Protest to Remove a Product from the Market: A
Model of an Informal Election for a Monopoly
Tomoya Tajika*June 13, 2024

Abstract
This study builds a model of a protest aiming at a ﬁrm to remove its product causing
negative externalities. The ﬁrm and the consumers are uncertain about the value and theextent of feeling guilty about consumers’ purchasing the product, but the consumers receive noisy signals. The ﬁrm cares about protests when it reveals the unpopularity of its
product. In our model, price is a key for information aggregation. With a higher price,
the utilities from the situation where the product is on sale are close between consumers
receiving good and bad news about the product quality. Then, the cost of participatingin the protest becomes close between these consumers, and the protest is uninformative
about consumers’ received signals. We show that with an endogenized price, the protest
is uninformative when the consumers’ signal is highly precise but is informative when
the signal is less accurate. This suggests that consumers’ ignorance may contribute to
the protest’s success.
Keywords: Protest, boycotts, information aggregation, ethical voters, pricing
JEL Classiﬁcation: D42, D72, D81, D82

1. Introduction
Firms sometimes face protest campaigns to remove their products from the market due to negative externalities (Egorov and Harstad, 2017). Negative externalities include pollution, workplace conditions (e.g., the use of child labor and poor working conditions), and political statements (e.g., fostering discrimination against minorities).¹ Although protest campaigns for ﬁrms’
products are common, one may doubt their eﬀects. Firms are not obligated to discontinue product sales even when many people participate, as protest participants need not be customers.
Despite this concern, protest campaigns still impact ﬁrms’ decisions as they may reveal information about the product’s popularity. People would be reluctant to participate in campaigns
aiming for product discontinuation if the product is precious. In contrast, if the product is lessvaluable, the negative externality exceeds the product value, which promotes participation in
the protests. Then, protest campaigns are a signal of product popularity. Once poor popularity
is revealed, the ﬁrm decides to discontinue the product. This study investigates this motivation
for protest campaigns and shows when protests are informative by building a game-theoretic
model.
In our model, a continuum of common-interested consumers decides whether to participate
in protest campaigns to remove a ﬁrm’s product with a negative externality. The ﬁrm and consumers observe public signals about the protest’s turnout, which is a sum of actual turnout andnoise followed by a normal distribution. Observing the public signal, the ﬁrm decides whether* College of Economics, Nihon University, Kanda-Misaki cho 1-3-2, Chiyoda Tokyo, 101-8360.
Email: tajika.tomoya@nihon-u.ac.jp¹Many protest activities are aiming for product discontinuation, E.g., protests for Nike for labor violations (e.g.,“Protesters call on NYU Bookstore to cut ties with Nike”, Washington Square News, Oct 13, 2023), a threat against the
publication of a book (“Japan ﬁrm nixes translation of U.S. book questioning trans surgery”, Japan Times, Dec. 6,2023), and so on.

1to continue the sale. The ﬁrm’s objective is to maximize its proﬁt. The ﬁrm decides whether
to discontinue the product, observing the protest’s turnout. We assume the ﬁrm is uncertain
about the product’s value for consumers. The value includes the guilty feelings that purchasing
a product causes negative externalities. Further, continuing product sales is costly. Then, if theprotest is suﬃciently informative and informs the ﬁrm that the product is not popular, the ﬁrm
discontinues the sales. The consumers are also uncertain about the product value but receive
noisy (and binary) signals about it.² Therefore, the protest has a function of information aggregation. Once the ﬁrm decides to continue the sale, the consumers decide whether to purchasethe product. This action aﬀects the value of protest campaigns because consumers are reluctant
to participate in the protest if the purchase brings much utility. The price of the product is thena key factor in consumers’ decisions to participate in the protest. We assume that the ﬁrm commits to a price before the protest campaign.
When the price is given, we show that the protest can be informative when the price is intermediate. Otherwise, the protest is no longer informative. The intuition is as follows. If the priceis too low, the ﬁrm gives up continuing the product sale irrelevantly to the protest turnout. On
the contrary, with a high price, the consumers gain less value from purchasing products even if
they receive good signals. Then, consumers’ beneﬁts from discontinuing product sales become
similar levels irrelevant to whether the signals are good or bad. Then, the protest turnout becomes less informative about the consumers’ received signals.
The above ﬁnding has implications for the ﬁrm’s pricing strategy. Suppose the consumers’
signals are suﬃciently precise. Then, the ﬁrm prefers to set a high price so that only consumers
receiving good signals will purchase the product. However, this makes the protest uninformative. In summary, consumers’ highly precise information makes the protest campaign uninformative. Conversely, suppose the consumers’ signals are less precise. In that case, even consumers receiving good signals have less willingness to pay. So, the ﬁrm prefers to set a price sothat both types (i.e., receiving good and bad signals) of consumers purchase. In this case, consumers receiving good signals obtain an information rent, while ones with bad signals do not
get the rent from product purchases. This discriminates the motivations for protest participation
between consumers receiving good and bad signals. Then, the protest becomes informative. To
underscore the novelty of our ﬁndings, we note that the informativeness of individual signals
and that of the protest may have negative correlations, a departure from previous studies. For
instance, while the model in Ekmekci and Lauermann (2022) shares similarities with ours, itfocuses on a protest against a government where the price is not a variable. In their case, they
demonstrate that the protest becomes more informative as individual signals become more precise.
Our results seem to depend on the decision ﬂow, in which the ﬁrm commits to a price before the consumers decide on participation. Hence, we also analyze the situation when the ﬁrm
decides the price after protest turnout is observed. The diﬀerence arises in the motivation ofparticipation in the protest. Participation in an informative protest signals the unpopularity ofthe product to the ﬁrm; the ﬁrm lowers the price of its product, but it also beneﬁts the consumers.
However, even in this twist, a similar result continues to hold. With higher signal precision, theprotest is more likely to be uninformative. On the contrary, with intermediate signal precision,
it becomes informative.
Our result suggests that consumers’ ignorance contributes to the successful protest campaigns
against the ﬁrm to withdraw its product.

²We can interpret this as a model of endogenized social norm. That is, each consumer cares about the norm if amajority of consumers care about purchasing the product causing negative externalities; otherwise, not. E.g., as a similar
setting, Fischer and Huddart (2008) consider players caring average norms.

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1.1. Related Literature
Many studies have investigated the consumers’ motivation to participate in protests, welfare
consequences, and aggregation of dispersed information. Among the studies, our study contributes to the literature on information aggregation when the threshold of the voting outcome
is endogenous. Battaglini (2017) studies a model of protest where the government’s decision ismade after observing protest turnout. The model of Ekmekci and Lauermann (2022) is closer
to ours in the sense that voters are a continuum.³
Our study also related to studies of boycotts. Some studies (e.g., Diermeier and Van Mieghem,2008, Delacote, 2009) model a boycott as a discrete public good game with a ﬁxed threshold. In
this modeling, as Delacote (2009) discusses, boycott campaigns have a weak point: While theparticipation of consumers who have a higher willingness to pay is more signiﬁcant to boycott,they have less incentive to participate. In our study, however, this point is signiﬁcant in making
the protest informative and signaling to the ﬁrm that their product is unpopular.
There are many other models of boycotts. For example, Egorov and Harstad (2017) built amodel of a boycott as a war of attrition. Baron (2001, 2003) considers an activist launchinga boycott and analyzes the eﬀects on strategic CSR behaviors. Several studies consider a market of credence goods and model a boycott as an instrument to make ﬁrms behave to beneﬁt
consumers. Feddersen and Gilligan (2001) and Innes (2006) consider a market with moralconcerned consumers and an activist. Feddersen and Gilligan (2001) consider a boycott as asignaling by the activist, while Innes (2006) shows that boycotts can arise under symmetric
information. Miyagiwa (2009) considers consumers who purchase less as they become more
suspicious of ﬁrms’ bad behavior. Glazer et al. (2010), Heijnen and Made (2012), and Peck(2017) consider boycotts as signaling of their moral concerns by consumers. In our study, protest
campaigns can be considered to signal moral concerns by interpreting the product’s value as adegree of (not) concern for morality. The main diﬀerence is that our study assumes consumers
are also uncertain about their moral concerns, and a problem of information aggregation arises.
Finally, we remark that our model is related to a study of turnout in a large election. Although
it is well known that almost all electorates abstain from voting in costly voting models, Feddersen and Sandroni (2006) provide a model to explain high turnout by considering a groupwise utilitarian electorate. In their model, the electorate, referred to as an ethical voter decides
whether to abstain from voting to maximize the total utilities of people who support the samecandidates. Ali and Lin (2013) show that reputation-concerned electorates behave as ethicalvoters. Ekmekci and Lauermann (2022) also consider ethical voters in a model of an informal
election, including protests.

2. ModelA ﬁrm sells a product to a continuum of consumers with unit demand. The population is normalized to 1. The marginal cost of production is 𝑤𝜅 ≥ 0. The utility of purchasing the productis 𝑤𝑣, where 𝑣 ∈ {0, 1}. We refer to 𝑣 as product quality4 and 𝑤 as the scale parameter. The priorprobability of 𝑣 = 1 is assumed 12 for notational simplicity.
Selling the product has a negative externality for the consumers. Consumers gain disutility
𝜍 > 0 when the product is sold. Consumers can conduct a protest campaign to stop the sale ofthe product. Participating in the campaign costs 𝑐𝑖 for consumer 𝑖. 𝑐𝑖 is identically and indepen³Relatedly, Correa (2022) considers a dynamic model of protest with a continuum of people. Battaglini et al. (2020)
conduct an experiment of a model similar to Battaglini (2017) and show that information sharing among the groupenhances information aggregation.
4𝑣 includes a psychological factor like feeling guilty about purchasing a product having a negative externality.

3

dently distributed by function 𝐹 among consumers. Then, when the product is sold at price 𝑤𝑝,
the consumer’s utility is as follows (𝜒 is the indicator function).
𝑢𝑖 = 𝑤 ⋅ (𝑣 − 𝑝) ⋅ 𝜒(purchase) − 𝜍 − 𝑐𝑖 ⋅ 𝜒(participate in protest).
Before participating in the protest campaign, each consumer gains information about the
product quality, 𝑣. The consumer receives signal 𝜃 ∈ {𝜃𝐻 , 𝜃𝐿 }. If 𝑣 = 1, each consumer receives
𝜃𝐻 with probability 𝜇 ∈ ( 12 , 1) and 𝜃𝐿 with probability 1 − 𝜇. If 𝑣 = 0, each receives 𝜃𝐿 withprobability 𝜇. Then, 𝜃𝐻 indicates that 𝑣 is likely to be high.
After the protest campaign, the ﬁrm and consumers observe a public signal, indicating theprotest campaign’s scale. Let the actual participation ratio be 𝜏. The ﬁrm and consumers observe 𝑡 = 𝜏 + 𝜀, where 𝜀 is a normal noise with 0 means and 𝜎2 variance. Let Φ be the standard
normal distribution function and 𝜑 be its density.
Observing the public signal 𝑡, the ﬁrm decides whether to discontinue the product. We assume that continuing product sales cost 𝑤𝐾 > 0. When the product continues to be sold, theconsumers decide whether to purchase the product.
The following ﬁgure summarizes the decision ﬂow.

Figure 1: Timeline of the decision ﬂow
Finally, the decision to participate in the protest follows the setting of Ekmekci and Lauermann (2022) and Feddersen and Sandroni (2006). That is, each consumer behaves to maximize
the overall utility of the consumers (referred to as ethical consumer). In other words, each consumer participates if and only if the marginal eﬀect of participation on expected utility exceeds
the participation cost.

3. Equilibrium with a given price3.1. After public signal observation
3.1.1. Consumer’s choiceThis section considers the condition of turnout when consumers purchase the product after public signal observation. Consider the consumer receiving signal 𝜃. The consumer purchases if
the expected product quality exceeds the price, that is, Pr(𝑣 = 1 | 𝜃, 𝑡) ≥ 𝑝. By abusing notation,
we write 𝑃𝜃 (𝑡) = Pr(𝑣 = 1 | 𝜃, 𝑡). The expected value of product quality isPr(𝑡 | 𝑣 = 1)𝜃
𝑃𝜃 (𝑡) =.
Pr(𝑡 | 𝑣 = 1)𝜃 + Pr(𝑡 | 𝑣 = 0)(1 − 𝜃)Let 𝜆𝑣 be the true turnout when the realized quality is 𝑣 ∈ {0, 1}. Then, 𝑃𝜃 (𝑡) is rewritten as𝑡−𝜆𝑃𝜃 (𝑡) =𝜑( 𝜎 1 )𝜃
𝑡−𝜆𝑡−𝜆𝜑( 𝜎 1 )𝜃 + 𝜑( 𝜎 0 )(1 − 𝜃).

By using this formula, when we assume 𝜆0 > 𝜆1 , inequality 𝑃𝜃 (𝑡) > 𝑝 is rewritten as𝑝𝜃
𝜆0 + 𝜆1 𝜎2 ln( 1−𝜃 ) − ln( 1−𝑝 )𝑡<+≕ 𝑇𝜃 (𝑝).
22𝜆0 − 𝜆1

4

In other words, 𝑇𝜃 (𝑝) is the supremum turnout with which the type-𝜃 consumers purchasethe product. The following lemma shows that the type-𝐻 consumers are likelier to purchase theproduct than the type-𝐿 consumers.
Lemma 1. If 𝜆0 > 𝜆1 , 𝑇𝐿 (𝑝) < 𝑇𝐻 (𝑝) for any 𝑝.
3.1.2. The ﬁrm’s choice
Given the consumer’s choice observing public signal 𝑡, this section considers the condition
when the ﬁrm discontinues the product sale. We focus on a cutoﬀ equilibrium; the ﬁrm discontinues the product sale if and only if 𝑡 > 𝑇 . Now we calculate 𝑇 .
Suppose that 𝑇𝐿 (𝑝) < 𝑇 < 𝑇𝐻 (𝑝). Then, at 𝑡 = 𝑇 , only the H-type consumer purchases the
product. In this case, the ﬁrm’s proﬁt isΠ𝐻 ≔ 𝑤([Pr(𝑣 = 1 | 𝑇 )𝜇 + Pr(𝑣 = 0 | 𝑇 )(1 − 𝜇)](𝑝 − 𝜅) − 𝐾).
This is because, when 𝑣 = 1, the population is H-type is 𝜇 while that is 1 − 𝜇 when 𝑣 = 0. As
the ﬁrm’s decision is independent of 𝑤, we omit the notation 𝑤 for the ﬁrm’s proﬁt hereafter.
𝐾
Note that Pr(𝑣 = 1 | 𝑇 )𝜇 + Pr(𝑣 = 0 | 𝑇 )(1 − 𝜇) ∈ [1 − 𝜇, 𝜇]. Therefore, if 𝑝 − 𝜅 ≥ 1−𝜇, the𝐾
expected proﬁt is no less than 0. Then, the ﬁrm always keeps the sale. In contrast, if 𝜇 ≥ 𝑝 −𝜅, the expected proﬁt is no more than 0. Then, the ﬁrm discontinues the sale.
𝐾
Otherwise 1−𝜇>𝑝−𝜅> 𝐾𝜇 . In this case, the ﬁrm discontinues the sale if and only if𝐾
𝜇− 𝑝−𝜅ln( 𝐾2𝜆 + 𝜆1 𝜎
𝑇≤ 0
+
2
2

𝑝−𝜅 −(1−𝜇))𝜆0 − 𝜆1≕ 𝑇𝑓 (𝑝).

The following lemma summarizes the discussion.
Lemma 2. Let Π𝐻 be the least turnout signal that the ﬁrm discontinues the sale when the H𝐾
type purchases, but the L-type does not. Then, if 𝑝 − 𝜅 > 1−𝜇, Π𝐻 > 0. If 𝐾
𝜇 > 𝑝 − 𝜅, Π𝐻 <𝐾
𝐾
0. If 1−𝜇 > 𝑝 − 𝜅 > 𝜇 , Π𝐻 ≤ 0 if and only if 𝑡 ≤ 𝑇𝑓 (𝑝)．
Although we focus on the case only the H-type purchases when the ﬁrm discontinues the
product, the remaining cases are discussed in the following sections.

3.2. Before public signal observation
Before the public signal observation, the consumer chooses whether to participate in the protest.
This section examines the condition and its informativeness regarding product quality.
We focus on the cutoﬀ strategy: each type-𝜃 consumer participates in protest if and only if themarginal beneﬁt of participation exceeds its cost. Following Ekmekci and Lauermann (2022),
we consider ethical consumers: they maximize the aggregate utility of the same type of consumers written in the following formula.56
𝑇 − d𝜏

∫(𝑤[𝑃𝜃 (𝑡) − 𝑝]+ − 𝜍)𝜉𝜃 (𝑡) d𝑡,
𝑡−𝜆

𝑡−𝜆

where we deﬁne [⋅]+ = max{⋅, 0} and 𝜉𝜃 (𝑡) = 𝜎1 (𝜃𝜑( 𝜎 1 ) + (1 − 𝜃)𝜑( 𝜎 0 )). d𝜏 represents a marginal increase in the participation rate. In this formula, the ﬁrm applies a cutoﬀ
5Note that assuming this utility does not aﬀect the purchasing decision as it does not aﬀect the other consumers’
utility and the ﬁrm’s decision.
6 Our results continue to hold even when the consumer maximizes social welfare. See Section 6.

5

criterion: discontinuing the product if and only if 𝑡 > 𝑇 . A consumer’s participation increases
turnout by d𝜏, which also increases their costs by 𝑐𝑖 d𝜏. Then, consumer 𝑖 participates in theprotest if and only if(𝜍 − 𝑤[𝑃𝜃 (𝑇 ) − 𝑝]+ )𝜉𝜃 (𝑇 ) ≥ 𝑐𝑖 .
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
marginal benefit of participation

Now let 𝑐𝜃 = (𝜍 − 𝑤[𝑃𝜃 (𝑇 ) − 𝑝]+ )𝜉𝜃 (𝑇 ) be the cutoﬀ of type-𝜃 consumer. Then, we havethat
𝜆1 = 𝜇𝐹(𝑐𝐻 ) + (1 − 𝜇)𝐹(𝑐𝐿 ),𝜆0 = (1 − 𝜇)𝐹(𝑐𝐻 ) + 𝜇𝐹(𝑐𝐿 ).
Therefore, 𝜆 ≔ 𝜆0 − 𝜆1 = (2𝜇 − 1)(𝐹(𝑐𝐿 ) − 𝐹(𝑐𝐻 )). This value plays a critical role inshowing the existence of an informative equilibrium.

3.3. Equilibrium characterization with given priceThis section characterizes the condition for the existence of an informative equilibrium. To thisend, we deﬁne the equilibrium using the optimal actions discussed in earlier sections. Let denote 𝜆 = 𝜆0 − 𝜆1 . Then, as 𝐹(𝑐) ∈ [0, 1], |𝜆| ≤ 2𝜇 − 1. Now, we deﬁne the equilibrium of thegame where the price is given.
Deﬁnition 1. (𝑇 , 𝑐𝐻 , 𝑐𝐿 ) is an equilibrium with given price 𝑝 if(i) 𝐸[profit | 𝑡] < 0 if and only if 𝑡 > 𝑇 .
(ii) 𝑐𝜃 = (𝜍 − 𝑤[𝑃𝜃 (𝑇 ) − 𝑝]+ )𝜉𝜃 (𝑇 ).
Note that if 𝑐𝐻 = 𝑐𝐿 , which implies 𝜆0 = 𝜆1 . Then, the public signal 𝑡 is independent of 𝑣.
We call this case uninformative.
Deﬁnition 2. An equilibrium is informative if 𝑐𝐿 > 𝑐𝐻 , and is uninformative if 𝑐𝐿 = 𝑐𝐻 .
It’s important to note that the uninformative equilibrium is a constant in our model. If the
ﬁrm sets 𝑇 = ∞ or 𝑇 = −∞, the protest does not aﬀect whether the product sale continues.
This leads to both types of consumers having indiﬀerent incentives. Conversely, if the protest
is uninformative, the ﬁrm’s decision remains unchanged regardless of the turnout. This implies
that 𝑇 = ∞ or 𝑇 = −∞.
Therefore, it is crucial to reiterate that our focus is on when an informative equilibrium exists. Our discussion is guided by the following proposition, which characterizes the condition
concerning 𝑝 for the existence of an informative equilibrium. This proposition is a key tool in
our analysis (the proof is relegated to the Appendix).
𝐾+(1−𝜇)𝜅

Proposition 1. Let 𝑃 ≔ 𝜇 𝜇(1−𝜇)+(2𝜇−1)𝐾 . (a) If 𝑝 ≥ max{𝜇, 𝑃} or 𝑝 ≤ 𝐾 + 𝜅, no informativeequilibrium exists． (b) Suppose that 𝐾 + 𝜅 < 𝑝 < min{𝜇, 𝑃}, 𝑓 (0) > 0 and 𝐹(0) > 0. Then, if
𝑤 is suﬃciently large, an informative equilibrium exists.
Here, 𝑃 is a threshold of 𝑝: 𝑇𝐻 (𝑝) > 𝑇𝑓 (𝑝) if and only if 𝑝 < 𝑃. 𝑇𝐻 (𝑝) > 𝑇 = 𝑇𝑓 (𝑝) implies
that at the threshold of the ﬁrm, on the condition that the ﬁrm gives up to continue the productsale, the 𝐻-type still has the incentive to purchase. This makes the 𝐻-type reluctant to participate in the protest, which is the source of the diﬀerence to the 𝐿-type consumer’s behavior.
6

In contrast, if 𝑇𝑓 (𝑝) ≥ 𝑇𝐻 (𝑝), the 𝐻-type never purchases the product, and the incentives ofparticipating in the protest are indiﬀerent as both 𝐻- and 𝐿-types consumers do not gain fromthe product. In this case, the protest is uninformative.

4. Firm’s optimal pricing
This section focuses on the most informative equilibrium for each price the ﬁrm sets and investigates when informative equilibrium exists by endogenizing the price. All proofs of this section
are relegated to the Appendix.

4.1. Benchmark
As a benchmark, this section considers the model without protest. In the benchmark case, theH-type’s expected product value is 𝐸[𝑣] = 𝜇, and that of the L-type consumers is 𝐸[𝑣] = 1 −𝜇. If the ﬁrm keeps the product sale, the optimal price is 𝑝 = 𝜇 or 𝑝 = 1 − 𝜇. The expected𝜇−𝜅
proﬁt when 𝑝 = 𝜇 is 2 − 𝐾, and that when 𝑝 = 1 − 𝜇 is 1 − 𝜇 − 𝜅 − 𝐾.

4.2. Optimal price.
Now, we consider the ﬁrm’s optimal pricing with the protest. First, we deﬁne the ﬁrm’s proﬁt.
As the ﬁrm keeps product sale unless 𝑡 < 𝑇𝑓 (𝑝), the ﬁrm’s expected proﬁt is
𝑇𝑓 (𝑝) − 𝜆1
𝑇 (𝑝) − 𝜆1
𝑇 (𝑝) − 𝜆1
) − Φ( 𝐿))(𝜇(𝑝 − 𝜅) − 𝐾) + Φ( 𝐿)(𝑝 − 𝜅 − 𝐾)]𝜎𝜎𝜎
𝑇𝑓 (𝑝) − 𝜆0
𝑇 (𝑝) − 𝜆0
𝑇 (𝑝) − 𝜆0
+ Pr(𝑣 = 0)[(Φ() − Φ( 𝐿))((1 − 𝜇)(𝑝 − 𝜅) − 𝐾) + Φ( 𝐿)(𝑝 − 𝜅 − 𝐾)].
𝜎𝜎
𝜎

Π(𝑝) = Pr(𝑣 = 1)[(Φ(The following proposition shows that the optimal price is 𝑝 = 𝜇, in which case the protest is
uninformative.
𝜇−𝜅Proposition 2. Suppose that 2 > max{𝐾, 1 − 𝜇 − 𝜅}. Then, for suﬃciently large 𝜎 > 0,𝑝 = 𝜇 is the optimal price and protest is uninformative for any 𝜆0 , 𝜆1 .
Note that this result is irrelevant to the size of 𝑤 as 𝑤 aﬀects the size of 𝜆1 , 𝜆0 , but they
are bounded by [0, 1]. Therefore, as Proposition 1 shows, an informative equilibrium exists if
𝑝 < {𝜇, 𝑃}. An informative equilibrium also beneﬁts the ﬁrm as it discontinues the product andsaves ﬁxed cost 𝐾 if it observes high 𝑡, which implies 𝑣 = 0 is likely enough.
However, the proﬁt is smaller than that when 𝑝 = 𝜇 if 𝜎 is suﬃciently large. In this case, thenoise of turnout is large enough, and therefore, even when the protest is informative, the beneﬁt
of gaining information is small. Then, rather than gaining information by setting a lower price,𝑝 = 𝜇 yields a greater proﬁt.
The above observation depends on whether an uninformative equilibrium with 𝑝 = 𝜇 yields
a suﬃcient proﬁt. If not, an informative equilibrium leads to a higher proﬁt, as the following
proposition shows.
Proposition 3. Take suﬃciently large 𝑤 so that informative equilibrium exists. Suppose that𝜇−𝜅1 − 𝜇 − 𝜅 > 𝐾 > 2 . Then, at the ﬁrm’s optimal price, the equilibrium is informative.
By Propositions 2 and 3, we observe that the resulting equilibrium is uninformative if 𝜇 is
high, while it is informative when 𝜇 is small. Corollary 1 summarizes the result, and Figure 2
illustrates it.
Corollary 1. Suppose that 𝜅 < 12 and 𝐾 ∈ ( 1−2𝜅, 12 − 𝜅). Take suﬃciently large 𝜎 and a suf3
ﬁciently large 𝑤 (it depends on 𝜎) to guarantee the existence of an informative equilibrium.
7

Then, there exists 𝜇∗ , 𝜇, 𝜇 with 𝜇∗ > 𝜇 > 𝜇 such that the equilibrium is uninformative if 𝜇 ∈(𝜇, 𝜇∗ ),7 and the equilibrium is informative if 𝜇 < 𝜇.

Figure 2: Classiﬁcation of equilibria
This result implies that individual consumers’ signal informativeness may negatively correlate with the protest’s informativeness. This contrasts sharply with Ekmekci and Lauermann(2022), who shows that each person’s signal informativeness positively correlates with theprotest campaign’s informativeness.
The main diﬀerence between their and our models is the existence of a price-setting ﬁrm.
In the models of Ekmekci and Lauermann (2022) and many protest studies, citizens campaign
against the policy-setting government, which aﬀects only the policy decision. However, the ﬁrm
makes two-dimensional decisions in our model: whether to discontinue the product sale andprice. With a higher informativeness of each individual signal, consumers are more willing topay, which leads to a greater proﬁt for the ﬁrm by setting a higher price. This makes the protest
uninformative by reducing the purchase incentive of 𝐻-type.

5. Uncommitted priceIn the previous section, the ﬁrm is supposed to commit to a price 𝑝 before the public signal isobserved. This section considers the case in which the ﬁrm decides the price after the public
signal observation. The timeline is depicted in Figure 3.

Figure 3: Timeline for the uncommitted price caseAs a crucial diﬀerence to the committed price case, the turnout aﬀects the price in the uncommitted case. With a greater turnout, if it is informative, it reduces the expected value of the
product. Then, the ﬁrm sets a lower price. This makes the 𝐻-type consumers participate in the7

The restriction 𝜇 < 𝜇∗ is needed because Proposition 2 is the statement when 𝜇 is ﬁxed.

8

protest more than the 𝐿-type consumers, as the 𝐻-type consumers earn more information rents
when purchasing. This makes the equilibrium likelier to be uninformative.
Now, we investigate how our conclusion changes. The previous section shows that with higher
𝜇, the equilibrium is likelier to be uninformative. Even in the uncommitted price case, a similar
conclusion is drawn. The following is a formal analysis. All proofs of the following section are
relegated to the Appendix.

5.1. Formal analysis of the uncommitted price caseTo focus on the existence of an informative equilibrium, assume that 𝜆0 > 𝜆1 . Note that evenin the uncommitted price case, the consumers’ choice of product purchase is unchanged. Thus,
type-𝜃 consumer purchases the product if and only if 𝑡 < 𝑇𝜃 (𝑝).
Given this strategy, the ﬁrm’s proﬁt is⎧−𝐾
if 𝑡 > 𝑇𝐻 (𝑝){{Π = ⎨[Pr(𝑣 = 1 | 𝑡)𝜇 + Pr(𝑣 = 0 | 𝑡)(1 − 𝜇)](𝑝 − 𝜅) − 𝐾 if 𝑡 ∈ (𝑇𝐿 (𝑝), 𝑇𝐻 (𝑝)]{{𝑝 − 𝜅 − 𝐾
if 𝑡 ≤ 𝑇𝐿 (𝑝).
⎩This implies that the optimal price is either 𝑝 = 𝑃𝐻 (𝑡) or 𝑃𝐿 (𝑡), and then, we need to compare the following equations.
Π∗𝐻 (𝑡) ≔ [Pr(𝑣 = 1 | 𝑡)(2𝜇 − 1) + 1 − 𝜇)](𝑃𝐻 (𝑡) − 𝜅) − 𝐾
Π∗𝐿 (𝑡) ≔ 𝑃𝐿 (𝑡) − 𝜅 − 𝐾.
−1

Also, the ﬁrm discontinues the product sales rather than selling it with 𝑝 = (𝑇𝐿 ) (𝑡) if 𝑝 −−1
𝜅 < 𝐾. In other words, (𝑇𝐿 ) (𝑡) < 𝐾 + 𝜅. Equivalently, 𝑡 > 𝑇𝐿 (𝐾 + 𝜅).
Below, to simplify the discussion, we assume 𝜅 = 0.
Lemma 3. Assume that 𝜅 = 0. Then, there are 𝜇∗ , 𝜇∗∗ with 𝜇∗ < 𝜇∗∗ such that (a) If 𝜇 >𝜇∗∗ , there is 𝑡 ̂ < 𝑇𝐿 (𝐾) such that Π∗𝐻 (𝑡) < Π∗𝐿 (𝑡) if and only if 𝑡 < 𝑡.̂ (b) If 𝜇 < 𝜇∗∗ , Π∗𝐻 (𝑡) <Π∗𝐿 (𝑡) for each 𝑡 < 𝑇𝐿 (𝐾). (c) 𝑇𝐿 (𝐾) → ∞ if 𝜇 < 𝜇∗ , and 𝑇𝐿 (𝐾) → −∞ if 𝜇 > 𝜇∗ .
Given this ﬁrm’s behavior, we consider the consumers’ incentive to participate in the protest.
We ﬁrst consider the case (a): there is 𝑡 ̂ < 𝑇𝐿 (𝐾) such that Π∗𝐿 (𝑡) > Π∗𝐻 (𝑡) if and only if 𝑡 < 𝑡.̂
Let 𝑇 ∗ be the solution to Π∗𝐻 (𝑡) = 0 with respect to 𝑡. This implies that at the optimal strategy,
̂ sets price 𝑃𝐻 (𝑡) when 𝑡 ∈ (𝑡,̂ 𝑇 ∗ ), and when 𝑡 >the ﬁrm sets price 𝑃𝐿 (𝑡) when 𝑡 ∈ (−∞, 𝑡),𝑇 ∗ , discontinues the product sale.
Under this ﬁrm’s behavior, the utility of the H-type is−𝜍 ∫𝑇 ∗ − d𝜏𝑇 ∗ − d𝜏𝜉𝐻 (𝑡) d𝑡 + 𝑤 ∫

̂
𝑡−𝑑𝜏

̂
𝑡−𝑑𝜏

+𝑤 ∫

[𝑃𝐻 (𝑡) − 𝑃𝐻 (𝑡 + d𝜏)]𝜉𝐻 (𝑡) d𝑡.

[𝑃𝐻 (𝑡) − 𝑃𝐿 (𝑡 + d𝜏)]𝜉𝐻 (𝑡) d𝑡.

Here, 𝑃𝐻 (𝑡) is the expected value of the product while 𝑃𝐿 (𝑡) is the price. A marginal increaseof turnout d𝜏 aﬀects the price. As the ﬁrm set price 𝑝 = 𝑃𝐻 (𝑡) if 𝑡 > 𝑡,̂ type-𝐻 consumers’ theinformational rent 𝑃𝐻 (𝑡) − 𝑃𝐿 (𝑡) vanishes. Therefore, the decision to discontinue the productonly aﬀects the externality term 𝜍.
The diﬀerentiation concerning d𝜏 is the marginal gain by participation. Therefore, they participate in the protest if and only if 𝑐𝐻 ≥ 𝑐𝑖 , where𝑇∗𝑡̂′ (𝑡)]𝜉 (𝑡) d𝑡 − [𝑃 (𝑡)̂ − 𝑃 (𝑡)]𝜉
′𝑐𝐻 ≔ 𝜍𝜉𝐻 (𝑇 ∗ ) + 𝑤[∫ [−𝑃𝐻𝐻𝐻𝐿 ̂
𝐻 (𝑡)̂ + ∫ [−𝑃𝐿 (𝑡)]𝜉𝐻 (𝑡) d𝑡].
𝑡̂9Note that 𝑃𝐿′ (𝑡) < 0. Then, the third term of the above equation is the gain from price manipulation by increasing turnout. In contrast, with a high turnout, the ﬁrm gives up on making all
types of consumers purchase the product, and then, type-𝐻’s informational rent vanishes. This
makes type 𝐻 consumers reluctant to participate in the protest, captured by the second term.
Now∗ we compare with 𝐿-type consumers’ utility. The utility of the L-type iŝ𝑇 − d𝜏𝑡−𝑑𝜏−𝜍 ∫𝜉𝐿 (𝑡) d𝑡 + 𝑤 ∫[𝑃𝐿 (𝑡) − 𝑃𝐿 (𝑡 + d𝜏)]𝜉𝐿 (𝑡) d𝑡 as the ﬁrm extracts type-𝐿 consumers’ full surplus. Then, the type-𝐿 consumers participate in the protest if and only if𝑡̂

𝑐𝐿 ≔ 𝜍𝜉𝐿 (𝑇 ∗ ) + 𝑤 ∫ [−𝑃𝐿′ (𝑡)]𝜉𝐿 (𝑡) d𝑡 ≥ 𝑐𝑖 .
We compare the relation between 𝑐𝐻 and 𝑐𝐿 with high 𝜇 cases. As we focus on a large 𝑤
case, we consider the coeﬃcients of 𝑤 in 𝑐𝐿 − 𝑐𝐻 , that is𝑇∗𝑡̂′ (𝑡)]𝜉 (𝑡) d𝑡 + [𝑃 (𝑡)̂ − 𝑃 (𝑡)]𝜉
𝐷 ≔ ∫ [−𝑃𝐿′ (𝑡)](𝜉𝐿 (𝑡) − 𝜉𝐻 (𝑡)) d𝑡 − ∫ [−𝑃𝐻𝐻𝐻𝐿 ̂
𝐻 (𝑡)̂

𝑡̂
𝜇Suppose that 2 > 𝐾. Then 𝑇 ∗ → ∞ as 𝜎 → ∞.

Proposition 4. Suppose that 𝜇 > 𝜇∗∗ . Then, for suﬃciently large 𝜎, 𝐷 < 0.
Proposition 4 implies that the protest becomes uninformative with high 𝜇. This is because
this inequality 𝑐𝐻 > 𝑐𝐿 is gained by the supposition that 𝜆0 > 𝜆1 , which means that 𝑐𝐿 > 𝑐𝐻 .
This is a contradiction. So, the equilibrium should be uninformative. This is a similar feature
to the committed price case. The intuition is the following. With a high 𝜇, the ﬁrm continues
to sell even when 𝑡 is suﬃciently high, in which case, while the L-type gives up to purchase,the H-type purchases. Then, the H-type’s incentive for price manipulation is much higher thanthat of the L-type. In contrast, the eﬀect on the informational rent is small as the ﬁrm is likely
to focus on selling only to the H-types when 𝜇 is high enough. This motivates the H-type toparticipate in the protest more than the L-type.
Next, we consider the case (b): for each 𝑡 < 𝑇𝐿 (𝜅 + 𝐾), Π∗𝐿 (𝑡) > Π∗𝐻 (𝑡). This implies thatfor any 𝑡 < 𝑇𝐿 (𝜅 + 𝐾), the ﬁrm set price 𝑃𝐿 (𝑡) and discontinues the product sale if 𝑡 > 𝑇𝐿 (𝜅 +𝐾). To simplify the notation, we denote 𝑇∗ = 𝑇𝐿 (𝜅 + 𝐾).
In this case, The utility of the H-type is𝑇∗ − d𝜏−𝜍 ∫𝑇∗ −𝑑𝜏

𝜉𝐻 (𝑡) d𝑡 + 𝑤 ∫[𝑃𝐻 (𝑡) − 𝑃𝐿 (𝑡 + d𝜏)]𝜉𝐻 (𝑡) d𝑡.

The diﬀerentiation concerning d𝜏 is the marginal gain by participation. Therefore, they participate in the protest if and only if𝑇∗

𝑐𝐻 ≔ 𝜍𝜉𝐻 (𝑇∗ ) − 𝑤[𝑃𝐻 (𝑇∗ ) − 𝑃𝐿 (𝑇∗ )]𝜉𝐻 (𝑇∗ ) + 𝑤 ∫ [−𝑃𝐿′ (𝑡)]𝜉𝐻 (𝑡) d𝑡 ≥ 𝑐𝑖 .
𝑇 − d𝜏𝑇 −𝑑𝜏

𝜉𝐿 (𝑡) d𝑡 + 𝑤 ∫ ∗Similarly, the utility of the L-type is −𝜍 ∫ ∗d𝜏)]𝜉𝐿 (𝑡) d𝑡. Then, they participate in the protest if and only if[𝑃𝐿 (𝑡) − 𝑃𝐿 (𝑡 +𝑇∗

𝑐𝐿 ≔ 𝜍𝜉𝐿 (𝑇∗ ) + 𝑤 ∫ [−𝑃𝐿′ (𝑡)]𝜉𝐿 (𝑡) d𝑡 ≥ 𝑐𝑖 .
We also compare 𝑐𝐻 with 𝑐𝐿 and thus, focus on the coeﬃcient of 𝑤 in 𝑐𝐿 − 𝑐𝐻 ,𝑇∗𝐷̃ ≔ ∫ [−𝑃𝐿′ (𝑡)](𝜉𝐿 (𝑡) − 𝜉𝐻 (𝑡)) d𝑡 + [𝑃𝐻 (𝑇∗ ) − 𝑃𝐿 (𝑇∗ )]𝜉𝐻 (𝑇∗ )Proposition 5. Suppose that 𝜇 ∈ (𝜇∗ , 𝜇∗∗ ). Then, for suﬃciently large 𝜎, 𝐷̃ > 0.

10

𝑐𝐻 > 𝑐𝐿 implies that the H-type is likelier to participate in the protest. Then, the equilibrium
becomes informative with a large 𝑤. Intuition is the following. With lower 𝜇, the ﬁrm gives up
selling even with a small 𝑡. Then, the incentive for price manipulation becomes smaller, and theeﬀect of losing informational rent dominates that incentive.
In conclusion, the informative equilibrium is more likely with intermediate 𝜇 cases, although
not in high 𝜇 cases. Therefore, the protest’s informativeness is non-monotonic to the individual
informativeness, which is a similar conclusion to the committed price case.

6. Conclusion and Discussions
This study examines when the protest campaign informs about the consumers’ values for the
product, which determines the ﬁrm’s decision. We show that the consumers’ individual signal
precision does not necessarily contribute to the protest’s informativeness. One implication of
our result is that consumers’ ignorance may contribute to the success of protest campaigns.
In the remainder of this study, we discuss the assumptions in our model.
Objective of participation in protest We have assumed that consumers consider their type’s
aggregate utility when deciding whether to participate in the protest. However, as in Ali and
Lin (2013), if this motivation comes from a reputation concern, it is more plausible to assume
that consumers consider the expected social welfare. Let’s consider a scenario where each consumer’s decision to participate in the protest is based on maximizing the expected social welfare.
The consumer’s belief, represented by 𝜃 ∈ {𝜃𝐻 , 𝜃𝐿 }, inﬂuences their expected social welfare
as follows.
𝑇 − d𝜏

𝜃∫(𝑤[𝑃𝜃 (𝑡) − 𝑝]+ − 𝜍)𝜉𝜃 (𝑡) d𝑡 + (1 − 𝜃) ∫𝑇 − d𝜏

(𝑤[𝑃𝜃′ (𝑡) − 𝑝]+ − 𝜍)𝜉𝜃′ (𝑡) d𝑡,

where 𝜃′ ∈ {𝜃𝐻 , 𝜃𝐿 } with 𝜃′ ≠ 𝜃. In this case, the H-type participates in the protest if and only∗ ≔ 𝜇𝑐 + (1 − 𝜇)𝑐 while the L-type does if and only if 𝑐 ≤ 𝑐∗ ≔ (1 − 𝜇)𝑐 +if 𝑐𝑖 ≤ 𝑐𝐻
𝐻
𝐿
𝑖
𝐻
𝐿
𝜇𝑐𝐿 . Then, as long as 𝑐𝐿 > 𝑐𝐻 , the cutoﬀ point is lower for the L-type consumers. Then, we
can apply a similar discussion to show the same statements.
Concerns for negative externalities In our model, a concern for the product value plays
an important role, but the concern for the negative externality itself does not. If the size of theconcern regarding the negative externalities is suﬃciently large, the participation rate of H- andL-type consumers becomes closer, which makes the protest uninformative. This observation
continues to hold even when concerns for the externality are distinct among consumers as theﬁrm does not care about the externality itself. In contrast, if the product quality includes guilty
feelings about consuming the product causing negative externality, it would be correlated to thesize of concerns for the negative externality itself. In this case, our conclusion remains true eventhough the concerns for negative externality are high.
Technical assumptions Our results depend on some technical assumptions: noise followed
by a normal distribution whose variance is high enough, and consumers’ signals are binary.
Considering a suﬃciently high noise is crucial for the committed price case. The ﬁrm beneﬁts
from informative equilibrium if the noise variance is small enough. The ﬁrm can save the ﬁxed
cost 𝐾 when 𝑣 = 0. In this case, if 𝑣 = 0, the ﬁrm discontinues the sale. If 𝑣 = 1, the optimal
price can be high but less than 𝜇 for the existence of an informative equilibrium. An interestingobservation is that if 𝜇 is small enough, the price becomes higher compared with the case where
the protest is absent, where the price is set so that both H- and L-type consumers buy. Due to theinformative protest, the ﬁrm can ignore the case that the product is unpopular. Another observation is that although the protest revealed almost complete information, the price is bounded
above by individual signal precision, 𝜇, to make the protest informative, by Proposition 1. This11

observation is important because it seems optimal for the ﬁrm to set a price close to 1 as theprotest reveals whether 𝑣 = 1 or 𝑣 = 0. In fact, if doing so, the H-type’s motivation becomesclose to the L-type’s, and the protest becomes uninformative. Then, even if the individual signal
precision is high, it does not necessarily beneﬁt the consumers, as it only increases the price.

Appendix
A. Proofs
A.1. Proof of Proposition 1
Before the proof of Proposition 1, we prepare the following lemma.
Lemma 4. Deﬁne 𝑃 be the larger solution to1−𝜇 1−𝑝𝑝 =𝜇𝐾𝜇− 𝑝−𝜅
𝐾
𝑝−𝜅 −(1−𝜇)with respect to 𝑝. Then,𝐾
𝑇𝐻 (𝑝) < 𝑇𝑓 (𝑝) if and only if 𝑝 > 𝑃 and 𝑇𝐿 (𝑝) < 𝑇𝑓 (𝑝) if and only if 𝑝 > 𝑃. Further, 1−𝜇+𝐾𝜅 > 𝑃 > 𝑃 > 𝜇 + 𝜅.
𝐾
𝐾
Proof of Lemma 4. (a) If 𝑝 ≥ 1−𝜇+ 𝜅, 𝑇𝑓 (𝑝) = ∞ by Lemma 2. If 𝑝 < 1−𝜇+ 𝜅, 𝑇𝑓 (𝑝) <𝑇𝐻 (𝑝) if and only if𝐾
𝜇− 𝑝−𝜅

𝜇 1−𝑝

𝐾𝑝−𝜅 −(1−𝜇)< 1−𝜇 𝑝 , which is equivalently 𝑝 > 𝑃. Further, a simple cal-𝐾
culation shows 1−𝜇+ 𝜅 > 𝑃.

𝐾(b) If 𝑝 < 𝐾
𝜇 , by Lemma 2, 𝑇𝑓 (𝑝) = −∞. Consider the case where 𝑝 ≥ 𝜇 . Note that 𝑇𝐿 (𝑝) <𝑇𝑓 (𝑝) if and only if𝐾
𝜇− 𝑝−𝜅

𝐾
𝑝−𝜅 −(1−𝜇)>1−𝜇 1−𝑝𝑝 , equivalently, 𝑔(𝑝) > 0, where 𝑔 is a convex quadratic𝜇function (illustrated in Figure 4). Then, 𝑇𝐿 (𝑝) < 𝑇𝑓 (𝑝) if and only if 𝑝 > 𝑃. The relation 𝑃 >𝐾
𝜇 + 𝜅 is shown by the fact that𝐾
𝜇− 𝑝−𝜅

𝐾
𝑝−𝜅 −(1−𝜇)>1−𝜇 1−𝑝𝐾
𝑝 fails if 𝑝 = 𝜇 + 𝜅 and larger 𝑝 satisﬁes𝜇𝑇𝑓 (𝑝) > 𝑇𝐻 (𝑝) > 𝑇𝐿 (𝑝).

Figure 4: Illustration of 𝑔∎
Proof of Proposition 1. (1) If 𝑝 ≥ 𝑃, as long as 𝑡 ≤ 𝑇𝐻 (𝑝), 𝑡 ≥ 𝑇𝑓 (𝑝). At 𝑡 > 𝑇𝐻 (𝑃), 𝐻 typedoes not purchase the product. Then, the ﬁrm gives up to sell the product. This implies that theﬁrm’s cutoﬀ point is 𝑇 = 𝑇𝐻 (𝑝). In this case, the equilibrium condition is12

𝜆0 − 𝜆1 = (2𝜇 − 1)(𝐹(𝑐𝐿 ) − 𝐹(𝑐𝐻 ))𝑐𝜃 = 𝜍𝜉𝜃 (𝑇𝐻 (𝑝))𝜇𝑝𝜆 + 𝜆1 𝜎2 ln( 1−𝜇 − ln( 1−𝑝 ))𝑇𝐻 (𝑝) = 0+22
𝜆0 − 𝜆1𝜉𝜃 (𝑇 ) =𝑇 (𝑝) − 𝜆0
𝑇 (𝑝) − 𝜆11
[𝜃𝜑( 𝐻) + (1 − 𝜃)𝜑( 𝐻
)].
𝜎
𝜎
𝜎
𝜇
𝑝
2 ln( 1−𝜇 − ln( 1−𝑝 ))Consider the case that 𝑝 ≥ 𝜇. Then, 𝜎2𝜆0 −𝜆1
ric and single-peaked around 0, we show that
𝑇 (𝑝)−𝜆0𝜑( 𝐻 𝜎≤ 0 when 𝜆0 ≥ 𝜆1 . As 𝜑 is symmet-

𝑇 (𝑝)−𝜆1) ≤ 𝜑( 𝐻 𝜎).

Note also that
2𝜇 − 1
𝑇 (𝑝)−𝜆𝑇 (𝑝)−𝜆⋅ [𝜑( 𝐻 𝜎 0 ) − 𝜑( 𝐻 𝜎 1 )] ≤ 0.
𝜎
Therefore, 𝜆0 ≥ 𝜆1 implies 𝑐𝐿 ≤ 𝑐𝐻 , which shows no existence of an informative equilibrium.
(2) Consider the case 𝑝 ∈ (𝑃, min{𝑃, 𝜇}). Then, if 𝜆0 > 𝜆1 , 𝑇𝐿 (𝑝) < 𝑇𝑓 (𝑝) < 𝑇𝐻 (𝑝) for any
𝑐𝐻 , 𝑐𝐿
Recall that the following equations characterize the equilibriums:
𝜆 ≔ 𝜆0 − 𝜆1 = (2𝜇 − 1)(𝐹(𝑐𝐿 ) − 𝐹(𝑐𝐻 )),𝑐𝐿 − 𝑐𝐻 = 𝜍 ⋅𝑐𝜃 = (𝜍 − 𝑤[𝑃𝜃 (𝑇𝑓 (𝑝)) − 𝑝]+ )𝜉𝜃 (𝑇𝑓 (𝑝)),𝐾𝜇− 𝑝−𝜅

ln( 𝐾
2

𝜆 𝜎
𝑇𝑓 (𝑝) − 𝜆1 = +2
2𝑝−𝜅 −(1−𝜇))𝜆𝐾
𝜇− 𝑝−𝜅

ln( 𝐾
2

𝜆 𝜎
𝑇𝑓 (𝑝) − 𝜆0 = − +22𝑝−𝜅 −(1−𝜇)𝜆𝜑(𝑃𝜃 (𝑇𝑓 (𝑝)) =𝜉𝜃 (𝑇𝑓 (𝑝)) =,),𝑇𝑓 (𝑝)−𝜆1)𝜃𝜎𝑇𝑓 (𝑝)−𝜆1
𝑇𝑓 (𝑝)−𝜆0𝜑( 𝜎 )𝜃 + 𝜑( 𝜎 )(1 − 𝜃),

𝑇𝑓 (𝑝)−𝜆1
𝑇𝑓 (𝑝)−𝜆01[𝜃𝜑( 𝜎 ) + (1 − 𝜃)𝜑( 𝜎 )].
𝜎This shows that 𝑐𝐿 and 𝑐𝐻 are depend on 𝜆, but not on 𝜆0 , 𝜆1 individually. Therefore, theequilibrium is a ﬁxed point of a self-map 𝜁 : 𝜆 ↦ (2𝜇 − 1)(𝐹(𝑐𝐿 ) − 𝐹(𝑐𝐻 )). We can easily
verify that 𝜁 is continuous.
By the deﬁnition of 𝑃 and 𝑃, 𝑃𝐻 (𝑇𝑓 (𝑝)) > 𝑝, and 𝑝 > 𝑃𝐿 (𝑇𝑓 (𝑝)). This implies that𝑐𝐻 = (𝜍 − 𝑤[𝑃𝐻 (𝑇𝑓 (𝑝)) − 𝑝])𝜉𝐻 (𝑇𝑓 (𝑝))𝑐𝐿 = 𝜍𝜉𝐿 (𝑇𝑓 (𝑝)) > 0.
Then, 𝑐𝐿 − 𝑐𝐻 = 𝜍(𝜉𝐿 (𝑇𝑓 (𝑝)) − 𝜉𝐻 (𝑇𝑓 (𝑝))) + 𝑤[𝑃𝐻 (𝑇𝑓 (𝑝)) − 𝑝]𝜉𝐻 (𝑇𝑓 (𝑝)) ≥ 0 for large
𝑤. Note that as 𝜆 → 0, 𝑇𝑓 (𝑝) → ∞, which implies that 𝑐𝐿 → 0 and 𝑐𝐻 → 0. Then, take 𝜆∗ > 0
suﬃciently small so that (2𝜇 − 1)(𝐹(𝑐𝐿 ) − 𝐹(𝑐𝐻 )) ≈ 𝑓 (0) ⋅ (𝑐𝐿 − 𝑐𝐻 ) ⋅ ℎ(𝜆∗ ) for some function ℎ. By taking large 𝑤 so that 𝑓 (0) ⋅ (𝑐𝐿 − 𝑐𝐻 ) ⋅ ℎ(𝜆∗ ) > 𝜆∗ . This implies that 𝜁(𝜆∗ ) =13

(2𝜇 − 1)(𝐹(𝑐𝐿 ) − 𝐹(𝑐𝐻 )) > 𝜆∗ . As 𝜁(𝜆) < 𝜆 for 𝜆 > 2𝜇 − 1, the intermediate value theorem
implies the existence of a ﬁxed point of 𝜁, and the ﬁxed point is a positive value. This showsthe existence of an informative equilibrium.
(3) Consider the case 𝑝 < 𝑃. This implies that the ﬁrm prefers to give up selling if 𝑡 > 𝑇𝐿 (𝑝).
If 𝑡 ≤ 𝑇𝐿 (𝑝), both H and L types purchase the product. This implies the ﬁrm’s proﬁt is 𝑝 − 𝜅.
If 𝑝 − 𝜅 > 𝐾, the give up point is 𝑇 = 𝑇𝐿 (𝑝). Then, by the deﬁnition of 𝑇𝐿 (𝑝),𝑐𝐻 = (𝜍 − 𝑤[𝑃𝐻 (𝑇𝐿 (𝑝)) − 𝑝])𝜉𝐻 (𝑇𝑓 (𝑝))𝑐𝐿 = 𝜍𝜉𝐿 (𝑇𝐿 (𝑝)) > 0.
A similar way to case (2) shows the existence of informative equilibrium.
If 𝑝 − 𝜅 ≤ 𝐾, the ﬁrm also prefers to give up at 𝑡 ≤ 𝑇𝐿 (𝑝) and therefore, 𝑇 = −∞. Then, theequilibrium becomes uninformative.
∎

A.2. Proof of Proposition 2
Before we prove Proposition 2, we provide the following lemma.
Lemma 5. 𝜇 > 𝑃 if and only if

𝜇−𝜅
> 𝐾.
2

The proof immediately follows from the deﬁnition of 𝑃.
𝜇−𝜅
Now we proceed to the proof of Proposition 2. Suppose that 2 > max{𝐾, 1 − 𝜇 − 𝜅}. This𝜇−𝜅implies that 𝜇 > 2−𝜅. Note that if 𝑝 = 𝜇, as 2 > 𝐾, 𝑝 > 𝑃 by Lemma 5. By Proposition 1,
3𝜇−𝜅
the equilibrium is uninformative and guarantees the proﬁt 2 − 𝐾.
Below, we examine the existence of an informative equilibrium that yields a proﬁt larger than𝜇−𝜅− 𝐾. As we consider an informative equilibrium, by Proposition 1, 𝑝 < 𝜇. First, consider2𝜇−𝜅
the case that 𝑝 < 𝑃. Note that the proﬁt is at most 𝑝 − 𝜅 − 𝐾. As we assume 2 > 1 − 𝜇 − 𝜅,𝑝 > 1 − 𝜇. This implies that1−𝑝1−𝜇ln() < 0.
𝑝𝜇Now we consider the limit 𝜎 → ∞. Note that as 𝜆0 − 𝜆1 = (2𝜇 − 1)(𝐹(𝑐𝐿 ) − 𝐹(𝑐𝐻 )) ∈[0, 2𝜇 − 1] ⊂ [0, 1].8 This shows that
𝑇𝐿 (𝑝) − 𝜆1 𝜎1−𝑝1−𝜇< ln() → −∞𝜎2𝑝
𝜇𝑇𝐿 (𝑝) − 𝜆01
𝜎1−𝑝1−𝜇<+ ln() → −∞
𝜎2𝜎 2𝑝𝜇Then, the ﬁrm’s proﬁt is at most ( 12 )(𝑝 − 𝜅) − 𝐾 at the limit. This is smaller thanas 𝑝 < 𝑃 < 𝜇.
Next, we consider the case that 𝑝 > 𝑃.

𝜇−𝜅
−𝐾
2

Claim 1. Suppose that 𝜇 > 𝑝 > max{ 12 , 𝑃}. Then, Π(𝑝) < [( 12 )(𝑝 − 𝜅) − 𝐾] for suﬃciently
large 𝜎.
Proof of Claim 1. Suppose that 𝑝 > 𝑃. Let 𝐷(𝜎) = Π(𝑝) − [( 12 )(𝑝 − 𝜅) − 𝐾], which is calculated as

Note that the value of 𝜎 can be taken independent of 𝑤 as the value of 𝑤 aﬀects only 𝜆. Here, 𝜆 is bounded by values
that are independent of 𝑤.
814𝑇 (𝑝) − 𝜆1
𝑇 (𝑝) − 𝜆1Pr(𝑣 = 1)[(Φ( 𝐻) − 1)(𝜇(𝑝 − 𝜅) − 𝐾) + Φ( 𝐿)(1 − 𝜇)(𝑝 − 𝜅)]𝜎𝜎
𝑇 (𝑝) − 𝜆0
𝑇 (𝑝) − 𝜆0
+ Pr(𝑣 = 0)[(Φ( 𝐻) − 1)((1 − 𝜇)(𝑝 − 𝜅) − 𝐾) + Φ( 𝐿
)𝜇(𝑝 − 𝜅)].
𝜎𝜎

Note that by 𝑝 ∈ (1 − 𝜇, 𝜇),
𝜎2
𝜆 𝜎2
𝜇
𝜇
𝑝
𝑝
𝑇𝐻 (𝑝) − 𝜆1 = +[ln( 1−𝜇 ) − ln( 1−𝑝 )] >[ln( 1−𝜇 ) − ln( 1−𝑝 )]
2 2𝜆
2
2
𝜎2
1
𝜆 𝜎
𝜇
𝜇
𝑝
𝑝
𝑇𝐻 (𝑝) − 𝜆0 = − +[ln( 1−𝜇 ) − ln( 1−𝑝 )] >[ln( 1−𝜇 ) − ln( 1−𝑝 )] −
2 2𝜆
2
2
2
2
𝜎
1
𝜆 𝜎
1−𝜇
1−𝜇
𝑝
𝑝
𝑇𝐿 (𝑝) − 𝜆1 = +[ln( 𝜇 ) − ln( 1−𝑝 )] <[ln( 𝜇 ) − ln( 1−𝑝 )] +
2 2𝜆
2
2
2
2
𝜎
𝜆 𝜎
1−𝜇
1−𝜇
𝑝
𝑝
𝑇𝐿 (𝑝) − 𝜆0 = − +[ln( 𝜇 ) − ln( 1−𝑝 )] <[ln( 𝜇 ) − ln( 1−𝑝 )].
2
2 2𝜆
By these equations, we can verify that as 𝜎 → ∞, 𝑇𝐻 (𝑝) − 𝜆𝑣 → ∞ and 𝑇𝐿 (𝑝) − 𝜆𝑣 → −∞.
𝑇 (𝑝)−𝜆𝑇 (𝑝)−𝜆Then, Φ( 𝐻 𝜎 𝑣 ) − 1 → 0 and Φ( 𝐿 𝜎 𝑣 ) → 0.9 Then, lim𝜎→∞ 𝐷(𝜎) = 0. Now we verify
the sign of 𝐷 for suﬃciently large 𝜎. To this end, we calculate 𝐷′ (𝜎), which is1𝜆1
𝜆𝐷′ (𝜎) = 𝜑(𝐴𝐻1 )( 𝐵𝐻 − 2 )(𝜇(𝑝 − 𝜅) − 𝐾) + 𝜑(𝐴𝐿1 )( 𝐵𝐿 − 2 )(1 − 𝜇)(𝑝 − 𝜅)
𝜆
𝜆
2𝜎
2𝜎
1
𝜆
1
𝜆
+𝜑(𝐴𝐻0 )( 𝐵𝐻 + 2 )((1 − 𝜇)(𝑝 − 𝜅) − 𝐾) + 𝜑(𝐴𝐿0 )( 𝐵𝐿 + 2 )𝜇(𝑝 − 𝜅),
𝜆
𝜆
2𝜎
2𝜎
where
𝑇𝜃 (𝑝) − 𝜆𝑣
, 𝑣 ∈ {0, 1},𝜎𝑝𝜃𝐵𝜃 = ln() − ln(
),
1−𝜃
1−𝑝

𝐴𝜃𝑣 =

and 𝐵𝐻 > 0 > 𝐵𝐿 as 𝜇 > 𝑝 > 12 > 1 − 𝜇.
Note that𝜑(𝐴𝐻1 )
1−𝜇
1−𝜇
𝑝
𝑝
𝜎
2
= exp([ 𝜆𝜎 − 𝜆 ln( 𝜇 )]𝜎 ln( 1−𝑝 )) > exp(−𝜎 ln( 𝜇 ) ln( 1−𝑝 )),𝜑(𝐴𝐿0 )
𝜑(𝐴𝐻0 )
1−𝜇
1−𝜇
𝑝
𝑝
𝜎
2
= exp(− 𝜆ln( 𝜇 )[ 𝜆𝜎 + 𝜎 ln( 1−𝑝 )]) > exp(−𝜎 ln( 𝜇 ) ln( 1−𝑝 )),𝜑(𝐴𝐿0 )𝜑(𝐴𝐿1 )
1−𝜇𝑝
= exp(ln( 1−𝑝 ) − ln( 𝜇 )),𝜑(𝐴𝐿0 )
𝜑(𝐴

)

𝜑(𝐴

)

Then, as 𝜇 > 12 and 𝑝 > 12 , 𝜑(𝐴𝐻1) → ∞, and 𝜑(𝐴𝐻0) → ∞ as 𝜎 → ∞. Note also that (𝜇(𝑝 −𝐿0𝐿0𝜅) − 𝐾) + ((1 − 𝜇)(𝑝 − 𝜅) − 𝐾) = 𝑝 − 𝜅 − 2𝐾 > 0. Using these facts, dividing 𝐷′ (𝜎) by
𝜑(𝐴𝐿0 )is positive for suﬃciently large 𝜎. As lim𝜎→∞ 𝐷(𝜎) = 0 and 𝐷′ (𝜎) > 0, 𝐷(𝜎) < 0 for
𝜆suﬃciently large 𝜎. Therefore, Π(𝑝) < [ 12 (𝑝 − 𝜅) − 𝐾] for suﬃciently large 𝜎.
∎
1Then, we conclude that for each 𝑝 < 𝜇, the proﬁt is at most 2 (𝑝 − 𝜅) − 𝐾 when 𝜎 is suﬃciently
large. Therefore, 𝑝 = 𝜇 is the optimal price, and the equilibrium is uninformative.
∎𝜇𝑝𝑝) − ln( 1−𝑝)| > |ln( 1−𝜇) − ln( 1−𝑝)|. This also implies that
Note that 𝜇 > 12 and 𝑝 > 12 implies that |ln( 1−𝜇𝜇𝑇 (𝑝)−𝜆𝑇 (𝑝)−𝜆Φ( 𝐻 𝜎 𝑣 ) converges more slowly than Φ( 𝐿 𝜎 𝑣 ). This intuition works in the following analysis.
915

A.3. Proof of Proposition 3
The proof consists of the following two claims, which hold under the assumption of Proposition 3.
Claim 2. If informative equilibrium does not exist, the expected proﬁt is at most 0.
Claim 3. By setting 𝑝 = 1 − 𝜇, the equilibrium is informative. The expected proﬁt is no lessthan (1 − 𝜇 − 𝜅 − 𝐾) 12 in this case.
Proof of Claim 2. By Proposition 1, the situation is either 𝑝 > min{𝜇, 𝑃} or 𝑝 < 𝜅 + 𝐾. If
𝑝 > 𝜇, as 𝑃𝐻 (𝑇 ) = 𝜇 < 𝑝 at the uninformative equilibrium, the proﬁt is 0 as the H type neverpurchases. If there is an uninformative equilibrium with 𝑝 ∈ (𝑃, 𝜇), as 𝑃𝐻 (𝑇 ) = 𝜇 > 𝑝 > 1 −𝜇−𝜅
𝜇, the proﬁt is 2 − 𝐾 < 0 as only the H type purchases.
At the uninformative equilibrium with 𝑝 < 𝜅 + 𝐾, the proﬁt is also 0 as the ﬁrm gives up
selling.
∎
Proof of Claim 3. First, consider the case where 1 − 𝜇 ≤ 𝑃. As 𝑝 = 1 − 𝜇 > 𝐾 + 𝜅, 1 − 𝜇 <𝜇 and 1 − 𝜇 < 𝑃 < 𝑃, an equilibrium exists and it is necessarily informative by Proposition 1.
Note that 𝑝 = 1 − 𝜇 ≤ 𝑃 implies 𝑇 = 𝑇𝐿 (𝑝). In this case, all consumer buys the product if
𝜆 +𝜆𝑡 < 𝑇 , and otherwise, never. Further, if 𝑝 = 1 − 𝜇, 𝑇𝐿 (𝑝) = 1𝜎 0 . Then, the expected proﬁt is111−𝜇−𝜅−𝐾[ Φ(𝑥) + Φ(−𝑥)](1 − 𝜇 − 𝜅 − 𝐾) =
,
2
2
2
𝜆 −𝜆

where 𝑥 = 02𝜎 1 .
Next, consider the case that 1 − 𝜇 > 𝑃. In this case, at 𝑡 = 𝑇 , only H type purchases the
product.
𝜇−𝜅
By assumption, 𝐾 > 2 implies that 𝑃 > 𝜇 > 1 − 𝜇, and as 1 − 𝜇 > 𝐾 + 𝜅, Proposition 1
implies that an equilibrium exists and it is necessarily informative at 𝑝 = 1 − 𝜇. Further, theexpected proﬁt is
1−𝜇−𝜅
× [𝜇Φ(𝑥 + 𝑧′ ) + (1 − 𝜇)Φ(𝑥) + (1 − 𝜇)Φ(−𝑥 + 𝑧′ ) + 𝜇Φ(−𝑥)]2−[𝜇Φ(𝑥 + 𝑧′ ) + (1 − 𝜇)Φ(−𝑥 + 𝑧′ )]𝐾(1)

𝐾
𝜇− 𝑝−𝜅
𝜎′where 𝑧 = 4𝑥 ln( 𝐾).
𝑝−𝜅 −(1−𝜇)𝜆 +𝜆1−𝜇−𝜅−𝐾
As 𝑇 > 𝑇𝐿 (𝑝) = 0 2 1 , the expected proﬁt (1) is greater than =
> 0.
2

∎
Proof of Proposition 3. By Claims 2 and 3, and assumption 1 − 𝜇 − 𝜅 > 𝐾, a price that
leads to an informative equilibrium yields a positive proﬁt, which is larger than the proﬁt of any
uninformative equilibrium. Then, the optimal price leads to an informative equilibrium.
∎

A.4. Proofs in Section 5
Proof of Lemma 3. First note that for suﬃciently small 𝑡, Π∗𝐿 (𝑡) > Π∗𝐻 (𝑡). This is because
for suﬃciently small 𝑡 ≈ −∞, 𝑃𝜃 (𝑡) ≈ 1 and therefore Π∗𝐿 (𝑡) ≈ 1 > 𝜇 ≈ Π∗𝐻 (𝑡).
𝜆 +𝜆
𝜆 −𝜆
Let 𝑎 = exp( 0𝜎2 1 (𝑡 − 0 2 1 )). Then, 𝑃𝜃 (𝑡) is written as𝑡−𝜆𝑃𝜃 (𝑡) =𝜑( 𝜎 1 )𝜃
𝑡−𝜆𝑡−𝜆𝜑( 𝜎 1 )𝜃 + 𝜑( 𝜎 0 )(1 − 𝜃)

16

=

1
1 + 1−𝜃𝑎𝜃,and also
1.
1+𝑎To simplify the notation, let denote 𝑃(𝑡) = Pr(𝑣 = 1 | 𝑡). By diﬀerentiation, if 𝜆0 > 𝜆1 , wehave
𝜆 −𝜆 1−𝜃2𝑃𝜃′ (𝑡) = − 0 2 1𝑎(𝑃𝜃 (𝑡)) < 0,𝜃𝜎Pr(𝑣 = 1 | 𝑡) =and each proﬁt has the following derivative, and the sign is negative as 𝜇 > 12 and 𝑃𝐻 (𝑡) >𝑃𝐿 (𝑡) > 𝜅.
𝑃 (𝑡)
1
1
Note that we can write 𝑃𝐻 (𝑡) = 1−𝜇and 𝑃𝐿 (𝑡) =. This implies that 𝑃𝐻(𝑡) and𝜇1+ 1−𝜇 𝑎

1+ 𝜇 𝑎𝑃(𝑡)are increasing in 𝑡.
𝑃𝐿 (𝑡)

𝐿

Now we calculate the ratio of the proﬁts:Π∗𝐻 (𝑡)𝑃 (𝑡)= (𝑃(𝑡)(2𝜇 − 1) + (1 − 𝜇)) 𝐻>1⟺∗
Π𝐿 (𝑡)𝑃𝐿 (𝑡)(2)
2

𝑍(𝑎) ≔ (1 − 𝜇)𝜇2 𝑎2 − (2𝜇 − 1)(1 − (1 − 𝜇)𝜇)𝑎 − 𝜇(1 − 𝜇) > 0
Note that the left-hand side is a convex quadratic function of 𝑎, and 𝑍(0) < 0. Also note that1−𝜇1−𝜇as 𝑃𝐿 (𝑡) > 𝐾, 𝑎 < [ 𝐾1 − 1] 𝜇 . A calculation shows that 𝑍([ 𝐾1 − 1] 𝜇 ) is decreasing in 𝜇,
1−𝜇and let 𝜇∗∗ be the solution that 𝑍([ 𝐾1 − 1] 𝜇 ) = 0. This implies that if 𝜇 < 𝜇∗∗ , Π∗𝐻 (𝑡) <Π∗𝐿 (𝑡) for each 𝑡. On the contrary, if 𝜇 > 𝜇∗∗ , there is 𝑡 ̂ < 𝑇𝐿 (𝐾) such that Π∗𝐻 (𝑡) < Π∗𝐿 (𝑡) if
and only if 𝑡 < 𝑡.̂(c) As 𝑇𝐿 (𝐾) is the solution of 𝑃𝐿 (𝑡) = 𝐾, That is,
𝜆 + 𝜆1𝜎2
1−𝜇𝑇𝐿 (𝐾) = 0+ln([ 𝐾1 − 1] 𝜇 )2𝜆0 − 𝜆11−𝜇

Let 𝜇∗ be the solution to ( 𝐾1 − 1) 𝜇 = 1. This exists if 𝐾 < 12 . Then, if 𝜇 < 𝜇∗ , 𝑇𝐿 (𝐾) →1−𝜇∗∞. In contrast, if 𝜇 > 𝜇∗ , 𝑇𝐿 (𝐾) → −∞. We can also verify that 𝑍([ 𝐾1 − 1] 𝜇∗ ) < 0, that is,𝑍(1) < 0. This implies that 𝜇∗ < 𝜇∗∗ .
∎𝜆0 −𝜆1𝜆1 +𝜆0
Proof of Proposition 4. Let 𝑎 = exp( 𝜎2 [𝑡 − 2 ]). Note that𝑃𝜃 =1
1 + 1−𝜃
𝑎
𝜃

𝜇
1−𝜇
− 𝜇
1−𝜇
𝑃𝐻 (𝑡) − 𝑃𝐿 (𝑡) =𝑎,𝜇1−𝜇[1 + 1−𝜇 𝑎][1 + 𝜇 𝑎]

𝜆0 − 𝜆1 1 − 𝜃
2𝑎[𝑃𝜃 (𝑡)] > 0.
𝜃𝜎22𝜇 − 1𝑡−𝜆𝑡−𝜆𝜉𝐿 (𝑡) − 𝜉𝐻 (𝑡) =[𝜑( 𝜎 0 ) − 𝜑( 𝜎 1 )]𝜎
2𝜇 − 1 𝑡−𝜆0
𝜆 −𝜆
𝜆 +𝜆
=
𝜑( 𝜎 )[1 − exp(− 0𝜎 1 [𝑡 − 0 2 1 ])]
𝜎
1
𝑡−𝜆
𝑡−𝜆
𝜉𝐻 (𝑡) = [𝜇𝜑( 𝜎 1 ) + (1 − 𝜇)𝜑( 𝜎 0 )].
𝜎
As shown in the proof of Lemma 3, Π∗𝐻 (𝑡) = Π∗𝐿 (𝑡) if and only if 𝑍(𝑎) = 0, which is deﬁned
𝜆 +𝜆
𝜆 −𝜆
in (2) . Therefore, when we write 𝑎 ̂ = exp( 0𝜎2 1 (𝑡 ̂ − 1 2 0 )), 𝑍(𝑎)̂ = 0. Note also that 𝑍 is−𝑃𝜃′ (𝑡) =17

independent of 𝑡 other than 𝑎. Then, even 𝜎 → ∞, 𝑎 ̂ is a ﬁnite value, which implies that 𝑡 ̂ ∈Θ(𝜎2 ).10 Further,̂𝜉𝐻 (𝑡)=
𝜉𝐻 (𝑡)̂

𝑡+𝑡−2𝜆𝜇 exp((𝑡 ̂ − 𝑡) 2𝜎2 1 ) + (1 − 𝜇) exp((𝑡 ̂ − 𝑡 + 𝜆0 − 𝜆1 )𝜇 + (1 − 𝜇) exp( 𝜎12 (𝜆0 − 𝜆1 )(𝑡 ̂ −̂
𝑡+𝑡−(𝜆0 +𝜆1 ))2𝜎2

𝜆0 +𝜆1
))
2

.

𝜉 (𝑡)

̂

As 𝜎𝑡2 converges to a real number, for each 𝑡 with |𝑡| < |𝑡|̂ + 𝑧 for some constant 𝑧, 𝜉𝐻 (𝑡)̂ → ∞.
𝐻
𝜉 (𝑡)In contrast, if 𝑡 < 𝑡,̂ 𝜉𝐻 (𝑡)̂ → 0 as 𝑡 ̂ → −∞. By Lemma 3, 𝑡 ̂ → −∞ as 𝜎 → ∞. Then, dividing
𝐻
𝐷 by 𝜉𝐻 (𝑡)̂ yields that𝑡̂𝑇∗𝜉 (𝑡) − 𝜉𝐻 (𝑡)′ (𝑡)] 𝜉𝐻 (𝑡) d𝑡 + [𝑃 (𝑡)̂ − 𝑃 (𝑡)].
∫ [−𝑃𝐿′ (𝑡)] 𝐿
d𝑡 − ∫ [−𝑃𝐻
(3)
𝐻
𝐿 ̂
𝜉𝐻 (𝑡)̂
𝜉𝐻 (𝑡)̂
𝑡̂
𝜇Note also that 𝜉𝐿 (𝑡) < 𝜉𝐻 (𝑡) for suﬃciently small 𝑡 when 𝜆0 > 𝜆1 . Also note that by 2 >𝐾, 𝑇 ∗ → ∞ as 𝜎 → ∞. Therefore, as 𝜎 → ∞, the value of (3) diverges to −∞. This shows that𝐷 < 0.
∎
Proof of Proposition 5. Dividing 𝐷̃ by 𝜉𝐻 (𝑇∗ ) yields that𝑇∫ ∗ [−𝑃𝐿′ (𝑡)](𝜉𝐿 (𝑡) − 𝜉𝐻 (𝑡)) d𝑡+ [𝑃𝐻 (𝑇∗ ) − 𝑃𝐿 (𝑇∗ )]𝜉𝐻 (𝑇∗ )(4)

As 𝜇 > 𝜇∗ , lim𝜎→∞ 𝑇∗ = −∞ by Lemma 3. Therefore, the ﬁrst term of (4) converges to𝜉 (𝑇 ) − 𝜉𝐻 (𝑇∗ )−𝑃𝐿′ (𝑇∗ ) 𝐿 ∗ ′.
𝜉𝐻 (𝑇∗ )Note that𝑇 −𝜆𝜉𝐿 (𝑇∗ ) − 𝜉𝐻 (𝑇∗ )=𝜉′𝐻 (𝑇∗ )

2

𝑇 −𝜆

2

−(2𝜇 − 1)(exp(− 12 ( ∗ 𝜎 0 ) ) − exp(− 12 ( ∗ 𝜎 1 ) ))
𝑇 −𝜆

𝑇 −𝜆

2

𝑇 −𝜆

𝑇 −𝜆

2

𝜇 ∗𝜎2 1 exp(− 12 ( ∗ 𝜎 1 ) ) + (1 − 𝜇) ∗𝜎2 0 exp(− 12 ( ∗ 𝜎 0 ) )𝜆 −𝜆

𝜆1 +𝜆0)))2= 𝑇 −𝜆
𝑇 −𝜆
𝜆 −𝜆
𝜆 +𝜆
𝜇 ∗𝜎2 1 + (1 − 𝜇) ∗𝜎2 0 exp( 0𝜎2 1 (𝑇∗ − 1 2 0 ))(2𝜇 − 1)(1 − exp( 0𝜎2 1 (𝑇∗ −𝑇

As shown in Lemma 3, 𝜎∗2 converges to a ﬁnite value. Then, the above value converges to aﬁnite value.
Note also that𝜆 −𝜆 𝜇1−𝑃𝐿′ (𝑡) = 0 2 1
×
> 0.
1−𝜇 1
𝜎
𝜇
𝜇 2
𝑎 + 2 1−𝜇 + ( 1−𝜇 ) 𝑎𝜆 −𝜆𝑎 = exp( 0𝜎2 1 (𝑡 −𝜆1 +𝜆0))2𝑇As 𝜎∗2 converges to a ﬁnite value, 𝑎 is also a ﬁnite value. Then, −𝑃𝐿′ (𝑇∗ ) → 0. Then, (4)reduced to [𝑃𝐻 (𝑇∗ ) − 𝑃𝐿 (𝑇∗ )], which is calculated as
𝜇
1−𝜇
− 𝜇
1−𝜇
𝑃𝐻 (𝑡) − 𝑃𝐿 (𝑡) =𝑎,𝜇1−𝜇(1 + 1−𝜇 𝑎)(1 + 𝜇 𝑎)
10

𝑔(𝑡)

Θ is Landau symbol; that is 𝑔(𝑡) ∈ Θ(𝑓(𝑡)) implies that lim𝑡→∞ 𝑓(𝑡) ∈ ℝ \ {0}18

As 𝑎 is a ﬁnite value, lim𝜎2→∞ 𝑃𝐻 (𝑇∗ ) − 𝑃𝐿 (𝑇∗ ) > 0. This concludes that 𝐷̃ > 0.

∎

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