DP2016-17

Welfare Effects of Endogenou s
Informa tion Acqu isition andDisclosure in Duopoly Ma rkets
Kazunori MIWA
March 30, 2016

Welfare Eﬀects of Endogenous
Information Acquisition and Disclosurein Duopoly Markets
Kazunori Miwa†
March 2016†

Research Institute for Economics and Business Administration (RIEB), Kobe University, 2-1 Rokkodai, Nada, Kobe 657-8501, Japan. Email: miwa@rieb.kobe-u.ac.jp

1

Welfare Eﬀects of Endogenous Information Acquisition andDisclosure in Duopoly Markets

Abstract: This paper investigates the interaction between firms’ information acquisition decisions and disclosure of internally acquired information
in a Cournot duopoly market. The results are as follows. Given that the
precision of firms’ private information is constant, mandatory disclosure of
information about the industry-wide demand uncertainty can enhance social
welfare. However, when the precision of firms’ private information is endogenously determined, mandatory disclosure is not always desirable. This isbecause when disclosure is mandated, firms acquire less precise informationcompared to the case where acquired information is not disclosed, and hence
their internal information environments are deteriorated. This can lead to“unintended consequences,” such that disclosure regulation strictly decreasessocial welfare on the whole.
Keywords: Information Acquisition; Disclosure; Duopoly; Social Welfare
JEL Classification: L13; M41; M48

2

1 Introduction
Accounting researchers recognize that management accounting systems (internal reporting) and financial accounting systems (external reporting) are closely linked (e.g.,
Hemmer and Labro, 2008; Taipaleenmäki and Ikäheimo, 2013; Zimmerman, 2014). However, as pointed out by Hemmer and Labro (2008), little is known about the interaction
between them. Basically, external reporting or, more broadly, information disclosure,is monitored by regulatory agencies such as the Securities and Exchange Commission(SEC), Financial Accounting Standards Board (FASB), and International AccountingStandards Board (IASB), mainly in an attempt to protect decision makers in the capital market. One indication of this is that if external reporting (disclosure) is closely
tied to management accounting (internal information) systems, then exogenously forcedchanges in disclosure rules might impact not only the information disclosed to outsideusers but also the information needed for firms’ internal use. If so, regulatory agencies
should consider the eﬀect of disclosure regulation on firms’ internal information environments and the resulting market outcomes in order to achieve the intended goal of
the regulation. In fact, several recent studies suggest that exogenous changes in disclosure regulation also alter firms’ internal information environments and aﬀect subsequent
managerial decisions (e.g., Graham, Hanlon, and Shevlin, 2011; Shroﬀ, 2014; Shroﬀ,
Verdi, and Yu, 2014).
This paper adds to the theoretical insights in the above discussion by investigating
the interaction between firms’ information acquisition decisions and disclosure of theinternally acquired information. Formally, this paper builds upon a standard duopoly
model with uncertainty. It is well known that product market competition aﬀects firms’
disclosure decisions (e.g., Vives, 1984; Gal-Or, 1985; Darrough, 1993; Raith, 1996; Suijs
and Wielhouwer, 2014). This paper extends the model by endogenizing firms’ internal
information acquisition decisions. In other words, a firm can choose the precision ofprivate information before disclosing it. Under this setting, this paper attempts toanswer the following questions. How does disclosure aﬀect firms’ information acquisition
decisions? How does the interaction between information acquisition and disclosure3

aﬀect social welfare? By answering these questions, the goal of this paper is to improve
our understanding of how regulatory arrangements for external reporting aﬀect firms’internal information environments and to evaluate the eﬀectiveness of the disclosure
regulation.
Duopolistic settings are appropriate for investigating the interaction between information acquisition and disclosure.1 Once privately acquired information is disclosed, arival firm can also observe and strategically use it. If the disclosed information is also
useful for the rival firm and improves its decision, then the disclosing firm might suﬀer acompetitive disadvantage compared to the case in which information is withheld. Thisimplies that disclosure will decrease the benefit obtained from acquiring costly information through the management accounting system. In addition, if a firm can free ride tosome extent on the information disclosed by the other firm, then the free-riding firm’s
incentive to acquire costly private information through the management accounting system might diminish. Conversely, if disclosure can change the rival’s behavior so thatthe disclosing firm benefits from revealing its private information, the firm might invest
more in its internal information system and disclose the acquired private information.
In this manner, it is possible that disclosure aﬀects the potential costs or benefits of
information acquisition depending on how the rival firm strategically responds to thedisclosed information.
Notably, this paper focuses on the setting where two firms compete in quantities, that
is, Cournot competition under industry-wide demand uncertainty. This is because previous analytical studies show that although Cournot duopolists commit to nondisclosureof private information related to industry-wide uncertainty, mandatory disclosure canenhance social welfare (e.g., Darrough, 1993; Suijs and Wielhouwer, 2014). This paper
considers whether the regulatory implications suggested by the previous studies are valid
when firms’ private information is endogenously determined.
The main results are as follows. Given that the precision of firms’ private informationis constant, disclosure of information on industry-wide demand uncertainty can enhance
1

In the context of the capital market, Pae (1999) analyzes the problem of information acquisition anddisclosure.

4

social welfare. Because firms have no incentives to voluntarily disclose their private
information ex ante in this case, this implies that disclosure regulation is eﬀective andsocially desirable. This result is the same as that of prior studies and indicates that
disclosure per se has a positive impact on social welfare (e.g., Suijs and Wielhouwer,2014). However, in the setting where firms’ private information is endogenously determined, mandatory disclosure is not always desirable. This is because disclosure has
a negative eﬀect on firms’ incentives for acquiring private information through the internal information system. That is, when disclosure is mandated, firms acquire lessprecise information compared to the case where acquired information is not disclosed,
and hence, their internal information environments are deteriorated. This leads to ineﬃcient production decisions, which in turn, have a negative impact on social welfare.
Taken together, mandating disclosure of privately acquired information has two eﬀects
on social welfare: (i) the direct positive eﬀect, which increases social welfare as shownin the prior studies, and (ii) the negative indirect eﬀect, which decreases social welfare
through the deterioration of the internal information environment of the firm (Ganuza
and Jansen, 2013). In fact, there exists a case where the negative indirect eﬀect dominates the positive direct eﬀect, that is, mandatory disclosure strictly decreases social
welfare on the whole. This result is in sharp contrast to the findings of prior studies andsuggests that disclosure regulation might fail to achieve the intended goal of enhancing
social welfare.
This paper contributes to the literature on disclosure in duopoly markets by examining the interaction between firms’ information acquisition and disclosure decisions.
Numerous analytical studies examine firms’ disclosure behavior in the presence of product market competition.2 However, most of the existing literature takes firms’ private
information as exogenous, and the interaction between firms’ information acquisitionand disclosure decisions is ignored. Notable exceptions are Kirby (2004), Jansen (2008),
2

See, for example, Vives (1984), Gal-Or (1985), Kirby (1988), Vives (1990), Darrough (1993), Sankar(1995), Raith (1996), Clinch and Verrecchia (1997), Pae (2000), Pae (2002), Arya and Mittendorf(2007), Hughes and Williams (2008), Arya, Frimor, and Mittendorf (2010), Bagnoli and Watts(2010), Corona and Nan (2013), Suijs and Wielhower (2014), Bagnoli and Watts (2015), and Hughes
and Pae (2015).

5

and Ganuza and Jansen (2013). They simultaneously consider the problem of information acquisition and disclosure in a duopoly/an oligopoly. Kirby (2004) analyzes astochastic oligopoly model and shows that mandatory disclosure requirements can create
a potential opportunity loss for firms. Kirby (2004) assumes that firms act cooperatively
in their choices of information acquisition and disclosure. Jansen (2008) also analyzes
a stochastic duopoly model, but he focuses on strategic disclosure, that is, firms decide
whether to disclose their private information or not after receiving the information. This
paper is most closely related to that of Ganuza and Jansen (2013). They mainly analyze a Cournot-type duopoly model under firm-specific cost uncertainty, and show that
disclosure increases firms’ incentives to acquire private information.
Focusing on the eﬀectiveness of the disclosure regulation, I provide additional insights
by identifying the eﬀects that disclosure has on firms’ internal information acquisition
decisions. Prior studies show that although Cournot duopolists prefer to not discloseindustry-wide demand information ex ante, mandatory disclosure requirements can enhance social welfare. This paper argues that it is unclear whether mandatory disclosure
can surely enhance social welfare. This is because mandatory disclosure might have anegative eﬀect on firms’ incentives to acquire costly private information. In fact, thispaper identifies an important condition, namely, that social welfare is strictly reducedby mandatory disclosure.
This paper also has implications for disclosure regulation. Recent disclosure requirements make an attempt to expand firms’ transparency and align their external andinternal information environments. For example, Management Discussion & Analysis(MD&A) disclosure requires firms to provide information that enables outside investors
to “see the firms through the eyes of management” (SEC, 2003). As a second example, the Statement of Financial Accounting Standards (SFAS) 131 and International
Financial Reporting Standard (IFRS) 8 adopt the so-called “management approach” insegment reporting and require externally reported segments to be defined based on thefirm’s internal reporting structure. However, the analysis suggests that expanding firms’transparency through aligning externally disclosed information with that for firms’ inter-

6nal use is not necessarily desirable. In particular, it is possible that mandatory disclosureof firms’ private information has a negative impact on the incentives for acquiring information for firms’ internal use, and ultimately deteriorates firms’ internal information
environments. This might lead to ineﬃcient managerial decisions and decline of social
welfare.
The remainder of this paper is organized as follows. Section 2 describes the model
setup. In section 3, I analyze the model and provide the equilibrium strategies of thefirms. In section 4, I examine the eﬀect on social welfare and discuss the desirability of
disclosure regulation. Section 5 summarizes the paper. All proofs are presented in the
appendix.

2 ModelConsider a single-period product market where two firms (i, j = 1, 2, i ̸= j) compete inquantities, that is, Cournot duopoly market.3 The utility function of the representative
consumer is assumed as follows.
U (q1 , q2 ) = α(q1 + q2 ) −β 2(q1 + q22 + 2γq1 q2 ),2(1)

where α and β are parameters, qi ≥ 0 is the quantity of firm i’s product, and 0 <γ < β is the degree of substitution between firm 1’s and firm 2’s products. Note that
U is quadratic, (strictly) concave, and symmetric in q1 and q2 (Vives, 1984). The∑
representative consumer maximizes U (q1 , q2 ) − 2i=1 pi qi , where pi is the price of firm i’s
product. Firm i has a constant marginal cost, denoted by ci . Without loss of generality,
I assume that β = 1 (hence, γ ∈ (0, 1)) and ci = 0.
The first-order condition of the consumer’s maximization problem gives the following

3

The basic structure and notations of the model are based on Darrough (1993), Arya and Mittendorf(2007), and Suijs and Wielhouwer (2014).

7

inverse demand function:pi = α − qi − γqj .

(2)I introduce uncertainty into the demand intercept.4 More specifically, following Darrough (1993) and Suijs and Wielhouwer (2014), I assume thatα = ᾱ + ∆α1 + ∆α2 ,(3)where ᾱ is a constant, and ∆α1 and ∆α2 are identically, independently, and normally
distributed with mean zero and variance s > 0.
Each firm i acquires a costly and noisy signal about ∆αi . The signal is denoted by ξi
and is expressed as
ξi = ∆αi + εi ,(4)where εi is normally distributed with mean zero and variance ei > 0.5 The informationacquisition cost, denoted by Ki , is assumed to be linear in 1/ei . Equivalently, Ki can be
written as
Ki (ηi ) =kηi,s(1 − ηi )(5)

where ηi = s/(s + ei ) and k > 0 is a constant.6 Given that E[∆αi |ξi ] = ηi ξi , ηi represents
the sensitivity of the conditional expectation to the realized value of signal ξi . If ηi islow, signal ξi is not so valuable for firm i in the sense that the conditional expectation
4

Most theoretical studies that analyze firms’ disclosure decisions in duopoly/oligopoly markets consider
two types of uncertainty: demand uncertainty and cost uncertainty. In other words, firms haveprivate information about market demand or their own production costs. In Cournot competition,
however, the key distinction is between industry-wide versus firm-specific information, not between
demand versus cost information (Christensen and Feltham, 2003, Ch.15).
5
Following previous studies, despite the normality assumption about ∆αi and εi , I ignore the possibilityof negative quantities. See, for example, Vives (1984, p.77, footnote 2), Darrough (1993, p.541,
footnote 15), and Suijs and Wielhouwer (2014, p.232. footnote 8).
6This expression of the information acquisition cost is the same as that used by Hauk and Hurkens(2001).

8remains largely unaltered from the expected value of the prior. In contrast, if ηi is high,firm i relies more on the signal to update the prior belief. ηi ∈ [0, 1) because ei > 0.
Hereafter, I refer to ηi as the precision of the signal and work with ηi .7
Each firm i decides whether to disclose the acquired signal ξi . In order to make theanalysis tractable, firm i’s disclosure policy is represented by θi ∈ [0, 1]. In particular,
firm i chooses probability θi : signal ξi acquired by firm i is truthfully and publicly
disclosed with probability θi , and concealed with probability 1 − θi .8
The timeline of the model is as follows. First, each firm i chooses its own disclosure
policy θi . Second, each firm i chooses the precision of the signal. As stated above, acquiring signal ξi , the precision of which is ηi , costs Ki (ηi ). Then, each firm i receives signal
ξi , which is publicly disclosed or concealed according to the precommitted disclosure
policy θi . After observing the signal, each firm i simultaneously chooses the quantity of
product qi . Note that signal ξj of firm j is also observed by firm i, if disclosed. Finally,
products are sold on the market and each firm i earns the profit, denoted by πi . In
terms of the amount, the final payoﬀ for firm i, denoted by Πi , is calculated by subtracting the information acquisition cost Ki (ηi ) from the profit earned on sales πi , thatis, Πi = πi − Ki (ηi ).
In the sequence of events, both firms are assumed to commit themselves to disclosure
policy before they acquire and observe the signal. Suijs and Wielhouwer (2014) pointout that the precommitment assumption is justified in several settings. For example,
it is likely that firms cannot strategically reconsider the disclosure policy on a case-bycase basis in settings where disclosure requires considerable changes in the information
technology (IT) infrastructure or governance structure. In addition, the analysis of thisex ante setting makes it possible to examine the regulatory implications clearly.
7

This definition about the precision of the signal is consistent with that of Ganuza and Jansen (2013).
The information criterion of Ganuza and Jansen (2013), called Integral Precision, is based on “theprinciple that an information structure ... is more informative (more precise) than another if it
generates more dispersed conditional expectations”(p.850).
8
Ganuza and Jansen (2013) also use this formulation for the disclosure policy of the firm for thesame reason as I do. See, Ganuza and Jansen (2013, p.850, footnote 9). As we will see in thefollowing sections, corner solutions (i.e., θi = 0 or 1) arise according to the types of information andcompetition in voluntary disclosure settings. That is, in equilibrium, the signal acquired by eachfirm i is fully disclosed or completely concealed.

9I assume that both firms are risk neutral and maximize their own expected payoﬀs.
In the next section, I will solve the model backward.

3 Equilibrium Strategies
3.1 Quantities of Products
Each firm chooses the quantity of product in order to maximize the expected profit conditional on available information.9 There are three possibilities: (i) both firms disclose,(ii) only one of the two firms discloses, and (iii) neither firm discloses.
3.1.1 If both firms disclose
Consider case (i): both firms disclose. In this case, both firms can condition theirproduction decisions on both signal values. Hence, firm i’s quantity can be denoted byqi (ξi , ξj ) and firm i chooses its quantity in order to maximize the expected profit.10
max E[(ᾱ + ∆αi + ∆αj − qi (ξi , ξj ) − γqj (ξj ,ξi ))qi (ξi , ξj )|ξi , ξj ],qi (ξi ,ξj )i, j = 1, 2,i ̸= j.

(6)

The first-order condition of the maximization problem of (6) gives the following best
response function for firm i.
E[ᾱ + ∆αi + ∆αj − γqj (ξj , ξi ) − 2qi (ξi , ξj )|ξi , ξj ] = 0⇒1
qi = (ᾱ + E[∆αi |ξi ] + E[∆αj |ξj ] − γE[qj (ξi , ξj )|ξi , ξj ]]).
2

(7)

Note that if we substitute the above first-order condition into the objective function (6),the expected profit of firm i can be calculated as the square of its quantity. The best

910

Note that the information acquisition cost is sunk at this stage.
It is assumed that a firm cannot add noise or bias to its private signal when disclosing it. In addition,
it is assumed that a firm knows the precision of a rival firm’s signal if it is disclosed.

10response function of each firm yields the following optimal quantity choice.
qidd (ξi , ξj ) =1(ᾱ + ηi ξi ).
2+γ

(8)

As superscript d denotes disclosure, the two superscripts dd in (8) denote disclosure byfirm i and firm j, respectively. Similarly, I use the superscript ϕ to denote nondisclosure. The ex ante expected profit of firm i can be calculated using the law of iterated
expectations and the fact that the expected profit of firm i is the square of its quantity.
E[πidd ] = (E[qidd (ξi , ξj )])2 + Var[qidd (ξi , ξj )]=1(ᾱ2 + sηi + sηj ).
2(2 + γ)(9)3.1.2 If only one firm discloses
Next, consider case (ii): only one of the two firms discloses. I assume that firm i discloses
its signal and firm j does not. Then, firm i’s quantity is only conditional on its own
signal whereas firm j can choose its quantity using both signals. The maximization
problem for each firm is as follows.
max E[(ᾱ + ∆αi + ∆αj − qi (ξi ) − γqj (ξj , ξi ))qi (ξi )|ξi ],(10)

max E[(ᾱ + ∆αi + ∆αj − qj (ξj , ξi ) − γqi (ξi ))qj (ξj , ξi )|ξi , ξj ].

(11)

qi (ξi )qj (ξj ,ξi )By applying the same procedure as in case (i), solving the problems denoted by (10)and (11) yields the following optimal quantity choice and the ex ante expected profit of11each firm.
1(ᾱ + ηi ξi ),
2+γ
1
2+γ
qjϕd =(ᾱ +ηj ξj + ηi ξi ),2+γ21
E[πidϕ ] =(ᾱ2 + sηi ),(2 + γ)2(2 + γ)212(ᾱ+E[πjϕd ] =sηj + sηi ).
(2 + γ)24
qidϕ =

(12)
(13)
(14)
(15)

3.1.3 If neither firm discloses
Finally, consider case (iii): neither firm discloses. Each firm chooses its quantity conditional on its own signal because the rival firm’s signal cannot be observed.
max E[(ᾱ + ∆αi + ∆αj − qi (ξi ) − γqj (ξj ))qi (ξi )|ξi ],qi (ξi )i, j = 1, 2,i ̸= j.

(16)

Solving the problem denoted by (16) for each firm yields the following optimal quantity
choice and the ex ante expected profit of firm i.
2+γ
1
(ᾱ +
ηi ξi ),2+γ2
1
(2 + γ)22E[πiϕϕ ] =(ᾱ+sηi ).
(2 + γ)24

qiϕϕ =(17)(18)

Combining the above three cases shows that the ex ante expected profit of firm i when
it chooses the optimal disclosure policy can be expressed as follows.
E[πi ] =θi (θj E[πidd ] + (1 − θj )E[πidϕ ])+ (1 − θi )(θj E[πiϕd ] + (1 − θj )E[πiϕϕ ]).

(19)

Firm i chooses the precision of the signal and disclosure policy in order to maximize theexpected profit in (19), taking into account the information acquisition cost.

123.2 Information Acquisition and Disclosure
Now, consider the optimal information acquisition and disclosure policy for each firm.
In the model, the firm’s information acquisition decision is treated as its choice of signal
precision. The maximization problem of firm i is as follows.
max E[Πi ] = E[πi ] − Ki (ηi ).

(20)

ηi

Assuming that the optimal signal precision, denoted by ηi∗ , is determined by the interior
solution, the following first-order condition is satisfied.11
θi

s
s
k
+
(1
−
θ
)
−
= 0.
i
(2 + γ)24 s(1 − ηi∗ )2(21)

From the first-order condition in (21), the disclosure policy of the rival firm, θj , does not
aﬀect firm i’s information acquisition choice. One can obtain the following proposition
about the interaction between information acquisition decision and disclosure policy byapplying the implicit function theorem.
Proposition 1. The optimal signal precision of firm i, ηi∗ , is negatively aﬀected by its
own disclosure policy, θi , and is independent of the rival firm’s disclosure policy, θj .
Proposition 1 shows that disclosure has a negative impact on firms’ activities to acquire
the disclosed information. In other words, firms acquire more (less) precise information
when the acquired information is concealed (disclosed).
Again using the implicit function theorem, the following comparative statics results
can be readily obtained.
Corollary 1. The optimal signal precision of firm i, ηi∗ , is1. negatively aﬀected by the parameter of information acquisition cost, k,
2. positively aﬀected by the magnitude of uncertainty, s, and11

Note that the second derivative of the objective function is −2k/s(1 − ηi )3 ; thus, the second-ordercondition is satisfied because ηi ∈ [0, 1).

13

3. negatively aﬀected by the degree of substitution between its own and the rival’s
products, γ, only when θi > 0.
The first and second results in Corollary 1 are intuitive. When each firm chooses
signal precision, it must face a trade-oﬀ between the costs and benefits of acquiring
the signal. Therefore, it is obvious that the lower the information acquisition cost, themore precise the signal acquired by the firm. In a similar way, the firm can obtain more
benefit from the acquired signal under large uncertainty because the signal is helpful tomitigate the uncertainty and improve the firm’s production decision. Thus, the larger
the uncertainty, the more precise the signal acquired by the firm. The third resultin Corollary 1 is related to the disclosure policy. In fact, when firm i precommits tonondisclosure strategy, that is, θi = 0, the degree of substitution between the two firms’
products has no eﬀect on the optimal signal precision.
Given that ηi = ηi∗ , firm i chooses the optimal disclosure policy.
max E[Πi (ηi∗ )] = E[πi (ηi∗ )] − Ki (ηi∗ ).

(22)

θi

It is easy to show that the objective function in (22) is decreasing in θi when ηi∗ isdetermined by the interior solution. By contrast, when ηi∗ = 0, firm i’s expected profit
does not depend on θi . Thus, the following proposition is derived.
Proposition 2. If ηi∗ ∈ (0, 1), the optimal disclosure policy of firm i is θi∗ = 0, that is,nondisclosure. If ηi∗ = 0, any disclosure policy is indiﬀerent for firm i.
This result is consistent with the findings of earlier studies, which demonstrate that
Cournot duopolists prefer not to disclose industry-wide information ex ante (e.g., Darrough, 1993; Suijs and Wielhouwer, 2014).
The intuitions of Propositions 1 and 2 are explained as follows. Suppose that η ∗ ∈(0, 1) and firm i obtains signal ξi > 0. This means that the market demand is likely to
be larger than the prior expectation. Thus, firm i responds to the signal to increase its
product quantity. The inverse is true if the signal value is negative, that is, ξi < 0: firm
i decreases its product quantity.

14

If firm i discloses signal ξi > 0, then the rival firm j can observe it and also increase
its quantity. Because the quantities are strategic substitutes in a standard Cournot
competition, this means that firm i increases its quantity but the increase in the quantity
is less than that in the nondisclosure case.12 Similarly, if firm i obtains and discloses ξi <0, firm i decreases its quantity but the quantity is more than that in the nondisclosurecase because firm j also decreases its quantity by observing ξi < 0. This implies that
if firm i discloses its private information, then the variance of its quantity decreases.
However, the profit of firm i is convex in its quantity; the ex ante expected profit of firmi becomes larger as the variance increases. Taken together, disclosure decreases the exante expected profit by decreasing the variance of the quantity. Hence, each firm has no
incentive to voluntarily disclose its private information.
Now, consider the decisions about information acquisition. As stated above, the more
precise the signal of firm i, the more heavily firm i relies on the signal and adjusts its
quantity. This increases the variance of the quantity, from the ex ante perspective, andthus increases the ex ante profit of firm i. However, this benefit from acquiring a more
precise signal decreases if the acquired signal is disclosed because the rival firm j canalso use the disclosed signal and adjust its quantity in the same direction. Thus, thevariance of the quantity and the ex ante profit of firm i decreases. This is why disclosure
has a negative eﬀect on firms’ information acquisition decisions.

4 Welfare Analysis
In this section, I examine how the interaction between information acquisition anddisclosure aﬀects social welfare. Following Suijs and Wielhouwer (2014), I consider
consumer surplus and total surplus as measures of social welfare. The utility function of the representative consumer is given by (1). I define consumer surplus as∑
CS = U (q1 , q2 ) − 2i=1 pi qi and total surplus as TS = CS + Π1 + Π2 . By substituting theinverse demand function given by (2) into the expression of the consumer surplus and
12See, Bulow, Geanakoplos, and Klemperer (1985) about strategic substitutes and complements.

15

taking expectations, the expected consumer surplus can be expressed as follows.13
ECS =θi (θj ECSdd + (1 − θj )ECSdϕ )+ (1 − θi )(θj ECSϕd + (1 − θj )ECSϕϕ ),

(23)

where
1+γ
(ᾱ2 + sηi + sηj ),(2 + γ)21+γ(2 + γ)22
ECSdϕ =(ᾱ+sη+sηj ),i(2 + γ)28(1 + γ)1+γ(2 + γ)22
ECSϕd =(ᾱ+sηi + sηj ),(2 + γ)28(1 + γ)
(2 + γ)2(2 + γ)2
1+γ
2
(ᾱ
+
sη
+
sηj ).
ECSϕϕ =i
(2 + γ)28(1 + γ)8(1 + γ)ECSdd =

(24)
(25)
(26)
(27)

As in the former section, superscript d and ϕ mean disclosure and nondisclosure, respectively. Similarly, the expected total surplus is given by
ETS =θi (θj ETSdd + (1 − θj )ETSdϕ )+ (1 − θi )(θj ETSϕd + (1 − θj )ETSϕϕ ),

(28)

ETS

dd

2
∑
3+γ
2
=
(ᾱ + sηi + sηj ) −Ki ,(2 + γ)2
i=1

(29)

ETS

dϕ

2
∑
3+γ
3(2 + γ)22=(ᾱ + sηi +sηj ) −
Ki ,(2 + γ)28(3 + γ)
i=1

(30)

ETS

ϕd

2
∑
3(2 + γ)2
3+γ
2
(ᾱ +
sηi + sηj ) −Ki ,=(2 + γ)28(3 + γ)
i=1

(31)

ETS

ϕϕ

2
∑
3+γ
3(2 + γ)23(2 + γ)2
2
=
(ᾱ +
sηi +
sηj ) −
Ki .
(2 + γ)28(3 + γ)8(3 + γ)
i=1

(32)

where

13

The derivations of the expected consumer surplus are presented in the appendix.

16In preparation for the following analysis, first consider the eﬀect of private signal precision on social welfare. Diﬀerentiating the expected consumer surplus and the expected
total surplus with respect to firm i’s signal precision gives the following lemma.
Lemma 1. Both the expected consumer surplus and the expected total surplus increase
with firm i’s signal precision, ηi .
Now I examine the eﬀect of disclosure on social welfare. As a benchmark, consider thecase where the precision of each firm’s private signal is constant, namely, η1 = η2 = η̄.
An easy calculation leads to the following lemma.
Lemma 2. If the precision of each firm’s private information is constant, that is, η1 =
η2 = η̄, both the expected consumer surplus and the expected total surplus are (weakly)
increasing functions of firm i’s disclosure policy, θi .
Notably, excepting the trivial case where firms’ private signals have no information
content, that is, η̄ = 0, disclosure enhances both the expected consumer surplus and the
expected total surplus. This indicates that disclosure per se has a positive impact onsocial welfare, which is consistent with the result of Suijs and Wielhouwer (2014).
However, the situation changes when the precision of each firm’s private informationis endogenously determined. In this case, following Ganuza and Jansen (2013), the eﬀect
of disclosure on the expected consumer surplus can be decomposed as follows.
∂ECS ∂ECS ∂ηi∗ ∂ECS ∂ηj∗
∂ECS
=
+
+
.
∂θi
∂θi
∂ηi ∂θi∂ηj ∂θi(33)

The first term of (33) exhibits the direct eﬀect of disclosure on the expected consumer
surplus. Assuming that η ∗ is determined by the interior solution, this direct eﬀect ispositive, that is, ∂ECS/∂θi > 0 as in Lemma 2. On the other hand, the second term of(33) expresses the indirect eﬀect in the sense that this captures the eﬀect of disclosureon the expected consumer surplus through the change in firm’s information acquisition
decision caused by disclosure. As shown in Lemma 1, the expected consumer surplus
is an increasing function of firm i’s signal precision, that is, ∂ECS/∂ηi > 0. However,17

Proposition 1 shows that disclosure negatively aﬀects firm i’s optimal signal precision,
namely, ∂ηi∗ /∂θi > 0. Thus, as a whole, the indirect eﬀect becomes negative. The third
term of (33) is zero because ∂ηj∗ /∂θi = 0, as shown in Proposition 1. Taken together,disclosure has two eﬀects on the expected consumer surplus: the positive direct eﬀect
and the negative indirect eﬀect.
The eﬀect of disclosure on the expected total surplus can also be decomposed asfollows.
∂ETS∂ETS ∂ETS ∂ηi∗ ∂ETS ∂ηj∗
=
+
+
.
∂θi
∂θi
∂ηi ∂θi∂ηj ∂θi(34)

By applying the same procedure, one can immediately show the following. First, assuming that η ∗ is determined by the interior solution, disclosure has a positive direct eﬀect
on the expected total surplus because ∂ETS/∂θi > 0. Second, however, disclosure also
has a negative indirect eﬀect because ∂ETS/∂ηi > 0 but ∂ηi∗ /∂θi < 0. The last term of(34) is zero because ∂ηj∗ /∂θi = 0. The above results can be summarized as the following
proposition.
Proposition 3. Disclosure of the signal that is privately and expensively acquired byeach firm has two types of eﬀects on the expected consumer surplus and expected total
surplus: (i) the direct positive eﬀect, which increases them and (ii) the negative indirect
eﬀect, which decreases them.
Proposition 3 indicates that if the positive direct eﬀect dominates the negative indirect
eﬀect, then disclosure regulation will be eﬀective and socially desirable. This is because,
as shown in Proposition 2, each firm has no incentive to voluntarily disclose its own
private signal but disclosure enhances social welfare.
In order to evaluate the eﬀectiveness of the disclosure regulation, I focus on the two
extreme cases: (i) perfect disclosure (θ1 = θ2 = 1, denoted by superscript D) and (ii)nondisclosure (θ1 = θ2 = 0, denoted by superscript N ).
Consider the optimal signal precision in the case of perfect disclosure. From (21), the18

first-order condition is given by∂E[Πi ]
k
s
−
= 0.
=
2
∂ηi θi =θj =1 (2 + γ)s(1 − ηi )2(35)

Because ηi ∈ [0, 1),

ηiD =


√

 s−(2+γ) k
s


0 if

2

s
if k ≤ (2+γ)
2,

(36)

s2
< k.
(2+γ)2

Similarly, for the case of nondisclosure, the first-order condition iss∂E[Πi ]k= −= 0,
∂ηi θi =θj =0 4 s(1 − ηi )2

(37)

and

ηiN =

 √

 s−2 k
s


0 if

2

if k ≤ s4 ,

(38)

s2
< k.
4

Apparently, the optimal precision of the signal that firm i acquires is always higher under
nondisclosure than under perfect disclosure except for the trivial case where ηiD = ηiN =0. In addition, notice that ηiD = ηjD and ηiN = ηjN owing to the symmetric structure of
the game.
Assuming that the optimal signal precision is determined by the interior solution, firm
i’s expected profit, expected consumer surplus, and expected total surplus in the caseof perfect disclosure are calculated as follows, respectively.
√
√
2
(2s
−
(2
+
γ)
k)(s
−
(2
+
γ)
k)
ᾱ
+
,
E[ΠD
i ] =
2
2
(2 + γ)(2 + γ) s

ECSD =

√
1+γ
2
(ᾱ
+
2(s
−
(2
+
γ)
k)),
(2 + γ)2

19

(39)

(40)

√
2
3
+
γ
2k(2
+
γ)
−
2s
k(2 + γ)(4 + γ)
ETSD =(ᾱ2 + 2s) +.
2(2 + γ)(2 + γ)2 s(41)

Similarly, firm i’s expected profit, expected consumer surplus, and expected total surplus
in the case of nondisclosure are calculated as follows, respectively.
√ 2
2
ᾱ
k)
(s
−
2
E[ΠN
+
,
i ] =
2
(2 + γ)
4s

(42)

√
2
(2
+
γ)
(s
−
2
1
+
γ
k)
(ᾱ2 +ECSN =),
2(2 + γ)4(1 + γ)√
3 + γ 2 3s2 + 8k − 10s k
ᾱ +
.
ETS =
(2 + γ)2
4s
N

(43)

(44)

Notice that when k > s2 /4, the optimal precision of the signal becomes ηiD = ηjD =ηiN = ηjN = 0. Thus, the signal has no information content and disclosure does not
aﬀect social welfare. Consider the case where k ≤ s2 /4. Previous studies show thatmandatory disclosure resolves the prisoner’s dilemma and increases both firms’ expected
profits when their products are not good substitutes (e.g., Darrough, 1993; Suijs andWielhouwer, 2014). However, the comparison between the equations (39) and (42)indicates that nondisclosure is still desirable for both firms even if their products are notgood substitutes. In particular, the condition regarding whether disclosure regulation isdesirable from the perspective of each firm is described as follows.
√(4 − 4γ − γ 2 )2 s2
N
2
−
2
and
k
<
,
E[ΠD
]
>
E[Π
]
if
0
<
γ
<
2
i
i
16(1 − γ)2 (2 + γ)2
√2 )2 s2
s2
0 < γ < 2 2 − 2 and (4−4γ−γ
2 (2+γ)2 ≤ k ≤ 4 ,16(1−γ)NE[ΠDi ] ≤ E[Πi ] if√
2 2 − 2 ≤ γ < 1 and k ≤ s2 .
4

(45)

(46)

More salient results can be viewed with regard to the expected consumer surplus andexpected total surplus. Proposition 3 argues that disclosure has a positive direct eﬀect

20

and a negative indirect eﬀect on the expected consumer surplus and the expected total
surplus, respectively. The analysis demonstrates that there exists a situation where thenegative indirect eﬀect dominates the positive direct eﬀect, that is, disclosure decreasessocial welfare on the whole. In particular, the conditions are as follows.
(4 + 4γ − γ 2 )2 s2,4(2 + γ)2 (2 + 3γ)2s2
(4 + 4γ − γ 2 )2 s2≤
k≤,4(2 + γ)2 (2 + 3γ)24

ECSD > ECSN

if k <(47)

ECSD ≤ ECSN

if

(48)

ETSD > ETSN

if k <(49)

ETSD ≤ ETSN

if

(50)

(12 − 4γ − 3γ 2 )2 s2,
4(6 − γ)2 (2 + γ)2(12 − 4γ − 3γ 2 )2 s2
s2
≤
k
≤
.
4(6 − γ)2 (2 + γ)2
4These results contrast sharply to those of previous studies, which treat firms’ private
information as exogenous. As shown in Lemma 1, if the precision of the private signal isconstant, then disclosure always increases both the expected consumer surplus and the
expected total surplus. However, if the precision of the private signal is endogenously
determined, disclosure has a negative impact on the signal precision and deteriorates
firms’ internal information environments. This negative impact of disclosure on signal
precision decreases the expected consumer surplus and expected total surplus. In somesituations, this negative indirect eﬀect dominates the positive direct eﬀect, and disclosure
regulation leads to “unintended consequences,” such that it decreases social welfare on
the whole.

5 Conclusion
Although some researchers argue that management accounting systems (internal reporting) and financial accounting systems (external reporting) are closely linked, little
is known about the interaction between them. This paper provides theoretical insights
into this topic by investigating the interaction between firms’ information acquisition21

decisions and disclosure of the internally acquired information. In particular, this paper builds upon a standard duopoly model with uncertainty and attempts to answer
the following questions. How does disclosure aﬀect firms’ information acquisition decisions? How does the interaction between information acquisition and disclosure aﬀect
social welfare? By answering these questions, I hope to improve our understanding of
how regulatory arrangements for external reporting aﬀect firms’ internal information
environments and to evaluate the eﬀectiveness of the disclosure regulation.
Specifically, this paper focuses on the setting where two firms compete in quantities,
that is, Cournot competition under industry-wide demand uncertainty. Previous analytical studies show that although Cournot duopolists commit to nondisclosure of private
information relating to industry-wide uncertainty, mandatory disclosure can enhancesocial welfare. However, the analysis shows that mandatory disclosure is not always desirable in the setting where firms’ private information is endogenously determined. Thisis because disclosure has a negative eﬀect on firms’ incentives for acquiring private information through internal information systems. That is, when disclosure is mandated,firms acquire less precise information compared to the case where acquired information is
not disclosed, and hence, their internal information environments are deteriorated. Thiscan lead to “unintended consequences,” such that disclosure regulation strictly decreasessocial welfare. In particular, this paper identifies the two eﬀects on social welfare caused
by mandating disclosure: (i) the direct positive eﬀect, which increases social welfare asshown in the prior studies, and (ii) the negative indirect eﬀect, which decreases social
welfare through the deterioration of firms’ internal information environments. In fact,
there exists a case where the negative indirect eﬀect dominates the positive direct eﬀect,that is, mandatory disclosure decreases social welfare on the whole. This result is in
sharp disagreement with that of prior studies and suggests that disclosure regulation
might fail to achieve the intended goal of enhancing social welfare.
This paper suﬀers from several limitations. For example, in the model analysis, I
assume that a firm cannot add noise or bias to its private signal when disclosing it. I
also assume that a firm knows the precision of a rival firm’s signal if it is disclosed.

22

These assumptions are somehow restrictive and might be relaxed in future research inorder to make the settings more realistic.

Appendix
Derivation of equilibrium strategies for Cournot competition withindustry-wide uncertainty
First, consider the case where both firms disclose their private signals. In this case,both firms choose their quantities by observing signals ξi and ξj . Thus, firm i’s optimalquantity is denoted by qidd (ξi , ξj ). Because information acquisition cost is sunk whenfirm i chooses its quantity, it solves the following maximization problem.
max E[πidd |ξi , ξj ]qi (ξi ,ξj )= (ᾱ + E[∆αi |ξi ] + E[∆αj |ξj ] − qi (ξi , ξj ) − γqj (ξj , ξi ))qi (ξi , ξj ).

(A.1)

The first-order condition is given by∂
E[πidd |ξi , ξj ]=0
∂qi (ξi , ξj )qi (ξi ,ξj )=qidd (ξi ,ξj )⇒ ᾱ + E[∆αi |ξi ] + E[∆αj |ξj ] − 2qidd (ξi , ξj ) − γqj (ξj , ξi ) = 0.

(A.2)

Observe that substituting (A.2) in the objective function yields
E[πidd |ξi , ξj ] = (qidd (ξi , ξj ))2 .

(A.3)

The second-order condition is satisfied as follows.
∂2
E[πidd |ξi , ξj ] = −2 < 0.
∂(qi (ξi , ξj ))2

(A.4)

23

From the first-order condition, one can obtain the following best response function foreach firm,1qi (ξi , ξj ) = (ᾱ + E[∆αi |ξi ] + E[∆αj |ξj ] − γ q̂j (ξj , ξi )),2
1
qj (ξj , ξi ) = (ᾱ + E[∆αi |ξi ] + E[∆αj |ξj ] − γ q̂i (ξi , ξj )),
2

(A.5)
(A.6)

where q̂j (q̂i ) is firm i’s (firm j’s) conjecture of the quantity produced by firm j (firm i).
Notice thatE[∆αi |ξi ] = E[∆αi ] +Cov[∆αi , ξi ]s
(ξi − E[ξi ]) =ξi = ηi ξi .
Var[ξi ]s + ei(A.7)

Assume the following linear (or strictly speaking, aﬃne) strategy for each firm (Raith,
1996, Prop.3.1).
qi (ξi , ξj ) = Ai0 + Ai1 ξi + Ai2 ξj ,(A.8)

qj (ξj , ξi ) = Aj0 + Aj1 ξj + Aj2 ξi .

(A.9)

Notice that qi (ξi , ξj ) = q̂i (ξi , ξj ) and qj (ξj , ξi ) = q̂j (ξj , ξi ) because both firms can observe
signals ξi and ξj . From equations (A.5) to (A.9), one can obtain2(Ai0 + Ai1 ξi + Ai2 ξj ) = ᾱ + ηi ξi + ηj ξj − γ(Aj0 + Aj1 ξj + Aj2 ξi ),(A.10)

2(Aj0 + Aj1 ξj + Aj2 ξi ) = ᾱ + ηj ξj + ηi ξi − γ(Ai0 + Ai1 ξi + Ai2 ξj ).

(A.11)

Comparing both sides of (A.10) and (A.11), and solving the simultaneous equations
yields the following optimal quantity choice for each firm.
1(ᾱ + ηi ξi + ηj ξj ),2+γ
1
qjdd (ξj , ξi ) =(ᾱ + ηj ξj + ηi ξi ).
2+γ
qidd (ξi , ξj ) =(A.12)(A.13)

Now, consider the expected profit before observing the signals. Using the law of24

iterated expectations and equation (A.3),
E[πidd ] = E[E[πidd |ξi , ξj ]]= (E[qidd (ξi , ξj )])2 + Var[qidd (ξi , ξj )].

(A.14)

Notice thatE[qidd (ξi , ξj )] =

1
ᾱ,
2+γ

(A.15)

and1(ηi2 Var[ξi ] + ηj2 Var[ξj ])2(2 + γ)1=(sηi + sηj ).
(2 + γ)2

Var[qidd (ξi , ξj )] =(A.16)Thus,1(ᾱ2 + sηi + sηj ),(2 + γ)21
E[πjdd ] =(ᾱ2 + sηj + sηi ).
(2 + γ)2E[πidd ] =(A.17)(A.18)

Applying similar procedures helps us derive the optimal quantities and expected profits
for the other cases. Next, consider the case where firm i discloses ξi but firm j withholds
ξj . Then, firm i can only use signal ξi and firm j can use both signals ξi and ξj . Firm
i’s optimal quantity is denoted by qidϕ (ξi ) and firm j’s optimal quantity is denoted by
qjϕd (ξj , ξi ). The first-order conditions for the profit maximization problems yield thefollowing best response functions.
1qi (ξi ) = (ᾱ + E[∆αi |ξi ] − γ q̂j (ξj , ξi )),2
1
qj (ξj , ξi ) = (ᾱ + E[∆αi |ξi ]E[∆αj |ξj ] − γ q̂i (ξi )).
2

25(A.19)(A.20)

Assume the following linear strategies.
qi (ξi ) = Ai0 + Ai1 ξi ,(A.21)

qj (ξj , ξi ) = Aj0 + Aj1 ξj + Aj2 ξi .

(A.22)

Because firm i cannot observe the realized value of ξj , its conjecture about firm j’s
quantity isq̂j (ξj , ξi ) = Aj0 + Aj1 E[ξj ] + Aj2 ξi = Aj0 + Aj2 ξi .

(A.23)

Therefore,2(Ai0 + Ai1 ξi ) = ᾱ + ηi ξi − γ(Aj0 + Aj2 ξi ),(A.24)

2(Aj0 + Aj1 ξj + Aj2 ) = ᾱ + ηi ξi + ηj ξj − γ(Ai0 + Ai1 ξi ).

(A.25)

Solving the simultaneous equations yields the following optimal quantities.
1(ᾱ + ηi ξi )
2+γ
(
)
1
2+γ
ϕd
qj (ξj , ξi ) =ᾱ +ηj ξj + ηi ξi .
2+γ2

qidϕ (ξi ) =(A.26)(A.27)The expected profit before observing the signal(s) can be calculated as follows.
E[πidϕ ] = (E[qidϕ (ξi )])2 + Var[qidϕ (ξi )]=1(ᾱ2 + sηi ),(2 + γ)2(A.28)

E[πjϕd ] = (E[qjϕd (ξj , ξi )])2 + Var[qjϕd (ξj , ξi )]()
1(2 + γ)22=ᾱ +sηj + sηi .
(2 + γ)2
4(A.29)

Finally, consider the case where both firms do not disclose their signals. In this case,26

each firm can observe only the realization of its own signal. Thus, the optimal quantityof each firm is denoted by qiϕϕ (ξi ) and qjϕϕ (ξj ). The first-order conditions for the profit
maximization problems yield the following best response functions.
1qi (ξi ) = (ᾱ + E[∆αi |ξi ] − γ q̂j (ξj )),2
1
qj (ξj ) = (ᾱ + E[∆αj |ξj ] − γ q̂i (ξi )).
2(A.30)(A.31)

Assume the following linear strategies.
qi (ξi ) = Ai0 + Ai1 ξi ,(A.32)

qj (ξj ) = Aj0 + Aj1 ξj .

(A.33)

Because each firm cannot observe the realization of the rival’s signal, the conjectures are
as follows.
q̂j (ξj ) = Aj0 + Aj1 E[ξj ] = Aj0 ,(A.34)q̂i (ξi ) = Ai0 + Ai1 E[ξi ] = Ai0 .

(A.35)

Hence,
2(Ai0 + Ai1 ξi ) = ᾱ + ηi ξi − γAj0 ,(A.36)

2(Aj0 + Aj1 ξj ) = ᾱ + ηj ξj − γAi0 .

(A.37)

Solving the simultaneous equations yields the following optimal quantities.
)
(
1
2+γ
ηi ξi ,
ᾱ +
2+γ
2
)
(
2+γ
1
ϕϕ
ηj ξj .
qj (ξj ) =
ᾱ +
2+γ
2

qiϕϕ (ξi ) =(A.38)(A.39)

27

The expected profit before observing the signal can be calculated as follows.
E[πiϕϕ ] = (E[qiϕϕ (ξi )])2 + Var[qiϕϕ (ξi )]()
1(2 + γ)2
2
=
ᾱ +
sηi ,
(2 + γ)24E[πjϕϕ ] = (E[qjϕϕ (ξj )])2 + Var[qjϕϕ (ξj )]()
1(2 + γ)2
2
=
ᾱ +
sηj .
(2 + γ)2
4(A.40)(A.41)

Proof of Proposition 1
The ex ante expected profit of firm i can be expressed as follows.
E[πi ] =θi (θj E[πidd ] + (1 − θj )E[πidϕ ])+ (1 − θi )(θj E[πiϕd ] + (1 − θj )E[πiϕϕ ]).

(A.42)

Firm i chooses the precision of signal ξi in order to maximize its ex ante expected profit.
max E[πi ] − Ki (ηi )ηi= E[πi ] −kηi.
s(1 − ηi )(A.43)

If we assume interior solution ηi∗ , then the following first-order condition holds.
∂(E[πi ] − Ki (ηi ))=0∂ηi
ηi =ηi∗
s
s
k
⇒ θi
+
(1
−
θ
)
−
= 0.
i
(2 + γ)24 s(1 − ηi∗ )2(A.44)

Observe that the second-order condition is satisfied as follows.
2k
∂2
=
−
(E[πi ] − Ki (ηi ))
< 0.
2
∂ηi
s(1 − ηi∗ )3
ηi =ηi∗28(A.45)

We defineF ≡ θissk+ (1 − θi ) −.
2(2 + γ)4 s(1 − ηi∗ )2(A.46)

The first-order condition requires F = 0. Applying the implicit function theorem,∂ηi∗∂F/∂θi=−∂θi∂F/∂ηi∗γ(4 + γ)s2 (1 − ηi∗ )3=−8k(2 + γ)2< 0,(A.47)∂F/∂θj∂ηi∗=−∂θj∂F/∂ηi∗= 0.

(A.48)

Proof of Corollary 1
Applying the implicit function theorem,
∂ηi∗
∂F/∂k
=−
∂k
∂F/∂ηi∗1 − ηi∗=−2k< 0,(A.49)

∂ηi∗
∂F/∂s
=−
∂s
∂F/∂ηi∗(1 − ηi∗ )[4k(2 + γ)2 + 4s2 (1 − ηi∗ )2 + γ(4 + γ)s2 (1 − ηi∗ )2 (1 − θi )]=8k(2 + γ)2 s> 0,(A.50)

29

∂ηi∗
∂F/∂γ
=−
∂γ
∂F/∂ηi∗s2 (1 − ηi∗ )3 θi=−k(2 + γ)3
≤ 0,with equality if and only if θi = 0.

(A.51)

Proof of Proposition 2
Direct calculation shows thatγ(4 + γ)sηi∗∂(E[πi ] − Ki (ηi )|ηi =ηi∗ ) = −< 0.
∂θi4(2 + γ)2

(A.52)

Derivation of the expected consumer surplus
From the definition of the consumer surplus,
1
CS = α(q1 + q2 ) − (q12 + q22 + 2γq1 q2 ) − p1 q1 − p2 q2 .
2

(A.53)

Substituting the inverse demand function into the above equation,
1
CS = (q12 + q22 + 2γq1 q2 ).
2(A.54)Taking expectations for both sides of the above equation,
1
ECS = (E[q12 ] + E[q22 ] + 2γE[q1 q2 ]).
2(A.55)

30

Consider the case where both firms disclose their private signals. Then,1
ECSdd = (E[(qidd (ξi , ξj ))2 ] + E[(qjdd (ξj , ξi ))2 ] + 2γE[qidd (ξi , ξj )qjdd (ξj , ξi )])2
1= (E[πidd ] + E[πjdd ] + 2γE[qidd (ξi , ξj )qjdd (ξj , ξi )])2(A.56)

Observe thatE[qidd (ξi , ξj )qjdd (ξj , ξi )]
[
]
1
1
=E
(ᾱ + ηi ξi + ηj ξj )(ᾱ + ηj ξj + ηi ξi )2+γ2+γ
= E[(qidd (ξi , ξj ))2 ] (= E[(qjdd (ξj , ξi ))2 ])= E[πidd ] (= E[πjdd ]).

(A.57)

Therefore,
ECSdd =1+γ(ᾱ2 + sηi + sηj ).
(2 + γ)2(A.58)

Next, consider the case where firm i discloses ξi but firm j withholds ξj .
1
ECSdϕ = (E[(qidϕ (ξi ))2 ] + E[(qjϕd (ξj , ξi ))2 ] + 2γE[qidϕ (ξi )qjϕd (ξj , ξi )])2
1= (E[πidϕ ] + E[πjϕd ] + 2γE[qidϕ (ξi )qjϕd (ξj , ξi )])2(A.59)

Observe thatE[qidϕ (ξi )qjϕd (ξj , ξi )])
]
[
1
2+γ
1
(ᾱ + ηi ξi )(ᾱ +ηj ξj + ηi ξi )
=E
2+γ
2+γ
2
1
1
=
ᾱ2 +
ηi2 Var[ξi ]22(2 + γ)(2 + γ)1=(ᾱ2 + sηi ).
(2 + γ)2

31(A.60)

Therefore,
ECSdϕ =

1+γ
(2 + γ)2
2
(ᾱ
+
sη
+
sηj ).
i(2 + γ)28(1 + γ)(A.61)

Finally, consider the case where both firms do not disclose their private signals.
1
ECSϕϕ = (E[(qiϕϕ (ξi ))2 ] + E[(qjϕϕ (ξj ))2 ] + 2γE[qiϕϕ (ξi )qjϕϕ (ξj )])2
1= (E[πiϕϕ ] + E[πjϕϕ ] + 2γE[qiϕϕ (ξi )qjϕϕ (ξj )])2(A.62)

Observe thatE[qiϕϕ (ξi )qjϕϕ (ξj )])
[
]
1
2+γ
1
2+γ
=E
(ᾱ +
ηi ξi )(ᾱ +ηj ξj )
2+γ
2
2+γ
2
1
=
ᾱ2 .
(2 + γ)2(A.63)

Therefore,
ECSϕϕ =

(2 + γ)2(2 + γ)2
1+γ
2
(ᾱ
+
sη
+
sηj ).
i(2 + γ)28(1 + γ)8(1 + γ)(A.64)

Proof of Lemma 1
For the expected consumer surplus, direct calculation shows that∂ECS((2 + γ)2 + (4 + 4γ − γ 2 )θi )s
=
> 0.
∂ηi
8(2 + γ)2

(A.65)

For, the expected total surplus, direct calculation shows that∂ETS
(2 + γ)2 (3s2 (1 − ηi )2 − 8k) + (12 − 4γ − 3γ 2 )s2 (1 − ηi )2 θi
=
> 0,
∂ηi
8(2 + γ)2 s(1 − ηi )2(A.66)

because when k > (3s2 (1 − ηi )2 )/8, ηi does not have the interior solution and ηi∗ = 0.

32

Proof of Lemma 2
Diﬀerentiating the expected consumer surplus and the expected total surplus with respect to firm i’s disclosure policy θi given that η1 = η2 = η̄,∂ECS(4 + 4γ − γ 2 )sη̄=
≥ 0,
∂θi ηi =ηj =η̄
8(2 + γ)2
with equality if and only if η̄ = 0.

(A.67)(12 − 4γ − 3γ 2 )sη̄
∂ETS
=
≥ 0,
∂θi ηi =ηj =η̄
8(2 + γ)2
with equality if and only if η̄ = 0.

(A.68)

Proof of Proposition 3
Proposition 3 is already proved by Proposition 1, Lemma 1, and Lemma 2.

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