MANAGERIAL AND DECISION ECONOMICS
Manage. Decis. Econ. 32: 493–504 (2011)
Published online 5 September 2011 in Wiley Online Library
(wileyonlinelibrary.com) DOI: 10.1002/mde.1550
Voluntary Quality Disclosure under
Price-Signaling Competition
Fabio Caldieraroa,†, Dongsoo Shinb,* and Andrew Stiversc,{
a
Department of Marketing and International Business, University of Washington, Seattle, Washington, USA
b
Department of Economics, Santa Clara University, Santa Clara, California, USA
c
Center for Food Safety and Applied Nutrition, US Food and Drug Administration, College Park, Maryland, USA
We analyze an oligopolistic competition with differentiated products and qualities. The quality of a product is not known to consumers. Each firm can make an imperfect disclosure of its
product quality before engaging in price-signaling competition. There are two regimes for
separating equilibrium in our model depending on the parameters. Our analysis reveals that,
in one of the separating regimes, price signaling leads to intense price competition between
the firms under which not only the high-quality firm but also the low-quality firm chooses
to disclose its product quality to soften the price competition. Copyright © 2011 John Wiley
& Sons, Ltd.
1. INTRODUCTION
Since Akerlof (1970), asymmetric information has
long been considered an important factor that causes
inefficiencies in markets. It is well understood by
now that when consumers cannot observe product
qualities, firms can use price as a signaling device for
their product quality—the low-quality firms have an
incentive to pretend that their products are high-quality,
and hence the high-quality firms need to distort their
prices to deter imitating behavior by the low-quality
firms. Alternatively, to reduce such distortions associated with asymmetric information, firms may choose
to subject their products or services to some process
that credibly discloses their qualities, if such a disclosure is feasible.
A credible information disclosure would involve
using an independent quality evaluation process with
public announcement of found quality, or advertisement
*Correspondence to: Santa Clara University, Economics, Santa
Clara, California, United States. E-mail: [email protected]
†
E-mail: [email protected]
{
E-mail: [email protected]
Copyright © 2011 John Wiley & Sons, Ltd.
with potential penalty for misrepresentation in the
presence of truth-in-advertising. For instance, some
agencies, such as Association of Home Appliance
Manufacturers and International Association of Air
Cleaner Manufacturers, have a ‘voluntary certification
program’ that air purifying manufacturers can choose
to participate at a cost. These agencies evaluate the participants’ products and make the quality information
public. Another example is the disclosure through special interest magazines—a manufacturer may submit
its product to such magazine’s review process, so that
the magazine’s assessment of the product can be published in an article, and consequently, made available
to readers.1
An interesting observation is that manufacturers
with products that are seemingly inferior in quality
sometimes chooses to disclose their product qualities.
Moreover, such products sometimes are priced higher
1
For example, for magazines such as Maximum PC or PC World,
the manufacturers must submit their products if they want to have
their products rated.
494
F. CALDIERARO ET AL.
than other competing products whose disclosed
qualities are higher. For example, according to the
Association of Home Appliance Manufacturers ratings,
model 30378 produced by Hunter ($180) outperformed
model 201 produced by Blueair ($299) in all quality
categories.2 Also, in a recent LCD evaluation by Maximum PC (September 2006), Dell’s 2407WFP ($850)
received a rate of 9 (where 10 is the best), whereas
HP’s LP2465 ($1000) received a rate of 4. Why would
a firm with a relatively inferior product with a higher
price disclose information on its product quality?
This seemingly puzzling phenomenon merits a closer
examination.
The objective of this paper is to provide a rational
for firms’ choices of quality disclosure that accommodates the phenomena described earlier. We look at an
oligopolistic industry in which each firm chooses price
as well as whether to withhold or advance information
about the quality of their products through a disclosure
channel. We develop a two-stage game in which two
firms compete with products that are differentiated
both horizontally (design or brand preferences) and
vertically (quality). To illustrate our point in a parsimonious setting, we assume that one firm produces a
high-quality product, whereas the product of the
other firm is low-quality. In the first stage, the firms
choose whether or not to disclose their product qualities simultaneously. A firm’s quality disclosure allows
a fraction of consumers to learn about its product quality, whereas the remaining fraction of the consumers
stay uninformed at the end of the first stage. At the
second stage, the firms engage in price competition
and may use price to signal their product qualities to
the uninformed consumers. To be sure, we want to
make it clear that quality disclosure in the first stage,
which is our issue of interest, is not a signaling device.
On the contrary, it is an instrument that a firm may use
to reduce the need of costly distortions to signal in the
second stage (when products are finally offered for
sale in the market).
We examine two separating equilibrium regimes in
our model: High-price signaling (commanding exclusivity) and low-price signaling (commanding large
market share).3 First, we find that the high-quality firm
has an incentive to disclose information so as to
2
See the website, http://cadr.org. Price information is from the internet retailer Amazon. Other websites also provide similar price
ranges for these products.
3
See for example, Caminal and Vives (1996) and Helloffs and
Jacobson (1999) among others for theoretical analyses and empirical
evidences for both cases.
Copyright © 2011 John Wiley & Sons, Ltd.
reduce the price distortion. Second and more interestingly, we find that the low-quality firm may also have
an incentive to disclose the quality of its product.
More specifically, quality disclosure by both highquality and low-quality firms occurs in a regime where
the high-quality firm distorts its price ‘downward.’ In
this regime, without quality disclosure the price competition (because of signaling effort) becomes very
intense. Thus, by increasing the proportion of the informed consumers in the first stage, the low-quality
firm can soften price competition in the second stage.
That is, by giving up some uninformed consumers in
the first stage (by making them informed), the lowquality firm can reduce the ‘battleground’ for the
remaining uninformed consumers in the second
stage. This, together with strategic complementarity
of prices, makes the low-quality firm better off by
unveiling its quality information to the market.
Our result suggests that a firm with relatively lower
quality product may have an incentive to voluntarily
disclose its quality information and that such a disclosure takes place with a high price of the product
(higher than the price of competing high-quality
products). According to our analysis, this can be the
case when a high-quality firm can produce at a lower
cost. In high-tech industries, for example, producing
a higher quality product at a lower cost is not very
uncommon. As Gibson, Goldenson and Kost (2006)
report, quality improvement and cost reduction are often paired with each other in such industries. In fact,
according to their report using Capability Maturity
Model Integration standard, companies’ quality improvement by 48% is accompanied by cost reduction
by 34%.
1.1. Literature Review
There is a long literature studying firms’ strategies to
signal private information to competitors or consumers. A large body of this literature, however, copes
with price as a signaling device for a monopolist.
Some papers, such as Milgrom and Roberts (1982),
Bagwell and Ramey (1988), and Shieh (1993) consider situations in which firms can produce higher
quality products at a lower cost. They show that a firm
with an advantage may distort the price downward to
signal its cost advantage or higher quality of the product. Other papers, such as Wolinsky (1983), Tellis and
Wernerfelt (1987), and Bagwell and Riordan (1991)
studied situations in which a higher quality is associated with a higher cost. In these papers, a firm may
signal the high quality of its product by distorting the
Manage. Decis. Econ. 32: 493–504 (2011)
DOI: 10.1002/mde
495
VOLUNTARY QUALITY DISCLOSURE
price upward. Our study encompasses both the cases
of high and low costs of producing high-quality
products.
There are papers that examine different ways to
convey signals. Wernerfelt (1994) introduces brand
externality as a signaling device. Balachander and
Srinivasan (1994) demonstrate that choice of product
line also conveys information about a firm’s competitive advantage. Lutz (1989) studies the usage of warranties to send positive signals to the market. Hellofs
and Jacobson (1999) provide empirical evidence that
both low market share (exclusivity) and high market
share can convey quality information. In this regard,
we point out that low (high) price signaling strategies
in our model are associated with high (low) market
share of a signaling firm.
Recently, researchers started paying more attention
to signaling in oligopolistic industries. Hertzendorf
and Overgaard (2001), and Fluet and Garella (2002)
consider price-advertising signals in duopoly, and
identify conditions for separating equilibria, and for
successful misrepresentation by a low-quality firm.
Yehezkel (2008) extends Hertzendorf and Overgaard
(2001) by introducing a fraction of informed consumers. This paper is different from ours in two ways.
First, in his study, the fraction of informed consumers
is exogenous, whereas in our paper, the fraction of informed consumers is determined endogenously by the
firms’ disclosure strategies. Second, as in Hertzendorf
and Overgaard (2001), the products are only differentiated vertically in his paper. In our paper, the firms’
products are differentiated both vertically and horizontally, raising situations in which the low-quality firm
discloses its quality information if the disclosure cost
is not too large.
Other papers focus on quality disclosure in oligopoly. Board (2009) shows that oligopolistic competition
may cause the unraveling mechanism to fail, and thus
the low-quality firm may be more incline to withhold
product-quality information. Levin, Peck and Ye
(2009) consider the possibility that firms may form a
cartel and shows that the cartel has higher incentive
to disclose the low-quality of a product because it
helps the cartel to differentiate its product line. Hotz
and Xiao (2010) show that firms may have an incentive to withhold quality information because disclose
can increase the demand elasticity of the products,
which in turn intensifies price competition between
firms. In these papers, unlike ours, signaling does not
play a role in the firms’ disclosure decision.
Lastly, there are studies related to credible disclosure versus signaling. In this branch of research, the
Copyright © 2011 John Wiley & Sons, Ltd.
paper by Moraga-Gonzales (2000) is more connected
to ours. The author considers a model in which firms
could endogenously inform a fraction of the consumers through some form of informative advertising.
This feature is similar to credible information disclosure in our paper. However, the paper does not
consider the effect of information revelation on equilibrium prices. Therefore, in his model, information
disclosure only occurs in a pooling equilibrium and
only by the high-quality firm. We contribute to the
literature by identifying that when there is competition, information disclosure occurs also in a signaling
equilibrium and that both the high-quality and the lowquality firms may have incentives to do so. Fishman
and Hagerty (2003) and Daughety and Reinganum
(2008) also study both disclosure and signaling in
a unified model. In Fishman and Hagerty (2003),
disclosure and signaling are complements, whereas
in our paper, they are substitutes. In Daughety and
Reinganum (2008), unlike in ours, if disclosure takes
place there is no need for signaling because disclosure
informs all consumers.
The rest of the paper is organized as follows. We
present the model in the next section. The game in
the second stage (price-signaling competition between
firms) is analyzed in section 3. In section 4, we analyze the game in the first stage (choice of information
disclosure) using the results from section 3. We gather
concluding remarks in section 5.
2. MODEL SETUP
We consider a two-stage game with two single-product
firms indexed by i 2 I = {1, 2}. For notational purposes,
we employ j 2 I (j 6¼ i) to designate ‘the other firm.’
The products are differentiated both horizontally
and vertically. In the first stage, the firms choose
whether to disclose their vertical quality, and in the
second stage, they choose prices. We focus on competition between firms with different qualities and
assume that firm 1 produces ‘high-quality’ products
and firm 2 produces ‘low-quality’ products. This perfect negative correlation is employed for expositional
purpose and our qualitative results hold as long as
there is some negative correlation between the types
of the firms. Negative correlation in qualities arises
from consumers’ relative preferences over product
qualities (relative preferences are known to be a
frequent phenomenon; one example is ‘positional
goods’). We assume that the firms have better knowledge about the industry than the consumers, and hence,
they know their competitor’s product quality. Consumers
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F. CALDIERARO ET AL.
cannot observe the quality of a product, but they have
a prior belief regarding a firm’s product quality that is
Pr(firm i = high-quality) = 1/2.
In the first stage, each firm can credibly disclose
information about its product quality at a cost g.
The choice of disclosure is denoted by xi 2 {0, 1},
with xi = 0 (withhold the information) and xi = 1 (disclose the information at the cost g). Disclosures by
the firm are imperfect—we denote by l(X) the probability that a consumer becomes informed of the firms’
product qualities when such information is disclosed,
where X = x1 + x2, with l(0) = 0 and 0 < l(1) < l(2) <1.
In other words, when a disclosure message is sent, the
consumers stochastically get informed of such a message. The disclosure technology is exogenous and
not improvable by the firms. The probability that a
consumer remains uninformed is strictly positive, but
as X (=x1 + x2) increases, a consumer becomes more
likely to be informed.4 Throughout this paper, we interpret l(X) as ‘the fraction of informed consumers’
at the end of the first stage. To simplify our analysis,
we assume that the 1 l(X) of consumers are informed of neither the disclosed quality information
nor the firm’s disclosure decisions.5
In the second stage, firm 1 (high quality) and firm 2
(low quality) choose p1 and p2 respectively, with
horizontally and vertically differentiated demand
Di(p1, p2, ai). The parameter ai 2 {aL, aH}, with
aH = aL + Δa and Δa > 0. For the fraction l of consumers (informed at the end of the first stage), a1 = aH
and a2 = aL. For the fraction 1 l of consumers (uninformed at the end of the first stage), ai are replaced by
the expected value E(ai) = mi(p1, p2)aH + (1 mi(p1,
p2))aL, where mi(p1, p2) is the posterior belief about
the probability that firm i is selling a high quality
good. Without price-signaling, only the fraction l of
consumers will be able to distinguish the two firms.
The remaining 1 l of consumers can only use their
priors and the information from the pair of prices to
update their posteriors about quality. Demand for firm
i’s product is described below:
Di ðp1 ; p2 ; ai Þ ¼ E ðai Þ pi þ bpj
¼ lai þ ð1 lÞ mi aH þ ð1 mi ÞaL
pi þ bpj ;
where b < 1 is the cross-price effect on demand. The
own-price effect is normalized to 1. Note that @ Di/@ pi
4
See Grossman (1981) and Hirshleifer and Teoh (2003) for a similar
setup.
5
In reality, for consumers who are not reached by the disclosed
information, finding out which firm disclosed its product quality
information would be extremely costly.
Copyright © 2011 John Wiley & Sons, Ltd.
pi < 0, @ Di/@ pj > 0, and @ Di/@ ai > 0 (again, the index j 2 I denotes ‘the other firm’). Expected profits
for the firms are then given by the expression:
pi ðp1 ; p2 ; mi Þ ¼ ðpi ci ÞDi ðp1 ; p2 ; E ðai ÞÞ gxi ;
where ci is the marginal production cost of firm i.
To guarantee a strictly positive demand, we assume
that aL > ci. We assume that these specifications are
common knowledge.
We close this section by summarizing the timing of
the game. In the first stage, firm 1 and firm 2 decide
whether or not to disclose their quality information
(choice of x1, x2 2 {0, 1}). Accordingly, the proportion
of informed consumers (l) is realized at the end of the
first stage. At the second stage, the firms engage in
price-signaling competition. In the subsequent sections, we analyze the game by backward induction.
3. ANALYSIS OF THE SECOND STAGE:
CHOICE OF PRICES
In the second stage, we look at the firms’ pricing strategies. The results in this section will be used in section
4 where we discuss information disclosure by each
firm before price-signaling competition. Recall that
1 l of consumers remain uninformed at the end of
the first stage, and thus, if profitable, a firm will price
so as to signal its product quality to the uninformed
consumers.
The building block of our analysis is as follows.
First, assuming that firm 1’s signaling equilibrium
price, denoted by pH, exists (later, we show that pH
exists), we define the consumer belief structure. Second, we derive firm 2’s price in an equilibrium
denoted by pL. Third, we identify two separating
regimes and construct a set of separating price pH in
each regime. Finally, we show that such set in each
separating regime is non-empty, depending on parameters, and select the least costly signaling price pH.
We proceed by defining a separating equilibrium
applying the Perfect Bayesian Equilibrium framework.
Definition 1.
A separating equilibrium entails {p1, p2} = {pH, pL}
and mi(p1, p2) such that:
(i) If pi = pH and pj 6¼ pH, then mi(p1, p2) = 1 and
mj(p1, p2) = 0,
(ii) If pi = pH and pj = pH, then mi(p1, p2) =
mj(p1, p2) = 1/2,
Manage. Decis. Econ. 32: 493–504 (2011)
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VOLUNTARY QUALITY DISCLOSURE
(iii) If pi 6¼ pH and pj 6¼ pH, then mi(p1, p2) =
mj(p1, p2) = 1/2,
(iv) pH 2 argmaxp1 p1(p1, pL, m1(p1, pL)),
(v) pL 2 argmaxp2 p2(pH, p2, m2(pH, p2)),
(vi) m1(pH, pL) = 1 and m2(pH, pL) = 0.
The explanation for the equilibrium is as follows.
Condition (i) states that, if firm i is pricing according
to the high-quality product and firm j is not, then consumers believe that firm i is the high-quality firm and
firm j is the low-quality firm. According to conditions
(ii) and (iii), if both firms are choosing the signaling
price (=pH), or both firms choose different prices from
the one that the high-quality firm would choose (6¼pH),
then consumers cannot tell a firm’s product quality
and revert to their priors (and believe that both firms
have the same probability of having high-quality).
Condition (iv) specifies that pH is the optimal price
of a high-quality firm given that firm 2 chooses pL.
Likewise, condition (v) specifies that pL is firm 2’s optimal price given that firm 1 chooses pH. Finally, condition (vi) says that consumers beliefs are confirmed
by the separating outcome—the consumers believe
that the product is high-quality when the firm charges
pH and that the product is low-quality when the firm
charges pL. This beliefs system is in-line with those
adopted in previous contributions, and our results still
hold with other reasonable beliefs systems. We will
further discuss this issue in the concluding section.
Definition 1 implies that a pair of prices in a separating
outcome must satisfy the conditions in the following lemma.
Lemma 1.
For a separating price profile the conditions below are
necessary:
p1 ðpH ; pL ; 1Þ ≥p1 ð^p 1 ; pL; 1=2Þ; 8^p 1 ≠ pH ;
p2 pH ; pL ; 0 ≥ p2 pH ; pH ; 1=2 ;
pL 2 argmaxp2 p2 pH ; p2 ; 0
The above expressions are standard incentive compatibility conditions in signaling models. The first
requires that firm 1 (high quality) prefers the signaling
price pH to some other price that allows mimicry by
firm 2 (low quality). The second requires that firm
2’s profit from being truthfully perceived as the low
quality is higher than its profit from pretending to be
the high quality. Lastly, the third expression requires
that firm 2 will optimally respond as if its quality information was fully revealed.
With these definitions, we characterize each firm’s
optimal price and payoff. As often occurs in signaling
Copyright © 2011 John Wiley & Sons, Ltd.
497
games, both separating and pooling equilibria may
emerge. Furthermore, it is also possible that the
high-quality firm’s best response price automatically
allows separation without distortion. Because such
outcomes are trivial situations, we focus on separating
equilibria with price distortions to make our point.
3.1. Separating Equilibrium
To find the separating equilibrium in the model, first
we obtain the strategy of firm 2, the low-quality firm
(Lemma 2 below), then we determine the set of candidates for pH, the separating equilibrium price for
firm 1, the high-quality firm (Lemma 3 below). Then,
we show that such set is non-empty, depending on
parameters (Lemma 4 below). Lastly, we obtain the
separating equilibrium price pH from the set of candidates.
Lemma 2.
In a separating equilibrium, the strategy of firm 2 is:
aL þ bpH þ c2
:
pL ¼ p2 pH ¼
2
Proof
See Appendix A.
We proceed by finding candidates for firm 1’s
separating price in equilibrium (candidates for pH).
Recalling Lemma 1, firm 1 wants to separate itself
from firm 2 (low quality). We use the superscript S
in pSi to denote separating outcomes and the superscript N in pNi to denote non-separating outcomes.
In a separating outcome, the profit for firm 1 (the
high quality) is given by:
p1 S pH ; pL ; 1 ¼ pH c1 aH pH þ bpL gx1 :
When firm 2 is truthfully recognized as the lowquality firm, its price is as in Lemma 2:
pL ¼
aL þ bpH þ c2
:
2
Thus, the profit for firm 1 in a separating equilibrium is:
p1 S pH ; pL pH ; 1 ¼ pH c1
aL þ bpH þ c2
H
H
a p þb
gx1 :
2
(1)
Manage. Decis. Econ. 32: 493–504 (2011)
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F. CALDIERARO ET AL.
If firm 1 deviates to a price p^ 1 6¼ pH while p2 = pL,
then consumers believe with m1 = 1/2 that firm 1 is the
low quality. In such a case, its profit would be:
p1 N p^ 1 ; pL ; 1=2 ¼ ð^
p 1 c1 Þ
aH þ aL
aL þ bpH þ c2
laH þ ð1 lÞ
^
p1 þ b
gx1 :
2
2
2
(4)
Firm 1’s best outcome in this case is obtained by
maximizing the above expression with respect to ^p 1 .
Such maximization yields:
L H L H ^ 1 p p ; p p ; 1=2
p1 N p
¼
2
ð2 þ bÞaL 2c1 þ bc2 þ b2 pH þ ð1 þ lÞΔa
4
gx1 :
(2)
In order to determine the set of candidates for pH
that satisfies the first condition in Lemma 1, we
define h(pH) to be the difference p1 S ðpH Þ p1 N ðpH Þ
using the expressions in (1) and (2). We verify
whether h(pH) is positive or negative:
aL þ bpH þ c2
h pH ¼ pH c1 aH pH þ b
2
ð2 þ bÞa 2c1 þ bc2 þ b p þ ð1 þ lÞΔa
4
L
2 H
2
:
= H, then Lemma 1 is violated, and firm 1
If pH 2
would prefer not to separate itself from firm 2.
Lemma 1 also indicates that with firm 1’s signaling
price pH, firm 2 would rather choose p2 = pL and
be truthfully identified as the low quality than attempt
to mimic firm 1 by choosing p2 = pH. In a separating
outcome, firm 2’s profit can be written as:
p2
p ; p ; 0 ¼ pL c2 aL pL þ bpH gx2 :
H
Using the expressions in (3) and (4), we determine
another set of candidates for pH that satisfy the second
condition in Lemma 1. As before, we define l(pH)
to be the difference p2 S ðpH Þ p2 N ðpH Þ and verify
whether the expression is positive or negative:
ðaL þ bpH þ c2 Þ2
l pH ¼
4
pH c2
laL þ ð1 lÞ
aH þ aL
ð1 bÞpH :
2
Function l(pH) is convex and quadratic in pH
with two real roots. Let these roots be p2 and p2 , where
p2 < p2 . These roots satisfy the second condition in
Lemma 1 with equality. We let:
o
n
L ¼ pH : l pH ≥ 0 ¼ pH : pH ≤ p2 or pH ≥
p2 :
= L, then Lemma 1 is violated, and firm 2
If p 2
would prefer to mimic firm 1.
Because Lemma 1 needs to be satisfied, our discussion indicates that all candidates for pH must be in the
intersection H∩L. Thus, we have the next lemma.
H
Function h(pH) is concave and quadratic in pH with
two real roots. Let these roots be p1 and p1 , where p1 <
p1 . These roots satisfy the first condition in Lemma 1
with equality. We let:
o
n
H ¼ pH : h pH ≥0 ¼ pH : p1 ≤ pH ≤ p1 :
S
If firm 2 mimics firm 1, while firm 1 is playing pH,
then firm 2’s profit would be:
p2 N pH ; pH ; 1=2
a þa
¼ pH c2 laL þ ð1 lÞ H L pH þ bpH gx2 :
Lemma 3.
A separating
n equilibriumo can only be supported by
H
p 2 S p : p1 ≤p≤p2 or
pH 2 S fp : p2 ≤p≤p
1 g:
Proof.
The proof directly
follows from the fact that
H ¼ pH : p1 ≤pH ≤p1 and
L ¼ pH : pH ≤p2 or pH ≥p2 . ■
L
Again, the optimal price for firm 2 in a separating
equilibrium is given by Lemma 2. Substituting for
the expression for pL in firm 2’s profit function, we
obtain:
ðaL þ bpH þ c2 Þ2
gx2 :
p2 S pH ; pL pH ; 0 ¼
4
Copyright © 2011 John Wiley & Sons, Ltd.
(3)
Lemma 3 says that the region of potential separating prices comes from two distinct ranges, S and S.
It can be easily seen that when pH 2 S, firm 1 is sending a signal by setting a high price, and when pH 2 S,
firm 1 is sending a signal by setting a low price. In
other words, when the high quality firm prices
too high or too low, it is not profitable for the low
quality firm to mimic the high quality firm.
Manage. Decis. Econ. 32: 493–504 (2011)
DOI: 10.1002/mde
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VOLUNTARY QUALITY DISCLOSURE
Lemma 4.
If c1 is sufficiently larger (smaller) than c2, then SðSÞ is
non-empty.
Proof.
See Appendix B. ■
To focus on separating outcomes, we assume that
the parameters satisfy the conditions in Lemma 3.
Figure 1 below presents the two sets for pH candidates:
S and S. In panel (a) of Figure 1, S 6¼ ∅ and S ¼ ∅, and
thus only a low price can signal high quality. As
pointed out in Lemma 4, this is the case when the
high-quality firm’s production cost is significantly
lower than the low-quality firm’s production cost.
Here, the high-quality firm can afford to distort its
price downward to signal, whereas it is too costly for
the low-quality firm to match the lower price. In panel
(b) on the other hand, S ¼ ∅ and S 6¼ ∅, thus only a
high price can signal high quality. This is the case
when the high-quality firm’s production cost is significantly higher than the low-quality firm’s production
cost. It is not profitable for firm 2 to match firm 1’s upward-distorted price because the own-price has a larger
impact on the demand than the cross-price (b < 1). That
is, firm 2 must cope with a large decrease in quantity
demanded to mimic firm 1’s high price. Consequently,
it is less profitable for firm 2 to mimic firm 1.
From now on, pH 2 S refers to ‘a high-price separating equilibrium,’ and pH 2 S refers to ‘a low-price
in signaling games,
separating equilibrium.’ As usual
p ¼
ð2 bÞðaL þ c2 Þþ ð1 lÞΔa þ
our model yields multiple Perfect Bayesian Equilibria
with separation. As will become clear later, however,
we note that the firms’ strategy in period 1 is not affected by multiplicity of equilibria in period 2. That
is, for any pH 2 S or pH 2 S in period 2, our results
for the disclosure strategiesin period 1 hold. Thus,
in what follows, we select the following pH without
loss of generality for each regime.
pH ¼ inf ðSÞ and pH ¼ supðSÞ:
(5)
The expression for pH in (5) reflects the idea that
firm 1 prefers to deviate its price as little as possible
from the one that satisfies the first order condition.
Again, the results in the first period hold for any pH
in the sets S or S:
We are now ready to determine the firms’ outcomes
in these two separating regimes. For convenience, we
introduce the following notations for each separating
equilibrium:
S H
L
S
S
S
H
L
1 p1 p ; p ; 1 and p p1 p ; p ; 1 ;
p
1
S
S H
L
S
S
H L
2 p2 p ; p ; 0 and p p2 p ; p ; 0 ;
p
2
L
where pL ¼ a
þb
pH þc2
2
H
p L ¼aL þbp þc2
2
and by Lemma 2.
By solving l(pH) = 0 for pH, we obtain the exact signaling prices: pH and p H :
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1 lÞΔa f2ð2 bÞ½aL ð1 bÞc2 þ ð1 lÞΔa g
;
ð 2 b Þ2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð2 bÞðaL þ c2 Þ þ ð1 lÞΔa ð1 lÞΔa f2ð2 bÞ½aL ð1 bÞc2 þ ð1 lÞΔa g
H
:
p ¼
ð2 bÞ2
H
(a)
(b)
l( pH)
l( pH)
p
S
p
S
h( pH)
h( pH)
Figure 1. Set of candidates for PH.
Copyright © 2011 John Wiley & Sons, Ltd.
Manage. Decis. Econ. 32: 493–504 (2011)
DOI: 10.1002/mde
500
F. CALDIERARO ET AL.
Using these expressions for the equilibrium price
in the two separating regimes, we obtain the
following profit expressions. In a high-price separating equilibrium,
bðaL þ bpH þ c2 Þ
S
H
L
H
1 ¼ ð
p c1 Þ a þ Δa p þ
p
2
and
2
ðaL þ b
pH c2 Þ
:
4
S2 ¼
p
Similarly, in a low-price separating equilibrium,
S
p ¼
1
1
H
b aL þ bp
þ c2
Aand
p H c1 @aL þ Δa p þ
2
pS ¼
2
0
H
aL þ bp
c2
4
H
2
:
In the following section, we proceed to the analysis
of the first stage in which each firm decides whether to
disclose its quality information.
4. ANALYSIS OF THE FIRST STAGE: CHOICE
OF INFORMATION DISCLOSURE
Using the profit expressions at the second stage,
we now analyze the firms’ optimal information
disclosure strategies at the first stage. Recall that
l is the proportion of informed consumers at the end
of the first stage and l(X) is strictly increasing in X,
where X = x1 + x2. Thus, we can see each firm’s disclosure strategy simply by looking at the effect of l on pi.
In other words, the sign of @pi/@l determines xi 2 {0, 1},
given that g is small enough. If @pi/@l > 0, then firm i
will disclose its product quality at the first stage (xi = 1
at the cost g). If @pi/@l < 0, then firm i will withhold
its quality information (xi = 0). If g is too large, then
it is clear that xi = 0. Below, we proceed to analyze
the two separating equilibrium regimes.6
l at the end of the first stage, the lower the need for
firm 1 to distort its prices upward. Thus, firm 1 benefits from disclosing information (x1 = 1) because this
increases l, which in turn reduces the upward distortion in p1 (pH decreases). However, firm 2 (the lowquality) becomes worse off as l increases. Again, as
l becomes larger, firm 1’s price becomes lower as firm
1 can reduce its upward price distortion. Thus, with
x2 = 1, not only more consumers are informed of firm
2’s low quality (at the end of the first stage), but also
firm 2’s price in equilibrium goes down as a response
to firm 1’s lower price (at the second stage). Together,
these effects diminish firm 2’s profit. Consequently,
firm 2 has an incentive to withhold the information
about its product quality (x2 = 0).
The above discussion is summarized in the following proposition.
Proposition 1:
In a high-price separating equilibrium, the high-quality
information of its true quality
Sfirm discloses
@
p1 =@l > 0 provided that g is not too large, while
the low-quality
information of its true
S firm withholds
quality @
p2 =@l < 0 .
Proof
See Appendix C. ■
4.2. Low-Price Separating Equilibrium
In this regime, firm 1 (the high quality) distorts its
price downward (instead of upward) to signal its
product quality. That is, for firm 1, the higher the l
at the end of the first stage, the lower the need for
downward price distortion. Thus, as in the high-price
separating regime, firm 1 benefits from disclosing
information (which increases l). Unlike in the previous
regime, however, firm 2 (the low quality) also benefits
from voluntarily disclosing its quality information
(contributing to an increase in l). We present our result
in the proposition below.
4.1. High-Price Separating Equilibrium
In a high-price separating equilibrium, firm 1 (the
high-quality) distorts its price upward so that consumers can identify its product quality. The higher the
In a non-separating equilibrium, firms do not distort their prices,
and therefore it is straightforward to verify that only the high-quality
firm discloses (certifies) its quality.
6
Copyright © 2011 John Wiley & Sons, Ltd.
Proposition 2
In a low-price separating equilibrium, not only does
the high-quality
firm discloses information of its true
quality @ pS =@l > 0 , but the low-quality firm also
1
chooses to do so @ pS =@l > 0 provided that g is
2
not too large.
Manage. Decis. Econ. 32: 493–504 (2011)
DOI: 10.1002/mde
VOLUNTARY QUALITY DISCLOSURE
Proof
See Appendix D. ■
Note that, in the low-price separating equilibrium,
the signaling price by firm 1 ( pH ) is smaller than the
low-quality firm’s price (pL). Therefore, information
disclosure by the low-quality firm is accompanied
by pL > pH . Again, when the high-quality firm can
produce at a lower cost, it will aggressively distort
its price downward to convince the uninformed
consumers at the end of the first stage. This leads to a
strategic effect that decreases the payoff of the lowquality firm in two ways. The lower price by the
high-quality firm decreases the demand of the lowquality firm, and it also forces the low-quality-firm
to respond with a lower price. As a result, the highquality firm’s downward price distortion reduces the
low-quality firm’s profit.
By disclosing its quality information to the market
(x2 = 1), firm 2 increases the proportion of informed
consumers, which leads to smaller downward distortion
in firm 1’s price (pH increasese), thus allowing firm 2
These effects together positively
to increase its price.
contribute to p2, and consequently, firm 2 also has
an incentive to disclose the information about its
product quality.
5. CONCLUDING REMARKS
In this paper, we have studied an oligopoly in which
two competing firms have the opportunity to reveal
information about their product qualities to a
fraction of consumers. Firms use their prices as the
signaling device for the remaining uninformed consumers. Two separating equilibrium regimes have
been identified. Our analysis suggests that not only
a high-quality firm but also a low-quality firm may
prefer to disclose the true quality of its product. The
price distortion for signaling by the high-quality
firm can either improve the payoff of the lowquality firm (as in the case in which high price
signals high quality) or diminish the payoff of the
low-quality firm (as in the case in which low price
signals high quality). In the latter case, the lowquality firm has an incentive to advance the true quality of its product to the market, so as to relax price
competition.
The results of our model have significant practical
relevance. In numerous industries, disclosing quality
information through some trusted mechanism is a
possibility that cannot be ignored. For instance, a large
Copyright © 2011 John Wiley & Sons, Ltd.
501
number of different products and services, such as
hotels, motor oil, desktop computers, glass, homenetworking equipment, to name a few, have the possibility of being assessed and certified by an expert body
that reports on the products intrinsic characteristics.
Managers responsible for these types of products
have to carefully consider whether the strategy of
information disclosure is worth for their products.
As our model suggests, the answer is dependent not
only on the product quality in itself but also on the
competitive landscape that ensues as a result of
choice of disclosure. Information disclosure is a way
to effectively widen the vertical quality difference
between the goods and thus soften competition.
Our result hinges on the following assumptions.
First, we adopted Bertrand competition with differentiated products—the products are differentiated not only vertically, but also horizontally.
For example, in Hertzendorf and Overgaard
(2001), the products are differentiated only vertically,
and thus, the quantity demanded for the low-quality
firm is zero if the low-quality firm’s price is higher
than its competitors. In such setting, our results do
not hold. Second, our result requires some degree of
(but not necessarily perfect) negative correlation.
Third, we considered price as the only signaling instrument. As mentioned in introduction, different
signaling devices could be introduced to the model.
For instance, dissipative advertising could be considered. Such advertisement can affect the price distortions. However, unless this additional instrument
completely removes the price distortion, that is, some
amount of price distortion still remains, our qualitative
results still hold.
We employed consumer beliefs that generate
regions of separating prices, without some other
exogenous reference points. Although this type of
belief system is often adopted in the signaling literature (e.g., Hertzendorf and Overgaard 2001, and
Daughety and Reinganum 2008), in practice, consumers may have various beliefs. This could allow the
consumers to infer the quality of the products according to some different rules, thus creating the possibility of different equilibria. For instance, consumers
could believe that whenever a price exceeds some
threshold, the product has high quality. If one knows
that, in certain situations, consumers make different
inferences on product qualities, then our results need
to be contextualized.
Our results imply that, when high-price signals quality, information disclosure is beneficial to both firms
and consumers. On the other hand, when low-price
Manage. Decis. Econ. 32: 493–504 (2011)
DOI: 10.1002/mde
502
F. CALDIERARO ET AL.
signals quality, disclosure enhances firm’s profits, but it
decreases consumers welfare, as a result of the higher
prices in equilibrium. Therefore, from the social welfare
standpoint, there is a clear trade-off.
Such a trade-off provides an insight on policy implication. Conventional wisdom may suggest that the
government should mandate quality disclosure as a
means to protect consumers. If the policy makers
are pro-consumers, then they may refrain from
such a mandate. However, if the policy makers
are concerned with total welfare of the market,
then they need to examine the industry more
carefully before implementing a mandatory disclosure
policy.
APPENDIX
A. Proof of Lemma 2
In a separating equilibrium, firm 2 has no incentive to distort its price to mimic firm 1, and hence, it chooses P2 to
maximize: pS2 ¼ ðp2 c2 ÞðaL p2 þ bpH Þ gx2 : The
first order condition with respect to p2 with a simple
L
H
rearrangement gives: p2 ðpH Þ ¼ a þbp2 þc2 : ■
B. Proof of Lemma 4
We show that, first, p1 p2 > 0 when c1 c2 is large
enough, and then p p > 0 when c2 c1 is large
1
2
enough. Because h(pH) is a quadratic function with
two real roots, solving h(pH) = 0 for pH gives p1 and
p , where p1 > p 1 . Similarly, l(pH) = 0 gives p2 and
1
2
2
p2 > p . Using the values for p1 and p2 from
p , where h(pH) = 0 and l(pH) = 0, respectively, we have:
2ðc1 c2 Þ 4 b2 þ 2 2ð1 þ l þ 2lb bÞ b2 Δa
p1 p2 ¼
2 2
4b
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
8ð1lÞΔa 4 b2 ð2 þ bÞaL 2 b2 c1 þbc2 b2 ð1 þ lÞ2ð3 þ lÞΔa
þ
2
4 b2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1 lÞΔa f2ð2 bÞ½aL ð1 bÞc2 þ ð1 lÞΔa g
:
2
4 b2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
8ð1 lÞΔa 4 b2 ð2 þ bÞaL 2 b2 c1 þ bc2 b2 ð1 þ lÞ 2ð3 þ lÞΔa
2
4 b2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1 lÞΔa f2ð2 bÞ½aL ð1 bÞc2 þ ð1 lÞΔa g
2
4 b2
captures the second and third terms with the square
roots. To check whether p1 p2 could be positive,
The term Y is positive when: d ≥
we first inspect Y:
0, where
d 8ð1 lÞΔa fð4 b2 Þ½ð2 þ bÞaL ð2 b2 Þc1
þ bc2 Þ b2 ð1 þ lÞ 2ð3 þ lÞΔa Þ; and
ð1 lÞΔa f2ð2 bÞ½aL ð1 bÞc1 þ ð1 lÞΔa g:
These are the expressions inside the square root
terms of Y . By solving this expression with equality
(d = 0) for c1, we find that the two square root
terms cancel out when:
c1 ¼
ð7 þ 4bÞaL þ ð1 þ 3bÞ
c2
8 4b2
þ
f46 þ 18l b½1 l þ 8bð1 þ lÞgΔa
:
8ð8 6b2 þ b2 Þ
The value for c1 necessarily implies that c1 > c2, because the first term is greater than c2, and the second term
is positive. However, if c1 > c2, then, it can be easily
> 0: Thus, when c1 c2 (>0) is large
checked that X
> 0: and Y > 0, which imply that
enough, we have X
p1 p2 > 0:
Similarly, using the values for p and p from
1
2
h( pH ) = 0 and l( pH ) = 0, respectively, we have the
following expression:
2ðc1 c2 Þ 4 b2 þ 2 2ð1 þ l þ 2lb bÞ b2 Δa
2 2
4b
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
8ð1 lÞΔa 4 b2 ð2 þ bÞaL 2 b2 c1 þ bc2 b2 ð1 þ lÞ 2ð3 þ lÞΔa
þ
:
2
4 b2
ffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1 lÞΔa f2ð2 bÞ½aL ð1 bÞc2 þ ð1 lÞΔa g
2
4 b2
p p ¼
2
1
Again, we let
We let
¼
X
¼
Y
2ðc1 c2 Þð4 b2 Þ þ 2½2ð1 þ l þ 2lb bÞ b2 Δa
ð4 b2 Þ2
captures the first term outside the square roots and
Copyright © 2011 John Wiley & Sons, Ltd.
X¼
2ðc1 c2 Þð4b2 Þþ2½2ð1þlþ2lbbÞb2 Δa
ð4 b2 Þ2
captures the first term outside the square roots and
Manage. Decis. Econ. 32: 493–504 (2011)
DOI: 10.1002/mde
503
VOLUNTARY QUALITY DISCLOSURE
in this case, the firm is distorting its price upward,
captures the second and third terms with the square
roots. To see that p p ¼ X þ Y > 0 when c1 c2
2 1 (>0) is large enough, we take the extreme values: c1 = 0
(the minimum possible c1) and c2 = aL (the
maximum possible c2). With these parametric values,
we have:
X ¼ 2a ð4 b Þ þ 2½2ð1 þ l þ 2lb bÞ b Δa ; and
ð4 b 2 Þ2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
8ð1 lÞΔa fð4 b2 Þ½ð2 þ bÞaL þ baL b2 ð1 þ lÞ 2ð3 þ lÞΔa g
Y¼
ð4 b2 Þ2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1 lÞΔa f2ð2 bÞ½aL ð1 bÞaL þ ð1 lÞΔa g
:
ð4 b2 Þ2
L
2
@
pS1
@
pH
< 0.
Next, we determine the sign of
@
pH
@l :
8
9
>
>
L
ð
2b
Þ
½
a
ð
1b
Þc
þ
ð
1l
ÞΔ
>
>
2
a
>
>
>
>
<
=
ðA1Þ
Δa 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2Δa ð1lÞð2bÞ½aL ð1bÞc2 þ½ð1lÞΔa 2 >
>
>
>
>
>
>
>
:
;
ff
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
8ð1 lÞΔa 4 b2 ð2 þ bÞaL 2 b2 c1 þ bc2 b2 ð1 þ lÞ 2ð3 þ lÞΔa
Y ¼
2
4 b2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1 lÞΔa f2ð2 bÞ½aL ð1 bÞc2 þ ð1 lÞΔa g
2
4 b2
@
pH
¼
@l
ðA2Þ
:
ð 2 b Þ2
2
It can be easily seen that X > 0: For Y; again, we
the square root terms
look at the expressions inside
of Y: Let
o 8ð1 lÞΔa fð4 b2 Þ½ð2 þ bÞaL þ baL b ð1 þ lÞ 2ð3 þ lÞΔa g; and
@
pS
t ð1 lÞΔa f2ð2 bÞ½a ð1 bÞa H
@
pS2
@
pS2
pH
@
@l ¼ @
@l :
pH
2bðaL þb
pH c2 Þ
> 0; be4
pH
seen above, @
@l < 0 .
the profit of firm 2. By the chain rule,
@
pS2 ð
pH ;pL ð
pH Þ;0Þ
@
pH
We note that
¼
cause a > c2. Furthermore, as
L
@
pS
p
implies that @pH =@l < 0. Therefore, @l1 ¼ @pH1 @
@l >
0; meaning that firm 1’s profit increases with l.
Next, we analyze the effect of quality disclosure on
L
2
L
Notice that expressions (A1) and (A2) in the RHS
of the above equation are positive, thus the entire expression in the curly brackets is negative, which
Thus,
@
pS2
@l
@
pS
p
¼ @pH2 @
@l < 0: ■
H
þð1 lÞΔa g:
From o t, we have the following expression:
D. Proof of Proposition 2
ð1 lÞ½f46 þ 18l b½1 l þ 8bð1 þ lÞgΔa
@ pS
þ2ð8 þ 7bÞð4 b2 ÞaL Δa > 0 ð∴Y > 0Þ
and thus X þ Y > 0 ■
Again,
@
pS1
We prove Proposition 1 by showing that @l > 0
(which implies that when g is small enough, x1 = 1)
and
@l
< 0 (implying that x2 = 0). Because
@
pSi
@l
¼
@
pSi
@
pH
@
pS1
@
pH
@pH
; we first determine the sign of . In this regime,
firm 1 (high quality) is setting a price different from
the first order condition so as to guarantee that consumers can distinguish between firms. Let p1FOC denotes
the undistorted First Order Condition price for firm 1.
FOC
1
We know that @p
. Because profits
@l ¼ 0 when p1 = p1
are a strictly concave function in own prices, it must
@l
be the case that
@
pS1
@
pH
< 0 when p >
H
@ pS
i
@l
@ pS
¼ @ pHi @ pH
@l
, and we begin the proof by
signing the effect of disclosure on the profit of firm
C. Proof of Proposition 1
@
pS1
@ pS
1 and 2
As in the previous proof, we show that @l>0
@l>0
(implying that when g is small enough, x1 = x2 = 1).
pFOC
.
1
Copyright © 2011 John Wiley & Sons, Ltd.
Because,
1. We first determine the sign of
@ pS
1
@ pH .
In this regime,
the firm 1 is playing a price different from the first
order condition so as to guarantee that consumers
can distinguish between the firms. Let p1FOC denotes
the undistorted First Order Condition price for firm
1
1. When p1 = p1FOC, we have @p
@p1 ¼ 0. Because profits
are an strictly concave function in own-prices, it must
@ pS
1
be the case that @ pH>0
when pH < pFOC
. In this regime,
1
the firm is distorting its price downward, and thus
@ pS
1 .
@ pH >0
Manage. Decis. Econ. 32: 493–504 (2011)
DOI: 10.1002/mde
504
F. CALDIERARO ET AL.
Next, we determine the sign of
8
9
>
>
ð2bÞ½aL ð1bÞc2 þð1lÞΔa
>
>
>
>
>
>
<
=
ðB1Þ
Δa 1 þpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
L
2Δa ð1lÞð2bÞ½a ð1bÞc2 þ½ð1lÞΔa >
>
>
>
>
>
>
>
:
;
ff
@ pH
¼
@l
@ pH
@l :
ðB2Þ
:
ð2 þ bÞ2
We let W = {(2b)[aL (1b)c2] + (1l)Δa}2, the
expression in the term (B1), and let Z = 2Δa(1 l)
(2 b)[aL (1 b)c2] + [(1 l)Δa]2, the expression
inside the square root (B2). It is straightforward to
show that W Z = {(2 b)[aL (1 b)c2]}2 > 0. Thus
(B1)/(B2) > 1, which implies that
@ pS1
@l
¼
@ pS1
@ pH
@ pH
@l
@ pH
@l>0
. Therefore
> 0:
Next, we analyze the effect of quality
disclosure on the profit of firm 2. Again by
the chain rule,
@ pS pH ;pL pH ;0
2 @ pH
¼
@ pS2
@l
@ pS2
@ pH
:
¼ @ pH @l
2b aL þbpH c2
4
> 0; because aL > c2.
Furthermore, as seen above,
@ pS
2
@l
¼
@ pS
2
@ pH
@ pH
@l
We note that
@ pH
@l
> 0 . Thus,
> 0: ■
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DOI: 10.1002/mde