External competition between HKP and SZP
According to Cournot price competition model, each port who provides heterogeneous services regards the price of its competitor as an established price and changes its price in the same direction with its competitor in order to maximise its profits.
As the two ports provide heterogeneous services, they have different demand functions. Therefore, we need to firstly embody the heterogeneity in the demand function. In this study, utility function of port consumers is applied to extract the demand functions of HKP and SZP.
The consumers are denoted with the parameter
ϕ and
ϕ ∈ [
n,
m], where 0 <
n <
m, which is uniformly distributed with density equal to 1. Then preference
U of consumer
ϕ is described by the expected utility function
Ui(
ϕ,
Pi) =
ϕui −
Pi. Let
\( \overline{\phi} \) indicates the consumers with indifferent preference on port services. By solving
Uh(
ϕ,
Ph) =
Us(
ϕ,
Ps). We can obtain the
\( \overline{\phi} \) as:
$$ \overline{\phi}=\frac{P_h-{P}_s}{u_h-{u}_s} $$
(1)
Now we assume the demand functions of HKP and SZP as:
$$ {Q}_h=m-\overline{\phi}=m-\frac{P_h-{P}_s}{u_h-{u}_s} $$
(2)
$$ {Q}_s=\overline{\phi}-n={\frac{P_h-{P}_s}{u_h-u}}_s-n $$
(3)
To simplify the calculation process, we assume
k =
uh −
us which indicates the difference of consumer’s utility toward the services provided by HKP and SZP, then we have:
$$ {Q}_h=m-\frac{P_h-{P}_s}{k} $$
(4)
$$ {Q}_s=\frac{P_h-{P}_s}{k}-n $$
(5)
With the demand functions, we can formulate the profits of HKP and SZP as:
$$ {\prod}_h={Q}_h\times \left({P}_h-{C}_h\right)-F{C}_h $$
(6)
$$ {\prod}_s={Q}_s\times \left({P}_s-{C}_s\right)-F{C}_s $$
(7)
The profit of HKP can be further described as:
$$ {\prod}_h=m{P}_h-m{C}_h-F{C}_h+\frac{P_h{P}_s-{P}_h^2+{P}_h{P}_h-{P}_s{C}_h}{k} $$
(8)
To maximise the profit of HKP, we take the partial derivative
Ph for (8) and it should be equal to 0:
$$ \frac{\partial {\prod}_h\left({P}_h,{P}_s\right)}{\partial {P}_h}=m+\frac{P_s+{C}_h-2{P}_h}{k}=0 $$
(9)
Then the optimal (profit-max) price function of HKP under inter-port competition should be:
$$ {P}_h=\frac{mk+{P}_s+{C}_h}{2} $$
(10)
Substitute it into demand function, we can find the optimal throughput function for HKP is:
$$ {Q}_h=m-\frac{mk-{P}_s+{C}_h}{2k} $$
(11)
Similarly, we can obtain the optimal price function for SZP, which is:
$$ {P}_s=\frac{P_h+{C}_s- nk}{2} $$
(12)
Then the optimal throughput function for SZP is:
$$ {Q}_s=\frac{P_{h-}{C}_s+ nk}{2k}-n $$
(13)
By solving the eq. (
10) and (
12) together, we can obtain the optimal prices for HKP and SZP respectively, which are also the equilibrium prices under inter-port competition:
$$ {P}_h=\frac{\left(2m-n\right)k+2{C}_h+{C}_s}{3} $$
(14)
$$ {P}_s=\frac{\left(m-2n\right)k+{C}_h+2{C}_s}{3} $$
(15)
Equation (
14) indicates a positive correlation between
Ph and
k. It means when the difference of customer utility towards HKP and SZP narrows, HKP has to reduce its price to improve the attractiveness to consumers.
Substitute
Ph and
Ps into (4) and (5), we can also obtain the equilibrium throughput for HKP and SZP as:
$$ {Q}_h=\frac{\left(2m-n\right)k+{C}_s-{C}_h}{3k} $$
(16)
$$ {Q}_s=\frac{\left(m-2n\right)k+{C}_h-{C}_s}{3k} $$
(17)
Internal competition between MTL and the HIT/ACT/CHT collaboration
In this section, we explore the intra-port competitive equilibrium of HKP of case 1 and case 2 scenarios, respectively.
Case 1: Price competition (Edgeworth model)
When Qh > CAa, according to Edgeworth model, there is no stable equilibrium price in the market and the market price will fluctuate between the monopoly market price and the perfect competitive market price. Therefore, we calculate the monopoly market price Pm (upper limit) and the perfect competitive market price Pc (lower limit). As the HIT/ACT/CHT collaboration and MTL produce homogeneous products, we assume that all terminals in HKP have the same cost function and cost structure.
Firstly, we assume the market demand curve of HKP is linear as:
In a monopoly market, with only one container terminal operator being the sole supplier for the whole market, the market demand curve is the operator’s demand curve. To maximise its profit, the marginal revenue (MR) of this operator must be equal to its marginal cost (MC). MR and MC are the derivatives of total revenue (TR) and total cost (TC) respectively:
The
MR can be calculated as:
$$ MR={(TR)}^{\prime }={\left({P}_0Q-b{Q}^2\right)}^{\prime }={P}_0-2 bQ $$
(19)
The
MC can be calculated as:
$$ MC={(TC)}^{\prime }={\left(F{C}_h+{C}_hQ\right)}^{\prime }={C}_h $$
(20)
We can obtain the throughput level in the monopoly market with eq. (
19) and (
20):
$$ {Q}_m=\frac{P_0-{C}_h}{2b} $$
(21)
Then the monopoly price of HKP is:
$$ {P}_m={P}_0-b\times \frac{P_0-{C}_h}{2b}=\frac{P_0+{C}_h}{2} $$
(22)
While in a perfect competitive market, the market price Pc must be equal to marginal cost (MC), e.g. Pc = Ch.
Under case 1, there is no equilibrium, alternatively, the equilibrium price Ph fluctuates between Pm and Pc, that is,. \( {C}_h<{P}_h<\frac{P_0+{C}_h}{2} \).
Case 2: capacity competition (Stackelberg leadership model)
When
Qh < CAa, according to Stackelberg Leadership model, the leader knows before the event that the follower observes its action and the follower cannot perform a non-Stackelberg follower action in future. In HKP, the leader (the HIT/ACT/CHT collaboration) makes decision first and the follower (MTL) follows. If the HIT/ACT/CHT collaboration adjusts its supply (cargo type, container throughput level, etc.), the market price (e.g. cargo and container handling price, berthing charges, shipping price) will change, which in turn prompt MTL to change its respective throughput to maintain profit maximisation (Von Stackelberg
1934). In this case, individual terminal operators can make decisions on throughput expansion to achieve economies of scale and thus reduce their production costs, investing in service efficiency improvement and attracting more customers.
Since the demand function of HKP is linear, the profit of the follower (MTL), ∏
l, is given as:
$$ {\prod}_l=\left(P-{C}_h\right)\times {Q}_l-F{C}_h $$
(23)
It can be further described as:
$$ {\prod}_l={P}_0{Q}_l-b{Q}_l^2-b{Q}_a{Q}_l-{C}_h{Q}_l-F{C}_h $$
(24)
To maximise the profit of MTL, we take the partial derivative
Ql and it should be equal to 0:
$$ \left({\prod}_l\right)\hbox{'}={P}_0-2b{Q}_l-b{Q}_a-{C}_h=0 $$
(25)
The reaction function of the follower, MTL, is:
$$ {Q}_l=\frac{P_0-b{Q}_a-{C}_h}{2b} $$
(26)
Then, the profit of the leader (the HIT/ACT/CHT collaboration), ∏
a, can be described as:
$$ {\prod}_a=\frac{P_0{Q}_a}{2}-\frac{b{Q}_a^2}{2}-\frac{C_h{Q}_a}{2}-F{C}_h $$
(27)
To maximise the profit of the HIT/ACT/CHT collaboration, we take the partial derivative
Qa and it should be equal to 0:
$$ \left({\prod}_a\right)\hbox{'}=\frac{P_0}{2}-b{Q}_a-\frac{C_h}{2}=0 $$
(28)
The optimal throughput of the leader, the HIT/ACT/CHT collaboration, is:
$$ {Q}_a=\frac{P_0-{C}_h}{2b} $$
(29)
The optimal throughput of the follower, MTL, is:
$$ {Q}_l=\frac{P_0-{C}_h}{4b} $$
(30)
Based on Stackelberg Leadership model, for intra-port competition, the throughput level of HKP is a constant value as follows:
$$ {Q}_h\hbox{'}={Q}_a+{Q}_l=\frac{3{P}_0-3{C}_h}{4b} $$
(31)
Therefore, substituting
Qh’ into the demand function of HKP, the internal equilibrium price
Ph’ can be obtained:
$$ {P}_h\hbox{'}=\frac{P_0+3{C}_h}{4} $$
(32)
Interconnection between internal competition and external competition
To better understand the interconnection between intra-port and inter-port competition, the last step is to analyse both competitions simultaneously. As shown in Fig.
3,
Qh = CAa is the boundary which determines the structure of intra-port competition. By inputting
Qh = CAa into eq. (
16),
kb (the
k which can achieve
Qh = CAa) can be obtained:
$$ {k}_b=\frac{C_h-C{}_s}{2m-n-3C{A}_a} $$
(33)
If 0 <
k <
kb, the leader-follower relationship exists between the HIT/ACT/CHT collaboration and MTL. According to eq. (
31), the internal equilibrium throughput of HKP is a constant value (
Qh’), which is not necessary to be the equilibrium throughput determined by inter-port competition (
Qh). Thus, there is only one level of heterogeneity (
ke) falling within the range 0 <
k <
kb can achieve the overall equilibrium for both intra-port and inter-port competition. We can solve this
ke by equating the internal equilibrium price (
Ph’) and the external equilibrium price (
Ph):
$$ {k}_e=\frac{C{}_h+3{P}_0-4{C}_s}{8m-4n} $$
(34)
The overall equilibrium for both intra-port and inter-port competition can only be achieved when k = ke within the range 0 < k < kb. However, the level of heterogeneity (k) is not determined by the container terminal operators in HKP but multiple factors, including but not limited to government policy, efficiency of customs clearance, service quality, etc. Since container terminal operators cannot move k to ke in short term, the overall equilibrium does not exist at the time that k ≠ ke.
If
kb <
k < +∞, there is no stable internal equilibrium price in the market and the market price will fluctuate between the monopoly market price and the perfect competitive market price. According to Edgeworth model, when MTL increases its price, the HIT/ACT/CHT collaboration will also increase price to compete with MTL. Then the total throughput of HKP will decrease and vice versa. Such the decrease and increase of HKP throughput will trigger the response from SZP, only when the optimal throughput of HKP is equal to the external equilibrium throughput level
Qh, can the competitive equilibrium for both inter-port competition and intra-port competition be achieved. This equilibrium can be represented by eq. (
35):
$$ {Q}_h=\frac{\left(2m-n\right)k+{C}_s-{C}_h}{3k}={Q}_a+{Q}_l $$
(35)
From eq. (
35), there will be two cases when the equilibrium can be reached:
Case (i). The HIT/ACT/CHT collaboration and MTL set different prices to make the sum of their throughput reach Qh., and Case (ii), they set an uniform price at Pa = Pl = Ph to reach Qh.
In reality,
Case (i) is less likely to be achieved because of the non-transparent port fee (price) mechanism and the independent operation strategy in HKP. First of all, there are different price levels for categorised consumers, e.g. special rate for consumers purchasing in bulk. It is hard for them to reflect the market change in short time. Also, due to their independent operation strategies, it is difficult for the HIT/ACT/CHT collaboration and MTL to collaborate and cooperate in the sense of better allocation of resources to provide more efficient services. Considering the rapid increase in the service efficiency of SZP, the heterogeneity between HKP and SZP is decreasing recently and thus widening the gap in throughput level between HKP and SZP. The infeasibility of case (i) is proved by the historical data of the throughput levels of HKP and SZP. Before 2019, the HIT/ACT/CHT collaboration and MTL adapted independent pricing strategy and operation strategy. However, below table indicates that these strategies are not efficient to protect HKP’s position against SZP, because the throughput level of HKP is decreasing continuously and the gap in throughput between HKP and SZP is becoming wider and wider. This data may prove that it is difficult for HKP to achieve the equilibrium in case (i) Table
2.
Table 2Ranking of Container Ports of World
HKP | 23,117 | 22,352 | 22,226 | 20,073 | 19,813 | 20,770 | 19,596 |
SZP | 22,941 | 23,278 | 24,037 | 24,205 | 23,979 | 25,209 | 25,736 |
Compared to
Case (i),
Case (ii) is more feasible that a coalition can be formed. It can at least achieve two advantages. First, easy to response to the external competition through setting an uniform price, in order to maximise the overall throughput. Second, forming a coalition may allow both the HIT/ACT/CHT collaboration and MTL to reduce cost through reducing surplus operation, sharing resource, etc. This also to some content, relieves the pressure from the decrease of heterogeneity between HKP and SZP. Based on our model, the best strategy for the container terminal operators in HKP to achieve profit maximisation is to collaborate and cooperate in the sense of better allocation of resources to provide more efficient port services and reduce the production cost. Wong et al. (
2018) propose a collaboration model for Hong Kong terminal operators to collaborate with each other to share their facilities, including berths, cranes, and yards. With this collaboration, vessels with high transshipment connections are allowed to berth within the same terminal to avoid unnecessary ITT. On 8th January, 2019, Hong Kong International Terminals Limited, Modern Terminals Limited, COSCO-HIT Terminals (Hong Kong) Limited, and Asia Container Terminals Limited announced the formation of the “Hong Kong Seaport Alliance”, a joint agreement designed to deliver more efficient services offering to carriers. In fact, the “Hong Kong Seaport Alliance” was approved and gradually implemented from 1st April (Hutchison Ports HIT
2019). Since this paper mainly focuses on qualitative analysis, the strategy of forming coalition applied by the terminals under HKP is expected to validate our model.
However, the accurate upper level of k is not +∞, because both the monopoly price in Edgeworth model and the extreme case that the maximum capacity cannot satisfy Qh will affect it:
On the one hand, according to Edgeworth model, the market equilibrium price of HKP (Ph) is limited to a range between the upper limit (monopoly price) and lower limit (perfect competitive market price), i.e. \( {C}_h<{P}_h<\frac{P_0+{C}_h}{2} \).
We can obtain the range of
k if the
Ph in Edgeworth model is substituted by eq. (
14):
$$ \frac{C_h-{C}_s}{2m-n}<k<\frac{1}{2}\frac{3{P}_0-2{C}_s-{C}_h}{2m-n} $$
(36)
As mentioned above, the applicable range of Edgeworth model is:
$$ {k}_b<k<+\infty $$
(37)
When Edgeworth model is applied, the range of
k should also meet the range set by eq. (
36). Thus, we consider the constraints of eq. (
36) and (
37) simultaneously and derive the range of
k for the application of Edgeworth model as:
$$ {k}_b<k<\frac{1}{2}\frac{3{P}_0-2{C}_s-{C}_h}{2m-n} $$
(38)
On the other hand, if the equilibrium throughput level (Qh), which is determined by external competition, outnumbers the maximum capacity of HKP, the equilibrium will not be achieved. In the long term, HKP will lose its position. Since Qh is only determined by heterogeneity k and they are positively correlated, the k (kt) which can produce the highest Qh, where Qh = CAa + CAl, is supposed to be the other upper limit of k.
From equation
Qh = CAa + CAl, a new upper level of
k (
kt) can be calculated and a new applicable range of
k is derived as eq. (
39):
$$ {k}_b<k<{k}_t=\frac{C_h-{C}_s}{2m-n-3\left(C{A}_a+C{A}_l\right)} $$
(39)
Comparing the two upper levels of
k in eqs. (
38) and (
39), we find that in Edgeworth model, the applicable range of
k, where the equilibrium exists, is determined by the maximum capacity of HKP (
CAa + CAl). The results of characterization (case 1 and case 2) are as follows:
Case 1:
$$ {\mathrm{CA}}_{\mathrm{a}}+{\mathrm{CA}}_{\mathrm{l}}>\frac{C_h\left(6m-3n\right)+{P}_0\left(3n-6m\right)}{\left(3{C}_h+6{C}_s-9{P}_0\right)} $$
(40)
The applicable range of
k:
$$ {k}_b<k<\frac{1}{2}\frac{3{P}_0-2{C}_s-{C}_h}{2m-n} $$
(41)
Case 2:
$$ {\mathrm{CA}}_{\mathrm{a}}+{\mathrm{CA}}_{\mathrm{l}}<\frac{C_h\left(6m-3n\right)+{P}_0\left(3n-6m\right)}{\left(3{C}_h+6{C}_s-9{P}_0\right)} $$
(42)
The applicable range of
k:
$$ {k}_b<k<{k}_t=\frac{C_h-{C}_s}{2m-n-3\left(C{A}_a+C{A}_l\right)} $$
(43)
As k represents the difference of consumers’ utility towards the services provided by HKP and SZP, when k = +∞, the nature of services provided by HKP and SZP is completely different and different needs of consumers can be fulfilled. In this way, competition will not exist. In contrast, when k = 0, the services of HKP and SZP are homogenous and it becomes a Bertrand game. Under this circumstance, all consumers will choose the port with lower price, i.e. SZP in our case.