Optimal Internal Control Regulation

Stefan F. Schantl

University of Melbourne

and

Alfred Wagenhofer

University of Graz

Abstract: Regulators increasingly rely on regulation of firms’ internal controls (IC) to

prevent accounting fraud. This paper considers a welfare-maximizing regulator’s task to set

the optimal tightness of an IC standard. We study an economy with strategic and compliant

owners who sell their firms to investors who then make investment decisions based on the

financial report. We show that IC standard tightness and the cost of non-compliance are

complements if the cost is below a threshold and are substitutes if the cost becomes higher.

The complementarity arises because a minimum non-compliance cost is necessary to induce

strategic owners to invest in IC effectiveness. Once the non-compliance cost grows further,

the optimal standard declines to avoid inducing too much costly investment in the IC system.

The greater the proportion of strategic owners, the more effective is IC regulation and the

higher is the welfare-maximizing IC standard.

Keywords: Internal controls, internal control standards, compliance, enforcement, regulation,

financial reporting quality.

JEL: D60, M41, M48

9 August 2017

1

1. Introduction

Securities regulators, such as the United States Securities Exchange Commission (SEC),

increasingly rely on regulation of firms’ internal controls over financial reporting to prevent

fraud and increase investment efficiency. For example, in its financial report 2015 the SEC

emphasizes “insufficient internal controls” as one area of particular attention in pursuing its

goal of enhancing accounting standard compliance (SEC 2015, p. 35). The Securities

Exchange Act 1934 requires the management of listed firms to maintain internal controls over

financial reporting to prevent misstatements in financial reports. The Sarbanes-Oxley Act

(SOX), as amended by the Dodd-Frank Act, mandates that a firm’s management assesses the

effectiveness of internal controls and discloses significant deficiencies and material

weaknesses. In addition, the auditor is required to verify management’s disclosures.

Regulations in many other countries mandate similar requirements for implementing effective

internal controls over financial reporting.

However, internal control (IC) regulation imposes significant implementation and

maintenance costs on all firms subject to the regulation, even on firms with a low accounting

fraud risk. For example, after the enactment of SOX many firms decided to go dark (Leuz,

Triantis, and Wang 2008) or go private (Engel, Hayes, and Wang 2007) indicating that SOX

imposed significant compliance costs on firms. Further Krishnan, Rama, and Zhang (2008)

provide evidence showing that a large part of compliance costs caused by SOX is attributable

to the establishment of sophisticated internal control processes.1

This paper studies (i) under which conditions IC regulation is welfare-enhancing and

(ii) how tight an IC standard should be set given a compliance mechanism to enforce such a

standard. We consider a regulator whose objective is to maximize social welfare, which

comprises the economic wealth created by firms less the direct costs of implementing and

maintaining IC systems. Considering both the benefits and costs of IC regulation is important

1 Substantial compliance costs were the main reason that smaller firms were later exempt from SOX.

2

to assess how particular regulation affects financial reporting quality and investment behavior

and to provide guidance on how IC standards should be set.

We consider an economy with firms that differ in their productivity following, among

others, Schwartz (1997), Dye (2002), and Lu and Sapra (2009). Firms are sold for

generational reasons to new investors in the capital market. Investors use the financial reports

to update their beliefs about firm productivity, which is important for their investment

decision in a growth opportunity the firm offers. Anticipating their investment, they determine

the price they offer to acquire a firm. Financial reporting therefore has a direct effect on

investment and welfare, which implies that financial reporting quality is an important factor in

this economy.

There are two types of firm owners, owners that are compliant with standards and

regulations and others that are strategic and choose whether to comply or to deviate in order

to maximize their expected wealth. The strategic owners may engage in costly manipulation

of the financial report to achieve a higher price for their firm. The likelihood of success of

manipulation decreases in the effectiveness of the IC system they have previously

implemented. Strategic owners can also choose not to comply with IC regulation.

We establish that strategic owners always underinvest in IC effectiveness relative to a

given IC standard. This result is reminiscent of a “shadow standard” in the sense of Dye

(2002). IC underinvestment by strategic owners is driven by two factors: (i) compliance costs

and (ii) the benefit from the increased success probability of future manipulation. The latter

factor can be so strong that strategic owners do not implement an IC system at all. Once the

costs of non-compliance with IC regulation (reputation or IC enforcement costs) exceed a

certain threshold, strategic owners will invest in the effectiveness of the IC system. Through

its effect on the implemented IC system, a higher IC standard increases equilibrium reporting

quality, the transaction price, and investment efficiency. Surprisingly, it does not necessarily

reduce owners’ manipulation effort because owners trade off the cost of manipulation with

the benefits that arise from greater sensitivity of the price and the reduced cost of direct

enforcement that follows from the greater reporting quality. We also find that direct public

3

enforcement has a reinforcing effect on IC effectiveness through mitigating IC

underinvestment incentives, which in turn alleviates the necessity for stringent IC regulation.

We then determine the welfare-maximizing tightness of the IC standard. We emphasize that

IC standards and the non-compliance costs imposed by IC enforcement are inherently linked.

When strategic owners’ IC underinvestment incentives are stronger than the non-compliance

costs from enforcement of the IC standard, IC regulation is strictly welfare inefficient because

strategic owners do not implement an IC system, while it (unnecessarily) imposes compliance

costs on compliant firms.

Given that the proportion of compliant to strategic owners is sufficiently small and

when IC non-compliance costs lie beyond a certain threshold, the optimal IC standard

prescribes the maximal effectiveness; and if they increase further, there is another threshold

beyond which it becomes optimal to lower the IC standard since the induced compliance costs

become very large. Intuitively, setting the highest IC standard complements the effect of

“intermediate” costs of non-compliance from IC enforcement to induce strategic owners to

invest in an IC system. In contrast, if the proportion of compliant to strategic owners is small

either no IC regulation or an IC standard that is not too high is welfare-optimal. The reason is

that compliant owners’ compliance costs represent costs of IC regulation without a

corresponding benefit and too tight an IC standard would lead to excess costs that are socially

inefficient.

Our paper has important implications for regulators. For example, the SEC’s current

strategy is to enforce IC regulation through external auditors and reporting of IC weaknesses

and deficiencies (Chan 2016) and relies, to a large extent, on firms’ reputation to comply with

the regulation. The combined evidence provided by Ashbaugh-Skaife, Collins, Kinney, and

LaFond (2008), Beneish, Billings, and Hodder (2008), and Doyle, Ge, and McVay (2007a) is

in line with the argument that firms with reported IC weaknesses publish lower quality

financial reports and face a higher cost of capital. However, the reporting of IC weaknesses by

managers and auditors as required under SOX Sections 302 and 404 reduce firms’ incentives

to underinvest in ICs, but does not fully eliminate these incentives. One implication of our

4

model therefore is that without proper IC enforcement that contributes to outweighing

strategic owners’ IC underinvestment incentives by imposing sufficiently large non-

compliance costs, any IC regulation is strictly welfare inefficient as it imposes excessive

compliance costs on firms with low manipulation risk. A second implication of our paper is

that the optimal tightness of IC standards crucially depends on the composition of firms in an

economy: when firms’ corporate governance and culture is generally law abiding, too tight IC

standards lead to excess compliance costs on the large proportion of compliant firms, thus

reducing social welfare.

Our paper has also implications on the relation between IC standard tightness and IC

enforcement. Conventional wisdom suggests that a tighter IC standard requires a stronger IC

enforcement implying that stronger enforcement allows the regulator to set a tighter standard.

Our analysis shows that this intuition is incomplete: the optimal tightness of an IC standard

and the strength of IC enforcement are complements if non-compliance costs are low, but

substitutes if they are large. Low enforcement does not provide sufficient incentives to

implement an effective IC system; hence, it is only costly to compliant owners and not

efficient. Increasing enforcement induces strategic owners to implement an effective IC

system, but the cost to compliant owners is still so strong that the optimal IC standard is set

not too high at first, but then increases in enforcement. Finally, if enforcement becomes

sufficiently strong, the regulator optimally reduces the standard to trade off greater investment

efficiency with higher IC implementation costs. With these results, our paper contributes to a

better understanding of the factors that determine the desirable tightness of IC standards.

2. Related literature

This paper is related to the literature in auditing and corporate governance which studies

the effectiveness of internal controls to mitigate manipulation of financial reports. The

literature on the endogenous choice of IC systems2 typically assumes that the manager or the

2 Models with exogenous internal control effectiveness include, e.g., Smith, Tiras, and Vichitlekarn (2000) and

Marinovic (2013).

5

owner of a firm invests into the effectiveness of an IC system before the manager observes

private information and chooses a level of manipulation of a financial report. Then an auditor

performs IC and substantial testing before releasing the financial report to outside investors

who base their pricing and investment decisions on that information. Pae and Yoo (2001)

identify conditions under which an increase in auditor liability can exacerbate an owner’s IC

underinvestment incentives and lower audit quality. Patterson and Smith (2007) study the

effect of an auditor’s report of IC weaknesses and find that when managers are penalized for

non-compliance with IC regulation, they implement more effective ICs and manipulate less,

although it can reduce the auditor’s effort to testing the IC system, which ex ante increases

audit risk. Chan (2016) studies the effects of the disclosure of IC weaknesses if the accounting

information is used not only for valuation but also to incentivize the manager. He assumes

that the owner implements the IC system and shows that the disclosure of IC weaknesses

increases investment in ICs and can strictly increase his expected payoff. These papers model

a strategic auditor as an enforcement mechanism, but take IC regulation as given, whereas the

present paper investigates whether it is welfare-optimal to impose IC regulation on firms in

the first place.

Other literature investigates additional forces that affect investment in IC systems in the

absence of auditing. Ewert and Wagenhofer (2017) find that a more conservative accounting

system enhances the incentives of managers to invest in more effective IC systems, thus

curbing their own benefit of manipulation. Gao and Zhang (2016) show that a manager’s

observable investment in an IC system creates a positive externality for peer firms because

managers manipulate more if they expect other firms to manipulate more. Since firms do not

internalize this effect, they ex ante choose a less effective IC system. Gao and Zhang use this

insight to argue that regulation of IC system choice can strictly improve social welfare. In

contrast, we show that IC regulation can be strictly welfare decreasing.

Our paper is also related to the literature on optimal regulation to ensure compliance

with accounting standards. For example, Friedman and Heinle (2016) study a cash diversion

problem in which an insider diverts cash, which is harmful to outsiders and overall impairs

6

social welfare. A social planner chooses the strictness and uniformity of a regulation, where

the regulation resembles exogenously given IC regulation that detects cash diversion with

some probability. They also assume that firms fully implement the regulation. The main focus

of Friedman and Heinle (2016) is costly lobbying by firms to influence the social planner

when choosing the regulation and they show that uniform regulation gives rise to a free-rider

problem, which overall reduces distortions from lobbying. In contrast, in our paper firm

owners strategically decide whether to comply with the IC regulation, and this decision is

important for the assessment of IC regulation’s welfare-effectiveness.

Other papers specifically study the setting and enforcement of accounting standards

directly, whereas the present paper studies IC regulation as a compliance mechanism to

enforce existing accounting standards. Dye (2002) studies the setting of an accounting

standard that requires classifying events into binary categories, where the manager can

manipulate the outcome of the classification at a cost. He finds that firms apply a “shadow

standard,” that is different to the officially stated standard and considers how sophisticated or

naïve standard setters set the classification threshold. Laux and Stocken (2017) study two

instruments that jointly influence financial reporting, a classification standard and ex post

enforcement. In their model, the manager engages in costly research and development and

discovers a project and the classification accounting system provides information about the

project to investors who decide whether to finance it. The standard and its enforcement affect

manipulation attempts and investment in R&D. Schantl (2017) establishes that the penalties

incurred by firms from regulatory enforcement can yield an inversely U-shaped association

between the tightness of the accounting standard and firm value as assessed by outside

investors. He further shows that more informative financial reporting can coexist with a lower

firm value. Gao and Zhang (2017) find that tightening auditing standards can reduce

incentives of auditors to acquire professional expertise.

7

3. Model

Production technology and reporting system

The economy consists of two types of firms, which differ in their level of productivity.

Productivity is a result of the availability of special core competences, such as technical

expertise, know-how and organizational and process knowledge, and it determines a

comparative advantage (or disadvantage) of the firm in undertaking projects. With equal

probability, productivity can be high ( = 1) or low ( = 0). Productivity affects the value of

existing assets and that of a growth option. The growth option consists of the opportunity to

realize an investment project that yields an expected project return of 2 I from an

investment volume I. The production technology is similar to that used in Schwartz (1997),

Dye (2002), and Lu and Sapra (2009).

The firm’s shares are held by a representative owner who also acts as the top manager

of the firm (the “owner” or “previous owner”). The owner must sell the firm to investors for

life-cycle reasons before being able to realize the investment opportunity. For example,

investing requires an amount of capital the owner is unable to raise due to a short time

horizon.

Before selling the firm, the owner issues a mandatory financial report that is informative

about the firm’s productivity . The financial report is denoted by r ∈{rL, rH}, where rL

indicates a low and rH a high productivity type, respectively. Investors determine the price of

the firm’s shares based on the financial report r and also use the report to decide on the

investment volume I.3 The capital market is perfectly competitive and investors hold rational

expectations about the firm’s productivity. They consequently price the firm’s shares

conditional on report r at the value of the investment project. The share price is denoted by

P(r), where PL ≡ P(rL) and PH ≡ P(rH).

3 We thus assume that the new investors do not immediately learn the productivity after acquiring control of the

firm. A reason may be that productivity is multidimensional and hidden in the organization, such that an in-depth

understanding of productivity only develops over time.

8

To focus on manipulation of financial reports, we assume that the preexisting

accounting system generates a signal that perfectly informs the current owner about the firm’s

productivity . This assumption implies that the applicable accounting standards are “perfect”

in the sense that they allow and require different accounting for different productivity.

If owners were to report productivity truthfully, they would issue a financial report rH

if = 1 and rL if = 0. However, owners can manipulate the financial report and report some r

≠ at a personal cost. For example, an owner may not impair existing assets if productivity

turns out to be low.

We assume two types of owners, compliant or non-compliant (“strategic”) owners to

capture the fact that some firms are more prone to deviating from legal requirements than

others.4 Possible reasons are individual differences of decision makers and corporate culture.

The portion of strategic owner types is q and of compliant types 1 – q, where q ∈ (0, 1], and

the type is unrelated to the firm’s productivity . Each owner in the economy privately learns

her actual type ex ante, but is unable to credibly communicate this fact. Compliant types obey

regulations and standards as mandated in the regulatory regime and always report the firm’s

productivity truthfully. Strategic owners decide whether to report truthfully or to deviate to

maximize their expected utility. Any enforcement mechanism is therefore concerned about

the decisions made by strategic owners.

Owners are risk neutral and their utility equals the expected selling price P less potential

enforcement penalties (see below). Because PH > PL (this will be shown to hold in

equilibrium), strategic owners have an incentive to manipulate r only if the firm has low

productivity ( = 0). We model manipulation as the probability b ∈ (0, 1] with which strategic

owners bias the report so as to report rH after observing = 0; the success of manipulation

depends on the effectiveness of an internal control system as will be discussed below. If = 1,

the accounting system already reports rH and there is no benefit of manipulation. To

4 This assumption allows us to assess the costs of uniform regulation on different types of firms. See also

Patterson and Smith (2007) for a similar assumption.

9

manipulate with success probability b, owners incur a private and unobservable disutility of

kb2/2, where k > 0 parameterizes the disutility.

Internal control system

Regulation can require firms to establish, operate, and maintain an IC system, which

assures the quality of financial reporting. The main function of the IC system is to correct

errors and to detect manipulation in the financial report and, by doing so, to assist in ensuring

compliance with the accounting standard. We assume a regulator sets or adopts IC regulation

which prescribes a certain level of IC effectiveness which we shall refer to as IC standard and

which is denoted by S ∈ [0, 1]. S = 0 is equivalent to no IC regulation. Our regulator can be a

single securities regulator who is also in charge of setting and enforcement of the IC standard,

or it can adopt or delegate these activities to another public or private institution. To comply

with IC regulation, firms should implement an IC system with effectiveness s ≥ S. A firm is

non-compliant with S if s < S; in this case an IC weakness arises.

Because we assume the underlying accounting system is perfect, errors only arise if

owners manipulate the report. The effectiveness of the IC system is a probability s ∈ [0, 1]

with which the system detects and corrects manipulation before the financial report is

published. After observing their types, owners choose the effectiveness s of the IC system

before observing productivity . A manipulation attempt b of the owner leads to a

manipulated report rH with probability Pr(rH | = 0) = (1 – s)b; manipulation is unsuccessful

with probability Pr(rL | = 0) = (1 – b) + sb.

Implementing an IC system with effectiveness s entails a private cost of cs2/4 to the

owner, where c > 0. Costs of complying with mandatory IC regulation can be high in practice,

as evidenced in studies such as, e.g., Engel, Hayes, and Wang (2007), Leuz, Triantis, and

Wang (2008), and Krishnan, Rama, and Zhang (2008). Compliant owners do not manipulate

the financial report and the optimal s = 0 because the IC system generates no benefit but is

costly. However, because these owners are compliant, they implement the standard s = S.

Strategic owners choose s by trading off costs and benefits of implementing s. In line with

prior literature on IC systems (e.g., Smith, Tiras, and Vichitlekarn 2000, Patterson and Smith

10

2007, Chan 2016), we assume that the implemented IC effectiveness s and the cost cs2/4 are

unobservable to the public, so that the investors directly infer neither s nor the owner’s type if

strategic owners choose s ≠ S.5

The regulator is benevolent and maximizes social welfare by choosing IC standard S.6

Social welfare comprises the expected wealth generated by undertaking the new investment

project and the cost cs2/4 of implementing an IC system with effectiveness s. We do not

include the owner’s disutility of manipulation kb2/2 in the social welfare measure because it is

socially undesirable. Any non-compliance costs such as penalties are direct wealth transfers

and do not affect social welfare.

Compliance mechanisms

Enforcement of the accounting standard relies on a direct regulatory enforcement

mechanism and the IC regulation. These mechanisms can be in the responsibility of a

securities regulator or other public or private enforcement bodies, or they can be left to market

forces, e.g., by prompting a reputation loss of the owner. The SEC is an archetypal example

of a regulatory institution that is in charge of the quality of financial reporting and has the

power to set standards and enforce them. It can perform certain tasks by itself, but can also

delegate them to other institutions that it oversees. It has also the power to inflict penalties for

non-compliance with accounting standards and for inadequate IC systems.

Regulatory enforcement of the accounting standard detects successful manipulation of a

financial report with some probability that is determined by both the probability of (random)

investigation and the probability of identifying manipulation. If a violation is detected, the

5 If the effectiveness or the cost of implementing the IC system were observable, investors would be able to

discern compliant and strategic owners. To avoid that, we assume the cost is a private cost; alternatively,

strategic owners who wish to deviate from S but remain undetected, can pay out the cost cS2/4 for some

unproductive purpose.

6 A related objective of a regulator is the (general) quality of financial reporting that is induced by regulation and

enforcement. This objective is implicit in our analysis as the information content of the financial report is

monotonic in the investment efficiency that determines social welfare.

11

enforcer inflicts a penalty on the firm. Both the detection probability and the penalty size

determine the expected penalty fA > 0, which we assume to be common knowledge. We refer

to fA also as the strength of direct public enforcement of accounting standards. Detected

manipulation can also lead to a restatement, but this information comes only after investors

acted based on the original financial report. Therefore, such enforcement lowers the owner’s

utility of manipulation in expectation.

We assume that even though regulatory enforcement ensues after the sale of the firm to

new investors, the penalty is borne by the original owner.7 That is, if some penalties are

inflicted onto the firm after acquisition, the new owners can reclaim the penalty from the

original owner who was responsible for any non-compliance with the accounting standard.

Acquisition contracts often contain contractual clauses with warranty for penalties for any

wrong-doings by the previous management.

In order to provide owners with incentives to consider the implementation of an IC

system we assume the existence of non-compliance costs that are privately incurred by an

owner whenever s < S. The cost of non-compliance can either be direct penalties imposed by

a regulatory enforcer or indirect costs either from contract provisions or a loss of reputation.8

We model the cost of non-compliance as a function of the deviation from the standard,

2

– / 4Cf S s , where fC > 0 captures the strength of the IC enforcement mechanism. There is

no non-compliance cost if s ≥ S. Much of our analysis is cast on the non-compliance cost

parameter fC, which for simplicity we also refer to as non-compliance cost or IC enforcement

cost.

7 Otherwise, rational investors would reduce the acquisition price by the conjectured penalties, which are a

constant in the owner’s expected utility that determines the manipulation effort in the first place. This would lead

to more complicated strategies without offering more economic insights. See Schantl (2017) for the valuation

implications of firm-incurred enforcement penalties.

8 In the current institutional setting in the U.S., a criminal offence is if the owner does not report IC weaknesses

as required under SOX Section 302 (civil provision) and Section 906 (criminal provision).

12

Sequence of events

Regulator chooses IC regulation with tightness S.

Nature determines the owner’s type (compliant, strategic) and owners learn their type.

Owners implement an IC system with effectiveness s.

Nature determines the firm’s productivity .

Owners observe a perfect accounting signal about productivity

Strategic owners engage in manipulation b = b() of the accounting signal.

Financial report r is publicly disclosed.

Firm is sold to new investors at market price P(r).

New investors invest I(r) in the growth project.

Enforcement of the accounting standard and the IC standard occur and non-compliance

costs are inflicted on the previous owner.

Payoff from investment project realizes and the firm is liquidated.

We discuss the significance of the model assumptions on our results and alternative

assumptions in Section 5.

4. Results

4.1. Internal controls and manipulation equilibrium

Given the sequential nature of the game, we initially take the tightness S of the IC

standard as fixed and examine the current owner’s IC effectiveness and manipulation

decisions and the inference of productivity by the investors (who then choose the acquisition

price and investment volume of the growth project). We derive the equilibrium conditions by

backward induction and begin with the final stage in the game, the investors’ investment and

pricing decisions after they observe the financial report r. Proofs are in the appendix.

Investors conjecture that compliant owners do not manipulate the financial report and

strategic owners attempt manipulation only if they observe low productivity, = 0, with

13

probability b , which is successful with probability ˆ ˆ(1 ).b s 9 A high report rH is indicative of

high productivity, whereas a low report rL is indicative of low productivity, so that PH > PL.

Conjecturing ˆ ˆH LP P , strategic owners have an incentive to report rH regardless of the actual

. If = 1, the report rH follows directly (and the owners have no incentive to underreport) and

if = 0 the owners have an incentive to engage in manipulation. This results in the following

beliefs by investors:

ˆ ˆ;1

Pr( 1| )ˆ1 1 )

,( ˆ

Hr b ssqb

(1)

and Pr( 1 0ˆ| ) .;Lr s

A report rL perfectly informs that = 0, whereas the probability that = 1 after rH is observed

equals ∈ [1/(1 + q), 1].

Given these beliefs, investors choose the investment volume in the growth project, I(r),

to maximize the value of the firm by maximizing

ˆ ˆ2Pr( 1| , ( ), ) ( ) ( )r b r s I r I r . (2)

for each report r = rL, rH. This implies I(rL) = 0 and 2 2ˆ ˆ( ) Pr( 1| , ( ), )H H HI r r b r s .

Note that the probability that the investment pays off, ˆ ˆPr( 1| , ( ), )r b r s , fully determines

the price because in our payoff function the best the investors can do is to “match” the

investment to the conjectured probability that the project yields a positive return.

In a competitive market, the price of the firm reflects the expected return from the future

investment opportunity, which implies

PL = 0

and 2

2

1

ˆ1 ˆ(1 )HP

qb s

> 0. (3)

This confirms the owners’ conjecture ˆ ˆ 0H LP P for all , [0,1]b s and the induced incentive

to manipulate only if productivity is = 0, which underlies the investors’ formation of beliefs.

9 A “hat” indicates the rational conjecture of the respective variable.

14

To choose the manipulation effort, strategic owners, upon observing = 0 maximize

their conditional expected utility based on the conjectured price ˆ ,HP

2

ˆPr( | 0)( )2

H H A

br P f k =

2

ˆ(1 )2

bs bY k (4)

where we use Y ≡ (PH – fA) and ˆ ˆ( )H AY P f for the conjecture of Y. The expected

incremental benefit of manipulation is ˆ(1 ) ,s Y which consists of the price differential

ˆ ˆ ˆH L HP P P less the expected penalty from detected manipulation fA, weighted by the

probability (1 – s) that the IC system does not correct manipulation. Manipulation overall

pays off only if ˆ(1 )s Y > 0, which requires that the enforcement of accounting standards is

not so strong as to deter manipulation directly. Hence, owners choose b = 0 if ˆ 0.Y If

ˆ 0,Y then the b that maximizes (4) equals

1 ˆmin (1 ) ,1b k s Y . (5)

Inserting b, owners’ conditional expected utility for = 0 becomes

1 2 2ˆ(1 ) ˆ if 0 (1 )2ˆ( 0, )

ˆ ˆ(1 ) if (1 )2

k s Ys Y k

EU Yk

s Y s Y k

for b < 1 and b = 1, respectively.

The owners determine IC effectiveness s before observing the accounting signal over

the firm’s productivity. Recall that the prior probability of = 1 is 1/2. Compliant owners

always comply with the standard, s = S. Strategic owners choose s to maximize the ex ante

expected utility,

2 2 2( )ˆPr( ) Pr( 0, ) Pr( 0)

2 4 4H H H A C

b s S sr P r f k c f

(6)

Due of the fact that b is bound from above by 1, this expected utility is

1 2 2 2 2

2 2

ˆ1 (1 ) ( )ˆ ˆ if 0 (1 )2 4 4 4ˆ( )

ˆ1 (1 ) 1 ( )ˆ ˆ if (1 ) ,2 2 2 4 4

H C

H

H C

k s Y s S sP c f s Y k

EU Ps Y s S s

P k c f s Y k

15

which yields

1 2

1 2

ˆˆmax 0, if 0 (1 )

ˆ

ˆ ifˆ

ma (1 )x 0, .

C

C

B

C

f S k Ys Y k

c f k

f S Y

c f

Ys

s Y k

(7)

An equilibrium in this game between the current owner and new investors is defined as

follows.

Definition: For a given S, a perfect Bayesian equilibrium (PBE) consists of the investors’

investment I(r) and pricing rule P(r), the owner’s manipulation strategy b, and IC

effectiveness s such that:

(i) Investment volume I(r) maximizes

ˆ ˆ2Pr( 1| , ( ), )r b r s I I for r = rL, rH;

(ii) Price P(r) satisfies

ˆ ˆ( ) 2Pr( 1| , ( ), ) ( ) ( )P r r b r s I r I r for r = rL, rH;

(iii) Manipulation b by the owners is as follows:

Compliant owners always choose b = 0;

Strategic owners choose b = 0 if = 1 and chooses b after observing = 0 to

maximize

2

ˆPr( | 0)2

H

br Y k ,

where ˆ ˆ( )H AY P f ;

(iv) Implementation of IC effectiveness s by the owners is as follows, where b is determined

according to (iii):

Compliant owners always choose s = S;

Strategic owners choose s to maximize

2 2 2( )ˆPr( ) Pr( 0, ) Pr( 0)

2 4 4H H H A C

b s S sr P r f k c f

.

16

In equilibrium, all conjectures equal the actual values, i.e., ˆ( ) ( ),I r I r ˆ( ) ( ),P r P r ˆ ,b b

and s s .

It is apparent from (5) that whether (1 )s Y is greater or less than k, is important

because it directly affects the equilibrium manipulation b. If (1 )s Y > k, then the

incremental benefit of manipulation exceeds its incremental cost factor, and strategic owners

always attempt to report rH (i.e., b = 1). On the other hand, if Y < 0, the expected costs from

direct public enforcement of accounting standards, outweighs the expected benefit so that

strategic owners never manipulate (b = 0). To focus on the more interesting cases for our

analysis, we restrict the strength of direct public enforcement fA as follows:

2

1

(1 )Af

q

, (8)

which ensures Y > 0 and b > 0 for any combination of manipulation and IC effectiveness

strategies. Notice that there is no similar natural boundary for fC because as long as fC < ∞

strategic owners will always set s < S (as we discuss later). Therefore, we state our results in

relation to k and fC to capture the level of compliance with the accounting standard and IC

regulation, respectively.10

The following proposition establishes existence of a unique equilibrium.

Proposition 1: For a given S, there exists a unique PBE with investors’ investment choice and

price determination

PL = I(rL) = 0 and

2

2

1( )

1 (1 )H H

sP I r

qb

, (9)

and strategic owners’ manipulation and IC effectiveness strategies {b, s} set out in Table 1,

where 0k and 0.Cf

10Alternatively, the boundary for k can also be restated as a boundary on fA because the threshold is k = PH – fA.

17

Table 1: Equilibrium manipulation and IC effectiveness (Proposition 1)

C Cf f C Cf f

k k

Case (a)

1

0

b k Y

s

Case (c)

1

1 2

1 2

1 2

(1 )C

C

C

C

k Y c f Sb

c f k Y

f S k Ys

c f k Y

k k

Case (b)

1

0

b

s

Case (d)

1

C

C

f S Y

b

sc f

Proposition 1 establishes the existence of two (unique) thresholds distinguishing the

four cases, k and .Cf As shown in Table 1, the four cases, (a) to (d), are characterized by

whether the strategic owners’ strategies b and s assume interior or boundary values because

both b and s are bound to be within [0, 1]. If condition (8) holds, the owners always

manipulate with some intensity (b > 0). The cost of manipulation captured by the disutility

scaling factor k can be so low that the owners always exert maximal manipulation effort (b =

1) (cases (b) and (d)). In contrast, when the owners’ disutility from manipulation is

sufficiently large, they choose an interior level of manipulation effort (cases (a) and (c)). The

second threshold relates to IC enforcement cost fC. If this cost is low, i.e.,   CCf f , then the

strategic owners do not implement an IC system (s = 0) (cases (a) and (b)); otherwise the

equilibrium IC effectiveness is interior (0 < s < 1) (cases (c) and (d)). The cases (c) and (d) are

those cases in which setting an IC standard has a positive effect on the information content of

the financial report and on firm value. We refer to these cases (and subcases thereof) in

subsequent analyses.

Figure 1 provides an example of the four cases that arise in the equilibrium. First note

that when { ,k k },C Cf f the thresholds separating case (b) from the adjacent cases (a) and

(d) are constants. If either k k or C Cf f holds, both thresholds are endogenously

determined as they both include endogenous variable , which depends on the induced b and

18

s, ( )Ck k f and ( ) .C Cf f k The information content of the financial report increases

in s and decreases in b; hence, increases towards the north-east in Figure 1.

Figure 1: Characteristics of the equilibrium for different costs of manipulation k and IC

enforcement costs fC

(Parameters: c = 3, fA = 1/6, q = 1, S = 1)

4.2. Properties of the equilibrium

In the following, we discuss main properties of the equilibrium. We start with an

observation on the implementation of IC regulation.

Lemma 1: Strategic owners always underinvest into the IC system as compared to the

prescribed IC standard S as long as S > 0, i.e., s < S.

Lemma 1 follows immediately from considering the equilibrium s in cases (c) and (d) of

Proposition 1. It is straightforward to see that the incentive to implement an IC system for

strategic owners arises from the IC non-compliance cost 2

– / 4Cf S s . Strategic owners

would never overinvest (s > S) because it is only the IC enforcement cost that induces an

s > 0. Therefore, if S = 0, the owners choose s = 0 and need not fear any cost because they

19

comply with S. If S > 0, the equilibrium s is strictly less than S in case (d) where k k ; in case

(c), where k k , we have s < S as long as c > 0 (which we assume throughout). Lemma 1

holds for any fC, and s approaches S if Cf .

Two factors drive the underinvestment in the IC system by a strategic owner. First, the

investment is directly costly to the owner. The costs comprise the direct cost of implementing

effectiveness s, cs2/4. These costs increase in fC and S, so setting S* = 1 provides the strongest

incentive to implement an s > 0. Second, underinvesting in the IC system increases the option

value of subsequent manipulation attempts. If   CCf f , this option is so valuable that

strategic owners do not implement any IC system (s = 0).

In the following, we state comparative statics analyses of the equilibrium for variations

of the main determining economic and enforcement factors. We begin with a variation of the

IC standard S, which is important to understand the effects of the optimal choice of the IC

standard, which we examine in the next subsection.

Proposition 2: Increasing the IC standard S induces

(i) an increase in the price PH; the increase is strict if s > 0;

(ii) an increase in IC effectiveness s; the increase is strict if s > 0;

(iii) a decrease or an increase in manipulation b; a necessary condition for an increase

is b < 1 and s > 0.

Proposition 2 confirms the intuition that a tighter IC standard (higher S) induces owners

to choose a more effective IC system s because higher S increases the cost of non-compliance.

This enhances the information content of the financial report and consequently improves

investors’ investment decision and the price they are willing to pay for the firm.11 However,

the induced higher price PH makes manipulation b incrementally more profitable, which again

dampens PH and works against the direct effect.

11 Recall that the information content is equivalent to the probability that a report rH indicates high productivity

, which is . The optimal investment volume is 2, and this is also the equilibrium price PH.

20

Proposition 2 (i) and (ii) state that an increase of S unambiguously increases the price

PH and IC effectiveness s. If s = 0 in equilibrium (cases (a) and (b) of Proposition 1), then a

variation of S has no effect, so s and PH are unchanged, which implies the increase is weak in

this case. If s > 0 in equilibrium (cases (c) and (d) of Proposition 1), s strictly increases in S,

which changes investors’ beliefs and results in higher investment volume and price PH.

However, the tension between the direct and indirect effects is apparent when it comes to the

equilibrium manipulation b. In case (d), the owners choose the maximal b = 1; hence,

b cannot be increased and there is no indirect effect.

In contrast, in case (c), where b and s are interior and react continuously to slight

changes in the parameters, Proposition 2 (iii) establishes that an increase of S can induce

either less or more manipulation b. The latter effect is counter-intuitive and occurs if

3

(1 )

(1 )A

qf

q

, given that fC and k are sufficiently low (see the proof). However, even if

manipulation increases the probability that it is successful, the success of manipulation

b(1 – s) unambiguously decreases because the effect of the increase in s outweighs an increase

in b on price. This implies that owners are less likely to incur the enforcement penalty from

direct public enforcement, which again creates a higher incremental benefit of manipulation.

This benefit, together with that from higher sensitivity of PH, can induce strategic owners to

increase b for higher S.

The next results provide more insights into the effects of the exogenous variables that

determine the economy and the compliance mechanisms in place.

Corollary 1: An increase in the proportion of strategic owners q induces

(i) a decrease in the price PH; the decrease is strict if b < 1 and/or s > 0;

(ii) an increase in IC effectiveness s; the increase is strict if b < 1 and/or s > 0;

(iii) a decrease in manipulation b; the decrease is strict if b < 1.

Corollary 1 states that the existence of more strategic owners decreases the investment

volume and reduces the price, which induces strategic owners to improve their ICs and reduce

their level of manipulation. The underlying reason is that investors anticipate a lower

information content of a favorable report rH the more strategic owners are present in the

21

economy. Because the price reaction becomes smaller, strategic owners’ benefit of

manipulation declines and mitigates manipulation incentives. It also leads to more compliance

with IC regulation because the option value from manipulation declines, so a higher s reduces

the IC enforcement cost in equilibrium. Although these effects on b and s again increase the

price PH, Corollary 1 (i) shows that the information content of a report rH and the

consequential price are still lower because the direct effect outweighs the indirect effects.

The next two results provide insights into the effects of a variation of the strength of the

compliance mechanisms.

Corollary 2: An increase in IC enforcement cost fC induces

(i) an increase in the price PH; the increase is strict if s > 0;

(ii) an increase in IC effectiveness s; the increase is strict if s > 0;

(iii) a decrease or an increase in manipulation b; a necessary condition for an increase is

b < 1 and s > 0.

The results in Corollary 2 are similar to those in Proposition 2 because the economic

trade-offs are the same. This suggests a substitutive relationship between the tightness of the

IC standard S and IC enforcement fC. Overall, stricter IC regulation induces implementation of

a more effective IC standard s, induces more efficient investing, a higher price PH, although

manipulation incentives can be higher or lower.

Corollary 3: An increase in the direct public enforcement strength fA induces

(i) an increase in the price PH; the decrease is strict if b < 1 and/or s > 0;

(ii) an increase in IC effectiveness s; the increase is strict if s > 0;

(iii) a decrease in manipulation b; the decrease is strict if b < 1.

Corollary 3 shows that stronger enforcement fA increases price PH and IC effectiveness

s, and it decreases manipulation effort b. The reason is that fA directly reduces the incremental

benefit of manipulation Y = (PH – fA) and, because of that, owners’ incentive to engage in

manipulation. Consequently, given an IC effectiveness s, they reduce b if b < 1 (cases (a) and

(c) of Proposition 1). It is interesting to observe that increasing fA has a positive indirect effect

on IC effectiveness s if s > 0 due to the fact that underinvesting in the IC system becomes less

22

beneficial if the benefit from manipulation declines. A higher s decreases the expected benefit

of manipulation and mitigates manipulation further. The two effects of a higher S on b and

s reinforce each other, which induces an increase of PH.

Taken together, Proposition 2 and Corollaries 2 and 3 establish that stricter IC

regulation and stronger direct public enforcement (at least weakly) improve the information

content of financial reports (as measured by ), which induces more efficient investment and

results in a higher price PH. At the same time, the results point towards a non-trivial

interaction of S, fA, and fC for determining manipulation, implementation of the IC system and

on the resulting price of the firm. In particular, direct public enforcement improves the

effectiveness of IC regulation by mitigating the IC underinvestment incentives of the owner,

which in turn reduces the necessity for stringent IC regulation. Therefore, public direct

enforcement partly substitutes for IC regulation.

4.3. Optimal internal control regulation

So far, we have taken the IC standard S as given. We now turn to optimal IC regulation,

that is the optimal tightness S of the IC standard for a given effectiveness of IC enforcement.

The benevolent regulator sets the standard S to maximize social welfare

W ≡ 2 2(1 )

2Pr( 1) ( ) Pr( ) ( )4

H H H

q S qsI r r I r c

. (10)

Social welfare comprises the expected wealth generated by the investors through investing

I(r) into the growth project, less the costs of establishing and maintaining the IC system with

effectiveness s,12 noting that with probability q owners are strategic and choose s and with

probability (1 – q) owners are compliant and implement the standard, s = S. As mentioned

earlier, we exclude strategic owners’ disutility of manipulation because it is not monetary and

12 Even though we assume these costs are private to avoid separation of the types, inherently these costs are

considered by the IC regulator as they affect all firms.

23

a socially undesirable activity. The non-compliance cost captured by 2

– / 4Cf S s is simply

a wealth transfer from owners to the enforcer and is irrelevant for social welfare.

To simplify the analysis, we set fA = 0.13 Using the equilibrium investment,

2( ) ,HI r and Pr(rH) = 1

1 (1 )qb s = , equation (10) is equal to

2 21(1 )

2 2

cW qs q S

. (11)

We begin with a setting in which all owners are strategic (q = 1), in which mandating an

IC standard S does not generate externalities to compliant owners, who would bear the full

cost cS2/4 without any benefit. For q = 1, social welfare equals

21.

2 2

csW

(12)

If the equilibrium s = 0 because the strength fC of the IC compliance mechanism is low

(cases (a) and (b) from Proposition 1), then S has no effect on social welfare regardless of the

amount. If fC is sufficiently high and c is sufficiently low (cases (c) and (d)), it is always

optimal for the regulator to set S = 1 to induce a greater s. However, the cost of implementing

a high s can be so high that it can be welfare maximizing to reduce S. The next proposition

states the optimal IC standard S for the case when IC regulation is most effective; that is,

when all owners are strategic (q = 1).

Proposition 3: Assume c > 2 and q = 1. The optimal IC standard S* and the induced IC

effectiveness s are as follows, where 0 C Cf f and the welfare-maximizing IC effectiveness

s* ∈ (0, S*);

(i) if C Cf f (cases (a) and (b)), then S* is arbitrary and s = 0;

(ii) if C C Cf f f (cases (c.1) and (d.1)), then S* = 1 and s ∈ (0, s*];

(iii) if C Cf f (cases (c.2) and (d.2)), then

* (0,1)S and s = s*;

13 We expect the main insights we derive from this constrained setting extend to 0 < fA < 1/(1 + q)2.

24

The proposition characterizes the optimal choice of S* and the induced s, and

particularly emphasizes how the equilibrium strategies depend on the non-compliance costs

captured by parameter fC. Similar to Proposition 1, the disutility of manipulation k

distinguishes cases b = 1 and b < 1. The assumption c > 2 is a sufficient condition that the

optimal S* can be less than 1.

Proposition 3 (i) states that the results from Proposition 1 (cases (a) and (b)) carry over

immediately. For IC regulation to generally be beneficial and thus improve investment

efficiency, it must be sufficiently tight to induce strategic owners to implement an IC system

with effectiveness s > 0. If C Cf f , then even the tightest IC standard S = 1 cannot induce s >

0. Therefore, any IC standard S*∈ [0, 1] is optimal.

The other statements of Proposition 3 refer to the cases (c) and (d) from Proposition 1

and distinguish two subcases in each case, (c.1) and (d.1), in which S* = 1, and (c.2) and (d.2),

in which 0 < S* < 1. The proof establishes the existence of a unique threshold Cf that

separates these cases, and Proposition 3 (iii) and (iv) state the optimal interior S*. When fC

exceeds threshold Cf , then strategic owners invest in the IC system (s > 0) enhancing

reporting quality and thus investment efficiency. That is, to induce s > 0 if fC is not very high,

the difference between the standard S* and s is set as large as possible. For “intermediate” fC

( )C C Cf f f the optimal tightness is S* = 1 and IC regulation induces an IC effectiveness s

which is weakly welfare-suboptimal, i.e., s ≤ s*.

If IC enforcement Cf exceeds the threshold Cf , the regulator can and will induce

strategic owners to implement the welfare-optimal IC effectiveness s*. Keeping S* = 1 beyond

Cf would incentivize owners to invest too much in the IC system because there is an optimal

level of IC efficiency due to the cost of implementing the IC system (captured by c).

Therefore, the regulator optimally sets S* < 1. The optimal standard S* decreases as Cf

increases to keep the total non-compliance costs *2

– / 4Cf S s constant such that strategic

owners always choose s = s* for any particular C Cf f .

Figure 2 illustrates the proposition for the case k < 1/4, which induces strategic owners

to always attempt to manipulate (b = 1). It depicts the behavior of optimal tightness S* and the

25

induced IC effectiveness s that the strategic owners implement as a function of S* and fC. S*

starts with an arbitrary level depicted by the shaded area, then “jumps” to S* = 1 (case (d.1))

and later declines (case (d.2)). The induced IC effectiveness s strictly increases if C C Cf f f

and then remains constant at its optimal level s* if C Cf f .

Figure 2: Optimal IC standard S* and induced IC effectiveness s for only strategic owners

(Parameters: q = 1, c = 3, k < 1/4)

Next, we consider the optimal IC standard for the more general case in which the

proportion q of compliant owners in the economy is less than 100 percent. In the limit, when

there are only compliant owners (q → 0), then the optimal IC standard S* = 0 because the only

purpose of IC regulation in our model is to detect accounting manipulation before investors

make their investment decision. Since compliant owners do not manipulate, the IC system has

no value. Proposition 4 generalizes the results of Proposition 3 for b = 1 when (0,1]q .

Proposition 4: Assume c > 2, k < 1/4, and q ≤ 1. The IC standard S* that maximizes social

welfare as defined under (11) is as follows, where 0 1q q and 0 :C C Cf f f

(i) :q q if C Cf f (case (b)), then S* = 0; if C C Cf f f (case (d.1)), then S* = 1; and if

C Cf f (case (d.2)), then * (0,1)S ;

S*

IC s

tandar

d S

and

ind

uce

d s

s

Cf Cf

IC enforcement cost fC

26

(ii) :q q q if C Cf f (case (b)), then S* = 0; if C C Cf f f (case (d.1)), then

* (0,1)S ; if C C Cf f f (case (d.2)), then S* = 1; and if C Cf f (case (d.3)), then

* (0,1)S ;

(iii) :q q if C Cf f (case (b)), then S* = 0; and if C Cf f (case (d)), then * (0,1)S .

The results in Proposition 4 are structurally similar to those in Proposition 3, but

somewhat more nuanced. The reason is that if q < 1, the regulator must also take into account

the cost of implementing S by the compliant owners, cS2/4. By definition, compliant owners

not only implement the standard S, but do not manipulate, so these costs are a direct welfare

loss. The total costs of mandating an S are

2 2(1 )4

cqs q S ,

which increases in (1 – q) because s < S.

The regulator trades off these costs against the welfare increase from inducing strategic

owners to implement an s > 0. Gross welfare for b = 1 is 1

1 (1 )

q s, and the benefit of

increasing s declines the lower the proportion of strategic owners q are in the economy. As

was already established earlier, there exists a lower threshold 0Cf that for C Cf f it is

optimal to forego any IC regulation (S* = 0). This result contrasts with Proposition 3 (i) that

stated multiple equilibria, because only the equilibrium with S* = 0 survives due to the strictly

positive cost of mandating any S > 0 for q < 1. Except for this fact, the functional form of the

IC standard S* described under Proposition 3 is similar as long as the proportion of strategic

owners is high ( ).q q The larger the proportion of strategic owners, the more beneficial is

IC regulation and, hence, it becomes useful already for relatively low enforcement levels.

If fC exceeds the threshold Cf , IC regulation becomes effective to induce an s > 0.

Proposition 4 describes the optimal S* for different combinations of q and fC. If q is

sufficiently small ( ),q q setting S = 1 is always too costly, so the regulator starts with

setting S* < 1 (Proposition 4 (iii)). For intermediate q ( )q q q there is a more subtle

trade-off between costs and benefits of IC regulation: Proposition 4 (ii) states that there is a

27

set of fC for which the maximal standard S* = 1 is optimal because it provides the strongest

incentives to strategic owners to implement a sufficiently strong IC system.

Similar to the case with q q , this occurs for fC slightly above Cf , but – interestingly –

there exists a range of fC ( )C C Cf f f for which S* < 1, before it increases to S* = 1 and

then falls to S* < 1 again for sufficiently high fC. That is, the function S* is non-monotonic in

fC. Intuitively, an increase of S* is a result of the relatively high cost of setting a high S*

incurred by compliant owners, so the regulator foregoes mandating IC regulation until it can

induce s > 0 for some S* < 1. If fC becomes higher, it pays to increase S* to induce a greater s,

despite the increasing costs. Consequently, the complementary relationship between IC

standard S* and fC in this case is a direct consequence of the existence of compliant owners.

Finally, if fC is sufficiently large, S* decreases in all cases in Proposition 4 to avoid too much

investment in the IC system.

Proposition 4 characterizes the optimal standard S* that efficiently induces strategic

owners to invest in IC effectiveness s > 0. It does not yet consider that social welfare from

abstaining from IC regulation altogether can dominate a setting in which the induced s is too

low to make IC regulation worthwhile. Social welfare under no IC regulation is simply

1

2 2(1 )q

if S* is set equal to 0 so that s = 0 because then there are no costs of IC

implementation. IC regulation that induces an s > 0, is preferred to no IC regulation if

2 * 21 1(1 )( )

2 1 (1 ) 4 2(1 )

cW qs q S

q s q

. (13)

Proposition 5: Assume that c > 2, k < 1/4, and q < 1. Relying on IC regulation is strictly

superior than no IC regulation for IC

C Cf f , where IC

C Cf f . Optimal IC regulation induces

an IC effectiveness *[ , ],

ICs s s where 0.

ICs

Proposition 5 establishes that IC regulation becomes optimal to no regulation only if

IC

C Cf f . Inducing strategic owners to implement an s > 0 requires setting S* > s, which

reduces social welfare by * 2(1 )( ) 0

4

cq S . From condition (13) it is straightforward to see

that any S > 0 is strictly suboptimal when C Cf f , and thus s = 0. Increasing fC therefore

28

leads to a jump of s from 0 to 0ICs once fC exceeds IC

Cf . Effectively, condition (13) shifts

the desirability of IC regulation to situations in which the benefit from inducing strategic

owners to implement a sufficiently effective IC system overcomes the burden this places on

compliant owners.

If fC becomes very large, the optimal S* and the induced s converge, and the maximum

value of the desirable s = s* becomes

* *2

lim limC Cf f

sq

sc

S

.

Note that s* increases in q and is therefore strictly lower for q < 1 than for q = 1 (as in

Proposition 3).

Figure 3 illustrates the optimal IC standard S* and the induced IC effectiveness s for

two different parameter values q = .8, .9; the other parameters are set equal to those in Figure

2 to facilitate comparability with q = 1.

Figure 3: Optimal IC standard S* and induced IC effectiveness s for different proportions

of strategic owners q (Parameters: c = 3, k < 1/4)

Figure 3 shows the jump of the induced IC effectiveness s at the threshold IC

Cf for each

q < 1 once fC increases. The threshold is sufficiently high that S* declines in the range in

29

which IC regulation is desirable, so that the shape described in Proposition 4 does not come to

bear in the examples. The figure shows another, less obvious, difference between the cases of

q < 1 and q = 1, which is the dual function of S* in the area in which IC regulation is

desirable: Similar to the case of q = 1, we observe a substitution effect between fC and S*, but

for q < 1 the induced s still increases, so that S* does not decline as fast as it would if the

induced s was held constant.

Propositions 4 and 5 describe the optimal IC standard S* for cases in which the public

enforcement system is weak or the manipulation cost k is low so that strategic owners always

manipulate (b = 1). Although conceptually not different, the trade-off that determines the

optimal IC standard S* becomes more complicated if strategic owners choose an interior

manipulation effort b < 1. The reason is that the IC regulation has a direct effect on b, which

jointly with s determines the equilibrium investment efficiency and firm price.

We rephrase the relation between the optimal IC standard S* and the strength of IC

enforcement in the following corollary.

Corollary 4: The optimal IC standard S* and the IC enforcement fC are complements for low

fC and are substitutes for high fC.

Conventional wisdom suggests that setting a tighter IC standard requires a more

effective IC enforcement, which implies a complementary relation between the IC standard

and its enforcement. This intuition holds in a lower range of enforcement costs. That is,

without a minimum effectiveness of IC enforcement, the IC standard will be ineffective

regardless of how tight the standard is because strategic owners still choose s = 0, whereas

compliant owners do not change their (non-)manipulation strategy. Once the IC enforcement

cost exceeds a threshold, the optimal standard jumps to either IC standard S* = 1 or S* < 1

depending on whether q q or q q , respectively. For sufficiently large levels of the IC

compliance mechanism ( )C Cf f there exists a substitutive relationship to avoid high

compliance costs c for a small welfare increase.

30

5. Robustness and extensions

In this section, we discuss import model assumptions for our results and potential

extensions of the model that can address further aspects of financial reporting regulation.

Assets in place. We assume the accounting report provides information about the firm’s

productivity , which directly determines the marginal profitability of the growth project the

new investors have available through the expected project return of 2 .I We do not

specify through which signal this information is provided to investors. For example,

productivity information can be signaled through disclosures, such as research and

development, risk disclosures, or forecasts. Productivity information can alternatively surface

through the carrying amount of existing assets. If they are measured at cost, they may, or may

not, be impaired; if they are measured at fair value, the value signals their productivity

directly. For simplicity and in line with prior literature (e.g., Patterson and Smith 2007, and

Lu and Sapra 2009), we normalize existing assets to 0 (so their carrying amount does not

directly signal the productivity) and focus on the value of the growth option.

To allow for existing assets suppose, for example, that the carrying amount of the assets

is V0 + V, where V > 0. The new investors would still choose an optimal investment volume

of 2( )HI r , so the price of the firm will be 2

0HP V V , which is strictly monotonic in

, so our main results are structurally unaffected.

Assumptions regarding the impact of different enforcement mechanisms. We distinguish

between direct public enforcement of the accounting standard (through fA) and enforcement by

imposing IC regulation. We model direct enforcement as a mechanism that ex post identifies

manipulation and penalizes the firm in such a case. It is effective because the strategic owners

anticipate the effect of the ex post enforcement and this mitigates manipulation incentives.

Nevertheless, a correction of the financial report comes too late to affect investors’ decisions.

In contrast, the IC system is effective ex ante because it identifies and corrects manipulation

before the financial report is released.

An alternative assumption for how the IC system influences owners’ manipulation

incentives is to assume that a more effective IC system increases their private disutility of

31

manipulating the financial report; that is, a more effective IC system increases the parameter k

in our setting (e.g., Laux and Stocken 2012). Since we are interested in choosing optimal IC

regulation, using such a compact representation of the IC system effectiveness by k would not

allow us to capture the characteristics of IC regulation. In particular, a key driver for our

results is the owner’s decision to what extent to implement the IC standard. We could add an

effect of the IC tightness on k, but that is unlikely to change our insights.

Non-compliance costs. The incentive to comply with regulations arises through the

threat of penalties and other non-compliance costs, such as reputational damages, so the

functional form of these costs has an effect on some results. We assume that direct public

enforcement strength fA, which is comprised of the likelihood of investigation, the probability

of identifying manipulation and the penalty, is constant. This assumption is without loss of

generality because of the binary reporting space r that can be either rH or rL and manipulation

only in the form of overreporting occurs.

More critical is the assumption of the cost of non-compliance with the IC standard,

which is 2

– / 4Cf S s . This cost is assumed quadratic in the level of IC underinvestment

relative to the mandated tightness S. An implication of this function is that at the point of s =

S, marginally underinvesting incurs no cost, and this effect leads to the result in Lemma 1 that

strategic owners never comply with the IC standard (i.e., s < S).14 If one assumes, for

example, a constant cost of IC non-compliance, the strategic owners’ optimal choice of IC

system effectiveness would either be s = S or s = 0, but not an interior level. This would affect

the functional form of s and S, but not our main result that IC regulation can be strictly

welfare-decreasing if not sufficiently supported by IC enforcement or if set too low or too

high.

Other benefits of IC systems. We assume that the function of the IC system is to detect

and correct manipulation attempts of the financial report. While this is an important function,

14 Laux and Stocken (2017) find a similar result that compliance depends on the assumed sensitivity of

regulatory penalties.

32

in practice IC systems have other functions as well. One of these is to mitigate random errors

in financial reporting. We exclude this function by assuming the underlying accounting

system is perfect if no manipulation occurs. Another function of IC systems is to deter non-

compliance with regulations other than those that have a direct impact on financial reporting.

Such actions may include, e.g., fraud in the company and corruption, which damage the

firm’s economic profitability. If there exist benefits besides those from mitigating

manipulation of financial reporting, then owners have an incentive to invest a minimum level

of IC effectiveness. This level may still differ between compliant and strategic owners,

depending on whether the underlying issue is under the owner’s control or not. However,

strategic owners underinvest into the IC system even if it provides other benefits to increase

the success of financial reporting manipulation.

Alternative enforcement mechanisms. An extension of the present paper is to explicitly

consider additional enforcement and compliance mechanisms. For example, external auditing

provides assurance that financial reports are prepared in compliance with accounting

standards, thus mitigating manipulation by identifying manipulation directly or by testing the

effectiveness of the IC system. Auditors are strategic economic actors and decide on their

audit procedures based on expected costs and benefits. Therefore, introducing auditing can

affect the efficiency of other enforcement mechanisms.15

Auditors are also required to report on IC weaknesses they observe. If such disclosure

reveals the actual IC effectiveness s strategic owners implemented, the equilibrium discussed

in our paper would change because investors do not need to conjecture s. However, in that

case any compliance mechanism would simply inflict a penalty rather than undertaking its

own investigation. If the auditor’s disclosure only discloses the fact that s < S, but not the

particular s, investors would infer the owner’s type because in our binary setting compliant

owners implement s = S. Any other channels through which investors could distinguish the

15 For example, Ewert and Wagenhofer (2016) study a setting in which increasing enforcement strength leads to

a crowding out of the auditor’s work and, ultimately, even to lower quality financial reports.

33

two types of owners would create similar strategies by strategic owners to disguise that

information. For example, the regulator can require management to assess and report on the

effectiveness of the IC system, e.g., as is the case under Section 404 of SOX. We note that our

main results do not depend on whether investors learn the owner’s type; in particular, in

Proposition 3 we explicitly study optimal IC regulation if all owners are strategic.

Public observability of the implemented IC effectiveness s would affect the equilibrium

because investors would be relieved from conjecturing s. However, we believe observability

is an unrealistic assumption even though management and auditors report about IC

implementation and potential weaknesses because such disclosures are not highly

informative.

6. Conclusions

Internal controls are an important mechanism to assure the quality of financial

reporting, which is important to improve investment efficiency in the economy. Securities

regulators have continuously strengthened the legal requirements for firms to implement and

maintain internal controls (IC) over financial reporting. The effectiveness of ICs depends on

both IC standards and mechanism to ensure compliance with these standards. While most

analyses focus on the compliance side and take IC regulation as given, this paper specifically

analyzes optimal IC regulation from a regulator’s perspective. Since investment into IC

systems is costly to firms, the economic welfare generated in the capital market of mandating

IC regulation must outweigh these costs.

We develop an economic model to assess how IC regulation affects firms’ investment in

the effectiveness of IC systems, which mitigates manipulation of financial reporting. We

derive equilibrium IC effectiveness and manipulation, which jointly determine the quality of

financial reports that inform about individual firms’ productivity and the volume of

investment in new projects. Hence our setting explicitly links financial reporting quality to

investment efficiency and welfare in the economy. Our main findings are:

34

Strategic owners always underinvest in IC effectiveness relative to a prescribed IC

standard. The level of underinvestment is driven by the direct implementation costs of

IC systems and the option value of subsequent successful manipulation.

Implementation of a more effective IC system strictly increases financial reporting

quality and investment efficiency through mitigating successful manipulation, although

it can increase owners’ incentives to manipulate.

Increasing the tightness of the IC standard increases the effectiveness of implemented

IC systems only if the compliance mechanism is sufficiently strong (implying a

substitutive relationship).

The welfare-maximizing IC standard is very tight for low non-compliance costs and

decreases if non-compliance costs become larger (implying a complementary

relationship).

A larger proportion of strategic owners makes IC regulation more desirable.

More effective direct enforcement of accounting standards reinforces IC effectiveness

through mitigating IC underinvestment incentives.

With these results, this paper contributes to a better understanding of IC regulation. Our

results offer specific predictions about the efficiency of IC regulation, the tightness of IC

standards, and the level of compliance with the standards, which can inform empirical studies.

The results should also be of interest to regulators because the choice of welfare-optimal IC

standards should be considered jointly with the costs that are imposed in case of non-

compliance with IC regulation.

35

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37

Appendix

Proof of Proposition 1

The equilibrium strategies {b, s} derived in the text preceding Proposition 1 are:

(1 ) 0k s Y : 1(1 )b k s Y and

1 2

1 2max 0, C

C

f S k Ys

c f k Y

< S ≤ 1

(1 )k s Y : b = 1 and max 0, C

C

f S Ys

c f

< S ≤ 1.

It is easy to see that I(rL) = 0 and I(rH) that maximize investors’ expected utility in (2),

and the strategic owner’s choice of b s as determined in (5) and (7), respectively, satisfy the

second-order conditions for a maximum.

Assumption (8), 2

1

(1 )Af

q

, ensures that Y > 0 for any feasible b and s because in any

equilibrium

2

2 2

1 1

(1 )1 (1 )HP

qqb s

.

The conditions for the different cases are endogenous because they depend on PH and s.

In a first step, we show that there exist unique value ranges for k and fC which separate the

four cases. The value of k determines whether b is interior (b < 1) or a boundary solution (b

=1), whereas the value of fC determines whether s is interior (s > 0) or a boundary solution (s

= 0). For this it is sufficient to show that there exists a unique pair of parameter values of k

and fC for which b = 1 and s = 0.

Consider b = 1 and s = 0, which implies a constant 2

1

(1 )HP

q

. At the boundary, the

following set of equations must simultaneously hold for k and fC:

1 2( ) 1Ak f

resulting from equating both values for b from (5) and

1 2 2

1 2 2

( )0

( )

C A

C A

f S k f

c f k f

resulting from the first equation for s in (7). The solutions are the two parameter values

2

1

(1 )Ak f

q

and 2

1 1

(1 )C Af f

S q

. Since there is only one uniquely defined pair

38

(k, fC), there exist unique thresholds , 0Ck f that separate four cases. The same solutions

arise from the second equation for s in (7), which is

2(0

)AC

C

f S

c f

fs

.

Further note that, given any S, the partial derivatives of b and s in each case are

1 2

1 2 1 22

(1 ) ( ) (1 )0

C C C

C C

k Y c f S c f k Y c f Sb

k k c f k Y c f k Y

,

1 2 0b

k Y k Yk k

,

1 21 2

2 1 221

(1 )0C

C C C C

k Yf S k Ys

f f c f k Y

S

c f k Y

cS

,

and 2

0( )

C

C C CC

f S Y cS Y

c f c f

s

f f

.

This implies that there must exist two unique thresholds 0k and 0Cf such that

b < 1 if k k and b = 1 if k k ;

s > 0 if C Cf f and s = 0 if

C Cf f .

This establishes the four cases (a) to (d) depending on k and fC as shown in Table 1.

To prove uniqueness of the equilibrium in each case, we insert the respective solutions

for b and s into the condition defining . This results in a single equilibrium condition that

implicitly defines denoted by as follows:

2

10

1 (1 )qb s

,

and 2 1 (1 ) 1q s b

.

To demonstrate uniqueness, it is sufficient to prove that there exists only a single root of

the equilibrium condition in . We do this by deriving the limits of the condition and showing

monotonicity in . A unique implies uniqueness of all decision variables {b, s, I, P} because

each can be written as a function of as the only endogenous variable.

39

Case (a): The equilibrium condition is

1 1(1 )a k qY

with the following properties:

11

1(lim 1 11 0)a Ak q f

1

1

1/(1 )

1 1lim 1 0

(1 ) (1 )a

qAfk q

q q

1 3(1 2 ) 0a k q

.

It follows that there is a unique satisfying the equilibrium condition.

Case (b): The equilibrium is unique because 1

(1 )q

.

Case (c): The equilibrium condition is

12

1

1 2

(1 )1 C

c

C

c f Sk qY

c f k Y

.

Using the following algebraic simplification from rearranging the equilibrium condition,

2

1 2

(1 ) (1 )C

C

c f S kq

c f k Y Y

,

we derive the following properties:

12

1

1 21

(1 )lim 1 (1 ) 1 0

(1 )

Cc A

C A

c f Sf k q

c f k f

12

1

221/(1 )1

2

(1 )1 1lim 1 0

(1 ) (1 )1

(1 )

Cc A

q

C A

c f Sk q f

q qc f k f

q

2 1 2

1 2

2 (1 ) 41 1 0c

C

k Y

Y c f k Y

.

It follows that there is a unique satisfying the equilibrium condition.

Case (d): The equilibrium condition is

40

1

(1 )1 C

C

d

c f S Yq

c f

with the following properties:

1

1

(1 ) (li 1 1 0

1 )m C A

C

d

c f S f

cq

f

1

2

1/(1 )

1

(1 ) 1lim 1

(1

0(1

)

)

C

C

d

A

q

c f S f

c f

qq

q

3

12

0d

C

q

c f

.

It follows that there is a unique satisfying the equilibrium condition.

Proof of Proposition 2 and Corollaries 1 to 3

In the proof of Proposition 1 we state the equilibrium conditions for each of the four

cases and prove the existence and uniqueness of thresholds k and .Bf The proof of

Proposition 2 and the three Corollaries applies the implicit function theorem to determine the

signs of the total derivatives of with respect to {fA, fC, q, S} for each case separately.

Case (a): The partial derivatives of a are

2

0 0a a a

AA A

d

ff k df

q

0 0a a a

CC C

d

ff df

1 2 0 0a a adk Y

qq dq

0 0a a ad

SS dS

.

Case (b): For the case in which b = 1 and s = 0, 1

1 q

= constant so all derivatives are 0.

Case (c): The partial derivatives of c are

1 2

1 2

2 (1 ) (1 )0 0

(1 )

c c c

AA AC C

cS k S Y d

ff dfc f S c f k Y

41

1 2

1 2

2 (1 )0 0

(1 )

c c c

CC CC C

cS k Y d

ff dfc f S c f k Y

2

1 2

1 2

(1 )0 0c CB c c

C

c f S dk Y

qq c f k Y dq

2 (1 )0 0

(1 )

c C c c

C

f d

SS c f S dS

.

Case (d): The partial derivatives of d are

2

0 0d d d

AA C A

d

ff dc f f

q

2

20

( )0d d d

CC CC

cS Y

c f

dq

ff df

2 (1 )0 0

C

d dCd c f S Y

c f

d

qq dq

2

0 0d d dC

C

q d

SS dS

f

c f

.

The total derivatives for b with respect to x = {fA, fC, q, S} are determined by

db b b d

dx x dx

.

If b = 1 (cases (b) and (d)) a variation of fA, fC, q, and S has no effect. Hence, the statements

hold trivially true. Consider b < 1 next.

Case (a): The first order conditions of 1b k Y are:

11 1

1 32 0

1 2A A

db d kk k

df df k q

; 0

C

db

df ; 12 0

db dk

dq dq

; and 0

db

dS .

Case (c): The partial derivatives of 1

1 2

(1 )C

C

k Y c f Sb

c f k Y

are as follows:

1 2

21 2

(1 )0

C C

A C

c f S c f k Yb

f k c f k Y

1 1 2

21 2

(1 )0

C C

k Y cS S k Yb

f c f k Y

0b

q

42

1

1 20C

C

f k Yb

S c f k Y

1 1 2

21 2

2 (1 )0

C C

C

k c f S c f k Yb

c f k Y

.

It immediately follows that 0b

q

. The total derivative with respect to fA is

1 1 2

21 2

(1 )1 2 0

C C

A AC

k c f S c f k Ydb d

df dfc f k Y

where

1 2

11 2

2 1 2

1 2

(1 ) 41

1 2 1 2 02 (1 ) 4

1 1

C c

A

C

k Y

Y c f k Yd

df k Y

Y c f k Y

.

The total derivatives with respect to S and fC can be simplified to

1 12 2

1 22 (1 ) ( )c C

A

C

k fdbf

dS c f k Y

and

1 1 21

2 2

21 2

(1 )2 (1 ) ( )c

A

C C

k cS k S Ydbf

df c f k Y

.

Both signs are determined by the sign of 2 22 (1 ) ( )Af , which can be positive

or negative. Above we establish 0C

d

df

, and it can be shown that 0

d

dk

. has the

following properties:

, 1/( 31 )lim lim

(1 )

(1 )C Cf fA

k k q

qf

q

1lim li ) 0m (1 Ak

f

2 3 1 0d

d

.

The last inequality follows from the fact that ≥ 1/2. Let the joint thresholds be T

C Cf f

and Tk k . The limit

3

(1 )

(1 )A

qf

q

has the following properties with respect to fA:

30 3

(1 ) (1 )0

(1 ) (1 )lim

AA

f

q qf

q q

43

2 31/(1 )

3

(1 ) 20

(1 ) (1li

)m

Af qA

q qf

q q

3

(1 )0

(1 )A

A

qf

f

d

d q

.

That is, 3

(1 )

(1 )A

qf

q

> 0 if fA is sufficiently large. It follows that there exist three unique

thresholds T

C Cf f , Tk k , and 2(0,1/ (1 ) )T

Af q such that if Tk k k ,

T

C C Cf f f , and T

A Af f , , 0C

db db

dS df and , 0

C

db db

dS df otherwise.

To prove the effects of x = {S, fA, fC} on the equilibrium s, the total derivatives are

ds s s d

dx x dx

.

If s = 0 (cases (a) and (b)) a variation of fA, fC, q, and S has no effect. Hence, the statements

hold trivially true. Consider s > 0 next.

Case (c): The partial derivatives of

1 2

1 2

C

C

f S k Ys

c f k Y

are:

1

21 2

2 (1 )0

C

A C

k Y c f Ss

f c f k Y

,

1 2

21 2

(1 )0

C C

cS k S Ys

f c f k Y

, 0s

q

1 20C

C

fs

S c f k Y

and

1

21 2

4 (1 )0

C

C

k Y c f Ss

c f k Y

.

It follows immediately that 0ds

dq . Further the total derivatives are

1 1

21 2

2 (1 )0

Cc

A C

k Y c f Sds

df c f k Y

1 21 2

21 2

(1 ) 2 (1 )1 0c

C C

cS k S Yds

df Yc f k Y

1 2

1 2

2 (1 )1 0c C

C

fds

dS c f k Y Y

.

(d): The total derivatives of C

C

f S

fs

Y

c

are:

44

3

10

1 / ( )

2 1 1

2A AC C C Cc f c f c f

ds

f cq f

d

df d

32 20

1

2 1

( (2 /) ( ) )C CC C C C

cS Y cS Y

c f c f c f q

ds d

df d ff c

20

C

ds d

dqcq fd

3

)2

2

/ (0

1 / ( )

C C C

C C C

f f c f

c f c

ds d

dS dSf q c f

.

Proof of Proposition 3

Proposition 1 states the equilibrium b and s for any feasible S, so the optimal S must

fulfill the same conditions. For S to affect PH it must be that s(S) > 0. If S = 1 induces s = 0,

then any S < 1 also induces s = 0. In this case and since q = 1 the regulator is indifferent

regarding the optimal IC standard, S* ∈ [0, 1]. This will be shown to hold in the general case

in which (0,1]q .

In cases (c) and (d), s > 0 in equilibrium. Proposition 3 establishes a further case

distinction, subcases (c.1), (c.2), (d.1) and (d.2), which arise because S* is bound above by 1.

The existence and uniqueness of the thresholds that separate cases (c.1) and (d.1) are proven

using the same technique as in the Proof of Proposition 1. The thresholds that distinguish the

four cases are k and Cf using q = 1. For cases (c.1) and (d.1), we have S* = 1, because this is

the S that induces an s > 0. As shown in Proposition 1, there exists a unique boundary value

0Cf for S = 1, where s > 0 if C Cf f . We consider the cases with b = 1 and b < 1.

Case (d): If k k we have b = 1 and

2

C

C

f S

c fs

. Substituting the equilibrium strategies

into

21

2 2

csW

and maximizing with respect to S yields

2* (2

in)

m ,1C

C

cS

f

cf

.

Inserting *S into the function for s results in

2

C

C

f

c fs

if * 1S and

2*s

c

if

2* (2 )C

C

c fS

cf

.

45

Note that 2

*sc

implicitly defines the welfare-maximizing IC effectiveness. The

expressions for both s and S are strictly positive if C Cf f . However,

2(2 )C

C

c f

cf

can

become greater than 1, which is infeasible and requires a boundary condition.

Case (c): If k k we have 1

41

2 (1 )C

C

k c f Sb

c f k

and

4

4

1

1

C

C

f S ks

c f k

. Substituting the

equilibrium strategies into

21

2 2

csW

and maximizing with respect to S yields

1 4

1

*

4

3 2 2,1

2

(1 )max

(1 ) 2

C

C

k c c f

f cS

c k

.

Inserting *S into b and s gives

21

1 4

C

ckb

c f k

and

1

1

4

4

C

C

f ks

c f k

if * 1S ,

1 2

1 4

1 2(

2 2

1 )

(1 )

ck

c kb

c

and

1*

4

1 4

2

2 2(1 )

k

c cs

k

if

4

*

1

1 4

3 2 (1 )

(1 )

2

2 2

C

C

k c c f

f c cS

k

.

Note that

1*

4

1 4

2

2 2(1 )

k

c cs

k

implicitly defines the welfare-maximizing IC

effectiveness. Again, the expressions for b and s as well as S are strictly positive if C Cf f .

However, 1 4

1 4

3 2 (1 2

1 ) 2(2

) C

C

k c c f

f c c k

can become greater than 1, requiring a boundary

condition.

The fact that in both cases S must be bound above by 1 implies there must exist another

threshold for fC that distinguishes the areas where S is in the interior and at the boundary.

To prove that there exists a second unique threshold at which cases (c.1), (c.2), (d.1), and

(d.2) meet we simultaneously solve the following pair of equations that hold for case (c),

1 2

1 4

(1 )1 21

2 (1 ) 2

ckb

c c k

and

1 4

1 4

(1 )

(1 )

3 2 21

2 2

C

C

k c c fS

f c c k

,

for Cf and k, which yields

46

2

2

(5 2 )

1 2(1 ) 2C

cf

c

and

2 21 2(1 ) 2

1 2(1 )

ck

c

.

Both values are functions of a (unique) , both directly and indirectly through Y. The

boundary is

2

2

1(5 2 )

1 2(1 ) 42C

cf

c

for all (1/ 2,1) .

This implies the existence of a threshold C Cf f so that for

C C Cf f f S* is at the

boundary of 1. The point must lie on the existing threshold k since both points are derived

applying (1 )k s Y .

To prove that the thresholds are unique, we show that there exists a unique point- for

which both values for Cf and k apply. This yields the following solution for :

1 2(1 )

2 ( 1) (1 )2

c

c c

.

For c > 2 this always lies in (1/2, 1). The right-hand side has the following limits:

1

1 2(1 )

2 ( 1) 2 2( 1)lim

(1 )

c c

c c c

< 1,

and

1/2

1 2(lim

(1 )

1 ) 4

2 ( 1) 2 8 1

c c

c c c

> 1/2.

Furthermore, 4

2( 1) 8 1

c c

c c

. This implies uniqueness of the above point values in k and Cf ,

and further the existence and uniqueness of thresholds 0 C Cf f and (1 ) 0k s Y .

To prove the uniqueness of the equilibrium with endogenous IC standard S it is again

necessary to show the uniqueness of in each case. Cases (a) and (b) follow the same

procedure as the proof of Proposition 1. Cases (c.1) and (d.1) follow the the same procedure

as the proof of Proposition 1 cases (c) and (d) for S = 1, respectively. Uniqueness in cases

(c.2) and (d.2) is shown as follows:

Case (c.2): The equilibrium condition can be written as follows:

47

1 42

12

.2 1 4

11 12

2 2(1 )c

k

c c kk

.

The limits regarding are as follows:

2

1 2 2

1

.2 11 (l

2 )im 0c

c

c

k

kk c

1

.21/ 1

1

2

22

21 2

2 / 8

2 2 /i

2l m 0

8c

c k c

c k kc

k

.

Further note that it is possible to show that .2 0c

. It follows that there is a unique

satisfying the equilibrium condition.

Case (d.2): The equilibrium condition can be written as follows:

1 2 1

.2 (2 )d c .

The properties of this condition with respect to are as follows:

.21

( 1

1

)lim 0

(2 )d

c

c

.21/2

1lim 0

2(8 1)d

c

.2

2 2

21 0

(2 )

d c

c

It follows that there is a unique satisfying the equilibrium condition. Overall this proves the

uniqueness of the equilibrium underlying Proposition 3.

Proof of Proposition 4

We begin by deriving the optimal IC standard for case b = 1 and (0,1]q . The first-

order condition of W with respect to S is

1(1 ) 0

2

dW d dsc q S cqs

dS dS dS

.

The relevant derivatives are stated in the proof of Proposition 2 and can be simplified to

2

32

C

C

f

c

qd

dS f q

and

32

C

C

d f

c qS f

s

d

.

48

Inserting into the first-order condition results in

2

2 3

2

(1 )( )( 2 )

( )C C

C C C

c qS

c q q

f f

f c f c f q

.

The solution can again be greater 1, so derive the threshold for fC we set S equal to 1 and

solve for fC which has two roots:

23 32 2 2

2

3(1 )( 2 ( )( )

) (1 )( ) (1 )( )Cf c q c q c q c qc

q qq

cq q q

.

The roots are real if the term under the square root is non-negative, which is the case for

sufficiently high q. To derive the threshold q for which the root is positive for q q , rewrite

the root as qq , where

4 2 6 2(1 ) (1 ) (1 2 )q q q q c q c q .

0q implies that there is a unique solution for fC,

2 3

2

(1 )(

(

)

)C

c qf

c

q

q

c

q

,

which is strictly greater than 2

2

1

(1 )Cf

q

that determines the threshold of fC below

which strategic owners choose s = 0 even if the regulator were to mandate S = 1. In this case

is

3

3 2 2

(1 )( )

(1 )( ) ( ( ))

c q

q

c c q

c c c q q c c q

.

This is unique and lies in the feasible interval, and it increases in q under the assumptions

made (applying the implicit function theorem).

The following properties of q with respect to q can be derived:

4

1lim 0qq

2

0lim ( ) 0qq

c c

4 2 6 2 3(1 ) (1 2 ) 2(1 ) 0q

q c c q qq

3 2 54 6 (1 ) 2 (1 )(1 3 ) 0q

q q q c q q

.

49

It follows that there must exist a unique threshold (0,1)q such that when q q , then * 1S

for C Cf f . In contrast, when q q then there must exist a non-empty interval of fC where

* 1S .

For any q q , it can now be stated that the two roots regarding fC for which * 1S are

positive. However, the larger value is the already familiar threshold Cf . The lower value

represents another threshold which we shall denote by Cf . For some values [ ,1]q q

Cf

can lie below threshold Cf . To derive a second threshold in q denoted by q we set C Cf f :

23 3 232 2 2

2(1 )( ) (1 )( ) 2 ( )

((1 )( )

)c q c

cq q q qq c q c q

cq

q

.

We solve this equation for q and obtain only one solution in the feasible range [ ,1]q :

22 2 2 2 2 3 2 2 2

3

(2 1) 8 ( ) (2 1)

4

c c c c c cq

c

.

Note that in this case 1

(1 )q

. We can now define the following condition:

23 3 22 2

2 3 3 5 2 3

(1 ) 1 (1 ) 8 8 1 (1 )

4 (1 ) (1 ) (1 ) (1 ) (1 ) (1 )q

q q c c qc c c c q

c q q q q q q

.

q has the following limits with respect to q:

1

24 3 22 3 1 1

14 16

lim 04

qq

cc c c

c

0

22 3 2 21

1 8 8 1 04

lim qq

c c c c c cc

.

Further it can be shown that 0q

d

dq

. This proves existence and uniqueness of a threshold

( ,1)q q such that if q q , then C Cf f , and the optimal IC standard S* follows the one

stated in Proposition 3. Whenever [ , )q q q then C Cf f , which proves Proposition 4 (ii).

Next, we prove the existence and uniqueness of the thresholds , ,C C Cf f f . Consider Cf .

For q q , at threshold Cf a S* = 1 would only initiate s = 0. Hence, 2

2

1

(1 )Cf

q

, and it

immediately follows that Cf is unique. For q q , S* < 1 for which s = 0.

Cf solves

50

2 3(1 )( )(

)2

2 )

(

C C C

C

C C

f c f c f q

f

qf

c q

qf c

.

There exists a single root of fC in the feasible range. Threshold Cf can be written as follows:

2 3(1 ) (1 )( 3 8 )

2C

c q c q c cqf

q

q

Note that at this threshold1

(1 )q

implying the existence and uniqueness of threshold

Cf

in every case. Note that it can now be stated that as long as q > 0, C Cf f yields S* = 0.

Consider Cf that arises in case [ , )q q q . This threshold can be written as

2 2

2

23 3 23(1 )( ) (1 )( 2 ( )

( )) (1 )( )C

cf q q q qc q c q c q c q

c qq

.

It can be shown that 0Cf

and that the at which S = 1 and

C Cf f is unique implying

uniqueness of Cf .

Finally consider Cf , which is relevant if q q . It is equal to

2

2

3 22

32 3(1 )( 2) (1 )( ) (1 )( )( )( )

C c q c q c q c qc

f q qq

cq q q

.

It can be shown that 0Cf

and that the at which S = 1 and

C Cf f is unique, implying a

unique threshold Cf . It follows that S* = 1 when q q and (max{ , }, ]C C C Cf f f f and that

* (0,1)S in the following cases: q q and C Cf f , and [ , )q q q and ( , )C C Cf f f .

In a last step, we prove uniqueness of in all cases. In case (b) uniqueness is shown in

the proof of Proposition 1 and in case (d.1) it can be shown applying the same procedures as

in the proof of Proposition 3. Uniqueness of in cases (d.2) and (d.3) in which * (0,1)S can

be shown using similar procedures as before. We define the equilibrium condition as follows:

21

*

.3 1 1 Cd

C

f Sq

c f

,

where *

3

2

2

2

(1 )( )(

)

2 )

(C C

C C C

c qS

c q

f f

f c f c fq q

. The limits of .3d with respect to are

51

1*

1

( 1)l 1 1 0

1im 1 C

a

C

f Sq

c f

and

1*

1/(1 )

( 1/ (1 )) 1lim 1 0

(

11

1 )

Ca

qC

f S qq

c f q

.

The first-order condition with respect to can be simplified to

* 2* 2.2 3 (1 )

1 2 ( ) 02

d C

C C

f Sc qq S

c f c f

.

It follows that there exists a unique in cases (d.2) and (d.3). Taken together these results

prove uniqueness of the equilibrium in Proposition 4.

Proof of Proposition 5

IC regulation is preferred to no IC regulation if

2 21 1

(1 )2 1 (1 ) 4 2(1 )

cW qs q S

q s q

.

The limits of W with respect to ( , )C Cf f are,

2

0

1 1lim lim (1 )

2(1 ) 4 2(1 )C Cf f s

cW W q S

q q

for any 0S

* *

* 2

*

1 1lim lim ( )

4 2(1 )2 1 (1 )Cf s S

cW W S

qq S

,

where the last inequality results from the fact that the first-order condition leads to 2

* * qs S

c

. Because W is concave in s and 0

C

ds

df , there must exist a unique threshold

( , )IC

C Cf f . Therefore, S* = 0 when IC

C Cf f and S* > 0 otherwise. IC

C Cf f implies that

there must also exist a minimum level of induced IC effectiveness *(0, ]ICs s .

Finally, it is never beneficial to induce an s > s*, which is the welfare-maximizing s if fC

is so high that all strategic owners comply with S*.