Discussion Paper No. 829

DOES SELF-REGULATION OF
ADVERTISEMENT LENGTH
IMPROVE CONSUMER WELFARE?
Taisuke Matsubae
Noriaki Matsushima

January 2012

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Does self-regulation of advertisement length improve consumer welfare?∗Taisuke Matsubae†
Competition Policy Research Center, Japan Fair Trade Commission
Noriaki Matsushima‡
Institute of Social and Economic Research, Osaka University
January 18, 2012

Abstract
In Japan, TV platforms regulate themselves as to the length of the advertisements they air. Using
modified Hotelling models, we investigate whether such self-regulation improves consumer and social
welfare or not. When all consumers choose a single TV program (the utility functions of consumers
satisfy the standard ‘full-coverage’ condition), self-regulation always reduces consumer welfare. It
improves social welfare only if the advertisement revenue of each platform is not small and the cost
parameter of investments in improving the quality of TV programs is small. When some consumers
have outside options (the standard ‘full-coverage’ condition is not satisfied), self-regulation can benefit
consumers because it increases the number of consumers who watch TV programs.

Keywords: self-regulation, program quality, advertisement volume, consumer surplus, heterogeneous
consumerJEL Classification codes: L13, L41, L82∗ We are grateful to Noriyuki Doi, Hiroyuki Odagiri, Yosuke Okada, and participants in Workshop in Competition Political
Research Center for discussion and comments. We especially would like to thank Koki Arai and Ayako Suzuki for their constructive discussion and comments. The views expressed herein are those of the authors and should not be interpreted as those of
the Japan Fair Trade Commission.
† Taisuke Matsubae, Competition Policy Research Center, Japan Fair Trade Commission, 1-1-1 Kasumigaseki, Chiyoda,
Tokyo 100-8987, Japan. E-mail: fwin6544@nifty.com‡ Corresponding author: Noriaki Matsushima, Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka,
Ibaraki, Osaka 567-0047, Japan. Phone: (81)-6-6879-8571. Fax: (81)-6-6879-8583. E-mail: nmatsush@iser.osaka-u.ac.jp1

1

Introduction

Most nations regulate advertisements in the broadcasting industry (Anderson (2007)). There are restrictions on both the length and content of advertisements. Length restrictions are widespread in developed
nations, with the exception of Japan and the US. It is not obvious whether length restrictions on advertisements are harmful to consumers. Advertisers on TV programs would like to communicate with theirpotential customers who are watching the programs. Those communications stimulate consumption andcan benefit social welfare. TV viewers, however, would like to enjoy the program content without commercial interruptions. Those interruptions are harmful from the viewpoint of welfare. It is this trade-off
that we investigate here. Especially, we take into account self-regulation of the length of advertisementsin the Japanese broadcasting industry.
In the Japanese broadcasting industry, there exists a policy of self-regulation (called “Jisyu-Kisei” atJapanese) on the length of advertisements although length restrictions are not officially imposed on TV
programs as they are in the US. To explain this self-regulation, we mention the basic composition of advertisements in television. In general, a television or radio advertisement is called commercial message (CM).
TV programs are funded by the fees obtained from CMs. In Japan, CMs are distinguished between “spotCM” and “time CM.” A spot CM is an advertisement during a station break and is usually broadcasted
for 15 seconds. A time CM is an advertisement by a sponsor of a TV program. The National Associationof Commercial Broadcasters in Japan (NAB) has imposed maximum lengths of time CMs depending on
the length of the TV program.1 For example, if the length of a TV program is greater than 5 minutes andless than 10 minutes, the maximum length of all advertisements in total is 2 minutes (see Table 1). This
guideline is self-regulated by the Japanese broadcasting industry.
Before we explain the model structure in this paper, we briefly describe the broadcasting industry
in Japan (see Fig 1). Sponsors (private and/or public firms, etc.) offer broadcast advertisements to aplatform and they pay fees for their advertisements to the platform. The platform produces the program
1 The NAB is a private non-profit organization whose membership consists of 201 commercial broadcasters in Japan. It wasestablished as a voluntary organization on July 20, 1951, by 16 commercial radio companies. The NAB has been a non-profit
incorporated association since April 21, 1952. Its membership now comprises all terrestrial radio and television broadcasters aswell as some satellite audio and television broadcasters (this information is available on the NAB website, http://nab.or.jp/).

2

The length of a TV program
5 minutes
10 minutes
20 minutes
30 minutes
40 minutes
50 minutes
60 minutes

The maximum length of all advertisements in total
1 minute
2 minutes
2.5 minutes
3 minutes
4 minutes
5 minutes
6 minutes

Table 1: The criteria imposed by NAB for advertisement lengthwith the advertisement fees.2 The platforms supply TV programs to consumers. The advertising fee often
depends on the time the program airs, the length of this advertisement, the expected number of viewersof the program, and so on. Platforms often take the following position on self-regulation: “In the absence
this self-regulation, consumer welfare will diminish because the length of advertisements per program
would rapidly increase. This self-regulation can be a beneficial commitment not to increase the length ofadvertisements.” We consider whether or not this claim is plausible.

Sponsors

Platforms

Viewers (Consumers)
Figure 1: The image of this market.

We present a theoretical model to discuss how self-regulation in a free-to-air broadcasting industry
affects the quality of TV programs, consumers, and social welfare. Basically, we follow the model setting
in Gabszewicz et al. (2001, 2004). We incorporate self-regulation and heterogeneity of consumer pref2 In Japan, intermediaries (advertisement agencies including “Dentsu” and “Hakuhodo”) usually coordinate the transactions

between sponsors and platforms.

3erences for advertisements into the model used by Gabszewicz et al. (2001, 2004). The model setting
is as follows. Consumers are distributed on a line segment as in Hotelling (1929). Two platforms arerespectively located at one of the edges in the line segment. The location of each consumer represents
his/her preference for the platforms. Each platform determines the quality of its program and the length ofadvertisements. Those two factors respectively generate additional gains and losses for consumers. Eachplatform incurs costs to improve its TV program. An increase in the length of advertisements on a platform
decreases the number of viewers. Some consumers take into account only the quality of TV programs or
only the length of advertisements in each platform. That is, there are three types of consumers; those
who take into account (i) quality and advertisement, (ii) only quality, and (iii) only advertisement. The
revenue of each platform is proportionally correlated to the total number of viewers times the length of theadvertisements (Gabszewicz et al. (2001, 2004)). The maximum advertisement length in each platform istaken to be the maximum allowed under self-regulation. The platforms set the level of self-regulation soas to maximize their joint profits.
We show that self-regulation always reduces consumer welfare.3 We also show that it improves social
welfare only if per-advertisement revenue is not small and the cost parameter of quality investment for
TV programs is small. Under these necessary conditions, each platform excessively engages in quality
investment if they did not impose self-regulation. Self-regulation works as a credible commitment not toinvest more. This limits the quality investment costs, although the benefits to consumers decrease. The
former positive effect can dominate the latter negative one. We also show that the advertisement regulation
imposed by a benevolent regulator can decrease consumer welfare even though it improves social welfare.
The study of advertisement in broadcasting industries has been extensive in the past decade.4 Mostof these contributions focus on the combination of advertisement and horizontal product differentiation
among private platforms in two-sided markets: Gabszewicz et al. (2004) study relationship between
program substitutability and the degree of advertisement aversion; Anderson and Coate (2005) consider
whether or not the advertisement level is socially optimal; Gantman and Shy (2007) focus on the effects of
3 An extended setting shows a slightly different result.
4 See the comprehensive surveys by Anderson (2007) and Anderson and Gabszewitcz (2006) about advertisement in the media.

4the improvement in advertisement quality. Peitz and Valletti (2008) compare the levels of advertisement
intensities and program differentiation under pay-tv and free-to-air; Crampes et al. (2009) consider theeffects of advertisement on entry in the media industry. González-Maestre and Martı́nez-Sánchez (2010)
consider the role of a publicly owned platform and program quality in the free-to-air broadcasting industry.
González-Maestre and Martı́nez-Sánchez (2011) consider the role of the endogenous choice of quality
programs in a mixed duopoly market.
In the context of adverting regulation, Anderson (2007) consider the regulation of television advertisement by public authority. Stühmeier and Wenzel (2010) study the effects of symmetric and asymmetricadvertisement regulation (advertisement cap) on competition for viewers and advertisers in a duopoly
framework where a public and a private broadcaster compete. While those papers consider a regulated
advertisement level that is a maximizer of social welfare, we focus on the self-regulated advertisementlevel, in which this level is a maximizer of joint profits among platforms.
The rest of the paper is organized as follows: Section 2 presents a model and the main result. Section
3 extends the basic model; we allow consumers to not watch TV programs. Section 4 concludes the paper.

2 The model with various types of consumers
We assume that two platforms are each located at one extreme of the linear market of length 1. In thislinear market, there exist tree types of consumers (see Fig. 2). Each consumer can choose either platform
1 or 2.
We define each type of consumer as follows.
Type 1: There are consumers indexed by x ∈ [0, 1] and distributed uniformly along the linear market. The
mass of consumers is 1. The utility of consumer x if she or he watches platform i ∈ {1, 2} is given as½
u(qi , ai , x) =µ + q1 − ka1 − tx
µ + q2 − ka2 − t(1 − x)

if i = 1,
if i = 2,

where µ is a positive constant and sufficiently large (this assumption ensures that each consumer chooses
one of the platforms), qi is the content quality of platform i, ai is the volume of advertisement, and t
represents the degree of substitutability between the two platforms. We assume that t is a constant positive5

real number and that k is the parameter representing the disutility per unit of advertisement and is aconstant positive real number.
Type 2: There are consumers indexed by y ∈ [0, 1] and distributed uniformly along the linear market. The
mass of consumers is λa We assume that λa is a positive real number. The utility of consumer y if (s)hewatches platform i ∈ {1, 2} is given as½µ − ka1 − tyu(ai , y) =µ − ka2 − t(1 − y)if i = 1,
if i = 2.

Type 3: There are consumers indexed by z ∈ [0, 1] and distributed uniformly along the linear market. The
mass of consumers is λq . We assume that λq is a positive real number. The utility of consumer z if (s)hewatches platform i ∈ {1, 2} is given as½µ + q1 − tzu(qi , z) =µ + q2 − t(1 − z)

if i = 1,
if i = 2.

Consumers in Type 1 get (dis)utility from both the quality of the program and the volume of advertisement, consumers in Type 2 get (dis)utility only from the volume of advertisement, and consumers in
Type 3 get utility only from the quality of the program.
P2

P1

D= 1

xn

Type 1
Type 2ya

D= λaType 3

wq

D= λq
1

0

Figure 2: Image of this model

We begin to calculate each location of indifferent consumers with respect to each type. The locationof indifferent type 1 consumers, xn , is given as,
xn (a1 , a2 , q1 , q2 ) =t + q1 − q2 − ka1 + ka2
.
2t

(2.1)

The location of indifferent type 2 consumers, ya , is given as,ya (a1 , a2 ) =t − ka1 + ka2
.
2t
6

(2.2)

The location of indifferent type 3 consumers, zq , is given as,zq (q1 , q2 ) =t + q1 − q2
.
2t

(2.3)

The gross profit of each platform consists of the advertisement revenue. We assume that one unitof view for advertisement generates the revenue γ , a positive constant. This assumption means that theadvertisement market is perfectly competitive (see Gabszewicz et al. (2001, 2004)). The total units ofview for advertisement shown in platform i is the volume of advertisement times the number of audiences:
Q1 (a1 , a2 , q1 , q2 ) = a1 (xn (a1 , a2 , q1 , q2 ) + λa ya (a1 , a2 ) + λq wq (q1 , q2 ))Q2 (a1 , a2 , q1 , q2 ) = a2 (1 − xn (a1 , a2 , q1 , q2 ) + λa (1 − ya (a1 , a2 )) + λq (1 − wq (q1 , q2 ))).
Each platform can improve the quality of its program, qi , by investments. We assume that the investment
cost for platform i is φ q2i , where φ is a positive real number. The profits of platforms 1 and 2 are given by

π1 (a1 , a2 , q1 , q2 ) = γ Q1 (a1 , a2 , q1 , q2 ) − φ q21 ,(2.4)

π2 (a1 , a2 , q1 , q2 ) = γ Q2 (a1 , a2 , q1 , q2 ) − φ q22 .

(2.5)

We consider two cases: (i) the platforms do not self-regulate on their advertisement volumes; (ii)the platforms impose self-regulation on their advertisement volumes. In the first case, the game runs asfollows. First, each platform simultaneously sets the quality of its program, qi . Second, given the valuesof qi (i = 1, 2), they simultaneously set the volumes of advertising, ai (i = 1, 2). In the second case, thegame runs as follows. First, the platforms determine the upper bound of ai , ar , to maximize their joint
profits. Second, each platform simultaneously sets the quality of its program, qi . Third, given the valuesof qi (i = 1, 2) and ar , they simultaneously set the volumes of advertising, ai ∈ [0, ar ] (i = 1, 2). We solve
the games by backward induction.

2.1 The case of no self-regulation in advertising
We consider a case in which each platform can freely choose the volume of advertising.

7

In the second stage, given the qualities of their programs, the reaction function of platform i derived
by its first-order condition and its optimal volume of advertising are given ast(1 + λa + λq ) + (1 + λq )(qi − q j ) + k(1 + λa )a j,
2k(1 + λa )3t(1 + λa + λq ) + (1 + λq )(qi − q j ), i, j = 1, 2, i ̸= j.
→ ai (qi , q j ) =3k(1 + λa )ai (a j ) =Substituting ai (i = 1, 2) into the profit functions, we solve the optimal qualities of their programs. We
assume thatφ > γ (1 + λq )2 /(18kt(1 + λa )) ≡ φ non .
Note that under this condition, the second-order conditions are satisfied with respect to the profit functions.
The reaction function of platform i derived by its first-order condition and the optimal quality ofplatform i’s program, q∗i , are,
3t γ (1 + λq )(1 + λa + λq ) − γ (1 + λq )2 q j,
18kt(1 + λa )φ − γ (1 + λq )2
γ (1 + λq )(1 + λa + λq )→ q∗i =.
6k(1 + λa )φqi (q j ) =Substituting q∗i (i = 1, 2) into ai (qi , q j ), we have the optimal advertising volume of platform i:a∗i =t(1 + λa + λq ).
k(1 + λa )We study the relationship between q∗i and λa . We illustrate how the optimal quality of program andthe optimal volume of advertising change when the number of consumers of type 2, λa , changes.
Proposition 2.1.
(i) An increase in λa (an increase in the mass of Type 2 consumers) lowers program quality.
(ii) An increase in λa (an increase in the mass of Type 2 consumers) lowers the volume of advertising.
The intuition behind this proposition is as follows. Type 2 consumers only care about the volume ofadvertising. An increase in the mass of Type 2 consumers means that consumers become more elastic
8

with regard to their dislike for advertising. To meet changes consumer taste, each platform decreases its
volume of advertising. As a result, the per-consumer revenue of each platform from advertising decreases
and then the incentive of each platform to engage in quality investment diminishes.
We study the relationship between q∗i and λq . We illustrate how the optimal quality of a program andthe optimal volume of advertising change when the number of consumers of Type 3, λq , changes.
Proposition 2.2.
(i) An increase in λq (an increase in the mass of Type 3 consumers) improves program quality.
(ii) An increase in λq (an increase in the mass of Type 3 consumers) increases the volume of advertising.
The intuition behind this proposition is as follows. Type 3 consumers only care about program quality.
In other words, those consumers do not take into account the volume of advertising. An increase in themass of Type 3 consumers means that consumers become less elastic in relation to changes in advertisingvolume. The increase in λq allows the platforms to increase their advertising volumes. As a result, theper-consumer revenue of each platform from advertising increases and then the incentive of each platform
to improve program quality improves.
We study consumer and social welfare under q∗ and a∗ . The entire consumer surplus (CS) is the total
consumer surplus for each type. Each consumer surplus of Type 1, Type 2, and Type 3 is denoted as CSn ,CSa , and, CSq , respectively. Each consumer surplus is calculated as:
Z x∗

Z 1

n

CSn∗ = µ +(q∗1 − ka∗1 − tz)dz + (q∗2 − ka∗2 − t(1 − z))dz,
0
xn∗
¶
µ
Z y∗
Z 1
a
∗
∗
∗
(ka1 + tz)dz − (ka2 + t(1 − z))dz ,CSa = λa µ −
0
y∗a
¶
µ
Z w∗
Z 1
q
CSq∗ = λq µ +(q∗1 − tz)dz + (q∗2 − t(1 − z))dz .
w∗q0

Note that CS = (2.6) + (2.7) + (2.8). Hence, CS is given as
CS∗ = µ (1 + λa + λq ) +(1 + λa + λq )(2γ (1 + λq )2 − 15kt(1 + λa )φ ).
12k(1 + λa )φWe call CS∗ the consumer surplus without self-regulation of advertising.
9

(2.6)
(2.7)
(2.8)

The total platforms’ profits (π ∗ = π1∗ + π2∗ ) are given as,π∗ =γ (1 + λa + λq )2 (18kt(1 + λa )φ − λ (1 + λq )2 ).
18k2 (1 + λa )2 φ

We call π ∗ the total profit without self-regulation for advertising.
We now calculate social welfare, W , which is defined as the sum of both platforms’ profits and thetotal consumer surplus, π ∗ +CS∗ :W ∗ = π ∗ +CS∗ .
We call W ∗ social welfare without self-regulation for advertisement.

2.2 The case of self-regulation for advertisement
We consider the case in which the volumes of advertisement are regulated by the programmers. We
assume that the volumes of advertisement must be lower than or equal to ar . We first solve the case by
assuming that ar is binding for the platforms in the third stage. After we derive the full game, we confirm
the condition in which ar is really binding in equilibrium.
We suppose that ar is binding for the platforms in the third stage. Anticipating the third stage outcome,we determine their respective qualities. The first-order condition of platform i with respect to qi and theoutcome are given as:

γ (1 + λq )ar − 4t φ qi
ar γ (1 + λq )= 0 → qi =
.
2t
4t φ

(2.9)

Given the quality of programs, we derive the optimal self-regulated advertisement level, a∗r . The joint
profits are given asµπ1 + π2 = γ ar (1 + λa + λq ) − 2φar γ (1 + λq )4t φa∗r is given asa∗r =4t 2 (1 + λa + λq )φ.
γ (1 + λq )2

10

¶2
.

We consider the condition such that the a∗r is binding for the platforms. This condition means that theadvertising level without self-regulation is higher than that with self-regulation. Thus, we consider theregion of φ such that a∗ − a∗r > 0. The region is given asφ<γ (1 + λq )2≡ φ̄ .
4kt(1 + λa )In this region, the optimal quality of the program under self-regulation for platform i, q∗ir is given as,q∗ir =t(1 + λa + λq ).
1 + λqWe state how the optimal qualities of programs and the volumes of advertisement change when eachtype of consumers increases. We omit the proof because the proof can be shown by using the previous
method.
Proposition 2.3.
(i) An increase in λa (an increase in the mass of Type 2 consumers) increases program quality.
(ii) An increase in λa (an increase in the mass of Type 2 consumers) increases the volumes of advertising.
(iii) An increase in λq (an increase in the mass of Type 3 consumers) decreases program quality.
(iv) An increase in λq (an increase in the mass of Type 3 consumers) decreases the volumes of advertising.
The properties of this proposition are quite different from those of the previous two propositions. We
summarize the two propositions in Table 2. We explain the mechanism behind this proposition.
They cooperatively determine the upper bound of ai at ar . In other words, they do not compete in
the advertisement levels. This implies that the demand of Type 2 consumers for platform i is fixed at

λa /2 (i = 1, 2) under all quality and advertisement levels. An increase in λa just enhances the demand of
consumers for both platforms. Because the demand of Type 2 consumers is fixed by their self-regulation,
they have an incentive to raise the regulation level ar (per consumer advertisement revenue). The increasein ar enhances the incentives of both platforms to improve the qualities of their programs.

11

a∗i
a∗r
q∗i
q∗ir

Increase in type 2 consumers↓ (Prop.2.1.(ii))↑ (Prop.2.3.(ii))↓ (Prop.2.1.(i))↑ (Prop.2.3.(i))

Increase in type 3 consumers↑ (Prop.2.2.(ii))↓ (Prop.2.3.(iv))↑ (Prop.2.2.(i))↓ (Prop.2.3.(iii))

Table 2: Summary of Propositions 2.1., 2.2., and 2.3.
This mechanism does not work in cases (iii) and (iv). Type 3 consumers do not take into account theupper bound of the advertisement level, ar . In other words, the self-regulation does not affect the choicesof those consumers. An increase in λq just enhances the demand of consuemrs for both platforms. To
meet this demand expansion, they would have a high incentive to enhance the qualities of their programs
if they did not impose the upper bound of the advertisement level. They anticipate that the demand
expansion does not expand the total demand for their programs. From the viewpoint of their joint profits,
the aggressive efforts to improve their program qualities have only a cannibalization effect on their profits.
To mitigate their aggressive incentives, the platforms lower the upper bound of the advertisement level, ar .
The decrease in ar diminishes the incentives of both platforms to improve the qualities of their programs.
Under φ < γ (1 + λq )2 /(4kt(1 + λa )), platform i’s profit is given as,

πir∗ =
Profitability of self-regulationt 2 (1 + λa + λq )2 φ.
(1 + λq )2

We check the condition under which self-regulation is profitable for theplatforms, πir∗ − πi∗ > 0. A simple calculation leads to the following inequality:³√ ´
3 − 5 γ (1 + λq )2φ<≡ φ̄ non .
12kt(1 + λa )Note that, φ̄ non < φ̄ . That is, a binding constraint a < ar does not always improve the profits of theplatforms. We assume that the following condition is satisfied with respect to φ throughout this section.5

φ non < φ < φ̄ non .
5 φ non means that this region of φ differs from region of φ in section 3.

12

We calculate the consumer surplus with self-regulation. Substituting a∗r and q∗ir for (2.6), (2.7), and(2.8), and summing to those, we have
CSr∗ = (1 + λa + λq )µ +t(1 + λa + λq )(3γ (1 + λq )2 − 16kt(1 + λa )φ ).
4γ (1 + λq )2The total profit of the platforms with self-regulation is given as,πr∗ =2t 2 (1 + λa + λq )2 φ.
(1 + λq )2As a result, social welfare with self-regulation is Wr∗ = CSr∗ + πr∗ .
¡¢t(1 + λa + λq ) 3γ (1 + λq )2 − 8t(2k(1 + λa ) − γ (1 + λa + λq ))φ∗Wr = (1 + λa + λq )µ +.
4γ (1 + λq )2

2.3

Self-regulation vs. no self-regulation

We compare between the consumer surplus with self-regulation and that without self-regulation. Let △CS
be CS∗ −CSr∗ . A simple calculation leads to the following proposition:
Proposition 2.4. △CS is a positive. Thus, the consumer surplus without self-regulation is higher thanthat with self-regulation.
Self-regulation reduces the costs incurred by watching advertisements. This is beneficial. Because
self-regulation in itself does not expand the total demand for programs, the benefit is not so strong. On
the other hand, it reduces the incentives of the platforms to engage in quality investments (q∗ir < q∗i ). The
latter negative effect dominates the former positive one.
We compare the difference between the social surplus without self-regulation and that with selfregulation. Note that the profit of each platform under self-regulation is higher than that under no selfregulation in φ ∈ (φ non , φ̄ non ). Let △W be W ∗ −Wr∗ . We characterize this difference by γ and φ .
Proposition 2.5.
(i) If 0 < γ < 11k(1 + λa )/(1 + λa + λq ), △W is positive.
(ii) If 11k(1 + λa )/(1 + λa + λq ) < γ ;13

(a) If φ non < φ < φ ∗ , where φ ∗ is the minimum solution such that △W = 0, △W is negative;(b) If φ ∗ < φ < φ̄ non , △W is positive.
The self-regulation improved social welfare compared with non-self-regulation when γ /k was not
small and φ was small enough. This is because each platform excessively engages in its investment if φ issmall enough. Self-regulation controls their overinvestment problem. This is beneficial from the viewpointof social welfare. Note that the overinvestment problem is not so serious when γ is small enough. A small

γ means that per-viewer revenue is small. This implies that the returns from quality investments are small.
That is, under a small γ , the platforms do not have strong incentives to engage in quality investments.

2.4

Comparison of social welfare: self-regulation and welfare maximization

We consider the volume of advertising that maximizes social welfare. In this case, the platforms mustchoose the advertising levels determined by a benevolent government. To solve the optimal volume ofadvertising, we calculate the equilibrium quality levels of their programs given the regulated advertisinglevels (defined it by ag ). The optimal quality of each program q∗∗i (a) is given byq∗∗i (ag ) =ag γ (1 + λa ).
4t φ

We substitute the optimal quality into π1 , π2 , and CS. Let the social welfare function be denoted as
W (ag ) = π1 + π2 +CS, whereW (ag ) = π1 + π2 +CS¡¢
ag γ 8t 2 φ (1 + λa + λq ) − γ (1 + λq )2 ag=
8t 2 φÃ¢!
¡−t 2 (1 + λa + λq )φ + ag γ (1 + λq )2 − 4kt(1 + λa )φ+ (1 + λa + λq )µ +
.
4t φ
∗∗
We evaluate a∗∗g ∈ arg maxW (ag ). Let ag be the optimal volume of advertising. Differentiating W (ag )with respect to ag , and solving the derivative with respect to a, we have:¡¡¢¢t −4kt(1 + λa )φ + γ (1 + λq )2 + 4t(1 + λa + λq )φ
∗∗
ag =
.
γ 2 (1 + λq )2From the result, we have the following proposition:
14

Proposition 2.6.
∗(i) If γ < (k(1 + λa ))/(1 + λa + λq ), a∗∗g is higher than ar .
∗(ii) If γ > (k(1 + λa ))(1 + λa + λq ), a∗∗g is lower than ar .

When γ is small, per-unit revenue is small. The incentives of quality investments are weak. The
benevolent government has an incentive to encourage the platforms to invest more. A higher value of ag
is a way to increase per-unit revenue from advertisement.
∗∗
We consider CS at a∗∗g . Substituting ag for (2.6), (2.7), and (2.8) and summing up the results, we

can derive the consumer surplus (denoted by CS∗∗ ). We compare CSr∗ with CS∗∗ , leading to the following
proposition:
Proposition 2.7. CS∗∗ is higher than CSr∗ .
The benevolent government takes into account not only the total profit but also the consumer surplus,
although the platforms take into account only their profits when they self-regulate. The former objective
fixes the distortion of self-regulation.
We compare the difference between the profits of the platforms in the cases of self-regulation andwelfare-maximizing regulation.
Proposition 2.8. π ∗∗ is lower than πr∗ .
This result is just the flip-side of the previous proposition. From the propositions, the self-regulationof advertisement by the platforms improves only their own profits.

2.5

The comparison between social welfare in the absence of self-regulation and that in
welfare-maximizing regulation

The consumer surplus is lower under conditions of self-regulation than under those with welfare-maximizing
regulation. This begs the question. Is welfare-maximizing regulation is better for consumers than no regulation? We compare the difference between the consumer surplus under welfare-maximization regulationand that without regulation. The result leads to the following proposition:
15

Proposition 2.9.
(i) If 5(1 + λa )k/(2(1 + λa + λq )) < γ , CS∗ > CS∗∗ .
(ii) If 49(1 + λa )k/(22(1 + λa + λq )) < γ ≤ 5(1 + λa )k/(2(1 + λa + λq )),(a) if φ non < φ < φ ⋆ , CS∗ > CS∗∗ ;(b) if φ ⋆ < φ < φ̄ non , CS∗ < CS∗∗ , where φ ⋆ is φ such that CS∗ = CS∗∗ .
(iii) If γ ≤ 49(1 + λa )k/(22(1 + λa + λq )), CS∗ < CS∗∗ .
This is related to the explanation for Proposition 2.6. When γ is small, per-unit revenue is small. The
incentives to improve quality are weak. A benevolent government has an incentive to encourage platforms
for investments. A higher value of ag is a way to increase per-unit revenue from advertising. This improves
the consumer surplus.

3 Consumers have outside options
We consider the case in which some consumers have outside options. There are two types of consumers
who can choose not to watch TV programs. In this market, there are three types of consumers (see Figs.
3 and 4).
P1P2Type 1

Pi

D=1
0

xn

Type 1 − i

D=λ
0

1

yi

Figure 4: Demand of Type 1 − iFigure 3: Demand of Type 1

Type 1 consumers are the same as those in the previous section. We introduce two types of consumers
that were not considered in the previous sections. We call those types ‘Type 1-i’ (i = 1, 2). Type 1-i isformally defined as follows.
Type 1-i: There are consumers indexed by yi ∈ [0, ∞) and distributed uniformly along the ‘half-line’ as
depicted in Figure 4. The mass of consumers is λ . This is a positive real number. The utility of consumer
16

yi if she or he watches platform i ∈ {1, 2} is given asu(qi , ai , yi ) = v + qi − kai − tyi ,where v is the reservation utility for watching a program but is not large. If her or his utility is negative,
she or he does not watch program i. We consider the location of indifferent consumers, yi , such that either
watching platform i or not watching platform i is indifferent:yi =v + qi − kai.
txn is the location of indifferent Type 1 consumers such that watching either platform 1 or platform 2 isindifferent. Hence, the total units of view for advertising shown in platform i is the volume of advertisingtimes the number of consumers:
Q1 (a1 , a2 , q1 , q2 ) = a1 (xn (a1 , a2 , q1 , q2 ) + λ y1 (a1 , q1 )),Q2 (a1 , a2 , q1 , q2 ) = a2 (1 − xn (a1 , a2 , q1 , q2 ) + λ y2 (a2 , q2 )).
Therefore, the profits of platform i are given by

πi = γ Qi (a1 , a2 , q1 , q2 ).

3.1

The case without self-regulation for the volume of advertising

We consider a case where each platform can freely choose a volume of advertising.
We begin to calculate the equilibrium qualities of programs and the volumes of advertising in thismarket. Given the qualities of their programs, the first-order condition of platform i isai =2λ v + t + (1 + 2λ )qi − q j + ka j.
2k(1 + λ )Each volume of advertisement is as follows:ai (qi , q j ) =(3 + 4λ )(t + 2vλ ) + (1 + 8λ + 8λ 2 )qi − (1 + 2λ )q j.
k (3 + 16λ + 16λ 2 )17

We now consider the situation in which each platform can improve the quality of its programs. We assume
that the investment cost of platform i is φ q2i , where φ is a positive real number. The profit of platform i is
as follows:

πi (q1 , q2 ) = γ Qi (ai (q1 , q2 ), a j (q1 , q2 ), q1 , q2 ) − φ q2i , i, j = 1, 2, j ̸= i.

(3.1)

We assume that the second-order conditions of the platforms are satisfied:¡¢2
γ (1 + 2λ ) 1 + 8λ + 8λ 2
< φ.
2kt(1 + 4λ )2 (3 + 4λ )2The first-order condition of platform i isqi (q j ) =γ (1 + 2λ )(1 + 8λ + 8λ 2 )((3 + 4λ )(t + 2λ v) − (1 + 2λ )q j ).
2kt(1 + 4λ )2 (3 + 4λ )2 φ − γ (1 + 2λ )(1 + 8λ + 8λ 2 )2The optimal quality of platform i, q∗i , is
q∗i =

¡
¢
γ (1 + 2λ )(t + 2vλ ) 1 + 8λ + 8λ 2.
2 (kt(1 + 4λ )2 (3 + 4λ )φ − γλ (1 + 2λ ) (1 + 8λ + 8λ 2 ))Hence, the optimal advertisement volume of platform i, a∗i , is given asa∗i =t(1 + 4λ )(3 + 4λ )(t + 2vλ )φ.
kt(1 + 4λ )2 (3 + 4λ )φ − γλ (1 + 2λ ) (1 + 8λ + 8λ 2 )Note that we use the same symbol as in the previous section for the optimal program quality and thevolume of advertising.
Under a∗i and q∗i , yi is given as¢
1 + 8λ + 8λ 2.
2 (kt(1 + 4λ ) (3 + 4λ )φ − γλ (1 + 2λ ) (1 + 8λ + 8λ 2 ))2k(1 + 4λ )(3 + 4λ )((1 + 2λ )v − t)φ + γ (1 + 2λ )
y∗i =
2

¡

Note that there exist Type 1-i consumers who watch platform i (yi > 0) if v > t/(1 + 2λ ).
We study the welfare properties under q∗ and a∗ . The consumer surplus of Type 1 (resp. 1-i) is denoted
as CSn (resp. CSo ). Each consumer surplus is calculated as¡¢γ (1 + 2λ )(2t + t λ + 4vλ ) 1 + 8λ + 8λ 2 − kt(1 + 4λ )(3 + 4λ )(8λ v + (5 + 4λ )t)φ
∗
CSn =
,
4 (kt(1 + 4λ )2 (3 + 4λ )φ − γλ (1 + 2λ ) (1 + 8λ + 8λ 2 ))
¡
¡
¢
¡
¢ ¢2
t γ (1 + 2λ ) 1 + 8λ + 8λ 2 + 2k((1 + 2λ )v − t) 3 + 16λ + 16λ 2 φ
∗
CSo =
.
4 (kt(1 + 4λ )2 (3 + 4λ )φ − γλ (1 + 2λ ) (1 + 8λ + 8λ 2 ))2
18

(3.2)
(3.3)

CS∗ = CSn∗ +CSo∗ (see (3.2) and (3.3)). The total platforms’ profits (π ∗ = π1∗ + π2∗ ) are given as,
³ ¡
¢2
¡
¢2 ´
γ (1 + 2λ )(t + 2vλ )2 φ 2kt 3 + 16λ + 16λ 2 φ − γ (1 + 2λ ) 1 + 8λ + 8λ 2.
π∗ =2 (kt(1 + 4λ )2 (3 + 4λ )φ − γλ (1 + 2λ ) (1 + 8λ + 8λ 2 ))2
We now calculate social welfare, W , defined as the sum of profits and consumer surplus, W ∗ = CS∗ + π ∗ .

3.2 The case of self-regulation of advertising volume
We consider the case in which the volumes of advertising are self-regulated. Hence, we assume that thevolumes of advertising are restricted to common ar for both platforms. Each platform chooses only thequality level of its program given ar . The first-order condition of πi with respect to qi is given as∂ πi ar γ + 2ar γλ − 4qit φ
=
= 0.
∂ qi
2t

(3.4)

Solving (3.4) with respect to qi , we obtain, for i = 1, 2,qi =ar γ (1 + 2λ ).
4t φGiven the qualities of their programs, we solve the self-regulated advertising level, a∗r :a∗r =4t(t + 2vλ )φ.
16kt λ φ + (1 − 4λ 2 )γ

We only consider the case in which self-regulation is binding, that is, a∗r < a∗ . We have the conditions asfollows.

φ<γ (1 + 2λ )2 (3 + 8λ ).
4kt(1 + 4λ )(3 + 4λ )We call the region of φ the ‘binding condition’. Under this condition, the optimal qualities of the programs
under self-regulation, q∗r , are:q∗r =γ (1 + 2λ )(t + 2vλ ).
16kt λ φ + (1 − 4λ 2 )γ

We restrict the region of φ such that φ satisfies both the second-order conditions and the binding conditionin this section. Thus,¡¢2γ (1 + 2λ ) 1 + 8λ + 8λ 2γ (1 + 2λ )2 (3 + 8λ )<=: φ̄ .
φ :=φ<
(2kt(1 + 4λ )2 (3 + 4λ )2 )4kt(1 + 4λ )(3 + 4λ )19

We calculate each consumer surplus with self-regulation. The consumer surplus of Type 1 (resp. 1-i)
is denoted as CSr n (resp. CSro ) in the case of self-regulation.
∗
CSrn
=µ−
∗
CSro
=

16kt(t + 4t λ + 2vλ )φ − γ (1 + 2λ )(3t + 2t λ + 8vλ ),4 (16kt λ φ + γ (1 − 4λ 2 ))(4kt(2λ v − t)φ + γ (t + v)(1 + 2λ ))2t (16kt λ φ + γ (1 − 4λ 2 ))2

.

(3.5)
(3.6)

∗ +CS∗ .
We denote the consumer surplus with self-regulation as CSr∗ := CSrnro

3.3

Comparison of welfare: self-regulation and non-self-regulation

We compare the consumer surplus of self-regulation with that without self-regulation. Let △CS be thedifference between CS∗ and CSr∗ , △CS ≡ CS∗ −CSr∗ . We characterize the relationship between consumer
welfare in the two cases by φ .
Proposition 3.1.
(i) If φ < φ < φ ∗ , △CS is positive, where φ ∗ is the minimum value that satisfies CS∗ = CSr∗ . Thus, theconsumer surplus in the case of non-self-regulation is higher than that in the case of self-regulation.
(ii) If φ ∗ < φ < φ̄ , △CS is negative.
Proposition 3.1 is different from proposition 2.7. If there exist consumers with outside options,the consumer surplus may be improved by self-regulation of advertising. In the previous section, selfregulation in itself does not expand the total demand for the programs. In this setting, self-regulation
expands the total demand for the programs. This is also a benefit to consumers. Because the platforms
aggressively engage in quality investment when φ is small, self-regulation significantly reduces the investment levels when φ is small. That is, when φ is small, the negative effect dominates the positive
one. Note that, self-regulation does not always reduce the incentives of the platforms to engage in quality
investments.

20

4Concluding remarks

In many countries, there are restrictions on both the length and content of advertisements. In the Japanese
broadcasting industry, there are self-regulation guidelines (called “Jisyu-Kisei” in Japannease) on thelength of advertisements, although length restrictions are not officially imposed on TV programs as they
are in the US. We theoretically investigate how the Japanese broadcasting industry’s self-regulation of theadvertising length affects consumer and social welfare.
The results and the policy implications are as follows. When all consumers choose a single TV
program (the utility functions of consumers satisfy the standard ‘full-coverage’ condition), then selfregulation always reduces consumer welfare. It improves social welfare only if the advertisement revenueof each TV platform is not small and the cost parameter of improving the quality of TV programs is small.
When some consumers have outside options (the standard ‘full-coverage’ condition is not satisfied), theself-regulation can benefit consumers because it increase the number of consumers who watch TV programs. The results mean that self-regulation does not work as a useful tool to improve consumer welfare
but does work as a collusive device to maintain the profits of TV platforms. This implies that we need toinspect broadcasters’ cooperative conduct even when it seems to benefit consumers.
Although the current model does not include competition for gathering advertisers, we can extend it
to that with this kind of competition. That is, we can investigate a broadcasting industry with two-sided
competition ( Rochet and Tirole (2003) and Armstrong and Wright (2007)). Although we expect that thisextension would complicate the analysis, this would give us interesting insights. This is a considerable
future research.

References
Anderson, S. P. (2007) “Regulation of Television Advertisement” in The Economic Regulation of Broadcasting Markets Evolving Technology and Challenges for Policy, edited by Paul Seabright and Jorgenvon Hagen, Cambridge University Press.

21

Anderson, S. P. and S. Coate (2005) “Market Provision of Broadcasting: A Welfare Analysis” Review of
Economic Studies 72(4), 947–972.
Anderson, S. P. and J. J. Gabszewicz (2006) “The Media and Advertisement: A Tale of Two-sided Markets,
in Victor A. Ginsburgh and David Throsby, Editors,” in Handbook of the Economics of Art and Culture,
Volume 1, Elsevier B.V.
Armstrong, M. and J. Wright (2007) “Two-sided Markets, Competitive Bottlenecks and Exclusive Contracts” Economic Theory 32(2), 353–380.
Crampes, C., C. Haritchabalet and B. Jullien (2009) “Advertisement, Competition and Entry in Media
Industries” Journal of Industrial Economics 57(1), 7–31.
Gabszewicz, J. J., D. Laussel, and N. Sonnac (2001) “Press Advertisement and the Ascent of the ‘PenséeUnique” European Economic Review 45(4-6), 641–651.
Gabszewicz, J. J., D. Laussel, and N. Sonnac (2004) “Programming and Advertisement Competition in
the Broadcasting Industry” Journal of Economics and Management Strategy 13(4), 657–669.
Gantman, N. and O. Shy (2007) “Broadcasting Competition and Advertisement Quality: Two-sided Market Approach” mimeo.
González-Maestre, M. and F. Martı́nez-Sánchez (2010) “The Role of Program Quality and Publicly-owned
Platforms in the Free to Air Broadcasting Industry” Working Paper. Serie AD 2010-19.
González-Maestre, M. and F. Martı́nez-Sánchez (2011) “Platform Quality and Advertisement Regulationin Broadcasting Markets” mimeo.
Hotelling, H. (1929) “Stability in Competition” Economic Journal 39(153), 41–57.
Peitz, M. and T. M. Valletti (2008) “Content and Advertisement in the Media: Pay-tv versus Free-to-air”
International Journal of Industrial Organization 26(4), 949–965.

22

Rochet, J.-C., J. Tirole (2003) “Platform Competition in Two-Sided Markets” Journal of the European
Economic Association 1(4), 990–1029.
Stühmeier, T. and T. Wenzel (2010) “Regulating Advertisement in the Presence of Public Service Broadcasting” mimeo.

23

Technical Appendix
A Proofs in Section 2
Proof of Proposition 2.1
We begin to prove (1). We partially differentiate the optimal quality of a program with respect to λa andget

γλq (1 + λq )
∂ q∗i
=−
< 0.
∂ λa
6k(1 + λa )2 φ
We prove (2). We partially differentiate the optimal volume for advertising with respect to λa and get
t λq
∂ a∗i
=−
< 0.
∂ λa
k(1 + λa )2Proof of Proposition 2.2
We begin to prove (1). We partially differentiate the optimal quality of a program with respect to λq andget
2γ + γλa + 2γλq
∂ q∗i
=
> 0.
∂ λq
6kφ + 6kλa φ
We prove (2). We partially differentiate the optimal volume for advertising with respect to λq and get

∂ a∗i
t
=
> 0.
∂ λq k(1 + λa )Proof of Proposition 2.4
Calculating △CS, we get¡¢(1 + λa + λq ) γ 2 (1 + λq )4 − 12kt γ (1 + λa )(1 + λq )2 φ + 24k2t 2 (1 + λa )2 φ 2.
6kγ (1 + λa )(1 + λq )2 φ

If γ 2 (1 + λq )4 − 12kt γ (1 + λa )(1 + λq )2 φ + 24k2t 2 (1 + λa )2 φ 2 > 0, then △CS > 0.
We focus on this numerator. Note that this numerator is a quadratic function of φ . This function isconvex because the second derivative with respect to φ is positive. The discriminant is positive. Solving
this equation with respect to φ , we get the lower root φ ∗ as follows:√
12kt γ (1 + λa )(1 + λq )2 − 3kt γ (1 + λa )(1 + λq )2∗.
φ :=48k2t 2 (1 + λa )2
24

³
√ ´
Note that, we have imposed the upper bound of φ at φ̄ non = 3 − 5 γ (1 + λq )2 /12kt(1 + λa ). We
estimate the difference between φ̄ non and φ ∗ . The difference is given as³√

φ̄ non − φ ∗ =√ ´
3 − 5 γ (1 + λq )212kt(1 + λa )< 0.

As a result, in the imposed region of φ , the numerator is a positive. We simply illustrate the idea of theproof (Fig. 5).

This range occurrs
in this region.

φ∗

φ̄non
φ

non

Figure 5: The image of this proof

Proof of Proposition 2.5△W is given as1

×

18k2 γ (1 + λa )2 (1 + λq )2 φ¡
(1 + λa + λq ) γ 2 (1 + λq )4 (3k(1 + λa ) − γ (1 + λa + λq )) − 18kt γ (1 + λa )(1 + λq )2 (2k(1 + λa )¢− γ (1 + λa + λq ))φ + 36k2t 2 (1 + λa )2 (2k(1 + λa ) − γ (1 + λa + λq ))φ 2 .

25

We focus on the numerator of this fraction. If the numerator is positive, then △W is also positive. Thus,
social welfare with self-regulation is lower than that without self-regulation. Let A(φ ) be this numerator¡A(φ ) := (1 + λa + λq ) γ 2 (1 + λq )4 (3k(1 + λa ) − γ (1 + λa + λq )) − 18kt γ (1 + λa )(1 + λq )2 (2k(1 + λa )¢− γ (1 + λa + λq ))φ + 36k2t 2 (1 + λa )2 (2k(1 + λa ) − γ (1 + λa + λq ))φ 2
Note that A(φ ) is a quadratic function with respect to φ . The coefficient of φ 2 is positive if and only if
2k(1 + λa )/(1 + λa + λq ) > γ . The extreme value of A(φ ) is φ ′ = γ (1 + λq )2 /4kt(1 + λa ). φ ′ is larger thanφ̄ non . Therefore, we have the following lemma:
Lemma A.1.

1. If 2k(1+ λa )/(1+ λa + λq ) > γ , then A(φ ) is minimized when φ ′ = γ (1+ λq )2 /4kt(1+λa ). Thus, this function is monotonically decreasing in φ ∈ [φ non , φ̄ non ].
2. If 2k(1 + λa )/(1 + λa + λq ) < γ , then A(φ ) is maximized when φ ′ = γ (1 + λq )2 /4kt(1 + λa ). Thus,
this function is monotonically increasing in φ ∈ [φ non , φ̄ non ].
We find that A(φ ) is monotonically decreasing or increasing in the imposed region of φ .
We prove in turn the case of (1) in proposition 2.5. We begin to consider the case in which A(φ ) is aconvex function. Thus, 2k(1 + λa )/(1 + λa + λq ) > γ . Substituting φ̄ non for A(φ ), we get(1 + λa + λq )kγ 2 (1 + λa )(1 + λq )4 .
Clearly, the equation is a positive. Note that φ ′ is larger than φ̄ non . By the lemma A.1 if 2k(1 +λa )/(1 + λa + λq ) > γ , then △W is positive.
Next, we consider the case in which A(φ ) is a concave function. Hence, substituting φ non for A(φ ),we have1− γ 2 (1 + λq )4 (−11k + γ − 11kλa + γλa + γλq ).
9
If γ < 11k(1 + λa )/(1 + λa + λq ), then the above equation is positive. Therefore, if 2k(1 + λa )/(1 +λa + λq ) < γ < 11k(1 + λa )/(1 + λa + λq ), then △W is positive. From those facts, if 11k(1 + λa )/(1 +λa + λq ) > γ , then △W is positive. Therefore, in the region of γ , social welfare without self-regulation ishigher than that with self-regulation.
26

We prove the case of (2) in Proposition 2.5. If 11k(1 + λa )/(1 + λa + λq ) < γ , then △W is negative at

φ non . On the other hand, when 11k(1 + λa )/(1 + λa + λq ) < γ , △W is positive at φ̄ non and monotonicallyincreasing in φ by lemma A.1. Thus, at φ such that △W = 0, the sign of △W changes from negative
to positive. Thus, in φ non < φ < φ ∗ , social welfare with self-regulation is higher than that without selfregulation. Otherwise, social welfare with self-regulation is lower than that without self-regulation. We
illustrate the sketch of proof with simple figures (Figs. 6, 7, and 8).
A

A

φ′

φ φ̄ φ′φ φ̄

Figure 6: 2k(1 + λa )/(1 + λa + λq ) > γ : sketchof (1)

Figure 7: 2k(1 + λa )/(1 + λa + λq ) < γ : sketchof (1)

A

φ∗

φ
φ′

φ̄

Figure 8: 2k(1 + λa )/(1 + λa + λq ) > γ : sketch of (2)Proof of Proposition 2.6
We begin to prove (1). Hence, we show that a∗∗ − a∗ is positive under the condition.

27

a∗∗ − a∗ is given as,t(k(1 + λa ) − γ (1 + λa + λq )) 4t 2 (k(1 + λa ) − γ (1 + λa + λq ))φ−.
kγ (1 + λa )γ 2 (1 + λq )2This equation is a linear function of φ and a downward slope under the condition. For any φ , this equationis positive. We can similarly prove (2).

Proof of Proposition 2.7
We show that CS∗∗ −CSr∗ is a positive. CS∗∗ −CSr∗ is given as¡¢2
γ (1 + λq )2 − 4kt(1 + λa )φ> 0.
4γ 2 (1 + λq )2 φ

Proof of Proposition 2.8
We show that π ∗∗ − πr∗ is a negative. The difference between π ∗∗ and πr∗ is¡¢2
γ (1 + λq )2 − 4kt(1 + λa )φ<0−
8γ 2 (1 + λq )2 φ

Proof of Proposition 2.9
The difference between CS∗ and CS∗∗ is given as follows.
CS∗ −CS∗∗ =1
12kγ 2 (1 + λa )(1 + λq )2 φ©
× −γ 2 (1 + λq )4 (3k − 2γ + 3kλa − 2γλa − 2γλq )+24kt γ (1 + λa )(1 + λq )2 (k − γ + kλa − γλa − γλq )φª−48k2t 2 (1 + λa )2 (k − γ + kλa − γλa − γλq )φ 2 .

The sign of this equation is equivalent to that of the numerator. Hence, we focus on the numerator. Notethat the numerator is a quadratic function with respect to φ . Let F(φ ) be the numerator. The following
lemma represents the property of F(φ ).
Lemma A.2. F(φ ) is monotonically increasing or decreasing in the region of φ < (γ (1 + λq )2 )/(4kt(1 +λa )).
28

Proof. Solving F ′ (φ ) = 0 with respect to φ , the extreme point φ ∗ of F(φ ) is given as:φ∗ =γ (1 + λq )2.
4kt(1 + λa )φ∗
We begin to prove (1) of the proposition. By the lemma, we have the fact that the numerator isa monotone function in the region of φ ∈ (0, (γ (1 + λq )2 )/(4kt(1 + λa )). Note that φ is in the region of[φ non , φ̄ non ] in this section, and φ̄ non < φ ∗ . Therefore, F(φ ) is a monotone function for any φ in[φ non , φ̄ non ].
Substituting φ non into F(φ ), we getF(φ non ) =1 2
γ (1 + λq )4 (22γ (1 + λa + λq ) − 49k(1 + λa )).
27

If k < (22γ (1 + λa + λq ))/(49(1 + λa )), then the equation is positive. Substituting φ̄ non into F(φ ), wehave
1F(φ̄ non ) = γ 2 (1 + λq )4 (2γ (1 + λa + λq ) − 5k(1 + λa )).
3
If k < (2γ (1 + λa + λq ))/(5(1 + λa )), then the equation is positive.
F(φ non ) and F(φ̄ non ) are positive if and only if k < (2γ (1 + λa + λq ))/(5(1 + λa )). Because F(φ ) is amonotone function, CS∗ > CS∗∗ if k < (2γ (1 + λa + λq ))/(5(1 + λa )).
We prove, in turn, (2) of the proposition. Next, in (2γ (1 + λa + λq ))/(5(1 + λa )) < k < (22γ (1 + λa +λq ))/(49(1 + λa )), F(φ ) is positive in the region of (φ non , φ ⋆⋆ ) but is negative in the region of (φ ⋆⋆ , φ̄ non ),where φ ⋆⋆ is φ such that CS∗ = CS∗∗ . Thus, if k is in ((2γ (1 + λa + λq ))/(5(1 + λa )), (22γ (1 + λa +λq ))/(49(1 + λa ))), then CS∗ > CS∗∗ in the region of(φ non , φ ⋆ ) and CS∗ < CS∗∗ in the region of (φ ⋆ , φ̄ non ).
We lastly prove (3) of the proposition. In the case of k > (22γ (1 + λa + λq ))/(49(1 + λa )), F(φ ) is
negative in the assumed region of φ . Therefore, CS∗ < CS∗∗ .

B

Proofs in section 3

Proof of Proposition 3.1
We begin to prove the case of (1).
29

△CS is given as
△CS =

1
³
´
2
4t (γ − 4γλ 2 + 16kt λ φ ) (−γλ − 10γλ 2 − 24γλ 3 − 16γλ 4 + 3kt φ + 28kt λ φ + 80kt λ 2 φ + 64kt λ 3 φ )2× ( (t + 2vλ )2
¡
¡
¢
¡
¢
¢
× γ 2 (1 + 2λ )2 1 + 8λ + 8λ 2 − 4kt γ (1 + 2λ ) 3 + 17λ + 16λ 2 φ + 8k2t 2 (1 + 4λ )(3 + 4λ )φ 2|{z}=Ā(φ )¢¡× ( − γ λ (1 + 2λ ) 1 + 8λ + 8λ 2
|
{z
}
2

2

=b

¢
− 2kt γ (1 + 2λ ) −3 − 22λ − 38λ 2 + 32λ 3 + 64λ 4 φ + 8k2t 2 λ (1 + 4λ )(3 + 4λ )(3 + 8λ )φ 2 )) ) .
|
{z
}
¡

=b′

Let b − b′ be B(φ ). Note that both Ā(φ ) and B(φ ) are quadratic functions with respect to φ . We
can characterize △CS by the conditions of the signs of both Ā(φ ) and B(φ ), since the denominator and(t + 2vλ )2 are positive. Hence, we evaluate the conditions of the signs of both Ā(φ ) and B(φ ).
Lemma B.1. The function Ā(φ ) is monotonically decreasing in φ ∈ [φ , φ̄ ].
Proof. When we differentiate Ā(φ ) with respect to φ , we get¢¡−4kt γ (1 + 2λ ) 3 + 17λ + 16λ 2 + 16k2t 2 (1 + 4λ )(3 + 4λ )φ .
Let the differential coefficient of the first order be Ā′ (φ ). Moreover, the differential coefficient of thesecond order of Ā(φ ) is
16k2t 2 (1 + 4λ )(3 + 4λ ).
′′Thus, the second-order condition is a positive. Let the differential coefficient of the second order be Ā (φ ).
Solving Ā′ (φ ) = 0 with respect to φ ,¡¢γ (1 + 2λ ) 3 + 17λ + 16λ 2
φ =.
4kt (3 + 16λ + 16λ 2 )′′′Note that φ ′ ∈ argmin Ā(φ ) since Ā (φ ) > 0. Calculating φ̄ − φ ′ , we get−

3γλ (1 + 2λ ).
4kt(1 + 4λ )(3 + 4λ )Thus, φ̄ < φ ′ . Moreover, since Ā(φ ) is convex, Ā(φ ) is monotonically decreasing in the region of φ .
30

Lemma B.2. The function B(φ ) is monotonically increasing in φ ∈ [φ , φ̄ ].
Proof. When we differentiate B(φ ) with respect to φ , we get¢¡−2kt γ (1 + 2λ ) −3 − 22λ − 38λ 2 + 32λ 3 + 64λ 4 + 16k2t 2 λ (1 + 4λ )(3 + 4λ )(3 + 8λ )φ .
Let the differential coefficient of first order be B′ (φ ). Moreover, the differential coefficient of second orderof B(φ ) is
16k2t 2 λ (1 + 4λ )(3 + 4λ )(3 + 8λ ).
′′Thus, second order condition is positive. Let the differential coefficient of second order be B (φ ). Solving
B′ (φ ) = 0 with respect to φ ,¡¢γ (1 + 2λ ) −3 − 22λ − 38λ 2 + 32λ 3 + 64λ 4
φ =.
8kt λ (3 + 8λ ) (3 + 16λ + 16λ 2 )
′′

′′

′′

Note that φ ′ ∈ argmin B(φ ) since B (φ ) > 0. Calculating φ − φ , we get¡¢
γ (1 + 2λ ) 9 + 126λ + 738λ 2 + 2336λ 3 + 4000λ 4 + 3328λ 5 + 1024λ 6.
8kt λ (1 + 4λ )2 (3 + 4λ )2 (3 + 8λ )′′Thus, φ > φ . Moreover, since B(φ ) is convex, B(φ ) is monotonically increasing in the region of φ .

B(φ)
Ā(φ)

φ′

φ

φ′
φ̄

Figure 10: Sketch of lemma 4

Figure 9: Sketch of lemma 3
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Substituting φ and φ̄ for Ā(φ ), we get¡¢¡¢
γ 2 (1 + 2λ )3 1 + 8λ + 8λ 2 11 + 116λ + 472λ 2 + 1024λ 3 + 1152λ 4 + 512λ 5
A(φ ) =,
(1 + 4λ )3 (3 + 4λ )3¡¢γ 2 (1 + 2λ )2 3 + 22λ + 49λ 2 + 32λ 3A(φ̄ ) = −.
2(1 + 4λ )(3 + 4λ )Since a discriminant of A(φ ) is a positive, sign(Ā(φ )) ̸= sign(Ā(φ̄ )), and convex, there exists φ ∗ ∈ [φ , φ̄ ]such that Ā(φ ) = 0.6 Let φ ∗ be φ ∗ .
Also, substituting φ and φ̄ for B(φ ), we get
B(φ ) =
B(φ̄ ) =¡¢¡¢γ 2 (1 + 2λ )3 1 + 8λ + 8λ 2 9 + 147λ + 932λ 2 + 2968λ 3 + 5120λ 4 + 4480λ 5 + 1536λ 6,(1 + 4λ )3 (3 + 4λ )3¡¢¡¢γ 2 (1 + 2λ )2 3 + 9λ + 8λ 2 3 + 22λ + 40λ 2 + 32λ 3.
2(3 + 4λ )By both B(φ ) and B(φ̄ ) are positives, and the above lemma B(φ ) is positive in the region of φ .
In the region of [φ , φ ∗ ], both Ā(φ ) and B(φ ) are positives. Thus, △CS is positive.
We prove the case of (2). By the logic of the proof for (1), in the region of [φ ∗ (Ā), φ̄ ], Ā is a negative
and B is a positive. Therefore, △CS is a negative.

Ā(φ)φ∗ (Ā)

B(φ)

φ̄
φ

Figure 11: Sketch of Proposition 3.1

6 sign(·) maps from the value to the sign. For example, sign(5) = +.

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C Comparison of programs quality of non-self-regulation with self-regulation
Proposition C.1. In three types model in section 2, q∗i > q∗r for all φ ∈ (φ non , φ̄ non ).
Proof. We evaluate φ such that q∗i = q∗r . The φ is given asφ̃ =γ (1 + λq )2.
6kt(1 + λa )If φ̃ > φ , then the difference is positive. Note that the difference between q∗i and q∗r is decreasing functionof φ . Since φ̃ is larger than φ̄ non , for any φ in the region the difference is positive.
Proposition C.2. In two types of models in section 3(i) q∗i > q∗r for all φ ∈ (φ , φ̃ ).
(ii) q∗i < q∗r for all φ ∈ (φ̃ , φ̄ ).
Proof. We evaluate φ such that q∗i = q∗r . The φ is given asφ̃ =γ (1 + 2λ )(1 + 8λ (1 + λ )).
2kt(3 + 4λ (5 + 4λ ))If φ̃ > φ , then the difference is positive. Note that the difference between q∗i and q∗r is a decreasing
function of φ . Since φ is smaller than φ̃ and φ̄ is larger than φ̃ , on φ̃ the difference changes to negativefrom positive.
Note C.3. In two types of models in section 3, φ̃ > φ ∗ . Thus, in the region of φ in Proposition 3.1 - (i),q∗r < q∗i .
Proof. We can proof this statement by using lemma B.1. Since, as with φ ∗ , A(φ ) changes to negative frompositive; if the sign of A(φ̃ ) is negative, then φ ∗ < φ̃ . A(φ̃ ) is given by−(γ + 2γλ )2 (1 + 8λ (1 + λ ))(3 + 2λ (11 + 4λ (5 + 4λ ))).
(3 + 4λ (5 + 4λ ))2Since this value is negative, we get φ ∗ < φ̃ .

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