2007・1
Privatization and entries of foreign enterprises in adifferentiated industry
Toshihiro Matsumura
Noriaki Matsushima
Ikuo Ishibashi

Privatization and entries of foreign enterprises in adifferentiated industry
Toshihiro Matsumura
Institute of Social Science, the University of Tokyo
Noriaki Matsushima∗
Graduate School of Business Administration, Kobe University
Ikuo Ishibashi
Department of Economics, Aoyama Gakuin University
January 9, 2007

Abstract
We investigate whether or not privatization is beneficial from the viewpoint of social
welfare in a monopolistic competition model. We discuss the relationship between thewelfare effects of privatization and the degree of foreign direct investment in the private
sector, which is an important problem in developing countries and in transition economies
such as China and Central and Eastern European countries. We find that, in the long run,privatization of a public firm is more likely to improve welfare when the country depends
on foreign capital in the private sector, whereas the opposite tendency exists in the shortrun.

JEL classification numbers: H42, L13, C72
Key words: mixed oligopoly, privatization, product differentiation, foreign firms, free
entry

∗

Correspondence author: Noriaki Matsushima, Graduate School of Business Administration, Kobe University, Rokkodai 2-1, Nada-Ku, Kobe, Hyogo 657-8501, Japan. Phone: +81-78-803-6981. E-mail:nmatsush@kobe-u.ac.jp1

1

Introduction

Since the 1980s, we have observed a worldwide wave of privatization. Nevertheless, public firms still exist, and many of them compete with private firms in private goods markets
in developed, developing, and former communist countries. Competition between public andprivate firms existed, or still exists, in a range of industries, including the airline, rail, telecommunications, natural gas, electricity, steel, and overnight-delivery industries, as well as services
including banking, home loans, health care, life insurance, hospitals, broadcasting, and education.1
Recently, studies of “mixed markets”, involving both private and public enterprises, havebecome increasingly popular.2 Most existing works have conducted short-run analyses, inwhich the number of firms is given exogenously. Anderson, de Palma, and Thisse (1997) is apioneering work that provides a long-run analysis in mixed markets. They use a model withmonopolistic competition. They show that, in the short run, the privatization of a public
firm never improves welfare, whereas, in the long run, privatization may or may not improvewelfare.3
Most existing works on mixed oligopoly, as well as that by Anderson, de Palma, and Thisse(1997), investigate the competition between public and domestic private firms. In real world
economies, however, competitors of public firms are not limited to domestic private firms. For
1

See, e.g., Matsushima and Matsumura (2003).

2

This interest in mixed oligopolies is because of their importance to the economies of Europe, Canada, andJapan. Although they are less significant in the United States, there are some examples of mixed oligopolies suchas the packaging and overnight-delivery industries. See Bös (1986, 1991) and Nett (1993) for excellent surveysof mixed oligopolies. The idea of mixed oligopoly dates to at least Merrill and Schneider (1966). Recently,
the literature on mixed oligopoly has become richer and more diverse. For example, Ishibashi and Matsumura(2006) investigate R&D competition between public and private sectors. Bárcena-Ruiz and Garzón (2006) andOhori (2006) analyze environmental policies. Mujumdar and Pal (1998) consider tax effects. Bárcena-Ruizand Garzón (2003) discuss a merger problem. Pal (1998) and Matsumura (2003a) discuss endogenous role.
Cremer, Marchand, and Thisse (1991), Li (2006), and Matsushima and Matsumura (2003) analyze endogenous
product differentiation.
3

For other papers on the long-run effect in mixed markets, see Futagami (1999) and Matsumura and Kanda(2005).

2

example, the New Zealand government set up a state-owned public bank to compete against
private foreign banks. Similarly, when the government of Brazil bargained with the Swiss
medical company Roche, it used a public medical institution as a potential competitor in thedomestic market. Électricité de France and Gas de France also compete against foreign private
firms in the EU energy markets. Especially in former communist transitional countries such
as Eastern Europe countries and China, many national firms compete against both foreignand domestic private firms.4
In this paper, we consider foreign competitors explicitly and investigate how the presenceof foreign competitors affects the efficiency of privatization. We extend the mixed oligopoly
model of Anderson, de Palma, and Thisse (1997) to the case of foreign competitors. We findthat the existence of foreign firms increases the welfare loss from the privatization of a public
firm in the short run. In the long run, the existence of foreign firms causes the privatization
to be more likely to improve welfare. If all private firms are foreign, the privatization of apublic firm always improves welfare.
The result is noteworthy. When a public firm competes against foreign firms, the public
firm lowers its price to enlarge the consumer surplus and to reduce the welfare loss, whichstems from the profits of the foreign firms. In fact, in the short run, the social value of the
public firm increases in the presence of foreign competitors. In the long run, however, thisproperty does not hold. In equilibrium, social welfare is equal to consumer surplus plus the
profit of a public firm because the profits of private firms are zero (as a result of free entry).
The consumer surplus depends on product diversity. As discussed above, the public firm
aggressively sets a lower price under foreign competition, resulting in a smaller number of
entries. This loss increases in the presence of foreign competitors, and privatization is always
beneficial when all competitors are foreign.
The relationship between the number of domestic and foreign firms and privatization is4

For competition between public and foreign private firms see Fjell and Pal (1996), Pal and White (1998),

Matsumura (2003b), and Matsushima and Matsumura (2006).

3

related to the literature on foreign direct investment (FDI) into transition economies such as
Central and Eastern European countries (CEECs). In our paper, in the long-run equilibrium,
privatization induces the entry of foreign and domestic firms. That is, privatization has apositive impact on FDI into the host country. Using dynamic panel data methods, Carstensen
and Toubal (2004) examine the determinants of FDI into CEECs. They show that both thelevel of privatization measured by private market share and the actual method of privatization
have considerable positive impacts on the decision to invest in CEECs. Our results are
consistent with the empirical evidence of the positive impact on the FDI decision.5
The remainder of this paper is organized as follows. In Section 2, we explain the basic
setting of mixed oligopoly. Section 3 conducts a short-run analysis. It demonstrates thatprivatization never improves welfare and that welfare loss from privatization is larger when thecompetitors are foreign. Section 4 conducts a long-run analysis. It shows that privatization
always improves welfare when all private firms are foreign. Section 5 discusses how costdifferences between public and private firms affect our results. Section 6 is the conclusion.

2

The model

We first describe a market equilibrium in which all firms are privately owned. There are n + 1
variants of a differentiated product. Market demand and consumer benefits are determinedby a representative consumer’s utility function:
Ã

!

n+1
X ρ
1
xi + x0 ,U (x1 , . . . , xn+1 ; x0 ) = ln
ρ
i=1

(1)

where xi denotes the quantity of variant i, x0 , the quantity of the numeraire, and ρ ∈ (0, 1), theparameterized consumer’s preference for variety. The restriction on ρ ensures that products
are substitutes, with a high ρ corresponding to a low preference for variety; when ρ → 1, allgoods are perfect substitutes, while preferences have the Cobb–Douglas form for ρ → 0. The
5

Carstensen and Toubal (2004) also point out that the South Eastern European Countries, Bulgaria andRomania, were unsuccessful in attracting FDI during the first half of 1990s, and then began to attract investors
only after they changed to foreign-oriented privatization policies in the late 1990s.

4budget constraint of the consumer is:n+1
X

pi xi + x0 = Y,

(2)

i=1

where pi is the price of variant i, and Y , the consumer’s income. Maximizing Eq. (1) subject
to Eq. (2) yields the demand functions for the (n + 1) variants:p−λ−1i, i = 1, . . . , n + 1,x̄i = Pn+1−λ
j=1 pj(3)

where λ ≡ ρ/(1 − ρ). Hence, λ is an inverse measure of consumer preference for variety: thesmaller the former, the stronger the latter. When λ = 0, the variants are considered to be
fully differentiated; when λ → ∞, all the variants are regarded as perfect substitutes, andthe consumer(s) buys the variant with the lowest price. Because total expenditure on the
differentiated product is 1, we assume throughout that Y > 1 in order to have an interior
solution to the representative consumer’s maximization problem.
We now consider the supply side. Firm i produces variant i at a constant marginal cost cand bears a fixed cost F , which is sunk in the short run. These costs are assumed to be thesame across firms so that all firms will be equally efficient, independent of their ownership. In
other words, we ignore the cost difference between public and private firms, as did Anderson,de Palma, and Thisse (1997). Many (but not all) empirical works show that a public enterprise
performs less efficiently than a private enterprise. In Section 5, we discuss how cost differences
between public and private firms affect the results.6 Hence, firm i’s profits are:πi = (pi − c)x̄i − F, i = 1, . . . , n + 1.

(4)

In a case in which all firms are private, it is well known that there exists a unique Nashequilibrium when the (n + 1) firms compete in prices. The equilibrium is symmetric and isgiven by:
·

¸

µ

¶

(n + 1)(1 − ρ)n+1
p =c 1+
=c 1+
.
nρ
nλ
∗

6

(5)

For discussions of endogenous cost differences, see Bös and Peters (1995), Corneo and Rob (2003), Ishibashi

and Matsumura (2006), Matsumura and Matsushima (2004), and Nett (1993, 1994).

5

The equilibrium outputs are the same for all firms, and the output per firm x∗ is equal to:
x∗ =

n
ρ
.
n + 1 (n + 1 − ρ)c(6)

The total surplus of the variants is:
Π∗ =

n+1
X

µ

πi∗ = (n + 1)

i=1

¶

1−ρ
−F .
n+1−ρ

To remove the case of natural monopolies, we assume that:
F <

1−ρ
.
2−ρ

In equilibrium, consumer surplus is given by the indirect utility function:V∗ =1ln[(n + 1)(1−ρ) (p∗ )−ρ ] + Y − 1,ρ(7)where p∗ is given by (5). Finally, the social surplus evaluated at equilibrium is:W∗ = V ∗ +n+1Xαi πi∗ ,(8)i=1

where αi ∈ [0, 1] is the ownership share of domestic investors for firm i.
Now, we suppose that there is free entry and exit in the industry. This can be viewed
as the long-run equilibrium whereby no costs are sunk. The equilibrium number of firms is
obtained from the zero-profit condition:
ne =

1−ρ
+ ρ,
F

(9)

while the corresponding equilibrium price is:
pe =

3

c
.
ρ(1 − F )(10)

The effect of privatization in the short run

To consider the effects of privatization, we characterize the equilibrium in which one firm
is public and n firms are private. We let firm n + 1 be the public firm and look for a price6equilibrium at which the public firm maximizes social surplus. The government holds all sharesin firm n + 1, so αn+1 = 1. For simplicity, we assume that αi = α for all i ∈ {1, 2, ..., n}..
The first-order conditions for the private firms are identical so that all private firms charge
the same price p̂, which is different from the price p̃ elected by the public firm. The objectivefunction of the public firm is written as follows:
W = V (p̂, p̃) + αnπ̂(p̂, p̃) + π̃(p̂, p̃),

(11)

where V (p̂, p̃) is indirect utility, π̂(p̂, p̃) = (p̂ − c)x̂ − F is the profit of a private firm, andπ̃ = (p̃ − c)x̃ − F is the profit made by the public firm evaluated at prices (p̂, p̃).
To derive the equilibrium strategies of the private firms, we define that X̂ ≡ P̂ /p̂ andX̃ ≡ P̃ /p̃ with:
P̂ ≡

p̂−λ
p̃−λ
,
P̃
≡
.
np̂−λ + p̃−λnp̂−λ + p̃−λ(12)

Let us suppose that supplier i charges pi , the private firms, p̂, and the public firm, p̃. The
demand function for firm i can then be rewritten as X̂i = Pi /pi with Pi defined by:
Pi ≡

p−λ
i
, i = 1, . . . , n.
p−λ
+
(n
−
1)p̂−λ + p̃−λi(13)Differentiating firm i’s profit with respect to pi , setting pi equal to p̂ and equating to zero
yields the following implicit equation for p̂:"p̂ = c 1 +1λ(1 − P̂ )

#

,

(14)

where P̂ is Eq. (13) evaluated at pi = p̂.
The first-order condition of the public firm is:∂W∂ p̃

∂V (p̂, p̃)∂ π̂(p̂, p̃) ∂ π̃(p̂, p̃)
+ αn
+
∂ p̃
∂ p̃
∂ p̃
∂ x̃
∂ x̂
+ (p̃ − c)= −x̃ + x̃ + αn(p̂ − c)
∂ p̃
∂ p̃
∂ x̂
∂ x̃
= αn(p̂ − c)+ (p̃ − c)
= 0.
∂ p̃
∂ p̃
=

(· .· Roy’s identity)

From the first-order condition and Eq. (12), we have the following lemma:
7(15)

Lemma 1 The public firm sets its price, which is lower than that of the private firms.
Proof: From Eq. (12), we have:
∂ x̂
∂ p̃

=

λp̂−λ−1 p̃−λ−1
P̂ P̃
=λ
,
(np̂−λ + p̃−λ )2
p̂p̃

∂ x̃
∂ p̃

=

−P̃ (nλP̂ + 1)−p̃−2λ − n(λ + 1)p̃−λ p̂−λ=.
p̃2 (np̂−λ + p̃−λ )2p̃2

Substituting them into Eq. (15), we have:p̂ − cp̃ − c=p̂
p̃

Ã

1 + nλP̂
αnλP̂

!

.

This equation implies Lemma 1.

(16)
Q.E.D.

As mentioned in Anderson, de Palma, and Thisse (1997), the public firm takes into accountconsumer surplus, while the private firms do not. The public firm lowers its price to increase
consumer surplus.
We now compare the price configuration above to the equilibrium price in Eq. (5) in the
fully private oligopoly. From Lemma 1, we have the following lemma:
Lemma 2 The equilibrium price of the private firm is lower in mixed oligopoly than that inprivate oligopoly where all firms are private.
Proof: See Appendix.
We now discuss an effect of privatization of the public firm. Welfare after privatization is
given by: (8) by substituting αi = α for i ∈ {1, 2, ..., n} and αn+1 = 1.7 Lemmas 1 and 2
imply p∗ > p̂ > p̃ ≥ c. That is, when there is a public firm in a differentiated oligopoly, allfirms charge a lower price than they do in the pure market. As mentioned in Anderson, dePalma, and Thisse (1997), the allocation of consumers across variants is no longer optimal
because the price of a public firm is lower than that of private firms. An answer is given in
the following proposition.
7

If foreign investors rather than domestic investors purchase the privatized firm, all of our results hold true

as long as the privatized firm is correctly priced.

8

Proposition 1 Social surplus falls in the short run if the public firm is privatized.
Proof: See Appendix.
An implication of the proposition is that the existence of a public firm is always socially beneficial in a differentiated oligopoly with a fixed number of firms. A similar result is presented
by Anderson, de Palma, and Thisse (1997) in the case where α = 1.. Our Proposition 1 showsthat their result holds regardless of α.
We then take a closer look at the relationship between α and the effect of privatization of
the public firm. From Eqs. (14) and (16), we have the following lemma:
Lemma 3 As α increases, p̂ and p̃ increase.
Proof: See Appendix.
The value of α represents the relative importance of private firms’ profits from the viewpointof social welfare. For instance, if the value is zero, the objective of the public firm is tomaximize consumer surplus plus its own profit and then it sets its price equal to marginalcost. As the public firm takes into account the profits of the private firms (the value of α
increases), the price of the public firm increases.
From Proposition 1 and Lemma 3, we find that, as the value of α increases, the pricesof the private and public firms increase, and the profits of the firms increase. The former
increment is a negative effect from the viewpoint of social welfare. In the latter increment,
the increase in the domestic firms’ profits is larger than that in the mixed oligopoly casebecause each firm’s profit is larger than that in the mixed oligopoly (π̃ < π ∗ , see Lemmas 1
and 2). The discussion leads to the following proposition:
Proposition 2 As α increases, the welfare loss from privatization decreases.
Proof: See Appendix.
The public firm sets its price at a lower level than the private firms because it takes consumer
surplus into account. As the share of the foreign firms (1 − α) increases, the consumer surplus
becomes more important than the producer surplus, so the public firm cuts its price. Through9the strategic interaction between public and private firms, the prices of private firms fall. In
other words, the public firm’s pricing becomes tougher when (1 − α) is large, and it effectively
restricts the surplus belonging to the foreigner. Therefore, the difference between the twomarket configurations spreads, and we derive Proposition 2. This result states that the social
value of the public firm increases in 1 − α (the share of foreign ownership). So in the shortrun, foreign investments in the private sector make the privatization of the public firm lessbeneficial for domestic welfare. As we show in the next section, however, this does not holdin the long run. In the long run, foreign investments in the private sector increase the benefitof privatization of the public firm.

4

The effect of privatization in the long run

In this section, we consider the effect of privatization in the long run. Suppose that private
firms enter until they earn zero profits. Let ñ be the number of private firms at the free-entry
equilibrium involving one public firm. At this equilibrium, all private firms charge the same
price p̂e , while the public firm sets a price p̃e . Moreover, we have:
Lemma 4 At the free-entry equilibrium, the private firms charge the same price regardless ofthe presence of a public firm: p̂e = pe .
Proof:

Assume that there is a public firm in the market. For each private firm i, thefirst-order condition is:µ¶

1− ρx̂i = 0.
ρ − 1 + (pi − c)pi
Substituting x̂i into the zero-profit condition, we have the same price as Eq. (10).

Q.E.D.

Note that the numbers of firms are not the same in the two market configurations. Whenne firms set pe , the private firms make zero profits. From Lemmas 2 and 4, ñ + 1 < ne , thatis, fewer firms are in business when one of them is a public firm.
The private firms’ profits being zero by the free-entry condition, the social surplus at the10free-entry equilibrium with a public firm can be rewritten as:W̃ e = V (p̂e , p̃e ) + π̃ e (p̂e , p̃e ).
Because the social surplus at the free-entry equilibrium with private firms is given only by
W e = V (pe ), we have:µW̃ e − W e =¶1ln[ñ(x̂e )ρ + (x̃e )ρ ] + π̃ e (p̂e , p̃e ) −
ρ

µ

¶

1
ln[ne (xe )ρ ] ,ρ

(17)where x̂e and x̃e denote a private or a public firm’s demand at the preprivatization free-entry
equilibrium, while xe is a private firm’s demand at the postprivatization case.
Proposition 3 (i) At the free-entry equilibrium, the social gain from privatization is equal to
the loss made by the public firm before it is privatized, and (ii) privatization improves welfare
if α is sufficiently close to 0.
Because p̂e = pe by Lemma 4, the zero-profit condition implies that each private

Proof:

firm sells the same quantity: x̂e = xe . Accordingly, it follows from Eq. (3) that:p̂−λ−1p̂−λ−1=
ne p̂−λñp̂−λ + p̃−λ⇒ ne = ñ +µ e ¶−λ

p̃
p̂e

.

(18)

Using Eq. (3), we have:µ e ¶ρ
x̃

x̂e

Ã

=

p̃−λ−1p̂−λ−1!ρµ e ¶−λ

=

p̃
p̂e

.

Substituting it into Eq. (17), we have:ÃeW̃ − W

e

=
=

"

Ã

1
ln (x̂e )ρ ñ +ρµ e ¶−λ !#

p̃
p̂e

!
e

e

+ (p̃ − c)x̃ − F

µ

−

¶

1
ln[ne (xe )ρ ]
ρ11ln [(x̂e )ρ ne ] + (p̃e − c)x̃e − F − ln[ne (xe )ρ ]ρρ= (p̃e − c)x̃e − F, (. .. Eq.(18) and x̂e = xe ).
Therefore, Proposition 3(i) holds.
From (16), we have that p̃ is sufficiently close to c when the value of α is sufficiently close
to 0. Because F is positive, the public firm has a deficit. From Proposition 3(i), we have11

Proposition 3 (ii).

Q.E.D.

The second result contains rich implications. As opposed to the short run, privatization
improves welfare if the presence of foreign capital in the private sector is sufficiently large. At
first glance, under the presence of foreign private firms, the existence of a public firm enhances
welfare because the public firm lowers its price to enlarge consumer surplus and to reduce thewelfare loss, which stems from the profits of the foreign firms. This effect in fact works in the
short run (Proposition 2). In the long run, however, this is not true. In equilibrium, social
welfare is equal to consumer surplus plus the profit of the public firm because the profits ofprivate firms are zero (because of free entry). Consumer surplus depends on product diversity,that is, the number of firms is important for consumers. Because the public firm lowers its
price, the number of entrants is lower than the case where the public firm privatizes. As theshare held by foreigners increases, the loss of diversity induced by the smaller entry becomessignificant because the price of a public firm becomes lower. This is why privatization improves
welfare if α is close to zero.

5

Inefficiency of the public firm

In previous sections we ignore the cost difference between public and private firms. In reality,
a public enterprise often performs less efficiently than a private enterprise. In this section, we
assume that the public firm is less efficient. We assume that c̃ > c, where c̃ is the marginal
cost of the public firm.

5.1

Short run

Most of the calculus is similar to that in the former setting. We only show the difference
between these settings.
We first derive the first-order condition of the public firm. This is related to Eq. (15) andas follows:∂W∂ p̃= αn(p̂ − c)

∂ x̂
∂ x̃
+ (p̃ − c̃)
= 0.
∂ p̃
∂ p̃

12

(19)

As in the proof of Lemma 1, we can rewrite Eq. (19) as follows:p̂ − cp̃ − c̃=p̂
p̃

Ã

1 + nλP̂

!

.

αnλP̂

(20)

The first-order conditions of the private firms are presented in (14), that is,"p̂ = c 1 +1#λ(1 − P̂ )

.

(14’)

From (14’) and (20), we have the following Lemma:
Lemma 5 The price of the public firm p̃ is equal to those of the private firms p̂ if and only
if:
·

¸

(n + 1)(n + 1 + (1 − α)nλ)c ≡ cp .
c̃ = 1 +nλ(n + 1 + nλ)(22)p̃ > p̂ > p∗ , if and only if c̃ > cp . p̃ < p̂ < p∗ , if and only if c̃ < cp .
From (14’) and (20), we also have the following Lemma:
Lemma 6 Social welfare decreases as c̃ increases.
From Lemmas 5 and 6, we have the following proposition:
Proposition 4 Privatization of the public firm enhances social welfare if and only if c̃ > c̃∗(c < c̃∗ < cp ).
Proof: From lemma 5, when c̃ = cp , p̃ = p̂ = p∗ . If the public firm is privatized, then prices
do not change, but the efficiency of the public firm improves. We find that privatization
improves welfare when c̃ = cp . As shown in Proposition 1, privatization reduces welfare whenc̃ = c. From lemma 6, as the value of c̃ increases, social welfare decreases. There exists
the value of c̃ = c̃∗ in which privatization is indifferent from the viewpoint of social welfare.
Therefore, Proposition 4 holds.

Q.E.D.

13

5.2

Long runIn this subsection, we consider the effect of privatization in the long run when the public firm
is less efficient. Lemma 4 in the former section also holds in this case. Moreover, Proposition
3 also holds. The proof is made if we replace c with c̃ in the proof of Proposition 3. Obviously,
given the costs of private firms, an increase in the cost of the public firm reduces its profits;thus, under Proposition 3 privatization of the public firm is more likely to improve welfare
under cost difference between public and private firms.

6

Concluding remarks

In this paper, we extend the mixed oligopoly model of Anderson, de Palma, and Thisse (1997)
to a case with foreign competitors. We find that the social value of the public firm increases
under the presence of foreign competitors in the short run. On the contrary, in the long run,privatization of a public firm is more likely to improve welfare when the competitors of the
public firm are foreign than when they are domestic. It is noteworthy that, when all private
firms are owned by foreigners, privatization always improves welfare.
This result indicates that the time inconsistency problem becomes more serious when
private competitors are foreign. The government has an incentive to privatize the public
firm so as to stimulate new entry of foreign firms. After the entry, however, the government
has an incentive to nationalize the privatized firm again. Expecting this renationalization,
the entry of foreign firms is restricted. The market integration increases the possibility ofinternational competition, and public firms are more likely to face competition against foreign
firms. Under these conditions, it is more important to make a commitment not to renationalize
the privatized firm.
In this paper, we assume that the public firm is a welfare maximizer. Although this assumption is very popular among the models of mixed oligopoly,8 other approaches also exist.9
8See, among others, De Fraja and Delbono (1989), Fjell and Pal (1996), Pal (1998), and Matsumura (1998).

9

See, e.g., Fershtman (1990), Futagami (1999), and Sappington and Sidak (2003).

14

Deviation from this welfare-maximizing assumption and the application of other approaches
to this problem remain topics for future research.
[2007.1.9 792]

15

Appendix
Proof of Lemma 2:

From Lemma 1, p̃ is smaller than p̂. We rearrange P̂ in Eq. (12):
P̂ =

1
1
1
=
<
.
−λ
λ
n + (p̃/p̂)n + (p̂/p̃)
n+1

Using the inequality, we rearrange p̂ in Eq. (14):"p̂ = c 1 +#1λ(1 − P̂ )

·

<c 1+

¸

·

¸

1
1+n
=c 1+
= p∗ .
λ(1 − 1/(n + 1))
λnp̂ in the implicit equation in Eq. (14) is smaller than p∗ .

Q.E.D.

We now show that social welfare is decreasing with respect to

Proof of Proposition 1:

p̂. If this is true, from Lemma 2, the proposition holds. Differentiating W in Eq. (11) withrespect to p̂, we have:
∂W
∂ p̂

=

∂V
∂ π̂ ∂ π̃
+ αn
+
∂ p̂
∂ p̂
∂ p̂

∂ x̂
∂ x̃
+ (p̃ − c)
∂ p̂
∂ p̂
∂ x̂
∂ x̃
= −(1 − α)nx̂ + αn(p̂ − c)+ (p̃ − c) .
∂ p̂
∂ p̂
= −nx̂ + αnx̂ + αn(p̂ − c)

Because p̃ maximizes W in Eq. (11), Eq. (15) holds, so that:
∂W
∂ p̂

∂ x̂
∂ x̂ ∂ x̃ ∂ x̃
− αn(p̂ − c)/∂ p̂∂ p̃ ∂ p̂ ∂ p̃·¸
αn(p̂ − c) ∂ x̂ ∂ x̃ ∂ x̂ ∂ x̃−∂ x̃/∂ p̃∂ p̂ ∂ p̃∂ p̃ ∂ p̂= −(1 − α)nx̂ + αn(p̂ − c)<"=αn(p̂ − c)(λ + 1)P̂ P̃
−(n(λ + 1)p̃−λ−2 p̂−λ + p̃−2λ−2 )/(np−λ + p̃−λ )2p̂2 p̃2

Therefore, Proposition 1 holds.
Proof of Lemma 3:#< 0.
Q.E.D.

We now show that dp̂/dα > 0 and dp̃/dα > 0. p̂ and p̃ are determinedby Eqs. (14) and (16). When α changes, p̂ and p̃ also change. To identify the changes, we
differentiate Eqs. (14) and (16).
16The total differentiation of Eq. (14) is:"#λP̂ P̃ dp̃
λ(1 − P̂ )2 dp̂
λP̂ P̃
=
+
.
p̃ dα
p̂
c
dα

(23)

The coefficients of dp̃/dα and dp̂/dα are positive. Their signs are the same.
The total differentiation of Eq. (16) is:λ(p̂ − c)P̃ − c(1 + nλP̂ ) dp̂
c(1 + nλP̂ ) − λ(p̃ − c)P̃ dp̃
p̂ − c
+
=
.
2
2
dα
dα
αp̂
p̂ (1 + nλP̂ )p̃ αnλP̂

(24)

Substituting dp̃/dα in Eq. (23) into the left-hand side of Eq. (24), we have:"λ(p̂ − c)P̃ − c(1 + nλP̂ )

"

>

+

p̂2 (1 + nλP̂ )−1p̂2 (1 + nλP̂ )+Ã1p̃αnλP̂c(1 + nλP̂ ) − λ(p̃ − c)P̃p̃2 αnλP̂
1+p̂

λ(1 − P̂ )2

!#

·

p̃

Ã

λP̂ P̃

λP̂ P̃
λ(1 − P̂ )2
+
p̂
c

!#

dp̂
dα

dp̂
dα

cP̂ P̃

· ·( . c(1 + nλP̂ ) − λ(p̃ − c)P̃ > c(1 + nλP̂ ) − λ(p̂ − c)P̃ )
=

1

"

Ã

"

Ã

1
1 1 + nλP̂− 2+ ·p̂
p̃ αnλP̂
1 + nλP̂

1 λ(1 − P̂ )2+p̂
cP̂ P̃

!#

!#

dp̂
dα

=

1
1 p̃(p̂ − c)
− 2+ ·p̂p̃ (p̃ − c)p̂
1 + nλP̂=(p̃ − c)(p̂ − c)(p̂ − c) λ(1 − P̂ )2 dp̂
·−++(p̃ − c)p̂2 (p̃ − c)p̂2 (p̃ − c)p̂dα
1 + nλP̂
cP̂ P̃=1

1 λ(1 − P̂ )2+p̂
cP̂ P̃

1
1

1 + nλP̂""

dp̂ · ·
( . Eq. (16))
dα##p̂ − p̃(p̂ − c) λ(1 − P̂ )2 dp̂
·.
+(p̃ − c)p̂2 (p̃ − c)p̂dα
cP̂ P̃

The coefficient of dp̂/dα is positive. The right-hand side of Eq. (24) is positive. Therefore,dp̂/dα is positive. We also find that dp̃/dα is positive.
Proof of Proposition 2:

Q.E.D.

Let W ∗ and W̃ be the welfare after and before privatization,

respectively.
dW̃
∂ W̃ dp̂∂ W̃ dp̃
=
+
+ nπ̃
dα
∂ p̂ dα∂ p̃ dαdW ∗= nπ ∗ .
dα

(25)
(26)

From Proposition 1 and Lemma 3, we have that the first and second terms in (25) are negative.
Because competition is more aggressive before privatization (see Lemmas 1 and 2), it is obviousthat π̃ < π ∗ . Under these conditions, (25) minus (26) is negative.
17

Q.E.D.

Proof of Lemma 5:

When p̂ = p̃, P̂ = 1/(n + 1) (see Eq. (12)). Taking this property into

account, we solve Eq. (20) and then have Eq. (22).
Before we show the proof of the rest of the Lemma, we provide several properties related
to the proof. P̂ in Eq. (12) is larger than 1/(n + 1) if and only if p̃ > p̂. From Eqs. (5) and(14’), we find that if p̃ > p̂, then p̂ > p∗ , otherwise p̃ < p̂ < p∗ .
We now show that if c̃ < cp , then p̃ < p̂ < p∗ . We employ the proof by refutation. Supposethat p̂ > p∗ when c̃ < cp . We arrange Eq. (20) as follows:H ≡p̂ − c p̃ − c̃
−
p̂
p̃

Ã

1 + nλP̂
αnλP̂

!

.

If H is negative, the public firm sets a lower price. When the public firm sets its price atp̃ = p̂, H is:
µ

H =
<
<

¶

p̂ − c p̂ − c̃ 1 + nλ/(n + 1)−p̂p̂
αnλ/(n + 1)µ¶
pp̂ − c p̂ − c 1 + nλ/(n + 1)−(· .· c̃ < cp )p̂p̂αnλ/(n + 1)µ¶p∗ − c p∗ − cp 1 + nλ/(n + 1)(· .· p∗ < p̂)−p∗p∗αnλ/(n + 1)

= 0.
Because H < 0, the public firm sets its price at p̃ < p̂. This contradicts the fact that if p̃ < p̂,p̂ < p∗ . Therefore, when c̃ < cp , p∗ > p̂ > p̃.
Proof of Lemma 6:

Q.E.D.

We first show that as the value of c̃ increases, the values of p̂ and p̃increase. Total differentials of the first-order conditions of the public and the private firms

18are:10
1
p̂

Ã

!

´
αnλP̂ c λP̃ (p̃ − c̃)
1 ³
1 + nλP̂−
dp̂ + 2 λP̃ (p̃ − c̃) − c̃(1 + nλP̂ ) dp̃ +dc̃ = 0,
p̂
p̃
p̃
p̃
|

³

{z

}

the S.O.C of the public firm´p̃ (1 − P̂ )2 p̂ + cP̂ P̃ dp̂ − cP̂ P̃ p̂ dp̃ = 0.
Substituting the second equation into the first, we have the coefficient of dp̂ in the firstequation as follows:
1cP̂ P̃ p̂ ·p̂Ã

= cP̂ P̃Ã

αnλP̂ c λP̃ (p̃ − c̃)−p̂p̃

αnλP̂ c c̃(1 + nλP̂ )−p̂p̃= −cP̂ P̃ (1 + (1 − α)nλP̂ ) −

!

!

−

+

(λP̃ (p̃ − c̃) − c̃(1 + nλP̂ ))((1 − P̂ )2 p̂ + cP̂ P̃ )p̃[c̃(1 + nλP̂ ) − λP̃ (p̃ − c̃)](1 − P̂ )2 p̂p̃[c̃(1 + nλP̂ ) − λP̃ (p̃ − c̃)](1 − P̂ )2 p̂
< 0.
p̃

Because the coefficient of dc̃ is positive, p̃ and p̂ increase as the value of c̃ increases.
We now show that social welfare decreases as c̃ increases.
First, we consider the relationship between social welfare, p̃, and c̃. We now suppose thatc̃ increases from cl to ch and that p̃l (resp. p̃h ) is the optimal value when c̃ = cl (resp. c̃ = ch ).
Denote social welfare as W (p̃, c̃). We can easily derive the following inequalities:
W (p̃l , cl ) > W (p̃h , cl ) > W (p̃h , ch ).
The first inequality holds because p̃l is the optimal price when c̃ = cl and the second holds
because of the efficiency of the public firm. Therefore, the increment in c̃ enhances p̃ and thenreduces welfare.
10

coefficient of dp̂

nλP̂
p̂

=

µ

αc
− λP̃
p̂

µ

p̂ − cp̃ − c̃
α
−
p̂
p̃

¶¶

Using Eq. (20), we arrange the equation:coefficient of dp̂

=

nλP̂
p̂

=

1
p̂

µ

µ

αc
− λP̃
p̂

µ

p̃ − c̃nλP̂ p̃1

λP̃ (p̃ − c̃)
αnλP̂ c
−
p̂
p̃

19

¶

.

¶¶

.

Second, we consider the relationship between social welfare, p̂, and c̃. As shown in the
proof of Proposition 1, social welfare decreases as p̂ increases. As the value of c̃ increases, thevalue of p̂ increases. Therefore, the increment in c̃ enhances p̂ and then reduces welfare.
From the two effects related to the increment in c̃, we find that social welfare decreases as
the value of c̃ increases.

Q.E.D.

20

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