Developing a two-country model of international mixed oligopoly, this note makes clear the determinant of trade patterns. We give a simple formula to predict bilateral patterns of trade which relates the degree of a country's privatization and the trading country''s competitiveness. If a semi-public firm is not sufficiently privatized in a country, this country exports the non-competitive good.

Kenji Fujiwara

English

Research Papers in Economics

Trade Patterns in an International Mixed Oligopoly 
Kenji Fujiwara
School of Economics, Kwansei Gakuin University
Abstract
Developing a two-country model of international mixed oligopoly, this note makes clear the
determinant of trade patterns. We give a simple formula to predict bilateral patterns of trade
which relates the degree of a country's privatization and the trading country's
competitiveness. If a semi-public firm is not sufficiently privatized in a country, this country
exports the non-competitive good.
Citation: Fujiwara, Kenji, (2006) "Trade Patterns in an International Mixed Oligopoly." Economics Bulletin, Vol. 6, No. 9 pp.
1-7
Submitted: July 28, 2006.  Accepted: September 1, 2006.
URL: http://economicsbulletin.vanderbilt.edu/2006/volume6/EB-06F10016A.pdf1 Introduction
Along recent movements of multilateral and regional trade liberalization, privatization
of public rms has also been an important policy debate. It is commonly recognized
that privatizing public rms improves eciency and welfare in an economy and hence is
regarded as one of the strong policy instruments. Indeed, privatization of public rms
has been in place around the world such as in China and Japan.
Given these observations and facts, there has been an accumulating literature on the
welfare analysis of privatization. The basic framework to do it is a model of mixed
oligopoly in which a public rm competes with several private rms. In a seminal
work, Matsumura (1998) and Matsumura and Kanda (2005) construct a model of mixed
oligopoly within which the public rm maximizes a weighted sum of social welfare and
prots, whereas private rms behave as a prot maximizer. Thus, an increase in the
weight on the public rm's prot is regarded as privatization. Mastsumura's (1998) con-
tribution lies in his conclusion that neither full privatization nor full nationalization is
the best policy, i.e., only partial privatization achieves social optimum.
Matsumura's (1998) model has recently been applied to incorporate foreign competi-
tors by a few papers.1 Chang (2005) and Chao and Yu (2006) extend the Matsumura
model to a segmented market model of international trade and compute the optimum
tari in the presence of privatization. These extensions are rather natural in view of a
large literature on optimal trade policies under oligopoly.
On the other hand, this paper constructs a two-country model of international oligopoly
to consider a positive aspect. In particular, we conne our interest to the determination
of trade patterns in an international oligopoly in which one public rm in a country com-
petes with private rms in the trading country. We shall demonstrate that the relative
degree of privatization to the number of foreign rms plays a decisive role.
The distinction between our model and that of Chang (2005) and Chao and Yu (2006)
is now briey made clear. First, while they assume that the home country's market is
segmented from the foreign market, they are completely integrated in our treatment.
Second, the above predecessors ignore the possibility of free entry among the private
rms. On the contrary, taking into account that free entry and exit make any private
rm's prot zero, we seek the direction of bilateral trade.
The rest of the paper is organized as follows. Section 2 sets out a basic model of
international mixed oligopoly and describes the world equilibrium. Section 3 makes use
of this model to determine the short-run trade pattern such that all rms earn positive
prots. Then, allowing for the long-run equilibrium with free entry among private rms,
Section 4 further explores the determinant of trade. Section 5 concludes the paper.
1Pal and White (1998) also develop a model of international mixed oligopoly, but they allow for no
possibility of partial privatization.
22 A Model
The model employed is a simple two-country oligopolistic model of international trade.
Two countries, Home and Foreign, are engaging in free trade and share the identical pref-
erence. In what follows, all Foreign variables are distinguished by attaching an asterisk
(*) to them. Both countries produce two goods, an oligopolized good and a competitive
good, which serves as a numeraire. Both goods are produced from only one primary
factor, labor, which is fully employed and in inelastic supply. Without loss of general-
ity, one unit of labor produces the same unit of numeraire, from which the wage rate is
internationally equalized and xed to unity.
On the other hand, the production of oligopolized good is subject to increasing returns.
Concretely, its cost function is specied by
cx + f; c > 0; f > 0;
where x is the output of non-competitive good. That is, the cost for producing x units of
non-competitive good requires cx units of variable input and f units of xed input. This
technology is shared by all oligopolistic rms irrespective its location.
Let us turn to the specication of the demand side. Each country's representative
consumer has the following utility function:
u = aq  
q2
2
+ z; a > 0; (1)
where u is the utility level, q the consumption of oligopolized good, and z that of nu-
meraire. Due to the quasi-linearity of preference, utility maximization subject to the
budget constraint yields the demand function of non-competitive good as:
q = a   p:
Exactly the same argument applies to the Foreign consumer. Thus, assuming that there
is one public rm in Home and n  2 private rms in Foreign, the world market-clearing
condition under free trade becomes
q + q
 = 2(a   p) = x +
X
x

j;
where the summation in Foreign is taken in the closed interval [1;n]. Inverting this
condition, the inverse demand function in the integrated world market is obtained as
p = a  
x +
P
x
j
2
: (2)
Making use of the above preliminary results, each country's consumer surplus is com-
puted as a function of total supply:
CS  aq  
q2
2
  pq
=
1
8

x +
X
x

j
2
: (3)
3And each rm's prot is now dened as follows.
 
 
a   c  
x +
P
x
j
2
!
x   f (4)


i 
 
a   c  
x +
P
x
j
2
!
x

i   f: (5)
Following Matsumura (1998) and Matsumura and Kanda (2005), the Home public
rm is presumed to maximize the weighted sum of prot and domestic welfare. This
implies that the public rm seeks to maximize
 + (1   )(CS + )
=  + (1   )CS;  2 [0;1]; (6)
where  = 1 (resp.  = 0) means full privatization (resp. nationalization) of the public
rm.
Invoking that all the Foreign rms maximize prots by choosing its output, each rm's
rst-order condition for objective maximization is derived as
a   c  
 + 3
4
x  
 + 1
4
x
 = 0 (7)
a   c  
1
2
x  
n + 1
2
x
 = 0: (8)
This completes the introduction of the basic setup.
3 Short-Run Trade Patterns
On the basis of the previous section's model, we are ready to establish the trade pattern
proposition. First of all, we pay attention to the short-run equilibrium such that neither
entry nor exit in the oligopolsitic industry is prohibited. Then, the Nash equilibrium
outputs are obtained from (7) and (8):
x
N =
2[n(1   ) + 2](a   c)
2(n + 1) +  + 1
(9)
x
N =
2( + 1)(a   c)
2(n + 1) +  + 1
: (10)
Since the quasi-linear preference has been assumed, Home exports the non-competitive
good if and only if its aggregate output exceeds Foreign's aggregate output, i.e., x > nx.
Combining this condition with (9) and (10), we have now established the rst main
proposition:
4Proposition 1. Suppose neither entry nor exit in the oligopolistic market. Then, Home
exports the non-competitive good if and only if  < 1=n.
Proof. As mentioned earlier, Home becomes an exporter of the oligopolistic good if and
only if x > nx. Substituting (9) and (10) into this inequality and rearranging, one can
obtain
n
(1   ) + 2 > n
(1   );
which immediately yields  < 1=n. Q. E. D.
Proposition 1 may give a very useful criterion for predicting the trade pattern. Only a
simple comparison between the degree of privatization in Home and the competitiveness,
i.e., the inverse of the number of private rms, in Foreign enables us to see the trade
pattern. According to it, Home tends to export the oligopolized good either if Home's
public rm is insuciently privatized or if Foreign's competitiveness measured by 1=n
is not so strong, or both hold.
4 Long-Run Trade Patterns
In the foregoing section, we have focused on the short-run equilibrium in which all
oligopolistic rms earn positive prots due to no entry into the oligopolistic sector. How-
ever, it is no wonder that at least any private rm earns zero prot in the long-run since
free entry makes each private rm's prot driven to zero.2 This section is devoted to
describing such a long-run outcome and deriving the trade pattern there.
To this end, let us calculate the equilibrium prot of each private rm. Summing up
(9) and (10), the world supply of non-competitive good is
n + n
x
 =
4(n + 1)(a   c)
2(n + 1) +  + 1
: (11)
Substituting (11) into the denition of the Foreign rm's prot (5) yields its equilibrium
prot:

 =
"
a   c  
2(n + 1)(a   c)
2(n + 1) +  + 1
#
2( + 1)(a   c)
2(n + 1) +  + 1
  f
= 2
"
( + 1)(a   c)
2(n + 1) +  + 1
#2
  f: (12)
2Matsumura and Kanda (2005) also characterizes such a long-run equilibrium in a closed economy
context.
5Setting (12) to zero and solving for n, the free entry number of private rms is
explicitly derived as
n
 =
( + 1)
h
(a   c)
q
2
f   1
i
  2
2
: (13)
With the help of the information so far, we can establish:
Proposition 2. Suppose free entry in Foreign's oligopolistic industry such that any
Foreign rm's prot is driven to zero. Then, the necessary and sucient condition for
Home to export the non-competitive good is
 <
2
(a   c)
q
2
f   1
(14)
Proof. Our starting point is the condition of  < 1=n or equivalently n < 1 in order
for Home to export the oligopolized good. Substituting (13) into this condition, we have
n
 =
( + 1)
h
(a   c)
q
2
f   1
i
  2
2
< 1;
which is simplied to (14). Q. E. D.
The above proposition sheds light on the importance of increasing returns for the
determination of trade patterns. To understand this, suppose that f = 2(a   c)2. Then,
the right-hand side in (14) diverges to innity and hence Home necessarily exports the
oligopolized good. Proposition 2 also provides a simple formula to determine the trade
pattern since all we need to do is the comparison between the degree of privatization ()
and the index which can be easily calculated by substituting the level of xed costs (f).
5 Concluding remarks
The purpose of this paper has been to apply a model of international mixed oligopoly
to a positive aspect of international trade with particular attention to trade patterns.
Assuming no cost asymmetries between countries, the interaction between the degree
of privatization in Home and the number of rms, i.e., competitiveness, in Foreign has
played a key role.
Proposition 1 gives a simple criterion for determining trade patterns. If Home's degree
of privatization is less than the inverse of the number of rms in Foreign, Home exports
the oligopolized good and imports the numeraire good. While Proposition 1 rests on
the assumption of the xed number of rms, Proposition 2 relaxes it and reconsiders the
determinant of trade patterns in a free entry equilibrium. If the xed cost is suciently
6large, i.e., under large economies of scale, Home is likely to export the non-competitive
good. These results well t into some reality of modern trade between the developed and
developing countries such as the United States and China.
We now give a possible direction of future research. As mentioned above, we allow
for no cost heterogeneity between countries, which seems unacceptable in view of the
modern world economy. Thus, it is needed to incorporate such a asymmetric cost into
the present model and reformulate an oligopolistic Ricardian model to evaluate the trade
patterns in terms of technology.
References
[1] Chao, C. and E. S. H. Yu (2006), \Partial privatization, foreign competition, and
optimum tari", Review of International Economics, 14, 87-92.
[2] Chang, W. W. (2005), \Optimal trade and privatization policies in an international
duopoly with cost asymmetry", Journal of International Trade and Economic Devel-
opment, 14, 19-42.
[3] Matsumura, T. (1998), \Partial privatization in mixed duopoly", Journal of Public
Economics, 70, 473-483.
[4] Matsumura, T. and O. Kanda (2005), \Mixed oligopoly at free entry markets", Jour-
nal of Economics, 84, 27-48.
[5] Pal, D. and M. D. White (1998), \Mixed oligopoly, privatization, and strategic trade
policy", Southern Economic Journal, 65, 264-281.
7

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Mixed oligopoly, privatization, and strategic trade policy&quot;,

Optimal trade and privatization policies in an international duopoly with cost asymmetry&quot;,

Partial privatization in mixed duopoly&quot;,

Partial privatization, foreign competition, and optimum tari&quot;,

Trade Patterns in an International Mixed Oligopoly

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Trade Patterns in an International Mixed Oligopoly

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