Is Social Welfare Increased By Private Firm Entry？
From a coopetition viewpoint

University of Hyogo graduate school
doctoral course in economics
Takeshi Yoshikawa

Introduction
a semi-public monopoly where theMany studies on a mixed oligopoly
mainly deal with competition between

market is monopolized by a semi-public

public and private firms, the effects offirm, 2) a mixed duopoly with one

privatization on market performance,semi-public firm and one private firm inand endogenous timing1.

the market, and 3) a mixed oligopoly

In reality, however, many firms

with one semi-public firm and two pri-choose to partake not only in competitivevate firms in the market.

but also in cooperative behaviors. For

Two effects appear from the emer-

example, when new communication

gence of private firms; 1) the market size

goods are developed, mobile phone com-is larger in a semi-public monopoly thanpanies first invest in advertising to form

in a mixed duopoly if there exists aa market for the new products, and thenhighly privatized semi-public firm in thethey compete with each other in themarket, and 2) the social welfare ismarket. This situation is called coopeti-greater in a mixed duopoly than in ation2.

mixed oligopoly if there exists a highly

The objective of this paper is toprivatized semi-public firm in the market

look into the effects on social welfare of

because the change in the sum of thethe emergence of private firms in aequilibrium cooperative behavior levels

market monopolized by semi-public

triggered by private firms entering thefirms by using a coopetition model. The

market has a positive effect (an increase

following three cases are considered: 1)

in the sum of the equilibrium output level) rather than a negative effect (an increase in the total cost).

See De Fraja, G., and Delbono, F.
(1989), Pal, D. (1998), White (1996), Fjell,K., Pal, D. (1996), Anderson, S. P., dePalma, A. and Thisse J. -F. (1997), Matsumura (1998), Nishimori and Ogawa(2002), Haruna (2005).
2 See Brandenburger and Nalebuff(1997), Krishnamurthy (2000), Deardenand Lilien (2001), Ngo and Okura (2008).
11ModelWe assume homogeneous goods

assumed to be the same for all the firms.

markets and compare the following threeHence, the total cost function is given bycases: 1) a semi-public monopoly whereC i  cqi  k i2

the market is monopolized by asemi-public firm, 2) a mixed duopoly

fori  1,2,3 .(2)Firm 2 and firm 3 denote privatewith one semi-public firm and one pri-firms, which maximize their profits. Us-vate firm in the market, and 3) a mixeding (1) and (2), the profit function isoligopoly with one semi-public firm andgiven by

two private firms in the market.

 i  Pqi  Ci

Firms existing in the market play afor i  2,3, (3)two-stage game. In the first stage, theand a semi-public firm is denoted byfirms choose their own cooperative be-firm 1. Using (2) and (3), the objective

havior levels. The cooperative behavior

function isU 1  W  (1   ) 1 ,level is denoted by k i for i  1,2,3 ,(4)where W , which is the sum of the con-which increases the market size. Thesumer surplus CS and the producer

overall market inverse demand function

surplus PS , represents the social welfare.

is given by

The social welfare W is thus given by

P  A Q,W  CS  PS.
(5)
The parameter   [0,1] can be in-(1)where Q is the total supply and A
represents the market size and is theterpreted in two ways; it represents thesum of a (a positive constant number)weight of the government’s participation

and K (the total cooperative behavior

in firm 1. It can be regarded as the im-level). Hence, A is formulated

port level attributed to the government’s

by A  a  K .

objective, and the government can con-

In the second stage, the firms

trol it by regulating the shareholding
ratio. Therefore, we can view  as thechoose their output levels q i

degree of privatization; i.e.   0 signi-

for i  1,2,3 , representing their compet-

fies that firm 1 is solely concerned about

itive behavior. The cost of expending co-its profit while   1 means that firm 1
aims to maximize social welfare regard-

operative behavior is denoted by k i . The2

less of its profit. Table 1 ex-marginal cost of production denoted bypresses A , CS , PS , and W in the threec is assumed to be a  c and is alsocases.

2Comparisons
reducing the elasticity of the reaction

This section scrutinizes the inequa-function.

lity relationships of the three cases toelaborate the effects of the emergence ofHence, k1

sm

private firms on social welfare.

is the greatest of thethree cases and K md is greater thanComparison of K

K mo when a highly privatizedsemi-public firm exists in the market.

This subsection deals with the in-Comparison of consumer surplus

equality relationships of K (the totalsum of the equilibrium cooperative behavior levels) in the three cases3.

This subsection considers the inequality relationships of the consumer

Finding 1. The inequality relationships

surpluses.

of K in the three cases.

K mo  K md  k1sm

Finding 2. The inequality relationships(0    0.184), (54)of the consumer surpluses.

CS sm  CS md  CS moK mo  k1sm  K md (0.184    0.293), (55)

. (58)The emergence of private firms

k1sm  K mo  K md (0.293    0.489), (56)increases the total supply, thus boosting
the consumer surplus.

k1sm  K md  K mo (0.489    1). (57)Comparison of producer surplus
The intercepts of reaction functions
is greater in a mixed duopoly than in aThis subsection concerns the in-mixed oligopoly and the slop of reaction

equality relationships of the producer

functions of semi-public firm is smaller

surpluses4in a mixed duopoly than in a mixed oligopoly, when there exists a highly priva-Finding 3. The inequality relationships

tized semi-public firm in the market, asof the producer surpluses.

shown in (54). On the other hand, thePS mo  PS md  PS sm (0    0.114), (59)slop of reaction functions of private firm
is smaller in a mixed duopoly than in aPSmo  PSsm  PSmd (0.114   0.301), (60)

mixed oligopoly. That is, participation
private firm in a mixed duopoly market,3PS mo  PS md  PS sm (0.301    1). (61)Rounded to three decimal places.

43

Rounded to three decimal places.

The inequality relationship of theWhen there exists a highly priva-

total sum of the equilibrium cooperative

tized semi-public firm in the market, asbehavior level is K  K
when there
exists a highly privatized semi-publicshown in (62), the inequality relationshipfirm in the market, as shown by (59). In aCS sm  CS md  CS mo and the inequality

mixed duopoly the positive effects, suchrelationship of the producer surplus isas the increase in the market size are

PS mo  PS md  PS sm , and the inequality

smaller than the negative effects, such as

relationship of the total sum of the equi-

the increase in the sum of the firm’s total

librium cooperative behavior levels ismomdof the consumer surplus iscosts. On the other hand, in a mixed oli-

K mo  K md  k1sm . We can thus say that

gopoly the positive effects are smaller

the change in the sum of the equilibrium

than the negative effects.

cooperative behavior levels caused by theHence, we conclude that the emergence of private firms has two effects onentrance of private firms has a positive

the producer surplus. 1) The entrance ofeffect (an increase in the equilibrium

two private firms always decreases theoutput level) rather than a negative ef-

producer surplus, and 2) if there exists afect (an increase in the total cost) if there

highly privatized semi-public firm in theexists a highly privatized semi-publicmarket, the producer surplus is smaller

firm in the market.
Therefore, the social welfare isin a semi-public monopoly than in asmaller in a mixed oligopoly market

mixed duopoly.

made up of two private firms and a

Comparison of social welfare

highly privatized semi-public firm thanin a mixed duopoly market composed of aThis subsection considers the in-private firm and a highly privatized

equality relationships of the social wel-semi-public firm. If a less privatized ra-fare. Figure 4, which shows the rela-ther than highly privatized semi-public

tionships of W , W
 , yields Finding 45.

firm exists in the market, the social wel-smmdand Wmoto

fare will increase in either case.
We conclude that deregulation re-Finding 4. The inequality relationships

sults in diminishing the social welfareof the social welfare.

when a highly privatized semi-public
firm exists in the market, as in the caseW sm  W mo  W md (0    0.014), (62)of a private firm entering a mixed duopoly market. However, it should be notedW sm  W md W mo (0.014  1). (63)that deregulation increases the social
welfare when a less privatized[Figure 4 insertions]5semi-public firm exists in the market.

Rounded to three decimal places.
4Conclusions
Using a coopetition model, we havebrium output level) rather than a nega-shown two effects of the emergence oftive effect (an increase in the total cost).

private firms: 1) the market size is larger

We conclude that deregulation re-in a semi-public monopoly than in asults in diminishing the social welfare

mixed duopoly if a highly privatizedwhen a highly privatized semi-publicsemi-public firm is in the market, and 2)

firm exists in the market, as in the casethe social welfare is greater in a mixedof two private firms entering a

duopoly than in a mixed oligopoly if asemi-public monopoly market.

highly privatized semi-public firm is in

We don’t refer the relationship of the op-

the market, because the change in the

timal value of αto the number of private

sum of the equilibrium cooperative be-firms existing in the market. Extending

havior levels triggered by private firms

our model to this direction remains forentering the market has a positive effectfuture research.

(an increase in the sum of the equili-5

Figure 4
The relation of W to  in the three cases

WW moW md
W sm

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