Information Revelation in Competitive Markets
Maxim Ivanovy
McMaster University
October 2010
Abstract
This paper analyzes a market with multiple sellers and horizontally di¤erentiated
products. We investigate the sellers’incentives to reveal product relevant information
that a¤ects the buyer’s private valuation. The main …nding is that if the number of
sellers is su¢ ciently large, there is a unique symmetric equilibrium with full information
disclosure. Thus, unlike the results by Lewis and Sappington (1994) and Johnson
and Myatt (2006) for monopoly, which state that the monopolist reveals either full
information or no information, intense competition results in a single extreme with
respect to information disclosure. We show that the market is always ine¢ cient, but
the magnitude of ine¢ ciency converges to zero at a high rate as competition intensi…es.
JEL classi…cation: C72, D43, D82, D83
Keywords: Information revelation, competition, market, di¤erentiated products
1
Introduction
The main question addressed by this paper is how intensity of market competition a¤ects the
amount of information released by sellers. In general, many real life situations serve as vivid
illustrations of voluntary information disclosure in markets with multiple products, where
sellers strategically reveal information about their products to potential buyers. Consider, for
instance, movie trailers, informative commercials on TV, screen shots of computer games,
free trial versions, etc. That is, in addition to providing a good, the supplier serves as a
source of information, especially if the product is new or sophisticated.
Though information about products is controlled by suppliers, it is assessed by consumers,
who know how closely the speci…c characteristics of the product match their preferences.
In other words, an important feature of information is that it often re‡ects a valuation
of the product by the individual buyer, but does not characterize the product quality.
For example, valuations of movies or computer games vary substantially across di¤erent
I am grateful to Kalyan Chatterjee, Andrei Karavaev, René Kirkegaard, Vijay Krishna, Fernando Leiva,
Marco Ottaviani, Marek Pycia, Lixin Ye, and seminar participants at the University of Pittsburgh and Brock
University for helpful comments. I also owe special thanks to two anonymous referees and the Associate Editor
for their suggestions. Misty Ann Stone provided invaluable help with copy editing the manuscript. This work
was partially supported by SSHRC Standard Research Grant 410-2010-1776. All mistakes are my own.
y
Department of Economics, McMaster University, Kenneth Taylor Hall, 1280 Main Street West, Hamilton,
ON, Canada L8S 4M4. Email: [email protected]. Tel: (905) 525-9140x24532, fax: (905) 521-8232.
1
consumers. Similarly, many characteristics of cars, e.g., design, power, size, weight, cannot
be measured on the quality scale.1 By releasing the product relevant information, the supplier
a¤ects the consumers’private values, but he cannot precisely predict its impact on the buyers’
willingness to purchase the product. That is, revealing information creates a lottery over
di¤erent types of buyers, depending on whether the properties of the product match their
needs. In general, if the buyer privately updates her valuation after obtaining information
from the seller, then the role of information is dual: it helps the seller to segment the market
and, at the same time, it provides the informational rent to consumers. In addition to this
trade-o¤, if the market is not monopolistic, then each seller must take into account the role of
information in winning competition against rivals, which is also a¤ected by their information
disclosure decisions.
These observations raise several questions that form the central focus of our paper. How
much information is released by sellers in oligopolistic markets? How are the sellers’disclosure
and pricing policies a¤ected by the intensity of competition? Is it possible to reach full
e¢ ciency in the market, and if not, what is the source and the magnitude of the ine¢ ciency?
In this paper, we address the above questions by analyzing a price-setting model of
competition with a representative consumer and multiple sellers who o¤er distinct and
substitutable products. Sellers simultaneously compete for a buyer over two dimensions. Each
seller determines the precision of the signal about his product, which is privately observed
by the consumer, and sets the price. The consumer wants to purchase an indivisible unit
of any good. After observing signals and prices from all sellers, the consumer updates her
expectations about the products and buys the product with the highest expected net payo¤.
Because each seller competes for a single-unit demand, we use the number of sellers as a
measure of market competitiveness throughout the paper. The model is equivalent to both a
common values situation with multiple consumers, who have perfectly correlated signals and
a continuum of consumers with independent realizations of product values and conditionally
independent signals. This setup is equally applicable to situations such as job search, where
employers play a role of sellers who possess information about characteristics of their jobs
and compete for a candidate by o¤ering a wage.
Our results can be summarized as follows. First, we identify scenarios in which all sellers
in the market fully reveal information to the buyer. In particular, we demonstrate that
a su¢ ciently competitive market results in full disclosure of information by all suppliers.
Second, we show that full information revelation does not guarantee full market e¢ ciency.
However, as the number of sellers increases, the magnitude of the ine¢ ciency converges to
zero at a rate faster than exponential. Thus, even though our environment di¤ers from the
standard Bertrand model, the market becomes similar to fully revealing competition with
the unique symmetric price that tends to the marginal cost. Finally, in order to apply the
results derived in the paper to a speci…c market, we provide criteria for determining whether
competition is su¢ ciently intense to ensure full information disclosure.
Related Literature. Our framework is closely related to the paper by Lewis and
Sappington (1994), who …rst investigate the incentives of a monopolist to reveal product
relevant information. They consider the situation in which the monopolist controls the
quality of buyer’s information by choosing the probabilities with which the buyer receives
an informative signal or pure noise. Lewis and Sappington (1994) demonstrate that the
trade-o¤ between two di¤erent aspects of information disclosure— segmenting the market
1
For instance, heavy cars are generally safer, but they consume more gas.
2
and endowing the buyer with the informational rent— strictly favors either full information
disclosure or revealing nothing.2 We depart from their work in that we allow for multiple
sellers, each of whom determines the quality of information about his product. By analyzing
the role of competition among sellers, we show that this additional factor magni…es the
bene…ts of segmenting the market relative to the costs of providing the informational rent
to the buyer. As a result, only the perfectly informative outcome survives as competition
becomes su¢ ciently …erce.
Johnson and Myatt (2006) consider the problem of the monopolist with a wider class of
buyers’ information structures. In particular, they introduce the rotation order of ranking
the quality of the buyer’s information, which we also use in our paper. Johnson and Myatt
(2006) characterize the conditions that guarantee the monopolist’s preference for extreme
qualities of information. They also analyze the case of Cournot oligopoly and study the
e¤ects of increased competition on the relationship between demand dispersion (which is
given exogenously) and …rm pro…tability. Our model di¤ers from their in two respects. First,
we consider the price-setting model similar to Shaked and Sutton (1982, 1983) and Moscarini
and Ottaviani (2001), in which sellers produce distinct and substitutable products. Second,
each seller in our environment is able to strategically a¤ect the distribution of the buyer’s
valuations and, hence, the demand dispersion for his product by changing the precision of
buyer’s information. A combination of these di¤erences allows us to gain insight into strategic
interactions between sellers through varying the quality of their information. As a result, the
sellers’preferences with respect to demand dispersion in our model are in contrast with those
by Johnson and Myatt (2006). In their model, if a …rm dislikes any local increase in dispersion,
that …rm will continue to dislike increased dispersion when the number of competitors rises,
which is the opposite of the results we obtain. As a starting point, we consider the situation,
in which the monopolist dislikes any dispersion in demand. However, when the number of
sellers increases, all sellers eventually prefer the highest possible dispersion in demand.
Moscarini and Ottaviani (2001) investigate price competition in the duopoly market
with private buyer’s information. They analyze the in‡uence of a common prior and buyer’s
private information on pricing decisions of the sellers. Damiano and Li (2007) extend the
model of Moscarini and Ottaviani (2001) by allowing the sellers to control the precision of
the buyer’s private information about their products. These models di¤er from ours in two
regards. First, we do not limit the number of sellers to two. As a result, we are able to
estimate the relationship between the intensity of competition and the sellers’disclosure and
pricing policies. Second, the space of product values in the mentioned works is binary, as
is the signal space for the buyer. Together, these imply that full information disclosure is
possible even with two sellers (Damiano and Li 2007), a property that does not generally
hold with a richer signal space. Moreover, even a larger number of sellers does not necessarily
result in full revelation. We demonstrate that the particular properties of the distribution of
values a¤ect the number of sellers that rules out full disclosure.3
2
This trade-o¤ crucially depends on the willingness of the average consumer to purchase the product and
on the variance of buyers’valuations. The demand function with a high variance of buyers’values increases
the attractiveness of segmenting the market by releasing information and serving only high-value consumers.
In a recent paper, Saak (2006) demonstrates that if the monopolist can control the precision of the buyer’s
private valuations in an arbitrary way, then he prefers to let the buyer know only whether her valuation is
above or below the unit production cost.
3
Formally, the game tree of our model is as follows. First, a seller i = 1; :::; N chooses a point— a
disclosure-price pair f i ; pi g— in the space [0; 1] R. By choosing i 2 [0; 1], the seller picks a distribution
F i (si jvi ) of the buyer’s signals si conditional on the product value vi from the family of distributions
3
The issue of the endogenous quality of information has also acquired attention in the
auction design literature. Bergemann and Pesendorfer (2007) focus on the seller’s problem of
maximizing the expected revenue by specifying the price and costless information structure
for each bidder. They demonstrate that there is no full disclosure for any number of bidders,
since it is optimal to discriminate among bidders by specifying coarse and non-symmetric
information structures. As the number of bidders goes up, the optimal information structures
converge to the fully informative ones. In our model, however, full information revelation
can be reached with a …nite number of players. Board (2009b) and Ganuza and Penalva
(2010) consider settings in which the auctioneer can improve the precision of the bidders’
information about their private values and in which the quality of information is identical
across bidders. In the case of costless information, they show that it is optimal for the seller
to release all information if the number of bidders exceeds some cut-o¤ level. Though there
is a similarity between their results for auctions and ours for the market of di¤erentiated
goods in terms of information obtained by the prospective buyer(s), the incentives of the
seller(s) to release information are driven by di¤erent forces.4 Also, it is important to note
that even if the buyer obtains perfectly precise information in our model, the market will
not necessarily be fully e¢ cient. This is due to allocative ine¢ ciency stemming from the
symmetric price, which is bounded away from the unit production cost.
Finally, our work is related to, but is separated from a large strand of the literature
on vertical product di¤erentiation. In this literature, each seller knows the value of his
product, or quality, for consumers.5 This di¤erence in the availability of information about
z = fF (sjv) j 2 [0; 1]g ordered by the quality of information . Also, sellers do not observe the decisions
of rivals. Second, the value of each product vi 2 [0; 1] and the signal about this value si are randomly drawn
according to the exogenous distribution of product values G (vi ) and the conditional distribution of signals
N
F i (si jvi ), respectively. Finally, the buyer observes triples f i ; pi ; si gi=1 and buys a single product at the
o¤ered price from the set f0; :::; N g, where 0 stands for not buying anything. Thus, our setup extends the
model by Johnson and Myatt (2006) to the number of sellers N > 1. Also, we depart from Moscarini and
Ottaviani (2001) by endogenizing the precision of buyer’s information. In addition, our model di¤ers from
Moscarini and Ottaviani (2001) and Damiano and Li (2007) by considering a continuous space of product
values and signals and allowing N > 2.
4
According to Ganuza and Penalva (2010): “...increasing precision has two e¤ects on the price: it increases
the willingness to pay by the winning bidder, which increases the price, but it also increases informational
rents, which lowers the price. Eventually, when the number of bidders is su¢ ciently high, the e¤ect on
e¢ ciency overwhelms the e¤ect on informational rents and information becomes valuable to the auctioneer.”
In the market of di¤erentiated products, an increase in the precision of information released by a single
seller also increases the informational rent. Given the same price of the product, a larger informational rent
increases the chance of not selling the product if its value is below the price. In addition, a higher precision
of information results in a bigger spread of the product’s posterior valuations. This increases the probability
that the product of this seller has the highest valuation for the buyer. When the number of sellers is high,
the second e¤ect overwhelms the …rst one.
5
The …rst papers that investigated the seller’s incentives to reveal the product’s quality are due to
Grossman and Hart (1980), Grossman (1981), and Milgrom (1981). Recent papers on the quality disclosure
include, for example, Levin, Peck and Ye (2005), who consider costly information signaling with horizontally
di¤erentiated products under duopoly and monopoly. Cheong and Kim (2004) examine the e¤ect of
competition on the …rms’ incentives to disclose quality when information disclosure is costly. Dye and
Sridhar (1995) investigate the role of competition in disclosure of information about the expected pro…tability
of the company’s cash ‡ows. Other studies consider extended information structures in which sellers can
have information about products of their competitors (Board 2009a) or when each market participant
possesses some private information (Daughety and Reinganum 2007). Daughety and Reinganum (2007,
2008) investigate markets in which the product quality may be signaled via prices. Stivers (2004) examines
incentives for information disclosure in a market with vertically di¤erentiated products, when buyers may
4
buyers’tastes crucially a¤ects the sellers’motives with respect to information disclosure. In
particular, even in the simplest setup of a single seller, the seller’s incentives to let the buyer
access the information about the value of the product known to the seller (its quality) can
be opposite to those in the situation when the consumer learns her private value.6
The rest of the paper is as follows. Section 2 describes the formal model. Motivating
examples are highlighted in Section 3. Section 4 provides the general analysis and presents
the main results about sellers’pricing decisions and information disclosure. Section 5 analyses
the market e¢ ciency. Section 6 concludes the paper.
2
The model
Consider a market with a …nite number N of sellers, who compete for a single consumer (she)
through selling N di¤erentiated and indivisible products, and where each seller (he) produces
one product. The consumer makes a mutually exclusive purchase among these substitutable
goods in the sense that she either buys exactly one unit from one of the sellers or makes no
purchase. The consumer’s private values of the products fvi gN
i=1 are drawn independently
from a distribution G(v), which has a positive and di¤erentiable density g(v) on [0; 1]. All
values are net of the sellers’costs, which are identical across sellers. However, each seller has
full control over information about his product, which implies that the buyer cannot access
the information about vi prior to interaction with seller i. That is, all products are ex-ante
identical with expected values v e = E[v].
The timing of the game is as follows. First, sellers simultaneously compete for a buyer over
two dimensions. Seller i o¤ers a price pi for his product and decides how much information
to reveal to the buyer. In particular, seller i chooses the quality i of the signal s i about the
product characteristics, but not the signal itself.7 For example, if the signal s i = vi + "i is
a sum of the true value vi and the noisy component "i , then the quality of the signal can be
represented by i = 1= "2i , where "2i is the noise variance. As a matter of notation, we use
si instead of s i .
Second, after observing all triples f i ; si ; pi gN
i=1 the buyer purchases the product j that
gives her the highest non-negative expected payo¤ ! sj
pj , where !j = ! sj = E vjsj
is the product’s posterior valuation. If ! sj
pj < 0, the buyer does not make a purchase.
Information structure. For simplicity, suppose that the signal quality i 2 [0; 1] ; 8i,
where i = 0 and i = 1 imply the perfectly uninformative and perfectly informative signals,
respectively. By choosing the signal quality, a seller i a¤ects the distribution function of
be unaware of the existence of information.
6
For example, consider a random variable v distributed uniformly over the unit interval. If v represents
the product quality for consumers, then the seller would disclose it almost surely. Otherwise, he cannot
charge more than the expected value E [v] = 1=2. However, if v > E [v], then the seller would reveal this
information. Thus, the seller could potentially hide information only if v
E [v], which, in turn, would
decrease the consumer’s expected value. Repeating this argument iteratively results in the full information
disclosure for any positive value. In contrast, if v is the buyer’s private value, then revealing information and
setting the optimal monopoly price pM = 1=2 brings the seller expected payo¤ 1=2 1=2 = 1=4, since the
buyer would buy the product only if v > pM . In contrast, the seller can extract the total surplus 1=2 by not
revealing information and setting the price at E [v].
7
For example, by changing the length of a movie trailer, the movie production company can a¤ect the
quality of information available to a potential viewer, or i . It is not, however, able to see the viewer’s
impression of the trailer, or s i .
5
Figure 1: Concentration of the density g (!)
Rotation of the distribution G (!)
posterior values of his product,
G i (!) =
Z
fsi :!(si )
dF i (si ) ;
!g
where F i (:) is the marginal distribution of the signal si .
We assume that G (!) is continuously di¤erentiable in for > 0, and has a positive
and di¤erentiable density g (!) on the support [! ; ! ]. To rank the family of distributions
G i (!i ) with respect to the quality of signals, notice …rst that by partially revealing or
distorting information, a seller cannot shift the buyer’s taste toward his product, since the
average valuation of a product is the same:
E [!i ] = Esi [E [vjsi ]] = E [v] = v e ; 8 i .
(1)
However, by changing the signal quality i , the seller a¤ects the spread of the buyer’s
posterior valuations. In order to capture the e¤ect of a change in the spread of G i (!) due
to varying the signal’s quality, we apply the rotation order introduced recently by Johnson
and Myatt (2006) for the monopolistic setup.8
De…nition 1 The family of distributions G (!) is rotation-ordered if, for each , there
exists a rotation point ! 0 , such that ! ? ! 0 () @G@ (!) 7 0; ! 2 (! ; ! ).
The main idea behind the rotation ordering is that a less informative signal has a smaller
in‡uence on updating the buyer’s prior information and, hence, stochastically shifts her
posterior valuations toward her prior expected value. That is, g (!) is concentrated around
v e , and the slope of the distribution G (!) locally rotates around some point within ! ; !
8
Johnson and Myatt (2006) provide two examples of information structures that can be ranked according
to this order. The …rst example is the “truth-or-noise” technology that returns a signal that is equal to v
with a probability or a random indistinguishable draw z from G (v) with a probability 1
. In the other
example, the value v is drawn from the normal distribution N ; 2 . However, the consumer observes the
conditionally unbiased signal s from the distribution N v;
6
1
2
.
(see Fig. 1). It is straightforward to show that rotated distributions with the same average
are also ordered by the second-order stochastic dominance.9
Because = 0 implies that G (!) is fully concentrated at ! = v e , it follows that !00 = v e .
We assume that the distribution of posterior values is not signi…cantly a¤ected by small
changes in the quality of information. That is, ! and ! are continuous in , and ! 0 and
! are di¤erentiable in some neighborhood of = 0.
3
Examples
We start by considering a few motivating examples that demonstrate the role of competition
in the sellers’decisions regarding information revelation and highlight the general intuition
behind the main results below. By examining situations with one, two, and three sellers,
we demonstrate that the sellers’ incentives to reveal information increase monotonically
as competition intensi…es, and change from one extreme to another in terms of information
disclosure. In particular, the examples will show that the monopolist never reveals any useful
information. In the case of duopoly, the sellers reveal information partially. Finally, the
market with three sellers results in full information revelation.
For simplicity, we assume that the buyer’s valuations are distributed uniformly on [0; 1].
Also, the quality of signals is extreme, i.e., the buyer either learns a product’s value precisely
or gets a fully uninformative signal. First, consider the case of the monopolist. If the seller
discloses information, then in any incentive-compatible mechanism, the seller cannot obtain
G(v)
< v (Myerson 1981). Hence, the seller’s
more than the virtual value of a good v 1 g(v)
expected pro…t is strictly below the expected value of the good v e . In contrast, the seller can
obtain v e by not revealing information and setting the price pM = v e .
Introducing a second seller in the market dramatically a¤ects the sellers’decisions about
information disclosure. To see this, suppose …rst that both sellers reveal information about
their products. p
Given these disclosure policies, it is optimal for each seller to charge the
same price p = 2 1 ' 0:414.10 This results in the sellers’expected pro…ts ' 0:172. In
contrast, if one of the sellers does not disclose information about his product and charges
the same price, his expected pro…t increases to 0:207. Thus, full information revelation
is not sustainable in equilibrium. There is also no equilibrium, in which both sellers hide
information. By contradiction, if both sellers do not disclose information, then their products
are ex-ante identical to the buyer. That is, the market transforms into the classic Bertrand
competition with the unique price at zero, which results in the zero expected pro…ts for both
sellers. However, each seller can guarantee a positive expected pro…t by revealing information
and charging a small positive price p" . In this case, the buyer will prefer the product of this
seller if v p" > v e , which occurs with a positive probability. As a result, only mixed-strategy
equilibria can exist in the two-seller case.11
Finally, consider a market with three sellers. In this case, there exists an equilibrium
such that all sellers reveal information, charge price p ' 0:322, and receive expected pro…t
9
For example, this follows from Proposition 6.D.2 (Mas-Colell et al. 1995).
Actually, it is the unique symmetric optimal price under full disclosure policies.
11
We provide an implicit characterization of a symmetric mixed-strategy equilibrium with two sellers and
outline its properties in Supplementary Online Appendix A. In particular, we show that any equilibrium
involves mixing with respect to the disclosure/non-disclosure policy. The price of the disclosing seller is
deterministic, whereas the non-disclosing seller mixes over prices according to some continuous distribution
on a single interval. Also, all prices are bounded from 0.
10
7
Figure 2: Sharing the market for di¤erent disclosure policies of seller 1
' 0:104. In contrast to the duopoly, hiding information by any seller and adjusting the
price optimally results in a lower expected pro…t of 0:082.
To explain the intuition behind these examples, we start with a simple case of the
duopolistic market, in which both sellers reveal information. Information disclosure increases
the dispersion of buyer’s posterior valuations across products. Hence, the fully revealing
market is characterized by substantial product di¤erentiation, which relaxes competition. A
combination of product di¤erentiation and a small number of competitors results in a high
probability of selling the product by a particular seller, which, in turn, allows sellers to set
prices far above the marginal cost.12 However, to sell the product, its valuation must be also
above the price. Since market power allows sellers to set relatively high prices, the sellers’
bene…ts of revealing information are damaged by the fact that the buyer is fully informed
about both products. Therefore, she does not make a purchase if the product’s value is below
its price. We refer to this as the informational rent e¤ect. For example, for the seller 1 this
e¤ect is graphically represented by area C in the left part of Fig. 2.
However, given that the competitor reveals information, one of the sellers, say, seller
1 may increase the overall probability of selling the product by not revealing information
and leaving the buyer with the expected valuation regarding his product. Then, the new
probability of selling this product is determined by the probability mass in area A0 in the
right part of Fig. 2. The reason is that having a small number of competitors implies that,
with a su¢ ciently high probability, the buyer would still prefer the product of this seller
if the net value of the competing product is below the net expected value. At the same
time, hiding information eliminates the risk of not selling the product if its true value is
below the price. In other words, it removes the informational rent e¤ect while preserving
product di¤erentiation. Together, these two factors relax the sellers’ incentives to disclose
12
v1
For, say, seller 1, this probability is determined by the probability mass in the area above line v2 =
p1 + p2 in the left part of Fig. 2.
8
information, which breaks full information revelation in the market.
If competition becomes …ercer, the non-revealing policy signi…cantly reduces the chance
of selling the product. To see this, notice …rst that the market with multiple products is
equivalent to a duopolistic market, which consists of the product of this seller and the most
valuable product o¤ered by competitors. In this environment, the expected value of seller
1’s product is much lower than that of the best competitor’s. Thus, the probability of sale is
low when not disclosed, since the highest value among rivals is almost surely larger than the
ex ante mean. Therefore, the concern of no sale outweighs the concern of giving too much
information rent to the buyer, which results in full information disclosure in equilibrium.13
The above intuition can be reinterpreted for the case of a continuum of buyers. A fully
revealing market is characterized by high segmentation since each seller serves only the
“loyal”consumers who prefer his product to those of competitors. If the number of sellers is
small, then each seller controls a large segment of the market. Moreover, this segment does not
change signi…cantly if a single seller decides to hide information about his product and leave
consumers with their expectations about its value. The reason for this is that a large share
of consumers still would prefer the product with the expected value to other products with
known valuations. In other words, the trade-o¤ between the bene…ts of segmenting the market
and the losses due to providing the informational rent to the buyer is basically una¤ected by
the presence of competitors. Thus, each seller has the incentive to hide information without
losing his market segment.
Intense competition essentially in‡uences this trade-o¤ via the market segmentation. In
particular, hiding information results in a loss of the most attractive share of buyers, who
would choose this product over the others if they were better informed about it. In addition,
as a variety of products with known values increases, the share of consumers who prefer the
product with unknown characteristics (that is, they expect that the product value is v e )
shrinks.14 That is, hiding information eliminates a large piece of the seller’s segment of the
market in favor of the rivals. Thus, lack of information moves the product into a “low value”
niche of the market relative to the best product among competing ones.
4
Analysis
We approach our primary goal of showing the existence of the fully revealing equilibrium
in two steps. Since the candidate equilibrium disclosure-price pairs involve full revelation,
the …rst step is to …nd the sellers’ pricing decisions for the given disclosure policies. This
issue is solved by demonstrating that it is optimal for each seller to set a symmetric price
pSN , hereafter, a market price, and characterize its asymptotic properties as a function of
13
Technically, even though not revealing information expands the set of values, which results in selling seller
1’s product (area A0 is bigger than A), the total mass of values is mainly concentrated in the complement
area B 0 . That is, hiding information results in a loss of the possibility of serving the high-value consumer,
who would prefer seller 1’s product to all others (area A \ B 0 ). In contrast, revealing information and,
thus, increasing the variance of ex-post values of the product is the only possibility of attracting the buyer
with a high realization of the product’s value and sell the product. In this situation, the role of the buyers’
informational rent is minor.
14
For example, in the case of two sellers, if one seller hides the information and generates the buyers’
valuations at v e , the share of consumers, who prefer this product to the competing one, is determined by
area A0 (see Fig. 2), which is equal to v e . Similarly, in the case of N sellers, the share of such consumers is
given by the N –dimensional box V = fvi ; i = 1; :::; N jvj 2 [0; 1] and vi 2 [0; v e ] ; i 6= jg. As N increases, the
N 1
volume of this box (v e )
decreases.
9
the number of sellers. Second, we prove that for the identical disclosure-price pairs 1; pSN
of rivals, no single seller can pro…tably deviate by choosing a di¤erent pair f ; pg if N is
su¢ ciently large.
4.1
Pricing with full information disclosure
In considering the strategic interaction among the sellers in the example above, one can
see that the informational component of the sellers’decisions cannot be separated from the
prices. The reason is that prices play a dual role. First, they in‡uence market segmentation
by changing the net value of each product relative to those o¤ered by competitors. Second,
they determine the magnitude of the informational rent e¤ect.15 Thus, we start the general
analysis by investigating the seller’s decisions over this dimension contingent on the fact that
all sellers fully disclose information. Given the regularity condition, we show the existence of
the unique symmetric price for an arbitrary number of sellers and investigate its magnitude
as the number of sellers increases.
First, consider the case of the monopolist, who sells the product to the buyer with
a valuation v
G [0; 1]. Let pM be the monopoly price, i.e., the solution to the
pro…t-maximization problem
max
x
(x) = max (1
x
G (x)) x
(2)
The …rst-order condition for this problem is
1
where (v) =
satisfy
g(v)
1 G(v)
G (x)
xg (x) = (1
G (x)) (1
x (x)) = 0,
is the hazard rate function of the distribution G (v). Thus, pM must
pM (pM ) = 1.
(3)
Throughout the paper, we assume the following regularity condition.
Condition 1 The density function g (v) is log-concave.
This condition implies that G (v) possesses two useful properties. First, G (v) has the
increasing hazard rate (IHR) or, equivalently, the log-concave survival function 1 G (v).16
As a result, the monopoly pro…t (x) is quasi-concave in x, so pM is uniquely determined
by (3). Moreover, we show below that this property implies that the symmetric price in the
competitive market is below pM . Second, G (v) is log-concave (Bagnoli and Bergstrom 2005).
If the number of sellers exceeds one, the IHR property may be insu¢ cient to prove
the existence of a symmetric price pSN . In order to resolve this issue, we employ Condition
1. Denote (x; ; p) the expected pro…t of a single seller who sets price x and quality
of information , given that the other sellers set price p and reveal full information. For
simplicity, we use (x; p) instead of (x; ; p) if = 1. Then, (x; p) is determined by
(x; p) = P (x; p) x,
15
Graphically, a change in prices shifts the “market segmenting”line v2 = v1 p1 +p2 in Fig. 2. In addition,
this change a¤ects the areas of vi pi ; i = 1; 2;e.g., area C, which re‡ects the informational rent e¤ect.
16
Though Condition 1 is more restrictive than the IHR property, it holds for a large class of standard
distributions in the subspace of parameters that imply the IHR property. These are, for instance, beta,
gamma, Weibull, and power distributions. For other examples, see Bagnoli and Bergstrom (2005).
10
where P (x; p) is the probability of selling the product. It is determined by
P (x; p) = GY (p) (1
G (x)) +
Z1
(1
G (y
p + x)) dGY (y) ,
(4)
p
where GY (y) = GN 1 (y) is the distribution of the maximal value y = max fv2 ; :::; vN g across
the competing products j = 2; :::; N .
The …rst component in (4) re‡ects the fact that each seller may become the monopolist
if the values of all other products are below p, which occurs with probability GY (p). With
probability 1 GY (p), the seller has to compete against the most valuable product o¤ered
by competitors. Hence, he sells the product only if v x y p, or v y p + x. Taking
the average over possible values of y gives the expected pro…t.
Notice that the seller’s expected pro…t is a convex combination of the quasi-concave
functions, which is generally not quasi-concave. However, Condition 1 resolves this issue.
Denote a price vector fpi gN
i=1 a vector of market prices, such that each pi a best-response
price to the prices of the other sellers, given the …xed disclosure policies of all sellers. Then,
Condition 1 implies that there exists a vector of market prices (Caplin and Nalebu¤ 1991).
Moreover, the lemma below shows the existence of the unique symmetric market price and
outlines its boundary properties for the full disclosure policies. All proofs can be found in
the Appendix.
Lemma 1 If all sellers fully disclose information, then there exists a unique symmetric
market price pi = pSN ; i = 1; :::; N , which is (1) less than the monopoly price, (2) strictly
positive, and (3) converges to 0 at the rate 1=N .
The proof is based on the aggregation theorem by Prékopa (1973), which is also used by
Caplin and Nalebu¤ (1991). We show that the pro…t function (x; p) has a unique maximizer,
because it is strictly quasi-concave in x. In order to characterize the asymptotic properties
of pSN , we demonstrate that it is bounded by sequences that have rate of convergence 1=N
unconditionally on the shape of G (v).
An important implication of the above lemma is that …ercer competition reduces the
magnitude of the informational rent e¤ect due to two factors. First, a lower price directly
reduces the chance of not selling a product with a relatively low value. Second, before making
a decision about a purchase, the buyer always prefers the product that gives her the highest
net value, and buys this product if this value is non-negative. In the case of a symmetric price,
this is equivalent to preferring the product with the highest value. However, the distribution
function of the best product shifts toward higher values as the number of sellers goes up,
which additionally decreases the probability of not selling the product to the informed buyer.
This is reminiscent of the “winner curse” e¤ect in the auction theory. Since a seller of a
particular product cannot observe the buyer’s signal, then the fact that his product wins
against competing ones raises the product’s expected value.17
17
Graphically, this means that for a given area C in Fig. 2, the probability mass in the area decreases as
the number of sellers goes up.
11
4.2
Information disclosure
This section investigates the sellers’ incentives to reveal information and states the main
results about information disclosure. We start with a condition about the family of
distributions G (!).
Condition 2 1
(!) ! is strictly pseudo-monotone in ! for all
> 0.18
The strict pseudo-monotonicity of 1
(!) ! is a technical assumption and is weaker
19
than the IHR property. It is a su¢ cient condition for the uniqueness of the monopoly price
pM if the buyer’s valuations are distributed according to G (!) (Ivanov 2009). Another
useful implication of this condition is related to the competitive market with full disclosure.
It guarantees that there are no bene…ts of charging the price above the monopoly one, if a
single seller deviates to another disclosure policy. That is, suppose initially that all sellers
fully reveal information, so that there is a unique symmetric market price pSN according to
Lemma 1. Now, if some seller chooses the disclosure policy < 1 and the optimal price p
for this policy, then it follows that p < pM .
The examples above demonstrate that if the number of sellers in the market is small, then
full information disclosure is not feasible. This is because in a market with few suppliers the
magnitude of the informational rent e¤ect is unambiguously bigger. First, the fully revealing
market is characterized by a high market price, which decreases the chance of selling a
product. Second, each seller cannot sell his product if one of the following events occurs:
the product’s value is below the price or the buyer’s payo¤ is lower than that from another
product. Thus, if one seller, for instance, hides information completely, then the buyer will
value his product at the mean. This eliminates the risk of the …rst event without raising
signi…cantly the risk of the second event. As a result, the probability of selling the product
increases.
However, if there are many competitors in the market, then the above logic does not work.
Hiding information generates the product value v e . The chance of selling such a product is
small, since the maximal valuation across other products is likely to be higher.20 That is,
the probability of the second event increases enormously. Moreover, competition pushes the
prices down, which leaves even less freedom for a potential deviator to attract the consumer
by o¤ering her a lower price. In other words, it is better to reveal the information hoping
that the value will be high. This intuition is shaped formally in the following theorem, which
is the main result of the paper.
Theorem 1 There is N0 such that for all N
N0 , there exists a unique symmetric
equilibrium, in which all sellers fully disclose information.
A decrease in the sellers’incentives to hide information in response to …ercer competition
can be explained from a di¤erent angle. Consider the monopoly that sells the product to a
18
A function ' ( ) is strictly pseudo-monotone if for every
and 0 6= ; ' ( ) ( 0
)
0 implies
0
'( )(
) < 0 (Hadjisavvas et al. 2005). Equivalently, ' ( ) 0 implies ' ( 0 ) < 0 for all 0 > .
19
A typical example of the pair of distributions fG (v) ; F (sjv)g that generates G (!), which satis…es
this condition is the “truth-or-noise” speci…cation (Lewis and Sappington 1994; Johnson and Myatt 2006).
Under this speci…cation, the signal s is equal to the true value v with probability . With probability 1
,
s is an indistinguishable independent draw from G(v). If G (v) satis…es Condition 1, Condition 2 holds as
well.
20
Instead of hiding information completely, revealing information partially forms a distribution of posterior
values G (!) that is concentrated around v e . Thus, the argument is still true.
0
12
buyer, who has an outside option. If the value of this option is small, then the monopolist
has no incentive to disclose information since it just endows the buyer with the informational
rent. However, when the outside option becomes more attractive (in particular, if it exceeds
the mean value), the monopolist will reveal information hoping that the buyer will value
his product higher than the outside option. Now, if we consider a particular seller in the
competitive market, then the maximal value across the products of competitors is the buyer’s
outside option. In other words, all sellers play a role of the buyer’s outside option for each
other. Thus, as the number of sellers goes up, the value of an outside option (stochastically)
grows, thereby, forcing the sellers to reveal full information.
Technically, by decreasing the quality of information about his product, the seller rotates
the distribution of expected valuations so that the density becomes concentrated around the
mean. This decreases the degree of di¤erentiation among products. In particular, it reduces
the chance of selling the object by making the tail of the distribution G (!) thinner. Since
the product has to compete against a product with a value Y , which is likely to be high,
the chance of selling the product falls. Notice that this intuition holds true, even when all
other competitors disclose information only partially. In this case, it is still pro…table for any
seller to reveal information completely, which guarantees the uniqueness of the symmetric
equilibrium.
Conditions for full disclosure. The intuition above helps characterize the necessary
conditions for full information disclosure. These conditions then can be used to verify if
competition is su¢ ciently strong to motivate the sellers to disclose all available information.
Also, these conditions allow us insight into the main properties of the distributions of
consumer’s values, which determine the trade-o¤ between the bene…ts of segmenting the
market and providing the informational rent to the buyer.
Lemma 2 Suppose pSN is the symmetric market price for the full disclosure policies. If either
S
1 GN (pS
N ) pN
1
S
e
N 1
S
,
or
(b)
p
v
and
G
p
, then there
(a) pSN
v e and GN 1 (v e )
N
N
N
N
ve
is no fully revealing symmetric equilibrium in the market with N sellers.
The above lemma illustrates the relationship between the properties of the distribution of
values and the magnitude of the informational rent e¤ect. For a given distribution, the …rst
pair of the conditions in Lemma 2 is violated as the number of sellers increases. Note …rst
that according to Lemma 1, pSN converges to 0 at the rate 1=N and, hence, eventually falls
below v e . Second, the left-hand side of the inequality declines at an exponential rate, whereas
the right-hand side has the rate of convergence 1=N . Formally, GN 1 (v e ) re‡ects the seller’s
bene…ts from extracting the informational rent from the buyer. This is the probability of
selling the product in the case of not disclosing any information and setting the same price
pSN v e . The right-hand side of the inequality is an upper bound on the seller’s probability
of selling the product in the case of full disclosure.21 Therefore, …erce competition reduces
the sellers’incentives to hide information.
Given a …xed number of sellers, the conditions in the lemma are likely to hold if the
density is concentrated around low values or, in other words, if the buyer’s informational
rent is su¢ ciently large. In this case, not revealing any information generates a deterministic
valuation v e , whereas full information revelation induces a random value v that is likely to
21
If all sellers fully disclose information and set the symmetric market price pSN , the probability of selling
1 GN (pS
N)
for a single seller is equal to
. This converges to N1 as N increases and, therefore, pSN falls to 0.
N
13
be below v e . This decreases the chance of selling the product. At the same time, there are
no incentives to reveal information anticipating that the value of the product will exceed the
best competing product since the values of competing products are also likely to be low. In
contrast, if the density is skewed to the right, then the magnitude of these factors becomes
smaller and it is possible to reach full disclosure with two sellers only.22
In this light, it is natural to ask how the cut-o¤ number of sellers N0 that sustains
full information disclosure depends on the properties of the distribution of values. In other
words, is there an upper bound on this cut-o¤ across all distributions? As shown by Board
(2009b) in an auction context, the answer to this question is negative. This is because for
any …nite number of bidders there is a distribution with a su¢ ciently large probability mass
at low values and, thus, su¢ ciently large informational rents that revealing information is
not pro…table. If there are no restrictions on G (v), a similar argument can be applied to
the market of di¤erentiated products. The Supplementary Online Appendix B to this paper
demonstrates that for the family of power distributions G (v) = v and an arbitrary N ,
there is a su¢ ciently small such the second pair of the conditions in Lemma 2 is satis…ed.
However, allocating a large probability mass near zero implies that the density must decline
rapidly at some point. This violates the IHR property and, thus, Condition 1.23 Due to the
restriction on G (v), the existence of the upper bound on N0 across all distributions with
log-concave densities remains an open question. The interpretation of the lemma above may
be seen more clearly in the duopoly market.
Corollary 1 If v e is no less than the median value and the monopoly price, then there is no
fully revealing symmetric equilibrium in the market with two sellers.
In the duopoly market, the inequality G (v e ) 1=2 can be rewritten as v e v m , where
v m = G 1 (1=2) is the median value. Because the market price is always below the monopoly
price by Lemma 1, it follows that the …rst pair of the conditions in Lemma 2 holds. The
implication of this corollary is that there is no fully revealing equilibrium for all symmetric
distributions in the duopoly market. For this class of distributions, it can be shown that
Condition 1 guarantees that the monopoly price is below the expected value, which is equal
to the median value.
Lemma 3 If g (v) is symmetric, then there is no fully revealing equilibrium in the market
with two sellers.
Intuitively, for any distribution, the probability mass below the mean value can serve as
an approximate measure of the informational rent e¤ect. The probability mass above the
mean measures the attractiveness of the competing product in the case of hiding information.
This is because by not disclosing information, the seller guarantees that the valuation of his
product is equal to the mean value. Given the fact that the competitor reveals information,
the probability mass above the mean determines the probability of the competing product
having a higher value than that of seller 1. Following this logic, the above result demonstrates
that if weights of these two factors are the same, the informational rent e¤ect on the sellers’
incentives dominates that of competition with the other seller. In other words, by hiding
information, the probability of selling the product to an uninformed consumer dominates
the probability of losing her to the seller with a better product.
v)
Consider g (v) = 1 exp(
= 2, the conditions in Lemma 2 do not hold if N > 1.
exp(
) ; v 2 [0; 1]. If
However, if = 8, then the density is concentrated near 0, and there is no full disclosure with even 3 sellers.
23
In fact, the density of the power distribution is not log-concave for < 1.
22
14
5
Market e¢ ciency
In this section, we relate the quality of information that is endogenously determined by the
market to market e¢ ciency. Market e¢ ciency is measured by the relative di¤erence between
the total surplus, i.e., a sum of the consumer surplus and the sellers’expected pro…ts, and
social welfare measured by the expected value of the most valuable product. Full e¢ ciency
can be reached if we introduce a central planner who forces all sellers to reveal full information
and set the price at the unit cost level.
In the case of the a monopoly, equilibrium is socially optimal. Even though the expected
payo¤ of the buyer is zero, the monopolist’s pro…t is equal to the expected value of the
good, which coincides with social welfare. If the number of sellers exceeds one, the market
is not e¢ cient. If competition is not strong, there exists both informational and allocative
ine¢ ciencies. First, since sellers do not reveal information precisely, there is an e¢ ciency loss
because the buyer may not buy the most valuable product. Second, product di¤erentiation
implies that pricing at unit cost is not an equilibrium because unconditionally on competitors’
decisions, each seller can guarantee positive pro…t by full disclosure and setting a small
positive price. As a result, there is a chance that the buyer will not purchase a product if its
value is below the price. However, if the market is competitive enough, then sellers disclose all
information and the informational component of ine¢ ciency disappears. At the same time,
product di¤erentiation still allows the sellers to set the positive prices that sustain allocative
ine¢ ciency. It is natural to ask how fast the market converges to the social optimum as the
number of sellers grows. The following lemma addresses this issue. We show that the welfare
losses due to the decision of an unsatis…ed buyer to leave the market decrease faster than
exponentially.
Lemma 4 As the number of sellers goes up, the ine¢ ciency of the market converges to zero
at the rate 1=N N .
Thus, the magnitude of ine¢ ciency goes down at a very fast rate as the number of the
sellers grows. This is because of two factors. First, the higher number of sellers implies the
highest value across products stochastically grows. Second, competition decreases the market
price, which tends to the marginal cost at the rate 1=N . An implication of the above result is
that more severe competition results in the convergence of the market to standard Bertrand
competition not only in terms of the symmetry of the price and the perfect quality of buyer’s
information, but also in terms of e¢ ciency, even though the model’s settings are essentially
di¤erent from the classic setup.
6
Conclusion
In this paper, we provide a possible explanation of di¤erent behavior of …rms with
respect to revealing information that can a¤ect buyers’private valuations of products. We
demonstrate that the …rms’incentives to provide such information crucially depend on the
…erceness of competition in the market. Starting with the analysis of the non-disclosure
case of a monopoly, we show that …rms never reveal full information in su¢ ciently
uncompetitive markets, and eventually disclose full information as the market becomes
su¢ ciently competitive. Thus, full information revelation is an endogenously determined
attribute of su¢ ciently competitive markets only. This result demonstrates that competition
15
re…nes two possible extreme choices of the monopolist, which prefers to reveal either full
information or no information (Lewis and Sappington 1994; Johnson and Myatt 2006).
Second, as the number of sellers increases, the market structure becomes similar to Bertrand
competition. That is, even though all products are di¤erentiated, sellers charge the symmetric
price that converges to the marginal cost as the number of sellers goes up. Since there are
no informational losses, the price is the only source of ine¢ ciency, the magnitude of which,
however, virtually disappears as competition intensi…es.
In this light, an interesting question for further exploration is the robustness of the number
of sellers, sustaining full information revelation, to modi…cations of the model’s settings. In
particular, given that all sellers prefer to fully disclosure information, then will it be pro…table
for them to reveal information if the distribution of valuations becomes more disperse or the
production costs increase? In general, as shown by Johnson and Myatt (2006), there is no a
single answer to this question even in the case of a monopolist, and the answer is positive
only if additional assumptions on the family of ordered distributions of posterior valuations
and the shape of cost function are imposed. In the competitive market, the situation is even
more complicated because of the pricing decisions of competitors. Under full information
disclosure, all sellers are likely to set a higher market price as a result of a larger variance
of valuations or a higher production cost. This increases the buyer’s informational rent and
creates incentives for a single seller to capitalize on this rent by not revealing information.
Though the magnitude of this e¤ect is insigni…cant if the market is su¢ ciently competitive,
it is ambiguous if the number of sellers is relatively small.
Appendix
In this section, we provide the proofs of Lemmas 1–4, Theorem 1, and Corollary 1. First, we outline
the properties of sellers’ pricing decisions given that all sellers fully disclose information. In this
case, the FOC with respect to the price for the problem of a single seller is
0
x (x; p)
(x; p) =
= P (x; p) + xP 0 (x; p) = 0.
From (4), the FOC function is:
(x; p) = GY (p) (1
G (x)
xg (x)) +
Z1
1
G (y
p + x)
xg (y
p + x) dGY (y)
p
=
Z1
1
G (max fx; y
p + xg)
xg (max fx; y
p + xg) dGY (y)
(5)
0
=
Z1
(1
G (max fx; y
p + xg)) (1
x (max fx; y
p + xg)) dGY (y) :
0
In symmetric equilibria, x = p, which gives the necessary condition for p:
(p) =
(p; p) = GY (p) (1
G (p)
pg (p)) +
Z1
(1
G (y)
pg (y)) dGY (y) = 0.
p
Claim 2
(x; p) has the following properties:
(0; p) > 0; 8p, and
16
(x; p) < 0; x > pM ; 8p.
(6)
Proof First, we have
(0; p) = GY (p) +
Z1
(1
G (y
p)) dGY (y) = GY (p) + 1
GY (p)
p
Z1
G (y
Z1
p) dGY (y) = 1
p
1
(x; p) =
(1
p) dGY (y) > 0:
p
Second, Condition 1 implies
pM (pM ) = 0; x > pM and
Z1
G (y
(x) is increasing. This leads to 1
G (max fx; y
p + xg)) (1
x (max fx; y
x (max fx; y
p + xg) <
p + xg)) dGY (y) < 0; x > pM :
0
Claim 3 There exists a unique pSN
Proof Because
(0) =
Z1
(p) is continuous,
(1
pSN = 0.
pM , such that
N 1
G (y)) dG
(pM ) < 0 by Claim 2, and
(y) =
0
Z1
GN
1
(y) dG (y) =
GN (y) j10
1
=
> 0,
N
N
0
it follows that
pSN = 0 for some pSN pM .
We prove that pSN is unique by replacing …rst GY (x) with an arbitrary distribution F (x), which
has a positive and continuous density f (x) on [0; 1]. Note that
g 0 (x)
g(x)
x + (x) x
0. Rewrite
(p) = F (p) (1
G (p)
0 (x)
=
g 0 (x)(1 G(x))+g(x)2
(1 G(x))2
0 gives
(p) as
pg (p)) +
Z1
pg (p)) +
Z1
(1
G (y)
yg (y) + yg (y)
pg (y)) dF (y)
(7)
p
= F (p) (1
G (p)
(1
G (y)
yg (y)) dF (y) +
p
Z1
(y
p) g (y) dF (y) .
p
Then,
0
(p) = f (p) (1
Z1
G (p)
pg (p))
g (y) dF (y) =
F (p) g (p) + pg 0 (p) + g (p)
F (p) g (p)
f (p) (1
Z1
g 0 (p)
+1
F (p) g (p) p
g (p)
p
p
Since pSN
pM , we have pSN
pSN
g 0 pSN S
p +1
g pSN N
This means
0
pSN < 0, so
pM (pM ) = 1 and
g 0 pSN S
p +
g pSN N
(p) intersects p
pSN pSN
0 only from above.
17
0.
G (p)
g (y) dF (y) .
yg (p))
Proof of Lemma 1. By Claim 3, there is a unique symmetric price pSN , which satis…es the
necessary condition (6). To prove that pSN is the symmetric optimal price under full disclosure
policies of all sellers, it is su¢ cient to show that P (x; p) is log-concave for x > 0. In this case, the
FOC function
1 P 0 (x; p)
+
(x; p) = P (x; p) x
x
P (x; p)
intersects x 0 once.
To prove the log-concavity of P (x; p), we employ the theorem by Prékopa (1973) about the
preservation of log-concavity by integration. In particular, let f (x; y) be a function of n+m variables,
where x is an n component and y is anR m component vectors. If f is log-concave in Rn+m and A
is a convex subset of Rm , then h (x) = A f (x; y) dy is log-concave in Rn .
From (4), P (x; p) can be written as
P (x; p) =
Z1
1
G (max fx; y
p + xg) dGY (y) =
0
Z1
min f1
G (x) ; 1
G (y
p + x)g gY (y) dy,
0
where gY (y) = G0Y (y) = (N 1) GN 2 (y) g (y). Note that the log-concavity of g(y) means the
log-concavity of G(y) (Bagnoli and Bergstrom 2005). Thus, gY (y) is log-concave as a product
of log-concave functions. In addition, the IHR property of G (x) implies that 1 G (x) and 1
G (y p + x) are log-concave in (x; y). Finally, the minimum of log-concave functions is log-concave.
This implies that f (x; y) = min f1 G (x) ; 1 G (y p + x)g gY (y) is log-concave in (x; y) and
so is P (x; p) as a function of x.
To characterize the asymptotic properties of pSN , rewrite (p) as
(p) = L1 (p) + L2 (p) + L3 (p) ,
where
L1 (p) = GN
L2 (p) =
Z1
1
1
(p) (1
G (p)
G (y) dGN
1
pg (p)) ;
(y) = GN (p)
GN
1
(p) +
GN (p)
, and
N
1
p
L3 (p) =
p
Z1
N 1
g (y) dG
(y) =
p (N
1)
p
Z1
g (y)2 GN
2
(y) dy:
p
Combining all components of
(p) leads to
(p) =
1
GN (p)
N
p (p) ,
where
(p) = g (p) G
N 1
(p) + (N
1)
Z1
g 2 (y) GN
2
(y) dy =
p
= g (p) GN
1
(p) +
Z1
g (y) dGN
p
1
(y) =
Z1
0
18
g (max fy; pg) dGY (y) :
Then, (p)
!2[0;1]
pSN =
1
By the same argument, (p)
1
pSN = 0 imply
g = min g (!) and
GN pSN
N
GN pSN
1
<
.
Ng
Ng
1
1
pSN
GN pSN >
g = max g (!). Also, for any b 2 (0; 1), we have 1
!2[0;1]
b for a su¢ ciently large N . This leads to pSN >
1 b
Ng .
To proceed to other results, we prove the following claim …rst.
Claim 4 Consider a continuous function ' (y; t) ; y 2 [0; 1] ; t 2 [t; t] and a distribution function
G (y), such that: 1) there exists an interval [a; b] [0; 1], such that, for any t 2 [t; t] ; ' (y; t) > 0,
if y 2 (a; b); and ' (y; t) 0, if y 2 [b; 1]; and 2) G (y) has a continuous density, which is positive
R1
on [a; b]. Then there exists N , such that ' (y; t) dGN (y) > 0 for all t and N N .
0
Proof Because q (y; t) = ' (y; t) g (y) is continuous, there exists q1 =
an interval [c1 ; c2 ], such that a < c1 < c2 < b and let q2 =
min
y2[c1 ;c2 ];t2[t;t]
max
y2[0;a];t2[t;t]
jq (y; t)j. Take
q (y; t). Clearly, q2 > 0. Now,
the integral can be written as
I (N ) =
Z1
N
' (y; t) dG (y) = N
0
=N
Z1
' (y; t) g (y) G
N 1
(y) dy = N
0
Za
q (y; t) GN
1
(y) dy + N
Z1
q (y; t) GN
1
(y) dy
0
Z1
q (y; t) GN
1
(y) dy:
a
0
Then,
I1 (N ) = q1 GN
1
(a) a <
Za
q 1 GN
1
Za
(y) dy
0
I2 (N ) = q2 G
N 1
(c1 ) (c2
q (y; t) GN
1
(y) dy, and
0
c1 ) <
Zc2
q2 G
N 1
(y) dy
c1
Z1
q (y; t) GN
1
(y) dy:
a
As N increases, I1 (N ) converges to zero at the faster rate than I2 (N ). Since I2 (N ) > 0, this
implies I (N ) > I1 (N ) + I2 (N ) > 0 for a su¢ ciently large N .
Proof of Theorem 1. Consider the problem of seller 1, when N 1 competitors fully reveal
information and charge the price pSN . The seller maximizes the pro…t function
x;
; pSN
=x
Z1
1
G
max x; y
pSN + x
dGN
1
(y)
0
with respect to the pair ( ; x).
First, we will prove that for an arbitrary > 0, the optimal price xN ! 0 as N ! 1. That is,
@ (x; ;pS
N)
for a given x > 0, x x; ; pSN =
< 0 for a su¢ ciently large N , where
@x
x
x;
; pSN
=
Z1
1
G
max x; y
pSN + x
0
19
xg
max x; y
pSN + x
dGN
1
(y) .
G (!). It must satisfy the FOC24
Let xM be the price of the monopolist for demand 1
1
xM
xM = 0 if xM > ! , and
1
xM
xM
(8)
0 if xM = ! ,
Employing (8) and Condition 2 leads to
1
G
pSN + x
max x; y
= 1
G
max x; y
1
G
xM
xg
pSN + x
1
xM
max x; y
1
xM
x
'1 y; pSN =
1
G
max x; y
which is continuous in y and pSN . Because pSN
max x; y
0; x
where the …rst inequality is strict for y > pSN . Hence,
If x < xM , then consider the function
x
pSN + x
pSN + x
xM ,
x; ; pSN < 0 for x
pSN + x
xg
max x; y
xM .
pSN + x
,
! 0 by Lemma 1, there exist a su¢ ciently small
N !1
pSN +
x y
+ x > xM if y > xM x + and
> 0 and a su¢ ciently large N 0 , such that y
x and
N
N 0 (or, equivalently, if pSN
). This implies '1 y; pSN > 0; y 2 xM x + ; !
S
S
'1 y; pN
0; y 2 [!
x; ! ] for all pN 2 [0; ]. Thus, all conditions of Claim 4 are satis…ed.
S
Then, there exists N 0 such that
N1 = max N 0 ; N 0 .
x x; ; pN > 0; N
Now, consider the FOC function with respect to . By the envelope theorem,
xN ; ; pSN =
S
S
@ (xN ; ;pN )
d (xN ; ;pN )
=
that gives
d
@
xN ;
; pSN
=
xN
Z1
@
G
@
max xN ; y
pSN + xN
dGN
1
(y) :
0
@G (!)
Note that ! = !
! 0 > 0; > 0. Otherwise, ! 0 = ! leads to @
< 0; ! 2 ! ; ! .
0
Thus, G 0 (!) > G (!) if < that contradicts (1).
! ve
If lim ! > !00 = v e , then xN ! 0 and pSN ! 0 imply that for " < lim 2 , there is
!0
!0
N !1
N !1
N2 , such that max pSN ; xN < ". That is, jtN j = pSN xN < " if N N2 . Hence, it follows that
".
y tN = y pSN + xN > y " > ! 0 if y > ! 0 + " and y tN y + jtN j < y + " < ! if y < !
This implies
' (y; tN ) =
' (y; tN )
@
@
G (y tN ) =
G y pSN + xN > 0; y 2 ! 0 + "; !
@
@
0; y 2 [!
"; ! ] , for all tN 2 [ "; "] .
" ; and
Applying Claim 4 results in
xN ; ; pSN > 0 for > 0 and N N0 = max N2 ; N .
If lim ! = !00 , then the di¤erentiability of ! 0 and ! in some neighborhood of
!0
= 0
implies that
is 0 o2 (0; 1), such that !
! 0 if and only if
0 . Take
n ! there
0
! 0 !0 0
0 ! 0
; 1 ! 0 . Since " <
, following the same logic as above results in
" < min
2
2
max pSN ; xN < " and
xN ; ; pSN > 0 for all
note …rst that @@ G (!) = 0; ! > ! . Denote
!
G=
sup
2[0;
24
0 ];(x;p)2[0;"]
2
Z x+p
>
0
and a su¢ ciently large N . If
@
G (max fx; y
@
p + xg) dG (y) :
0
In general, if ! > 0, the monopolist’s solution may be boundary, i.e., xM = ! .
20
0,
It follows that
S
!
xN ; ; pSN
=
ZxN +pN
xN
@
G
@
max xN ; y
pSN + xN
dGN
1
(y)
0
xN +pS
N
!
Z
xN
@
G
@
pSN + xN
max xN ; y
dGN
1
(y)
0
!
<"
xN +pS
N
Z
@
G
@
max xN ; y
pSN + xN
(N
1) GN
2
(y) dG (y)
0
!
< " (N
1) GN
2
!
xN + pSN
0
S
ZxN +pN
@
G
@
pSN + xN
max xN ; y
dG (y)
0
" (N
2
(!
0
+ ") G:
xN ; ; pSN by the Taylor’s formula around
Expanding
xN ; ; pSN =
where ~ 2 [0;
1) GN
0 ].
= 0 gives
~
; ~; pSN
xN
x0 ; 0; pSN +
;
Because
x0 ; 0; pSN = GY v e
x0 + pSN x0 < GY (v e + ") v e = GN
1
(v e + ") v e ;
we have
xN ; ; pSN
~
; ~; pSN
xN
x0 ; 0; pSN +
N 1
<G
e
e
0
1) GN
(v + ") v + " 0 G (N
2
(!
0
+ ") ;
0;
where the right part of the inequality converges to 0 exponentially. This is because " < 1
leads to v e + " < ! 0 + " < 1 that, in turn, implies G (v e + ") < G (! 0 + ") < 1.
At the same time,
pSN ; pSN
=
1
GN pSN S
pN
N
1
GN pSN
g
2
!
0
1
;
N2
converges to 0 as N 2 . That is, there is N0 such that
pSN ; pSN >
xN ; ; pSN for all N
N0
and
0.
Since no seller can deviate if all competitors reveal full information, and x = pSN is the best
response to the same price of competitors, it follows that the symmetric strategy pro…le f i ; pi gN
i=1 =
1; pSN constitutes an equilibrium. To see that the identi…ed equilibrium is unique in the class of
symmetric equilibria as N ! 1, notice that all arguments in the proof hold if the distribution
GN 1 (y) is replaced by another distribution GN 1 (y) as long as it has a positive density on a
non-degenerated interval. Thus, for any other symmetric strategy of competitors pSN ; , where
> 0, the integral in
x; ; pSN is taken over GN 1 (y). Therefore, using the same technique as
above, it follows that
xN ; ; pSN > 0, and since < 1, a single seller can pro…tably deviate
by disclosing information of quality > . In addition, = 0 cannot be an equilibrium, since it
implies that all products are ex-ante identical, i.e., both the price level pSN and pro…ts must be zero.
21
This is not an equilibrium, since each seller can obtain the positive expected pro…t by revealing
information and charging a su¢ ciently small price ".
Proof of Lemma 2. By contradiction, suppose there is a fully revealing symmetric equilibrium
with the price pSN , which results in the pro…t of each seller:
GN pSN S
1
pN < pSN .
N
N
1
pSN ; pSN = P pSN ; pSN pSN =
First, consider the case of pSN v e and GN 1 (v e ) N1 . If, say, seller 1 does not reveal information
and charges the price x 2 pSN ; v e , his pro…t becomes
d
= GY v e
x + pSN x,
(9)
since he sells the product if Y < v e x + pSN . By setting x = pSN , the seller gets
Then, GY (v e ) = GN 1 (v e ) N1 implies
d
Second, if pSN
v e and GN
d
= GN
1 S
p >
N N
= GY (v e ) pSN
1
1
S
1 GN (pS
N ) pN
N
ve ,
pSN
pSN v e >
= GY (v e ) pSN .
pSN ; pSN .
setting x = v e results in the seller’s pro…t
GN pSN S
pN =
N
1
d
pSN ; pSN ,
which means that the seller can bene…cially deviate from the full disclosure policy.
1
e
v m , where
Proof of Corollary 1. If N = 2, the condition G (v e )
2 is equivalent to v
1
1
e
S
=G
v , it follows that pN pM by Claim 3, and the
2 is the median value. Thus, if pM
…rst pair of conditions in Lemma 2 holds.
vm
Proof of Lemma 3. If g (v) is symmetric on [0; 1], then v e = v m = 12 . According to Corollary
1, it is su¢ cient to show that pM
v e . The log-concavity of g (v) implies that it is unimodal,
so that it reaches the maximum g at x = 21 , where g
1. Otherwise, we have the contradiction:
R1
R1
g (v) dv
gdv < 1.
0
0
The FOC function for the monopoly’s pro…t
M
(x) =
0
M
(x) = 1
G (x)
which intersects the x axis once, because
lim x (x) = 1. Then,
M
(x) = (1
xg (x) = (1
G (x)) x is
G (x)) (1
x (x)) ,
(x) = x (x) is strictly increasing,
(0) = 0, and
x!1
M
which implies pM
(v e ) = 1
G (v e )
1
g
2
1
2
v e g (v e ) =
1
2
1
= 0,
2
ve.
Proof of Lemma 4. By Theorem 1, as the number of sellers N increases, it results in full
information disclosure. Thus, the market ine¢ ciency is determined by
M EN
SWN T SN
=
=
SWN
R1
vdGN (v)
0
R1
vdGN (v)
pS
N
R1
vdGN
0
22
(v)
=
pRS
N
vdGN (v)
0
R1
0
vdGN
;
(v)
where T SN =
R1
vdGN (v) is the total surplus, and SWN =
S
S
ZpN
ZpN
vdGN (v) = pSN GN pSN
R1
vdGN (v)
GN (v) dv < pSN GN pSN ;
0
0
and
vdGN (v) is the social welfare. Then,
0
pS
N
we have
R1
v e . Expanding G (v) by Taylor’s formula results in
0
G pSN = g0 v +
g 0 (~
pN ) S
pN
2
2
g0 pSN +
gs S
p
2 N
2
,
where g0 = g (0) ; p~N 2 0; pSN , and gs = max jg 0 (v)j. Thus,
M EN =
where O(x) has an order x. Since pSN 2
pSN g0 pSN
ve
1 b 1
Ng ; Ng
N
+O
pSN
2N
,
, M EN converges to 0 at the rate 1=N N .
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