follows. First, intermediaries simultaneously choose their search
environments. Second, each seller chooses a price for its product. Third, the consumer chooses an
intermediary. She then selects evaluation depth and breadth. As before, the choices of depth and breadth
are simultaneous. At last, she evaluates products at the selected depth and purchases the best one based
on her evaluation. This timing implies that given a chosen intermediary, the consumer chooses evaluation
depth and breadth and selects the best product in the same way as in section 2, as given in Lemma 1. We
therefore start our analysis from stage three where the consumer chooses one intermediary at which to
shop.
The consumer’s intermediary choice depends on her expected net benefit of product evaluation at
each intermediary,
£
$½
, Á
; Ä
%
½
ln$ Á
% $1 Ä
%Á
$½
%
where Ä
is the search environment on intermediary , (½
, Á
) is the consumer’s evaluation plan on
intermediary which can be obtained from Lemma 1, and
is the equilibrium price on intermediary .
As before, we treat sellers on an intermediary as a priori symmetric and therefore focus on equilibria
where all the sellers on a given intermediary charge the same price. Because the consumer chooses an
intermediary before evaluation, her intermediary choice depends on her rational expectations of the
equilibrium prices.
To capture the differentiation among intermediaries, we assume that consumers (unit mass with
unit demand) are uniformly distributed on a standard Hotelling line of unit length, with two
intermediaries located at each extreme (without loss of generality, intermediary 1 and 2 are respectively
located at location 0 and 1). Horizontal differentiation between the two intermediaries may reflect, for
81
example, website style and layout. A consumer’s location on the line reveals her ideal preference and she
has disutility if her location does not match that of the chosen intermediary. Let Ó denote the linear travel
cost per location unit traveled. Larger Ó implies greater heterogeneity in consumers’ tastes towards the
intermediaries (or one can view Ó as the indicator of intermediaries’ horizontal differentiation). Thus, the
consumer located at ‘I0,1J has total disutility Ó
for intermediary 1 and Ó$1 %
for intermediary 2.
Thus, this consumer’s expected surplus of evaluating and buying at intermediary 1 and 2 are
respectively given by
£
$½
, Á
; Ä
% Ó
(4a)
£
$½
, Á
; Ä
% Ó$1 %
(4b)
Assume that the consumer chooses the intermediary which gives her larger expected surplus.
43
To derive consumer demand for each intermediary, suppose that the consumer located at ¥ is
indifferent between the two intermediaries and the market coverage is full, £
$½
, Á
; Ä
% Ó¥
£
$½
,
Á
; Ä
% Ó$1 ¥%
. This implies that consumers located at ‘I0, ¥% choose intermediary 1 because (4a)
is larger and consumers located at ‘$¥, 1J choose intermediary 2 because (4b) is larger. Let Õ
denote
consumer demand for intermediary . Consumer demand for each intermediary is given by
Õ
¥
Ö
6
$”
6
,Â
6
;‰
6
%VÖ
2
$”
2
,Â
2
;‰
2
%
×
and Õ
1 ¥
Ö
2
$”
2
,Â
2
;‰
2
%VÖ
6
$”
6
,Â
6
;‰
6
%
×
. (5)
We now consider the game played by the competing sellers of each intermediary. A seller’s
profit is determined by its price, and also by the overall consumer demand at its intermediary which was
previously assumed to be fixed in section 2. To determine the symmetric equilibrium prices, we focus on
seller of intermediary by assuming that it sets price at
and other sellers at this intermediary sets
price at
Ø
. The expected profit of seller is given by
k
$1 Ì%Õ
O
Â
/
l
Q
*
$+,
./
%/À
/
1
*
$+,
./
%/À
/
1
D$Â
/
V%*
$+,
.
Ø
/
%/À
/
1
(6)
43
We assume that is sufficiently large so that either (4a) or (4b) is positive for any . This ensures that (1) all
consumers are induced to participate in search and (2) competition between intermediaries is maintained for large
travel costs.
82
where Ì‘$0,1% denotes the referral fees charged by intermediary . Two points about (6) should be
emphasized. First, the only difference between expressions (6) and (2) is the extra term Õ
, capturing the
consumer traffic at intermediary . Second, as argued earlier, Õ
is affected by the equilibrium prices (see
(5)). That is, any seller’s deviation from a potential symmetric equilibrium cannot affect Õ
. Thus, seller
on intermediary does not internalize Õ
when choosing its price
, implying that travel cost (Ó) does
not directly affect seller ’s choice of price.
These two points also imply that sellers’ prices at a given
intermediary are identical as given in Lemma 2.
Intermediary chooses its search environment Ä
to maximize its expected profit:
k
ÌÕ
. (7)
It is important to note that intermediary internalizes its overall consumer demand Õ
when choosing Ä
,
which is the main distinction from section 2. Thus, travel cost affects equilibrium prices only through
intermediaries’ incentives, a result established in the following proposition.
Proposition 3:
(i) When intermediaries are relatively differentiated (Ó/
cdÙ
2
,6f
c,Ù
2
Nln$%
c
c,6
*
2
V
R), the
equilibrium outcomes are the same as in Proposition 1.
(ii) Otherwise, in equilibrium, the intermediary minimizes search costs in the search
environment $Ä
A
1%. The symmetric equilibrium price is PO
c
c,6
Q.
This proposition illustrates that when intermediaries are relatively more differentiated than sellers,
large Ó/, they do not have to aggressively compete for consumers by providing very helpful search
environment that greatly lowers search costs. This allows intermediaries to focus on protecting sellers
from fierce competition just as the monopoly intermediary does in section 2 and we obtain the same
outcome as indicated in Proposition 1. Alternatively, when intermediaries are relatively less differentiated
than sellers, low Ó/, they compete more aggressively to attract consumers by providing very helpful
83
search environments, although doing so intensifies competition among sellers. In fact, in this case,
because Ä
A
1 consumers incur no search costs and evaluate all sellers at the chosen intermediary.
Therefore, unlike the monopoly case, the number of sellers affects the equilibrium prices.
The number of sellers on an intermediary also plays a role in determining the search environment
through its impact on the threshold for Ó/. As indicated in Proposition 3, this threshold is Ushaped in .
For intermediate levels of , intermediaries provide the monopoly search environment (Ä
A
1%. When
the number of sellers is very small or very large, however, intermediaries have a greater incentive to
provide perfectly helpful search environment (Ä
A
1%. Specifically, as becomes very small,
consumers optimally evaluate all sellers, even with positive search costs (See Lemma 1). By removing
search costs, a competitive intermediary attracts additional consumers without adding competitive
pressure on sellers’ prices. On the other hand, when becomes very large, intermediaries enter a
prisoner’s dilemma. That is, by jointly maintaining Ä
A
1, intermediaries earn higher profits. But,
unilaterally lowering search costs, an intermediary can attract consumers to its platform by ensuring them
more surplus (better fitting seller at reduced search costs).
5. Conclusion
This paper has examined strategic design considerations of online shopping intermediaries’ search
environments. We defined a search environment as means to control consumers’ evaluation costs.
Because these intermediaries receive revenue from competing thirdparty sellers, they must balance
improvements in consumers’ search benefits with sellers’ profit incentives. We showed that an
intermediary’s optimal search environment includes positive search costs, but only up to a point. If
consumers face too much search costs, they can scale back the depth of their evaluation of sellers’
products – a regime of partial product evaluation. Our model indicated that it is optimal for the
intermediary to provide search environment that embeds sufficiently low evaluation costs to ensure
consumers evaluate products at full depth. In addition, our findings showed that for heterogeneous
consumers the intermediary should provide helpful enough search environment to just guarantee all
84
consumers (including the consumers with low evaluation abilities) fully evaluate products, a result
reinforcing the importance of full evaluation in the intermediary’s design objective. We also showed that
competitive intermediaries can be induced to provide more helpful search environment in order to attract
shoppers to their platforms, but only if they are sufficiently undifferentiated.
Our research is only a first attempt at understanding the strategic factors that affect the design of
the search environment at online shopping intermediaries. We focused on the strategic role that consumer
search costs play in consumers’ ability to find the right product and thirdparty sellers’ ability to set
profitable prices. In so doing, we omitted several strategic variables that intermediaries may consider.
One such factor is the type of thirdparty sellers the intermediary allows to sell on its platform. We
focused on symmetric sellers of identical ex ante quality. Given that consumers typically differ in the
taste for quality, shopping intermediaries may find it profitable to provide a search environment with a set
of vertically differentiated sellers. Another interesting factor in the design of the search environment
arises when thirdparty sellers provide multiple products. In fact, most of these sellers on intermediaries
carry several products, trying to seek better fits to consumers’ idiosyncratic tastes. This provokes the
possibility that consumers’ search decisions involve two dimensions – across firm and within firm (Liu &
Dukes 2013). Because sellers do not compete in price among their own products, search costs may play a
different role for intermediary profitability. In addition, it might be interesting to study intermediary’s
strategic consideration of search environment design in price dimension, especially in the situation where
there are many sellers selling the identical products. We believe research on these issues can help us
better understand the design of search environments at online shopping intermediaries.
85
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89
APPENDIX 1
The appendix contains all proofs for all propositions, lemmas, and corollaries of the first chapter
“Consideration Set Formation: The Case of Withinfirm & Acrossfirm Evaluation Costs”.
Proof of Lemma 1
A consumer’s maximization of (3) subject to ̂
, 1, … , implies the following Lagranian
Ú$,
, … ,
#
, Û
, … , Û
#
%
ln ∑ expI$
/
% ln
J
! $ % ∑
∑ Û
$
%
.
We ignore integer constraints and assume that for an optimal sample plan $ K, ̂
!% there exist Û
Ç
0,
such that Û
Ç
$̂
% 0 for 1, … , K, and
ÜÚ
Ü#
0,
ÜÚ
Ü@
.
0 for all 1, … , K .
We first show that the optimal number of products ̂
evaluated at each considered firm is identical across
firms. Define the benefit of sample plan $,
!
#
%, gross of evaluation costs, as Ý$
,
, … ,
#
%
ln ∑ expI$
/
% ln
š
J
š
!. Suppose ̂
and Û
0 for all . Then «Ú/«
0 implies
«Ý/«
for all . The only solution has ̂
̂
Þ ̂ #
. Denote the common value as ̂. Now
consider the boundary case in which the consumer evaluates all products at the firms: ̂
for all .
Then for all , «Ý/«
Û
Ç
, for some Û
Ç
0. Now suppose the asymmetric case by assuming that
̂
̂ š
for some pair , �. Then
«Ú/«
0 ß d̂
f
1
2
1
6
V
O
W
P
6
P
2
Q Ý$̂
, … ̂ # K
% and
«Ú/«
š
0 ß $̂ š
%
1
2
1
6
V
O
WDà
Í
P
6
P
2
Q Ý$̂
, … ̂ # K
%.
Since
/
1, we have ̂
̂ š
, which is a contradiction. Therefore, for any , the consumer’s
optimal sample plan has the same number of products ̂ evaluated at each of the firms. Using the
simplification that
Ý$, , … %
ln
ln ,
we find the explicit solution to (3) by maximizing
90
ln
ln $ % ,
with respect to $, % subject to . This yields
ß
̂
P
2
W
O
UVW
P
6
VP
2
Q
K
P
6
VP
2
UVW
j , and ß á
̂
K
P
6
UVWDWX
j .
Note that ̂ and K are weakly decreasing and increasing in . Thus, the optimal sampling plan can be
expressed as in the statement of the lemma.■
Proof of Lemma 2
Using the argument of Anderson & de Palma (1992), hereafter (AdP), we establish that any firm finds it
optimal to charge a single price for all its products. Suppose consumers adopt the sampling plan $, %.
Because consumers do not observe firms’ strategies and their sampling plans cannot respond to
deviations from any equilibrium, firms’ pricing game in our setting is nearly identical to the one analyzed
in (AdP). The only difference is the fact that consumers may not evaluate all firms ( % and products
offered by each firm ( ). Nevertheless, each product from any firm is inspected with equal
probability. Therefore, firms’ pricing incentives in our setting are identical to those in (AdP). Hence, as in
(AdP)
for all firms 1, … , and all products 1, … , .
Now suppose consumers expect all firms to play $, % and implement the sampling plan $, %. We start
by supposing firm deviates in the product line choice with
and all other firms . Maximizing (5) with
respect to
subject to (4) implies
/$1 (
%. (A.1)
Similarly,
/$1 (%, (A.2)
where (
$ 1%( 1. An argument similar to proof of the Lemma (p. 266) of Anderson & de Palma
(1992) establishes that $
, % is a unique equilibrium to the game in which firm sells
products and all
other firms sell .
91
Using the expressions (1) and (2), profits to firm when it sells
products and all other firms sell ,
simplifies to
k
$
, % O
#
l
Q O
P
6
#V
Q Ž
,
.
1
6
Γ$
, % &
,
where,
Γ$
, %
_
`
a
`
b
$
/%
1
6
1
2
i
,
$
/%
1
6
1
2
i
$/%
1
6
1
2
i
1 i ,
j
Denote r$%
#
2
V#
#
2
V#D
and “$%
#
#V
. If
, then ½k
/½
0 with
implies
O
P
2
ml
Q r $% and
“$%. An argument similar to Anderson & de Palma (1992), Proposition
1 (p. 268) proves $, % is a unique equilibrium to this game.
If
, , then ½k
/½
& 0. Since ½k
/½
0 for
it is optimal for firm to set
.
Therefore, and
“$% is the unique equilibrium
The case
leads to the outcome for
, .
The case
leads to the outcome for
, .
Hence, for a given sampling plan $, %, the unique equilibrium as expressed in the lemma. ■
Proof of Lemma 3
The boundary n is determined by the equation
P
2
ml
o
#
A
2
V#
A
#
A
2
V#
A
D
p O
UVW
P
6
VP
2
Q O
P
2
W
Q,
where
A
$
%/$ %
It implies,
& O
UVW
P
6
VP
2
Q O1
#
A
2
V#
A
Q,
Solving explicitly, there is one solution n in real value.
92
It is straightforward that n 0 since otherwise the righthand side of the expression above would be
negative. Also n , since otherwise n 0 , which is debunked above. Thus, 0 n .
Parameterize the equilibrium by :$
A
$%,
A
$%,
A
$%,
A
$%%. By definition of n
^$, %
P
2
lm
r $%
P
2
W
O
UVW
P
6
VP
2
Q ̂$%,
is satisfied for K $% $
%/$ %
/$ %. Thus,
A
$ n %
P
6
VP
2
UVW q
P
6
UVW qDW qX
A
$W q%
,
A
$ n %
A
$ n %
P
2
lm
r$
A
$ n% %
P
2
lm
O
UVW q
P
6
VP
2
Q,
A
$ n %
“$
A
$ n %%.
At n, both the consumers and firms simultaneously operate at the corner solution and satisfy the
conditions for an interior optimum. For all ] n , exactly one of either the consumer or the firms operate
at their respective interior optimum.
Let n . By Lemma 1:
A
$%
P
6
VP
2
UVW q
A
$ n% .
This implies
&
rd
A
$%f
&
rd
A
$ n %f
n
o
n
p
o
p
A
$%
A
$%
where the first inequality is due to the fact that r$% is nondecreasing in , the equality by definition of
n, the strict inequality by assumption ( n), the last two inequalities by Lemmas 1 and 2, respectively.
Hence, by virtue of the strict inequality, firms optimally operate on the boundary of their profit
maximization when n .
Finally, let n .
We show that the equilibrium sampling plan (
A
$%,
A
$%% cannot be the interior solution to consumers’
optimal consideration set formation problem. Suppose it were. Then
A
$%
P
6
VP
2
UVW
, and
A
$%
P
2
W
O
UVW
P
6
VP
2
Q.
93
We now can write
A
$ n %
P
6
VP
2
UVW q
P
6
VP
2
UVW
A
$%,
which follows from Lemma 1and
A
A
in equilibrium. Hence,
A
$ n %
A
$%. This last fact implies
rd
A
$%f r$
A
$ n%% . Thus,
P
2
W
O
UVW
P
6
VP
2
Q
A
$%
A
$%
P
2
lm
rd
A
$%f
P
2
lm
rd
A
$ n %f
A
$ n %
P
2
W q
O
UVW q
P
6
VP
2
Q,
where the inequality holds from Lemma 2. The order of the two extreme expressions contradicts n.
Therefore, the equilibrium sampling plan cannot be an interior solution to the consumers’ consideration
set formation problem. ■
Proof of Proposition 1
By Lemma 3, n implies the equilibrium is determined by the interior solution to the consumers’
consideration set formation problem. Thus,
A
and
A
are expressed as indicated. We must have
A
A
in any equilibrium and Lemma 2 implies the expression for
A
.
Also by Lemma 3, n implies the equilibrium is determined by the interior solution to the firm’s
profit maximization. Thus,
A
P
2
lm
r$
A
%, where $
A
,
A
% must be the boundary solution to the
consumers’ optimal consideration set problem when ̂$
A
%
A
. Lemma 2 implies the expression for
A
.
■
Proof of Corollary 1
The impact of changes , on
A
and k
A
are determined simply through the changes in the equilibrium
sampling plan. That is,
Üã
A
ÜW
“
Ê
$
A
% O
Ü#
A
ÜW
Q and
Üä
A
ÜW
P
6
l
“
Ê
$
A
% O
Ü#
A
ÜW
Q & O
ÜX
A
ÜW
Q, (A.3)
For n , the comparative statics of
A
,
A
, and
A
with respect to follow directly from inspecting the
expression given in Proposition 1. However, the impact of on k
A
is ambiguous.
For n , because the consumer’s optimal sample plan is characterized by an implicit function, we must
first solve the comparative statics on
A
and
A
simultaneously.
94
For , the pair equations
Ü#
A
ÜW
VP
6
$UVWDWX
A
%
2
O
A
1
ÜX
A
ÜW
Q, and
ÜX
A
ÜW
P
2
lm
r
Ê
$
A
% O
Ü#
A
ÜW
Q,
imply «
A
/«, «
A
/« 0.
(A.3) implies Äå$«
A
/«% Äå$«
A
/«%.
Finally, evaluating the comparative statics of k
A
is direct for .
Üã
A
ÜW
0
ÜX
A
ÜW
ß
Üä
A
ÜW
0,
For n , the comparative statics of
A
,
A
and
A
with respect to follow from directly inspecting the
expression implied in Proposition 1.
The comparative statics of k
A
with respect to can be deduced from evaluating (A.3) with the
equilibrium expressions from Proposition 1. That is,
Üä
A
ÜU
P
6
l
“
Ê
$
A
% O
#
UVW
Q
m
W
O
P
2
P
6
VP
2
Q 0 æ
l
O
#
A
#
A
V
Q
m
W
O
P
2
P
6
Q,
For n ,
Ü#
A
ÜU
VP
6
$UVWDWX
A
%
2
O1
ÜX
A
ÜU
Q, and
ÜX
A
ÜU
P
2
lm
r
Ê
$
A
% O
Ü#
A
ÜU
Q,
imply «
A
/«, «
A
/« 0.
(A.3) implies Äå$«
A
/«% Äå$«
A
/«%
Finally, evaluating the comparative statics of k
A
is direct for using (A.5) and the above results.
Specifically,
Üã
A
ÜU
0
ÜX
A
ÜU
ß
Üä
A
ÜU
0. ■
Proof of Corollary 2
Form Lemma 2, ̂$% decreases in
A
(“
Ê
0). However, an increase in
increases the consumer’s
benefit from considering more firms (“$
A
% 0) from Lemma 1. Thus, the comparative statics of
A
and
k
A
A
/ &
A
with respect to
both have a direct effect and an indirect effect.
Specifically,
95
”ã
A
”P
6
“$
A
%
çèé
D
“Ë$
A
% O
”#
A
”P
6
Q
çê ê ê êèê ê ê êé
V
, (A.4)
or
”ã
A
”P
6
0
æ
P
6
UVW
A
$
A
1% when n .
For k
A
, note that
affects not only
A
and
A
, but also
A
. The total impact can be seen by the following:
”ä
A
”P
6
0 æ
”ã
A
”P
6
& O
”X
A
”P
6
Q. (A.5)
Specifically,
Üä
A
ÜP
6
l
g
#
A
#
A
V
P
6
UVW
O
#
A
V
Q
h & O
P
2
W
Q O
UVW
$P
6
VP
2
%
2
Q,
or
Üä
A
ÜP
6
0
æ
P
6
UVW
A
$
A
1%
mlP
2
$UVW%W
O
#
A
V
#
A
Q when n .
For n, the comparative statics of
A
with respect to
follow directly from inspecting the expression
given in Proposition 1.
”X
A
”P
6
0 if n implies
”ã
A
”P
6
0 ß
”ä
A
”P
6
0.
and it also implies
may decrease price but increase profit.
For n,
Ü#
A
ÜP
6
UVWDWX
A
P
6
$UVWDWX
A
%
2
O
ÜX
A
ÜP
6
Q, and
ÜX
A
ÜP
6
P
2
lm
r
Ê
$
A
% O
Ü#
A
ÜP
6
Q,
imply «
A
/«
, «
A
/«
0.
Rearranging the terms of (A.4), it can be shown that
Üã
A
ÜP
6
0 æ
P
6
$UVWDWX
A
%
2
D
1
6
1
2
†ëØ$9
A
%
c…
#
A
$#
A
V%
UVWDWX
A
.
The fact that and rË$% are positive implies that
P
6
UVWDWX
A
A
$
A
1%, (A.6)
96
is sufficient for the positive derivative. Using the expression for
A
in Proposition 1 and noting that
A
P
6
UVWDWX
A
, it can be shown that (A.6) holds for all
A
2, which is assumed.■
Proof of Corollary 3
The impact of changes in
on
A
and k
A
are determined simply through the changes in the equilibrium
sampling plan. That is,
Üã
A
ÜP
2
“
Ê
$
A
% O
Ü#
A
ÜP
2
Q and
Üä
A
ÜP
2
P
6
l
“
Ê
$
A
% O
Ü#
A
ÜP
2
Q & O
ÜX
A
ÜP
2
Q,
For n , the comparative statics of
A
,
A
, and
A
with respect to
follow directly from inspecting
the expression given in Proposition 1.
For k
A
,
Üä
A
ÜP
2
P
6
l
“
Ê
$
A
% O
UVW
Q
m
W
O
UVW
P
6
VP
2
Q O1
P
2
P
6
VP
2
Q 0
æ
l
O
#
A
#
A
V
Q
m
W
,
For n , because the consumer’s optimal sample plan is characterized by an implicit function, we must
first solve the comparative statics on
A
and
A
simultaneously.
For
, the pair of equations
Ü#
A
ÜP
2
VP
6
$UVWDWX
A
%
2
O
ÜX
A
ÜP
2
Q, and
ÜX
A
ÜP
2
lm
r$
A
%
P
2
lm
r
Ê
$
A
% O
Ü#
A
ÜP
2
Q,
imply «
A
/«
0 «
A
/«
.
(A.3) implies Äå$«
A
/«
% Äå$«
A
/«
%.
For
,
Üä
A
ÜP
2
P
6
l
“
Ê
$
A
% O
Ü#
A
ÜP
2
Q & O
ÜX
A
ÜP
2
Q,
ÜX
A
ÜP
2
g
W
l
O
#
A
#
A
V
Q
&h,
which is positive when the squaredbracketed term is positive since «
A
/«
0. ■
Proof of Proposition 2
First note that the social optimum is independent of whether is greater or less than n. The maximizer of
(6) subject to is explicitly expressed as follows:
97
‰Š
W
§
mlWì
2
Vml$UVW%VWì
6
DWì
2
D
ƒ
µmlW$UVW%ì
2
Ddml$U—W%DWì
6
VWì
2
f
2
&©, (A.7)
‰Š
mlW
Æí4&$ %
$&$ %
%
&$ %
È. (A.8)
Using these expressions, we immediately can compute the expression for
‰Š
‰Š
in (7) and establish the
comparison with
A
A
when n . Showing that
‰Š
A
is equivalent to showing
mlì
2
Vml$UVW%VWì
6
DWì
2
DíµmlW$UVW%ì
2
D$ml$UVW%DWì
6
VWì
2
%
2
ml
W
ì
6
Vì
2
UVW
,
which reduces to the condition $2&$ %
%
0 and thus always holds. Finally, it must be the case
that
‰Š
A
since
‰Š
‰Š
A
A
and
‰Š
A
. ■
Proof of Proposition 3
The maximization of (6) implies the system of first order conditions:
‰Š
P
2
#
‹Œ
WDlm
and
‰Š
P
6
@
‹Œ
WDUVW
.
Then compare with the corresponding equilibrium values in the limit t 0
D
:
‰Š
A
P
2
UVWDW#
‹Œ
P
2
lm
r $
A
% t
P
2
lm
N1 r O
P
6
U
QR 0, (A.9)
since r $% 1 for all . Thus, there exists �
0 such that
‰Š
A
for all B I0, �
%. In addition,
‰Š
A
P
6
UVWDW@
‹Œ
P
6
UVWDWX
A
t 0.
For any B I0, �
%,
‰Š
A
implies
‰Š
A
. For the difference in total number of products
evaluated
‰Š
‰Š
A
A
Wtî
ï
ð ñ ñ ò
P
6
P
2
Ulm
P
6
P
2
Ulm
r O
P
6
U
Q
P
6
P
2
Ulm
o1 r O
P
6
U
Qp.
This limit expression is strictly positive for B I0, �
%. (See A.9).) This establishes part (i). To establish
part (ii), recall that the symmetric equilibrium outcome $
A
,
A
,
A
,
A
% is continuous in at n. By
Proposition 2,
‰Š
‰Š
A
A
at n. Therefore, there exists a �
n such that
‰Š
‰Š
A
A
for
all B $ �
, nJ . ■
98
APPENDIX 2
The appendix contains all proofs for all propositions, lemmas, and corollaries of the second chapter
“Consumer Search with Limited Product Evaluation”
Appendix A
Proof of Proposition 1
Starting from the last stage where a consumer takes � B $0,1% as given, she chooses an optimal
evaluation breadth by maximizing (3) with respect to , which yields the solution in (4). As long as
P
š
, the consumer evaluates at least one product. Substituting (4) into (3) yields the utility from each of
the evaluation modes: £
ž
and £
�
.
The condition 0 ∆ directly implies £
�
£
ž
and the consumer engages in full evaluation.
Otherwise, ∆
P
š
, the consumer prefers partial evaluation (£
ž
£
�
). ■
Proof of Corollary 1
We determine prices
ž
A
and
�
A
by maximizing (5) with respect to
and invoking symmetry, which
implies
A
šP
VšU/P
. The price
ž
A
is obtained by setting � 1. ■
Proof of Proposition 2
In (8), ; Imax
??#
�
; I“
$�
%J!J ; Imax
??#
�
$1 �
%•!J since we know ; I“$�
%J
$1 �
%•.
Fix and let Ý s max
??#
�
$1 �
%•!. We can write the cdf of Ý as
PrIÝ J ∏ Pr N
XV$Vš
.
%ö
š
.
R
∏ Ž
V*
,
e,d6,3
.
f÷
3
.
Ž
V ∑ *
,
e,d6,3
.
f÷
3
.
.
,
because
!
"
l
are i.i.d. standard extreme value random variables (or standard Gumbel).
Differentiating with respect to yields the pdf of Ý:
99
i $% Ž
V ∑ *
,
e,d6,3
.
f÷
3
.
.
ø∑
*
,
e,d6,3
.
f÷
3
.
š
.
ù.
The consumer’s objective (8) is therefore equivalent to
ú Ž
V ∑ *
,
e,d6,3
.
f÷
3
.
.
ø∑
*
,
e,$6,3
.
%÷
3
.
š
.
ù
u
Vu
½ ∑ �
.
We start with case (ii). First assume that Ž
V
. The solution �
A
P
U
Ž
V
1, all , is the only
symmetric solution to the first order condition of the interior maximization for (8) with respect to �
,
j ûÖd#, š
.
!
.56
9
f
ûš
8
ü
š
.
"š
ú Ž
V#*
,
e,$6,3%÷
3 V
e, $6,3 %÷
3
XVö
š
ý
Ž
V#*
,
e,$6,3%÷
3 V
e, $6,3 %÷
3
XVöVš
š
ý
u
Vu
½ 2�
0, 1, … , ,.
Under the symmetric depth condition, �
� for all , we can differentiate (8) with respect to and
solve the following:
j ûÖd#, š!
.56
9
f
û#
ü
š"š
A
0.
to obtain
A
Ž
. We now establish that the solution �
A
P
U
Ž
V
1, all , is a local optimum by
showing that the Hessian matrix þ ¶
Ü
2
ÖO#, š
.
!
.56
9
A
Q
Üš
/
Üš
8
·
,
is negative semidefinite, or equivalently, its
leading principal minors, denoted þ
, þ
, …, alternate in sign beginning with þ
 0. Note that at
any symmetric solution, all diagonal terms of þ are equal to :
j û
2
ÖO#, š
.
!
.56
9
A
Q
ûš
/
2
š
.
"š
A
and all off diagonal
terms are equal to
j
Ü
2
ÖO#, š
.
!
.56
9
A
Q
Üš
/
Üš
8
š
.
"š
A
,
. We compute both of these values below.
j û
2
Öd#, š
.
!
.56
9
f
ûš
/
2
š
.
"š
ú 2Ž
V#*
,
e, $6,3 %÷
3 V
e, $6,3 %÷
3
#X
š
N
ö
š
2
XV$Vš%ö
š
ý
R Ž
V#*
,
e, $6,3 %÷
3 V
e, $6,3 %÷
3
#X
š
O
ö
š
u
Vu
100
XV$Vš%ö
š
2
Q g
ö
š
XV$Vš%ö
š
2
Ž
V
e, $6,3 %÷
3
O
ö
š
XV$Vš%ö
š
2
Qh 2Ž
V#*
,
e, $6,3 %÷
3 V
e, $6,3 %÷
3
O
ö
š
XV$Vš%ö
š
2
Q
*
,
e, $6,3 %÷
3
š
2
*
,
e, $6,3 %÷
3 OV
÷
3
D
e, $6,3 %÷
3
2
Q
š
Ž
V#*
,
e, $6,3 %÷
3
*
,
e, $6,3 %÷
3
š
ý
*
,
e, $6,3 %÷
3 O
2÷
3
2
V
2$e, $6,3 %÷%
3
ý
Q
š
*
,
e, $6,3 %÷
3 OV
÷
3
D
e, $6,3 %÷
3
2
Q
š
2
*
,
e, $6,3 %÷
3 OV
÷
3
D
e, $6,3 %÷
3
2
Q
2
š
½ 2.
Let
XV$Vš%ö
š
. Then we can rewrite
j û
2
Öd#, š
.
!
.56
9
f
ûš
/
2
š
.
"š
ú YÆ2Ž
V#*
,
V
$• % Ž
V#*
,
V
$• %
$1 Ž
V
%È
u
Vu
ÆŽ
V#*
,
V
I2 4$• % $• %
J 2Ž
V#*
,
V
$• %$1 • %ÈZ
šD$Vš%ö
š
2
½ 2.
At the symmetric optimal �, 2�
A
2Ž
V
. This affords the simplification that
: O
š
A
Q ú YÆ2Ž
V#*
,
V
$• % Ž
V#*
,
V
$• %
$1 Ž
V
%È ÆŽ
V#*
,
V
I2
u
Vu
4$• % $• %
J 2Ž
V#*
,
V
$• %$1 • %ÈZ
š
A
D$Vš
A
%ö
š
A
½ 2Ž
V
P
š
A
N
$Vš
A
%ö
š
A
2Ž
V
R, (A1)
where
ú YÆ2Ž
V#*
,
V
$• % Ž
V#*
,
V
$• %
$1 Ž
V
%È Ž
V#*
,
V
I2 4$• %
u
Vu
$• %
J 2Ž
V#*
,
V
$• %$1 • %Z ½ , and
ú YÆ2Ž
V#*
,
V
$• % Ž
V#*
,
V
$• %
$1 Ž
V
%È Ž
V#*
,
V
I2 4$• %
u
Vu
$• %
J 2Ž
V#*
,
V
$• %$1 • %Z ½ .
We now turn to the crosspartial term. For any , we can write the generic cross partial derivative of
£$, �
!
"
#
% evaluated for the symmetric case as
101
j Ü
2
Öd#, š
.
!
.56
9
f
Üš
/
Üš
8
š
.
"š,
ú ÆŽ
V#*
,
V
$• %
2Ž
V#*
,
V
$• %$1 • %È
šD$Vš%ö
š
2
u
Vu
½
š
Mú ÆŽ
V#*
,
V
$• %
2Ž
V#*
,
V
$• %$1 • %È
u
Vu
½
$Vš%ö
š
ú ÆŽ
V#*
,
V
$• %
2Ž
V#*
,
V
$• %$1 • %È
u
Vu
½ S.
For simplicity,
P
š
A
NÏ
Ï
$Vš
A
%ö
š
A
R, (A2)
where
Ï
ú ÆŽ
V#*
,
V
$• %
2Ž
V#*
,
V
$• %$1 • %È
u
Vu
½ , and
Ï
ú ÆŽ
V#*
,
V
$• %
2Ž
V#*
,
V
$• %$1 • %È
u
Vu
½ .
Using (A1), one can directly conclude that : þ
 0 by showing
$Vš
A
%ö
š
A
2Ž
V
0.270671.
At
A
Ž
, that real values for
and
can be calculated (approximately 0.192489 and – 0.024982,
respectively) in order to confirm that
2Ž
V
and
0, which implies the above inequality. Hence,
þ
 : 0. We use the real values of Ï
and Ï
, (– 0.0301279 and 0.0102812, respectively) in (A2) to
establish that
þ
 ü
:
:
ü :
0
î.îî $î.îµ
µ Dš
A
%$î.
µµ Dš
A
%
$š
A
%
2
0,
þ

:
:
:
0
î.îîîîµ $î.î Dš
A
%dî.
D.µµ š
A
D$š
A
%
2
f
$š
A
%
ý
0,
both of which hold since �
A
0. Continuing in this manner, we evaluate the signs of the remaining
principal minors. Showing þ
µ
 0, þ
 0, þ
 0, þ
 0, and þ
 0 is the same as showing,
respectively
.
î
,
$Vî.îîDš
A
%$î.
µ
Dš
A
%Æî.
µ D.µµµ š
A
D$š
A
%
2
È
$š
A
%
0,
102
.µî
,
$Vî.îµµDš
A
%$î.
µ Dš
A
%$î.
µ Dš
A
%Æî.
D.µµ š
A
D$š
A
%
2
È
$š
A
%
0,
.î
,
$Vî.îîDš
A
%$î.
Dš
A
%Æî.
µ D.µ š
A
D$š
A
%
2
ÈÆî.µî D.µî šD$š
A
%
2
È
$š
A
%
0
.µî
,6
$Vî.îµDš
A
%$î.
î Dš
A
%$î.
Dš
A
%Æî.
î D.µµ š
A
D$š
A
%
2
ÈÆî.µî D.µµ šD$š
A
%
2
È
$š
A
%
0
.µî
,62
$Vî.îDš
A
%$î.µDš
A
%Æî.D.µµš
A
D$š
A
%
2
ÈÆî.
D.µ šD$š
A
%
2
ÈÆî.µD.µš
A
D$š
A
%
2
È
$š
A
%
0,
all of which hold since �
A
Ž
V
.
(ii) Now assume that 0 Ž
V
. Then, for any ,
j ûÖd#, š
.
!
.56
9
f
ûš
/
š
.
"š
0,
for �
1, 1, … , . Thus, �
A
1, is the locally optimal depth for all products sampled. Again,
under the symmetric depth condition �
�, all , we use the first order condition
j ûÖd#, š!
.56
9
f
û#
ü
š"
0,
to solve for
A
/.
Finally, we show that under the condition � � , the consumer does not find it optimal to add products
beyond
A
at a different depth. First note that it can be directly verified that the solution above satisfies
�
A
�. Now assume the consumer evaluates å 1 additional product at depth � B I� , �
A
%. She obtains
no benefit from doing so if the net benefit is negative (net evaluation costs exceed net evaluation benefit),
å�
; Nmax O �
A
!
"
#
A
�
!
#
A
D
#
A
D
QR ; Æmaxd �
A
!
"
#
A
fÈ .
For �
A
� �
; Nmax O �
A
!
"
#
A
�
!
#
A
D
#
A
D
QR ; Æmaxd �
A
!
"
#
A
fÈ
; Nmax O �
A
!
"
#
A
D
QR ; Æmaxd �
A
!
"
#
A
fÈ �
A
ln ¶
Ž
å
Ž
·
Thus, the net benefit is negative as long as the following holds
å�
�
A
ln O
*
2
D
*
2
Q.
103
Let �$å% s
P
U
Ž
V
ƒln O
*
2
D
*
2
Q /å. Then, for any evaluation depth � B Æ�$å%, �
A
f , the consumer obtains no
benefit from evaluating additional å products. Since �$å% is decreasing in å, the condition that � �
�$å 1% is sufficient to ensure that it is not beneficial for the consumer to deviate from the sample plan
$
A
, �
A
% by choosing to evaluate additional products beyond
A
at depth � B I� , �
A
%. ■
Proof of Lemma 1
The first order conditions for the maximization of (12) with respect to and imply the expressions
given in the lemma. It is directly verified that
Ü
2
Ö
Ü#
2
0. Also, N
Ü
2
Ö
3
Ü#
2
R N
Ü
2
Ö
3
Ü@
2
R N
Ü
2
Ö
3
Ü#Üš
R
�
O
P
6
P
2
�Q,
which is positive for � B $0,1J under the assumption that
. This establishes that solution (
š
,
š
%
constitutes a maximizer. ■
Proof of Proposition 3
To determine the equilibrium evaluation depth, maximize (12) with respect to � after substituting in the
expressions for $
š
,
š
% from the Lemma. This yields,
�
A
¸
P
6
P
2
ƒ
P
6
2
µP
2
2
U
6
*
2
U
2
if
U
6
P
2
*
2
$P
6
VP
2
%
1 if 0
U
6
P
2
*
2
$P
6
VP
2
%
j .
Plugging �
A
into $
š
,
š
%, one can obtain $
A
,
A
% as given in the statement of the proposition.
The symmetric equilibrium price is determined by maximizing (14) with respect to
and using the
symmetry condition. Substitute
A
,
A
and �
A
to obtain the expressions for
A
in the full and partial
evaluation regimes. ■
Appendix B (Additional Supporting Analyses)
B.1 Partial Product Evaluation with Discrete Number of Attributes
In the main analysis, we assumed that the depth of partial evaluation was continuous. Doing so facilitates
the use of calculus to derive optima and comparative statics results. However, it may be more natural to
interpret partial product evaluation as evaluation a subset of discrete set of product attributes. In this
appendix, we demonstrate that our continuous partial product evaluation model can be approximated by a
104
model in which each product has a discrete number of attributes and generate the utility formulation in
(3).
Specifically, suppose the product from firm has B ! attributes. We assert that the random
utility variable
can be represented by a finite sum of independent random variables, Y
Z
"
, each of
which represents the utility component from one of the attributes and has the same variance. The
following lemma establishes the assertion.
Lemma B.1 Let
be a random variable with extreme value distribution and B !. Then there
exists a set of independent random variables
!
"
, with following properties.
(i)
∑
" "
”
.
(ii) For each "̂ B 1, … , !, define ̂
$"̂% ∑
#̂ "
. Then ̂
$"̂% ƒ
#̂
"
”
and $ d̂
$"̂%f
O
#̂
Q $ $
%.
Proof: By the decomposability property of the extreme value distribution (EVD), for any
� B $0,1%,
"
”
�
“$�%, where the first term has EVD with the cumulative distribution
function Ž
*
,�/3
and is independent of “$�%. For 1, define
í1/ A
. For B 2, … , !
we construct
as follows. Let � ƒ
V
. Then the selfdecomposability property of
implies
that there exists a random variable, denoted “ oƒ
V
p, which is independent of
, such that
ƒ
V
A
“ oƒ
V
p
"
”
. Define the random variable
ƒ
A “ oƒ
V
p for 2, . . . .
Summing these independent random variables:
∑
"
"
”
ƒ
∑ ƒ
“ oƒ
V
p
"
"
”
ƒ
%ƒ
“ ¶ƒ
·
çê ê ê êèê ê ê êé
&
. 5
À
∑ ƒ
“ oƒ
V
p
"
105
Þ
"
”
ƒ
#̂
∑ ƒ
“ oƒ
V
p
"#̂D
, "̂ B 2, … , !,
Þ
“$1%
"
”
"
”
.
By construction of the random variables
!
"
, they are independent from each other and from
the random utility term
. And, as established above, their sum ∑
"
has the same
distribution as
. This establishes (i).
To establish part (ii), first note that an argument similar to that in (i) can be used to show
̂
$"̂% ƒ
#̂
"
”
. To establish the variance of this random variable, observe the general result that the
selfdecomposability property of
with EVD implies $ $
% $ d�
“$�%f
�
$ $
% $ d“$�%f , where “$�% '
. Using the definition of
above, it is readily seen
that for any 1, … ,
$ d
f $ (ƒ
A “ oƒ
V
p)
O1
V
Q $ $
% O
Q $ $
%.
Therefore,
$ d̂
$"̂%f ∑ $ $
%
#̂ "
O
#̂
Q $ $
%,
which establishes (ii). ■
To complete the discrete interpretation of our partial product evaluation model, suppose that a
consumer plans to evaluate "̂ B 1, … , ! attributes of each of the inspected products 1, … , . Our
construction requires that the consumer evaluates attributes 1, … , "̂!. Her total search costs of evaluating
products at the depth � í"̂/ is $"̂/ %, or �
. After a consumer incurs the evaluation cost to
examine "̂ attributes, her benefit is determined by her expected choice: ; Æmax
",…#
�
È � ln
•. Hence, with discrete attributes, we arrive at the same ex ante evaluation objective as given by (3) in
the main text.
B.2 Fixed Costs
106
In our specification of the main model, we assume the consumer incurred an evaluation cost that is
proportional to the square of depth, �
but pays no fixed evaluation cost (e.g. travel cost). In this section,
we extend the simple model of section 2 to include the fixed cost,
î
0, incurred by the consumer for
each product evaluated, in addition to the costs �
for evaluating a product at depth �. For the purpose
of tractability, we assume that
î
[ and [ 0.
44
With fixed costs, the consumers’ expected evaluation
benefit (given by expression (3)) becomes,
£
š
$% � ln$% • $�
[% (B1)
The consumer determines the number of firms to evaluate by maximizing (B1).
�
A
š
¤
P
$š
¤
2
D\%U
,
45
ž
A
P
$D\%U
for [ 0. To ensure the consumer at least evaluates one product, we assume the evaluation cost is
always less than a critical value ( ¥ s
š
¤
P
š
¤
2
D\
). Given constant prices and the optimal number of firms
evaluated, a consumer’s expected utility from the two evaluation modes is given by
£
�
�
�
ln g
š
¤
P
dš
¤
2
D\f U
h • �
�
,£
ž
ln N
P
$D\%U
R • .
The following lemma summarizes the consumer’s choice of evaluation mode and firms’ equilibrium
prices.
Lemma B.2: Consider the equilibrium when consumers choose between evaluation modes (P)atrial,
with �
�
B I�, 1%, � 0, and (F)ull, with �
ž
1.. Let ∆s O
š
¤
2
D\
š
¤
P
Q
3
¤
6,
O
P
D\
Q
6
6,3
¤
Ž
V
and 0 [
[
�
$�
�
% s
*
6,3
¤
Vš
¤
Vš
¤
*
6,3
¤
�
�
.
44
The results about the effects of
î
and on consumer’s (k, m) choice and firms’ pricing power in this paper
remain with a generalized fixed cost
î
0.
45
Note that
A
can either increase or decrease in �
�
. Specifically, when �
�
is small (�
�
B $0, √[J), the marginal
increase in evaluation benefit from an increase in �
�
exceeds the corresponding increase in evaluation costs because
the evaluation benefit is linear in �
�
while the evaluation costs are convex. This implies that the consumer tends to
evaluate more products. However, when �
�
is large (�
�
B $√[ ,1%), the reverse holds and consumers evaluate fewer
products in response to larger �
�
.
107
(i) Then consumers engage in full evaluation if 0 ∆ and engage in partial evaluation if
∆ ¥.
(ii) The competitive equilibrium prices (
A
šP
V
6
9
A
) in each evaluation mode are:
ž
A
P
V
$1
2
ï+%¨
1
,
�
A
š
¤
P
V
$3
¤
2
ï+%¨
3
¤
1
.
Proof: Similar to the proof of Proposition 1 and Corollary 1.
This lemma illustrates that partial product evaluation in equilibrium occurs only when [ is not too large.
46
The results of Lemma B.2 are consistent with those in Proposition 1 and Corollary 1 where fixed costs
are not present.
It is noteworthy that [
�
$�
�
% increases in �
�
, implying that there is a lower bound for the restricted depth
�
�
, �$[% 0. Intuitively, a consumer will not evaluate any product at a small partial depth if she has to
invest in a fixed cost [ before doing so. Similarly, in the more generalized model with unrestricted depth,
any fixed cost, [ 0, implies that no consumer in equilibrium would evaluate a product at an arbitrarily
small depth �. This provides an alternative interpretation of the minimum depth threshold, � 0,
assumed in the main text.
46
Our results also show that when the fixed cost is relatively large ([
�
[ �
�
/ �
), the consumer finds it
optimal to inspect fully if she considers it worth evaluating at all. Our model, in this situation, is consistent with the
conventional search literature in which partial product evaluation is not modeled.
108
APPENDIX 3
The appendix contains all proofs for all propositions, lemmas, and corollaries of the third chapter “Online
Shopping Intermediaries: The Strategic Design of Search Environments”
Proof of Lemma 1
Consumers simultaneously choose the optimal evaluation breadth and depth by maximizing (1) with
respect to Á and ½ B I0, 1J.
47
The first order condition yields the expressions for Á
¬
$Ä% and ½
Ç
$Ä% in
Lemma 1. Checking the Hessian matrix:
Ü
2
Ö$”,Â;‰%
ÜÂ
2
”P
Â
2
_
`
a
`
b
P
2
$V‰%U
Ž
V
0 Ä 1
P
U
Ž
V
$V‰%
2
U
2
P
1
P
U
Ž
V
Ä 1
P
Ul
P
l
2
1
P
Ul
Ä 1
j ,
Ü
2
Ö$”,Â;‰%
Ü”
2
2$1 Ä%Á
_
a
b
2$1 Ä%Ž
0 Ä 1
P
U
Ž
V
2 1
P
U
Ž
V
Ä 1
P
Ul
2$1 Ä% 1
P
Ul
Ä 1
j ,
Ü
2
Ö$”,Â;‰%
ÜÂÜ”
P
Â
2$1 Ä%½
_
a
b
P
*
2
0 Ä 1
P
U
Ž
V
$1 Ä% 1
P
U
Ž
V
Ä 1
P
Ul
P
l
2$1 Ä% 1
P
Ul
Ä 1
j .
One can easily verify that
Ü
2
Ö$”,Â;‰%
ÜÂ
2
Ü
2
Ö$”,Â;‰%
Ü”
2
N
Ü
2
Ö$”,Â;‰%
ÜÂÜ”
R
0 holds for 0 Ä 1
P
Ul
. Hence, the
solution to
ÜÖ
ÜÂ
ÜÖ
Ü”
0 is the interior maximum for Ä B N0,1
P
Ul
R.
Now suppose 1
P
Ul
Ä 1. Then
j ÜÖ
Ü”
ü
Â"*
2
2I1 $1 Ä%Ž
½/J 2 O1
*
2
l
½Q,
which is positive for all ½ B I0,1J and Ž
. Therefore, setting ½ 1 is required at any maximized
solution. In this case, the consumer optimal choice of Á must satisfy either
j ÜÖ
ÜÂ
ü
”",Â¯l
0 or
47
Liu & Dukes (2014) show that symmetric depth d is an optimal solution in the setting in which consumers are
permitted to evaluate products with different depths.
109
j ÜÖ
ÜÂ
ü
”",Â"l
0. In the first case, we must have Á
P
$V‰%U
. This solution for Á is a maximizer since
j Ü
2
Ö
ÜÂ
2
ü
”",Â"
1
$6,‹ %¨
0. In the second case, at the boundary Á , we have
j ÜÖ
ÜÂ
ü
”",Â"l
P
l
$1 Ä% 0
under the condition that Ä 1
P
Ul
. Hence, the boundary solutions O½ 1, Á
P
$V‰%U
Q and $½ 1, Á
% are both maximizing. ■
Proof of Lemma 2
Given Æ½
Ç
$Ä%, Á
¬
$Ä%È, we determine equilibrium price by maximizing (2) with respect to
and invoking
symmetry. It is straightforward to show
Üä
.
Üã
.
$1 Ì%
Â
l
N
,
.
$,
.
V%
”P
(
R 0,
implies ̂$Ä%
”
¬
$‰%P
V
6
Ã
Í
$‹%
. Substituting in the expressions for ½
Ç
$Ä% and Á
¬
$Ä% into ̂$Ä%yields the expression for
prices given in the statement of the lemma.
One can also check
Ü
2
ä
.
Üã
.
2
$1 Ì%
Â
l
N
,
.
$,
.
V%$,
.
V%
$”P %
2
,
.
$,
.
V%
”P
R.
Evaluating
Ü
2
ä
.
Üã
.
2
at any point where
Üä
.
Üã
.
0 yields
Ü
2
ä
.
Üã
.
2
$1 Ì%
Â
l
,
.
”P
0.
Thus,
̂$Ä% satisfies the seller’s S.O.C. for profit maximization.■
Proof of Proposition 1
The intermediary chooses Ä to maximize (3) subject to Ä B I0, 1J. That is, the profit maximizing level of
search aids is obtained by maximizing sellers’ price ̂$Ä%. From the expression for ̂$Ä% in Lemma 2,
̂$Ä% increases in Ä for Ä B N0, 1
P
U
Ž
V
Q, decreases in Ä for Ä B N1
P
U
Ž
V
, 1
P
Ul
R and is independent
from Ä for Ä B $1
P
Ul
, 1J. This yields the profit maximizing level of search aids Ä
A
as expressed in the
statement of the proposition. Plugging Ä
A
back into Æ½
Ç
$Ä%, Á
¬
$Ä%È and ̂$Ä% yields the result. ■
110
Proof of Proposition 2
We first determine the equilibrium prices of sellers. Assume that seller sets price at
while all other
sellers set price at . Under this condition, seller ’s expected profit is given by,
k
$1 Ì%
Â

l
*
+,
.
À

1
*
+,
.
À

1
D$Â

V%*
+,
À

1
Â
.
l
*
+,
.
À
.
1
*
+,
.
À
.
1
D$Â
.
V%*
+,
À
.
1
,
Taking the FOC of k
with respect to
and invoking symmetry yields the prices
P
Ã
.
,6
Ã
.
À
.
D
Ã

,6
Ã

À

.
Following a proof similar to that of Lemma 2, one can check that
Üä
.
Üã
.
V/
M
Â

l
N
,
.
$,
.
V%
”

P
(
Ð
R
Â
.
l
N
,
..
$,
..
V%
”
.
P
(
Ñ
RS,
where (
Ð
*
+,
.
À

1
*
+,
.
À

1
D$Â

V%*
+,
À

1
and (
Ñ
*
+,
.
À
.
1
*
+,
.
À
.
1
D$Â
.
V%*
+,
À
.
1
.
Evaluating
Ü
2
ä
.
Üã
.
2
at
P
Ã
.
,6
Ã
.
À
.
D
Ã

,6
Ã

À

yields
Ü
2
ä
.
Üã
.
2
V/
l
6
Ã

D
Ã
.
,6
Ã
.
À
.
”

Ã
.
,6
Ã
.
À
.
”

D
Ã

,6
Ã

”

P
O
Â

1Q
6
Ã
.
D
Ã

,6
Ã

À

”
.
Ã

,6
Ã

À

”
.
D
Ã
.
,6
Ã
.
”
.
P
O
Â
.
1Q,
which is negative because Á
Ñ
Á
Ð
1. Thus, the solution to the F.O.C. is a maximizer.
Next, we determine the consumers’ evaluation plan $½
Ð
, Á
Ð
% and $½
Ñ
, Á
Ñ
% for each interval illustrated in
Table 2.
When Ï Ž
,
(i) For Ä B N0, 1
P
*
2
ÒU
Q, one can verify from Table 2 that the evaluation depth and breadth for both
types of consumers are respectively given by O½
Ñ
P
*
2
Ò$V‰%U
, Á
Ñ
Ž
Q and O½
Ð
P
*
2
$V‰%U
, Á
Ð
Ž
Q
.
(ii) For Ä B N1
P
*
2
ÒU
, 1
P
*
2
U
Q, the evaluation plan for both types of consumers are respectively
given by O½
Ñ
1, Á
Ñ
P
$V‰%ÒU
Q and O½
Ð
P
*
2
$V‰%U
, Á
Ð
Ž
Q.
111
(iii) For Ä B N1
P
*
2
U
, 1
P
ÒUl
Q, the evaluation plan for both types of consumers are respectively
given by O½
Ñ
1, Á
Ñ
P
$V‰%ÒU
Q and O½
Ð
1, Á
Ð
P
$V‰%U
Q.
(iv) For Ä B N1
P
ÒUl
, 1
P
Ul
Q, the evaluation plan for both types of consumers are respectively given
by $½
Ñ
1, Á
Ñ
% and O½
Ð
1, Á
Ð
P
$V‰%U
Q.
(v) For Ä B N1
P
Ul
, 1R, the evaluation plan for both types of consumers are respectively given by
$½
Ñ
1, Á
Ñ
% and $½
Ð
1, Á
Ð
%.
Plugging $½
Ð
, Á
Ð
% and $½
Ñ
, Á
Ñ
% back into yields
_
`
`
`
a
`
`
`
b
P
2
$*
2
V%$V‰%U$ÒD%
Ä B N0, 1
P
*
2
ÒU
Q
P
2
PD$*
2
VVÒ%$V‰%U
Ä B N1
P
*
2
ÒU
, 1
P
*
2
U
Q
P
2
PV
$0ï6%$6,‹ %¨
2
Ä B N1
P
*
2
U
, 1
P
ÒUl
Q
P
2
2c,6
c
PV$V‰%U
Ä B N1
P
ÒUl
, 1
P
Ul
Q
Pl
lV
Ä B N1
P
Ul
, 1R ,
j
which increases in Ä B N0,1
P
*
2
U
R, decreases in Ä B O1
P
*
2
U
, 1
P
Ul
Q and is independent of Ä B
N1
P
Ul
, 1R. This yields the intermediary’s profit maximizing level of search aids in search environment,
Ä
A
1
P
*
2
U
.
Plugging Ä
A
back into the expressions for $½
Ð
, Á
Ð
% and $½
Ñ
, Á
Ñ
%yields
½
Ð
A
1, Á
Ð
A
Ž
and ½
Ñ
A
1, Á
Ñ
A
*
2
Ò
Ž
.
When Ï Ž
,
(i) For Ä B N0, 1
P
*
2
ÒU
Q, one can verify from Table 2 that the evaluation depth and breadth for both
types of consumers are respectively given by O½
Ñ
P
*
2
Ò$V‰%U
, Á
Ñ
Ž
Q and O½
Ð
P
*
2
$V‰%U
, Á
Ð
Ž
Q
.
112
(ii) For Ä B N1
P
*
2
ÒU
, 1
P
ÒUl
Q, the evaluation plan for both types of consumers are respectively
given by O½
Ñ
1, Á
Ñ
P
$V‰%ÒU
Q and O½
Ð
P
*
2
$V‰%U
, Á
Ð
Ž
Q.
(iii) For Ä B N1
P
ÒUl
, 1
P
*
2
U
Q, the evaluation plan for both types of consumers are respectively
given by $½
Ñ
1, Á
Ñ
% and O½
Ð
P
*
2
$V‰%U
, Á
Ð
Ž
Q.
(iv) For Ä B N1
P
*
2
U
, 1
P
Ul
Q, the evaluation plan for both types of consumers are respectively given
by $½
Ñ
1, Á
Ñ
% and O½
Ð
1, Á
Ð
P
$V‰%U
Q.
(v) For Ä B N1
P
Ul
, 1R, the evaluation plan for both types of consumers are respectively given by
$½
Ñ
1, Á
Ñ
% and $½
Ð
1, Á
Ð
%.
Plugging $½
Ð
, Á
Ð
% and $½
Ñ
, Á
Ñ
% back into yields
_
`
`
`
a
`
`
`
b
P
2
$*
2
V%$V‰%U$ÒD%
Ä B N0, 1
P
*
2
ÒU
Q
P
2
PD$*
2
VVÒ%$V‰%U
Ä B N1
P
*
2
ÒU
, 1
P
ÒUl
Q
P
2
c,6
c
PD$*
2
V%$V‰%U
Ä B N1
P
ÒUl
, 1
P
*
2
U
Q
P
2
2c,6
c
PV$V‰%U
Ä B N1
P
*
2
U
, 1
P
Ul
Q
Pl
lV
Ä B N1
P
Ul
, 1R
j ,
which increases in Ä B N0,1
P
*
2
U
R, decreases in Ä B $1
P
*
2
U
, 1
P
Ul
% and is independent of Ä B
N1
P
Ul
, 1R. This yields the intermediary’s profit maximizing level of search aids in search environment,
Ä
A
1
P
*
2
U
.
Plugging Ä
A
back into the expressions for $½
Ð
, Á
Ð
% and $½
Ñ
, Á
Ñ
%yields
½
Ð
A
1, Á
Ð
A
Ž
and ½
Ñ
A
1, Á
Ñ
A
Ž
.■
Proof of Proposition 3
We claim that there is a symmetric equilibrium in which both intermediaries set
113
Ä
A
¸
1
P
U
Ž
V
×
P
l d*
2
Vf
lV*
2
Nln$%
l
lV
*
2
V
R
1 0
×
P
l d*
2
Vf
lV*
2
Nln$%
l
lV
*
2
V
R .
j
To prove this claim we demonstrate directly that one intermediary, intermediary 1, cannot be more
profitable by deviating from Ä
A
given that the other intermediary, intermediary 2, chooses Ä
A
.
(i) Suppose
×
P
l d*
2
Vf
lV*
2
Nln$%
l
lV
*
2
V
R and consider any deviation Ä
] Ä
A
Ä
A
1
P
U
Ž
V
,
with the corresponding profits denoted by k n
$Ä
%. Any deviation Ä
B N0, 1
P
U
Ž
V
Q leads to profits
k n
$Ä
%
×
O
*
2
V
Q
/P
2
$V‰
6
%U
NÓ
*
2
$*
2
V%
P
2
$V‰
6
%U
P
*
2
V
R,
which we show is increasing on this interval. Specifically,
Üä q
6
Ü‰
6
0 as long as
×
P
P
*
2
$*
2
V%$V‰
6
%U
*
2
V
.
for all Ä
. We have,
×
P
l d*
2
Vf
lV*
2
Nln$%
l
lV
*
2
V
R
*
2
V
P
*
2
$*
2
V%$V‰
6
%U
*
2
V
.
where the first inequality holds by assumption, the second since Ž
, and the third for Ä
B
N0, 1
P
U
Ž
V
Q. Therefore, any deviation Ä
Ä
A
1
P
U
Ž
V
is not profitable.
Any deviation Ä
B O1
P
U
Ž
V
, 1
P
lU
Q leads to
k n
$Ä
% O
×
Q
/P
2
PV$V‰
6
%U
áÓ
P
2
PV$V‰
6
%U
ln N
P
$V‰
6
%U
R %
P
V
6
Ù
2
1.
This deviation is not profitable if
Üä q
6
Ü‰
6
0, which requires
×
P
i$Ä
% s
P
$V‰
6
%U
P
PV$V‰
6
%U
ln N
P
$V‰
6
%U
R
*
2
V
.
Since
×
P
l d*
2
Vf
lV*
2
Nln$%
l
lV
*
2
V
R Ž
*
2
V
i O1
P
U
Ž
V
Q, profits are decreasing near (and
to the right of) Ä
A
. Note that the function i $Ä
% is strictly increasing in Ä
. Therefore, the condition
114
k
A
k n$Ä
% at Ä
1
P
lU
, the right endpoint of the interval, is sufficient for k
A
k n$Ä
% at any Ä
B
O1
P
U
Ž
V
, 1
P
lU
Q. This condition is
k
A
/P
O
*
2
*
2
V
Q O
/P
×
Q O
l
lV
Q NÓ ln$ %
l
lV
O
*
2
*
2
V
Q 2R k n
O1
P
lU
Q,
which holds by our assumption
×
P
l d*
2
Vf
lV*
2
Nln$%
l
lV
*
2
V
R.
Any deviation Ä
B O1
P
lU
, 1R leads to a profit of
k n
$Ä
% O
/P
×
Q O
l
lV
Q NÓ ln$ %
l
lV
$1 Ä
%
P
*
2
V
R,
which is increasing in Ä
and therefore bounded above by k n$Ä
1%. The condition
×
P
l d*
2
Vf
lV*
2
Nln$%
l
lV
*
2
V
R directly implies that
k
A
/P
O
*
2
*
2
V
Q O
/P
×
Q O
l
lV
Q NÓ ln$ %
l
lV
P
*
2
V
R k n
$1% k n
$Ä
%,
for all Ä
B O1
P
lU
, 1R.
(ii) Now suppose
×
P
l d*
2
Vf
lV*
2
Nln$%
l
lV
*
2
V
R. We first consider any deviation Ä
B N0,1
P
U
e
V
Q.
This leads to a deviation profit of
k n
$Ä
%
/P
2
×U
N
$V‰
6
%$*
2
V%
R MÓ
P
2
U*
2
N
$V‰
6
%$*
2
V%
R N
l
lV
ln$ %RS.
Characterizing the shape of this deviation profit function depends on the relative level of Ó/. We argue
that k n
$Ä
% k
A
for different three levels of Ó/.
For 0
×
P
ln$%
l
lV
, the expression for the demand at intermediary 1,
Õ
2
×
MÓ
P
2
U*
2
$V‰
6
%$*
2
V%
N
l
lV
ln$ %RS 0.
So any deviation under this condition is not profitable.
For ln$%
l
lV
×
P
ln $%
l
lV
*
2
V
, the derivative «k n
/«Ä
has the following property:
j Üä q
6
Ü‰
6
ü
‰
6
"V
1
¨
*
,2
0
j Üä q
6
Ü‰
6
ü
‰
6
"î
.
115
Since the derivative is continuous, it means that any maximizer, Ä̂
, of k n
$Ä
% in N0,1
P
U
e
V
Q must solve
Üä q
6
Ü‰
6
0. This solution is expressed Ä̂
1
21
¨
MŽ
$Ž
1% N
×
P
ln$%
c
c,6
RS
V
and leads to profits
k n
$Ä̂
% d
Ù
2
31
f N
×
P
ln$%
l
lV
R M1
1
4
Nln$%
l
lV
RS.
Under the condition that
×
P
ln $%
l
lV
*
2
V
,
k n
$Ä̂
%
µ
O
*
2
/P
*
2
V
Q ¶
2
Ù
2
,6
56$l %V
c
c,6
D
2
Ù
2
,6
·
/P
$V/l %
,
where the last term is the profit k
A
the intermediary earns by sticking to Ä
A
1. Hence, no deviation
Ä
B N0,1
P
U
Ž
V
Q is profitable.
For ln $%
l
lV
*
2
V
×
P
l d*
2
Vf
lV*
2
Nln$%
l
lV
*
2
V
R, we have
Üä q
6
Ü‰
6
0 for all Ä
B N0,1
P
U
e
V
Q.
Thus,
k n
$Ä
% k n
O1
P
U
Ž
V
Q
/P
×
O
*
2
*
2
V
Q MÓ N
*
2
V
l
lV
ln$%RS k
A
,
for all Ä
B N0,1
P
U
Ž
V
Q. Hence, for any
×
P
l d*
2
Vf
lV*
2
Nln$%
l
lV
*
2
V
R, there is no profitable
deviation for any Ä
B N0,1
P
U
Ž
V
Q.
Now consider deviations Ä
B O1
P
U
Ž
V
, 1
P
lU
Q when Ä
A
1.
Intermediary 1’s profit is given by
k n
$Ä
%
/P
2
PV$V‰
6
%U
O
×
Q MÓ Nln O
P
$V‰
6
%U
Q
P
PV$V‰
6
%U
1 ln$%
l
lV
RS.
It can be shown that
Üä
6
Ü‰
6
0 æ
×
P
P
$V‰
6
%U
P
PV$V‰
6
%U
ln N
P
$V‰
6
%U
R
l
lV
ln$ % s i$Ä
%.
where i$Ä
% 0 is increasing on O1
P
U
Ž
V
, 1
P
lU
Q .
Suppose 0
×
P
Ž
*
2
V
l
lV
ln$ % i O1
P
U
Ž
V
Q. Then
Üä
6
Ü‰
6
0 for all Ä
B O1
P
U
Ž
V
, 1
P
lU
f . Thus,
116
k n
$Ä
% k n
O1
P
lU
Q
/P
O
l
lV
Q O1
P
×
Q
for all Ä
B O1
P
U
Ž
V
, 1
P
lU
Q. However, by choosing Ä
A
1, intermediary 1 earns k
A
/P
$V/l %
,
which exceeds k n
O1
P
lU
Q.
Suppose Ž
*
2
V
l
lV
ln$ %
×
P
l d*
2
Vf
lV*
2
Nln$%
l
lV
*
2
V
R. Then
Üä
6
Ü‰
6
0 near Ä
1
P
U
Ž
V
. In this case, k n
$Ä
% is bounded by either k n
OÄ
1
P
U
Ž
V
Q or k n
OÄ
1
P
lU
Q. We know
from above that both of these values are exceeded by the profit k
A
. Hence there is no profitable deviation
Ä
B O1
P
U
Ž
V
, 1
P
lU
Q.
Finally consider any deviation Ä
B O1
P
lU
, 1Q. This leads to profits given by
k n
$Ä
%
/P
×
O
l
lV
Q IÓ $1 Ä
%J,
which is obviously increasing in Ä
. Therefore, choosing Ä
A
1 gives intermediary 1 more profit than
any in Ä
B O1
P
lU
, 1Q.■
Abstract (if available)
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University of Southern California Dissertations and Theses
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Asset Metadata
Creator
Liu, Lin
(author)
Core Title
Essays on consumer product evaluation and online shopping intermediaries
School
Marshall School of Business
Degree
Doctor of Philosophy
Degree Program
Business Administration
Publication Date
03/11/2014
Defense Date
03/11/2014
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
competition,consumer search,game theory,Internet,OAIPMH Harvest
Format
application/pdf
(imt)
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Dukes, Anthony (
committee chair
), Yang, Sha (
committee chair
), Dutta, Shantanu (
committee member
), Selove, Matthew (
committee member
), Tan, Guofu (
committee member
)
Creator Email
linliu@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/uscthesesc3369048
Unique identifier
UC11297537
Identifier
etdLiuLin2292.pdf (filename),uscthesesc3369048 (legacy record id)
Legacy Identifier
etdLiuLin2292.pdf
Dmrecord
369048
Document Type
Dissertation
Format
application/pdf (imt)
Rights
Liu, Lin
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
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Tags
consumer search
game theory
Internet