# Optimization of the Pricing Strategies between Container Terminals under Deregulation

Authors: Gang Dong
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

According to the dual-track system implemented on port tariffs in past years, the vast majority of state-owned container terminals adopt the standard rates specified by China’s Ministry of Transport, while the container terminals of joint ventures are permitted to charge their stevedoring rate with a 20% float ratio up and down. The latest port reform was to improve the port tariff formation mechanism by speeding up the implementation of detailed list and public notice on port pricing. This paper analyses the optimization of the pricing strategies between container terminals under deregulation. Based on a two-stage noncooperative game theoretical model, the Nash equilibria of pricing strategy profiles between container terminals of one port under deregulation are derived. Although the price-matching strategy may be employed by the foreign-owned container terminal, which usually resulting in a total social welfare loss, the price-matching pricing strategy not being adopted by the state-owned container terminal will avert tacit collusion. Numerical simulation is applied to the case of Shenzhen Port.


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## Body

## 1. Instruction

With container transport becoming an essential component of the global supply chain, the importance of the container terminal industry for industrial activity, merchandise trade, globalized production processes, and economic growth cannot be overemphasized. Meanwhile, container terminals also have been under constant pressure to adapt to changes in the economic, institutional, regulatory, and operating landscapes [1].Since the late 1980s, the ownership structures of container terminals in many countries have been experiencing a transformation from public regulation to private operation. The size of a port or terminal is closely correlated with its efficiency, and some support exists for the claim that the transformation of ownership from the public to the private sector improves economic efficiency [2]. Because of the quantitative relationship that exists between the port ownership structure and port efficiency, a stochastic frontier model has been applied to show whether port privatization is a necessary strategy for ports to gain a competitive advantage [3]. Moreover, a direct correlation between the reform and the change in efficiency has been established to analyse the efficiency of the Spanish Port Authorities during the last three decades [4].Some studies have addressed the efficiency gains of port privatization from an institutional and operational perspective. The optimization of container terminal operations can be restructured, especially for large ports, thus allowing specialized private entities to concentrate on terminal operation and cargo handling services [5]. The potential efficiency gains from privatization are discussed, thus providing a special reference to the privatization of the transport infrastructure in developing countries [6]. There exist equilibria in which governments choose privatization, and the national welfare of each port country is higher relative to a situation where ports are public [7]. The two port regulation modes are compared, finding that the tariffs, port efficiency level, port service demand, and social welfare are higher under the decentralization mode [8]. The various patterns of port cooperation in connection with the governance structure of ports are compared, introducing a generalization of the port cooperation issue in conclusion [9]. However, in these studies, port tariffs are not centre stage.In China, many ports have rapidly improved their container-handling performance in recent years. Yap and Lam [10] unveil the competitive dynamics between the major container ports in East Asia by analysing their extent and intensity, thus concluding that the economic development of China’s interior provinces will continue to stimulate strong container-handling performance in its ports together with investments in terminal capacity and operational efficiency. Yeo et al. [11] present a structure for evaluating port competitiveness, insisting that Chinese ports threaten to oust Busan in Korea as the regional hub following the intensive state investments in port development. Song and Geenhuizen [12] apply panel data analysis in 1999-2010 of China to estimate the output elasticity of port infrastructure through the production function.The reasons why China can invest heavily in large-scale projects of port and container terminals are also addressed. Yang et al. [13] explore the various paths to find a direction that better suits China’s national conditions during the past 60 years, concluding that the reforms of port governance have had a more lasting positive effect on port throughput than physical investments. Zhuang et al. [14] use alternative duopoly games to model port competition, finding that the interport competition can lead to port specialization. Wu et al. [15] pay attention to how a port would respond if a rival port uses a type of capacity investment strategy, obtaining that investments in port capacity contribute greatly to the local government’s performance. Zhang [16] notes that the optimal incentive scheme is the same in the international landlord port financing model with profit sharing rents or mixed rents.This paper is motivated by the employment of a price-matching guarantee policy in retailing, which results either in price collusion or price discrimination in a duopoly. Edlin and Emch [17] find that price matching with entry creates greater welfare losses than monopolies in markets with a low ratio of fixed to marginal costs, using parameters from the US wholesale gasoline and air travel markets. Monika and Dhruv [18] examine the process to determine the likelihood that a consumer will claim a price-matching refund when he or she identifies a lower competitive price after the purchase. If consumers are heterogeneous with respect to firm loyalty, and a firm has more loyal customers than the other firm, the equilibrium matching policy and pricing strategy depend on the market conditions (Koh et al., [19]). Under the background of port tariff deregulation, if one port has multiple container terminals, which are state-owned, privately owned or joint-ventured, providing perfectly homogenous services or basically identical services to the shipper, which makes it possible for them to adopt price-matching strategies. However, while the price-matching guarantees seem applicable to container terminals of one port, it is not certain that they will employ price-matching strategies with the deregulation of port capital and port tariff.The structure of the game employed by this paper is similar to the one used by Dong et al. [20], in which the differential designed capacities of two container terminals of one port are investigated, and two container terminals maximize their profits separately according to whether they adopt the price-matching policy or not, drawing the conclusion that price-matching strategies facilitate tacit collusion between container terminals. This paper comprehensively takes account of the differential designed capacities, capital structures and tariff deregulation of the container terminals. Therefore, the container terminal subjected to foreign investment still maximizes its profits separately according to whether adopts the price-matching policy or not. In contrast, the state-owned container terminal, which can be considered as an economic entity with port authority of local government, should take its own profits and the surplus of shipper as the pricing objective. More importantly, the state-owned container terminal attempts to realize its social welfare maximization under the background of port tariff deregulation. At this moment, we ask whether the price-matching strategy is the only Nash equilibrium for the container terminals of one port. We also address how to avoid tacit collusion among container terminals of one port, which otherwise results in social welfare losses with port tariff deregulation.We organize the rest of the paper as follows. We first present the model basics under the circumstance of port tariff deregulation in Section2. The two-stage game analysis among the container terminals of one port is conducted in Section 3. In Section 4, we solve the Nash equilibrium and compare social welfare in the scenarios of adopting the price-matching strategy or not. In Section 5, we will verify some of the analytical results and further explore the managerial insights and policy implications through a case study of Shenzhen Port. Conclusions and directions for future research are summarized in Section 6.


## 2. Methods

In this section, we construct an analytical framework model for the two container terminals in the same port area. With the gradual deregulation of foreign investment into the port sector, the container terminals of one port are built by different capital, such as state-owned, foreign-owned, bank financing, self-collected funds, IPOs and so on. In addition, the number of major container terminals in mainland China further increased to 99, among which 59 container terminals involved foreign investment, and the proportion of foreign investment container terminals increased to 59.6% in 2016 (China Port Yearbook, [21]). For simplicity, we consider a port with two kinds of capital sources, which are represented by container terminals 1 and 2, and are denoted as being foreign-owned and state-owned, respectively.Considering an economy with a monopolistic sector with two firms, each one producing a differentiated good, and the consumers of the same type with a utility function separable and linear, following Singh and Vives [22], the quadratic and strictly concave utility function gives rise to a linear demand structure, the inverse demands of the container terminals are given by,(1)f1+t·q1s1=α1-β1q1-γq2(2)f2+t·q2s2=α2-β2q2-γq1where f is the tariff of the container terminal of one port, t is the parameter of waiting costs in the container terminal, s is the designed capacity of the container terminal, q is the output volume of the container terminal. α is the intercept of the reservation price for the container terminal’s service. β and γ are the sensitivity of the container terminal’s demand to its own generalized price and the other’s generalized price. The values of αi and βi for i=1,2 are positive. Since two container terminals are located in the same port, we assume that the container terminals are perfectly substitutable; i.e., α1=α2=α and β1=β2=γ=β. Without the loss of generality, we also assume that 0<s1≤s2.Located in the same region, the two container terminals face ocean carriers’ selection decisions to meet the original demand of the shipper or freight forwarder, such as terminal tariffs, congestion cost as well as total transport chain’s cost, connectivity, availability, immediacy, etc. For example, prompt response, 24 hour a day, seven days a week service and less waiting time service. In addition, professional and skilled labour in the operation of container terminal, network connectivity of feeder ports and transfer stations, the area and policy of free trade port, which can be divided into two categories: hardware factors include infrastructure and facility, denoted by designed capacity, the congestion cost of liner company starts to rise sharply when the actual container throughput exceeds 80% of rated capacity. The software factors contain the quality and skill of terminal worker and management personnel, terminal operation process system and automated terminal technology, as well as custom clearance process and maritime administrative inspection, reflected by the parametert.In addition, the generalized price that is assumed to take the sum of the container terminal’s tariff and waiting time costs, with the latter expressed by the ratio of container throughput to the designed capacity that indicates the container terminal’s efficiency, has been adopted in some studies to analyse port congestion costs (see De Borger and Van Dender, [23]; Basso and Zhang, [24]; Bae et al. [25]; Chen and Liu, [26]; Sheng et al. [27]).Therefore, the demands for container terminals’ services can be solved by Eqs. (1) and (2):(3)q1=s1βs2f2-βs2+tf1+attβs1+βs2+t(4)q2=s2βs1f1-βs1+tf2+attβs1+βs2+tContrary to the Cournot model, the Bertrand model is a good approximation of duopoly competition under the circumstance that it is more difficult to adjust the price than it is to adjust capacity/output (Cabral, [28]). The dual-track system has been implemented on port tariffs in China in past years. According to this pricing policy, the vast majority of state-owned container terminals adopt the standard rates specified by China’s Ministry of Transport, while the container terminals of joint ventures are permitted to charge their stevedoring rate with a 20% float ratio up and down. In order to transform the container terminal’s tariffs from mainly depending on government pricing to being based on market regulations, charging method of port tariffs was formulated by China’s Ministry of Transport and National Development and Reform Commission, which come into effect on March 1, 2016, and valid for 5 years. According to the method, the port tariffs of market regulation include operation lump sum rate, storage fee, custody fee, as well as the service charges of ship material, oil and gas supply, power supply, garbage receiving process and sewage disposal. Moreover, the container terminal can set different lump sum rates in accordance with providing differential service.The game is assumed to be played in two stages through backwards-induction. In the second stage, the two container terminals have to make a decision on their tariffs independently according to the pricing strategy that is adopted. Then, in the first stage, both container terminals decide simultaneously whether to adopt a price-matching strategy or not.


## 3. Optimizing Strategy

### 3.1. Price-Matching Strategy Is Employed by Neither Container Terminal

Both container terminals have identical and constant variable costs in consideration of locating within the same port, and the variable costs are normalized to zero on the supposition that basically the same equipment and related fees of energy consumption, truck drayage, handling labour, etc. exist for the two container terminals in one port area.Since neither the foreign-owned or state-owned container terminal has a conjecture about the other one, the container terminal owned by foreign capital will maximize its own profits as follows:(5)maxf1⁡π1f1=f1.s1βs2f2-βs2+tf1+attβs1+βs2+tBy taking the derivative, the first-order conditions (FOCs) and the second-order conditions (SOCs) are given by(6)∂π1∂f1=-2βs2+ts1tβs1+βs2+tf1+βs1s2tβs1+βs2+tf2+αs1βs1+βs2+t∂2π1∂f12=-2βs2+ts1tβs1+βs2+tTo solve container terminal 1’s optimization problem, we set the first-order condition equal to zero as(7)-2βs2+ts1tβs1+βs2+tf1+βs1s2tβs1+βs2+tf2+αs1βs1+βs2+t=0which leads to the following solution of the container terminal invested by foreign capital:(8)f1=βs2f2+αt2βs2+tWith regard to the container terminal established by state-owned capital, which can be considered as an economic entity where port authority is controlled by the local government, the pricing objectives of the state-owned container terminal not only include its own profits, similar to the foreign-owned container terminal, but also contain the surplus of the shippers.According to the classic quadratic utility function (see Singh and Vives, [22]), the continuum of shippers of the same type with a utility function can be expressed as follows:(9)Uq1,q2=α1q1+α2q2-β1q12+2γq1q2+β2q222To avoid social welfare loss with port tariff deregulation, the state-owned container terminal should charge its tariff to realize the social welfare maximization:(10)maxf2⁡wf2=αq1+q2-βq1+q222+π2-f2+t·q2s2q2Similarly, the first-order condition can be rewritten as(11)∂w2∂f2=-αs2βs1+βs2+t-βαs1+s2-f1s1-f2s2βs1+βs2+t∗-s2βs1+βs2+t-2t∗-βs1+ts2tβs1+βs2+tf2+βs1s2tβs1+βs2+tf1+αs2βs1+βs2+t∗-βs1+ttβs1+βs2+tBy setting it equal to zero, we obtain the following solution:(12)f1=2atβs1+tβs1+βs2+t2β2s12βs2+2t+β2s2t7s1+2s2+2t24βs1+3βs2+2tf2=at2βs1+tβs1+2βs2+2t2β2s12βs2+2t+β2s2t7s1+2s2+2t24βs1+3βs2+2tTherefore, the profits and social welfare for both container terminals are given, respectively, by(13)π11=4a2s1tβs1+t2βs2+tβs1+βs2+t2β2s12βs2+2t+β2s2t7s1+2s2+2t24βs1+3βs2+2t2(14)w21=a24β5s14s22+16β4s14s2t+28β4s13s22t+8β4s12s23t+12β3s14t2+72β3s13s2t2+73β3s12s22t2+28β3s1s23t2+4β3s24t2+40β2s13t3+110β2s12s2t3+76β2s1s22t3+16β2s23t3+44βs12t4+64βs1s2t4+20βs22t4+16s1t5+8s2t52β3s12s2+4β2s12t+7β2s1s2t+2β2s22t+8βs1t2+6βs2t2+4t3-2


### 3.2. Price-Matching is Employed by Both Container Terminals

According to price-matching strategies employed by foreign-owned and state-owned container terminals, the two container shippers should charge the same tariff to the shipper. Under these circumstances, each container terminal forecasts that one unilateral tariff cut below the other’s counterpart will be matched by the other.Therefore, the profits of a foreign-owned container terminal can be rewritten as(15)min⁡f1,f2-βs2+ts1tβs1+βs2+tmin⁡f1,f2+βs1s2tβs1+βs2+tmin⁡f1,f2+αs1βs1+βs2+tIn consideration of a fixed value of the state-owned container terminal’s tariff, the foreign-owned container terminal’s strategyf1>f2 has the same result as the strategy f1=f2. For this reason, we only analyse f1≤f2, for which the foreign-owned container terminal’s optimization problem becomes(16)maxf1:f1≤f2⁡π1f1=f1-s1βs1+βs2+tf1+αs1βs1+βs2+twhich is solved at f1∗=α/2. Otherwise, there exists f1∗=f2.Similarly, the first-order condition the state-owned container terminal of can be rewritten as(17)∂w2∂f2=-αs1+s2βs1+βs2+t-βs1+s2a-f2βs1+βs2+t∗-s1+s2βs1+βs2+t-2t∗a-f2βs1+βs2+t∗-s2βs1+βs2+twhich is solved as(18)f2∗=ats2-s1βs1+s22+2s2tIf the following inequality holds:(19)2βs2+tt-βs2-tβ≤s1≤s2Therefore, we obtainf1∗=f2∗=α/2. Otherwise, we can get f1∗=f2∗=ats2-s1/βs1+s22+2s2t.In the meantime, the profits and social welfare for both container terminals are given, respectively, by(20)π12=s1βs1+βs2+t·α22(21)w22=3βs1+s22+2t2s1+s22βs1+βs2+t2·α22


### 3.3. Price-Matching Is Only Employed by Foreign-Owned Container Terminal

At this very moment, the foreign-owned container terminal employs the price-matching strategy, while the state-owned container terminal does not.The profits of both container terminals can be expressed as(22)π1f1=min⁡f1,f2-βs2+ts1tβs1+βs2+tmin⁡f1,f2+βs1s2tβs1+βs2+tf2+αs1βs1+βs2+tπ2f2=f2-βs1+ts2tβs1+βs2+tf2+βs1s2tβs1+βs2+tmin⁡f1,f2+αs2βs1+βs2+tProposition 1.
It is necessary to havef1=f2 in any equilibrium of price-matching only employed by the foreign-owned container terminal.Proof.
On the supposition off1>f2, by taking the price-matching strategy employed by the foreign-owned container terminal into consideration, the social welfare of state-owned container terminal can also be rewritten as(23)maxf2⁡wf2=αs1+s2a-f2βs1+βs2+t-βs1+s2a-f2/βs1+βs2+t22-t·s2a-f2/βs1+βs2+t2s2
The first-order condition can be obtained by(24)-as1+s2βs1+βs2+t-βs1+s2a-f2βs1+βs2+t∗-s1+s2βs1+βs2+t-2t∗a-f2βs1+βs2+t∗-s2βs1+βs2+t=0
Thus, the second-order condition of it is given by(25)-βs1+s2βs1+βs2+t∗s1+s2βs1+βs2+t-2tβs1+βs2+t∗s2βs1+βs2+t<0
Evidently, the social welfare function of the state-owned container terminal is concave and maximized at(26)f2=αts2-s12ts2+βs1+s22
Given the state-owned container terminal 2’s optimal tariff isαts2-s1/2ts2+βs1+s22, we can obtain the foreign-owned container terminal’s profits as follows:(27)min⁡f1,f2-βs2+ts1tβs1+βs2+tmin⁡f1,f2+βs1s2tβs1+βs2+t∗αts2-s12ts2+βs1+s22+αs1βs1+βs2+twhich is maximized at(28)f1=βs2αts2-s1/2ts2+βs1+s22+αt2βs2+t
Therefore,(29)△f=f2-f1=-αs1tβs1+3βs2+2t2βs2+t2ts2+βs1+s22<0
It is a contradiction to the supposition off1>f2 in the equilibrium of the price-matching that is only employed by the foreign-owned container terminal.
Similarly, suppose thatf1<f2 in equilibrium of price-matching is only employed by the foreign-owned container terminal. The state-owned container terminal will maximize social welfare at(30)f2=2βs1+tβs1f1+αttβs2+2βs1+t2
Consequently, the foreign-owned container terminal’s profits are maximized at(31)f1=βs2f2+αt2βs2+t
Therefore, we can obtain the following inequality expression:(32)△f=f2-f1=αβs1t2βs2+t2β2s12βs2+2t+β2s2t7s1+2s2+2t24βs1+3βs2+2t>0
In conclusion, the price-matching strategy only employed by the foreign-owned container terminal has no effect on the equilibrium outcome in this game.


### 3.4. Price-Matching Is Only Employed by State-Owned Container Terminal

In contrast to the game of price-matching only employed by foreign-owned container terminal, the profits of both container terminals are, respectively, given by(33)π1f1=f1-βs2+ts1tβs1+βs2+tf1+βs1s2tβs1+βs2+tmin⁡f1,f2+αs1βs1+βs2+twf2=αq1+q2-βq1+q222+π2-min⁡f1,f2+t·q2s2q2The above functions can also be rewritten as(34)π1f1=f1-βs2+ts1tβs1+βs2+tf1+βs1s2tβs1+βs2+tf2+αs1βs1+βs2+twf2=ααs1+s2-f1s1-f2s2βs1+βs2+t-βαs1+s2-f1s1-f2s2/βs1+βs2+t22-t·αs1+s2-f1s1-f2s2/βs1+βs2+t2s2Proposition 2.
No price-matching equilibrium only employed by the state-owned container terminal satisfiesf2<f1.Proof.
On the supposition that there exists an equilibrium satisfyingf2<f1, the state-owned container terminal’s objective function is as follows:(35)maxf2:f2≤f1⁡ααs1+s2-f1s1-f2s2βs1+βs2+t-βαs1+s2-f1s1-f2s2/βs1+βs2+t22-t·αs1+s2-f1s1-f2s2/βs1+βs2+t2s2which is maximized at(36)min⁡2βs1+tβs1f1+αttβs2+2βs1+t2,f1
Iff1>α(2βs1+t)/3βs1+βs2+2t,f2=(2βs1+t)(βs1f1+αt)/tβs2+2βs1+t2<f1. Given f2<f1, foreign-owned container terminal’s profits can be rewritten as(37)f1-βs2+ts1tβs1+βs2+tf1+βs1s2tβs1+βs2+tf2+αs1βs1+βs2+twhich is maximized at(38)f1=βs2f2+αt2βs2+t
Thus, we have(39)f1=βs22βs1+tβs1f1+αt/tβs2+2βs1+t2+αt2βs2+twhich is solved at(40)f1=2αtβs1+tβs1+βs2+t2β2s12βs2+2t+β2s2t7s1+2s2+2t24βs1+3βs2+2t
That is,(41)f1-α2βs1+t3βs1+βs2+2t=-2aβ3s122βs1s2+s1t+4s2t+aβs1t2β2s22+2βs1t+5βs2t+2t23βs1+βs2+2t2β2s12βs2+2t+β2s2t7s1+2s2+2t24βs1+3βs2+2t
The value off1 above is less than α(2βs1+t)/3βs1+βs2+2t, which is a contradiction.Proposition 3.
Any tariff combination satisfyingαt/βs2+2t≤f1=f2≤α(2βs1+t)/3βs1+βs2+2t is a Nash equilibrium of the price-matching only employed by the state-owned container terminal.Proof.
The state-owned container terminal’s objective function(42)maxf2:f2≤f1⁡ααs1+s2-f1s1-f2s2βs1+βs2+t-βαs1+s2-f1s1-f2s2/βs1+βs2+t22-t·αs1+s2-f1s1-f2s2/βs1+βs2+t2s2is maximized at(43)min⁡2βs1+tβs1f1+αttβs2+2βs1+t2,f1.
If the following inequality holds,(44)2βs1+tβs1f1+αttβs2+2βs1+t2≥f1f1≤α2βs1+t3βs1+βs2+2t
Under the present circumstances, the state-owned container terminal has no motivation to deviate fromf2=f1 unilaterally.
On the other hand, the profit function of the foreign-owned container terminal is given by(45)f1-βs2+ts1tβs1+βs2+tf1+βs1s2tβs1+βs2+tf2+αs1βs1+βs2+t
With some algebra, the function is concave and is maximized at(46)f1=βs2f2+αt2βs2+t
Ifαt/(βs2+2t)≤f2, βs2f2+αt/2βs2+t≤f2. Thus, given f2≥αt/(βs2+2t), the foreign-owned container terminal will not raise its tariff. Otherwise, the profits of foreign-owned container terminal become(47)f1-s1βs1+βs2+tf1+αs1βs1+βs2+twhich is increasing when f1≤α(2βs1+t)/3βs1+βs2+2t<α/2.
Therefore, foreign-owned container terminal has no motivation to reduce its tariff either.Proposition 4.
There are no other price-matching equilibria only employed by the state-owned container terminal satisfyingf1=f2.Proof.
In consideration of an equilibrium satisfyingf1=f2>α(2βs1+t)/3βs1+βs2+2t, then (2βs1+t)(βs1f1+αt)/tβs2+2βs1+t2<f1=f2. Thus, from the proof of Proposition 3 above, we understand that the state-owned container terminal has a motivation to deviate to (2βs1+t)(βs1f1+αt)/tβs2+2βs1+t2 unilaterally.
On the other hand, iff1=f2<αt/(βs2+2t), βs2f2+αt/2βs2+t>f2=f1.
According to the profit function of the foreign-owned container terminal, which is concave, it is maximized at(48)βs2f2+αt2βs2+t
Therefore, the foreign-owned container terminal would be better off if it increases its tariff fromf1 to βs2f2+αt/2βs2+t.In brief, the equilibrium off1=f2=α(2βs1+t)/3βs1+βs2+2t incurs the highest profits and social welfare among all equilibria for both container terminals, which are given as follows:(49)π14=a2s12βs1+t3βs1+βs2+2t2(50)w24=a25βs12+6βs1s2+βs22+4s1t+2s2t23βs1+βs2+2t2


## 3.1. Price-Matching Strategy Is Employed by Neither Container Terminal

Both container terminals have identical and constant variable costs in consideration of locating within the same port, and the variable costs are normalized to zero on the supposition that basically the same equipment and related fees of energy consumption, truck drayage, handling labour, etc. exist for the two container terminals in one port area.Since neither the foreign-owned or state-owned container terminal has a conjecture about the other one, the container terminal owned by foreign capital will maximize its own profits as follows:(5)maxf1⁡π1f1=f1.s1βs2f2-βs2+tf1+attβs1+βs2+tBy taking the derivative, the first-order conditions (FOCs) and the second-order conditions (SOCs) are given by(6)∂π1∂f1=-2βs2+ts1tβs1+βs2+tf1+βs1s2tβs1+βs2+tf2+αs1βs1+βs2+t∂2π1∂f12=-2βs2+ts1tβs1+βs2+tTo solve container terminal 1’s optimization problem, we set the first-order condition equal to zero as(7)-2βs2+ts1tβs1+βs2+tf1+βs1s2tβs1+βs2+tf2+αs1βs1+βs2+t=0which leads to the following solution of the container terminal invested by foreign capital:(8)f1=βs2f2+αt2βs2+tWith regard to the container terminal established by state-owned capital, which can be considered as an economic entity where port authority is controlled by the local government, the pricing objectives of the state-owned container terminal not only include its own profits, similar to the foreign-owned container terminal, but also contain the surplus of the shippers.According to the classic quadratic utility function (see Singh and Vives, [22]), the continuum of shippers of the same type with a utility function can be expressed as follows:(9)Uq1,q2=α1q1+α2q2-β1q12+2γq1q2+β2q222To avoid social welfare loss with port tariff deregulation, the state-owned container terminal should charge its tariff to realize the social welfare maximization:(10)maxf2⁡wf2=αq1+q2-βq1+q222+π2-f2+t·q2s2q2Similarly, the first-order condition can be rewritten as(11)∂w2∂f2=-αs2βs1+βs2+t-βαs1+s2-f1s1-f2s2βs1+βs2+t∗-s2βs1+βs2+t-2t∗-βs1+ts2tβs1+βs2+tf2+βs1s2tβs1+βs2+tf1+αs2βs1+βs2+t∗-βs1+ttβs1+βs2+tBy setting it equal to zero, we obtain the following solution:(12)f1=2atβs1+tβs1+βs2+t2β2s12βs2+2t+β2s2t7s1+2s2+2t24βs1+3βs2+2tf2=at2βs1+tβs1+2βs2+2t2β2s12βs2+2t+β2s2t7s1+2s2+2t24βs1+3βs2+2tTherefore, the profits and social welfare for both container terminals are given, respectively, by(13)π11=4a2s1tβs1+t2βs2+tβs1+βs2+t2β2s12βs2+2t+β2s2t7s1+2s2+2t24βs1+3βs2+2t2(14)w21=a24β5s14s22+16β4s14s2t+28β4s13s22t+8β4s12s23t+12β3s14t2+72β3s13s2t2+73β3s12s22t2+28β3s1s23t2+4β3s24t2+40β2s13t3+110β2s12s2t3+76β2s1s22t3+16β2s23t3+44βs12t4+64βs1s2t4+20βs22t4+16s1t5+8s2t52β3s12s2+4β2s12t+7β2s1s2t+2β2s22t+8βs1t2+6βs2t2+4t3-2


## 3.2. Price-Matching is Employed by Both Container Terminals

According to price-matching strategies employed by foreign-owned and state-owned container terminals, the two container shippers should charge the same tariff to the shipper. Under these circumstances, each container terminal forecasts that one unilateral tariff cut below the other’s counterpart will be matched by the other.Therefore, the profits of a foreign-owned container terminal can be rewritten as(15)min⁡f1,f2-βs2+ts1tβs1+βs2+tmin⁡f1,f2+βs1s2tβs1+βs2+tmin⁡f1,f2+αs1βs1+βs2+tIn consideration of a fixed value of the state-owned container terminal’s tariff, the foreign-owned container terminal’s strategyf1>f2 has the same result as the strategy f1=f2. For this reason, we only analyse f1≤f2, for which the foreign-owned container terminal’s optimization problem becomes(16)maxf1:f1≤f2⁡π1f1=f1-s1βs1+βs2+tf1+αs1βs1+βs2+twhich is solved at f1∗=α/2. Otherwise, there exists f1∗=f2.Similarly, the first-order condition the state-owned container terminal of can be rewritten as(17)∂w2∂f2=-αs1+s2βs1+βs2+t-βs1+s2a-f2βs1+βs2+t∗-s1+s2βs1+βs2+t-2t∗a-f2βs1+βs2+t∗-s2βs1+βs2+twhich is solved as(18)f2∗=ats2-s1βs1+s22+2s2tIf the following inequality holds:(19)2βs2+tt-βs2-tβ≤s1≤s2Therefore, we obtainf1∗=f2∗=α/2. Otherwise, we can get f1∗=f2∗=ats2-s1/βs1+s22+2s2t.In the meantime, the profits and social welfare for both container terminals are given, respectively, by(20)π12=s1βs1+βs2+t·α22(21)w22=3βs1+s22+2t2s1+s22βs1+βs2+t2·α22


## 3.3. Price-Matching Is Only Employed by Foreign-Owned Container Terminal

At this very moment, the foreign-owned container terminal employs the price-matching strategy, while the state-owned container terminal does not.The profits of both container terminals can be expressed as(22)π1f1=min⁡f1,f2-βs2+ts1tβs1+βs2+tmin⁡f1,f2+βs1s2tβs1+βs2+tf2+αs1βs1+βs2+tπ2f2=f2-βs1+ts2tβs1+βs2+tf2+βs1s2tβs1+βs2+tmin⁡f1,f2+αs2βs1+βs2+tProposition 1.
It is necessary to havef1=f2 in any equilibrium of price-matching only employed by the foreign-owned container terminal.Proof.
On the supposition off1>f2, by taking the price-matching strategy employed by the foreign-owned container terminal into consideration, the social welfare of state-owned container terminal can also be rewritten as(23)maxf2⁡wf2=αs1+s2a-f2βs1+βs2+t-βs1+s2a-f2/βs1+βs2+t22-t·s2a-f2/βs1+βs2+t2s2
The first-order condition can be obtained by(24)-as1+s2βs1+βs2+t-βs1+s2a-f2βs1+βs2+t∗-s1+s2βs1+βs2+t-2t∗a-f2βs1+βs2+t∗-s2βs1+βs2+t=0
Thus, the second-order condition of it is given by(25)-βs1+s2βs1+βs2+t∗s1+s2βs1+βs2+t-2tβs1+βs2+t∗s2βs1+βs2+t<0
Evidently, the social welfare function of the state-owned container terminal is concave and maximized at(26)f2=αts2-s12ts2+βs1+s22
Given the state-owned container terminal 2’s optimal tariff isαts2-s1/2ts2+βs1+s22, we can obtain the foreign-owned container terminal’s profits as follows:(27)min⁡f1,f2-βs2+ts1tβs1+βs2+tmin⁡f1,f2+βs1s2tβs1+βs2+t∗αts2-s12ts2+βs1+s22+αs1βs1+βs2+twhich is maximized at(28)f1=βs2αts2-s1/2ts2+βs1+s22+αt2βs2+t
Therefore,(29)△f=f2-f1=-αs1tβs1+3βs2+2t2βs2+t2ts2+βs1+s22<0
It is a contradiction to the supposition off1>f2 in the equilibrium of the price-matching that is only employed by the foreign-owned container terminal.
Similarly, suppose thatf1<f2 in equilibrium of price-matching is only employed by the foreign-owned container terminal. The state-owned container terminal will maximize social welfare at(30)f2=2βs1+tβs1f1+αttβs2+2βs1+t2
Consequently, the foreign-owned container terminal’s profits are maximized at(31)f1=βs2f2+αt2βs2+t
Therefore, we can obtain the following inequality expression:(32)△f=f2-f1=αβs1t2βs2+t2β2s12βs2+2t+β2s2t7s1+2s2+2t24βs1+3βs2+2t>0
In conclusion, the price-matching strategy only employed by the foreign-owned container terminal has no effect on the equilibrium outcome in this game.


## 3.4. Price-Matching Is Only Employed by State-Owned Container Terminal

In contrast to the game of price-matching only employed by foreign-owned container terminal, the profits of both container terminals are, respectively, given by(33)π1f1=f1-βs2+ts1tβs1+βs2+tf1+βs1s2tβs1+βs2+tmin⁡f1,f2+αs1βs1+βs2+twf2=αq1+q2-βq1+q222+π2-min⁡f1,f2+t·q2s2q2The above functions can also be rewritten as(34)π1f1=f1-βs2+ts1tβs1+βs2+tf1+βs1s2tβs1+βs2+tf2+αs1βs1+βs2+twf2=ααs1+s2-f1s1-f2s2βs1+βs2+t-βαs1+s2-f1s1-f2s2/βs1+βs2+t22-t·αs1+s2-f1s1-f2s2/βs1+βs2+t2s2Proposition 2.
No price-matching equilibrium only employed by the state-owned container terminal satisfiesf2<f1.Proof.
On the supposition that there exists an equilibrium satisfyingf2<f1, the state-owned container terminal’s objective function is as follows:(35)maxf2:f2≤f1⁡ααs1+s2-f1s1-f2s2βs1+βs2+t-βαs1+s2-f1s1-f2s2/βs1+βs2+t22-t·αs1+s2-f1s1-f2s2/βs1+βs2+t2s2which is maximized at(36)min⁡2βs1+tβs1f1+αttβs2+2βs1+t2,f1
Iff1>α(2βs1+t)/3βs1+βs2+2t,f2=(2βs1+t)(βs1f1+αt)/tβs2+2βs1+t2<f1. Given f2<f1, foreign-owned container terminal’s profits can be rewritten as(37)f1-βs2+ts1tβs1+βs2+tf1+βs1s2tβs1+βs2+tf2+αs1βs1+βs2+twhich is maximized at(38)f1=βs2f2+αt2βs2+t
Thus, we have(39)f1=βs22βs1+tβs1f1+αt/tβs2+2βs1+t2+αt2βs2+twhich is solved at(40)f1=2αtβs1+tβs1+βs2+t2β2s12βs2+2t+β2s2t7s1+2s2+2t24βs1+3βs2+2t
That is,(41)f1-α2βs1+t3βs1+βs2+2t=-2aβ3s122βs1s2+s1t+4s2t+aβs1t2β2s22+2βs1t+5βs2t+2t23βs1+βs2+2t2β2s12βs2+2t+β2s2t7s1+2s2+2t24βs1+3βs2+2t
The value off1 above is less than α(2βs1+t)/3βs1+βs2+2t, which is a contradiction.Proposition 3.
Any tariff combination satisfyingαt/βs2+2t≤f1=f2≤α(2βs1+t)/3βs1+βs2+2t is a Nash equilibrium of the price-matching only employed by the state-owned container terminal.Proof.
The state-owned container terminal’s objective function(42)maxf2:f2≤f1⁡ααs1+s2-f1s1-f2s2βs1+βs2+t-βαs1+s2-f1s1-f2s2/βs1+βs2+t22-t·αs1+s2-f1s1-f2s2/βs1+βs2+t2s2is maximized at(43)min⁡2βs1+tβs1f1+αttβs2+2βs1+t2,f1.
If the following inequality holds,(44)2βs1+tβs1f1+αttβs2+2βs1+t2≥f1f1≤α2βs1+t3βs1+βs2+2t
Under the present circumstances, the state-owned container terminal has no motivation to deviate fromf2=f1 unilaterally.
On the other hand, the profit function of the foreign-owned container terminal is given by(45)f1-βs2+ts1tβs1+βs2+tf1+βs1s2tβs1+βs2+tf2+αs1βs1+βs2+t
With some algebra, the function is concave and is maximized at(46)f1=βs2f2+αt2βs2+t
Ifαt/(βs2+2t)≤f2, βs2f2+αt/2βs2+t≤f2. Thus, given f2≥αt/(βs2+2t), the foreign-owned container terminal will not raise its tariff. Otherwise, the profits of foreign-owned container terminal become(47)f1-s1βs1+βs2+tf1+αs1βs1+βs2+twhich is increasing when f1≤α(2βs1+t)/3βs1+βs2+2t<α/2.
Therefore, foreign-owned container terminal has no motivation to reduce its tariff either.Proposition 4.
There are no other price-matching equilibria only employed by the state-owned container terminal satisfyingf1=f2.Proof.
In consideration of an equilibrium satisfyingf1=f2>α(2βs1+t)/3βs1+βs2+2t, then (2βs1+t)(βs1f1+αt)/tβs2+2βs1+t2<f1=f2. Thus, from the proof of Proposition 3 above, we understand that the state-owned container terminal has a motivation to deviate to (2βs1+t)(βs1f1+αt)/tβs2+2βs1+t2 unilaterally.
On the other hand, iff1=f2<αt/(βs2+2t), βs2f2+αt/2βs2+t>f2=f1.
According to the profit function of the foreign-owned container terminal, which is concave, it is maximized at(48)βs2f2+αt2βs2+t
Therefore, the foreign-owned container terminal would be better off if it increases its tariff fromf1 to βs2f2+αt/2βs2+t.In brief, the equilibrium off1=f2=α(2βs1+t)/3βs1+βs2+2t incurs the highest profits and social welfare among all equilibria for both container terminals, which are given as follows:(49)π14=a2s12βs1+t3βs1+βs2+2t2(50)w24=a25βs12+6βs1s2+βs22+4s1t+2s2t23βs1+βs2+2t2


## 4. Comparison

Theorem 5.
The Nash equilibria are price-matching strategies employed by neither container terminal or only by the foreign-owned container terminal.Proof.
According to the payoffs of both container terminals for each game that are given above, as well as the assumption of (2βs2+tt-βs2-t)/β≤s1<s2, it is sufficient to show that π12>π14, w22<w21=w23, and w24<w21.
These inequalities can be shown as follows:(51)π12-π14=a2β2s1s1-s224βs1+βs2+t3βs1+βs2+2t2>0w22-w21w23=-a2βs24β6s16s2+8β6s15s22+4β6s14s23+16β5s16t+76β5s15s2t+104β5s14s22t+44β5s13s23t+8β5s12s24t+88β4s15t2+297β4s14s2t2+302β4s13s22t2+133β4s12s23t2+36β4s1s24t2+4β4s25t2+200β3s14t3+488β3s13s2t3+366β3s12s22t3+124β3s1s23t3+16β3s24t3+232β2s13t4+376β2s12s2t4+168β2s22s1t4+20β2s23t4+136βs12t5+112βs1s2t5+8βs22t5+32s1t682β4s13s2+2β4s12s22+4β3s13t+13β3s12s2t+9β3s1s22t+2β3s23t+12β2s12t2+21β2s1s2t2+8β2s22t2+12βs1t3+10βs2t3+4t42-1<0w24-w21=-a2βs128β6s14s22+32β5s14s2t+72β5s13s22t+20β5s12s23t+14β4s14t2+156β4s13s2t2+192β4s12s22t2+60β4s1s23t2+8β4s24t2+60β3s13t3+279β3s12s2t3+214β3s1s22t3+44β3s23t3+94β2s12t4+220β2s1s2t43+84β2s22t4+64βs1t5+64βs2t5+16t66β4s13s2+2β4s12s22+12β3s13t+29β3s12s2t+13β3s1s22t+2β3s23t+32β2s12t2+40β2s1s2t2+10β2s22t2+28βs1t3+16βs2t3+8t4-2<0Theorem 6.
The price-matching not being adopted by the state-owned container terminal averts tacit collusion.Proof.
It is sufficient to show that the collusive tariffs are equal to the equilibrium tariffs under the price-matching adopted by both container terminals, i.e.,f1=f2=α/2, which also solves the following optimization problem:(52)maxf1,f2⁡f1.s1βs2f2-βs2+tf1+attβs1+βs2+t+f2.s2βs1f1-βs1+tf2+attβs1+βs2+tTheorem 7.
The price-matching strategies employed by both container terminals result in total social welfare loss.Proof.
The total social welfare is the sum of both foreign-owned and state-owned container terminals’ profits and the consumer surplus, which can be written as(53)tw=w2+π1-f1+t·q1s1q1
If the price-matching strategy is employed by neither container terminal, the total social welfare under this circumstance is denoted astw1. However, if price-matching strategy is employed by both container terminals, we have q1=s1/βs1+βs2+t·α/2 and q2=s2/βs1+βs2+t·α/2, and the total social welfare in this game is labelled as tw2.
The total social welfare loss can be rewritten as(54)Δtw=tw1-tw2=a2β2s24β5s16s2+8β5s15s22+4β5s14s23+16β4s16t+52β4s15s2t+40β4s14s22t+12β4s13s23t+8β4s12s24t+56β3s15t2+97β3s14s2t2+62β3s13s22t2+69β3s12s23t2+36β3s1s24t2+4β3s25t2+56β2s14t3+58β2s13s2t3+102β2s12s22t3+100β2s1s23t3+16β2s24t3+8βs13t4+32βs12s2t4+88βs22s1t4+20βs23t4+24s1s2t5+8t5s2+s1s2-s18∗2β4s13s2+2β4s12s22+4β3s13t+13β3s12s2t+9β3s1s22t+2β3s23t+12β2s12t2+21β2s1s2t2+8β2s22t2+12βs1t3+10βs2t3+4t42-1>0
Ass1≤s2 is assumed, especially under the circumstance of (2βs2+tt-βs2-t)/β≤s1<s2, we can obtain that above expression is positive, which completes the proof.


## 5. Simulation Example

In this section, we will verify some of the analytical results and further explore the managerial insights and policy implications through a simulation example of Shenzhen Port, which reflects some different scenarios.


### 5.1. Background Information about the Case Study

Shenzhen Port is located in the Pearl River Delta and is spread along Shenzhen city’s 260 km coastline. It is separated by the New Territories and the Kowloon Peninsula of Hong Kong into two areas: the western port area and the eastern port area. The former mainly contains Shekou Container Terminal (SCT), and the latter refers to the Yantian International Container Terminal (YICT). As an important gateway for the import and export trade of China, Shenzhen Port is one of the hubs in the Pearl River Delta. It handles a container throughput of 25.21 (million TEUs) in 2017, with its volume increasing by 5.13 percent, thus ranking it as the third container port in the world for five consecutive years.Firstly, SCT was founded in 1989 with the registered capital of 0.6 billion Hong Kong dollars. As a wholly foreign-owned container terminal, China Merchants Holdings (International) Company Limited (CMHI) owns 100% of SCT. Headquartered in Hong Kong, CMHI’s is the flagship company of China Merchants Group. CMHI was listed on the Hong Kong Stock Exchange in 1992. Upon robust growth and increased market recognition, it became a Hong Kong Hang Seng Index constituent stock in September 2004. In 2017, the designed capacity of SCT reached 3.8 (million TEUs) and it handled a container throughput of 5.3 (million TEUs).Secondly, YICT was established in 1993. As a joint venture by the Shenzhen Yantian Port Group and Hutchison Port Holdings Trust (HPH Trust), the former is a state-owned enterprise affiliated with State-owned Assets Supervision and the Administration Commission of Shenzhen Municipal People’s Government. The latter is the first publicly traded container port business trust listed on the Singapore Exchange. It is affiliated with Hutchison Ports, who is the global leader in the container port industry based on throughput, and a subsidiary of CK Hutchison Holdings Limited. HPH Trust owns interests in the world class deep-water container port assets located in two of the world’s busiest container port cities by throughput at Hong Kong and Shenzhen. In 2017, the designed capacity of YICT is 8.8 (million TEUs), and it handled a container throughput of 12.7 (million TEUs).


### 5.2. Assumptions about the Parameters of the Model

First, according to background information about the case study, the designed capacities of container terminals are set ats1=3.8millionTEUsands2=8.8(millionTEUs), respectively.Second, following the survey conducted by Zheng and Negenborn (2014) and the related data of the port of Shenzhen in 2017, the average container dwell time in the port of Shenzhen is approximately 6 days, and the daily storage charge for a 20 feet dry container is 11 RMB. Considering the largest value that the volume-capacity can take is 3, the value oft is set at 22 RMB (11∗6/3).Third, the cross-price parameter is a measure of the sensitivity of the demand for one container port to changes in the other container port’s price. Luo et al. [29] set this value at 0.1 million TEUs/HK$. Combined with (3) and (4), we have a/βs1+βs2+t=3 and βs1s2/t(βs1+βs2+t)=0.1, which means that α=385.85 and β=8.46.All parameters values used in the application are summarized in Table1.Table 1
Parameters values in the case study.
Parameter Concept Value a Potential demand 385.85 s 1 Designed capacity 3.8 s 2 Designed capacity 8.8 t Waiting costs 22 β Tariff sensitivity 8.46


### 5.3. Comparison Analysis

After applying the values in Table1 to the model, we obtain Table 2.Table 2
Equilibrium of no price-matching and price-matching strategies.
Game f 1 (RMB) f 2 (RMB) q 1 (million TEUs) q 2 (million TEUs) π 1 (million RMB) π 2 (million RMB) Total welfare(million RMB) Subgame 1 99 133 11.88 13.91 1175 1849 5837 Subgame 2 193 193 5.70 13.20 1100 2547 5158From Table2, we find some interesting observations as follows.(i)
A higher terminal tariff causes a lower container throughput. In particular, the container throughput of the foreign-owned terminal falls faster.(ii)
A higher terminal tariff also causes more profit for the state-owned terminal and less profit for the foreign-owned terminal.(iii)
A higher terminal tariff eventually leads to the total social welfare decreasing more quickly.


### 5.4. Sensitivity Analysis

We perform a sensitivity analysis to gain further managerial insights. Specifically, the sensitivity analysis can be used to judge the effect trends of the container terminal throughput and the whole throughput as well as the total social welfare of the container terminals in one port if the designed capacity expands.We present the results in Table3.Table 3
Sensitivity analysis of increasing container terminal capacity.
Scenario s 1 q i (millionTEUs) q j (millionTEUs) π 1 (million RMB) π 2 (million RMB) Total welfare(million RMB) Case 1 3.8 5.70 13.20 1100 2547 5158 Case 2 4.3 6.25 12.78 1205 2466 5202 Case 3 4.8 6.76 12.39 1304 2390 5244 Case 4 5.3 7.24 12.02 1396 2318 5283 Case 5 5.8 7.69 11.67 1484 2251 5319 Case 6 6.3 8.12 11.34 1566 2187 5354 Case 7 6.8 8.52 11.03 1644 2127 5387 Case 8 7.3 8.90 10.73 1717 2070 5418 Case 9 7.8 9.26 10.45 1787 2016 5448 Case 10 8.3 9.61 10.19 1854 1965 5476 Case 11 8.8 9.93 9.93 1917 1917 5503In Table3, we increase the capacity of foreign-owned container terminal by 0.5 million TEUs at a time while keeping all other parameters the same as the base case that was illustrated in Table 1. From the base case (Case 1) to Case 11, with the designed capacity of foreign-owned container terminal expanding by 131.6%, the container throughput and profit of the foreign-owned container terminal continually increase 74.2% and 74.3%, respectively. In addition, the total social welfare of the port remains positive, thus increasing by 6.7%.Table3 shows that without external competition from other ports, the foreign-owned container terminal has an incentive to expand its capacity. If all other parameters are maintained, we find some interesting results as follows:(i)
A larger capacity of the foreign-owned container terminal leads to a higher container throughput for the container terminal, while that of the state-owned container terminal drops slightly. Consequently, the total container throughput remains increasing.(ii)
A larger capacity of the foreign-owned container terminal also leads to more profit for it and lower profit for the state-owned container terminal.(iii)
A larger capacity of the foreign-owned container terminal eventually causes the total social welfare to continually increase.


## 5.1. Background Information about the Case Study

Shenzhen Port is located in the Pearl River Delta and is spread along Shenzhen city’s 260 km coastline. It is separated by the New Territories and the Kowloon Peninsula of Hong Kong into two areas: the western port area and the eastern port area. The former mainly contains Shekou Container Terminal (SCT), and the latter refers to the Yantian International Container Terminal (YICT). As an important gateway for the import and export trade of China, Shenzhen Port is one of the hubs in the Pearl River Delta. It handles a container throughput of 25.21 (million TEUs) in 2017, with its volume increasing by 5.13 percent, thus ranking it as the third container port in the world for five consecutive years.Firstly, SCT was founded in 1989 with the registered capital of 0.6 billion Hong Kong dollars. As a wholly foreign-owned container terminal, China Merchants Holdings (International) Company Limited (CMHI) owns 100% of SCT. Headquartered in Hong Kong, CMHI’s is the flagship company of China Merchants Group. CMHI was listed on the Hong Kong Stock Exchange in 1992. Upon robust growth and increased market recognition, it became a Hong Kong Hang Seng Index constituent stock in September 2004. In 2017, the designed capacity of SCT reached 3.8 (million TEUs) and it handled a container throughput of 5.3 (million TEUs).Secondly, YICT was established in 1993. As a joint venture by the Shenzhen Yantian Port Group and Hutchison Port Holdings Trust (HPH Trust), the former is a state-owned enterprise affiliated with State-owned Assets Supervision and the Administration Commission of Shenzhen Municipal People’s Government. The latter is the first publicly traded container port business trust listed on the Singapore Exchange. It is affiliated with Hutchison Ports, who is the global leader in the container port industry based on throughput, and a subsidiary of CK Hutchison Holdings Limited. HPH Trust owns interests in the world class deep-water container port assets located in two of the world’s busiest container port cities by throughput at Hong Kong and Shenzhen. In 2017, the designed capacity of YICT is 8.8 (million TEUs), and it handled a container throughput of 12.7 (million TEUs).


## 5.2. Assumptions about the Parameters of the Model

First, according to background information about the case study, the designed capacities of container terminals are set ats1=3.8millionTEUsands2=8.8(millionTEUs), respectively.Second, following the survey conducted by Zheng and Negenborn (2014) and the related data of the port of Shenzhen in 2017, the average container dwell time in the port of Shenzhen is approximately 6 days, and the daily storage charge for a 20 feet dry container is 11 RMB. Considering the largest value that the volume-capacity can take is 3, the value oft is set at 22 RMB (11∗6/3).Third, the cross-price parameter is a measure of the sensitivity of the demand for one container port to changes in the other container port’s price. Luo et al. [29] set this value at 0.1 million TEUs/HK$. Combined with (3) and (4), we have a/βs1+βs2+t=3 and βs1s2/t(βs1+βs2+t)=0.1, which means that α=385.85 and β=8.46.All parameters values used in the application are summarized in Table1.Table 1
Parameters values in the case study.
Parameter Concept Value a Potential demand 385.85 s 1 Designed capacity 3.8 s 2 Designed capacity 8.8 t Waiting costs 22 β Tariff sensitivity 8.46


## 5.3. Comparison Analysis

After applying the values in Table1 to the model, we obtain Table 2.Table 2
Equilibrium of no price-matching and price-matching strategies.
Game f 1 (RMB) f 2 (RMB) q 1 (million TEUs) q 2 (million TEUs) π 1 (million RMB) π 2 (million RMB) Total welfare(million RMB) Subgame 1 99 133 11.88 13.91 1175 1849 5837 Subgame 2 193 193 5.70 13.20 1100 2547 5158From Table2, we find some interesting observations as follows.(i)
A higher terminal tariff causes a lower container throughput. In particular, the container throughput of the foreign-owned terminal falls faster.(ii)
A higher terminal tariff also causes more profit for the state-owned terminal and less profit for the foreign-owned terminal.(iii)
A higher terminal tariff eventually leads to the total social welfare decreasing more quickly.


## 5.4. Sensitivity Analysis

We perform a sensitivity analysis to gain further managerial insights. Specifically, the sensitivity analysis can be used to judge the effect trends of the container terminal throughput and the whole throughput as well as the total social welfare of the container terminals in one port if the designed capacity expands.We present the results in Table3.Table 3
Sensitivity analysis of increasing container terminal capacity.
Scenario s 1 q i (millionTEUs) q j (millionTEUs) π 1 (million RMB) π 2 (million RMB) Total welfare(million RMB) Case 1 3.8 5.70 13.20 1100 2547 5158 Case 2 4.3 6.25 12.78 1205 2466 5202 Case 3 4.8 6.76 12.39 1304 2390 5244 Case 4 5.3 7.24 12.02 1396 2318 5283 Case 5 5.8 7.69 11.67 1484 2251 5319 Case 6 6.3 8.12 11.34 1566 2187 5354 Case 7 6.8 8.52 11.03 1644 2127 5387 Case 8 7.3 8.90 10.73 1717 2070 5418 Case 9 7.8 9.26 10.45 1787 2016 5448 Case 10 8.3 9.61 10.19 1854 1965 5476 Case 11 8.8 9.93 9.93 1917 1917 5503In Table3, we increase the capacity of foreign-owned container terminal by 0.5 million TEUs at a time while keeping all other parameters the same as the base case that was illustrated in Table 1. From the base case (Case 1) to Case 11, with the designed capacity of foreign-owned container terminal expanding by 131.6%, the container throughput and profit of the foreign-owned container terminal continually increase 74.2% and 74.3%, respectively. In addition, the total social welfare of the port remains positive, thus increasing by 6.7%.Table3 shows that without external competition from other ports, the foreign-owned container terminal has an incentive to expand its capacity. If all other parameters are maintained, we find some interesting results as follows:(i)
A larger capacity of the foreign-owned container terminal leads to a higher container throughput for the container terminal, while that of the state-owned container terminal drops slightly. Consequently, the total container throughput remains increasing.(ii)
A larger capacity of the foreign-owned container terminal also leads to more profit for it and lower profit for the state-owned container terminal.(iii)
A larger capacity of the foreign-owned container terminal eventually causes the total social welfare to continually increase.


## 6. Conclusions

With gradual deregulation of foreign investment into its port sector, Sino-foreign joint ventures and even wholly foreign-owned terminals are permitted in China. In the meantime, the latest reform of China was to deregulate the port tariffs according to China’s Ministry of Transport and National Development and Reform Commission, which issued the method of port tariffs on July 12, 2017. Under this background, the container terminal operator of China can independently formulate specific tariffs no more than the upper bound of the government’s guidance, such as cargo dues, pilotage fees, tug fees, parking fees, special trimming charges, and oil fence charges. Moreover, the competitive service is charged based on governmental guidance and uniform pricing from market regulation. At this moment, although price-matching guarantees seem available to container terminals according to the port deregulation of the pricing policy, it is not clear whether they will employ price-matching policies or not, or whether the price-matching strategy is the only Nash equilibrium for the two container terminals of one port. We are also concerned with how to avoid tacit collusion between container terminals, which would otherwise result in social welfare losses in the process of port deregulation.In this paper, we focus on port deregulation’s influence on the price-matching strategy of container terminals in one port, which is mainly financed by foreign or state capital, using a two-stage noncooperative game theoretical model, and then solve the game through backwards-induction. The main findings are different from previous conclusions in that the only Nash equilibrium available is for both terminals to adopt the price-matching policy if the capacities of the two terminals are not equal, as well as price-matching strategies facilitating tacit collusion between container terminals. To address the differential designed capacity and port deregulation of the two container terminals in one port comprehensively, the main contribution of this paper to the intraport competition literature is that price-matching adopted by both container terminals is not the Nash equilibrium, even though the foreign-owned container terminal maximizes its own profit. Therefore, as an economic entity with port authority, the state-owned container terminal formulated its pricing strategy from the perspective of the local government, comprehensively considering its own profits and the surplus of the container shipper. The policy recommendation is that the price-matching strategy not employed by the state-owned container terminal can avert tacit collusion between the container terminals of one port. Due to raising the equilibrium container port tariff, the price-matching strategy employed by the container terminals of one port will eventually result in potential social welfare losses.The author recognizes that there is still space to further study this topic. Potential directions for future study include but are not limited to the following. First, we should further enrich the current optimizing model to consider the different marginal operation cost and technological spillover, as well as trade facilitation, revenue maximization and other objectives. Second, the case study could be better linked with various aspects of the model when further detailed data are able to be collected in the future, as it is usually a combination of commercial, governance, institutional, and social decisions. Third, more quantitative tests and calculations of the game in this case remain. Finally, the optimization of the pricing strategy of container terminal should be taken into the global supply chain consideration in future study.



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*Source: 1626429-2018-11-22.xml*