Analyzing the Impact of Government Subsidies on Carbon Emission Mitigation Considering Carriers’ Price-and-Service Competition and Green Shippers

Abstract

High operational costs discourage shipping carriers from adopting green technologies, thereby exacerbating carbon emissions. Government subsidies can address the financial predicament faced by carriers. However, previous studies have overlooked whether governmental subsidies can help mitigate carbon emissions with intensified competition in both price and service between carriers, alongside rising environmental consciousness from shippers. To fill in this gap, game-theoretic models have been developed to explore optimal strategies for each partner of a shipping supply chain under three scenarios. Optimal solutions are derived through model analysis, followed by numerical analysis. Our findings are as follows: (1) the provision of governmental subsidization is conducive to a significant decrease in carbon emission with carriers’ price-and-service competition and shippers’ green awareness; (2) freight prices, profits and social welfare are all negatively related to government subsidies in a certain price-competitive environment; (3) price competition intensity is not conducive to carbon emission reduction but can benefit prices and social welfare; and (4) both low-carbon preference and intensified service competition jointly benefit profits and social welfare but are detrimental to carbon emission reduction. Our paper provides several meaningful insights for governments and shipping companies in formulating emission reduction strategies, contributing to environmental benefits and supporting the achievement of sustainability goals.

1. Introduction

About 90% of global goods trade is conducted by shipping to facilitate the movement of goods and resources and to support cross-national supply chains and global economic integration. However, shipping is also one of the major sources of global greenhouse gas emissions. According to the International Maritime Organization (IMO), shipping accounts for about 2–3% of global greenhouse gas emissions. Clarksons Research also had an estimate that shipping’s global Greenhouse Gas (GHG) emissions will increase to 1046 million tonnes of CO₂ in 2024 [1] and are categorized as Scope 1 emissions, which are the most difficult to eliminate [2]. Without mitigation measures, shipping emissions will increase to 130% of 2008 levels by 2050 [3]. Failure to accelerate decarbonization could lead to higher costs, regulatory penalties and a loss of competitiveness as markets increasingly prioritize sustainability [4].

Promoting decarbonization has become an imperative globally and is the consensus of world economic development [5], and decarbonization of maritime shipping has become an important target for the government, port companies and shipping operators [6]. Some agencies have pledged to achieve “net zero” emissions by midcentury or thereabouts, and the IMO has set intermediate targets for emissions and zero-emission fuels [7]. EU “Fit for 55” has created a legal obligation to reduce EU emissions by at least 55% by 2030 (as compared to 1990 levels) and will introduce the FuelEU Maritime Regulation as of 1 January 2025, which may be complex to implement [4].

By 2050, to align with efforts to limit global warming [8], 35,000 existing ships must be retrofitted to zero-emission ones [9], and retrofit involves the adoption of low-carbon technologies, such as upgrading the power system, installing environmental protection equipment, etc. It is possible for ships to make investments to upgrade existing ships with new green technologies [10], but it would be a costly expenditure for carriers, resulting in a low retrofit rate for ships [6,11,12], and the lack of motivation for carriers to adopt green technologies contributes to the exacerbation of carbon emissions. From a long-term perspective, encouraging carriers to invest in green technologies is not only in line with sustainable development goals but also brings competitive advantages and better economic benefits.

Carriers not only compete on price but also on service, including reliability, on-time delivery, and safety [13], and faster transportation service enables the product to be supplied with higher quality [14]. Carrier service is becoming increasingly critical to a larger number and wider variety of shippers [15] and is becoming increasingly intense [13]. As customer demands from shippers for green services continue to rise, carriers are enhancing their competitiveness through diversified service offerings, and the competition is deemed to be the significant factor that impacts the investment in reduced carbon emissions. In addition, more and more shippers prefer green transportation service, and a growing body of research suggests that consumers are green aware [16,17]. Furthermore, shippers are increasingly prioritizing sustainability by aligning shipping supply chains with green objectives and partnering with carriers that demonstrate environmental responsibility. This growing demand for low-carbon logistics solutions is also driving competition among shipping companies to provide more sustainable services.

To decarbonize, government subsidies have become one of the main approaches to encouraging the extensive use of low-emission technologies [18,19,20] and have proven effective in reducing costs for ocean carriers [21,22]. With governmental subsidies, shipping carriers face reduced cost pressures, enabling them to invest in green technologies. This not only benefits individual carriers but is also crucial for global efforts to combat climate change. But in response to price-and-service competition between carriers and shippers’ green awareness, whether government subsidies can help reduce carbon emission is an intriguing topic that merits exploration.

Some studies, such as Wang and Zhu [23], analyzed the development of the maritime supply chain considering the competition among different carriers with aims to reduce emission. Liu and Wang [13] constructed models to investigate incentives of alliance and values considering service competition, but neither of them took into account the significant impact that subsidies can have. Lu et al. [14] and Yi et al. [24] highlighted price competition between shipping carriers in freight transportation, but they did not consider service competition among shipping carriers.

Although these studies have contributed valuable insights, the interplay between government intervention (e.g., subsidies) and carbon emission alleviation with competition in both price and service for carriers remains underexplored. Given the increasing role of policy incentives in shaping sustainable and competitive maritime logistics, understanding how subsidies influence the strategic interactions between shipping carriers is crucial. Moreover, service competition is also a key differentiator in modern shipping markets, yet existing models predominantly focus on price competition. Motivated by this, we want to fill this gap by addressing the following specific research questions:

(1)

Should governments provide subsidies to competitive carriers in terms of price and service to support their green technology investment?

(2)

Is the government subsidy beneficial to decarbonization in the price-and-service competitive environment?

(3)

How do competition intensities and environmental awareness impact prices, profits, carbon emission reductions and the total social welfare?

To answer the above questions, we construct game-theoretic models to clarify the interactions among port, government, carriers and shippers of the shipping supply chain, in which carriers are competitive in price and service and shippers are environmentally conscious. A comprehensive framework is formulated under three strategies, i.e., FP, GP and HP. One carrier is subsidized in Scenario FP and GP, while both are provided subsidies in Scenario HP. Optimal solutions for each partner under all scenarios are derived, followed by numerical analysis to verify the optimal results for different partners.

Contributions of our paper lie in the following aspects, which significantly distinguish it from the extant literature: (1) we consider the government subsidy policy, carriers’ price and service competition, and shippers’ environmental awareness in the model simultaneously; (2) the effectiveness of subsidies to mitigate carbon emission with carriers’ price and service competition is verified through models constructed, and carriers are encouraged to make the transition to promote environmental sustainability; (3) we also shed light on the impact of shippers’ environmental awareness, price and service competition intensity on carbon emission, profits, prices and social welfare; and (4) our research broadens the research thread in the topic and can provide policy guidelines for decision-makers, such as governments and carriers.

The remainder of the paper is organized as follows: Section 2 reviews the literature. Section 3 describes the problem, constructs the model in detail and analyzes equilibrium solutions, followed by a numerical analysis provided in Section 4. Section 5 concludes the research work. Proofs are provided in Appendix A.

2. Literature Review

Our paper broadly intersects with three significant domains of study as follows: government subsidies on ports and carriers, carbon emission control in shipping supply chains and competition in price and service among ocean carriers.

2.1. Government Subsidies on Ports and Carriers

The existing literature primarily focuses on the role of government subsidies in ports, shipping companies and carriers within inland waterway transportation and maritime supply chains [25,26,27,28]. Jasmi and Fernando [29] investigated the relationship between drivers such as carriers, green initiative and regulation of the maritime green supply chain management. Zhuge et al. [30] formulated subsidy design models to reduce vessel speed. Song et al. [31] constructed a Nash game to analyze the impact of government subsidies on shipping companies. Results indicated that government subsidies can improve supply chain profits.

A dynamic game model of a three-echelon maritime supply chain with government subsidies was presented to abate carbon emission in Huang et al. [21]. Findings show that government subsidies can improve greenness with carbon abatement technology. Chen et al. [32] established a bi-level decision-making model considering providing subsidies to ports with the achievement of carbon neutrality. Peng et al. [33] discussed two subsidy strategies widely adopted for ports. They found that two subsidy strategies can achieve the same effects on the application of shore power. Zhen et al. [34] used the Stackelberg game methodology to investigate optimal subsidies on the installation of shore power on ports to reduce carbon emission. Li et al. [35] considered government subsidy strategies on ports to reduce carbon emission. The findings indicated that an increase in subsidies can decrease unit investment cost.

Luo et al. [6] considered government subsidy policy to promote shore power usage with an aim to reduce carbon emission. They found that a subsidy policy is effective in the short term. Hu et al. [36] and Wang et al. [37] designed different subsidy schemes for carriers and revealed the benefits of subsidies from the shipping company. Wang et al. [19] designed subsidy schemes for shipping companies to coordinate inland river ports. The results demonstrated that the implementation of coordination through subsidies is applicable and valid. Zhong et al. [38] analyzed the effectiveness of shore power systems in reducing emissions and investigated the impact of incentive policies on sustainable port development. Different from the above literature, our study simultaneously considers government subsidies, carrier competition, green-conscious shippers, and carbon emission reduction, and the impact of government subsidies on carbon emission reduction under such a competitive environment remains underexplored.

2.2. Carbon Emission Control in Shipping Supply Chains

Ports and shipping companies are the main implementing entities to mitigate carbon emission. Technical low-carbon technologies such as shore power [38,39,40,41,42], electric propulsion (EP) systems [43,44], blockchain technologies [22,45,46], carbon tax and cap-and-trade [23,47,48] are often adopted to decrease carbon emission by ports and carriers. Yang et al. [49] and Jiao and Wang [40] established game models considering the adoption of shore power and low-sulfur fuel oil in the ports aiming to meet low-carbon strategies. Taking Shekou Container Terminal as an example, Wan et al. [50] analyzed the impact of three different emission reduction strategies. They found the shore power system has the highest emission reduction effect. Livaniou and Papadopoulos [51] explored the potential of LNG in replacing conventional marine fuels, namely heavy fuel oil and marine diesel oil, to mitigate emission in shipping. Artificial intelligence and automation for shoreside operations were evaluated by Tsolakis et al. [52] to investigate the environmental impact at ports.

Wang et al. [18] proposed a trilevel programming model to address government subsidies on the adoption of liquefied natural gas to implement green maritime transportation. Fan et al. [53] investigated the effect of sulfur emission on ports and employed models to evaluate the effect. The analysis showed that the emission control policies implemented were effective and further ensured ships’ compliance. Liu et al. [54] explored the impact of carbon tax policy and knowledge sharing on the carbon emission reduction technology of a shipping supply chain with market uncertainty. Kong et al. [55] constructed models of the maritime supply chain to abate carbon emission with the system dynamic method. Results proved that using shore power can be beneficial for reducing emissions. Wang et al. [37] made a bibliometric analysis of carbon emission reduction in ports. Zhuge et al. [30] investigated the design of ECA policies to mitigate carbon dioxide emissions in global shipping. They found that the proposed ECA policies can help reduce carbon emission. Wang [56] and Luo et al. [6] found government incentives can promote shore power usage to meet carbon emission goals and make shipping service more green-conscious. In summary, it can be seen that the carbon emission mitigation of a shipping supply chain is broadly explored in extant literature. However, the impact of government subsidies on carbon emission reduction, particularly under carriers’ price-and-service competition and green-conscious shippers, remains insufficiently studied.

2.3. Competition in Price and Service Between Ocean Carriers

Competitive shipping companies want to maximize profits and pursue bigger market share, leading to fiercer price and service competition between carriers. Gelareh et al. [57] addressed the competition between one liner shipping company and one incumbent in a hub-and-spoke network. Wang et al. [58] proposed three game theoretical models to explain the competitive behaviors between two liner container shipping carriers. Lee and Song [59] conducted a literature review on competition among maritime logistics operators for better responding to challenges. Liu and Wang [13] investigated the competition between two carriers in a one-to-two shipping service chain. Song et al. [60] considered service competition between two liner companies and built game models to depict liner alliances. The results indicated that liner companies may achieve a win–win outcome with a low-price level.

On the basis of competition between carriers, Wang and Zhu [23] constructed models to pursue optimal solutions with different power structures considering the impact of carbon tax policy. Wang et al. [22] explored the blockchain investment strategies in a competitive game model, and they found competition intensity would affect the price of shipping companies. Wang et al. [61] constructed a game theory framework in which two types of shipping alliances compete in capacity. The findings demonstrated that in a competitive market, a mixed-alliance structure is beneficial to increasing social welfare and consumer surplus. Despite these advancements, research on carbon emission mitigation with government subsidies in a competitive environment involving green-conscious shippers remains limited.

To date, research in this area remains relatively limited, and our study wants to fill in this gap and attempt to explore the impact of governmental subsidies on carbon emission reduction in a price-and-service competitive environment with eco-conscious shippers. Our paper also enriches theoretical research related to the government’s green subsidy policies. In addition, it is also beneficial to policymakers and carriers to have a better understanding of how governmental subsidies impact carbon emission reduction in a competitive environment and with green-conscious shippers.

3. The Model

3.1. Problem Description

Consider a shipping supply chain consisting of shippers, two competitive carriers, a port and the government (see Figure 1). As the service provider, the port is committed to providing better cargo tracking and more efficient transshipment [4]. For simplicity, the port’s operational costs are assumed to be zero. As receivers of cargo, shippers are environmentally sensitive and are heterogeneous in their green transportation preference. Carriers take responsibility for moving goods from the origin to the destination, ensuring safety and timeliness during transit. Additionally, they offer customer services, such as providing updates on transportation status, resolving any issues that arise during transit and maintaining communication with clients. Both carriers compete in price and service to expand consumer markets. The government acts as the Stackelberg leader, strategically determining and providing subsidies to the carriers to incentivize green technology adoption, and the government seeks to balance both economic efficiency and environmental sustainability, ensuring that the subsidies lead to optimal outcomes for the carriers and society as a whole. As the follower in the game, the carriers respond to these subsidies by making investments in green technologies aimed at reducing carbon emissions. Each carrier’s decision-making process is driven by the objective of maximizing its own profit while simultaneously considering the long-term benefits of social welfare. Thus, the interaction between the government and the carriers is strategically structured to balance the dual objectives of individual profit maximization and the enhancement of overall social welfare.

In the basic model, the sequence is as follows: first, the government decides to subsidize carriers and sets the subsidy ratio; second, carriers make decisions to invest in green technologies in price-and-service competition.

According to [45,62,63,64], we suppose demand functions are represented as follows:

$$q_1=a-p_1+k{p}_2+{\gamma }_{1}e+{\beta }_1\theta$$

(1)

$$q_2=a-p_2+k{p}_1+{\gamma }_{2}e+{\beta }_2\theta$$

(2)

where $a$ is the potential market share, $p_i$ is the freight price of carriers, $q_i$ is the demand, $e$ is carbon emission reduction, $\theta$ measures the service level provided by carriers, ${\gamma }_i$ ($0<{\gamma }_i<1$) represents green preference from shippers, and $k$ ($0<k<1$) is the intensity of competition between carriers. The service level intensity of carriers, represented by ${\beta }_i$ ($0<{\beta }_i<1$), varies and is differentiated by the services they provide.$k$ and ${\beta }_i$ elaborate price and service effects, thereby increasing the significance of maintaining competitive pricing and service offerings.

The total investment cost to decarbonize for carriers is $0.5u_i{e}^2$, with the government covering a ratio of $h_i$, which is common in relevant literature [23,35]. For convenience, all notations are listed in

Table 1. Notations.

Symbols	Description
$a$	Potential market share
$\theta$	Service level
$k$	Competition intensity between carriers
${\beta }_i$	Service level intensity from carriers
${\gamma }_i$	Degree of green preference from shippers
$e$	Carbon emission reduction
$p_i$	$\mathrm{Freight} \mathrm{price} \mathrm{of} \mathrm{carrier} i$$, i=\mathrm{1,2}$
$w$	Port service price
$q_i$	$\mathrm{Demand} \mathrm{for} \mathrm{carrier} i$$, i=\mathrm{1,2}$
$u_i$	$\mathrm{Subsidy} \mathrm{coefficient}, u_i>0$$, i=\mathrm{1,2}$
$h_i$	$\mathrm{Share} \mathrm{ratio} \mathrm{of} \mathrm{the} \mathrm{subsidy}, i=\mathrm{1,2}$
$S$	$\mathrm{Subsidy} \mathrm{by} \mathrm{the} \mathrm{government}, S>0$
${\pi }_i$	$\mathrm{Profit} \mathrm{of} \mathrm{carriers} \mathrm{or} \mathrm{ports}, i=\mathrm{1,2}$
${\pi }_p$	Profit of ports
$SW$	Social welfare

.

3.2. Analysis Scenario

Three different scenarios are elaborated in our paper, namely FP, GP and HP, each reflecting different approaches to carbon emission reduction strategies in the maritime industry. In Scenario FP, only Carrier 1 accepts government subsidies and adopts green technologies to reduce carbon emissions. This scenario focuses on the impact of targeted subsidies for a single carrier in the industry, encouraging innovation in green technologies. In Scenario GP, Carrier 2 is subsidized through government subsidies to invest in green technology adoption, which highlights the effect of policy interventions aimed at promoting sustainable practices in another segment of the market. In Scenario HP, both carriers receive subsidies from the government, enabling them to jointly invest in carbon emission abatement strategies. This scenario aims to explore the cumulative impact of policy support on multiple actors within the supply chain, fostering collective action towards sustainability and emission reduction. Each of these scenarios provides valuable insights into how government interventions can influence the adoption of green technologies and the overall reduction of carbon emissions in the maritime sector.

(1)

Scenario FP

In this situation, only Carrier 1 is subsidized to reduce carbon emission, and the government provides $S^{fp}$ to Carrier 1. Profits for different partners and social welfare are denoted as Equations (3)–(8).

$${\pi }_1^{fp}=\left(p_1^{fp}-w^{fp}\right)q_1^{fp}-0.5(1-h_1)u_1{{(e}^{fp})}^2$$

(3)

$${\pi }_2^{fp}=(p_2^{fp}-w^{fp})q_2^{fp}$$

(4)

$${\pi }_p^{fp}=w^{fp}(q_1^{fp}+q_2^{fp})$$

(5)

$$\begin{array}{cc}\hfill {CS}^{fp}& ={\int }_{p_1^{fp}}^{a+k{p}_2^{fp}+{\gamma }_1{e}^{fp}+{\beta }_1\theta }\left(a-p_1^{fp}+k{p}_2^{fp}+{\gamma }_1{e}^{fp}+{\beta }_1\theta \right)d{p}_1^{fp}\hfill \\ & ={\displaystyle \frac{{[a-p_1^{fp}+k{p}_2^{fp}+{\gamma }_1{e}^{fp}+{\beta }_1\theta ]}^2}{2}}\hfill \end{array}$$

(6)

$$S^{fp}=0.5h_1{u}_1{\left(e^{fp}\right)}^2$$

(7)

$$\begin{array}{c}{SW}^{fp}={\pi }_1^{fp}+{\pi }_2^{fp}+{\pi }_p^{fp}+{CS}^{fp}-S^{fp}\hfill \\ =p_1^{fp}{q}_1^{fp}+p_2^{fp}{q}_2^{fp}+{\displaystyle \frac{{[a-p_1^{fp}+k{p}_2^{fp}+{\gamma }_1{e}^{fp}+{\beta }_1\theta ]}^2}{2}}\hfill \\ -0.5u_1{\left(e^{fp}\right)}^2\hfill \end{array}$$

(8)

Equations (3)–(5) are profits for Carrier 1, Carrier 2 and the port, which are determined by freight prices, port service price and demands. In addition, the investment that Carrier 1 makes is inclusive in Equation (3). Equation (7) is the subsidy provided by the government. Equation (6) represents the consumer surplus, and social welfare, as defined in Equation (8), is the sum of the profits of carriers and the port, the consumer surplus and the subsidy provided by the government. Carriers and the port seek to maximize their profits, while the government’s objective is to maximize social welfare.

Taking first-order derivatives of ${SW}^{fp}$ on $p_1^{fp}$, $p_2^{fp}$ and $e^{fp}$, we can get optimal results such as $p_1^{fp\mathrm{*}}$, $p_2^{fp\mathrm{*}}$, $e^{fp\mathrm{*}}$, $q_1^{fp\mathrm{*}}$, $q_2^{fp\mathrm{*}}$ and ${SW}^{fp\mathrm{*}}$. Let ${\displaystyle \frac{{\partial SW}^{fp}}{\partial p_1^{fp}}}=0$ and ${\displaystyle \frac{{\partial SW}^{fp}}{\partial e^{fp}}}=0$; thus, $h_1^{\mathrm{*}}$, $w^{fp\mathrm{*}}$, ${\pi }_1^{fp\mathrm{*}}$, ${\pi }_2^{fp\mathrm{*}}$ and ${\pi }_p^{fp\mathrm{*}}$ can be obtained.

Table 2. Optimal results in Scenario FP.

Variable	Optimal Results
$p_1^{fp*}$	${\displaystyle \frac{k\{{\gamma }_1\left(a+{\beta }_1\theta \right)\left(k{\gamma }_1+{\gamma }_2\right)-({\gamma }_1^2-u_1\left)\right[a\left(1+k\right)+\theta (k{\beta }_1+{\beta }_2)\left]\right\}}{[\left(2k^2-2\right)\left({\gamma }_1^2-u_1\right)-{(k{\gamma }_1+{\gamma }_2)}^2]}}$
$p_2^{fp*}$	${\displaystyle \frac{{\gamma }_1\left(a+{\beta }_1\theta \right)\left(k{\gamma }_1+{\gamma }_2\right)-({\gamma }_1^2-u_1)[a\left(1+k\right)+\theta (k{\beta }_1+{\beta }_2\left)\right]}{[\left(2k^2-2\right)\left({\gamma }_1^2-u_1\right)-{(k{\gamma }_1+{\gamma }_2)}^2]}}$
$e^{fp*}$	${\displaystyle \frac{-2{\gamma }_1\left(a+{\beta }_1\theta \right)\left(k^2-1\right)+(k{\gamma }_1+{\gamma }_2)[a\left(1+k\right)+\theta (k{\beta }_1+{\beta }_2\left)\right]\}}{[\left(2k^2-2\right)\left({\gamma }_1^2-u_1\right)-{(k{\gamma }_1+{\gamma }_2)}^2]}}$
$q_1^{fp*}$	${\displaystyle \frac{\left(a+{\beta }_1\theta \right)[2u_1\left(1-k^2\right)-{\left(k{\gamma }_1+{\gamma }_2\right)}^2]+{\gamma }_1(k{\gamma }_1+{\gamma }_2)[a\left(1+k\right)+\theta (k{\beta }_1+{\beta }_2\left)\right]\}}{[\left(2k^2-2\right)\left({\gamma }_1^2-u_1\right)-{(k{\gamma }_1+{\gamma }_2)}^2]}}$
$q_2^{fp*}$	${\displaystyle \frac{u_1\left(1-k^2\right)\left[a\left(1-k\right)+\theta \left({\beta }_2-k{\beta }_1\right)\right]+\left({\gamma }_1+k{\gamma }_2\right)[a\left({\gamma }_2-{\gamma }_1\right)+\theta \left({\beta }_1{\gamma }_2-{\beta }_2{\gamma }_1\right)]}{[\left(2k^2-2\right)\left({\gamma }_1^2-u_1\right)-{(k{\gamma }_1+{\gamma }_2)}^2]}}$
${\pi }_1^{fp*}$	$\begin{array}{c}{\displaystyle \frac{{\{2u}_1\left(k^2-1\right)\left(a+{\beta }_1\theta \right)+(k{\gamma }_1+{\gamma }_2)\left[a\left({\gamma }_2-{\gamma }_1\right)+\theta \left({\beta }_1{\gamma }_2-{\beta }_2{\gamma }_1\right)\right]\}}{2{[\left(2k^2-2\right)\left({\gamma }_1^2-u_1\right)-{(k{\gamma }_1+{\gamma }_2)}^2]}^2}}\\ \times \{\left(k-1\right)\left[4a{u}_1\left(k+1\right)-a{\gamma }_1^2\left(k+2\right)\right]+{\gamma }_2\left[2a{\gamma }_2+a{\gamma }_1\left(1+3k\right)\right]+{\beta }_1\theta \left[4u_1\left(k^2-1\right)-{\gamma }_1^2\left(k^2-2\right)\right]+\theta [{\beta }_1{\gamma }_2(3k{\gamma }_1+2{\gamma }_2)-{\beta }_2{\gamma }_1(k{\gamma }_1+{\gamma }_2)\left]\right\}\end{array}$
${\pi }_2^{fp*}$	$\begin{array}{c}{\displaystyle \frac{{\{-au}_1\left(1+k\right){\left(1-k\right)}^2-a\left({\gamma }_2-{\gamma }_1\right)\left(k{\gamma }_2+{\gamma }_1\right)-{\beta }_1\theta \left[k\left(k^2{u}_1+{\gamma }_2^2\right)+\left({\gamma }_1{\gamma }_2-k{u}_1\right)\right]+{\beta }_2\theta [k\left(k{u}_1+{\gamma }_1{\gamma }_2\right)-(u_1-{\gamma }_1^2\left)\right]\}}{{[\left(2k^2-2\right)\left({\gamma }_1^2-u_1\right)-{(k{\gamma }_1+{\gamma }_2)}^2]}^2}}\\ \times \{3a{u}_1\left(k^2-1\right)+2ak{\gamma }_1\left({\gamma }_2-{\gamma }_1\right)+a{\left({\gamma }_1-{\gamma }_2\right)}^2+{\beta }_1\theta \left[u_1\left(3k+2\right)\left(k-1\right)+{\gamma }_2\left(2k{\gamma }_1-{\gamma }_1+{\gamma }_2\right)\right]+{\beta }_2\theta [-2k{\gamma }_1^2+\left(k-1\right)u_1+{\gamma }_1({\gamma }_1-{\gamma }_2)\left]\right\}\end{array}$
$S^{fp*}$	${\displaystyle \frac{\left(k{\gamma }_1+{\gamma }_2\right)\left\{a\left[u_1\left(1+k\right)+{\gamma }_2({\gamma }_2-{\gamma }_1)\right]+\theta \left[u_1\left(k{\beta }_1+{\beta }_2\right)+{\gamma }_1\left({{\beta }_1\gamma }_2-{\beta }_2{\gamma }_1\right)\right]\right\}\times \{a\left(1+k\right)\left[{\gamma }_2-(k-2){\gamma }_1\right]-{\beta }_1\theta \left(k^2{\gamma }_1-k{\gamma }_2-2{\gamma }_1\right)+{\beta }_2\theta (k{\gamma }_1+{\gamma }_2\left)\right\}}{2{[\left(2k^2-2\right)\left({\gamma }_1^2-u_1\right)-{(k{\gamma }_1+{\gamma }_2)}^2]}^2}}$
$h_1^{*}$	${\displaystyle \frac{(k{\gamma }_1+{\gamma }_2)[-a{u}_1\left(1+k\right)+a{\gamma }_1\left({\gamma }_1-{\gamma }_2\right)-\theta u_1\left(k{\beta }_1+{\beta }_2\right)+\theta {\gamma }_1({{\beta }_2\gamma }_1-{\beta }_1{\gamma }_2\left)\right]}{u_1\{a\left(1+k\right)\left[\left(k-2\right){\gamma }_1-{\gamma }_2\right]+{\beta }_1\theta \left[(k^2-2){\gamma }_1-{k\gamma }_2\right]+{\beta }_2\theta (k{\gamma }_1+{\gamma }_2\left)\right\}}}$
${SW}^{fp*}$	${\displaystyle \frac{-a^2\left[k{u}_1\left(k-2\right)+{\left({\gamma }_1-{\gamma }_2\right)}^2-3u_1\right]-2a{\beta }_1\theta \left[\left(k-2\right)\left(k+1\right)+{\gamma }_2^2-{\gamma }_1{\gamma }_2\right]-2a{\beta }_2\theta \left[-u_1\left(k+1\right)+{\gamma }_1^2-{\gamma }_1{\gamma }_2\right]-{\beta }_1^2\theta \left[\left(k^2-2\right)u_1+{\gamma }_2^2\right]-2{\beta }_1{\beta }_2\theta \left[k{u}_1+{\gamma }_1{\gamma }_2\right]+{\beta }_2^2{\theta }^2({\gamma }_1^2-u_1)]}{2[\left(2k^2-2\right)\left({\gamma }_1^2-u_1\right)-{\left(k{\gamma }_1+{\gamma }_2\right)}^2]}}$

shows optimal results under Scenario FP.

(2)

Scenario GP

In this scenario, Carrier 2 invests in decreasing carbon emission with subsidies from the government, and the government shares $h_2$ of the total investment $0.5(1-h_2)u_2{{(e}^{gp})}^2$. Profits for carriers and ports and social welfare are expressed as Equations (9)–(14).

$${\pi }_1^{gp}=(p_1^{gp}-w^{gp})q_1^{gp}$$

(9)

$${\pi }_2^{gp}=\left(p_2^{gp}-w^{gp}\right)q_2^{gp}-0.5(1-h_2)u_2{{(e}^{gp})}^2$$

(10)

$${\pi }_p^{gp}=w^{gp}(q_1^{gp}+q_2^{gp})$$

(11)

$$\begin{array}{cc}\hfill {CS}^{gp}& ={\int }_{p_2^{gp}}^{a+k{p}_1^{gp}+{\gamma }_2{e}^{gp}+{\beta }_2\theta }a-p_2^{gp}+k{p}_1^{gp}+{\gamma }_2{e}^{gp}+{\beta }_2\theta d{p}_2^{gp}\hfill \\ & ={\displaystyle \frac{{[a-p_2^{gp}+k{p}_1^{gp}+{\gamma }_2{e}^{gp}+{\beta }_2\theta ]}^2}{2}}\hfill \end{array}$$

(12)

$$S^{gp}=0.5h_2{u}_2{\left(e^{gp}\right)}^2$$

(13)

$$\begin{array}{c}{SW}^{gp}={\pi }_1^{gp}+{\pi }_2^{gp}+{\pi }_p^{gp}+{CS}^{gp}-S^{gp}\hfill \\ =p_1^{gp}{q}_1^{gp}+p_2^{gp}{q}_2^{gp}+{\displaystyle \frac{{\left[a-p_2^{gp}+k{p}_1^{gp}+{\gamma }_2{e}^{gp}+{\beta }_2\theta \right]}^2}{2}}\hfill \\ -0.5u_2{\left(e^{gp}\right)}^2\hfill \end{array}$$

(14)

Similarly, Equations (9)–(11) are profits of Carrier 1, Carrier 2 and the port, respectively. The port and carriers are profit maximizers. Equation (12) is the subsidy provided by the government. Equations (12) and (14) provide consumer surplus and social welfare in Scenario GP, and the government’s goal is the maximization of social welfare.

By solving first-order derivatives of ${SW}^{gp}$ on $p_1^{gp}$, $p_2^{gp}$ and $e^{gp}$, we can get $p_1^{gp\mathrm{*}}$, $p_2^{gp\mathrm{*}}$, $e^{gp\mathrm{*}}$, $q_1^{gp\mathrm{*}}$, $q_2^{gp\mathrm{*}}$ and ${SW}^{gp\mathrm{*}}$. Let ${\displaystyle \frac{\partial {\pi }_2^{gp}}{\partial p_2^{gp}}}=0$ and ${\displaystyle \frac{\partial {\pi }_2^{gp}}{\partial e^{gp}}}=0$; we can get $h_2^{\mathrm{*}}$, $S^{gp\mathrm{*}}$, ${\pi }_1^{gp\mathrm{*}}$ and ${\pi }_2^{gp\mathrm{*}}$.

Table 3. Optimal results in Scenario GP.

Variable	Optimal Results
$p_1^{gp*}$	${\displaystyle \frac{-{\gamma }_2\left(a+{\beta }_2\theta \right)\left(k{\gamma }_2+{\gamma }_1\right)+({\gamma }_2^2-u_2)[a\left(1+k\right)+\theta ({\beta }_1+k{\beta }_2\left)\right]}{[\left(2k^2-2\right)\left(u_2-{\gamma }_2^2\right)+{(k{\gamma }_2+{\gamma }_1)}^2]}}$
$p_2^{gp*}$	${\displaystyle \frac{k\{-{\gamma }_2\left(a+{\beta }_2\theta \right)\left(k{\gamma }_2+{\gamma }_1\right)+({\gamma }_2^2-u_2\left)\right[a\left(1+k\right)+\theta ({\beta }_1+k{\beta }_2)\left]\right\}}{[\left(2k^2-2\right)\left(u_2-{\gamma }_2^2\right)+{(k{\gamma }_2+{\gamma }_1)}^2]}}$
$e^{gp*}$	${\displaystyle \frac{2{\gamma }_2\left(a+{\beta }_2\theta \right)\left(k^2-1\right)-(k{\gamma }_2+{\gamma }_1)[a\left(1+k\right)+\theta ({\beta }_1+k{\beta }_2\left)\right]}{[\left(2k^2-2\right)\left(u_2-{\gamma }_2^2\right)+{(k{\gamma }_2+{\gamma }_1)}^2]}}$
$q_1^{gp*}$	${\displaystyle \frac{u_2\left(1-k^2\right)[a\left(k-1\right)-\theta \left({\beta }_1-k{\beta }_2\right)]+(k{\gamma }_1+{\gamma }_2)[a\left({\gamma }_2-{\gamma }_1\right)+\theta {\gamma }_2({\beta }_1-{\beta }_2\left)\right]}{[\left(2k^2-2\right)\left(u_2-{\gamma }_2^2\right)+{(k{\gamma }_2+{\gamma }_1)}^2]}}$
$q_2^{gp*}$	${\displaystyle \frac{\left(k{\gamma }_2+{\gamma }_1\right)\left[a\left({\gamma }_1{-\gamma }_2\right)+\theta \left({\beta }_2{\gamma }_1-{\beta }_1{\gamma }_2\right)\right]-2u_2(1-k^2)[a+{\beta }_2\theta )]}{[\left(2k^2-2\right)\left(u_2-{\gamma }_2^2\right)+{(k{\gamma }_2+{\gamma }_1)}^2]}}$
${\pi }_1^{gp*}$	$\begin{array}{c}{\displaystyle \frac{[a{u}_2\left(1+k\right){\left(1-k\right)}^2+a\left({\gamma }_1-{\gamma }_2\right)\left(k{\gamma }_1+{\gamma }_2\right)+\theta u_2\left(1-k^2\right)\left({\beta }_1-k{\beta }_2\right)+\theta (k{\gamma }_1+{\gamma }_2\left)\right({\beta }_2{\gamma }_1-{\beta }_1{\gamma }_2\left)\right]}{{[\left(2k^2-2\right)\left(u_2-{\gamma }_2^2\right)+{(k{\gamma }_2+{\gamma }_1)}^2]}^2}}\\ \times \{a{u}_2\left(1-3k^2\right)-2ak{\gamma }_2\left({\gamma }_1-{\gamma }_2\right)-a{\left({\gamma }_1-{\gamma }_2\right)}^2-{\beta }_1\theta \left[k\left(u_2-2{\gamma }_2^2\right)-u_2\right]-{\beta }_1\theta {\gamma }_2\left({\gamma }_2-{\gamma }_1\right)-{\beta }_2\theta u_2\left(3k+2\right)\left(k-1\right)-{\beta }_2\theta [{\gamma }_1^2+(2k-1){\gamma }_1{\gamma }_2\left]\right\}\end{array}$
${\pi }_2^{gp*}$	$\begin{array}{c}{\displaystyle \frac{\left\{2u_2\left(k^2-1\right)\right(a+{\beta }_2\theta )+\left(k{\gamma }_2+{\gamma }_1\right)[a\left({\gamma }_1-{\gamma }_2\right)-\theta ({\beta }_2{\gamma }_1-{\beta }_1{\gamma }_2)]}{{2[\left(2k^2-2\right)\left(u_2-{\gamma }_2^2\right)+{(k{\gamma }_2+{\gamma }_1)}^2]}^2}}\\ \times \left\{a{\left(1+k\right)[4u}_2\left(k-1\right)-k{\gamma }_2^2\right]+2\left(a+{\beta }_2\theta \right)\left({\gamma }_1^2+{\gamma }_2^2\right)+a{\gamma }_1{\gamma }_2(3k-1)-{\beta }_1\theta {\gamma }_2\left(k{\gamma }_2+{\gamma }_1\right)+4{\beta }_2\theta u_2\left(k^2-1\right)+k{\beta }_2\theta {\gamma }_2(3{\gamma }_1-k{\gamma }_2)\}\end{array}$
$S^{gp*}$	$\begin{array}{c}{\displaystyle \frac{\left(k{\gamma }_2+{\gamma }_1\right)[a{u}_2\left(1+k\right)+a{\gamma }_2({\gamma }_1-{\gamma }_2)+\theta {\gamma }_2\left({\beta }_2{\gamma }_1-{\beta }_1{\gamma }_2\right)+\theta u_2({\beta }_1{+k\beta }_2\left)\right]}{{2[\left(2k^2-2\right)\left(u_2-{\gamma }_2^2\right)+{(k{\gamma }_2+{\gamma }_1)}^2]}^2}}\\ \times \left\{-a(1+k)\left[\right(k-2){\gamma }_2+{\gamma }_1\right]+{\beta }_1\theta \left(k{\gamma }_2+{\gamma }_1\right)-{\beta }_2\theta \left(k^2{\gamma }_2-k{\gamma }_1-2{\gamma }_2\right)\}\end{array}$
$h_2^{*}$	${\displaystyle \frac{\left(k{\gamma }_2+{\gamma }_1\right)[-a{u}_2\left(1+k\right)-a{\gamma }_2\left({\gamma }_1-{\gamma }_2\right)+\theta {\gamma }_2\left({{\beta }_1\gamma }_2-{\beta }_2{\gamma }_1\right)-\theta u_2({\beta }_1+k{\beta }_2\left)\right]}{u_2\{a\left(1+k\right)\left[\left(k-2\right){\gamma }_2-{\gamma }_1\right]+\theta \left(k{\gamma }_2+{\gamma }_1\right)\left(k{\beta }_2-{\beta }_1\right)-2\theta {\beta }_2{\gamma }_2\}}}$
${SW}^{gp*}$	${\displaystyle \frac{a^2{u}_2\left[\left(k-3\right)(k+1)\right]+a^2{({\gamma }_1-{\gamma }_2)}^2+2a{\beta }_1\theta \left[{\gamma }_2\left({\gamma }_2-{\gamma }_1\right)-(1+k)u_2\right]+2a{\beta }_2\theta \left[u_2\left(k^2-k\right)+{\gamma }_1^2-{\gamma }_1{\gamma }_1-2u_2\right]+{\beta }_1^2{\theta }^2\left({\gamma }_2^2-u_2\right)-2{\beta }_1{\beta }_2\theta \left(k{u}_2+{\gamma }_1{\gamma }_2\right)+{\beta }_2^2\theta (k^2{u}_2+{\gamma }_1^2-2u_2)}{2\left[\left(2k^2-2\right)\left(u_2-{\gamma }_2^2\right)+{\left(k{\gamma }_2+{\gamma }_1\right)}^2\right]}}$

below shows the optimal results obtained.

(3)

Scenario HP

In this scenario, both carriers make investments in carbon emission abatement with subsidies from the government, and the total investment cost for carriers to reduce carbon emission is $0.5u_3{{(e}^{hp})}^2$. Profits and social welfare are expressed as Equations (15)–(20).

$${\pi }_1^{hp}=\left(p_1^{hp}-w^{hp}\right)q_1^{hp}-0.5(1-h_3)u_3{{(e}^{hp})}^2$$

(15)

$${\pi }_2^{hp}=\left(p_2^{hp}-w^{hp}\right)q_2^{hp}-0.5(1-h_4)u_3{{(e}^{hp})}^2$$

(16)

$${\pi }_p^{hp}=w^{hp}(q_1^{hp}+q_2^{hp})$$

(17)

$$\begin{array}{cc}\hfill {CS}^{hp}& ={\int }_{p_1^{hp}}^{a+k{p}_2^{hp}+{\gamma }_1{e}^{hp}+{\beta }_1\theta }(a-p_1^{hp}+k{p}_2^{hp}+{\gamma }_1{e}^{hp}+{\beta }_1\theta )d{p}_1^{hp}\hfill \\ & +{\int }_{p_2^{hp}}^{a+k{p}_1^{hp}+{\gamma }_2{e}^{hp}+{\beta }_2\theta }(a-p_2^{hp}+k{p}_1^{hp}+{\gamma }_2{e}^{hp}+{\beta }_2\theta )d{p}_2^{hp}\hfill \\ & ={\displaystyle \frac{{\left(q_1^{hp}\right)}^2+{\left(q_2^{hp}\right)}^2}{2}}\hfill \end{array}$$

(18)

$$S^{hp}=0.5h_3{u}_3{\left(e^{hp}\right)}^2+0.5h_4{u}_4{\left(e^{hp}\right)}^2$$

(19)

$$\begin{array}{c}{SW}^{hp}={\pi }_1^{hp}+{\pi }_2^{hp}+{\pi }_p^{hp}+{CS}^{hp}-S^{hp}\hfill \\ =p_1^{hp}{q}_1^{hp}+p_2^{hp}{q}_2^{hp}+{\displaystyle \frac{{\left(q_1^{hp}\right)}^2+{\left(q_2^{hp}\right)}^2}{2}}-u_3{\left(e^{hp}\right)}^2\hfill \end{array}$$

(20)

Equations (15)–(17) are profits for both carriers and the port. Equation (19) is the total subsidy provided by the government. Equations (18) and (20) represent consumer surplus and social welfare, respectively. Carriers and the port seek to maximize their profits, and the government is to maximize social welfare.

Taking first-order derivatives of ${SW}^{hp}$ on $p_1^{hp}$, $p_2^{hp}$ and $e^{hp}$, and letting ${\displaystyle \frac{{\partial \pi }_1^{hp}}{\partial p_1^{hp}}}=0$, ${\displaystyle \frac{{\partial \pi }_1^{hp}}{\partial e^{hp}}}=0$, ${\displaystyle \frac{{\partial \pi }_2^{hp}}{\partial p_2^{hp}}}=0$ and ${\displaystyle \frac{{\partial \pi }_2^{hp}}{\partial e^{hp}}}=0$, we can obtain the following optimal results, as shown in

Table 4. Optimal results in Scenario HP.

Variable	Optimal Results
$p_1^{hp*}$	${\displaystyle \frac{k[2a{u}_3\left(1-k^2\right)+a{\gamma }_1\left({\gamma }_2-{\gamma }_1\right)-\theta {\gamma }_1\left({\beta }_2{\gamma }_1-{\beta }_1{\gamma }_2\right)+2\theta {\beta }_2{u}_3(1-k^2\left)\right]}{2u_3{(k^2-1)}^2+(k^2-1)({\gamma }_1^2+{\gamma }_2^2)}}$
$p_2^{hp*}$	${\displaystyle \frac{k[2a{u}_3\left(1-k^2\right)-a{\gamma }_2\left({\gamma }_2-{\gamma }_1\right)-\theta {\gamma }_2\left({\beta }_1{\gamma }_2-{\beta }_2{\gamma }_1\right)+2\theta {\beta }_1{u}_3(1-k^2\left)\right]}{2u_3{(k^2-1)}^2+(k^2-1)({\gamma }_1^2+{\gamma }_2^2)}}$
$e^{hp*}$	${\displaystyle \frac{a\left({\gamma }_2+{\gamma }_1\right)+\theta \left({\beta }_1{\gamma }_1+{\beta }_2{\gamma }_2\right)}{2u_3\left(1-k^2\right)-({\gamma }_1^2+{\gamma }_2^2)}}$
$q_1^{hp*}$	${\displaystyle \frac{2a{u}_3\left(k+1\right){\left(k-1\right)}^2+a\left({\gamma }_1-{\gamma }_2\right)\left({k\gamma }_1+{\gamma }_2\right)+2\theta u_3\left(k^2-1\right)\left(k{\beta }_2-{\beta }_1\right)+\theta ({k\gamma }_1+{\gamma }_2)\left({\beta }_1{\gamma }_2+{\beta }_2{\gamma }_1\right)}{(k^2-1)[2u_3\left(k^2-1\right)+\left({\gamma }_1^2+{\gamma }_2^2\right)]}}$
$q_2^{hp*}$	${\displaystyle \frac{2a{u}_3\left(k+1\right){\left(k-1\right)}^2+a\left({\gamma }_2-{\gamma }_1\right)\left({k\gamma }_2+{\gamma }_1\right)+2u_3\left(1-k^2\right)\left({\beta }_2-k{\beta }_1\right)+\theta ({k\gamma }_2+{\gamma }_1)\left({\beta }_1{\gamma }_2-{\beta }_2{\gamma }_1\right)}{(k^2-1)[2u_3\left(k^2-1\right)+\left({\gamma }_1^2+{\gamma }_2^2\right)]}}$
${\pi }_1^{hp*}$	$\begin{array}{c}{\displaystyle \frac{\{2a{u}_3\left(1-k\right){\left(k+1\right)}^2+a\left({\gamma }_1-{\gamma }_2\right)\left({k\gamma }_1+{\gamma }_2\right)+2\theta u_3\left(1-k^2\right)\left({\beta }_1-k{\beta }_2\right)+\theta \left({k\gamma }_1+{\gamma }_2\right)\left({\beta }_2{\gamma }_1-{\beta }_1{\gamma }_2\right)\}}{{(k^2-1)}^2{[2u_3\left(k^2-1\right)+\left({\gamma }_1^2+{\gamma }_2^2\right)]}^2}}\\ \times \{2u_3\left(1-k^2\right)\left[a\left(1-k\right)+\theta \left({\beta }_1-k{\beta }_2\right)\right]+0.5a{k}^2{\gamma }_1\left({\gamma }_1+{\gamma }_2\right)-a{\gamma }_2^2+a{\gamma }_1\left({\gamma }_1-{\gamma }_2\right)\left(k-0.5\right)+0.5{\beta }_1\theta k^2\left({\gamma }_1^2-k{u}_3\right)-\theta {\beta }_1{\gamma }_2(k{\gamma }_1+{\gamma }_2)+0.5{\beta }_2\theta [k^2{\gamma }_1{\gamma }_2+2k{\gamma }_1^2+{\gamma }_1{\gamma }_2\left]\right\}\end{array}$
${\pi }_2^{hp*}$	$\begin{array}{c}{\displaystyle \frac{\{2a{u}_3\left(1+k\right){\left(1-k\right)}^2+\left({k\gamma }_2+{\gamma }_1\right)[a\left({\gamma }_2-{\gamma }_1\right)+\theta {\beta }_1{\gamma }_2-{\beta }_2\theta {\gamma }_1\left]{+2\theta u}_3\left(k^2-1\right)\left(k{\beta }_1-{\beta }_2\right)\right\}}{{(k^2-1)}^2{[2u_3\left(k^2-1\right)+\left({\gamma }_1^2+{\gamma }_2^2\right)]}^2}}\\ \times \{2a{u}_3\left(1+k\right){\left(1-k\right)}^2+ak{\gamma }_2\left(1+0.5k\right)\left({\gamma }_1+{\gamma }_2\right)-a\left({\gamma }_1^2-0.5{\gamma }_1{\gamma }_2+0.5{\gamma }_2^2\right)+0.5{\beta }_1\theta {\gamma }_2\left({\gamma }_1+k^2{\gamma }_1+2k{\gamma }_2\right)+\theta \left(1-k^2\right)\left(2{\beta }_2{u}_3-0.5{\beta }_2{\gamma }_2^2-2k{\beta }_1{u}_3\right)+{\beta }_2\theta {\gamma }_1({\gamma }_1+k{\gamma }_2\left)\right\}\end{array}$
$S^{hp*}$	${\displaystyle \frac{k\left[a\left({\gamma }_1+{\gamma }_2\right)+\theta \left({\beta }_1{\gamma }_1+{{\beta }_2\gamma }_2\right)\right]\times \{2a{u}_3\left(1+k^2\right)\left({\gamma }_1+{\gamma }_2\right)+a({\gamma }_1+{\gamma }_2\left)\right({{\gamma }_1-{\gamma }_2)}^2+2\theta u_3\left(1+k^2\right)\left({{\beta }_2\gamma }_1+{{\beta }_1\gamma }_2\right)+\theta ({\beta }_1{\gamma }_2-{\beta }_2{\gamma }_1)({\gamma }_1^2-{\gamma }_2^2)\}}{2(k^2-1){[2u_3\left(k^2-1\right)+\left({\gamma }_1^2+{\gamma }_2^2\right)]}^2 }}$
$h_3^{*}$	${\displaystyle \frac{u_3\left(k^2-1\right)\left[{\gamma }_1\left(2ak-{\beta }_1\theta \right)+{\beta }_2\theta \left(2k{\gamma }_1+{\gamma }_2\right)\right]+a\left({\gamma }_2-{\gamma }_1\right)\left[k\left(k{u}_3-{\gamma }_1^2\right)+\left({\gamma }_1{\gamma }_2-u_3\right)\right]+\theta {\gamma }_1({\beta }_2-{\beta }_1{\gamma }_2)(k{\gamma }_1+{\gamma }_2)}{\theta u_3(k^2-1)({\beta }_1{\gamma }_1+{\beta }_2{\gamma }_2)}}$
$h_4^{*}$	${\displaystyle \frac{a{k}^2{u}_3\left({\gamma }_1+2k{\gamma }_2\right)-ak{u}_3{\gamma }_2\left(k+2\right)+a\left(u_3+k{\gamma }_2^2{+\gamma }_1{\gamma }_2\right)\left({\gamma }_2{-\gamma }_1\right)+{\beta }_1\theta u_3\left({\gamma }_1+2k{\gamma }_2\right)\left(k^2-1\right)+\theta {\beta }_1{\gamma }_2^2\left({\gamma }_1+2{\gamma }_2\right)+\theta {\beta }_2{\gamma }_2{u}_3\left(1-k^2\right)-\theta {\beta }_2{\gamma }_1{\gamma }_2({\gamma }_1+k{\gamma }_2)}{\theta u_3(k^2-1)({\beta }_1{\gamma }_1+{\beta }_2{\gamma }_2)}}$
${SW}^{hp*}$	${\displaystyle \frac{-4{a^{2}u}_3\left(k^2+1\right)+a^2{\left({\gamma }_1-{\gamma }_2\right)}^2+4a\theta u_3({\beta }_1+{\beta }_2)\left(k^2-1\right)+2a\theta ({\beta }_1{\gamma }_2-{\beta }_2{\gamma }_1)({\gamma }_2-{\gamma }_1)+2{{\theta }^{2}u}_3\left(k^2-1\right)({\beta }_1^2+{\beta }_2^2)+{(\theta {\beta }_1{\gamma }_2-\theta {\beta }_2{\gamma }_1)}^2}{2\left(k^2-1\right)\left[2u_3\right(k^2-1)+({\gamma }_1^2+{\gamma }_2^2\left)\right]}}$

.

3.3. Equilibrium Analysis

Proposition 1.

If $u_1>{\gamma }_1^2+{\displaystyle \frac{{(k{\gamma }_1+{\gamma }_2)}^2}{2-2k^2}}$, then ${SW}^{fp}$ is a negative concave function of $p_1^{fp}$, $p_2^{fp}$ and $e^{fp}$.

See Appendix A.

Proposition 2.

If $u_2>{\gamma }_2^2+{\displaystyle \frac{{(k{\gamma }_2+{\gamma }_1)}^2}{2-2k^2}}$, then ${SW}^{gp}$ is negative jointly concave on $p_1^{gp}$, $p_2^{gp}$ and $e^{gp}$.

See Appendix A.

Proposition 3.

If $u_3>{\displaystyle \frac{{\gamma }_1^2+{\gamma }_2^2}{2-2k^2}}$, then ${SW}^{hp}$ is negative jointly concave on $p_1^{hp}$, $p_2^{hp}$ and $e^{hp}$.

See Appendix A.

Figure 2 indicates that carriers should adopt FP strategy when price competition intensity increases, as the subsidy coefficient is much higher; in this way, carriers can get more profits, and we can derive that $u_2>u_1>u_3$.

Lemma 1.

[Impact of price competition intensity on prices, carbon emission, profits and social welfare]

(1) 

$p_i^{*}$ increases in $k$ and satisfies $p_1^{gp*}>p_1^{hp*}>p_1^{fp*}$, $p_2^{fp*}>p_2^{hp*}>p_2^{gp*}$;

(2) 

$e^{*}$ decreases in $k$, and $e^{gp*}>e^{fp*}>e^{hp*}$;

(3) 

${\pi }_i^{*}$ varies in $k$, and ${\pi }_1^{fp*}>{\pi }_1^{hp*}>{\pi }_1^{gp*}$, ${\pi }_2^{gp*}>{\pi }_2^{hp*}>{\pi }_2^{fp*}$;

(4) 

${SW}^{*}$ increases in $k$, and when $k$ is lower than 0.588, ${SW}^{hp*}>{SW}^{gp*}>{SW}^{fp*}$; when $k$ belongs to [0.588, 0.704], ${SW}^{gp*}>{SW}^{hp*}>{SW}^{fp*}$; when $k$ is higher than 0.704, ${SW}^{gp*}>{SW}^{fp*}>{SW}^{hp*}$.

The intensity of competition can help increase freight price and social welfare and help decrease profits in most circumstances when too much of the operational costs are used to reduce carbon emission to meet shippers’ green preference. However, without subsidies and carriers’ unwillingness to use cleaner fuels, this will result in the harmful effect of increased carbon emissions.

Lemma 2.

[Impact of subsidy coefficient on prices, carbon emission, profits and social welfare]

(1) 

$p_i^{*}$ decreases in $u_i$, and $p_1^{gp*}>p_1^{hp*}>p_1^{fp*}$, $p_2^{fp*}>p_2^{hp*}>p_2^{gp*}$;

(2) 

$e^{*}$ decreases in $u_i$, and $e^{gp*}>e^{fp*}>e^{hp*}$;

(3) 

${\pi }_i^{*}$ decreases in $u_i$, and ${\pi }_1^{fp*}>{\pi }_1^{hp*}>{\pi }_1^{gp*}$, ${\pi }_2^{gp*}>{\pi }_2^{hp*}>{\pi }_2^{fp*}$;

(4) 

${SW}^{*}$ decreases in $u_i$, when $u_i$ is lower than 2.14, ${SW}^{gp*}>{SW}^{hp*}>{SW}^{fp*}$; when $u_i\in \left[\mathrm{2.14,12}\right]$, ${SW}^{hp*}>{SW}^{gp*}>{SW}^{fp*}$.

It is obvious that freight prices under different scenarios can be lowered with subsidies, and the adoption of less-carbon fuel can be beneficial to environmental sustainability, leading to abatement in carbon emission. Furthermore, subsidies can offset operational costs, which can lead to a decrease in profits and social welfare.

Lemma 3.

[Impact of service level on prices, carbon emission, profits and social welfare]

(1) 

$p_i^{*}$ increases in $\theta$, and $p_1^{gp*}>p_1^{hp*}>p_1^{fp*}$, $p_2^{fp*}>p_2^{hp*}>p_2^{gp*}$;

(2) 

$e^{*}$ decreases in $\theta$, and $e^{gp*}>e^{fp*}>e^{hp*}$;

(3) 

${\pi }_i^{*}$ increases in $\theta$, and ${\pi }_1^{fp*}>{\pi }_1^{hp*}>{\pi }_1^{gp*}$, ${\pi }_2^{gp*}>{\pi }_2^{hp*}>{\pi }_2^{fp*}$;

(4) 

${SW}^{*}$ increases in $\theta$, and ${SW}^{hp*}>{SW}^{gp*}>{SW}^{fp*}$.

Lemma 3 indicates that high-quality service level can lead to higher operational costs, prompting carriers to increase freight prices. Meanwhile, customer satisfaction and loyalty can be improved with high-quality service levels and can attract high-end customers to pay more for faster and more reliable transportation, which boosts the increase in profits and social welfare. Additionally, the increase in service level always generates higher speeds and more consumption in fuel, resulting in increased carbon emissions.

4. Numerical Analysis

Based on the above theoretical analysis, next, a numerical simulation analysis will be conducted to verify all propositions and lemmas. According to extant literature [65,66], the relevant parameter assumptions are set as follows: a = 20, $k\in [0,0.8]$, $u_i\in [0,12]$, ${\beta }_i\in [0,1]$, ${\gamma }_i\in [0,1]$, $\theta =10$.

As Figure 3a shows, with ${\gamma }_1=0.1$ and ${\gamma }_2=0.9$, the increase in competition intensity can help increase freight prices, while carriers also aim to provide better transportation service. This is because carbon emission abatement necessitates increased investments in cleaner fuels, and carriers must raise prices to offset the additional operational costs. Despite these cost pressures, higher investments in cleaner fuels consistently lead to greater carbon emission reductions, as depicted in Figure 3b. Therefore, government subsidies should be increased to encourage the adoption of cleaner energy sources in transportation.

In a competitive market, carriers may need to enhance service quality or provide additional value-added services, such as faster delivery and improved customer service. However, these efforts often result in reduced profit margins, as illustrated in Figure 3c. Notably, in Scenario GP, subsidized Carrier 2 can achieve positive profits due to market demand stimulation driven by service improvements. In contrast, the profit curve for Carrier 1 in Scenario FP remains stable despite competition. This stability is likely attributable to a shift in competitive dynamics from price-based competition to service differentiation, such as speed, reliability and customer service enhancements. Additionally, growing consumer awareness of environmental issues has intensified competition, prompting carriers to adopt greener technologies, such as carbon-neutral shipping, to appeal to environmentally conscious customers. This shift contributes to reduced carbon emissions and other environmental benefits, thereby enhancing overall social welfare, as depicted in Figure 3d.

When $k$ is set 0.1, it is obvious that freight price will be lowered with the increase in the subsidy coefficient, as Figure 4a shows. When the subsidy coefficient is less than value 2, freight price drops sharply for Carrier 1 in Scenario GP and Carrier 2 in Scenario FP and changes smoothly thereafter. This is because high operational costs are offset by subsidies. And similar changes happen to carbon emission reduction, as illustrated in Figure 4b. At first, in the GP scenario, carbon emission decreases sharply when $u_i$ is less than value 2.14, while others show a relatively smooth trend. This is because carriers use cleaner energies with subsidies from the government, which is beneficial for carbon emission reduction and aligns with reality. While all profit and social welfare curves under different scenarios also decline with the increase in the subsidy coefficient, this is because fierce competition between carriers forces them to enhance the service level and increase investments in carbon emission abatement.

With the increase in service level, from Figure 5, we can see that all curves of prices, profits, social welfare and carbon emission rise, which aligns with Lemma 3. In a highly competitive market, offering high-quality service can help reduce the risk of delays, damage or loss of goods during transportation. Such higher operational costs enable shippers to pay higher fees for these services with strong willingness. With a high-quality service level, carriers tend to use higher speeds or run without being fully loaded, leading to exacerbated carbon emissions. Meanwhile, high-quality service helps attract more customers and increase cargo volume. In addition, value-added services such as expedited shipping, special handling or customized solutions also help to increase overall revenue and social welfare.

As Figure 6a,b shows, profits for both carriers in different scenarios all increase with the increase in green preference from shippers. This is because, on the one hand, carriers make investments in cleaner technology adoption, reducing operational costs and ultimately boosting profitability. On the other hand, shippers’ green preferences indirectly push carriers to access policy-driven benefits, such as government subsidies. And green transport services can help carriers attract environmentally conscious shippers and earn more profits. Service intensity can also help increase carriers’ profits, as presented in Figure 6c,d. High-quality services enhance carriers’ reputation and brand value, making it a preferred choice for shippers. Furthermore, service competition often incentivizes carriers to improve operational efficiencies, contributing to increased profitability.

Figure 7a,c show the impact of green preference on social welfare and carbon emission reduction. When environmental awareness ${\gamma }_1$ and ${\gamma }_2$ increase, ${SW}^{*}$ and $e^{*}$ also increase. This is because shippers usually opt for more energy-efficient or environmentally friendly transport modes, and if subsidy policies are not properly designed, they might lead to inefficiencies or unintended consequences that increase emissions. Furthermore, encouraging carriers to invest in green technologies, in turn, might create more social benefits.

As the service level intensifies, it can lead to both an increase in carbon emissions and social welfare, as displayed in Figure 7b,d. When carriers improve service levels, it often means more frequent trips, longer routes or higher speeds, which increases fuel consumption and, therefore, carbon emissions. Furthermore, an improvement in the service level of carriers usually means faster and more reliable transportation, thus enhancing shippers’ satisfaction and utility and overall social welfare.

5. Conclusions and Discussion

5.1. Main Findings

Shipping activities are a major source of greenhouse gas emissions, making it essential to tackle the issue of carbon emission for shipping companies. The IMO sets regulations like the International Convention for the Prevention of Pollution from Ships (MARPOL), which aims to reduce shipping pollution emissions and mitigate the environmental impact. These regulations affect the economic behavior of shipping companies, often leading them to invest in cleaner technologies or fuel. However, high operational costs hinder shipping carriers from investing in green technologies, while government subsidies can help overcome these cost barriers. But in a price-and-service competitive environment, whether governmental subsidies can aid in carbon emission mitigation is a topic less commonly addressed in extant literature. Inspired by this, our paper considers a shipping supply chain consisting of a port, the government, two competitive carriers and shippers. Game-theoretic models are developed to depict shipping carriers’ green technology adoption decisions and the design of government policies. Ultimately, optimal results are derived and testified through numerical analysis. Our study yields several interesting results.

First, increased price competition intensity can help increase freight prices, carbon emission and social welfare under all scenarios. However, it may lead to reduced profits in some situations due to additional services provided by carriers. Second, governmental subsidies can help lower freight prices and carbon emission; meanwhile, they also result in a reduction in profits and social welfare. This indicates that a government subsidy strategy can benefit carbon emission mitigation. Third, under all scenarios, prices, profits, social welfare and carbon emissions increase with the rise in service level, given a determined competition coefficient when Carrier 1, Carrier 2 or both carriers are subsidized. Finally, when green preference and intensified service level increase, profits for different carriers, social welfare and carbon emission reduction also increase under three different scenarios. In summary, our findings can assist carriers in formulating subsidy policies that support carbon emission reduction efforts with shippers’ low-carbon preferences when carriers are competitive in price and service. And our paper can enrich the theoretical research by incorporating government subsidies, green environmental consciousness, carriers’ price and service competition within a shipping supply chain.

5.2. Managerial Implications

Our study serves as a first step in incorporating game theory to design subsidies in a price-and-service competitive environment with green shippers. It contributes to the extant literature as follows:

Firstly, we uncover that governmental subsidies play a substantial role in carbon emission abatement across all scenarios within a price-and-service competitive environment, particularly when shippers exhibit strong green awareness. These subsidies provide carriers with the financial incentives necessary to adopt greener technologies and practices, contributing to significant reductions in carbon emissions. However, the study also reveals that government subsidies have a negative impact on freight prices, profits, and social welfare, regardless of the specific scenario in which carriers operate. While subsidies can promote environmental sustainability, they may also distort market dynamics by increasing carriers’ operational costs. This can lead to higher freight prices and reduced profitability. Additionally, these distortions may limit the overall economic benefits to social welfare, highlighting the need for carefully designed policy interventions. Future research could explore how subsidy schemes can be optimized to mitigate these negative effects while still achieving the desired environmental outcomes.

Secondly, a higher service quality level enhances prices, profits, carbon emissions and social welfare under different scenarios. However, in a highly price-competitive market, this can lead to increased carbon emissions due to the greater resource consumption associated with providing higher-quality services. This finding offers valuable managerial insights for decision-makers, including governments and shipping companies, in formulating effective environmental protection policies. Specifically, it emphasizes the need to balance service quality improvements with strategies aimed at reducing carbon emissions. Policymakers should consider regulatory frameworks that encourage both higher service standards and sustainable practices, ensuring that quality enhancements do not come at the expense of environmental goals.

Thirdly, price competition between carriers is detrimental to carbon emission reduction and may result in a decline in profits under certain scenarios. However, it has a positive influence on freight prices and social welfare. Our study demonstrates that price competition complicates carriers’ decisions to adopt green technologies, as the pressure to reduce costs may limit the resources available for sustainability investments. This highlights the need for policymakers to design balanced policies that encourage green technology adoption while maintaining competitive market dynamics. Furthermore, the findings suggest that governments should consider integrating incentives for environmental practices within competitive frameworks to ensure that carbon emission reduction is not sacrificed for short-term economic gains.

Lastly, increased shippers’ green preferences and intensified service competition jointly improve profits and social welfare but lead to an increase in carbon emissions under all scenarios. This finding suggests that while carriers are incentivized to enhance service quality to meet consumers’ environmental demands, such efforts often involve higher resource consumption, contributing to greater carbon emissions. In a competitive market, carriers are committed to delivering high-quality services to satisfy consumers’ preferences, particularly in terms of sustainability. However, this focus on service excellence, while beneficial for social welfare and profits, may create unintended environmental consequences. Thus, policymakers need to carefully design regulatory frameworks that support green innovation without undermining service quality improvements. Balancing environmental goals with competitive market dynamics is essential to ensure that sustainability is integrated into the business model without causing negative trade-offs.

5.3. Limitations and Future Research

This article focuses on the effect of governmental subsidies on carbon emission reduction with carriers’ competition and shippers’ green environmental consciousness and, inevitably, has some limitations that warrant further exploration. First, future research can extend the discussion by examining the impact of government subsidies on ports and forwarders to reduce carbon emission with carriers’ price and service competition and shippers’ green preference in a shipping supply chain. Second, in a price-and-service competitive environment, governments may adopt alternative regulatory measures, such as carbon emission taxes or carbon quotas, to incentivize environmental sustainability. It is interesting to consider the effect of carbon emission taxes or carbon quotas to better understand their performance in terms of sustainability and marginal revenue.