Discussion Paper No. 864

PORT PRIVATIZATIONIN AN INTERNATIONAL OLIGOPOLY

Noriaki Matsushima
Kazuhiro Takauchi

February 2013

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Port privatization in an international oligopoly∗

Noriaki Matsushima†
Institute of Social and Economic Research, Osaka University
Kazuhiro Takauchi‡
Faculty of Economics, Management and Information Science,
Onomichi City University
February 10, 2013

Abstract
We investigate how port privatization affects port charges, firm profits, and welfare. Ourmodel consists of an international duopoly with two ports and two markets. When the unittransport cost is large, privatization of ports decreases the prices for port usage, althoughneither government has an incentive to privatize its port. The equilibrium governmental
decisions are inconsistent with the desirable outcome if the unit transport cost is not large
enough. The smaller country’s government is more likely to privatize its port, although thelarger country’s government is more likely to nationalize its port to protect its domestic
market.

Key words: Port; Privatization; Port charge; Oligopoly; Strategic trade policy.
JEL codes: L33; F12; R48; F13.
∗

The first author gratefully acknowledges financial support from a Grant-in-Aid for Basic Research(24530248, 24530285) from the Japan Society for the Promotion of Science. Needless to say, we are responsible
for any remaining errors.
†
Corresponding author: Noriaki Matsushima, Institute of Social and Economic Research, Osaka University,
Mihogaoka 6-1, Ibaraki, Osaka 567-0047, Japan. Phone: +81-6-6879-8571, Fax: +81-6-6879-8583, E-mail:nmatsush@iser.osaka-u.ac.jp‡
Kazuhiro Takauchi, Faculty of Economics, Management and Information Science, Onomichi City University,
1600-2 Hisayamada-cho, Onomichi-city, Hiroshima 722-8506, Japan.

11Introduction

Ports are an important factor in international trade because they directly influence trade costs
through cargo handling charges. About 90% of the international trade by the European Unionis performed by sea (González and Trujillo, 2008). The main trade costs include port tariffs,
cargo handling charges, and transit time in ports (Trujillo and Nombela, 1999; Limão and
Venables, 2001; Tongzon and Heng, 2005). The percentage of the cargo handling charge over
the total bill is about 70-90% (Trujillo and Nombela, 1999). That is, this charge is a very
important factor in international trade.
Recognizing the importance of port charges in international trade, practitioners and researchers have recently taken a closer look at the important determinants of port charges. For
decades, privatization policy has been recognized as one of the most important determinants ofport charges because the ownership structure of ports influences their efficiency and how ports’charges are determined. Following the example of the UK, many countries have moved—orare moving—toward the privatization of some public ports. In the UK, many major ports,
including Grimsby, Immingham and Teesport are company ports created by their own Act ofParliament (Baird and Valentine, 2007). Furthermore, port practitioners have provided manysound reasons for the move toward the privatization of public ports. The typical reasons are efficiency improvement and trade expansion (Baird, 2002). In fact, several empirical studies haveshown that port privatization improves efficiency and increases port trade volumes (Trujillo
and Nombela, 1999). As those practitioners implicitly believe, port privatization drasticallychanges port objectives. We can expect that the changes in port objectives through privatization will influence the port charges levied on many exporters through payments to shippers
that use ports. In other words, privatization influences the activities of exporters through thechanges in their payments to shippers who are intermediaries of those exporters. We therefore investigate how port privatization affects firm profits and welfare through changes in port
objectives. We illustrate how the difference between the objectives under the two different
organization regimes, privatization and nationalization, affects port charges, firm profits, and2social welfare.
We formulate a simple model that describes the effect of port privatization on social welfare.
The model is based on a two-country setting as in Brander and Krugman (1983). The firmin a country supplies its product to both countries, which implies that the two firms compete
in each country. The objective of a public port is to maximize domestic welfare, whereasthat of a private port is to maximize its own profit.1 We consider the following three-stage
game. First, each government independently decides whether to privatize its port. Second,
the ports independently set their port charges to maximize their objectives. Finally, the two
firms compete in quantity in the two countries simultaneously.
We show that depending on the per-unit transport cost, several types of organization mode
can appear in equilibrium. When the per-unit transport cost is low enough, two outcomes canappear in equilibrium: both ports are privatized or no port is privatized. When the per-unit
transport cost is intermediate, both ports are privatized in equilibrium. When the per-unit
transport cost is large, no port is privatized in equilibrium, although the port charge would
be lowered by the privatization of ports. This result implies that a governmental decision on
the ownership structure of ports in its country can be a strategic tool to control trade volume.
This provides a new insight in the context of transportation research and strategic trade policy.
We further investigate the welfare property of the equilibrium outcome and how market
size asymmetry affects governmental decisions concerning the organization modes of ports. We
show that the equilibrium governmental decisions are inconsistent with the socially desirable
organization modes of ports if the per-unit transport cost is not large enough. That is, thetwo governments are in a prisoners’ dilemma situation. We also show that the government
in the large country is more likely to nationalize its port to protect its domestic market.
Finally, we briefly discuss a case in which each port engages in cost-reducing activities after
the organization structure of each port is determined by each government.
1This follows the model assumption in Zhang and Zhang (2003), Mantin (2012), and Matsumura andMatsushima (2012), who discuss airport privatization.

3

Theoretically, our paper is closely related to two papers that consider the privatization
policy of airports in the context of airline competition (Mantin, 2012; Matsumura and Matsushima, 2012).2 They consider the following model setting. There are two airlines that supply
final products to passengers. Each airline company uses two airports to supply travel services.
Each airport monopolistically sets its airport charge. Because the consumer market consistsby the international flight services, the number of markets is one in these models. They show
the possibility that airport privatization appears in equilibrium. The driving force behind
their results is the rent-shifting effect of privatization, which our paper partially shares with
theirs. However, these studies do not consider reciprocal trade models, which implies thatthey do not capture the nature of strategic trade policy. Our paper is also related to thepapers that emphasize the market power of agents related to the international transport ofproducts (Francois and Wooton, 2001; Behrens and Picard, 2011; Kleinert and Spies, 2011).
Those papers mainly focus on shippers’ market power, but they do not explicitly consider theactivities of ports, which are gatekeepers for imports and exports.
Czerny et al. (2011) also discuss the effect of port privatization in a Hotelling spatialcompetition model with local markets (Takahashi, 2004). Two ports compete in a third
country market consisted by a Hotelling line. Each port also has a monopoly market in
its own country. The port owners compete in port charge for the usage of their ports, which
implies that the ports are substitute in the third country market and that the port charges are
strategic complements. In Czerny et al. (2011), because a nationalized port takes into accountthe consumer surplus in its country as well as its own profit, it is more likely to set a lower
port charge, which accelerates the competition between the two ports. Because a privatized
port does not take into account the consumer surplus in its country, privatization works as
a commitment not to set a lower port charge, which mitigate the competition between theports. This is why privatization can appear in their model. We believe that our paper is a2

Our paper is also related to the context of airline competition (Basso, 2008; Basso and Zhang, 2008; Haskel

et al., 2013; Zhang and Zhang, 2003, 2006). Zhang and Czerny (2012) provide an excellent survey concerning
recent research on airline and airport competition.

4

complement to Czerny et al. (2011).
The remainder of the paper is as follows. Section 2 presents the model. Section 3 provides
the main result. Section 4 extends the basic model. Section 5 concludes the paper. All proofs
are available in the Appendix.

2

Model

We consider a two-way oligopolistic trade model with two ports. There are two countries,
Home and Foreign, in the world. Each country has a port and a homogeneous manufacturing
firm (henceforth, we call it a firm). Each firm supplies its product to both markets. When afirm exports, it must use the two ports with payments and incur a per-unit shipping fee forcompetitive shippers. For simplification, no firm incurs any costs except the costs associatedwith transportation.
The inverse market demands in Home (H) and Foreign (F ) are1
pH = a − (qHH + qF H ), pF = a − (qF F + qHF ),bwhere pi is the price in country i, qji denotes the quantity supplied by a firm in j in market i(i, j = H, F ), and b is the market size of F compared to that of H. Without loss of generality,
we assume that b ≤ 1, that is, the market size in F is smaller than or equal to that in H.
Given the quantities supplied by the firms, the consumer surplus in i is given as

CSH =(qHH + qF H )2
(qF F + qHF )2, CSF =.
22b

τ (> 0) is the per-unit shipping fee and fi is the per-unit fee for the usage of the port
in i (i = H, F ).3 The owner of the port in i maximizes its objective, which depends onwhether its ownership structure is private or public (we explain this later). The total per-unit
3

Hummels and Skiba (2004) empirically show that the transport price is not iceberg type but per-unit type.

Hence, this setting is natural.

5

transportation cost, t, ist ≡ τ + fH + fF .
The marginal cost of firm i in i is zero, but that is t in j (i, j = H, L, i 6= j).
The profit of firm i (Πi ) is the sum of the profits in the two markets, πii = pi qii and
πij = (pj − t)qij ; that is,Πi ≡ πii + πij = pi qii + (pj − t)qij .
The owner of the port in i does not incur any operation cost. This port earns its profit
from the export from i to j and the import from j to i. The total profit of the port in i isgiven asRi ≡ fi (qij + qji ) = fi (qHF + qF H ).
There are two scenarios related to the objective of each port owner. In the first, it maximizes its own profit; in the second, it maximizes its domestic surplus. The former represents a
privatized port and the latter a nationalized port. When the owner of the port in i is a profit
maximizer, it maximizes Ri . When the owner of the port in i is a domestic welfare maximizer,
it maximizes
Wi = Ri + Πi + CSi .

(1)

This specification is used in the literature on airport privatization (e.g., Zhang and Zhang,2003; Basso and Zhang, 2008).
We consider the following four cases: I. Both owners are private; II. Only the owner of theport in H is private; III. Only the owner of the port in F is private; and IV. Both owners are
public.
The timing of the game is as follows: first, each government simultaneously determines theownership structure of the port. Second, each port simultaneously sets the fee fi to maximize
its objective which is determined in the first stage. Third, the firms simultaneously determine
their quantities in the two markets. We solve the game by backward induction.

6

3

Results

In this section, we derive the main results of this paper. To clarify the analysis, in this section,
we consider the case in which the market size in the two countries is symmetric, that is,b = 1. In the next section, we briefly discuss how market size asymmetry affects the ownership
structures of the ports.
The third stage

In the third stage, given the fee levels fH and fF , each firm simultaneously

sets its quantities in the two markets. As in the standard Cournot duopoly outcome, theequilibrium outcome in the third stage is given asqii (fH , fF ) =a + τ + fH + fF
a − 2(τ + fH + fF ); qji (fH , fF ) =
.
3
3

(2)

Substituting these into Πi , Ri , and CSi , we have Πi (fH , fF ), Ri (fH , fF ), and CSi (fH , fF ) in
the third stage.
The second stage

To discuss the outcome in the second stage, we derive the two reaction

functions of the port in i in the two scenarios: private and public.
When the port in i is private, the maximization problem is given by

max Ri (fH , fF ).
fi

From the problem, we have the reaction function of the port in i:fiP (fj ) =a − 2τ − 2fj
,
4

(3)

where the superscript P represents the case of privatization. This is negatively correlated toτ and fj . Each port earns its profit through the transaction of international shipments, which
are negatively correlated to the shipping fee (τ ): the number of international transactions
decreases as the shipping fee increases. The port fees are also negatively correlated with the7international transactions. A port must lower its port fee, given that another port sets a higher
fee, which decreases transactions.
When the port in i is public, the maximization problem is given by

max Ri (fH , fF ) + Πi (fH , fF ) + CSi (fH , fF ).
fi

From the problem, we have the reaction function of the port in i:
fiN (fj ) =2a − τ − fj
,
13

(4)

where the superscript N represents the case of nationalization. To understand the difference between the two reaction functions, we check the partial derivatives of Πi (fH , fF ) andCSi (fH , fF ) with respect to fi :∂πii (fH , fF ) ∂πij (fH , fF )∂Πi (fH , fF )2(a − 5(τ + fH + fF ))
=
,
+
=−
∂fi
∂fi
∂fi
9
|
{z
} |
{z
}
(+)

(5)

(−)

2(2a − (τ + fH + fF ))∂CSi (fH , fF )
=−
< 0.
∂fi
9

(6)

In the first equation, the positive sign represents the import prevention effect in market i andthe negative sign represents the export prevention effect in market j. Rises in consumer prices
through the increase in fi generate the two effects. The effect in each market is stronger as
the quantity supplied in the market becomes larger because the effect of a price increase isapplied to the total quantity supplied in each market. The shipping fee enhances the importprevention effect but diminishes the export prevention effect. This implies that the former
effect dominates the latter one when the shipping fee is large. In fact, the partial derivativesof Πi (fH , fF ) are positive if the transportation cost, which is the sum of the transportation
costs, is larger than a/5. In the second equation, a rise in the port fee only increases themarginal cost of the exporter, which reduces the consumer surplus. Because the marginal8

effect of fi on the consumer surplus is positively correlated with supply size, a higher shippingfee (τ ) weakens the negative effect of the increase in fi . To sum up, because of the effect of fi
on the profit and the consumer surplus, the shipping fee does not substantially influence thefee schedule of a nationalized port (fiN (fj )), meaning that τ does not substantially affect thevalue of the reaction function’s intercept.
The outcomes in the four subgames

Using the two types of reaction functions, we draw theequilibrium outcomes in the four subgames. fi (fj ) is the reaction function of the nationalized
port in i and fˆi (fj ) is that of the privatized port in i. The four intersections generated by thefour reaction functions are the equilibrium outcomes in the four subgames. The two curves
that pass the intersections generated by the pairs of fH and fF and fˆH and fˆF are the iso-social
surplus curves.
[Figures 1-3 here]
The first stage

The first stage decision of each government is to choose one of the reaction

functions. Solving the four outcomes, we derive the following proposition:
Proposition 1 (i) If 0 ≤ τ < a/46, (P, P ) and (N, N ) can appear in equilibrium. (ii) If
a/46 ≤ τ < a/11, only (P, P ) appears in equilibrium. (iii) If a/11 ≤ τ ≤ a/4, only (N, N )appears in equilibrium.
When τ is small enough, two outcomes can appear in equilibrium: both ports are privatizedor they are both nationalized. First, suppose that both ports are nationalized (point A in
Figure 1). The trade barrier to firm H is small enough, implying that the demand for eachport is large enough. Under the large demand, if a port is privatized, it sets a significantly
higher port fee. This implies that the negative effect of privatization is significant. In fact,
the privatization of port H substantially moves the reaction function of port H rightward,
which changes the intersection from A to B (Figure 1). Second, suppose that both ports are
privatized (point C in Figure 1). The trade barrier to firm H is higher because privatized ports
9set higher port fees. Given their organization forms, the nationalization of port H induces alarger rent shift from port H to port F because of the strategic substitutability of port fees
(note that the rent-shifting effect is the main concern in Matsumura and Matsushima (2012)
and Mantin (2012)). The negative effect of this rent shift dominates the positive effect ofnationalization on the firm profit and the consumer surplus. In fact, the privatization of port
H substantially moves the reaction function of port H leftward, which changes the intersection
from C to D (Figure 1).
When τ is intermediate, both ports are privatized in equilibrium. To understand the reason
nationalized ports are privatized, we first suppose that both ports are nationalized (point A
in Figure 2). The trade barrier to firm H is slightly larger than that in the previous case,
implying that the demand for each port is not so large. Under moderate demand, if a port isprivatized, it does not set a significantly higher port fee. This implies that the negative effectof privatization is not very significant. In fact, privatization of port H moderately moves thereaction function of port H rightward, which changes the intersection from A to B (Figure 2).
Second, suppose that both ports are privatized (point C in Figure 2). Under the symmetricmarket size assumption, the rent shift is relatively large, although the import prevention effectthrough nationalization is effective. The property of the rent shift is similar to that in theprevious case.
When τ is large, both ports are nationalized in equilibrium. First, suppose that both ports
are nationalized (point A in Figure 3). The privatization of port H moves the reaction functionof port H leftward, which changes the equilibrium point from A to B (Figure 3). Because of
the import prevention effect through nationalization, the reaction function of nationalized port
H is located on the right-hand side. Because fF (fH ) is downward sloping, through an increasein fF , privatization always generates a rent shift from country H to country F , which harmssocial welfare in country H. In fact, point B is located above the iso-welfare curve passing
through point A. Second, suppose that both ports are privatized (point C in Figure 3). The
nationalization of port H moves the reaction function of port H rightward, which changes the10equilibrium point from C to D (Figure 3). Through a decrease in fF , privatization generates
a rent shift from country F to country H, which benefits social welfare in country H. In fact,
point C is located below the iso-welfare curve passing through point D. To sum up, port
nationalization can be a barrier to international trade.
We briefly discuss the relation between the equilibrium organization structure and thewelfare ranking. The following proposition and Figure 4 summarize the relation.
Proposition 2 The welfare ranking between the privatization and nationalization regimes
changes twice as the shipping fee goes up: (i) If τ ≤ a/11, SWiP P ≤ SWiN N .

(ii) If

a/11 < τ < 55a/227, SWiP P > SWiN N . (iii) If 55a/227 ≤ τ ≤ a/4, SWiP P ≤ SWiN N ,where i = H, F .
[Figure 4 here]
In the first range (τ < a/11), privatization is more likely to appear in equilibrium althoughnationalization is preferable. As discussed in Mantin (2012) and Matsumura and Matsushima(2012), privatization of a port causes a rent shift from another country to its country althoughexpanding the total trade volume through the nationalization of ports is beneficial for bothcountries. That is, the temptation for the rent-shifting benefit through privatization causes
the prisoners’ dilemma.
In the second range (a/11 < τ < 55a/227), nationalization is more likely to appear inequilibrium, although privatization is preferable. As discussed earlier, when τ is not small, thenationalization of a port diminishes the total trade volume because of the import prevention
effect, although expanding the total trade volume through the privatization of ports is beneficial for both countries. That is, the temptation for import prevention through nationalization
causes the prisoners’ dilemma.
In the third range (55a/227 ≤ τ ≤ a/4), nationalization is more likely to appear in equilibrium, and it is preferable. When τ is large enough, exporters are inefficient firms in each11

market. As discussed in Lahiri and Ono (1988), eliminating inefficient firms improves welfare. This is because decisions concerning the organization of ports can be beneficial for bothcountries.

4

Extensions

We further discuss two cases in which the market sizes are heterogenous and in which ports
engage in cost-reducing activities.

4.1

Market size asymmetry

We take into account the heterogeneity of market sizes in the two countries. That is, we relax
the restriction on b in the previous section and assume that b ≤ 1. Figure 5 summarizes theresult.
[Figure 5 here]
The port in the larger country is more likely to be nationalized, while the one in the smaller
country is more likely to be privatized. The main reason for this tendency is that import prevention is more effective for the larger country. Privatized ports do not take into account thebenefit of import prevention for the larger country, but only maximize their port revenues. To
internalize the import prevention effect, the larger country must nationalize its port. Note thatin the non-trade models in Mantin (2012) and Matsumura and Matsushima (2012), an asymmetric privatization mode does not appear in equilibrium. This is because import prevention
does not matter in their models although it is one of the main factors in our model.

4.2

Cost-reducing investments

Finally, we briefly discuss a case in which each port engages in cost-reducing activities after
the organization structure of each port is determined by each government. This is also an important aspect of privatization policy for ports, as Trujillo and Nombela (1999) have observed
12that the privatization of ports in Chile has improved efficiency in terms of cargo handlingcosts.
Based on the basic model, we consider the following game. First, each government simultaneously determines the ownership structure of the port. Second, each port simultaneously
sets its effort level ei to reduce its per-unit operation cost c − ei and then incurs the effort cost
γe2i , where c and γ are positive constants.4 Third, observing the second stage outcome, eachport simultaneously sets the fee fi to maximize its objective which is determined in the firststage. Finally, the firms simultaneously determine their quantities in the two markets.
In the extended model, the nationalized port in the large country mainly focuses on theprotection of the domestic market through a higher port fee. Because the pricing scheduledecreases the trade volume between the countries, the incentive for the nationalized port toengage in cost-reducing activities is weak. In contrast, the privatized port in the small country
does not consider import protection, but focuses on the profit from its port fee. Because
increasing the trade volume is important for the privatized port, its incentive to engage incost-reducing activities is strong. Therefore, the efficiency level of the privatized port is higher
than that of the nationalized one under the case in which only the small country’s government
privatizes its port, which seems consistent with the finding in Trujillo and Nombela (1999).5
Proposition 3 Suppose that only the small country’s government privatizes its port. The
efficiency level of the privatized port is higher than that of the nationalized one.
We can numerically show parameter ranges in which only the small country’s government
privatizes its port in equilibrium, although we do not explicitly mention the problem becauseof the mathematical complexity. Figure 6 shows an numerical example in which the asymmetricprivatization mode appears in equilibrium.
[Figure 6 here]4

We appropriately set those exogenous parameters to secure the second-order conditions and the positive

per-unit operation costs of the ports.
5

In the Appendix, we calculate the equilibrium outcome under the subgame in which only the small country’s

government privatizes its port.

13Note that Matsumura and Matsushima (2012) consider the efforts of airports to reduce
airport congestions which generate negative externalities on consumer demand. Reducing
the negative externalities enhances the demand for airport usage. They show that under
an asymmetric privatization mode, the privatized airport has a stronger incentive to reduce
airport congestions. The logic behind the result is different from ours. In their model, therent-shifting effect of privatization, which leads to a higher airport charge of the privatizedairport, increases its marginal gain from the effort. In our model, import protection weakens
the incentive of the nationalized port to engage in cost reduction even though it sets a higher
port charge. Furthermore, an asymmetric privatization mode never appears in Matsumura
and Matsushima (2012), although it can appear as an equilibrium outcome in our model.

5Conclusion

Following the example of the UK, many countries have moved—or are moving—toward theprivatization of some public ports. Privatization influences the activities of exporters throughthe changes in their payments to shippers who are intermediaries of those exporters. We therefore investigate how port privatization affects firm profits and welfare through the changes in
the ports’ objectives. We formulate a simple model that describes the effect of port privatization on social welfare. The model is based on a two-country setting as in Brander andKrugman (1983). We assume that the objective of a public port is to maximize domestic
welfare, whereas that of a private port is to maximize its own profit.
We show that depending on the per-unit transport cost, several types of organization mode
can appear in equilibrium. When the per-unit transport cost is low enough, there can be two
equilibrium outcomes: either both ports are privatized or neither port is privatized. When the
per-unit transport cost is intermediate, both ports are privatized in equilibrium. When the
per-unit transport cost is large, no port is privatized in equilibrium, although the port charge
would be lower by the privatization of ports. This result implies that a governmental decisionon the ownership structures of ports in the country can be a strategic tool to control trade14

volume. This provides a new insight in the context of transportation research and strategic
trade policy. We also show that the equilibrium governmental decisions are inconsistent withthe socially desirable organization modes of ports if the per unit transport cost is not large
enough.
We further investigate the welfare property of the equilibrium outcome and how market
size asymmetry affects governmental decisions concerning the organization modes of ports. We
show that the government in the large country is more likely to nationalize its port to protect
its domestic market. Finally, we have briefly discussed a case in which each port engages incost-reducing activities after the organization structure of each port is determined by eachgovernment. We show that the efficiency level of the privatized port is higher than that of thenationalized one. This seems consistent with the finding in Trujillo and Nombela (1999).
We have used a simple oligopoly model to describe the effect of port privatization on
consumer and social welfare. Expanding the simple framework to monopolistic competition
models is a considerable work left to future research. Furthermore, we do not consider competition among international ports. As in the literature of airport competition (Pels et al., 2000)
and port competition (Czerny et al., 2011), the extension is a considerable work left to future
research.

References[1] Baird, A. (2002) “Privatization trends at the world’s top-100 container ports,” Maritime
Policy & Management, 29(3), pp. 271-84.
[2] Baird, A. and V. F. Valentine (2007) “Port privatization in the United Kingdom,” Researchin Transportation Economics, 17, pp. 55-84.
[3] Basso, L. J. (2008) “Airport deregulation: Effects on pricing and capacity,” International
Journal of Industrial Organization, 26, pp. 1015-1031.

15

[4] Basso, L. J. and A. Zhang (2008) “Sequential peak-load pricing: The case of airports andairlines,” Canadian Journal of Economics, 41, pp. 1087-1119.
[5] Behrens, K. and P. M. Picard (2011) “Transportation, freight rates, and economic geography,” Journal of International Economics, 85, pp. 280-91.
[6] Brander, J. and P. Krugman (1983) “A ‘reciprocal dumping’ model of international trade,”
Journal of International Economics, 15, pp. 313-21.
[7] Czerny, A., F. Höffler, and S. Mun (2011) “Port competition and welfare effect of strategic
privatization,” Presented in 25th Annual Meeting of the Applied Regional Science Conference.
[8] Francois, J. and I. Wooton (2001) “Trade in international transport services: The role ofcompetition,” Review of International Economics, 9, pp. 249-61.
[9] González, M. M. and L. Trujillo (2008) “Reforms and infrastructure efficiency in Spain’s
container ports,” Transportation Research Part A, 42, pp. 243-57.
[10] Haskel, J., A. Iozzi, and T. Valletti (2013) “Market structure, countervailing power andprice discrimination: The case of airports,” Journal of Urban Economics, 74, pp. 12-26.
[11] Hummels, D. and A. Skiba (2004) “Shipping good apples out? An empirical confirmationof the Alchian-Allen conjecture,” Journal of Political Economy, 112, pp. 1384-1402.
[12] Kleinert, J. and J. Spies (2011) “Endogenous transport costs in international trade,” IAW
Discussion Papers No. 74.
[13] Lahiri, S. and Y. Ono (1988) “Helping minor firms reduces welfare,” Economic Journal,
98, pp. 1199-1202.
[14] Limão, N. and A. J. Venables (2001) “Infrastructure, geographical disadvantage, transportcost, and trade,” World Bank Economic Review, 15, pp. 451-479.

16

[15] Mantin, B. (2012) “Airport complementarity: Private vs. government ownership andwelfare gravitation,” Transportation Research Part B, 46, pp. 381-88.
[16] Matsumura, T. and N. Matsushima (2012) “Airport privatization and international competition,” Japanese Economic Review, 63, pp. 431-450.
[17] Pels, E., P. Nijkamp, and P. Rietveld (2000) “Airport and airline competition for passengers departing from a large metropolitan area,” Journal of Urban Economics, 48, pp.
29-45.
[18] Takahashi, T. (2004) “Spatial competition of governments in the investment on public
facilities,” Regional Science and Urban Economics, 34, pp. 455-488.
[19] Tongzon, J. and W. Heng (2005) “Port privatization, efficiency and competitiveness: Someempirical evidence from container ports (terminals),” Transportation Research Part A, 39,
pp. 405-24.
[20] Trujillo, L. and G. Nombela (1999) “Privatization and regulation of the seaport industry,”
World Bank, Policy Research Working Papers, No. 2181.
[21] Zhang, A. and A. I. Czerny (2012) “Airports and airlines economics and policy: Aninterpretive review of recent research,” Economic of Transportation, 1, pp. 15-34.
[22] Zhang, A. and Y. Zhang (2003) “Airport charges and capacity expansion: Effects ofconcessions and privatisation,” Journal of Urban Economics, 53, pp. 54-75.
[23] Zhang, A. and Y. Zhang (2006) “Airport capacity an congestion when carriers have market
power,” Journal of Urban Economics, 60, pp. 229-247.

17

Appendix
Proof of Proposition 1: By using (3) and (4), we can derive the equilibrium port fees in thefour subgames. When both ports are privatized, we use (3). By using the equations, we havethe equilibrium port fees:
PP
fH= fFP P =a − 2τ
,6

where the subscript P P indicates the subgame in which both ports are privatized. Applying
similar procedures to the rest of the cases, we have the equilibrium port fees in the subgames
as follows:
PN
fH= fFN P =3(3a − 8τ )7a − 2τ
2a − τ
NP
NN, fH= fFP N =, fH
= fFN N =
,
50
50
14

where P N , N P , and N N respectively indicate the subgame in which only port H is privatized,that in which only port F is privatized, and that in which both ports are privatized. Substituting the outcome in the third stage (equation (2)) and the equilibrium port fees derived
above into the equation of the social surplus in (1), we have the following payoff matrix.
Table 1: The payoff matrix of the port privatization game (b = 1)
HF

P

N

P

65a2 − (26a − 35τ )τ
162
2 − (26a − 35τ )τ
65a
SW PHP =162

249a2 − 26(3a − 4τ )τ
625
2 − 8(4a − 7τ )τ
51a
SW PHN =125

N51a2 − 8(4a − 7τ )τ
125
2249a − 26(3a − 4τ )τ
P
SW N
H =
625

20a2 − 13(a − 2τ )τ
49
220a − 13(a − 2τ )τ
N
SW N
H =
49

SW PF P =

P
SW N
=
F

18

SW PF N =

N
SW N
=
F

From Table 1,(a − 11τ )(287a − 457τ ),101250(a − 11τ )(a − 46τ )SWHN N − SWHP N = SWFN N − SWFN P =.
6125

SWHP P − SWHN P = SWFP P − SWFP N =From the equations, we find that SWHP P > SWHN P if and only if τ < a/11 and that SWHN N ≤SWHP N if and only if a/46 ≤ τ < a/11.
If 0 ≤ τ < a/46, (P, P ) and (N, N ) can appear in equilibrium. If a/46 ≤ τ < a/11, only
(P, P ) appears. If a/11 ≤ τ ≤ a/4, only (N, N ) appears.

Q.E.D.

Proof of Proposition 2: From Table 1, simple algebra yields
SWiP P − SWiN N =−(a − 11τ )(55a − 227τ ).
7938

For the equation, we easily find that SWiP P − SWiN N > 0 if and only if a/11 < τ < 55a/227.
Thus, we obtain Proposition 2.

Q.E.D.

Equilibrium outcome under country size asymmetry:

We discuss the case of country size

asymmetry (i.e., 2/9 < b ≤ 1). In the third stage, given the fee levels fH and fF , each firm
simultaneously sets its quantities in the two markets. As in the standard Cournot duopoly
outcome, when the inverse demand function is pi = a − (qii + qji )/bi , the equilibrium outcomein the final stage is given asqii (fH , fF ) =bi (a − 2(τ + fH + fF ))bi (a + τ + fH + fF ); qji (fH , fF ) =
.
3
3

(7)

Note that bH = 1 and bF = b(≤ 1). Substituting these into πii , πji , Ri , and CSi , we havebi (a − 2(τ + fH + fF ))2bi (a + τ + fH + fF )2, πji (fH , fF ) =
,
(8)
9
9
fi (bi + bj )(a − 2(τ + fH + fF ))bi (2a − (τ + fH + fF ))2
Ri (fH , fF ) =, CSi (fH , fF ) =
.(9)
3
18

πii (fH , fF ) =19

From the maximization problem relevant to ownership structure, we obtain the port feesin each subgame. When all ports are privatized, we obtainfi (f−i ) =a − 2τ
a − 2τ − 2f−iPP(i = H, F, i 6= −i) ⇒ fH
= fFP P =,46

where fi (f−i ) is the reaction function of port i. When only port H is privatized, we obtaina − 2τ − 2fF(3b − 1)a + (2 − 3b)τ + (2 − 3b)fH, fF (fH ) =,4
4 + 9b3[(2 + b)a − 4(1 + b)τ ](9b − 2)a + 2(2 − 3b)τ
PN
⇒ fH
=
, fFP N =.
10(2 + 3b)10(2 + 3b)
fH (fF ) =When only port F is privatized, we obtain(3 − b)a − (3 − 2b)τ − (3 − 2b)fF
a − 2τ − 2fH, fF (fH ) =,9 + 4b4(9 − 2b)a − 2(3 − 2b)τ3((1 + 2b)a − 4(1 + b)τ )
NP
⇒ fH
=
, fFN P =.
10(3 + 2b)10(3 + 2b)
fH (fF ) =Finally, when all ports are nationalized, we obtain(3 − b)a − (3 − 2b)τ − (3 − 2b)fF,
9 + 4b(3b − 1)a + (2 − 3b)τ + (2 − 3b)fH,
fF (fH ) =4 + 9b(5b − 1)a + 2(2 − 3b)τ(5 − b)a − 2(3 − 2b)τ
NN
⇒ fH
=
, fFN N =.
14(1 + b)
14(1 + b)
fH (fF ) =20

From these results, we obtain the following results in the subgames.
5(12 + b)a2 − 2(3 + 10b)aτ + 5(3 + 4b)τ 2,
162
5(1 + 12b)a2 − 2(10 + 3b)aτ + 5(4 + 3b)τ 2
SW PF P =,162
(16 + 48b + 37b2 + b3 )a2 − 8(1 + b)3 aτ + 4(1 + b)2 (3 + 4b)τ 2
SW PHN =,10(2 + 3b)2
SW PHP =SW PF N =(4 + 88b + 235b2 + 171b3 )a2 − 2(1 + b)(2 + b)(4 + 9b)aτ + 4(1 + b)2 (4 + 9b)τ 2,
50(2 + 3b)2

P
SW N
H =

(171 + 235b + 88b2 + 4b3 )a2 − 2(1 + b)(1 + 2b)(9 + 4b)aτ + 4(1 + b)2 (9 + 4b)τ 2,
50(3 + 2b)2

(1 + 37b + 48b
P
=
SW N
F

2 + 16b3 )a2 − 8(1 + b)3 aτ + 4(1 + b)2 (4 + 3b)τ 2

10(3 + 2b)2
(39 + b)a2 − 2(9 + 4b)aτ + 4(9 + 4b)τ 2
N
SW N
,
H =
98
(1 + 39b)a2 − 2(4 + 9b)aτ + 4(4 + 9b)τ 2
N
SW N
=
.
F
98

,

Using the above outcome, we check the equilibrium outcome in the full game. First, giventhat the rival country chooses P , the welfare differences between choosing P and N are givenas[(3 − 4b)a + (3 + 8b)τ ][(153 + 216b + 88b2 )τ − (117 + 126b + 44b2 )a],4050(3 + 2b)2[(3b − 4)a + (8 + 3b)τ ][(88 + 216b + 153b2 )τ − (44 + 126b + 117b2 )a].
SW PF P − SW PF N =4050(2 + 3b)2P
SW PHP − SW N
H =From these equations, SWHP P − SWHN P ≥ 0 if and only ifτ≤(4b − 3)a.
3 + 8b(10)τ≤(4 − 3b)a.
8 + 3b(11)

SWFP P − SWFP N ≥ 0 if and only if

21Second, given that the rival country chooses N , the welfare differences between choosing P
and N are given as[(1 − 2b)a + (3 + 8b)τ ][(22 + 28b − 4b2 )τ − (2 − b2 )a],245(1 + b)3[(2 − b)a − (8 + 3b)τ ][(4 − 28b − 22b2 )τ − (1 − 2b2 )a]
N
NP
−
SW
=
SW N
.
F
F
245(3 + 2b)2
N
PN
SW N
H − SW H =From these equations, SWHN N − SWHP N ≥ 0 if and only if(2 − b2 )a(2b − 1)a(2b − 1)a(2 − b2 )a
≤
τ
≤
,
or
≤
τ
≤
.
22 + 28b − 4b2
3 + 8b
3 + 8b
22 + 28b − 4b2(12)

SWFN N − SWFN P ≥ 0 if and only if(2 − b)a(−1 + 2b2 )a(−1 + 2b2 )a(2 − b)a
≤τ ≤
,
or
≤τ ≤
.
8 + 3b
−4 + 28b + 22b2
−4 + 28b + 22b2
8 + 3b

(13)

From (10) to (13), we have the conditions that P P , N P , P N , and N N can appear as equilibrium outcomes. The following figure summarizes the threshold values of the conditions in(10), (11), (12), and (13).
τ /a
1/4
2−b
8+3b

0.20

4−3b
8+3b

0.15
0.10

2b−1
3+8b

2−b2
22+28b−4b2

4b−3
3+8b

0.05

−1+2b2−4+28b+22b2

0
2/9

b
0.4

0.6
The threshold curves
22

0.8

1

P P can appear if and only if both (10) and (11) hold. N P can appear if and only if neither(10) nor (13) holds. P N can appear if and only if neither (11) nor (12) holds. Note that weneed to include the case in which τ is equal to the threshold value in the cases of N P and
P N . N N can appear if and only if both (12) and (13) hold (see Figure 5).
Cost-reducing investments

We now show that the efficiency level of the privatized port ishigher than that of the nationalized one under the case in which only the small country’s
government (country F ’s government) privatizes its port. We not discuss the outcome in thestage in which the ports set their port charges. We focus on the subgame in which only the small
country’s government (the government in country F ) privatizes its port. The maximization
problems of the ports are given bymax πHH (fH , fF ) + πHF (fH , fF ) + R̃H (fH , fF , eH ) + CSH (fH , fF ),fH

max R̃F (fH , fF , eF ),fF

where πHH and πHF are in (8), CSH is in (9), and R̃i (fH , fF , ei ) is given byR̃i (fH , fF , ei ) =(fi − (c − ei ))(bi + bj )(a − 2(τ + fH + fF )).
3

From the problem, we have the equilibrium outcome in this stage (note that, bH = 1 andbF = b):(9 − 2b)a − 2(3 − 2b)(c − eF + τ ) + 24(1 + b)(c − eH ),10(3 + 2b)3(1 + 2b)a − 12(1 + b)(c − eH + τ ) + 2(9 + 4b)(c − eF )fF (eH , eF ) =.
10(3 + 2b)fH (eH , eF ) =Substituting the outcome into the objectives of the ports, we have the equilibrium outcome inthe subgame.
We now discuss the stage in which the ports set their effort levels. Each port sets its effort

23

level to maximize its objective. Solving the maximization problems, we have

eH

=

eF

=

eF − eH=(1 + b)(9 + 4b)((1 + 2b)a − 4(1 + b)(2c + τ )),2(25(3 + 2b)2 γ − 2(1 + b)2 (21 + 16b))12(1 + b)2 ((1 + 2b)a − 4(1 + b)(2c + τ )),2(25(3 + 2b)2 γ − 2(1 + b)2 (21 + 16b))(3 + 8b)(1 + b)((1 + 2b)a − 4(1 + b)(2c + τ ))> 0.
2(25(3 + 2b)2 γ − 2(1 + b)2 (21 + 16b))

Note that the equilibrium trade volume is positive if and only if (1 + 2b)a − 4(1 + b)(2c + τ ) > 0
and that the denominator is positive if the second-order conditions in the second stage aresatisfied. The third equation implies that the ex post efficiency level of the privatized port
is higher than that of the nationalized one under the case in which only the small country’s
government (country F ’s government) privatizes its port. Figure 6 shows an numerical example
in which only the small country’s government privatizes its port in equilibrium.

24

fF

0.18

D
`
f F H fH L

0.17`
SWH H f H , f F L

C

0.16

`
f H H fF L

fH H fF L
0.15

A

SWH H f H , f F L
B

fF H fH L

0.14

fH
0.13

0.14

0.15

0.16

0.17

0.18

Figure 1: The reaction functions and the iso-surplus curves (b = 1, τ = 0)

25

fF
0.160

D
0.155

`
f F H fH L
0.150

C

`
SWH H f H , f F L

0.145`
f H H fF L

fH H fF L

0.140

A

SWH H f H , f F L
fF H fH L
B

fH
0.135

0.140

0.145

0.150

0.155

0.160

Figure 2: The reaction functions and the iso-surplus curves (b = 1, τ = 1/20)

26

fF
0.140

0.138

0.136

`
f H H fF L

fH H fF L

B
SWH H f H , f F L

A

fF H fH L

0.134`
SWH H f H , f F L

C

D

0.132

`
f F H fH L

fH
0.130

0.132

0.134

0.136

0.138

0.140

Figure 3: The reaction functions and the iso-surplus curves (b = 1, τ = 1/10)

27

(P, P )(N, N )(P, P )

(N, N )

τ

1/46
SWiP P > SWiN N

SWiP P ≤ SWiN N
0

a/11

SWiP P≤ SWiN N
55a/227

τ

a/4

Figure 4: Equilibrium outcomes and welfare ranking under country size symmetry (b = 1)

28

τ /a
1/4
0.20

(N, N )

0.15
(N, P )

0.10
0.05

(N, N )

0
2/9

0.4

0.6

√
1/ 2 3/4 0.8(P, P )
(N, N )(P, P )b
1

Figure 5: Equilibrium outcomes under country size asymmetry (2/9 < b ≤ 1)

29

Γ
100
90
80
70
60
50
40
30
0.3 0.4 0.5 0.6 0.7 0.8 0.9

1bFigure 6: The region in which only the small country’s government privatizes its port (c = a/20,
τ = 0)

30