Vertical and Horizontal Competition between Public Firms and Privatization【Articles】

Vertical and Horizontal Competition
between Public Firms and Privatization
公企業間の垂直的・水平的競争
および民営化OSHIMA Kosuke
大

島

考

介

Contents
1. Introduction
2. The model
3. Horizontal competition
4. Vertical competition
5. Conclusion（Abstract）
The literature of studies on privatization using models of mixed oligopoly and public duopoly has a long history. The issues of these studies include partial privatization, multiple
countries or regions, diﬀerentiated goods, centralization and decentralization, and others.
Competitions in those models are basically horizontal ones, such as competition between local
public ﬁrms and between public and private ﬁrms. However, little attention has been paid tothe research on vertical competition in this area such as that between state-owned and local
public ﬁrms. The present paper ﬁrst sets up a simple two-city model of horizontal competition
and analyzes the equilibrium. We should note that the privatization in one city can change theobjective function of the other cityʼs public ﬁrm if its residents own the stock of the privatizedﬁrm. Then we modify the model into that of a vertical competition and investigate the results.
We ﬁnd that vertical and horizontal competitions between public ﬁrms can achieve the sameequilibrium although the reaction functions of the ﬁrms are diﬀerent. If the city-owned ﬁrm isprivatized a diﬀerent equilibrium is realized, but the welfare levels of the cities remain thesame. The model of the present paper can be used as a benchmark and may be extended in various ways. For example, one may assume that the cities are asymmetric in population or others.
Alternatively, one can consider diﬀerentiated goods, partial privatization, or public-private
partnerships.

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Vertical and Horizontal Competition between Public Firms and Privatization

1. Introduction
The literature of studies on privatization using models of mixed oligopoly and public duopoly has a long history. Mixed oligopoly is a market where a small number of ﬁrms, public andprivate, compete with each other. Public duopoly is a market with two public ﬁrms owned by
diﬀerent countries or local governments. While private ﬁrms maximize proﬁts, a public ﬁrm
acts in order to maximize (national or local) social welfare. If it is privatized it acts as a private ﬁrm and hence aﬀects the social welfare. Therefore it is important to consider how thegovernment should handle those public ﬁrms.
Since De Fraja and Delbonoʼs (1989) seminal paper many researchers focused on various
issues in this area. Matsumura (1998) took into account partial privatization in mixed duopoly
model and showed that partial privatization can be more desirable than full nationalization or
full privatization. Matsumura and Kanda (2005) considered the case where ﬁrms can enter themarket freely and they showed that the government should fully own the public ﬁrm. Fujiwara(2007) introduced product diﬀerentiation into the model and showed that partial privatization
is optimal except in extreme cases.1) Researchers such as Bárcena-Ruiz and Garzón (2005) andDadpay and Heywood (2006) considered two-country models and investigated strategic interactions between two countries in deciding whether to privatize their public ﬁrms. Han and Ogawa(2008), allowing for partial privatization, examined the impact of the market integration of two
countries and showed that the governments are less eager to privatize in the international
mixed oligopoly market than in a single-country framework. Mantin (2012) and Matsumura andMatsushima (2012) investigated public duopoly models where each country owns an airport anddecides whether to privatize it. In their models the strategic interaction between the airports
is strategic substitution. They showed that the two countries privatize their airports despite
higher welfare levels under non-privatization by both countries. That is, the governments face
the prisonerʼs dilemma. Oshima (2018) considered a diﬀerentiated mixed duopoly with partial
privatization in a two-city model where a public ﬁrm operate in one city and a private ﬁrm in
the other, and compared decentralized and centralized solutions. The result is that in mostcases the privatization level is higher under centralization (the state owns the public ﬁrm) thanunder decentralization (the city owns the public ﬁrm). In particular, under decentralization, thepublic ﬁrm is fully owned by the city government if the substitutability of diﬀerentiated goods
is relatively high.
Competitions in those models mentioned above are basically horizontal ones. They are, forexample, competitions between local public ﬁrms and between public and private ﬁrms. However, little attention has been paid to vertical competition in this area. Here the phrase
vertical competition refers to a competition between ﬁrms owned by diﬀerent levels of governments, such as national and local governments. An example of vertical competition would be
airports. One can imagine state-owned, local government-owned, and private airports competing in a certain region. One may also consider prefectural and municipal museums operating in『商学集志』第 92 巻第 1 号（’22. 6）+OSHIMA Kosuke氏.indd 2―2―2022/06/16 9:33:03

Vertical and Horizontal Competition between Public Firms and Privatizationa certain city. How the ﬁrms interact with each other can aﬀect the eﬃciency of the market.
We should also note that the privatization in one city changes the objective function of public
ﬁrm in the other city if its residents own the stock of the privatized ﬁrm. In the present paper
we compare horizontal and vertical competitions using a simple two-city model, which can be abenchmark for further research.
The rest of the paper is organized as follows. In section 2 we set up a model commonlyused for horizontal and vertical competitions. In section 3 we derive the equilibria of horizontal competition and evaluate the social welfare levels. Section 4 provides the equilibria ofvertical competition and a comparison with horizontal competition. Section 5 concludes.
2. The model
Suppose a country that consists of two cities, city 1 and city 2 with the same population.
The residents are homogeneous. We standardize the total population to unity, and hence thepopulation of each city is 1/2. Each city has a public ﬁrm. City 1ʼs ﬁrm (ﬁrm 1) and city 2ʼsﬁrm (ﬁrm 2) produce a homogeneous good. In Mantin (2012) and Matsumura and Matsushima(2012) the two airports are located in two diﬀerent countries, respectively, and airlines connect the two airports. Therefore, they are complements. In the present paper, however, thegoods produced by the two ﬁrms are (perfect) substitutes. Taking airports as an example, thetwo airports are located in neighboring cities, respectively, and airlines connect each of themwith foreign airports. They are competing with each other and hence are substitutes.
The proﬁt of a public ﬁrm belongs to the city which owns the ﬁrm. If the ﬁrm is privatized its stock is widely sold to residents around the country. Let us assume that the stock isevenly owned by the residents of two cities. Therefore the proﬁt of the privatized ﬁrm is distributed in half to the two cities.
The inverse demand function of the good is expressed as below:p  a  ( q1  q2 ),where ,

1

and

2

(1)

denote the price and quantities of the good produced by ﬁrms 1 and 2,

respectively. Parametershows the size of the market. From (1) the consumer surplus of thismarket is as follows:CS ( q1  q2 )2
.
2

(2)

Now we consider the producers. Suppose that the ﬁrms face the same technology and thecost function ,  1, 2, whereis the marginal cost, , andis the ﬁxed cost. As

shown in De Fraja and Delbono (1989) and others the increasing marginal cost works in favorof privatization. Hence we assume that the marginal cost is constant, which is a usual assumption in the literature.2) 3) Then the proﬁt of ﬁrm is as follows:―3―+OSHIMA Kosuke氏.indd 3『商学集志』第 92 巻第 1 号（’22. 6）2022/06/16 9:33:03

Vertical and Horizontal Competition between Public Firms and Privatization

 i  ( p  c ) qi  f , i  1, 2.
The social welfare of the country,(3), is deﬁned as the sum of consumer surplus and prof-

its of the two ﬁrms:
W  CS   1   2 .

(4)

Because the two cities are symmetric in consumption, consumer surplus for each city is/2.

Therefore the welfares of the two cities,

1

,

2

are deﬁned as follows;

W i  CS /2   i , i  1, 2,

no matter whether ﬁrms are privatized or not. If ﬁrm(5)
is privatized  /2 belongs to city

(  1, 2,  ). Assuming perfect foresight, however, city receives the same amount from residents in city when it sells the ﬁrmʼs stock.
We consider a two-stage game. In the ﬁrst stage the cities 1 and 2 choose whether toprivatize their public ﬁrms. In the second stage the ﬁrms determine the quantities supplied
simultaneously (that is, we consider a Cournot competition). Then we solve the game by backward induction.
In the second stage, if ﬁrm 1 is privatized, it maximizes its proﬁt  1. From (1) and (3) wehavemax  1  ( a  c  q1  q2 ) q1  f .

From the ﬁrst-order condition we have ﬁrm 1ʼs reaction function as follows:q1 a  c  q2
.
2

(6)

Similarly, if ﬁrm 2 is also privatized, one can obtain ﬁrm 2ʼs reaction function as below:q2 a  c  q1
.
2

(7)

On the other hand, if both ﬁrms are city-owned ﬁrm 1 maximizes

1. Therefore, from (1) ‒ (3)and (5) we havemax W 1 

( q1  q2 )2
 ( a  c  q1  q2 ) q1  f .
4

If, however, only ﬁrm 2 is privatized, ﬁrm 1 maximizesof city1

  2/2. That is, while the welfare

is expressed as (5) the incentive of the city is aﬀected if ﬁrm

is privatized and theresidents of city own half of the stock of ﬁrm . Therefore we havemax W 1 

2

『商学集志』第 92 巻第 1 号（’22. 6）+OSHIMA Kosuke氏.indd 42

q  3f( q1  q2 )2
 ( a  c  q1  q2 )  q1  2  
.
4
2 
2

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Vertical and Horizontal Competition between Public Firms and Privatization

Figure 1: Horizontal competition
From the ﬁrst-order condition we obtain public ﬁrm 1ʼs reaction function as follows: 2 a  2 c  q2
3
q1  
2(ac  q2 )
3

if firm 2 is city-owned,(8)
if firm 2 is privatized.

Similarly, we have public ﬁrm 2ʼs reaction function as below:
 2 a  2 c  q1
3
q2  
2(ac  q1 )
3

The intuition is that the public ﬁrm
owned, and let ﬁrm

if firm 1 is city-owned,(9)
if firm 1 is privatized.

produces less if ﬁrm

produce more. This is because city

is privatized than if it is city-

takes into account the interests of

its residents as shareholders of ﬁrm .
3. Horizontal competition
Let PP denote competition between private ﬁrms 1 and 2. Similarly, PG denotes competition between private ﬁrm 1 and public ﬁrm 2 while GG denotes that between public ﬁrms 1 and2. From (6) ‒ (9) one can draw reaction curbs of the two ﬁrms under GG, PG and PP as in
Figure 1.4) P1 and P2 represent the reaction curves of private ﬁrms 1 and 2, while G1 and G2
those of public ﬁrms 1 and 2. The points show the equilibria of the three types of competition.
Now let us see the equilibrium. Under PP, from (6) and (7) we have the equilibrium production levels as follows:―5―+OSHIMA Kosuke氏.indd 5『商学集志』第 92 巻第 1 号（’22. 6）2022/06/16 9:33:04

Vertical and Horizontal Competition between Public Firms and Privatization

Table 1: Payoﬀ matrix of horizontal competition
ﬁrm 2
P

G

P

2
( a  c )2  f ,92( a  c )2  f
913( a  c )2  f ,64

17( a  c )2  f
64

G

17
( a  c )2  f ,64

13( a  c )2  f
64

1( a  c )2  f ,41( a  c )2  f
4

ﬁrm 1

q1PP  q2PP 

where

1

and

2

ac
,
3

(10)

denote equilibrium production levels of ﬁrms 1 and 2 under PP. Substitut-ing (10) into (4) and (5) we have the levels of social welfare
and2

and regional welfares

1

as follows:
W PP 

4( a  c )2  2 f ,9

W 1 PP  W 2 PP 2
( a  c )2  f .
9(11)

Under PG, from (6) and (9) we have
q1PG 

ac,
4

q2PG 

ac
.
2

(12)

Substituting (12) into (4) and (5) yields
W PG 

1513( a  c )2  2 f , W 1 PG 
( a  c )2  f ,
32
64
W

2 PG

17
( a  c )2  f .

64

(13)

Under GG, from (8) and (9) we have
q1GG  q2GG 

ac
.
2

(14)

Substituting (14) into (4) and (5) yields
W GG 

1( a  c )2  2 f ,2

W 1 GG  W 2 GG 

1
( a  c )2  f .
4(15)

We can now compare the welfare levels under PP, PG, and GG. From (11), (13), and (15) wehave the following inequality:
W PP  W PG  W GG .
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(16)

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Vertical and Horizontal Competition between Public Firms and Privatization

One can see that GG is the most preferable as a whole in this simple model. From (11), (13),and (15) we can also make the payoﬀ matrix of horizontal competition as Table 1. Because
public ownership is the dominant strategy for both cities the Nash equilibrium is (G, G). Under
PG and GP the city which does not privatize its public ﬁrm is better oﬀ than under GG, but it
is not feasible. The assumption of public ﬁrmʼs objective function when the other cityʼs ﬁrm isprivatized, which was mentioned in the last section, aﬀects the payoﬀs under PG and GP. The
Nash equilibrium is unaﬀected, though.5)
4. Vertical competition
Suppose now that the national government owns one of public ﬁrms. Without loss of generality, we assume that the national government owns ﬁrm 1 and city 2 owns ﬁrm 2. That is,the vertical competition arises where the ﬁrms of national and local governments compete in
the market. The state-owned ﬁrm 1 maximizes the social welfare of the country,. Therefore,

from (2) ‒ (4) we havemax W ( q1  q2 )2
 ( a  c  q1  q2 )( q1  q2 )  2 f .
2From the ﬁrst-order condition we obtain ﬁrm 1ʼs reaction function as follows:
q1  a  c  q2 .

(17)

The objective function of state-owned ﬁrm 1, and hence its reaction function, is not aﬀected nomatter whether ﬁrm 2 is privatized or not. This is because ﬁrm 1ʼs objective is to maximize
the countryʼs social welfare, or the sum of total consumer surplus and total proﬁts, which is
not aﬀected by the privatization of ﬁrm 2.
Let NG denote competition between state-owned ﬁrm 1 and city-owned ﬁrm 2. Similarly, NP
denotes competition between state-owned ﬁrm 1 and private ﬁrm 2. From (7), (9), and (17) one
can draw reaction curbs of the two ﬁrms, ﬁrm 1 and ﬁrm 2, under NG and NP as in Figure 2.6)
Now let us see the equilibrium. Under NG, from (9) and (17) we have the equilibrium production levels as follows:
q1NG  q2NG 

ac
.
2

(18)

Substituting (18) into (4) and (5) we have the levels of social welfare and regional welfares asfollows:
W NG 

( a  c )2
( a  c )2
 2 f , W 1 NG  W 2 NG 
 f.
2
4

(19)

From (14), (15), (18), and (19) one can see that NG achieves the same equilibrium as GG
although ﬁrm 1ʼs reaction curb is diﬀerent from that of GG (see Figure 2).
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Vertical and Horizontal Competition between Public Firms and Privatization

Figure 2: Vertical competition

Under NP, from (7) and (17) we have the equilibrium production levels as follows:
q1NP  a  c ,q2NP  0.

(20)

Substituting (20) into (4) and (5) yields
W NP 

( a  c )2
( a  c )2
 2 f , W 1 NP  W 2 NP 
 f.
2
4

(21)

Therefore, from (15), (19), and (21) one can see that NP achieves the same welfare level as GG
and NG.
From (11), (13), (19), and (21) we have the payoﬀ matrix of vertical competition as Table 2.
There exist two Nash equilibria, (N, G) and (N, P) in this game. Firms 1 and 2 are, however,originally owned by national and city governments, respectively. That is, (N, G) is the statusquo. In addition, as one can see from (20), the private ﬁrm 2 gets crowded out by state-owned
ﬁrm 1 in the equilibrium (N, P). Therefore one can predict that city 2 will not privatize ﬁrm 2
if ﬁrm 1 is state-owned.
As far as the welfare level is concerned, the equilibria (N, G) and (N, P) achieve the same
welfare level and are Pareto eﬃcient. This is because the slope of the reaction function of thestate-owned ﬁrm is 1 and ﬁrms produce and sell goods at the marginal cost, . That is, there
is a trade-oﬀ between the amounts of goods produced by the two ﬁrms. This result holds even
if private ﬁrm 2ʼs reaction function is diﬀerent (e.g.,2 2 (  )/3 /2) and both ﬁrms pro-1

duce positive amounts of goods under NP.

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Vertical and Horizontal Competition between Public Firms and Privatization

Table 2: Payoﬀ matrix of vertical competition
ﬁrm 2
P

G

P

2
( a  c )2  f ,92( a  c )2  f
913( a  c )2  f ,64

17( a  c )2  f
64N

1( a  c )2  f ,41( a  c )2  f
4

1( a  c )2  f ,41( a  c )2  f
4

ﬁrm 1

5. Conclusion
The literature of studies on mixed oligopoly and public duopoly has a long history andvarious issues have been investigated such as partial privatization, multiple countries orregions, diﬀerentiated goods, centralization and decentralization, and others. While these studies basically investigated horizontal competitions, little attention has been paid to the researchon vertical competitions in this area.
In the present paper we ﬁrst considered horizontal competition using a simple two-citymodel. Assuming that city ʼs residents own half of ﬁrm ʼs stock if it is privatized, the objective function of city ʼs public ﬁrm is aﬀected so that it takes into account the interests of city
ʼs residents as shareholders of private ﬁrm . Then the dominant strategy for both cities ispublic ownership and the Nash equilibrium is (G, G). Next we supposed that the national government owns ﬁrm 1 and city 2 owns ﬁrm 2, and investigated the vertical competition. As theobjective of state-owned ﬁrm 1 is to maximize the social welfare of the country as a whole, its
objective function, and hence, its reaction function, is not aﬀected no matter whether ﬁrm 2 isprivatized or not. As a result, although the reaction function of ﬁrm 1 under NG (vertical competition) is diﬀerent from that under GG (horizontal competition), NG and GG achieve thesame equilibrium. Because the slope of ﬁrm 1ʼs reaction curb is 1 NP achieves the same welfare level as under NG and GG in equilibrium. In that case, however, the private ﬁrm 2 getscrowded out by state-owned ﬁrm 1 and hence city 2 will not privatize ﬁrm 2.
We found that in our simple model the governments do not sell their ﬁrms. This is true inmany cases of “privatization.” Recently, for example, the operation of many airports, state-runand local government-run, were left to private ﬁrms, while the governments still own those airports. In order to take into account complex realities, however, we need to consider various
extensions to the model.
For example, one may assume that the cities are asymmetric in population, productivity,
or others. It may be meaningful to suppose that the goods which the ﬁrms produce are diﬀerentiated. One can also introduce partial privatization into the model, where the public ﬁrm
maximizes the weighted average of social welfare and ﬁrmʼs proﬁt. Recently, however, participation of private ﬁrms in public works projects often takes the form of public-private―9―+OSHIMA Kosuke氏.indd 9『商学集志』第 92 巻第 1 号（’22. 6）2022/06/16 9:33:06

Vertical and Horizontal Competition between Public Firms and Privatization

partnerships, as in airports mentioned above. Therefore it may be worth introducing such an
approach into the model. These are left for future research.

Notes1) See also Anderson et al. (1997), Saha (2009) and Choi (2012) for product diﬀerentiation.
2) We also assume that ﬁrms cannot discriminate local consumers from consumers living in the other city.
3) In the present model we do not assume transportation costs to avoid complexities.
4) GP is symmetrical to PG. Therefore, in order to avoid complication we omit GP in Figure 1.
5) If the public ﬁrm 1 disregarded the proﬁt of private ﬁrm 2, the welfare levels of cities 1 and 2 under GP
would be 7 (  )2/25  and (  )2/5  instead of 17 (  )2/64  and 13 (  )2/64  . Under PG, thewelfare levels of cities 1 and 2 would be (  )2/5 and 7 (  )2/25  . Because 2/9  7/25 and 1/5 

1/4, (G, G) is the Nash equilibrium in this case, too. Since 17/64  7/25 and 13/64  1/5, however, theactual Nash equilibrium is closer to an opposite solution (P, P).
6) One should read “city-owned” in (8) and (9) as “state-owned” here.

References
Anderson, S. P., A. de Palma and J.-F. Thisse (1997) “Privatization and Eﬃciency in a Diﬀerentiated Industry,”Vol. 41, No. 9, pp.1635-1654.
Bárcena-Ruiz, J. C. and M. B. Garzón (2005) “International Trade and Strategic Privatization,”Vol. 9, No. 4, pp.502-513.
Choi, K. (2012) “Cournot and Bertrand competition with asymmetric costs in a mixed duopoly revisited,” MPRA
Paper, No. 37004
Dadpay, A. and J. S. Heywood (2006) “Mixed Oligopoly in a Single International Market,”Vol. 45, No. 4, pp.269-280.
De Fraja, G. and F. Delbono (1989) “Alternative Strategies of a Public Enterprise in Oligopoly,”Vol. 41, No. 2, pp.302-311.
Fujiwara, K. (2007) “Partial Privatization in a Diﬀerentiated Mixed Oligopoly,”Vol. 92,

No. 1, pp.51-65.
Han, L. and H. Ogawa (2008) “Economic Integration and Strategic Privatization in an International Mixed Oligopoly,”Vol. 64, No. 3, pp.352-363.

Mantin, B. (2012) “Airport complementarity: Private vs. government ownership and welfare gravitation,”Vol. 46, pp.381-388.
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pp.473-483.
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1, pp.27-48.
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Vertical and Horizontal Competition between Public Firms and Privatization
Oshima, K. (2018) “Diﬀerentiated mixed duopoly and decentralization in a two-city model,”Vol. 97, No. 4, pp.1425-1440.
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No. 1, pp.25-43.

（要旨）
混合寡占と公共複占のモデルを用いた民営化に関する研究には長い歴史がある。これらの研究の課題には，部分民営化，複数の国または地域，差別化された財，中央集権と地方分権などが含まれる。これらのモデルにおける競争は，地方公企業間の競争や公企業と民間企業間の競争など，基本的には水平的競争である。しかしこの分野では，国営企業と地方公企業間の競争のような垂直的競争に関する研究にはほとんど注意が払われてこなかった。本論文は，最初に水平的競争の単純な 2 都市モデルを構築し，均衡を分析する。ある都市での民営化は，他方の都市の住民が民営化された企業の株を所有している場合，他方の都市の公企業の目的関数を変えることに注意する必要がある。次に，モデルを垂直的競争に変更して結果を分析する。その結果，公企業の反応関数が異なるのにもかかわらず，公企業間の垂直的競争と水平的競争は同じ均衡を実現することが分かる。このとき都市所有の企業が民営化されても社会厚生水準は同じになるが，均衡は異なったものになる。本論文のモデルをベンチマークとして，様々な拡張を考えることができる。例えば，都市の人口等が非対称であると仮定してもよい。あるいは，差別化された財，部分民営化，または官民協働などを検討することができる。

― 11 ―+OSHIMA Kosuke氏.indd 11『商学集志』第 92 巻第 1 号（’22. 6）2022/06/16 9:33:07