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Information Economics and Policy 16 (2004) 553–577

The product market opportunity lossof mandated disclosure q

Alison J. Kirby *

School of Management, Boston University, 595 Commonwealth Avenue, Boston, MA 02215, USA

Accepted 15 June 2003

Available online 19 December 2003

Abstract

In the interests of protecting decision makers in the financial markets, the Securities and

Exchange Commission requires publicly traded companies to publicly disclose certain ac-

counting information. Such disclosure requirements however create a potential opportunity

loss. They may destroy firms� opportunities for implementing an alternative information ac-

quisition and exchange regime – one that would optimize the firms� product market profits.

Nevertheless, we show that despite the previous imposition of an opportunity loss, firms may

still favor future increased disclosure requirements. Finally, when information cost declines,

although firms� welfare may decrease, their desire for increased disclosure requirements always

strengthens.

� 2003 Elsevier B.V. All rights reserved.

JEL classification: D43; L13

Keywords: Mandated disclosure; Information sharing; Product market; Information cost

1. Introduction

Companies publicly disclose accounting information such as a balance sheet and

an income statement every quarter. Such disclosures are mandated and regulated

by the Securities and Exchange Commission (SEC), and used by (potential)

qHelpful suggestions were received from an anonymous referee.* Tel.: +1-617-353-2029; fax: +1-617-353-6667.

E-mail address: [email protected] (A.J. Kirby).

0167-6245/$ - see front matter � 2003 Elsevier B.V. All rights reserved.

doi:10.1016/j.infoecopol.2003.06.002

mail to: [email protected]

554 A.J. Kirby / Information Economics and Policy 16 (2004) 553–577

shareholders and bankers to make investment decisions. 1 These accounting dis-

closures are also observable by competitor firms and used in making decisions such

as setting production quantities and prices. 2 Consequently these disclosures may

have significant effects on a firm�s profits. The view taken here is that disclosure

regulation by the SEC/FASB may preclude firms in an industry from creating a

jointly optimal information sharing program, or from committing (explicitly orimplicitly) not to exchange information. As such, mandating disclosure standards

imposes an opportunity loss on firms� expected profits earned in the product

market.

The following research questions are addressed. In the absence of mandated

disclosures, what level of information acquisition and disclosure would firms op-

timally jointly select? How do these optimal levels of acquisition and disclosure

depend on industry parameters? What is the opportunity loss suffered when dis-

closure is instead mandated? How do firms view subsequent changes in disclosurelevels? How does diminishing information cost affect firms� welfare and firms� at-titudes toward additional disclosure requirements? Finally, can firms recover this

opportunity loss?

A stochastic oligopoly model is used to gauge the impact on firm expected

profits of changing information regimes and to address these questions. We as-

sume many different industries, parameterized by differing demand and cost co-

efficients, but each composed of identical firms. Each firm purchases costly

private information about an unknown industry-specific market demand param-eter. An industry�s information regime consists of all firms each acquiring mconditionally independent private observations about the unknown market pa-

rameter (via the installation of an internal management accounting system) and

subsequently each disclosing f randomly selected signals from the set of m. 3 For

example, revenues earned, order backlog data and revenue projections and

forecasts can all be considered noisy observations of future market demand.

The greater the proportion of these observations which are shared or disclosed

the more accurate are firms� post-disclosure information sets (with respect to theunknown market parameter), and the less private information there remains in

1 The SEC in turn delegates decision making authority regarding accounting specifics to a private body,

namely the Financial Accounting Standards Board (FASB). The SEC/FASB has a primary responsibility

of providing financial market decisionmakers with equal access to information so that the financial

markets can operate fairly and efficiently.2 Although this information describes past transactions, it also provides a signal about future conditions

of demand and cost structure, and may do so with differing levels of accuracy. For example, profit

disclosed simply as the change in owners� equity potentially provides a much less accurate signal about

past sales revenue and consequently likely future demand, than if both revenues and costs were disclosed

or if revenues were divided into product lines.3 m is used to designate the number of managerial/private accounting signals acquired and f to indicate

the number of financial/public accounting signals shared or disclosed. A firm can vary the level of

information it acquires by hiring more (or less) accounting personnel, and by investing in more (or less)

sophisticated information technology.

A.J. Kirby / Information Economics and Policy 16 (2004) 553–577 555

the market following disclosure, i.e. the greater the correlation between firms�post-disclosure information sets. 4

The analysis shows that the optimal level of mandated disclosure is either full

disclosure of all acquired information or no disclosure. Surprisingly, partial disclo-

sure (disclosure of some of the acquired information) is never optimal in this setting.

As expected, the optimal level of information acquired is a decreasing function of thecost of information. However this optimal level ranges discontinuously from no

information to the acquisition of perfect information, with the discontinuity oc-

curring as the optimal disclosure level switches from none to full. Thus the optimal

mutual disclosure regime may be industry specific: for some industries the optimal

disclosure regime may be the acquisition and full disclosure of a few observations.

For other industries the optimal disclosure regime may be the acquisition of many

observations with no subsequent disclosure. These results represent an uncon-

strained optimum in the problem of cooperative information regime choice on thepart of firms in a particular industry.

It is in this environment that we consider the impact of disclosure mandated by

the SEC/FASB. Such mandated disclosures compromise the ability of firms to im-

plement the optimal information regime, and insodoing create an opportunity loss.

In this way this research attempts to evaluate disclosure choices within the context of

other information and other information regimes. We show that this opportunity

loss consists of two components: the first due to the mandated disclosure level not

being the optimal full disclosure level, and the second due to full disclosure poten-tially not being optimal.

Further, we show that his notion of opportunity loss is helpful in formulating the

FASB/SEC�s problem by providing a well-defined way of comparing competitive

effects of a given disclosure level across industries. For industries in high information

cost environments, ceteris paribus, the impact of increasing disclosure levels is always

to exacerbate firms� opportunity loss, while for those in low information cost envi-

ronments, there is a desire for increased disclosure levels. For industries with inter-

mediate levels of the information cost parameter, firms prefer more stringentmandatory disclosure levels if the existing levels are low and less stringent if the

existing levels are high. Noteworthy, is that even in industries for which no disclosure

is actually optimal, we see that firms might still prefer increased disclosure

requirements.

We also consider the impact of changes (specifically reductions) over time in the

information cost parameter. We show that although the attitude toward increasing

disclosure requirements strengthens for firms in all industries as information cost

decreases, the actual welfare of firms in some industries will increase and othersdecrease.

4 Consideration is exogenously limited to information regimes which are (a) public in their disclosures,

(i.e. any disclosures are available to all other competitor firms in the industry), and (b) identical in the

characteristics of the information available to each firm within a particular industry, (i.e. information

systems are assumed to be symmetric).


Finally, we consider the possibility that firms within an industry may coopera-

tively be able to recover some of this opportunity loss. Full recovery is possible only

for those industries in which full disclosure is optimal and the mandated disclosure

level is below the optimal full disclosure level. In general, however, full recovery is

not possible, since if the mandated disclosure level is greater than zero, then the

optimal information regime choice is now a constrained version of the originalproblem. No recovery is possible for those industries in which full disclosure is

optimal, but the mandated disclosure level already exceeds the optimal full disclosure

level. Partial recovery however is feasible in some cases, either by acquiring and

sharing (disclosing) more information than mandated, or by acquiring additional

observations but keeping them private: in other words by implementing a partial

disclosure regime. This contrasts with the earlier result that partial disclosure is never

optimal.

The paper is organized as follows. A brief literature review completes this section.The product market game is presented in Section 2 and solutions derived for equi-

librium output and generic expected profit levels as functions of the information

regime. In Section 3 the jointly optimal information regime is derived in the absence

of mandated disclosure. In Section 4 we examine properties of the opportunity loss

created when the SEC/FASB mandates a particular level of disclosure. Concluding

remarks are given in Section 5.

The industrial organization literature has examined incentives for information

sharing. 5 Typically, expected equilibrium profits are compared for a limited set ofinformation sharing alternatives: sharing all or no private information. Sharing

noisy market demand information when cost functions are linear and firms are

Cournot competitors reduces expected firm profits. See Clarke (1983). 6 However,

Vives (1984) shows that expected profits are higher when sharing single signals in a

Cournot duopoly when the goods produced are complements. Kirby (1988) shows

that complete sharing is preferred to keeping information private when cost func-

tions are sufficiently convex. Since firms are better off as the accuracy or precision of

their post-disclosure information sets increases, and worse off as the correlationbetween firms� post-disclosure information sets increases, one might conjecture that

partial disclosure (which retains some private information for each firm) might be

the preferred disclosure regime.

Regarding partial information sharing previous research provides conflicting re-

sults. In a simplified setting, Novshek and Sonnenschein (1982) allowed for partial

information sharing but showed that equilibrium expected profits were undominated

only for the extremes of no disclosure and full disclosure of all observations and

furthermore that both extremes lead to identical levels of expected profits. By con-

5 See Novshek and Sonnenschein (1982), Clarke (1983), Vives (1984), Gal-Or (1985, 1986), Kirby (1988)

and Raith (1996).6 These results generally reverse when one considers cost information instead of market demand

information (see Shapiro, 1986), or alternatively if one considers Bertrand rather than Cournot

competition (see Vives, 1984; Gal-Or, 1986).


trast, Kirby (1993) showed that partial information sharing in the form of multiple

information sharing pools is optimal in some cases. 7

The question of information acquisition was examined by Fried (1984) using a

duopoly model which allowed for asymmetric, binary choices regarding costless

production and disclosure of cost information by the duopolists. Such duopolists are

motivated to disclose information, in turn requiring the costless acquisition of theinformation. Li et al. (1987) analyzed the case of endogenous research by Cournot

oligopolists (without the possibility of subsequent information sharing) and showed

inefficiencies relative to social welfare maximizing levels. The inefficiency was at-

tributed to a lack of information pooling. 8

Accounting researchers have used stochastic oligopoly models of information

sharing to examine the ex ante product market effects of alternative mandated ac-

counting disclosures. Hughes and Kao (1991) show that full disclosure (interpreted

as selective capitalization of R&D expenses) produces greater expected profits thandoes partial disclosure (interpreted as immediate writeoff). Feltham et al. (1992)

show that under conditions of Cournot competition and demand uncertainty,

multisegment firms receiving private information are worse off under line of business

reporting than under aggregate reporting, but that expected consumer surplus and

expected social welfare are greater under line of business reporting. 9

The current paper differs from those mentioned above in that it simultaneously

considers the problems of selecting information acquisition and information dis-

closure levels. Moreover, it views the product market effects of mandatory ac-counting disclosures in the context of existing information arrangements, and to this

end uses an opportunity loss notion of firm welfare.

2. Model

Consider an oligopolistic industry of n firms. All firms in the industry have

identical production cost functions, CðxiÞ ¼ cxi þ dx2i , where c and d are known, dis nonnegative, and xi is firm i�s output. 10 Firms operate in a market with price a

linear function of the total quantity produced: P ¼ a� bPn

i¼1 xi; bP 0; where b is

7 For example, firms in an industry of eight firms could share information by forming two pools of four

firms, where firms share their private information only with the three other firms in their pool.8 Fried and Sinha (2003) assume costly information by restricting the frequency with which firms can

acquire perfect information. They show that the greater the degree of correlation between firms� costs themore likely they are to collaborate on the timing of their information acquisition, assuming that they

subsequently share their information.9 Others (Newman and Sansing, 1993; Gigler, 1994; Hayes and Lundholm, 1996) have considered

disclosure decisions in the face of multiple audiences, such as the capital market and competitors, but have

addressing the issue of voluntary disclosure made after the receipt of information.10 This assumption of identical cost functions for all firms in an industry is most descriptive when firms

adopt similar production technologies, and operate at similar scales and with similar efficiency levels. As dvaries, the shape of the cost function varies from convex (d > 0) to linear (d ¼ 0) reflecting decreasing and

constant returns to scale.


constant and known by all firms, but a is an unknown parameter distributed

Nðla; r2aÞ.

11 All firms know this distribution and adopt it as their prior distribution

over a. 12

2.1. Private information acquisition

All firms in the industry invest in costly private information (management ac-

counting) systems of identical sophistication. This accounting system might generate

information about past revenues, which may in turn provide a noisy signal about the

unknown future market demand parameter, a. We further assume that this man-agement accounting system for firm i generates a set of m signals about a in the form

of the pre-disclosure (or pre-sharing) information set: fs1i ; s2i ; . . . ; smi g. Each signal is a

noisy representation of the true underlying demand parameter (net of the linear cost

term):

11 Th12 Wh

about

telecom

post B13 Ac14 Th

sji ¼ a� cþ mji :

The noise terms mji are independently and identically distributed Nð0; r2

mÞ. The noiseterms are independent across firms and are also independent of a. As the manage-

ment accounting system produces more independent observations about the market

demand parameter, firm i�s posterior estimate of that parameter becomes more ac-

curate. Define the accuracy Am of the management accounting system with respect tothe unknown market demand parameter as: 13

Am � r2a

r2a þ

r2mm

:

It is costly for the management accounting system to generate or acquire informativesignals on market demand. This information cost is $k per unit of accuracy of the

predisclosure information set, so that the overall cost of purchasing m signals is kAm,

and k is the cost of perfect information. This specification of the information cost

function is thus concave in the number of observations, and therefore exhibits a

decreasing cost per observation. 14

2.2. Information disclosure

Information sharing is modelled as the mutual truthful disclosure by all firms in

the industry to each other. Each firm randomly selects a subset of f of their m private

is parameter reflects the maximum size of the market.

ile accountants tend to focus on the proprietary nature of cost information, private information

market demand is also proprietary, particularly in industries where products are new (e.g. the

munications industry) or where consumer tastes have changed dramatically (e.g. the beef market

SI).

curacy is therefore the precision of an information system normalized to lie between zero and one.

us it is feasible to purchase perfect information at finite cost.


observations about future market demand produced by the management accounting

system. 15 Assuming n; f are common knowledge, this set of f disclosed observations

by each of n firms may be aggregated into an industry average:

15 If fi

misrep

low le

possib16 Th17 We

sF ¼Pn

i¼1

Pfj¼1 s

ji

nf; ð1Þ

where f 2 ½0;m�. Similarly, firm i�s remaining m� f privately retained observations

are reflected fully by their average:

sMi ¼Pm

j¼fþ1 sji

m� fð2Þ

and firm i�s post-disclosure information set becomes: Iiðm; f Þ ¼ fsF ; sMi g.16 Firm i

uses the information in Ii to select its output level as it competes in the product

market. A timeline of events follows.

Informationregime(m, f )

agreed upon

Firms purchasem observationsabout demand,

m ∈ [0, )

Firms disclosef of the m

observations,f

Firms selectoutputs,

x i(sMi , sF )

Profitsrealized

i∞ Π∈ [0,m]

In order to ultimately address the issue of optimal information regime selection,

we first solve for the optimal output strategies and the level of expected profits

generated under a given information regime, ðm; f Þ, implemented symmetrically by

all firms in the industry.

2.3. Equilibrium output strategies

In the game of output choice in the product market each firm independently se-

lects a level of output so as to maximize its expected profits, EPi, conditional on its

post disclosure information set Iiðm; f Þ, assuming that all other firms do the same. 17

Formally,

maxxiðIiÞ

EPið�jIiÞ � maxxiðIiÞ

E PðxiÞxi½ � CðxiÞjIi�; ð3Þ

maxxiðIiÞ

EPið�jIiÞ � maxxiðIiÞ

E a

""� b

Xj

xj

#xi � cxi � dx2i

��Ii#: ð4Þ

rms could strategically disclose (either by strategically selecting which observations to disclose or by

resenting them) then there would be incentives to select the lowest observations, so as to encourage

vels of production by the competition. We assume that a finite auditing cost could remove the

ility of any such strategic misrepresentation.

e notation Iiðm; f Þ is simplified to Ii when not misleading.

ignore the cost of information since the information set Iiðm; f Þ is exogenous at this stage.


The first order condition for this optimization problem is

18 See19 Str

output

Eða� cjIiÞ � 2ðbþ dÞxiðIiÞ � bðn� 1ÞEðbxj jIiÞ ¼ 0; ð5Þ
where xiðIiÞ is firm i�s level of output and Eðbxj jIiÞ is firm i�s conjecture about other
firms� output levels conditional on having observed the post-disclosure information

set. Conjecture the following optimal symmetric output strategy linear in both sig-nals, for all i:

xiðIiÞ ¼ p þ qF nfsF þ qMðm� f ÞsMi : ð6Þ
Incorporating different coefficients on the shared and private signals (qF ; qM ) allowsfor the possibility that the same value of the signal is reacted to differently depending
on whether it is publicly or privately observed despite its identical accuracy.The equilibrium output strategy as a function of the post-disclosure information

set is derived to be: 18

xiðIiÞ ¼ qMðm; f Þ ar2m

r2a

ðla

�� cÞ þ anfsF þ ðm� f ÞsMi

�; ð7Þ

where

qMðm; f Þ ¼1

2ðbþ dÞ r2mr2aþ nf

h iþ 2ðbþ dÞ þ bðn� 1Þ½ �ðm� f Þ

ð8Þ

and

a ¼ 2ðbþ dÞ2ðbþ dÞ þ bðn� 1Þ : ð9Þ

It is assumed that the mean of the demand intercept, la, is sufficiently greater thanthe linear cost parameter, c, that the optimal output is always positive. 19 Note that

the optimal output is indeed affected differently by disclosed and private observa-

tions. Comparing (6) and (7) reveals that qF ¼ aqM , and since a < 1, the optimal

output level is altered more by a signal which forms part of the private aggregate

than one which forms part of the disclosed aggregate, even though each signal is

equally accurate relative to the unknown demand parameter. Note also that the

expected output is

EðxiðIiÞÞ ¼la � c

2ðbþ dÞ þ bðn� 1Þ ð10Þ

and that the ex post expected profits for firm i (i.e. conditional on its information set)

are a convex function of equilibrium output levels:

PiðIiÞ ¼ ðbþ dÞðxiðIiÞÞ2: ð11Þ

Appendix A.

ictly speaking given the assumption of normal distributions there is always a chance of negative

.


2.4. Expected profits in the product market

Ex ante expected profits accruing to each firm, denoted EPiðm; f Þ, are calculatedby substituting the optimal output strategy into the profit function and taking ex-

pectations over all possible sets of observations, or equivalently by taking expecta-

tions of ex post expected profits.

20 Th

cooper

associa

behavi

EPiðIiÞ ¼ ðbþ dÞEðxiðIiÞÞ2; ð12Þ

EPiðIiÞ ¼ ðbþ dÞ½VarðxiðIiÞÞ þ ½EðxiðIiÞÞ�2�: ð13Þ

Note that the expected optimal output, while dependent on la � c, is independent ofm and f . The variance of the optimal output, however, while dependent on m and f ,is independent of la � c. Thus firm expected profit is composed of two terms, but

only the first is a function of the information regime parameters m and f . This firstcomponent is the information-related component of expected profits. As indicated

earlier, this model most likely predicts the behavior of firms in industries for whichla � c is sufficiently large that resulting optimal output levels are positive. Never-

theless, since the objective here is to make comparative statistics statements of ex-

pected profits over m and f , we need only consider the information-related

component of the expected profits under information regime ðm; f Þ. It is identical forall levels of la � c, and is given by

Information related component of

EPiðm; f Þ ¼ ðbþ dÞVarðxiðIiðm; f ÞÞÞ

¼ ðbþ dÞq2M r2a anf½

hþ ðm� f Þ�2 þ r2

m a2nf�

þ ðm� f Þ�i;

ð14Þ

where qMðm; f Þ and a are as defined above. For ease of reference, we refer to this

expression simply as expected firm profits, EPiðm; f Þ.

3. Optimal information regime (m*, f *)

The next step considers the derivation of the optimal information acquisition and

disclosure regime. Backward programming is used. First, for a given level of infor-

mation acquired, m, the optimal disclosure regime f �ðmÞ is identified. Second, the

optimal level of information acquisition m� is identified knowing that subsequently

f ¼ f �ðm�Þ will be implemented. This choice is made cooperatively by the firms – as

if by a single decisionmaker. 20

us our model exhibits an asymmetry in that firms act competitively in the product market, but

atively in their choice of information regime. This is consistent with legal guidelines for trade

tions. They may organize the exchange of information as long as it does not lead to collusive

or in the market.


3.1. Optimal disclosure level: f �(m)

Formally, we solve the following constrained optimization problem:

21 By

occurs

f ¼ �profit

set f0;22 Th

was ex

compa

desirab

are suffi23 See

maxf

EPiðf jmÞ;subject to f 2 ½0;m�

ð15Þ

and are able to prove the following.

Proposition 1. Given an exogenous level of information acquired by all firms, as pa-rameterized by the number of i.i.d. observations (m) acquired about the unknown de-mand parameter, the optimal disclosure policy as parameterized by the number of thoseobservations to be mutually disclosed by all firms is either all or none, i.e.f �ðmÞ 2 f0;mg. Furthermore, if d > ð<Þdcrit then full (no) disclosure is optimal, where

dcrit ¼b2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin½nmr2

a þ r2m�

½mr2a þ r2

m�

s � 1

!:

The proof is given in Appendix B. 21 Note that while partial disclosure of a subsetof observations is never optimal, the actual amount of disclosed information even

under full disclosure is always limited by the amount of information acquired and

therefore may be very small. The specific amount disclosed will now depend on the

level of information acquired which is a function of the cost of information and is

solved for in Section 4. The proposition also shows that which of the two infor-

mation disclosure extremes is preferred depends on industry parameters. 22

This ‘‘lack of an interior optimum’’ result is surprising. Prior research has ana-

lyzed partial disclosure in the form of information sharing sub-pools rather than inone large pool, and shown that an interior level of sharing may be optimal. 23 Under

the current form of partial disclosure however, partial disclosure is never optimal.

The intuition for this ‘‘lack of an interior optimum’’ result is that relative to public

disclosure of information, the drop in accuracy resulting from limited disclosure

outweighs the benefit resulting from the decrease in correlation between firms� in-formation sets and vice versa relative to the case of no disclosure.

way of example, if b ¼ 2, n ¼ 21, r2a ¼ 10, r2m ¼ 20 and m ¼ 2 then if d is 3 then the minimum

at f ¼ 2:31, which is greater than m and consequently f � ¼ 0. If d ¼ 30 then the minimum occurs at

0:04 and the optimal number of observations to be disclosed is two. If d 2 ½3:58; 25:18� then the

minimizing f lies in the interval ð0; 2Þ, and again the profit maximizing level of f is an element of the

2g.is part of the result is consistent with a result in Kirby (1988), where the set of information regimes

ogenously rather then endogenously restricted to the two disclosure extremes. This result can also be

red directly to Vives (1984) where the degree of product differentiation affects the relative

ility of disclosure levels. In his model under Cournot competition and demand uncertainty if goods

ciently not good substitutes then firms are better off sharing information than keeping it private.

Kirby (1993).


3.2. Optimal acquisition level: m�jf �(m)

The final step in solving for the optimal information regime is to derive the op-

timal number of observations to be privately acquired, so as to maximize equilibrium

expected product market profits net of the information acquisition cost:

maxm

EPiðm; f �ðmÞÞ�

� kr2a

r2a þ r2

m=m

� ��: ð16Þ

The solution to this problem is described by considering three ranges of the industry

cost parameter, d.

Case 1 (d � b2ðn� 1Þ). In this range d > dcrit for all values of m and therefore full

disclosure (i.e. f � ¼ m) is optimal for all levels of m (by Proposition 1) or equivalently

for all levels of the information cost parameter, k. Thus the specific information

acquisition problem for this case assumes f ¼ m in the profit function:

maxm

EPiðm;mÞ�

� kr2a

r2a þ r2

m=m

� �� max

m

a2r2a

4ðbþ dÞnmr2

a

nmr2a þ r2

m

� �� k

r2a

r2a þ r2

m=m

� ��; ð17Þ

EPiðm;mÞ is calculated by substituting f ¼ m into the expression for expected profits

in Eq. (14). The first and second order necessary conditions for a maximum reveal

that the optimal number of observations to be acquired when subsequently all aredisclosed, is

m�FULL ¼

r2m 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikðbþ dÞ

p� a

ffiffiffiffiffiffiffinr2

a

p� �r2a a

ffiffiffiffiffiffiffinr2

a

p� 2n

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikðbþ dÞ

p� � : ð18Þ

This optimal number of observations is feasible only if it is nonnegative. Con-

straining m�FULL to be positive requires the numerator and denominator to have the

same sign. This can be represented by imposing constraints on the cost of infor-mation, namely that k 2 ½k4ðdÞ; k5ðdÞ�, where:

k4ðdÞ ¼r2a

na2

4ðbþ dÞ ; ð19Þ

k5ðdÞ ¼nr2

aa2

4ðbþ dÞ : ð20Þ

Outside this range of k values, the first order derivative of expected profits with

respect to m becomes monotonically positive when k < k4ðdÞ and negative when

k > k5ðdÞ. Summarizing we have the following lemma.


Lemma 1. When dP b2ðn� 1Þ, if k 2 ½k4ðdÞ; k5ðdÞÞ then purchasing and fully dis-

closing m�FULL observations is optimal. If k < k4ðdÞ then purchasing and disclosing

perfect information is optimal, m�FULL ! 1. If kP k5ðdÞ then acquiring no private

information is optimal, m�FULL ¼ 0.

These regions are illustrated in Fig. 1.

Case 2 (d < b2ð ffiffiffi

np � 1Þ). In this range d < dcrit for all values of m and therefore no

disclosure is optimal (i.e. f � ¼ 0) for all levels of m (by Proposition 1) or equivalently

for all levels of the information cost parameter, k. Thus the specific information

acquisition problem for this case assumes f ¼ 0 in the profit function:

maxm

EPiðm; 0Þ�

� kr2a

r2a þ r2

m=m

� ��ð21Þ

� maxm

a2r2a

4ðbþ dÞmr2

a

ðmr2a þ r2

mÞmr2

a þ r2m

mr2a þ ar2

m

� �2"� kmr2

a

mr2a þ r2

m

#; ð22Þ

where EPiðm; 0Þ is calculated by substituting f ¼ 0 into the expression for expected

profits in Eq. (14). The first and second order necessary conditions for a maximumreveal that the optimal number of private observations to be acquired, m�

NO is a

solution to the following cubic equation representing the relevant portion of the first

order condition:

a2r2a

4ðbþ dÞ ðmr2a þ r2

mÞ2½ð2a� 1Þmr2

a þ ar2m� � kðmr2

a þ ar2mÞ

3 ¼ 0: ð23Þ

Let k6ðdÞ be the critical value of k that just generates m�NO ¼ 0 as the solution to this

first order condition. Thus

k6ðdÞ ¼ r2a=½4ðbþ dÞ�:

Production Cost, d

k7(d)

k5(d)

k6(d)

k4(d)

NO INFORMATION ACQUIRED

PERFECT INFORMATION ACQUIRED

k1(d)

m*FULL and FULL DISCLOSURE

)1(2

n–b

)3(2

n–b

)1(2

n–b

m*NO and NO DISCLOSURE

Information Cost, k

Fig. 1. Optimal information regime as a function of cost parameters.


Summarizing we have the following lemma.

Lemma 2. When d < b2ðffiffiffin

p� 1Þ, then nondisclosure of all privately acquired infor-

mation is optimal. If k < k6ðdÞ then purchasing m�NO private observations and not

disclosing is optimal. If kP k6ðdÞ then acquiring no private information is optimal.

Note that in this Case 2 parameter range (d < b2ð ffiffiffi

np � 1Þ) purchasing perfect

private information is never optimal. The production cost parameter d is sufficiently

small that the competitive benefits to improved knowledge of the market demand

parameter are too small to outweigh the costs of purchasing and sharing informa-

tion. For it to be worthwhile to acquire perfect information the information cost

parameter k would have to be less than a critical level such that oEPi=om > 0; 8m.This k1ðdÞ corresponds to m ! 1 being a solution to the first order condition.

Consequently:

k1ðdÞ ¼ r2aa

2ð2a� 1Þ=½4ðbþ dÞ�:

However in the Case 2 parameter range k1ðdÞ is always negative as seen in Fig. 1.

Case 3 (b2ð ffiffiffi

np � 1Þ < d < b

2ðn� 1Þ). For this intermediate range of production cost

parameter values Proposition 1 implies that the optimal disclosure level will be a

function of the amount of information purchased or equivalently a function of the

information cost parameter. If k is sufficiently large then full disclosure is the more

efficient form of investing in higher accuracy information. We define a function k7ðdÞas the set of points fðk; dÞg, such that if the cost of information is k for an industry

with production cost coefficient d, then the firms in that industry are indifferent

between the optimal full disclosure regime and the optimal no disclosure regime.Thus Case 2 results are applicable for k < k7ðdÞ and Case 1 results are applicable for

kP k7ðdÞ. This is summarized in the following lemma, and illustrated in Fig. 1.

Lemma 3. If b2ð ffiffiffi

np � 1Þ < d < b

2ðn� 1Þ, then the optimal acquisition and disclosure

choices are jointly functions of the cost of information and the production cost pa-rameter. If k1ðdÞ > 0 and k 2 ½0; k1ðdÞÞ then purchasing perfect information is optimal,and the issue of disclosure is moot. If k 2 ½k1ðdÞ; k7ðdÞ�, then purchase of m�

NO obser-vations and no disclosure is optimal. If k 2 ½k7ðdÞ; k5ðdÞ� then purchase of m�

FULL ob-servations and their full disclosure is optimal. Finally, if kP k5ðdÞ then purchase of noprivate information is optimal and again the issue of disclosure is moot.

3.3. Comparative statistics: information cost, k

For an intermediate value of d, Fig. 2 illustrates how the optimal acquisition anddisclosure choices interact for a specific industry as the information cost parameter

changes. If the cost of information is very high (i.e. k > k5ðdÞ), firms optimally ac-

quire no information and of course the disclosure issue is moot. For lower levels of

m*NO

m*FULL

k5(d) k7(d)

Optimal Number of Observations Acquired, m*

Cost of Information, k

Fig. 2. Optimal number of observations acquired under full and no disclosure.


information cost, (namely k7ðdÞ < k < k5ðdÞ) firms optimally purchase a few private

observations and mutually disclose them with all their competitors. Within this krange as the cost of information decreases progressively more observations are ac-

quired. When the cost decreases to k ¼ k7ðdÞ the firms in this industry are indifferent

between (a) purchasing m�NO observations and keeping them private, and (b) pur-

chasing only m�FULL observations but disclosing them publicly. Below this critical

level of information cost, firms purchase increasingly more information but opti-

mally do not share it.

In summary, as the cost of information, k, increases, ceteris paribus, the optimal

number of observations acquired, m�, decreases monotonically, but not continuously.If information cost is low, then under the optimal information regime information is

always kept private. However as information becomes more costly, the more eco-

nomical way of achieving higher accuracy is to publicly disclose and aggregate rather

than to acquire. Finally, beyond a certain level of information cost, no information is

acquired at the optimum. What is new here is that this decrease in information

optimally acquired, exhibits a discontinuity.

4. Opportunity loss of mandatory disclosure

The analysis so far has pursued the question of identifying the most efficient

information acquisition and sharing regime for firms to implement coopera-

tively, while nevertheless assuming that they act competitively in their productiondecisions. 24

24 Implementation of such an optimum would require a mechanism to guarantee the agreed upon levels

of acquisition and disclosure, since it has been shown that the optimum does not constitute a Nash

equilibrium in the level of disclosure selected. See Kirby (1998).


Into this scenario of firms cooperatively selecting an optimal information and

sharing regime, consider the entrance of disclosure regulators such as the SEC/

FASB. They mandate the disclosure level that firms must provide to decision makers

in the financial markets, and indirectly the information acquisition level. Such in-

formation being publicly disclosed is also available to competitors and furthermore,

is audited to ensure compliance with the prescribed disclosure levels. Intentionally or

not, the SEC/FASB has created a mutual information acquisition and sharing regime of

the exact form assumed in our model.

This raises several questions: What impact does the introduction of a mandatory

acquisition and disclosure regime have on firms� product market profits, relative to

what they might have implemented in the absence of mandatory disclosure? Once a

mandatory disclosure requirement is in place, how will firms react toward

strengthening mandatory disclosure requirements? As information costs decline over

time, what is the impact on firms� welfare and on their attitudes toward disclosure?Are there actions that firms can still take cooperatively to recover some of the op-

portunity loss? These questions are addressed in the following sections.

4.1. Impact of mandating disclosure: components of opportunity loss

Using the model from Section 2, we can evaluate the impact on firm i�s productmarket profits of the SEC/FASB mandating disclosure of f ¼ �m, which in turn

demands acquisition of m ¼ �m. There is a remote chance that the level of disclosure

mandated is exactly the optimal full disclosure level, �m ¼ m�FULL, and that industry

parameters are such that full disclosure itself is optimal, f � ¼ m. In this case, there

would be no opportunity loss caused by mandating this level of disclosure. More

generally, however, there will be an opportunity loss. This opportunity loss, OL,

from imposing ð�m; �mÞ is given by

OLið�mÞ ¼ net EPiðm�; f �Þ � net EPið�m; �mÞ; ð24Þ

where

net EPiðm; f Þ ¼ EPiðm; f Þ � kr2a

r2a þ r2

m=m

� �ð25Þ

is the effect of information regime ðm; f Þ on profits, net of the cost of information,

and EPiðm; f Þ is given in (14). The expression for the opportunity loss for firm i canbe decomposed into two components:

OL1 ¼ net EPiðm�; f �Þ � net EPiðm�FULL;m

�FULLÞ; ð26Þ

OL2 ¼ net EPiðm�FULL;m

�FULLÞ � net EPið�m; �mÞ: ð27Þ

The first component describes the lost profits when the level of mandated disclosureequals the optimal full disclosure level, �m ¼ m�

FULL. This component is positive only

when the optimal disclosure regime is no disclosure. The second component de-

scribes the lost profits of mandating �m rather than the optimal level of full disclosure


m�FULL. Further we can define a relative measure of welfare for firm i under man-

datory disclosure regime �m as the negative of the opportunity loss at �m:

25 d 2

Firm Welfareið�mÞ ¼ net EPið�m; �mÞ � net EPiðm�; f �Þ: ð28Þ

Proposition 2. Firm welfare weakly decreases with the imposition of mandateddisclosure.

Proof. This follows immediately from the definition of ðm�; f �Þ as the unconstrainedoptimum, and welfare being zero when no mandated disclosure regime exists. Fig.

3(a) illustrates an example of the opportunity loss imposed by mandating disclosure

at f ¼ m ¼ �m. In this example, since d 2 ½b2ðffiffiffin

p� 1Þ; b

2ðn� 1Þ� and k 2 ½k4; k7�, no

disclosure would have been optimal in the absence of mandated disclosure, implying

that OL1 > 0. By contrast, in Fig. 3(b), the cost parameters are such that full dis-

closure would have been optimal in the absence of mandated disclosure. Conse-

quently, OL1 ¼ 0. 25 This is interesting. While the imposition of mandatory

disclosures is typically viewed as being detrimental to firms� profits, it is so here not

due to lost proprietary information but due to lost information sharing possibilities.

�

4.2. Attitudes toward changing disclosure requirements

Having imposed a mandatory disclosure regime, regulators concerned about

welfare effects in the product market of changing disclosure requirements would be

interested in evaluating oWelfarei=o�m, which we refer to as the firm�s attitude towardchanging disclosure requirements. A positive (negative) slope reflects a positive

(negative) attitude to increases in mandated disclosure levels. Since only OL2 is afunction of �m, only the net expected full disclosure profit function is relevant in

anticipating firms� attitudes toward proposed changes in disclosure levels.

Proposition 3. If k < k4, then firms always prefer higher disclosure requirements. Ifk 2 ½k4; k5�, then firms prefer higher disclosure requirements only if �m < m�

FULL. Ifk > k5, then firms never prefer higher disclosure requirements.

Proof. This result is derived from conditions describing the shape of the full dis-closure net profit function (net EPðm;mÞ) as either being monotonically increasing,

having an interior optimum or being monotonically decreasing. �

This proposition holds regardless of whether firms in the industry jointly prefer

full disclosure of m�FULL or no disclosure of m�

NO, and even if there was no welfare loss

from implementing �m initially, (i.e. �m ¼ m�FULL ¼ f �).

½b2ð ffiffiffi

np � 1Þ; b

2ðn� 1Þ� and k 2 ½k7; k6�.

OL2

OL(m)

Net EΠi(m |f )

OL1

FULL

NO

mFULL* mNO

* mm

Net ΕΠi(m |f )

OL2 m+∆)(

OL2 m( )FULL

ΝΟ

mFULL∗ mNO

∗ mm + ∆m

FULL

NOOL1

OL2

Net EΠi(m |f )

mNO* mm

(a)

(b)

(c)

Fig. 3. (a) Opportunity loss at mandated disclosure level m when no disclosure is optimal

d 2 b2

ffiffiffin

p� 1ð Þ; b

2n� 1ð Þ

� �; k 2 k4; k7½ �

. (b) Opportunity loss at mandated disclosure level m when full

disclosure is optimal d 2 b2

ffiffiffin

p� 1ð Þ; b

2n� 1ð Þ

� �; k 2 k7; k6½ �

. (c) Opportunity loss at mandated disclosure

level m when no disclosure is optimal ðd < b2

ffiffiffin

p� 1ð Þ; k 2 k1; k4½ �Þ.



Corollary 1. Even in industries for which no disclosure of m�NO observations is optimal,

firms may still respond positively to the requirement to increase disclosure levels beyond�m.

This occurs, when �m < m�FULL, as in Fig. 3(a), and for all levels of �m in Fig. 3(c). 26

Despite globally preferring no disclosure of m�NO observations, locally, given that �m

has already been mandated, firms prefer an increase in mandated disclosure level to�mþ D over keeping the disclosure level at �m.

Corollary 2. Even in industries for which full disclosure of m�FULL observations is op-

timal, firms may still respond negatively to the requirement to increase disclosure levelsbeyond �m.

This is true when �m > m�FULL, as in Fig. 3(b).

In summary, even though firms in all industries suffer a welfare loss when man-

dated disclosure is implemented, industries are likely to differ in the attitude of their

firms towards subsequent changes in the level of mandated disclosure.

4.3. Decreasing information cost

Further to our discussion of the firms� opportunity loss caused by imposing a

mandated disclosure level, we consider the impact of expected declines in the in-

formation cost, k, on firm welfare and on firms� attitudes toward disclosure.

4.3.1. Impact on firm welfare

It is clear that reductions in the information cost will increase expected profits for

each company, given the existing acquisition and disclosure regime, �m, since now the

same level of information can be acquired at lower total cost. However, if we con-sider, as above, that firms� welfare in this setting is a function of the expected op-

portunity loss, then the impact of information cost reductions on firms� welfare is

more complex. First, a drop in k, will increase m�FULL and the profits at the optimal

full disclosure level. Second, a drop in k will also increase m�NO and the profits at this

optimal no disclosure level. The net effect on each firm�s welfare is not obvious.

Proposition 4. As k decreases, firm welfare at mandatory disclosure level �m, decreasesif �m < m�

FULL, when k 2 ½k4; k5�. Otherwise, firm welfare increases.

That decreases in information costs might make firms worse off in an opportunity

loss sense is surprising and yet intuitive. Consider, �m < m�FULLðkÞ. Since

om�FULL

ðkÞok < 0,

then as k decreases the optimal full disclosure level increases and moves progressively

further away from the stationary mandated disclosure level �m. Furthermore since the

whole net expected profit curve moves up as k declines, and more so for higher values

26 Fig. 3(c) is characterized by k 2 ½k1; k4� and d < b2ð ffiffiffi

np � 1Þ.


of m, the opportunity loss at �m increases, and welfare declines. The converse holds

when �m > m�FULLðkÞ.

For k > k5, since m�FULL is already zero, it is unaffected by further decreases in

information cost. Consequently the optimal level of full disclosure profits is also

unaffected by further decreases in k, and therefore firm welfare is affected only by the

impact of changes in k on the information cost component of profits for firm i ofpurchasing �m signals. Similar reasoning holds for the case when k < k4.

Observation 1. The extent of opportunity loss caused by imposing mandatory disclo-sure is affected not only by changes in the level of mandated disclosure, but also byexogenous changes in industry parameters, such as the information cost parameter, k.

This is an appealing idea. It implies that regulators wishing to economize on the

number of changes in mandatory disclosure requirements, might tend initially tomandate relatively high levels of disclosure with the argument that over time as kdecreases, such higher levels of disclosure would exhibit the regulators� acceptablewelfare consequences in the product market (as well as their desired consequences in

the financial markets).

4.3.2. Impact on attitude toward changing disclosure requirements

We now consider the impact of declining information cost on firms� attitudestoward changing the level of mandated disclosure.

Proposition 5. As the cost of information decreases, firms’ attitudes toward tighteningdisclosure requirements improve.

Proof.

o

okoWelfareð�m; �mÞ

o�m

!¼ �r2

mr2a

ð�mr2a þ r2

mÞ2: ð29Þ

As the information cost declines, the attitude toward increases in mandated disclo-

sure improves, either by turning more positive or less negative, (a) regardless of the

existing level of �m in relation to m�FULL, and (b) irrespective of whether the change in

information cost moves the optimal disclosure regime to f ¼ 0, implying that OL1

has become positive. In summary, even though welfare at a given level of disclosure

may increase or decrease as the information cost declines, the change in attitude offirms toward increases in disclosure levels is always positive. This derives directly

from the information cost term in the net expected profit function. �

4.4. Recovering the opportunity loss

Finally, we briefly consider the possibility that the opportunity loss may be re-

covered by cooperative actions taken by the firms.


4.4.1. Complete recovery

We have already seen that in those cases where �m < m�FULL some of the oppor-

tunity loss can be recovered as �m is increased, or as k exogenously increases.

Equivalently, some of the loss could be recovered by the firms orchestrating their

own supplemental information exchange arrangement, such that the total amount of

information purchased and exchanged (or disclosed) equals m�FULL. This would in-

volve additional purchase and exchange of ðm�FULL � �mÞ observations and would

reduce OL2 to zero. Thus a complete recovery of the opportunity loss imposed by

mandating disclosure levels is feasible, if k; d parameters are such that full disclosure

is optimal and therefore OL1 is zero. 27;28

4.4.2. No recovery

Conversely, no willful recovery of the opportunity loss may be feasible when full

disclosure is optimal but the level of mandated disclosure already exceeds the opti-

mal full disclosure level, �m > m�FULL.

29 However, as we saw in the previous sub-

section, some of the loss may be recovered spontaneously if k were to decrease.

4.4.3. Partial recovery

In the case where no disclosure is optimal, OL1 is positive, and regardless of the

level of �m, partial recovery at best is feasible. If �m < m�FULL, then, as above, OL2 can

be recovered completely by purchasing and exchanging m�FULL � �m additional ob-

servations. However, this still leaves OL1 in its entirety. A potentially more efficient

way to recover part of the total opportunity loss, when OL1 is positive, is for the

firms to all purchase � additional observations none of which are subsequently ex-

changed or disclosed. This produces a partial disclosure regime, in which �mþ �observations are purchased but only �m are disclosed, as mandated. The optimal level

of additional observations to purchase will be a function of the existing mandated

disclosure level, �m. The resulting partial disclosure regime does not necessarilyproduce higher firm welfare than if the firms had simply acted to drive OL2 to zero.

Observation 2. A partial disclosure regime always optimally recovers some of OL1

when d < b2ð ffiffiffi

np � 1Þ.

In this parameter region, the slope of partial disclosure profits with respect to � ispositive, when evaluated at � ¼ 0 and �m ¼ m�

FULL. While this condition on the first

order partial derivative is sufficient for a partial disclosure arrangement to be pre-ferred to the optimal full disclosure regime at �m, it is unlikely to be necessary.

27 This occurs when d > b2ðn� 1Þ;8k (Lemma 1), or when d 2 ½b

2ðffiffiffin

p� 1Þ; b

2ðn� 1Þ� and k 2 ½k7; k5�

(Lemma 3), for those cases where �m < m�FULL.

28 In this case the attitude toward additional increases in mandated disclosure is neutral as long as

�m < m�FULL, since the supplemental information acquisition and exchange could always be correspond-

ingly curtailed.29 For example, if d > b

2ðn� 1Þ (Lemma 1).


A final observation regarding recovery of the opportunity loss through a partial

disclosure regime follows.

Observation 3. A partial disclosure regime in which �m observations are disclosedpublicly and a further ��ð�mÞ observations are acquired by each firm and kept private cannever fully recover the opportunity loss caused by imposing mandated disclosure re-quirements of �m, �m > 0.

This follows directly from the choice of optimal partial disclosure regime being

constrained by f 2 ½�m;m�, where �m > 0. It is interesting in light of the earlier result in

Proposition 1 that partial disclosure is never optimal in the unconstrained infor-

mation regime choice problem.

5. Conclusion

The objective here has been to shed light on the SEC/FASB�s disclosure choice

problem, recognizing (a) that the phenomenon of mandated disclosure creates a

setting of mutual information sharing, and (b) that by mandating a disclosure level

the SEC/FASB imposes an opportunity cost on firms in the form of potentially

destroying those firms� ability to jointly create an optimal information acquisition

and disclosure regime. This has enabled the development of a welfare measure,that the SEC/FASB can use in understanding (a) welfare effects and (b) firms�attitudes toward changes in mandated disclosure levels. The results highlight the

notion that firms can be made better off with higher levels of disclosure and yet

still be considerably worse off than if mandated disclosures were removed entirely.

Furthermore, as information cost declines firms prefer increases in disclosure, even

though their welfare in opportunity loss terms may increase or decrease. The

larger picture from the SEC/FASB�s viewpoint however needs to trade off the

product market opportunity loss effects of mandated disclosure described herewith desired effects on other market participants, such as consumers and financial

decision makers. 30

Appendix A. Derivation of equilibrium output

Substituting (6) into (5), the first order condition can be written as

30 Pre

that hi

Eða� cjIiÞ � 2ðbþ dÞ p�

þ qF nfsF þ qMðm� f ÞsMi�

� bðn� 1Þ ph

þ qF nfsF þ qMðm� f ÞEðsMj jIiÞi¼ 0: ðA:1Þ

liminary analysis indicates that expected consumer welfare is increasing in both m and f , indicatinggher mandated levels of firm disclosure are always regarded favorably by consumers.


The two aggregate signals in firm i�s post-disclosure information set are both noisy

signals about the unknown parameter a� c. Bayesian updating by firm i results inthe posterior estimate Eða� cjIiÞ, where: 31

31 See32 Fir

Finally

Eða� cjIiÞ ¼sF

r2m=nfþ sMi

r2m=ðm�f Þ þla�cr2a

nfr2mþ m�f

r2mþ 1

r2a

; ðA:2Þ

Eða� cjIiÞ ¼ kAsF þ kBsMi þ kCðla � cÞ; ðA:3Þ

where

kA ¼ nfr2a

r2m þ ½nf þ ðm� f Þ�r2

a

; ðA:4Þ

kB ¼ ðm� f Þr2a


a

ðA:5Þ

and

kC ¼ r2m


a

: ðA:6Þ

Because firm i�s information set contains no information about the error terms

in the information privately retained by its competitors, updating by firm i aboutthe private information retained by its competitors is limited to the a� c term.

Thus:

EðsMj jIiÞ ¼ Eða� cjIiÞ: ðA:7Þ

Substituting Eqs. (A.3) and (A.7) into the first order condition (A.1) gives:

kAsF þ kBsMi þ kCðla � cÞ � 2ðbþ dÞ p�

þ qF nfsF þ qMðm� f ÞsMi�

� bðn� 1Þ p�

þ qF nfsF þ ðm� f ÞqM ½kAsF þ kBsMi þ kCðla � cÞ��¼ 0: ðA:8Þ

To solve for the optimal output strategy (i.e. identify p; qF and qM in Eq. (6)) this

first order condition must hold for all values of both signals. Thus the following

three conditions must hold: 32

kA � 2ðbþ dÞnfqF � bðn� 1ÞnfqF � bðn� 1Þðm� f ÞqMkA ¼ 0;

kB � 2ðbþ dÞðm� f ÞqM � bðn� 1Þðm� f ÞqMkB ¼ 0;

kCðla � cÞ � 2ðbþ dÞp � bðn� 1Þp � bðn� 1ÞqMkCðla � cÞ ¼ 0:

DeGroot (1970).

st, the coefficient of terms in sF must be zero. Second, the coefficient of terms in sMi must be zero.

, the constant term must also net to zero.


Simultaneously solving these equations for p; qF and qM and substituting into Eq.

(6) gives the equilibrium output strategy as a function of the post-disclosure infor-

mation set: 33

33 Fro

xiðIiÞ ¼ qM ar2m

r2a

ðla

�� cÞ þ anfsF þ ðm� f ÞsMi

�;

where

qMðm; f Þ ¼1

2ðbþ dÞ r2mr2aþ nf

h iþ 2ðbþ dÞ þ bðn� 1Þ½ �ðm� f Þ

ðA:9Þ

and

a ¼ 2ðbþ dÞ2ðbþ dÞ þ bðn� 1Þ : ðA:10Þ

Appendix B. Proof of Proposition 1

The proof shows that (a) a unique finite-valued stationary point exists for the

EPiðf jmÞ function, (b) that stationary point is a minimum, and (c) EPiðf ¼ mjmÞ >EPiðf ¼ 0jmÞ when d > dcrit.

(a) From Eqs. (14), (A.9) and (A.10) the first order partial derivative of expected

profits with respect to f is

oEPiðf jmÞof

¼ a2r2m f ðan� 1Þða2n� 2aþ 1Þ þ mð3a2n� 2an� 2aþ 1Þ þ aða2n� 1Þ½ �

4ðbþ dÞ a r2mr2aþ mþ f ðan� 1Þ

h i3 :

Since the numerator is linear in f there exists a unique finite-valued stationary point

whose f value may or may not be in the relevant range ½0;m�). Since the denominator

is cubic in f , the slope of the expected profit function will also tend to zero as f tends

to plus or minus infinity. Thus the expected profit function has three stationarypoints of which only one is finite-valued. Let the finite valued stationary point be

fstat.(b) Evaluating the second order derivative of the expected profit function at fstat

gives:

o2EPiðf jmÞof 2

��fstat

¼ a2r2mðan� 1Þða2n� 2aþ 1Þ

4ðbþ dÞ a r2mr2aþ mþ f ðan� 1Þ

h i3 ;

m Basar (1978) it is known that this strategy constitutes the unique Bayesian equilibrium.


which is always positive. Consequently, the finite stationary point is a minimum.

This guarantees that partial disclosure cannot maximize firms� expected profits. The

optimization of profit so far has not considered the constraint that f 2 ½0;m�.Consideration of this constraint implies that one of three situations may occur. If fstatlies below the range of feasible f values: ½0;m�, then the expected profit function is

positively sloped when f 2 ½0;m� and the optimal value of f is f � ¼ m. If fstat liesabove the range of feasible f values: ½0;m�, then the expected profit function is

negatively sloped when f 2 ½0;m� and the optimal value of f is f � ¼ 0. If fstat lieswithin the range of feasible f values: ½0;m�, then the expected profit function is U-

shaped when f 2 ½0;m� and the optimal value of f is f � 2 f0;mg, depending on in-

dustry parameters.

(c) If fstat 2 ½0;m� then f � 2 f0;mg. In this setting of exogenous m, which value f �

takes is determined by comparing expected profits at f ¼ 0 and f ¼ m:

EPiðf ¼ mjmÞ ¼ nma2r4a

4ðbþ dÞ½r2m þ nmr2

a�

and

EPiðf ¼ 0jmÞ ¼ ma2r4aðr2

amþ r2mÞ

4ðbþ dÞ½ar2m þ mr2

a�2:

The condition that EPiðf ¼ mjmÞ � EPiðf ¼ 0jmÞ > 0 requires that:

4d2ðr2m þ mr2

aÞ þ 4bdðr2m þ mr2

aÞ � b2ðn� 1Þ½ðr2m þ mðnþ 1Þr2

aÞ� > 0:

Since the left hand side of this inequality is a positive quadratic in d, it is positive forvalues of d outside its roots. Since d is assumed non-negative we have that:

d > � b2

1

"�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin½r2

m þ nmr2a�

½r2m þ mr2

a�

s #:

Thus f � ¼ m if

d > � b2

1

"�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin½r2

m þ nmr2a�

½r2m þ mr2

a�

s #:

Otherwise f � ¼ 0: �

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