2008-58
Should public sectors be complements of private
sectors?
石橋
松島

郁雄
法明

Should public sectors be complements of private sectors?∗
Ikuo Ishibashi†
Department of Economics, Aoyama Gakuin University
Noriaki Matsushima‡
Graduate School of Business Administration, Kobe University
December 1, 2008

Abstract
This paper discusses competition between high-quality private service providers
that maximize their own profits and a low-quality public service provider that maximizes social surplus. There are two heterogeneous consumer groups: those who demand only high-quality services and those who care little whether services are highor low-quality. The setting reflects the fact that some consumers feel dissatisfaction
with public service providers. We show that, under certain conditions, social welfare
is smaller when there is a public service provider than when there is not. The result
holds even though the efficiency of the public service is equal to that of the private
services.

JEL classification: H42, L13, I12
Key words: public versus private sectors, quality, differentiation, mixed markets.

∗
We would like to thank Hideki Konishi, Naoko Nishimura, Fumihiko Numao, Debashis Pal, Ryusuke
Shinohara, Ryuichi Tanaka, and seminar participants at Cincinnati University, Osaka University, Shinshu
University, and Tokyo Institute of Technology for their helpful comments. The authors gratefully acknowledge financial supports from Grant-in-Aid for Encouragement of Young Scientists from the Japanese
Ministry of Education, Science and Culture. Needless to say, we are responsible for any remaining errors.
†
Ikuo Ishibashi: Faculty of Economics, Aoyama Gakuin University 4-4-25 Shibuya, Shibuya-ku, Tokyo
150-8366 Japan. E-mail: ishibash@econ.aoyama.ac.jp‡
Noriaki Matsushima (Correspondence author): Graduate School of Business Administration, Kobe
University, 2-1 Rokkodai, Nada, Kobe, Hyogo 657-8501, Japan. Phone: +81-78-803-6981. Fax: +81-78803-6977. E-mail: nmatsush@kobe-u.ac.jp1

1

Introduction

In vertically differentiated mixed markets,1 it is often observed that public firms provide
lower-quality products at a low price. For example, in Japan, public nurseries provide
fewer convenient services concerning opening hours than private ones. In 2006, althoughmore than 80% of private nurseries were open more than 11 hours, fewer than 50% of the
public ones were open more than 11 hours (research on facilities for social welfare 2006;
Ministry of Health, Labor, and Welfare). Public nursery services are insufficient for someparents because of constraints concerning their working hours.2 In reality, although publicnurseries accept children less than 95% of their capacity, private ones accept children morethan 110% of their capacity even though the private nurseries are more expensive that the
public ones. These facts suggest that the services provided by the public sector are lessconvenient. Another example is the existence of public and private sport facilities (e.g.,gyms and tennis courts). Public sports facilities are frequently poorly equipped; however,they generally charge lower fees than their private counterparts.3 Health care systems areanother example in which public service providers are cheaper and of poorer quality.4 In
1

A market is called mixed when profit-maximizing private firms and a welfare-maximizing public firmcoexist. Mixed markets are widely observed in Europe, Canada, and Japan. One of the most importantquestions in studies of mixed markets is whether or not a public firm should be privatized. Because
the public firm’s aggressive behaviors often cause private firms’ undesirable reactions through market,
the answer to the question is not obvious even if public firms are properly operated without efficiency
problems such as moral hazard. See Bös (1986, 1991) and Nett (1993) for excellent surveys of mixed
oligopolies. Recently, the literature on mixed oligopoly has become richer and more diverse. For example,
Poyago-Theotoky (1998), Corneo and Rob (2003), Ma (2003), Matsumura and Matsushima (2004), andIshibashi and Matsumura (2006) investigate cost differences between public and private sectors. Pal (1998),
Matsumura (2003), and Lu (2006) discuss the endogenous role. Cremer et al. (1991), Matsushima andMatsumura (2003), Li (2006), and Lu and Poddar (2007) analyze endogenous product differentiation.
2

We do not want to suggest that the staff of public nurseries are inferior to those in private ones. The

emphasis is on the differences in the hours of service provided.
3

Glazer and Niskanen (1997) also mention that governmental facilities are often small and of poor

quality and that many services provided by such facilities are also provided privately, for example, as inpublic versus private schools and universities, public versus private medical care, and public mass transit
versus the use of privately owned automobiles.
4

Several researchers investigate the interaction between lower-quality health care service by public

sector and higher-quality health care service by private sector. The main concerns of them are summarized
as follows: quality and regulation (Barros and Martinez-Giralt (2002)), waiting list (Barros and Olivella2

many developed and developing countries, service providers in traditional public healthsystems have typically faced very weak incentives to respond to patient preferences andtheir perceptions of quality. One indication of the resulting quality deficiencies is the large
number of patients who choose to seek care in the private sector, despite the availability of
public sector services provided at no cost.5 For instance, when someone has a cold, he/she
has several choices for seeking treatment. First, the patient could go to a (private/public)
hospital; second, the patient could buy and consume cold medicine; and third, the patient
could rest and sleep. If the patient wanted more thorough treatment, he could go to a
private hospital. If the patient did not feel that more care was needed, he could go to a
public hospital, take cold medicine, or sleep more.
One rationale for the existence of public firms is that they are socially desirable, eventhough they are unprofitable. For example, when only a few consumers show concern for
quality, no private firm would offer lower-quality services for less.6
Although public firms seem to be socially desirable, we demonstrate in this paper that,in some situations, the establishment of public firms diminishes social welfare.
In more precise terms, we consider the following situation based on the work of Ishibashi
and Matsushima (2008). There are two heterogeneous consumer groups. One consistsof consumers who demand high-quality products. Low-quality products are worth little
to them (we call them quality conscious).7 The other consists of consumers who care(2005), March and Schroyen (2005)), and physician dual practice (Rickman and McGuire (1999), Gonzàlez(2004), Biglaiser and Ma (2007), Brekke and Sørgard. (2007)). Garcı́a-Prado and González (2007) provide
an excellent survey in the topic of physician dual practice. Those papers do not consider mixed oligopoly
models with those heterogeneous consumer groups discussed here.
5

The phenomenon is called as bypassing health provider in the literature of health economics. After

Akin and Hutchinson (1999) show it as an important phenomenon in Sri Lanka, several researchers investigate this kind of problems in some developing countries (see Leonard et al. (2002) and Gauthier andWane (2008)).
6

If there is an entry barrier, it is also possible that no private firm wants to produce lower-quality

products even if it can gain a positive profit. For example, when competition in the large high-end market
is more profitable than monopoly in the small low-end market, no private firms may enter the small low-end
market.
7

Alternatively, it is assumed that consumers in this group have sufficiently larger willingness to pay

for high-quality products than that for low-quality products.

3

little whether products are high- or low-quality. Therefore, they buy products with thelowest price (we call them price conscious). High-quality (resp. low-quality) products areproduced by private firms (resp. a public firm if it is established). In this setting, we showthat the establishment of a public firm worsens social welfare if the size of the former(quality conscious) consumer group is medium.
Ironically, the driving force of this result is the public firm’s aggressive behavior in
the lower-quality market. The welfare-maximizing public firm behaves aggressively inthe lower-quality market and the resulting price of lower-quality products falls substantially. This causes private firms to focus on quality-conscious consumers because no
profit-maximizing private firm has an incentive to sell its higher-quality products to priceconscious consumers in this situation. As a result, the price of higher-quality products
is kept relatively high, and the welfare loss for quality-conscious consumers who give up
buying higher-quality products becomes large. If the relative size of price-conscious consumers is small in comparison to that that of quality-conscious consumers, this negative
effect outweighs the positive effect of the low price set by the public firm for price-conscious
consumers.
We also note that, unlike many existing studies with standard results, we investigate
not only whether or not privatization is preferable from the viewpoint of social welfare but
also whether or not the nonexistence (or exit) of a public firm is preferable. We believethat these comparisons are important to learn that the establishment of public firms canbe harmful to social welfare.
In this paper, a theoretical contribution is made to studies of mixed markets. The technological assumptions required for our result (the nonexistence (or exit) of a public firm ispreferable) are different from those in studies of mixed markets, such as quadratic production cost (De Fraja and Delbono (1989)), variety expansion with free entry (Anderson etal. (1997)), excess R&D investments (Matsumura and Matsushima (2004)), and productpositioning (Matsushima and Matsumura (2006)). Since our paper does not include those
properties, we provide a new theoretical insight on the topic of mixed markets.

4The remainder of this paper is organized as follows. Section 2 presents the basic model.
Section 3 presents the main result. Section 4 concludes the paper.

2

Model

We consider an industry with two differentiated products (h and l). For convenience, wecall h and l high-quality and low-quality products, respectively. There are n private firmsthat produce h at a constant marginal cost normalized to zero and a public firm (firm0) that produces l at a constant marginal cost normalized to zero.8 Each private firm
maximizes its own profit, and the public firm maximizes social welfare.9 No fixed costis assumed for production. Let qi be firm i’s output level (i = 0, 1, . . . , n). In addition,
define q ≡ (q1 , . . . , qn ).
We assume that there are two groups of consumers, H (the high-end market) and L(the low-end market). For simplicity, we consider a polar case of the heterogeneity of
consumer groups. The consumers in H demand only h; that is, the quality of l is not at
all sufficient for the consumers in H (see Figure 1).
[Figure 1 here]
Let ph be the price of h. The demand function of this high-end market, DH (ph ), is given
by10

½
H

h

D (p ) =80
if ph ∈ (1, ∞),h
1 − p if ph ∈ [0, 1].

Even if there are private firms that produce l, our main results still hold. As we will see later, apublic firm with a constant marginal cost (normalized to zero) supplies l so that the price of l becomeszero whether or not there are private firms that produce l.
9In this paper, the government is not permitted to nationalize more than one firm. As pointed out by

Merrill and Schneider (1966), the most efficient outcome is achieved by the nationalization of all firms, if
nationalization does not change the costs of firms (i.e., no X-inefficiency in the public firm exists). The
need for the analysis of a mixed oligopoly lies in the fact that it is impossible or undesirable, for politicalor economic reasons, to nationalize an entire sector. For example, without competitors, public firms may
lose the incentive to improve their costs, resulting in a loss of social welfare. Thus, we do not consider the
possibility of nationalizing all firms.
10This demand function is derived by assuming that a typical consumer in H has the willingness to pay

x for product h, x is distributed uniformly on [0, 1], and the total population is 1.

5

The consumers in L are indifferent between h and l. In other words, the high quality of h(compared with l) is of no value to consumers in L. Let pl be the price of l. The demand
function of this low-end market, DL (pl ), is given by11½
Ll

D (p ) =0
if pl ∈ (a, ∞),lb(1 − p /a) if pl ∈ [0, a].

We assume 0 < a ≤ 1. Note that DL (pl ) is a linear demand function such that the highest
willingness to pay is given by a and the largest demand (at pl = 0) is given by b. Thus,(a, b) measures the relative market properties of the low-end market taking the high-end
market as a reference point.12
We describe how ph and pl are determined given the above consumers. As long asP
1 − ni=1 qi ≥ a(1 − q0 /b), the high-end and low-end markets are separated. In otherPwords, no consumer in L buys h. Therefore, ph is given by 1 − ni=1 qi , and pl is givenPby a(1 − q0 /b). If 1 − ni=1 qi < a(1 − q0 /b), the markets are connected. That is, someconsumers in L buy h. Because h and l are completely indifferent to consumers in L,P
ph = pl = a(1 + b − ( ni=1 qi + q0 ))/(a + b).
In summary, the prices are determined as follows.

n
n
X
X

aq0


1
−
q
if
qi ≤ 1 − a +
,
i

b
h
i=1
i=1
p (q, q0 ) =P
a(1 + b − ( ni=1 qi + q0 ))otherwise.
a+b

n
³
X

q0 ´
aq0


a
1
−
,
if
qi ≤ 1 − a +

b
b
l
i=1
p (q, q0 ) = a(1 + b − (Pn q + q ))

0

i=1 i

otherwise.
a+b
11

(1)

(2)

This demand function is derived by assuming that a typical consumer in L has the same willingness

to pay x for product h or for product l, x is distributed uniformly on [0, a], and the total population is b.
12

Although a ≤ 1 is assumed for simplicity, it seems reasonable to assume that quality-conscious

consumers tend to evaluate high-quality products at least as high as quality-unconscious consumers. As
for b, no upper bound is assumed. Therefore, our analysis can cover various demand structures on therelationship between the high-end and the low-end markets.

6Let πi (q, q0 ) is firm i’s profit function. For i = 1, 2, . . . , n, it can be expressed as follows.

n
n
X
X

aq0


(1
−
q
)q
if
qi ≤ 1 − a +
,
i i

b
i=1
i=1
πi (q, q0 ) =
(3)
P


a(1 + b − ( ni=1 qi + q0 ))
qi otherwise.
a+b
Social welfare, which is also the objective function of the public firm, can be expressed as:
 Z PnZ q0 ³i=1 qi

x´


dx
(1
−
x)dx
+
a
1
−


b
 0
0



Pn

aq0

if

i=1 qi ≤ 1 − a + b ,SW (q, q0 ) = Z(4)
Z Pn qi +q0

1−a

i=1
a(1 + b − x)dx
(1 − x)dx +


a+b

0
1−a




otherwise.

3

ResultIn the following subsections, we consider two polar cases: (1) a public firm produces
product l; and (2) no public firm exists. Comparing the two cases, we derive the main
result of this study.

3.1

Case I: A public firm in the low-end market

In this subsection, we assume that there is a public firm that maximizes social welfare in
the low-end market.
To show that no private firm wants to sell its product to the consumers in L, we firstconsider the best response of the public firm. From (4), we can easily derive the best
response of the public firm as follows:

n
X



b
if
qi ≤ 1,i=1
q0 (q) =nXqi otherwise.
1 + b −


(5)

i=1

From (3), we can also derive the best response of each private firm as follows:PnX
1 − j6=i qj

aq0


,
if
qi ≤ 1 − a +
,

2
b
i=1
qi (q0 , q−i ) =P
1 + b − j6=i qj − q0
otherwise.
2
7

(6)

From the equations, we have the following unique Nash equilibrium:(q0∗ , q ∗ ) = (b, q x ) , where q x ≡ (1/(n + 1), . . . , 1/(n + 1)).
Each private firm’s equilibrium profit is 1/(n + 1)2 . When the public firm exists, thedemand for each private firm is similar to that in which the consumers in L do not exist
because the public firm fully covers the demand of the consumers in L to maximize social
welfare. As a result, there is no room for the private firms to supply for the consumers in
L. We summarize this result as follows.
Lemma 1 If there is a public firm, the unique Cournot-Nash equilibrium is q0 = b, q x =(1/(n + 1), . . . , 1/(n + 1)). The profit of each firm and social welfare are:
πI ≡

3.2

1
ab + (1 + ab)n(n + 2)
, SWI ≡.
(n + 1)2
2(n + 1)2

Case II: No public firm in the low-end market

In this subsection, we consider a case in which the public firm exits from the low-end
market and n private firms can potentially sell to both groups of consumers. This casebecomes a simple Cournot oligopoly game with a kinked demand curve.
There are two candidates of Cournot equilibria:
q x and q y ≡ ((1 + b)/(n + 1), . . . , (1 + b)/(n + 1)).
This is caused by our kinked inverse demand function. q x (resp. q y ) is the CournotPequilibrium when the whole inverse demand function is given by px (q) = 1 − i=1 qi
P(resp. py (q) = a(1 + b − i=1 qi )/(a + b)) (see (1)). Therefore, we must check not only the
local optimality (i.e., first-order condition) but also the global optimality (i.e., deviations“beyond” the kinked point). After several calculations, we obtain the following lemmas(the calculations are described in the Appendix):
Lemma 2 q y = ((1 + b)/(n + 1), . . . , (1 + b)/(n + 1)) becomes a Cournot equilibrium ifa≥(2 − (n − 1)b)2
2
or b ≥.
(n + 1)(4 − (n − 3)b)n−1
8The profit of each firm and social welfare are:πIIy ≡a(1 + b)2
(1 − a)2 b + (a + b)(1 + ab)n(n + 2)
,
SW
≡
.
IIy
(n + 1)2 (a + b)2(a + b)(n + 1)2

Lemma 3 q x = (1/(n + 1), . . . , 1/(n + 1)) becomes a Cournot equilibrium ifa≤4.
(n + 1)(4 + (n + 1)b)

The profit of each firm and social welfare are:πIIx ≡

1
n2
,
SW
≡
.
IIx
(n + 1)2
2(n + 1)2

Lemma 2 says that q y becomes an equilibrium if the low-end market is sufficiently
profitable (i.e., a is relatively large for a given b) or large (i.e., b is higher than 2/(n − 1)).
The first condition is relatively easy to understand. Firms sell to the consumers in L
because the increase in sales outweighs the decrease in price. What the second condition
implies is slightly more complicated. Although a firm must decrease its quantity drasticallyP
to satisfy i=1 qi ≤ 1 − a (i.e., raising the market price by selling only to the consumers inPP
H, see (1)), it is impossible to satisfy i=1 qi ≤ 1−a given j6=i qj = (n−1)(1+b)/(n+1) ≥1 under b ≥ 2/(n − 1). Therefore, no firm has an incentive to raise the market price even
if the equilibrium market price is very low.
In general, the condition in Lemma 3 says the inverse. That is, firms sell only tohigh-end consumers if the low-end market is sufficiently unprofitable and/or inelastic.
It is noteworthy that this game has both equilibria under a certain range of (a, b)
because it is possible to satisfy both a ≥ (2 − (n − 1)b)2 /((n + 1)(4 − (n − 3)b)) anda ≤ 4/((n + 1)(4 + (n + 1)b)).

3.3

Comparison

Using the results obtained so far, we determine the condition under which social welfare
in case I is smaller than that in case IIy.
First of all, q y must be a Cournot equilibrium in the absence of the public firm.13
From Lemma 2, q y becomes a Cournot equilibrium if one of the following inequalities is
13

Otherwise, the supply of h does not depend on whether or not the public firm exists in the low-end

market. Therefore, the existence of the public firm is always preferable for social welfare.

9satisfied:a≥(2 − (n − 1)b)2
2
or b ≥.
(n + 1)(4 − (n − 3)b)n−1

Second, we need SWI < SWIIy . That is, case I is smaller than case IIy. Thiscondition can be rewritten as follows:
a<

1
.
2+b

Therefore, we obtain the following result.
Proposition 1 The nonexistence of the public firm can be beneficial from the viewpointof social welfare if

1, where aB (n) ≡aB (n) < a <
2+b

(2 − (n − 1)b)2(n + 1)(4 − (n − 3)b)
0

2
,
n−1
2
if b ≥.
n−1
if b <

After some calculations, we obtain that both 2/(n − 1) and (2 − (n − 1)b)2 /(n + 1)(4 −(n − 3)b) decrease with n.14 This implies that, as the number of private firms increases,
the interval [aB (n), 1/(2 + b)] expands. Therefore, the existence of the public firm is more
likely to worsen social welfare when there are many private firms in the high-end market.
Figure 2 shows the region in which this proposition holds (n = 2). It is noteworthy
that a < 1/(2 + b) does not hold for any b > 0 as long as a > 1/2.
[Figure 2 here]
Note that, given that there are multiple equilibria (the value of a is on the intermediatelevel mentioned above) in the absence of the public firm, the entry of the public does notalways harm the social welfare.
The rough intuition of Proposition 1 is as follows. As case I shows, the existence of apublic firm makes private firms focus on the high-end market. This change brings about
two effects: (i) the supply of the l product becomes b while (ii) the total supply of the h14

The partial differentials of (2 − (n − 1)b)2 /(n + 1)(4 − (n − 3)b) and 2/(n − 1) with respect to n arerespectively −8(1 + b)2 (2 − (n − 1)b)/(n + 1)2 (4 − (n − 3)b)2 and −2/(n − 1)2 . Both of them are negative.

10

product becomes n/(n + 1) from q y . Clearly, (i) is positive, and (ii) is negative from theviewpoint of social welfare.
However, if q y becomes an equilibrium in the absence of the public firm, we musttake the negative effect of (ii) into account. There are some consumers in H who wouldpurchase the h products if the public firm did not exist whether or not the welfare loss
from such consumers in H exceeds the welfare gain from consumers in L, who can obtainthe l product due to the entry of the public firm. The losses are imposed on a relatively
small number of consumers in H, and the gains are imposed on a relatively large numberof consumers in L. However, py (q y ) < px (q x ) implies that the total willingness to pay of
those consumers in H is higher than that of the consumers in L. Therefore, the total effectdepends on the demand structure that affects the impacts of (i) and (ii). The conditionsin Proposition 1 are those under which (ii) outweighs (i).

3.4

Privatization

In the previous subsection, we compare the case in which a public firm exists with that inwhich no public firm exists.
In this subsection, we consider a case in which the public firm is privatized and nprivate firms can potentially sell to both groups of consumers.
In this case, there are two candidates of Cournot equilibria:q z ≡ (q0z , q1z , . . . , qnz ), q w ≡ (q0w , q1w , . . . , qnw ),11+bb(i = 1, . . . , n), qjw ≡(j = 0, 1, . . . , n).
where q0z ≡ , qiz ≡
2
n+1
n+2
This is caused by our kinked inverse demand function. q z (resp. q w ) is the Cournot
equilibrium when ph > pl (resp. ph = pl ) (see (1)). Therefore, we must check not only the
local optimality (i.e., first-order condition) but also the global optimality (i.e., deviations“beyond” the kinked point). After several calculations, we obtain the following lemmas:
Lemma 4 q w = ((1 + b)/(n + 2), . . . , (1 + b)/(n + 2)) becomes a Cournot equilibrium ifb(3 − (n − 1)b)2
3< a, or b ≥.
(5 − (n − 3)b)((n + 1)b − 1)n−111The profit of each firm and social welfare are:πIIIw ≡SWIIIw ≡a(1 + b)2,
(n + 2)2 (a + b)
3a + 4b − 2ab + 4a2 b + 3ab2 + (a + b)(1 + ab)n(n + 4).
2(a + b)(n + 2)2

Lemma 5 q z = (b/2, 1/(n + 1), . . . , 1/(n + 1)) becomes a Cournot equilibrium ifa≤16.
(n + 1)(8 + (n + 1)b)

The profit of each private firm and social welfare are:πIIIz ≡

3ab + n(n + 2)(4 + 3ab)
1
, SWIIIz ≡.
(n + 1)28(n + 1)2

Lemma 4 says that q w becomes an equilibrium if the low-end market is sufficiently
profitable (i.e., a is relatively large for a given b) or large (i.e., b is higher than 3/(n − 1)).
The condition in Lemma 5 says the inverse. That is, private firms sell only to high-end
consumers if the low-end market is sufficiently unprofitable and/or inelastic. The valuesof a are such that both conditions in Lemmas 4 and 5 are satisfied. On the other hand, if
b is small, neither q w nor q z can become an equilibrium.
If q w appears in equilibrium, privatization is better than the exit of the public firmfrom the viewpoint of social welfare because the total quantity supplied when the public
firm is privatized is larger than that when the public firm exits. On the other hand, if q z
appears in equilibrium, the existence of the public firm is better than privatization fromthe viewpoint of social welfare because privatization just decreases the quantity suppliedby the former public firm.
From Lemma 4, we have the following proposition.
Proposition 2 Social welfare under privatization can be larger than that in which the
public firm exists if(n + 2)2 b,((n + 1)b − 1)((n + 1)b + 2n + 3)b(3 − (n − 1)b)2(5 − (n − 3)b)((n + 1)b − 1)
where aC (n) ≡ 0aC (n) < a <

12

3
,
n−1
3
if b ≥.
n−1
if b <

Figures 3a and 3b show the region in which this proposition holds (n = 2 and n issufficiently large). It is noteworthy that (n + 2)2 b/(((n + 1)b − 1)((n + 1)b + 2n + 3)) (line(vi)) converges to 1/(2 + b) (line (iii)) as the value of n increases.
[Figures 3a and 3b here]
When n is small, the exit of the public firm tends to be superior to privatization. The
privatized low-end firm lowers the price for low-end consumers, pl , and then the lower
price discourages the incentives of high-end firms to supply their products aggressively.
Those high-end firms supply their products only for high-end consumers. The price forhigh-end consumers, ph , becomes higher. When n is large, however, the discouragement
is not so significant because the larger number of high-end firms in itself enhances thetotal quantities supplied by those firms. The existence of the privatized firm additionally
increases the total quantities supplied by those firms.

4

Concluding remarks

This paper discusses competition between high-quality private service providers that maximize their own profits and a low-quality public service provider that maximizes social
surplus. There are two heterogeneous consumer groups: those who demand only highquality services and those who care little whether services are high- or low-quality. We
show that, under certain conditions, the social welfare is smaller when there is a public
service provider than when there is not. The result holds even though the efficiency of the
public service is equal to that of the private service.
In this paper, following the literature of mixed oligopoly, we assume that the public
firm tries to maximize social welfare and its production cost is as low as those of private
firms. In reality, however, public firms might have other objectives, and they might sufferfrom an efficiency problem, such as moral hazard. Even in such situations, most of ourarguments hold as long as a public firm tends to behave more aggressively in a market
than private firms. In this sense, we can say that our arguments are robust with respect
to those factors.
13

In this paper, the quality of services provided by public firms is exogenously given.
Investigating endogenous determinations of service qualities is an important matter. Thisis a relevant subject for future research.
[2008.12.1, 903]

14

Appendix
n private firms with no public firm

Before proofs of Lemmas 2 and 3, we firstdescribe the setting of the model with n private firms. We consider a case in which a
public firm does not exist and the n private firms can potentially sell to both groups ofconsumers. This is useful for proofs of Lemma 2 and 3.
We describe how the price ph is determined given the two groups of consumers. As
Plong as 1 − ni=1 qi ≥ a, no consumer in L buys h. Therefore, ph is given byph = 1 −If 1 −

n
X

qi .

i=1

Pn

i=1 qi < a, some consumers in L buy h. Because h and l are completely indifferent

to consumers in L, this means thatPa(1 + b − ni=1 qi ).
p =p =a+bhl

In summary, ph is determined as follows.

n
n
X
X



qi
if
qi ≤ 1 − a,
1 −
i=1
i=1
ph (q) =Pn a(1 + b − i=1 qi ) otherwise.
a+b
Let πi (q) be the profit function of firm i. For i = 1, 2, . . . , n, this can be expressed asfollows.

!
Ã
n
X



qi qi
 1−

if

n
X

qi ≤ 1 − a,πi (q) =

i=1
i=1
(7)
Pn


a(1
+
b
−
q
)q

i=1 i i
otherwise.
a+b
P
We solve the local optimal solutions in the two cases: (A) ni=1 qi is small, and (B) it

is large. The first-order conditions lead to(A) qiA (n) =(B) qiB (n) =

1
,
n+1

n
X
i=1

qiA (n) =n,n+1

π A (n) =1,(n + 1)2

if a ≤

1
, (8)
n+1

n
1+b X Bn(1 + b)a(1 + b)2,
qi (n) =, π B (n) =
,
n+1
n+1
(a + b)(n + 1)2i=1¾½
1 − nb, 0 . (9)if a ≥ max
n+1

15

From (4), in each case, social welfare is described as:
 Z Pni=1 qi

n2


(1
−
x)dx
=
,
case (A), 02(n + 1)2

 Z 1−aZ Pn qii=1a(1 + b − x)
SW (n) =
(1
−
x)dx
+
dx


a+b
0
1−a





(1 − a)2 b + (a + b)(1 + ab)n(n + 2)
, case (B).
=
2(a + b)(n + 1)2
Proof of Lemma 2(10)

We have to check whether the local optimal solution in (9) is also

globally optimal.
In case (B), given the quantities supplied by the other firms, if a firm sets a smaller
P
B
B
quantity qdB (n), which satisfies n−1
i=1 qi (n) + qd (n) ≤ 1 − a, the profit function is givenbyπdB (n) =µ¶(n − 1)(1 + b)
B1−
− qd (n) qdB (n).
n+1

(11)

When b ≥ 2/(n − 1), this is non-positive for any qdB (n) ≥ 0, and then the interior solutionin (9) is better for the deviating firm. When b < 2/(n − 1), the first-order condition leadsto
2 − (n − 1)b,
qdB (n) =2(n + 1)n−1XqiB (n) + qdB (n) =i=1

2n + (n − 1)b,2(n + 1)(2 − (n − 1)b)2
2 − (n − 1)b, (12), if a ≤2
4(n + 1)2(n + 1)P
B
otherwise, qdB (n) is the corner solution, qdB (n) = 1 − a − n−1i=1 qi (n), and the interior
πdB (n) =solution in (9) is better for the deviating firm. Moreover, if πdB (n) in (12) is smaller thanor equal to π B in (9), the case (B) is an equilibrium outcome. The condition under which
the solution in (9) is an equilibrium outcome is(2 − (n − 1)b)2a(1 + b)2
2≤
or b ≥.
4(n + 1)2
(a + b)(n + 1)2
n−1
The condition can be summarized as follows:(2 − (n − 1)b)2(n + 1)(4 − (n − 3)b)a ≥ aB (n) = 0Therefore, Lemma 2 holds.

2
,
n−1
2
if b ≥.
n−1

if b <(13)

Q.E.D.
16

Proof of Lemma 3

We have to check whether the local optimal solution in (8) is also

globally optimal.
In case (A), given the quantities supplied by the other firms, if a deviating firm sets aPAA
larger quantity, qdA (n), which satisfies n−1
i=1 qi (n) + qd (n) ≥ 1 − a, the profit function is
given by (see (7) and (8))πdA (n) =a(1 + b − (n − 1)/(n + 1) − qdA (n))qdA (n)
.
a+b

(14)

The first-order condition leads to
qdA (n) =n−1
2 + (n + 1)b X A
2n + (n + 1)b,qi (n) + qdA (n) =,2(n + 1)2(n + 1)i=1¾½
a(2 + (n + 1)b)2
2 − (n + 1)bAA, 0 ; (15), if a > ad (n) ≡ max
πd (n) =4(a + b)(n + 1)2
2(n + 1)
Pn−1 Aqi (n), and the interior
otherwise, qdA (n) is the corner solution, qdA (n) = 1 − a − i=1
solution in (8) is better for the deviating firm. That is, if a < aA
d (n), the interior solutionin (8) is an equilibrium outcome. Moreover, if πdA (n) in (15) is smaller than or equalto π A (n) in (8), the quantities in case (A) are optimal. The condition under which thesolution in (8) is an equilibrium outcome isa(2 + (n + 1)b)2
1≤
or a ≤ max
2
4(a + b)(n + 1)
(n + 1)2½¾
2 − (n + 1)b,0 .
2(n + 1)The latter condition is redundant. The condition under which case (A) is an equilibriumoutcome isa ≤ aA (n) ≡4.
(n + 1)(4 + (n + 1)b)

Therefore, Lemma 3 holds.
Proof of Lemma 4(16)Q.E.D.

We have to check whether the local optimal solution in Lemma 4

is also globally optimal.
Given the quantities supplied by the other firms, if a firm sets a smaller quantity qdw ,Pw
w
w
which satisfies n−1
i=1 qi + qd ≤ 1 − a + aq0 /b, the profit function is given byµ¶(n − 1)(1 + b)ww
πd = 1 −− qd qdw .
(17)
n+2
17

When b ≥ 3/(n − 1), this is non-positive for any qdw ≥ 0, and then the interior solution in
Lemma 4 is better for the deviating firm. When b < 3/(n − 1), the first-order condition
leads toqdw =n−1
3 − (n − 1)b X w
2n + 1 + (n − 1)b,qi + qdw =,2(n + 2)2(n + 2)
i=1

πdw =

b(3 − (n − 1)b)
(3 − (n − 1)b)2, if a ≤, (18)
2
4(n + 2)2(b(n + 1) − 1)otherwise, qdw is the corner solution, qdw = 1 − a −Pn−1 ww
i=1 qi + aq0 /b, and the interior

solution in Lemma 4 is better for the deviating firm. Moreover, if πdw in (18) is smaller
than or equal to πIIIw in Lemma 4, this is an equilibrium outcome. The condition underwhich the solution is an equilibrium outcome is(3 − (n − 1)b)2a(1 + b)2
3.
≤
or b ≥
4(n + 2)2
(a + b)(n + 2)2
n−1
The condition can be summarized as follows:b(3 − (n − 1)b)2(5 − (n − 3)b)((n + 1)b − 1)a ≥ aC (n) =


 0

3
,
n−1
3
if b ≥.
n−1

if b <

Therefore, Lemma 4 holds.
Proof of Lemma 5(19)Q.E.D.

We have to check whether the local optimal solution is also globally

optimal.
Given the quantities supplied by the other firms, if a deviating firm sets a largerPzzz
quantity, qdz , which satisfies n−1
i=1 qi + qd ≥ 1 − a + aq0 /b, the profit function is given byπdz =a(1 + b − (n − 1)/(n + 1) − b/2 − qdz )qdz
.
a+b

(20)

The first-order condition leads to
4 + (n + 1)bqdz =,4(n + 1)n−1
X

qiz + qdz =

i=1

πdz =

4n + (n + 1)b,4(n + 1)a(4 + (n + 1)b)2, if a > azd ≡ max
16(a + b)(n + 1)2
18½¾
4 − (n + 1)b, 0 ; (21)2(n + 1)otherwise, qdz is the corner solution, qdz = 1 − a/2 −Pn−1 zi=1 qi , and the interior solution isbetter for the deviating firm. That is, if a < azd , the interior solution is an equilibriumoutcome. Moreover, if πdA (n) in (21) is smaller than or equal to πIIIz in Lemma 5, thequantities q z are optimal. The condition under which the solution in Lemma 5 is anequilibrium outcome is1a(4 + (n + 1)b)2≤
or a ≤ max
2
16(a + b)(n + 1)
(n + 1)2½¾
2 − (n + 1)b,0 .
2(n + 1)The latter condition is redundant. The condition under which this is an equilibriumoutcome isa≤16.
(n + 1)(8 + (n + 1)b)

Therefore, Lemma 5 holds.

(22)
Q.E.D.

19

Product h

Consumer H

Consumer L[No public firm exists]

Product h

Product l

Consumer H

Consumer L[A public firm exists]
Figure 1: The market structure20

b

q x and q y can be equilibria.

5

4

3
(iii)
2

q y is a unique equilibrium.

(ii)

1
(i)
0

a
0.1

0.2

0.3

0.4

0.5

Figure 2: Parameter range within which the public firm can be harmful.
(Horizontal: a, Vertical: b)
Note: (i) a =4
1(2 − (n − 1)b)2, (ii) a =, (iii) a =.
(n + 1)(4 − (n − 3)b)(n + 1)(4 + (n + 1)b)
2+b

21

b
5

Pub

4

(iv)

Pri.(M)(v)(ii)
Pri.(M)
Exit(U)

2

1

Pri.(U)(i)
Exit(M)

Pub
0

(vi)

(iii)

Pri.(M)
3 Exit(M)Exit(U)

Pub
a

0.1

0.2

0.3

0.4

0.5

Figure 3a: A comparison of the three cases (n = 2).
(Horizontal: a, Vertical: b)
Note 1: (i) a =(2 − (n − 1)b)2
4
1, (ii) a =, (iii) a =,(n + 1)(4 − (n − 3)b)(n + 1)(4 + (n + 1)b)2+b(iv) a =b(3 − (n − 1)b)2
16, (v) a =,(5 − (n − 3)b)((n + 1)b − 1)
(n + 1)(8 + (n + 1)b)(vi) a =Note 2:Pri.(U):
Pri.(M):
Exit(U):
Exit(M):Pub.:(n + 2)2 b.
((n + 1)b − 1)(3 + 2n + (n + 1)b)

Privatization is superior to the existence of the public firm.
(q w appears in equilibrium (a unique equilibrium).)
Privatization can be superior to the existence of the public firm.
(Both q w and q z can appear in equilibrium (multiple equilibria).)
The exit of the public firm is superior to its existence.
(q y appears in equilibrium (a unique equilibrium).)
The exit of the public firm can be superior to its existence.
(Both q x and q y can appear in equilibrium (multiple equilibria).)
The existence of the public firm is superior to the rest of the cases.

22

b
5

4
(iii)
3

Pub

Exit(U)
Pri.(U)
(vi)

2

1
(i)
0

(iv)

Pub.
0.1

0.2

0.3

0.4

Pub. a0.5

Figure 3b: A comparison of the three cases (n is sufficiently large).
(Horizontal: a, Vertical: b)
Note 1: (i) a =4
1(2 − (n − 1)b)2, (ii) a =, (iii) a =,(n + 1)(4 − (n − 3)b)(n + 1)(4 + (n + 1)b)2+b(iv) a =b(3 − (n − 1)b)2
16, (v) a =,(5 − (n − 3)b)((n + 1)b − 1)
(n + 1)(8 + (n + 1)b)(vi) a =Note 2:Pri.(U):
Exit(U):Pub.:(n + 2)2 b.
((n + 1)b − 1)(3 + 2n + (n + 1)b)

Privatization is superior to the existence of the public firm.
(q w appears in equilibrium (a unique equilibrium).)
The exit of the public firm is superior to its existence.
(q y appears in equilibrium (a unique equilibrium).)
The existence of the public firm is superior to the rest of the cases.

23

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25