Synergies of Government Subsidies and Service Premium: A Game-Theoretic Analysis of Transport Mode Selection for Electric Vehicle Exports

Abstract

This paper investigates the coordination between logistics and policy decisions for electric vehicle (EV) exports under the Belt and Road Initiative. Focusing on the two modes—maritime shipping and the China Railway Express (CR Express)—along with government production subsidies, import tariffs, and service premium, a Stackelberg game model for a cross-border supply chain comprising a domestic manufacturer and an overseas retailer is constructed. The equilibrium outcomes under four scenarios formed by combining subsidy policies and transportation modes (Models NM, NR, GM and GR) are compared theoretically and numerically, with further evaluation of capacity constraints and power structures, as well as the robustness verification of the core findings. Results show that the CR Express mode exhibits a service-driven nonlinear cost pattern, where its service premium amplifies positive market responses. Its appeal to the manufacturer, however, is tightly constrained by fixed cost. Furthermore, government subsidies can overcome this barrier by synergizing with the service premium, turning the CR Express into a relatively advantageous strategy. Moreover, subsidy efficacy is conditional, depending heavily on the service premium level and logistics cost coefficient, leading to a proposed differentiated subsidy framework. This study offers a theoretical basis for corporate logistics strategy and targeted policy design.

1. Introduction

Against the urgent backdrop of the global energy transition and the imperative to address climate change, the electric vehicle (EV) industry has emerged as a pivotal force driving the transformation of the global transportation system [1,2]. As the world’s largest producer and consumer market for EVs, China is actively leveraging the Belt and Road Initiative to expand into overseas markets while maintaining robust domestic growth, thereby exporting green technologies and production capacity [3,4,5]. Data from the China Association of Automobile Manufacturers (CAAM) shows that from January to November 2025, China exported 1.633 million EVs to Belt and Road countries—a year-on-year surge of 83.5%—making this region a key engine for the industry’s global expansion. This progress benefits from top-down policy guidance and the trade facilitation arrangements established with partner countries.

However, the operation of China’s cross-border EV supply chains faces multiple challenges. Primarily, cross-border trade costs, most notably import tariffs, directly undermine product price competitiveness and erode supply chain profits [6]. Moreover, the strategic complexity of international logistics presents a significant trade-off, as exports predominantly rely on two modes, i.e., maritime shipping and the China Railway Express (CR Express) [7]. CR Express entails higher unit costs, yet its significantly shorter and more predictable transit times —for instance, approximately one-third that of maritime shipping—and lower delay risk, compared to maritime shipping’s exposure to port congestion and route disruptions, enhance supply chain reliability and resilience, underpinning its distinctive service premium [3,8]. Consequently, logistics decisions become a critical factor influencing market demand and overall supply chain performance. Finally, a pressing challenge lies in optimizing the efficacy of government subsidy policies. A key issue is the understanding of variations in the effectiveness of production subsidies across different transportation modes and their precise calibration to maximize incentives, which constitutes a crucial theoretical and practical imperative.

The existing literature has established a solid foundation by examining cross-border supply chain management [9,10,11,12,13,14], policy design [9,15,16,17,18,19] and logistics optimization [7,11,20,21,22,23,24,25]. A critical synthesis of these streams, however, is required to address the integrated challenges of EV exports. This endeavor demands a novel framework to model the strategic interplay among policies, costs, and logistics choices, as well as generalizable insights beyond context-specific simulations for actionable cross-market guidance.

To bridge these gaps, this paper aims to systematically investigate the following three core research questions:

(1)

How can a Stackelberg game model formalize and solve the supply chain members’ problem of optimal transportation mode selection in a policy-free environment, given the trade-offs between cost, service premium, and fixed investment?

(2)

How does the introduction of government production subsidies differentially affect pricing, logistics investment, demand, and profit distribution under the two transportation modes, thereby altering the supply chain’s equilibrium?

(3)

What is the interaction mechanism between subsidy policies and different transportation modes? How can a differentiated policy framework, based on key market parameters, be constructed to enhance subsidy efficiency and overall supply chain competitiveness?

To answer these questions, this paper focuses on a cross-border EV supply chain consisting of a domestic manufacturer and an overseas retailer, specifically examining the interactive effects between transportation mode choice (maritime shipping and CR Express) and government production subsidy policies within the typical context of exports to Belt and Road countries.

The innovative contributions of this paper are threefold. First, this study develops an integrated Stackelberg game framework that simultaneously incorporates government subsidies, tariff costs, and two heterogeneous transportation modes. Through systematic comparison across four scenarios (Models NM, NR, GM and GR), the model explicitly captures the strategic interdependence between policy instruments and operational decisions. Second, moving beyond qualitative description, this paper formally endogenizes the service premium of CR Express as a core demand driver, which theoretically reveals its nonlinear amplifying effect on profit and, more distinctively, uncovers a synergistic enhancement mechanism with government subsidies. Third, the analysis derives generalizable decision thresholds for critical parameters (e.g., service premium level, fixed cost), enabling a shift from scenario-specific results to a differentiated subsidy strategy framework with clear theoretical grounding. These thresholds provide novel, actionable decision support for enhancing the precision of both corporate logistics strategy and public policy, bridging a gap between theoretical modeling and practical implementation.

The remainder of this paper is organized as follows. Section 2 provides a literature review. Section 3 describes the problem and presents the basic assumptions. Section 4 constructs the game-theoretic models for the four scenarios and solves for their equilibria. Section 5 presents theoretical analysis, deriving and discussing propositions concerning the impact of key parameters, mode comparison, and the conditional effects of subsidies. Section 6 employs numerical experiments to validate the theoretical findings and visually demonstrate decision boundaries. Section 7 examines the effects of capacity constraints and power structure on supply chain equilibrium, with robustness tests conducted on the model’s core findings. Finally, Section 8 concludes the paper, summarizing managerial insights and suggesting directions for future research.

2. Literature Review

Our research synthesizes insights from and contributes to three streams of literature: (1) cross-border supply chains, (2) tariffs and government subsidies, and (3) international transportation modes. The following review situates our work within these areas, thereby clarifying its distinct contributions.

2.1. Cross-Border Supply Chains

Research on cross-border supply chains extensively examines the coordination challenges arising from operational and policy-induced frictions [9,10,11,12,13]. A primary focus has been on strategic interactions under asymmetric information and conflicting incentives. For instance, Chen and Xu (2024) analyze the “prisoner’s dilemma” in order timing within a co-opetitive supply chain, where retailers face a trade-off between securing first-mover advantages and acquiring demand information [11]. Similarly, pricing and channel coordination under tax burdens are critical. Yu et al. (2024) investigate a global manufacturer’s pricing dilemma when import taxes incentivize the emergence of unauthorized parallel trade channels, requiring a strategic balance between tax offset and market control [12]. Furthermore, supply risk mitigation is a central theme, with Cui et al. (2025) examining how a domestic manufacturer’s multi-sourcing strategy interacts with a foreign supplier’s decision on offering medium-quality components, highlighting the interplay between diversification benefits and competitive effects [13].

Another significant stream of literature investigates how external policy frameworks and emerging technologies reshape cross-border supply chain structures and decisions [14,26,27,28,29]. Policy instruments are shown to have complex, conditional effects. Wei et al. (2023) compare how different taxation models (e.g., destination-based vs. origin-based) influence optimal pricing and order quantities in a dual-channel supply chain, with tariffs consistently impacting wholesale prices [14]. Li et al. (2025) further explores the strategic interplay between carbon abatement policies (constraints, trading, tariffs) and import quotas, finding that the welfare-maximizing policy mix depends critically on quota levels [26]. Concurrently, the adoption of digital technologies like blockchain is a major research frontier. Studies such as Jiang et al. (2024) and Mishra et al. (2025) demonstrate blockchain’s role in enhancing transparency, trust, and coordination in dual-channel and cold chain contexts, respectively, though its viability depends on investment costs and consumer sensitivity [27,28]. Xie et al. (2025) extends this by analyzing the dynamic stability of pricing decisions under different blockchain adoption modes, revealing how adoption entities reshape profit distribution and system resilience [29].

2.2. Tariffs and Government Subsidies

A substantial body of literature examines the multifaceted impact of tariff policies on supply chain operations [14,30,31,32,33,34]. Studies reveal that the specific design of a tariff instrument significantly influences firm strategies. For instance, Niu et al. (2022) compare ad valorem and specific tariffs, showing that the preference of e-tailers for a particular type depends on their product quality segment, and that a “quality update dilemma” can distort pricing and sales [30]. Beyond the type, the level and structure of tariffs critically shape global sourcing decisions. Research demonstrates that higher tariffs can backfire on policies aimed at reshoring production. Chen et al. (2022) find that raising import tariffs might discourage local sourcing for a global manufacturer due to its integrated supply chain structure and the foreign supplier’s strategic response [31]. The impact is also evident in operational metrics; Wei et al. (2023) and Hu et al. (2022) both find that tariffs consistently exert downward pressure on wholesale prices and can erode overall supply chain profits [14,32]. Furthermore, tariffs are often studied in conjunction with other policy tools, such as carbon taxes or import quotas, indicating their role within a broader policy ecosystem that shapes complex operational trade-offs [33,34].

Concurrently, research on government subsidies, particularly for promoting EVs, provides insights into optimal policy design [9,15,16,17,18,19]. Studies explore the efficacy of different subsidy targets, such as subsidizing consumers versus charging infrastructure firms, and consider consumer behavioral factors like “EV anxiety” and environmental benefit in determining optimal subsidy levels [16,17]. A key and highly relevant stream within this literature investigates the interaction between subsidy and tariff policies. Several studies explicitly model this interplay. Yi and Wen (2023) examine a transnational green supply chain and concludes that government subsidies in the exporting country can effectively counteract the negative impact of tariffs imposed by the importing country on product greenness, prices, and profits [18]. Similarly, Fan et al. (2020) analyze the strategic game between domestic and imported EV manufacturers, finding that implementing combined subsidy and tariff policies can improve the profit of the domestic manufacturer and social welfare, especially when technology spillover is significant [9]. These studies confirm that subsidies and tariffs are not independent levers; their effects are interconnected, and a joint analysis is crucial for effective policymaking and strategic firm response. This body of work provides a foundation for our investigation into how production subsidies interact with tariffs in the specific context of transportation mode selection.

2.3. International Transportation Modes

A substantial stream of literature in operations research focuses on the operational optimization and comparative analysis of different international transportation modes [7,20,21,22,23,24,25]. Much of this work employs mathematical programming and simulation models to address routing, scheduling, and cost-efficiency challenges within multimodal networks. For instance, studies like Zhen et al. (2025) and Zhang et al. (2025) delve into the tactical choice between specific maritime shipping subtypes, such as container versus roll-on/roll-off shipping for automobiles or liner versus tramp shipping, analyzing their cost structures and optimal use under demand uncertainty [23,24]. Concurrently, research on port operations and shipping technologies, such as stowage planning and automated guided vehicle scheduling in container terminals, seeks to enhance the efficiency and resilience of the underlying logistics infrastructure [7,25]. These studies provide a foundational understanding of the cost, capacity, and operational flexibility characteristics inherent to different transportation methods, which are essential for modeling their strategic selection.

A distinct and rapidly growing body of research specifically examines the CR Express as a transformative logistics mode along the Belt and Road [3,8,35,36,37,38]. Scholars have investigated its economic viability, market development, and competitive positioning. Key themes include the role of government subsidies in its initial growth phase and the subsequent need for sustainable, market-driven operations [36]. Furthermore, studies assess the CR Express’s performance through lenses such as sustainable development, focusing on infrastructure reliability and trade facilitation [3], and through novel connectivity indices that compare its monetary and time-based costs directly against traditional maritime shipping [8]. This research collectively establishes CR Express as a strategically important, time-competitive alternative to maritime shipping, while also highlighting its unique cost structure and policy-dependent nature. However, it also identifies a gap in systematically modeling how its distinctive service advantage (e.g., speed and reliability) interacts with government subsidies within a coordinated supply chain decision-making framework, which is the focal point of our study.

2.4. Research Gaps

Based on the existing literature, this study aims to address the following key research gaps:

(1)

While existing literature valuably examines policy interactions and transportation choices separately, the tripartite interplay among subsidies, tariffs, and logistics mode selection remains less explored. Studies linking subsidy and tariff policies often treat logistics as an exogenous cost; Conversely, research on transport modes does not adequately incorporate the strategic policy environment when optimizing for efficiency. Therefore, a unified theoretical framework is needed to capture the equilibrium decisions arising from the strategic interactions between governmental policies and a firm’s choice between cost-driven (maritime shipping) and service-oriented (CR Express) transport.

(2)

Although the service advantages of the CR Express are widely acknowledged, its corresponding service premium is seldom formalized as an endogenous driver of market demand within quantitative models. Current approaches often treat this premium as a qualitative descriptor, a methodological choice that limits the ability to capture its intrinsic role in shaping consumer demand. Consequently, how this demand-enhancing effect synergizes with government subsidies to influence supply chain profitability and mode selection remains a critical yet under-analyzed dynamic.

(3)

Studies on transportation mode selection and policy design often rely on methodologies such as numerical simulation or case studies. While valuable for yielding context-specific insights, such approaches make it challenging to derive universal, analytical thresholds for key parameters (e.g., service premium level, fixed cost). This leaves a gap in providing a clear, generalizable decision boundary map, which is crucial for guiding firms’ logistics strategies and governments’ policy design across varied market conditions.

To address the identified methodological gaps, this study develops a game-theoretic framework that transcends the piecemeal approach. It enables integrated analysis of subsidies, tariffs, and transportation mode choices as interdependent equilibrium outcomes. Crucially, the service premium of the CR Express is formally parameterized in the demand function, facilitating the analysis of its quantitative interaction with policy levers. Unlike context-specific simulations, this analytical approach derives generalizable, closed-form thresholds that delineate clear decision boundaries for strategy and policy design.

3. Problem Description and Assumptions

To examine the interaction mechanism between government subsidies and transportation mode choice, this study constructs a bilateral monopoly structure consisting of a manufacturer and a retailer. The manufacturer, located in the home country, produces EVs, while the retailer, based in a country along the Belt and Road, is responsible for overseas sales. This two-echelon supply chain structure is illustrated in Figure 1.

The manufacturer sells products to the retailer at a free-on-board (FOB) price $w$, and the retailer subsequently sells them to overseas consumers at a retail price $p$. In addition to bearing the unit production cost $c_m$, the manufacturer is also responsible for product logistics. When exporting to countries along the Belt and Road, several transportation modes are typically available, such as railway, maritime shipping, multimodal transport, and road transport. Considering factors such as speed, cost, capacity, and applicability, the supply chain manager needs to determine the optimal logistics level $t$ to meet market requirements. This study examines two mainstream modes: maritime shipping and the CR Express. Maritime shipping offers lower unit costs but longer transit times, with consumers focusing primarily on the product itself, its price, and basic logistics timeliness. Although CR Express incurs higher unit costs, its faster and more reliable transportation service provides consumers with higher perceived value, thereby generating a service premium. Building on prior work [39,40], the manufacturer’s logistics cost function is specified as follows: under maritime shipping mode, it is $k\cdot t^2/2$, where $k>0$ is the logistics cost coefficient; under CR Express mode, there exists a fixed cost $F>0$ [3], which covers one-time or periodic expenditures such as terminal access fees, dedicated equipment leases, administrative setup, and contractual infrastructure charges, making the total logistics cost $F+k\cdot t^2/2$.

The logistics level $t$ directly influences product availability, delivery time, inventory management, transportation cost, and market responsiveness, thereby affecting market demand. This study assumes that the demand for EVs in countries along the Belt and Road is linearly influenced by the retail price $p$ and the logistics level $t$: product utility is negatively correlated with $p$ and positively correlated with $t$. Following existing research [41,42], the consumer utility functions under the two transportation modes are respectively defined as:

$$U_1=v-p+\beta \cdot t$$

(1)

$$U_2=v\cdot (1+\gamma )-p+\beta \cdot t$$

(2)

Here, $v$ represents the consumer’s basic perceived utility for the product, which follows a uniform distribution on the interval [0,1]; $\gamma >0$ denotes the service premium coefficient brought by the CR Express mode; and $\beta >0$ is the consumer’s sensitivity coefficient to the logistics level.

It is important to note that the service premium coefficient $\gamma$ is modeled as an exogenous parameter. This is because $\gamma$ represents the market’s aggregated valuation of CR Express’s systematic advantages—such as schedule reliability and infrastructure resilience—over maritime shipping. It thus captures the net effect of underlying factors including consumer preference for certainty and supply-chain predictability, as well as the competitive differentiation of CR Express in the market.

A consumer will purchase the product if and only if the utility $U>0$. Consequently, the market demand functions under the two modes can be derived as:

$$D_1=1-p+\beta \cdot t$$

(3)

$$D_2=1-{\displaystyle \frac{p-\beta \cdot t}{1+\gamma }}$$

(4)

Considering that most countries along the Belt and Road impose ad valorem import tariffs, the retailer must pay a tariff based on the import value, which includes the product price and the unit logistics surcharge (including freight, insurance and other additional costs), denoted as $c_r$. Given a tariff rate $\delta$, the tariff amount is $\delta \cdot (w+c_r)$. Some countries may implement tariff reduction policies to promote trade. Let the reduction degree be $\lambda$ ($0<\lambda <1$), then the actual tariff paid becomes $\delta \cdot (1-\lambda )\cdot (w+c_r)$. Furthermore, to boost exports, the home government may provide a unit production subsidy $\mu$ to the manufacturer, resulting in total government subsidy expenditure of $\mu \cdot D$ where $D$ represents the market demand. To focus on the economic mechanisms of tariff and subsidy policies, this study does not incorporate fluctuations in financial variables (e.g., exchange rates), or external operational shocks (e.g., political volatility, technical reliability issues, and infrastructure disruptions) in the model, thereby maintaining clarity and focus on the core economic logic.

The supply chain members engage in a Stackelberg game: the manufacturer, as the leader, first decides the logistics level $t$ and the FOB price $w$; the retailer, as the follower, determines the retail price $p$ after observing the manufacturer’s decisions. Both parties aim to maximize their own profits.

Based on the previous settings, this study constructs and compares the following four scenario models:

• Model NM (No subsidy and maritime shipping scenario): The manufacturer adopts maritime shipping without government production subsidy.
• Model NR (No subsidy and CR Express scenario): No government production subsidy is provided, and the manufacturer adopts the CR Express mode.
• Model GM (With subsidy and maritime shipping scenario): The government provides a unit production subsidy $\mu$, and the manufacturer adopts the maritime shipping mode.
• Model GR (With subsidy and CR Express scenario): With the unit subsidy $\mu$ in place, the manufacturer adopts the CR Express mode.

The superscripts ‘NM’, ‘NR’, ‘GM’ and ‘GR’ denote the four scenarios, respectively. The subscripts ‘M’ and ‘R’ refer to the manufacturer and the retailer, respectively. All key notations used throughout the paper are summarized in

Table 1. The main notations in this paper.

Symbol	Definition
$c_m$	Unit production cost
$c_r$	Unit logistics surcharge
$k$	Logistics cost coefficient
$\gamma$	Service premium coefficient brought by the CR Express mode
$\beta$	Consumer’s sensitivity coefficient to the logistics level
$F$	Fixed cost incurred by CR Express
$\delta$	Tariff rate
$\lambda$	Tariff reduction degree
$\mu$	Production subsidy provided by the government
$D^i$	Demand for EVs in Model $i$, $i\in \{NM,NR,GM,GR\}$
$w^i$	FOB price of EVs in Model $i$, $i\in \{NM,NR,GM,GR\}$
$t^i$	Logistics level in Model $i$, $i\in \{NM,NR,GM,GR\}$
$p^i$	Retail price of EVs in Model $i$, $i\in \{NM,NR,GM,GR\}$
${\pi }_j^i$	Profit of $j$ in Model $i$, $i\in \{NM,NR,GM,GR\}$, $j\in \{M,R\}$

.

4. Materials and Methods

4.1. Model NM (No Subsidy and Maritime Shipping Scenario)

In this scenario, the government does not provide production subsidies, and the manufacturer adopts the maritime shipping mode. The manufacturer’s profit is derived from the gross profit generated by selling EVs to the retailer, with the final profit being formed after deducting the logistics costs. The retailer’s profit, on the other hand, comes from the gross profit earned by selling EVs to end consumers, and the final profit is determined after paying the relevant import tariffs. The profit functions of the manufacturer and the retailer are given by:

$${\pi }_M^{NM}(w^{NM},t^{NM})=(w^{NM}-c_m)\cdot (1-p^{NM}+\beta \cdot t^{NM})-{\displaystyle \frac{1}{2}}k\cdot {(t^{NM})}^2$$

(5)

$${\pi }_R^{NM}(p^{NM})=(p^{NM}-w^{NM}-\delta \cdot (1-\lambda )\cdot (w^{NM}+c_r))\cdot (1-p^{NM}+\beta \cdot t^{NM})$$

(6)

This Stackelberg game is solved by backward induction. First, the retailer’s reaction function with respect to the retail price $p^{NM}$ is derived. Next, this reaction function is substituted into the manufacturer’s profit function to solve for the optimal FOB price $w^{NM*}$ and logistics level $t^{NM*}$. Finally, these optimal values are back-substituted to obtain the optimal retail price $p^{NM*}$, market demand $D^{NM*}$, and the optimal profits for both parties ${\pi }_M^{NM*}$ and ${\pi }_R^{NM*}$, which are listed as follows:

$$t^{NM*}={\displaystyle \frac{\beta \cdot \left(1-\delta \cdot (1-\lambda )\cdot c_r-(1+\delta \cdot (1-\lambda ))\cdot c_m\right)}{4\cdot k\cdot (1+\delta \cdot (1-\lambda ))-{\beta }^2}}$$

$$w^{NM*}=c_m+{\displaystyle \frac{2\cdot k\cdot \left(1-\delta \cdot (1-\lambda )\cdot c_r-(1+\delta \cdot (1-\lambda ))\cdot c_m\right)}{4\cdot k\cdot (1+\delta \cdot (1-\lambda ))-{\beta }^2}}$$

$$p^{NM*}=c_m+\delta \cdot (1-\lambda )\cdot (c_m+c_r)+{\displaystyle \frac{3\cdot (1+\delta \cdot (1-\lambda ))\cdot k\cdot \left(1-c_m-\delta \cdot (1-\lambda )\cdot (c_m+c_r)\right)}{4\cdot k\cdot (1+\delta \cdot (1-\lambda ))-{\beta }^2}}$$

$$D^{NM*}={\displaystyle \frac{2\cdot k\cdot \left(1-c_m-\delta \cdot (1-\lambda )\cdot (c_m+c_r)\right)}{4\cdot k\cdot (1+\delta \cdot (1-\lambda ))-{\beta }^2}}$$

$${\pi }_M^{NM*}={\displaystyle \frac{2\cdot k^2\cdot {\left(1-c_m-\delta \cdot (1-\lambda )\cdot (c_m+c_r)\right)}^2}{{(4\cdot k\cdot (1+\delta \cdot (1-\lambda ))-{\beta }^2)}^2}}$$

$${\pi }_R^{NM*}={\displaystyle \frac{4\cdot (1+\delta \cdot (1-\lambda ))\cdot k^2\cdot {\left(1-c_m-\delta \cdot (1-\lambda )\cdot (c_m+c_r)\right)}^2}{{(4\cdot k\cdot (1+\delta \cdot (1-\lambda ))-{\beta }^2)}^2}}$$

4.2. Model NR (No Subsidy and CR Express Scenario)

In this scenario, the government does not provide a subsidy, and the manufacturer adopts the CR Express mode, bearing a fixed logistics cost $F$. The profit functions of both parties are as follows:

$${\pi }_M^{NR}(w^{NR},t^{NR})=(w^{NR}-c_m)\cdot (1-{\displaystyle \frac{p^{NR}+\beta \cdot t^{NR}}{1+\gamma }})-{\displaystyle \frac{1}{2}}k\cdot {(t^{NR})}^2-F$$

(7)

$${\pi }_R^{NR}(p^{NR})=(p^{NR}-w^{NR}-\delta \cdot (1-\lambda )\cdot (w^{NR}+c_r))\cdot (1-{\displaystyle \frac{p^{NR}+\beta \cdot t^{NR}}{1+\gamma }})$$

(8)

Using backward induction, we derive the equilibrium results in Model NR, which are listed as follows:

$$t^{NR*}={\displaystyle \frac{\beta \cdot \left((1+\gamma )-c_m-\delta \cdot (1-\lambda )\cdot (c_m+c_r)\right)}{4\cdot k\cdot (1+\delta \cdot (1-\lambda ))\cdot (1+\gamma )-{\beta }^2}}$$

$$w^{NR*}=c_m+{\displaystyle \frac{2\cdot k\cdot (1+\gamma )\cdot \left((1+\gamma )-c_m-\delta \cdot (1-\lambda )\cdot (c_m+c_r)\right)}{4\cdot k\cdot (1+\delta \cdot (1-\lambda ))\cdot (1+\gamma )-{\beta }^2}}$$

$$p^{NR*}=c_m+\delta \cdot (1-\lambda )\cdot (c_m+c_r)+{\displaystyle \frac{3\cdot (1+\delta \cdot (1-\lambda ))\cdot k\cdot \left((1+\gamma )-c_m-\delta \cdot (1-\lambda )\cdot (c_m+c_r)\right)}{4\cdot k\cdot (1+\delta \cdot (1-\lambda ))\cdot (1+\gamma )-{\beta }^2}}$$

$$D^{NR*}={\displaystyle \frac{2\cdot k\cdot (1+\gamma )\cdot \left((1+\gamma )-c_m-\delta \cdot (1-\lambda )\cdot (c_m+c_r)\right)}{4\cdot k\cdot (1+\delta \cdot (1-\lambda ))\cdot (1+\gamma )-{\beta }^2}}$$

$${\pi }_M^{NR*}={\displaystyle \frac{2\cdot k^2\cdot {(1+\gamma )}^2\cdot {\left((1+\gamma )-c_m-\delta \cdot (1-\lambda )\cdot (c_m+c_r)\right)}^2}{{(4\cdot k\cdot (1+\delta \cdot (1-\lambda ))\cdot (1+\gamma )-{\beta }^2)}^2}}-F$$

$${\pi }_R^{NR*}={\displaystyle \frac{4\cdot (1+\delta \cdot (1-\lambda ))\cdot k^2\cdot {(1+\gamma )}^2\cdot {\left((1+\gamma )-c_m-\delta \cdot (1-\lambda )\cdot (c_m+c_r)\right)}^2}{{(4\cdot k\cdot (1+\delta \cdot (1-\lambda ))\cdot (1+\gamma )-{\beta }^2)}^2}}$$

4.3. Model GM (With Subsidy and Maritime Shipping Scenario)

In this scenario, the government provides the manufacturer with a unit production subsidy $\mu$, and the manufacturer adopts the maritime shipping mode. The profit functions at this point are as follows:

$${\pi }_M^{GM}(w^{GM},t^{GM})=(w^{GM}-c_m+\mu )\cdot (1-p^{GM}+\beta \cdot t^{GM})-{\displaystyle \frac{1}{2}}k\cdot {(t^{GM})}^2$$

(9)

$${\pi }_R^{GM}(p^{GM})=(p^{GM}-w^{GM}-\delta \cdot (1-\lambda )\cdot (w^{GM}+c_r))\cdot (1-p^{GM}+\beta \cdot t^{GM})$$

(10)

We use the backward induction method to derive the equilibrium results, listed as follows:

$$t^{GM*}={\displaystyle \frac{\beta \cdot \left((1-c_m+\mu )-\delta \cdot (1-\lambda )\cdot (c_m+c_r-\mu )\right)}{4\cdot k\cdot (1+\delta \cdot (1-\lambda ))-{\beta }^2}}$$

$$w^{GM*}=(c_m-\mu )+{\displaystyle \frac{2\cdot k\cdot \left((1-c_m+\mu )-\delta \cdot (1-\lambda )\cdot (c_m+c_r-\mu )\right)}{4\cdot k\cdot (1+\delta \cdot (1-\lambda ))-{\beta }^2}}$$

$$p^{GM*}=(c_m-\mu )+\delta \cdot (1-\lambda )\cdot (c_m+c_r-\mu )+{\displaystyle \frac{3\cdot (1+\delta \cdot (1-\lambda ))\cdot k\cdot \left((1-c_m+\mu )-\delta \cdot (1-\lambda )\cdot (c_m+c_r-\mu )\right)}{4\cdot k\cdot (1+\delta \cdot (1-\lambda ))-{\beta }^2}}$$

$$D^{GM*}={\displaystyle \frac{2\cdot k\cdot \left(1-\delta \cdot (1-\lambda )\cdot c_r-((1-c_m+\mu )-\delta \cdot (1-\lambda )\cdot (c_m+c_r-\mu )\right)}{4\cdot k\cdot (1+\delta \cdot (1-\lambda ))-{\beta }^2}}$$

$${\pi }_M^{GM*}={\displaystyle \frac{2\cdot k^2\cdot {\left(1-\delta \cdot (1-\lambda )\cdot c_r-((1-c_m+\mu )-\delta \cdot (1-\lambda )\cdot (c_m+c_r-\mu )\right)}^2}{{(4\cdot k\cdot (1+\delta \cdot (1-\lambda ))-{\beta }^2)}^2}}$$

$${\pi }_R^{GM*}={\displaystyle \frac{4\cdot (1+\delta \cdot (1-\lambda ))\cdot k^2\cdot {\left(1-\delta \cdot (1-\lambda )\cdot c_r-((1-c_m+\mu )-\delta \cdot (1-\lambda )\cdot (c_m+c_r-\mu )\right)}^2}{{(4\cdot k\cdot (1+\delta \cdot (1-\lambda ))-{\beta }^2)}^2}}$$

4.4. Model GR (With Subsidy and CR Express Scenario)

This scenario combines the government production subsidy with the CR Express transportation mode. The profit functions of the manufacturer and the retailer are as follows:

$${\pi }_M^{GR}(w^{GR},t^{GR})=(w^{GR}-c_m+\mu )\cdot (1-{\displaystyle \frac{p^{GR}+\beta \cdot t^{GR}}{1+\gamma }})-{\displaystyle \frac{1}{2}}k\cdot {(t^{GR})}^2-F$$

(11)

$${\pi }_R^{GR}(p^{GR})=(p^{GR}-w^{GR}-\delta \cdot (1-\lambda )\cdot (w^{GR}+c_r))\cdot (1-{\displaystyle \frac{p^{GR}+\beta \cdot t^{GR}}{1+\gamma }})$$

(12)

The model is solved using backward induction, yielding the optimal solutions for the retail price, FOB price, logistics level, and the profits of the cross-border supply chain members under Model GR, as shown:

$$t^{GR*}={\displaystyle \frac{\beta \cdot \left((1+\gamma )-\delta \cdot (1-\lambda )\cdot c_r-(1+\delta \cdot (1-\lambda ))\cdot (c_m-\mu )\right)}{4\cdot k\cdot (1+\delta \cdot (1-\lambda ))\cdot (1+\gamma )-{\beta }^2}}$$

$$w^{GR*}=c_m+{\displaystyle \frac{2\cdot k\cdot (1+\gamma )\cdot \left((1+\gamma )-\delta \cdot (1-\lambda )\cdot c_r-(1+\delta \cdot (1-\lambda ))\cdot (c_m-\mu )\right)}{4\cdot k\cdot (1+\delta \cdot (1-\lambda ))\cdot (1+\gamma )-{\beta }^2}}$$

$$p^{GR*}=(c_m-\mu )+\delta \cdot (1-\lambda )\cdot (c_m+c_r-\mu )+{\displaystyle \frac{3\cdot (1+\delta \cdot (1-\lambda ))\cdot k\cdot \left((1+\gamma )-\delta \cdot (1-\lambda )\cdot c_r-(1+\delta \cdot (1-\lambda ))\cdot (c_m-\mu )\right)}{4\cdot k\cdot (1+\delta \cdot (1-\lambda ))\cdot (1+\gamma )-{\beta }^2}}$$

$$D^{GR*}={\displaystyle \frac{2\cdot k\cdot (1+\gamma )\cdot \left((1+\gamma )-\delta \cdot (1-\lambda )\cdot c_r-(1+\delta \cdot (1-\lambda ))\cdot (c_m-\mu )\right)}{4\cdot k\cdot (1+\delta \cdot (1-\lambda ))\cdot (1+\gamma )-{\beta }^2}}$$

$${\pi }_M^{GR*}={\displaystyle \frac{2\cdot k^2\cdot {(1+\gamma )}^2\cdot {\left((1+\gamma )-\delta \cdot (1-\lambda )\cdot c_r-(1+\delta \cdot (1-\lambda ))\cdot (c_m-\mu )\right)}^2}{{(4\cdot k\cdot (1+\delta \cdot (1-\lambda ))\cdot (1+\gamma )-{\beta }^2)}^2}}-F$$

$${\pi }_R^{GR*}={\displaystyle \frac{4\cdot (1+\delta \cdot (1-\lambda ))\cdot k^2\cdot {(1+\gamma )}^2\cdot {\left((1+\gamma )-\delta \cdot (1-\lambda )\cdot c_r-(1+\delta \cdot (1-\lambda ))\cdot (c_m-\mu )\right)}^2}{{(4\cdot k\cdot (1+\delta \cdot (1-\lambda ))\cdot (1+\gamma )-{\beta }^2)}^2}}$$

4.5. Summary of the Results in the Four Scenarios

This section summarizes the optimal solutions of the four models (NM, NR, GM and GR) from Section 4.1, Section 4.2, Section 4.3 and Section 4.4 to facilitate an overall comparative analysis. To simplify the expressions, the following notation is introduced: Let $\phi =1+\delta \cdot (1-\lambda )$ represents the tariff cost amplification factor, let $\epsilon =\delta \cdot (1-\lambda )\cdot c_r$ represents the fixed tariff cost related to the logistics surcharge. Furthermore, to ensure the negative definiteness of the Hessian matrix for the profit functions, it is assumed that $\Delta =2\cdot k\cdot \phi -{\beta }^2>0$ (under Models NM and GM) and ${\Delta }^{\prime }=2\cdot k\cdot \phi \cdot (1+\gamma )-{\beta }^2>0$ (under Models NR and GR) always holds. The optimal solutions for each model are consolidated in

Table 2. The equilibrium in four models.

Outcome	NM	NR	GM	GR
$t^{*}$	${\displaystyle \frac{\beta \cdot \left(\begin{array}{l}1-\epsilon \\ -\phi \cdot c_m\end{array}\right)}{\Delta }}$	${\displaystyle \frac{\beta \cdot \left(\begin{array}{l}(1+\gamma )-\epsilon \\ -\phi \cdot c_m\end{array}\right)}{{\Delta }^{\prime }}}$	${\displaystyle \frac{\beta \cdot \left(\begin{array}{l}1-\epsilon \\ -\phi \cdot (c_m-\mu )\end{array}\right)}{\Delta }}$	${\displaystyle \frac{\beta \cdot \left(\begin{array}{l}(1+\gamma )-\epsilon \\ -\phi \cdot (c_m-\mu )\end{array}\right)}{{\Delta }^{\prime }}}$
$w^{*}$	$c_m+{\displaystyle \frac{2\cdot k\cdot \left(\begin{array}{l}1-\epsilon \\ -\phi \cdot c_m\end{array}\right)}{\Delta }}$	$c_m+{\displaystyle \frac{2\cdot k\cdot (1+\gamma )\cdot \left(\begin{array}{l}(1+\gamma )-\epsilon \\ -\phi \cdot c_m\end{array}\right)}{{\Delta }^{\prime }}}$	$(c_m-\mu )+{\displaystyle \frac{2\cdot k\cdot \left(\begin{array}{l}1-\epsilon \\ -\phi \cdot (c_m-\mu )\end{array}\right)}{\Delta }}$	$(c_m-\mu )+{\displaystyle \frac{2\cdot k\cdot (1+\gamma )\cdot \left(\begin{array}{l}(1+\gamma )-\epsilon \\ -\phi \cdot (c_m-\mu )\end{array}\right)}{{\Delta }^{\prime }}}$
$p^{*}$	$\begin{array}{l}\phi \cdot c_m+\epsilon \\ +{\displaystyle \frac{3\cdot \phi \cdot k\cdot \left(\begin{array}{l}1-\epsilon \\ -\phi \cdot c_m\end{array}\right)}{\Delta }}\end{array}$	$\begin{array}{l}\phi \cdot c_m+\epsilon \\ +{\displaystyle \frac{3\cdot \phi \cdot k\cdot \left(\begin{array}{l}(1+\gamma )-\epsilon \\ -\phi \cdot c_m\end{array}\right)}{{\Delta }^{\prime }}}\end{array}$	$\begin{array}{l}\phi \cdot (c_m-\mu )+\epsilon \\ +{\displaystyle \frac{3\cdot \phi \cdot k\cdot \left(\begin{array}{l}1-\epsilon \\ -\phi \cdot (c_m-\mu )\end{array}\right)}{\Delta }}\end{array}$	$\begin{array}{l}\phi \cdot (c_m-\mu )+\epsilon \\ +{\displaystyle \frac{3\cdot \phi \cdot k\cdot (1+\gamma )\cdot \left(\begin{array}{l}(1+\gamma )-\epsilon \\ -\phi \cdot (c_m-\mu )\end{array}\right)}{{\Delta }^{\prime }}}\end{array}$
$D^{*}$	${\displaystyle \frac{2\cdot k\cdot \left(\begin{array}{l}1-\epsilon \\ -\phi \cdot c_m\end{array}\right)}{\Delta }}$	${\displaystyle \frac{2\cdot k\cdot (1+\gamma )\cdot \left(\begin{array}{l}(1+\gamma )-\epsilon \\ -\phi \cdot c_m\end{array}\right)}{{\Delta }^{\prime }}}$	${\displaystyle \frac{2\cdot k\cdot \left(\begin{array}{l}1-\epsilon \\ -\phi \cdot (c_m-\mu )\end{array}\right)}{\Delta }}$	${\displaystyle \frac{2\cdot k\cdot (1+\gamma )\cdot \left(\begin{array}{l}(1+\gamma )-\epsilon \\ -\phi \cdot (c_m-\mu )\end{array}\right)}{{\Delta }^{\prime }}}$
${\pi }_M^{*}$	${\displaystyle \frac{2\cdot k^2\cdot {\left(\begin{array}{l}1-\epsilon \\ -\phi \cdot c_m\end{array}\right)}^2}{{\Delta }^2}}$	${\displaystyle \frac{2\cdot k^2\cdot {(1+\gamma )}^2\cdot {\left(\begin{array}{l}(1+\gamma )-\epsilon \\ -\phi \cdot c_m\end{array}\right)}^2}{{{\Delta }^{\prime }}^2}}-F$	${\displaystyle \frac{2\cdot k^2\cdot {\left(\begin{array}{l}1-\epsilon \\ -\phi \cdot (c_m-\mu )\end{array}\right)}^2}{{\Delta }^2}}$	${\displaystyle \frac{2\cdot k^2\cdot {(1+\gamma )}^2\cdot {\left(\begin{array}{l}(1+\gamma )-\epsilon \\ -\phi \cdot (c_m-\mu )\end{array}\right)}^2}{{{\Delta }^{\prime }}^2}}-F$
${\pi }_R^{*}$	${\displaystyle \frac{4\cdot \phi \cdot k^2\cdot {\left(\begin{array}{l}1-\epsilon \\ -\phi \cdot c_m\end{array}\right)}^2}{{\Delta }^2}}$	${\displaystyle \frac{4\cdot A\cdot k^2\cdot {(1+\gamma )}^2\cdot {\left(\begin{array}{l}(1+\gamma )-\epsilon \\ -\phi \cdot c_m\end{array}\right)}^2}{{{\Delta }^{\prime }}^2}}$	${\displaystyle \frac{4\cdot \phi \cdot k^2\cdot {\left(\begin{array}{l}1-\epsilon \\ -\phi \cdot (c_m-\mu )\end{array}\right)}^2}{{\Delta }^2}}$	${\displaystyle \frac{4\cdot \phi \cdot k^2\cdot {(1+\gamma )}^2\cdot {\left(\begin{array}{l}(1+\gamma )-\epsilon \\ -\phi \cdot (c_m-\mu )\end{array}\right)}^2}{{{\Delta }^{\prime }}^2}}$

.

A comparative analysis of the structures of the optimal solutions in Table 2 reveals the fundamental differences in the mechanisms driven by the two transportation modes and the subsidy policy. First, in terms of the driving logic, the maritime shipping models (NM and GM) exhibit a typical cost-linear-driven characteristic. The numerator of their optimal solutions contains only linear combinations of basic parameters, as can be seen, for instance, in the optimal retail price $p^{NM*}=\phi \cdot c_m+\epsilon +(3\cdot \phi \cdot k\cdot \left(1-\epsilon -\phi \cdot c_m\right))/\Delta$ and the corresponding demand $D^{NM*}=2\cdot k\cdot \left(1-\epsilon -\phi \cdot c_m\right)/\Delta$, which are composed solely of first-order terms in tariffs ($\delta$, $\lambda$), logistics cost ($k$), and unit costs ($c_m$, $c_r$). Conversely, the denominator is simply the feasibility condition $\Delta =2\cdot k\cdot \phi -{\beta }^2$ without any additional multipliers. This indicates that changes in prices, demand, and profits are directly and linearly proportional to changes in the underlying cost parameters.

In contrast, the CR Express models (NR and GR) exhibit a nonlinear pattern driven by the integrated effects of cost and service. The introduction of the service premium coefficient $\gamma$, through the $(1+\gamma )$ multiplier effect in the numerator, amplifies the linkage strength between costs and demand. Simultaneously, it introduces structural modifications in the denominator. Compared with the maritime shipping models (NM and GM), the denominator has been changed from $\Delta =2\cdot k\cdot \phi -{\beta }^2$ to ${\Delta }^{\prime }=2\cdot k\cdot \phi \cdot (1+\gamma )-{\beta }^2$, where ${\Delta }^{\prime }$ is obviously an expression containing $\gamma$. As a result, for the same change in costs, the increase in demand and profit under the CR Express mode can reach $(1+\gamma )$ times that of the maritime shipping mode. In the specific case where $\gamma =0$—implying consumers are solely price-sensitive—the optimal logistics levels, prices, and demand coincide under both transportation modes, leaving only a profit difference of F for the manufacturer and erasing any competitive advantage of the CR Express.

Furthermore, the subsidy policy operates differently and generates distinct cooperative effects between the two types of models. In the maritime shipping models (comparing Models NM and GM), the government subsidy $\mu$ achieves linear cost reduction merely by substituting for the unit cost ($c_m$ becomes $c_m-\mu$), leading to a linear cost reduction without altering the fundamental solution structure ($\Delta =2\cdot k\cdot \phi -{\beta }^2$ remains unchanged). Consequently, the subsidy’s benefits are transmitted primarily through the channel of price reduction to stimulate demand—a mechanism with limited conversion efficiency that tends to reinforce pure price competition.

Finally, in the CR Express models (comparing Models NR and GR), the subsidy policy and the service premium $\gamma$ exhibit a clear synergistic enhancement effect. The subsidy $\mu$ not only directly reduces the effective unit cost but, more importantly, by alleviating the constraints imposed by the fixed cost $F$ and high logistics investment, it allows the nonlinear demand-pull effect inherent in $\gamma$ to be fully unleashed. The profit and demand growth in Model GR significantly outpace that in Models GM and NR. Thus, for subsidy allocation, the critical insight is that competitive advantage is not transferred equally across modes. Superior leverage is achieved by pairing subsidies with the unique service-based advantages of the CR Express.

5. Analysis

5.1. Impact of Key Parameters

This subsection focuses on analyzing the effects of the logistics cost coefficient $k$ and the service premium coefficient $\gamma$ on the optimal solutions of the models, as detailed in Propositions 1 and 2.

Proposition 1 (Impact of the logistics cost coefficient $k$).

For Models NM, NR, GM, and GR, the partial derivatives of their optimal solutions with respect to $k$ and the differences between these derivatives satisfy the following relationships:

(1)  ${\displaystyle \frac{\partial {\mathrm{\Pi }}^{*}}{\partial k}}<0$;

(2) $\left|{\displaystyle \frac{\partial {\mathrm{\Pi }}_{NR}^{*}}{\partial k}}\right|<\left|{\displaystyle \frac{\partial {\mathrm{\Pi }}_{NM}^{*}}{\partial k}}\right|$, $\left|{\displaystyle \frac{\partial {\mathrm{\Pi }}_{GR}^{*}}{\partial k}}\right|<\left|{\displaystyle \frac{\partial {\mathrm{\Pi }}_{GM}^{*}}{\partial k}}\right|$.

• where ${\mathrm{\Pi }}^{*}$ represent an equilibrium solution under any of the models.

Proposition 1 reveals the comprehensive dampening effect of rising logistics costs on the performance of the cross-border EV supply chain. Specifically, an increase in $k$ directly leads the manufacturer to reduce the optimal logistics investment level $t^{*}$. To maintain market attractiveness, supply chain members lower product prices ($w^{*}$ and $p^{*}$). However, the combined effect of degraded logistics service and price adjustments ultimately causes the final demand $D^{*}$ to contract, thereby squeezing the profits of both the manufacturer and the retailer, ${\pi }_M^{\ast }$ and ${\pi }_R^{\ast }$. Furthermore, the CR Express models exhibit lower sensitivity to changes in $k$. The fundamental reason is that the consumer service premium $\gamma$ enhances product value perception and price tolerance, partially buffering the pressure from rising logistics costs and strengthening the supply chain’s cost resilience.

The government production subsidy $\mu$, by reducing the manufacturer’s net production cost, generally enhances the supply chain’s ability to withstand fluctuations in logistics costs (comparing Model GM with NM, and Model GR with NR). It is particularly noteworthy that the combination of the subsidy and the CR Express mode (Model GR) exhibits the greatest buffering capacity, as reflected in its smallest absolute partial derivative value. This suggests that in environments with volatile logistics costs, this combination is the optimal strategy for enhancing supply chain stability and risk resistance.

In the models, the service premium coefficient $\gamma$, as the core feature distinguishing the CR Express mode from maritime shipping, appears only in the consumer utility functions of Models NR and GR. Therefore, Proposition 2 specifically focuses on these two scenarios to systematically analyze the mechanism through which gamma influences the supply chain equilibrium solutions.

Proposition 2 (Impact of the service premium coefficient $\gamma$).

For Models NR and GR, the partial derivatives of their optimal solutions with respect to $\gamma$ and the differences between these derivatives satisfy the following relationships:

(1)  ${\displaystyle \frac{\partial w^{GR*}}{\partial \gamma }}({\displaystyle \frac{\partial w^{NR*}}{\partial \gamma }})>0$,  ${\displaystyle \frac{\partial t^{GR*}}{\partial \gamma }}({\displaystyle \frac{\partial t^{NR*}}{\partial \gamma }})<0$,  ${\displaystyle \frac{\partial p^{GR*}}{\partial \gamma }}({\displaystyle \frac{\partial p^{NR*}}{\partial \gamma }})>0$,  ${\displaystyle \frac{\partial D^{GR*}}{\partial \gamma }}({\displaystyle \frac{\partial D^{NR*}}{\partial \gamma }})>0$;

(2) ${\displaystyle \frac{\partial w^{GR*}}{\partial \gamma }}<{\displaystyle \frac{\partial w^{NR*}}{\partial \gamma }}$; ${\displaystyle \frac{\partial t^{GR*}}{\partial \gamma }}<{\displaystyle \frac{\partial t^{NR*}}{\partial \gamma }}$, ${\displaystyle \frac{\partial p^{GR*}}{\partial \gamma }}<{\displaystyle \frac{\partial p^{NR*}}{\partial \gamma }}$, ${\displaystyle \frac{\partial D^{GR*}}{\partial \gamma }}<{\displaystyle \frac{\partial D^{NR*}}{\partial \gamma }}$.

Proposition 2 reveals the promoting effect of the service premium $\gamma$ on supply chain performance under the CR Express mode and its underlying mechanism. An increase in $\gamma$ directly enhances consumers’ perceived value and willingness to pay for the product, enabling the manufacturer and the retailer to raise the FOB price $w^{*}$ and the retail price $p^{*}$ while still stimulating an expansion in market demand $D^{*}$. This ultimately leads to synchronized growth in profits for both supply chain parties. The seemingly counterintuitive finding that $\partial t^{*}/\partial \gamma <0$ reveals a profound economic trade-off between service premium ($\gamma$) and optimal logistics level ($t^{*}$). In this model, $\gamma$ represents the brand reputation inherently associated with the CR Express as a transport mode, which exogenously determines the baseline attractiveness of the service. In contrast, $t^{*}$ denotes the variable marginal cost incurred by the manufacturer to further enhance the customer experience. The two factors exhibit a substitutive relationship in creating customer value. When a high $\gamma$ already confers strong market premium to the product, additional investment in $t^{*}$ will encounter diminishing marginal returns. Therefore, driven by the objective of profit maximization, the firm rationally chooses to reduce its investment in $t^{*}$ and instead relies more on leveraging the inherent advantage of CR Express.

In addition, government production subsidies significantly weaken the marginal effect of the service premium coefficient $\gamma$ on supply chain decisions. Specifically, under subsidized conditions (Model GR), the impact of $\gamma$ on wholesale price $w^{*}$, logistics level $t^{*}$, retail price $p^{*}$ and market demand $D^{*}$ is less pronounced compared to the unsubsidized scenario (Model NR), i.e., $\partial w^{GR*}/\partial \gamma <\partial w^{NR*}/\partial \gamma$, $\partial t^{GR*}/\partial \gamma <\partial t^{NR*}/\partial \gamma$, $\partial p^{GR*}/\partial \gamma <\partial p^{NR*}/\partial \gamma$ and $\partial D^{GR*}/\partial \gamma <\partial D^{NR*}/\partial \gamma$. In terms of the underlying mechanism, subsidies attenuate the role of service premiums primarily through two channels. First, subsidies directly lower the effective cost for firms, thereby expanding the potential market demand base. This reduces the relative impact of a marginal increase in $\gamma$ on the already enlarged market. Second, subsidies reduce the manufacturer’s effective production cost ($c_m-\mu$), which alters the marginal return calculation for logistics investment. In the absence of subsidies, higher production costs encourage manufacturers to enhance product differentiation by improving logistics levels. Subsidies, however, partly substitute for this need for differentiation, rendering firms less compelled to respond logistically to an increase in the service premium $\gamma$.

5.2. Comparative Analysis of Different Transportation Modes

This subsection compares the equilibrium solutions under the maritime shipping (NM/GM) and the CR Express (NR/GR) modes, aiming to reveal the impact mechanism of the service premium and differences in transportation cost structures on supply chain decisions and performance. Let $\Delta {\mathrm{\Pi }}^{N*}={\mathrm{\Pi }}^{NR*}-{\mathrm{\Pi }}^{NM*}$ and $\Delta {\mathrm{\Pi }}^{G*}={\mathrm{\Pi }}^{GR*}-{\mathrm{\Pi }}^{GM*}$ represent the differences in the model’s optimal solutions, and let $T=((1+\delta \cdot (1-\lambda ))(c_m+\delta \cdot (1-\lambda )\cdot (c_r+c_m))$ and $T^{\prime }=((1+\delta \cdot (1-\lambda ))((c_m-\mu )+\delta \cdot (1-\lambda )\cdot (c_r+c_m-\mu ))$ represent the tariff and related cost structures, with a detailed analysis presented in Proposition 3.

Proposition 3. (Comparison of different transportation modes).

Across subsidy scenarios, the differences in equilibrium solutions between the maritime shipping models (NM/GM) and the CR Express models (NR/GR) are as follows:

(1) Scenario without subsidies (NM/NR): $\Delta t^{N*}>0$ if $T>T_0$, otherwise $\Delta t^{N*}<0$; and $\Delta w^{N*}>0$, $\Delta p^{N*}>0$, $\Delta D^{N*}>0$, $\Delta {\pi }_R^{N*}>0$; and $\Delta {\pi }_M^{N*}>0$ if $F<F_0$, otherwise $\Delta {\pi }_M^{N*}<0$.

(2) Scenario with subsidies (GM/GR): $\Delta t^{G*}>0$ if $T^{\prime }>T_0$, otherwise $\Delta t^{G*}<0$; and $\Delta w^{G*}>0$, $\Delta p^{G*}>0$, $\Delta D^{G*}>0$, $\Delta {\pi }_R^{G*}>0$; and $\Delta {\pi }_M^{G*}>0$ if $F<{F_0}^{\prime }$, otherwise $\Delta {\pi }_M^{G*}<0$.

• where $T_0={\displaystyle \frac{{\beta }^2}{4\cdot k}}$, $F_0=2k^2({\displaystyle \frac{{(1+\gamma )}^2{((1+\gamma -c_m)-\delta (1-\lambda )(c_r+c_m))}^2}{{(4k(1+\gamma )(1+\delta (1-\lambda ))-{\beta }^2)}^2}}-{\displaystyle \frac{{((1-c_m)-\delta (1-\lambda )(c_r+c_m))}^2}{{(4k(1+\delta (1-\lambda ))-{\beta }^2)}^2}})$ and ${F_0}^{\prime }=2k^2({\displaystyle \frac{{(1+\gamma )}^2{((1+\gamma +\mu -c_m)-\delta (1-\lambda )(c_r+c_m-\mu ))}^2}{{(4k(1+\gamma )(1+\delta (1-\lambda ))-{\beta }^2)}^2}}-{\displaystyle \frac{{((1+\mu -c_m)-\delta (1-\lambda )(c_r+c_m-\mu ))}^2}{{(4k(1+\delta (1-\lambda ))-{\beta }^2)}^2}})$.

The comparative analysis in Proposition 3 reveals that, under both scenarios without government subsidy (Models NR and NM) and with government subsidy (Models GR and GM), the CR Express transportation mode demonstrates systematic advantages over traditional maritime shipping in most supply chain performance metrics. Specifically, regardless of the presence of a subsidy, the equilibrium wholesale price $w^{*}$, retail price $p^{*}$, market demand $D^{*}$, and retailer profit ${\pi }_R^{*}$ under the CR Express mode are significantly higher than those under the maritime shipping mode. This advantage stems primarily from the service premium $\gamma$ generated by the faster and more stable logistics services provided by the CR Express. This premium directly enhances consumers’ willingness to pay, thereby allowing supply chain members to set higher prices and achieve greater market coverage. As a follower, the retailer can fully capitalize on the market expansion benefits brought by this premium, leading to an unconditional increase in its profit.

However, the analysis also uncovers two key conditional differences. First, the comparison results for the logistics level $t^{*}$ is not absolute but depends on system parameters. In the scenario without a subsidy, the logistics investment in the CR Express will be higher only when tariff-related costs $(c_m+\delta \cdot (1-\lambda )\cdot (c_r+c_m))$ are sufficiently high, satisfying $4\cdot k\cdot ((1+\delta \cdot (1-\lambda ))(c_m+\delta \cdot (1-\lambda )\cdot (c_r+c_m))>{\beta }^2$; otherwise, it may be lower than maritime shipping. The economic logic behind this finding is that high tariff costs intensify end-price pressure, motivating manufacturers to enhance product differentiation and support the price premium by improving the logistics service level of the CR Express. Second, the manufacturer’s profit advantage is strictly constrained by the inherent fixed cost $F$ of the CR Express. It is only more profitable for the manufacturer to adopt the CR Express when $F$ is below a critical threshold ($F<F_0$) determined jointly by market scale, service premium, and cost structure.

In summary, the CR Express holds significant potential for enhancing the overall market performance of the supply chain due to its service premium characteristic. However, its successful application depends on specific market conditions (e.g., tariff level $\delta$) and the firm’s cost structure (particularly fixed cost $F$). While the production subsidy $\mu$ provided by the government does not alter the fundamental direction of the aforementioned comparisons, it can enhance the attractiveness and feasibility of the CR Express mode to a certain extent. It achieves this by reducing the manufacturer’s net production cost, effectively lowering the threshold condition for achieving a logistics level advantage, and raising the critical $F$-value for manufacturer profitability ($F_0<{F_0}^{\prime }$).

5.3. Impact of Government Subsidies on Equilibrium Solutions

This subsection aims to investigate the impact of the government subsidy $\mu$ on the supply chain equilibrium solutions, with a particular focus on the moderating role played by the transportation mode (maritime shipping and CR Express). By comparing the partial derivatives of the equilibrium solutions in Models GM and GR with respect to $\mu$, we can reveal how the effectiveness of the subsidy is systematically constrained by key parameters such as the service premium and logistics costs. A detailed analysis is provided in Proposition 4.

Proposition 4. (Conditional effect of government subsidy $\mu$).

For Models GM and GR, the differences between these partial derivatives of their optimal solutions with respect to $\mu$ satisfy the following relationships:

(1)  ${\displaystyle \frac{\partial t^{GR*}}{\partial \mu }}<{\displaystyle \frac{\partial t^{GM*}}{\partial \mu }}$  if  $k>k_0$, otherwise  ${\displaystyle \frac{\partial t^{GR*}}{\partial \mu }}>{\displaystyle \frac{\partial t^{GM*}}{\partial \mu }}$;

(2)  ${\displaystyle \frac{\partial w^{GR*}}{\partial \mu }}>{\displaystyle \frac{\partial w^{GM*}}{\partial \mu }}$,  ${\displaystyle \frac{\partial p^{GR*}}{\partial \mu }}>{\displaystyle \frac{\partial p^{GM*}}{\partial \mu }}$,  ${\displaystyle \frac{\partial D^{GR*}}{\partial \mu }}>{\displaystyle \frac{\partial D^{GM*}}{\partial \mu }}$,  ${\displaystyle \frac{\partial {\pi }_R^{GR*}}{\partial \mu }}>{\displaystyle \frac{\partial {\pi }_R^{GM*}}{\partial \mu }}$;

(3) ${\displaystyle \frac{\partial {\pi }_M^{GR*}}{\partial \mu }}>{\displaystyle \frac{\partial {\pi }_M^{GM*}}{\partial \mu }}$, if $\gamma >{\gamma }_0$, otherwise ${\displaystyle \frac{\partial {\pi }_M^{GR*}}{\partial \mu }}<{\displaystyle \frac{\partial {\pi }_M^{GM*}}{\partial \mu }}$.

• where $k_0={\displaystyle \frac{{\beta }^2}{4\cdot ((1+\delta \cdot (1-\lambda ))(c_m+\delta \cdot (1-\lambda )\cdot (c_r+c_m))}}$ and ${\gamma }_0={\displaystyle \frac{{\beta }^2(2+c_m-\mu +\delta (1-\lambda )(c_m-c_r-\mu ))-4k(1+\delta (1-\lambda ))}{4k(1+\delta (1-\lambda ))-{\beta }^2}}+{\displaystyle \frac{F\cdot {(4k(1+\delta (1-\lambda ))-{\beta }^2)}^3}{8k^2(1+\delta (1-\lambda )){(1+c_m-\mu +\delta (1-\lambda )(c_m-c_r-\mu ))}^2}}$.

The core insight of Proposition 4 is that the efficacy of government subsidies is not constant but depends on the critical parameter environment of the supply chain operation, i.e., service premium $\gamma$ and logistics cost coefficient $k$. The service premium $\gamma$ is the decisive factor determining whether the CR Express mode can effectively translate subsidies into a competitive advantage. When $\gamma >{\gamma }_0$, the subsidy and the service premium generate synergy. This synergy significantly expands demand through price reductions, making the utilization efficiency of subsidies in the CR Express mode comprehensively surpass that in maritime shipping. Conversely, if the perceived timeliness is insufficient ($\gamma <{\gamma }_0$), the transmission efficiency of subsidies within the CR Express mode is lower.

Moreover, the logistics cost coefficient $k$ influences the intensity with which subsidies incentivize logistics investment. When logistics operations are efficient ($k<k_0$), subsidies can more effectively encourage the CR Express mode to elevate its logistics level. However, this incentivizing effect is weakened when logistics costs are high ($k>k_0$).

Based on the previous analysis, this part proposes a differentiated subsidy strategy framework based on parameter thresholds, as illustrated in Figure 2. This strategic diagram aims to provide policymakers with a clear decision-making pathway.

• Region I (High $\gamma$, Low $k$): In this region, the CR Express offers strong inherent advantages in a low-cost environment, making subsidies highly effective for driving market expansion and profit growth. Therefore, if the policy goal is to leverage this competitive edge for rapid market scaling, a higher subsidy intensity serves as a potent instrument. Managerially, this corresponds to exporting high-value, time-sensitive EVs (e.g., premium models) where reliability is paramount (high $\gamma$) and logistics are streamlined through scale and standardization (low $k$). Subsidies here directly boost a strategic supply chain segment.
• Region II (High $\gamma$, High $k$): Here, a significant service premium is substantially offset by high logistics costs, implying that direct subsidies alone have limited growth-stimulating effects. For long-term impact, policy should prioritize structural reductions in $k$ (e.g., via infrastructure or operational upgrades), with subsidies playing a short-term stabilizing role. This often involves shipping EVs to markets valuing speed but facing operational complexities, which inflate $k$. Co-investment in streamlining operations is thus a key precursor.
• Region III (Low $\gamma$, Low $k$): The market shows little sensitivity to CR Express service advantages, while the logistics system itself is cost-efficient. Consequently, subsidies for CR Express yield low returns, as they fail to address low consumer valuation ($\gamma$). Policymakers should therefore exercise caution—from a cost–benefit perspective, promoting maritime shipping may be a more efficient alternative for trade-volume objectives. This fits exports of low-margin, cost-competitive EVs where maritime shipping’s cost advantage is decisive. Thus, diverting this cost-sensitive EV traffic to the CR Express via subsidies would be economically inefficient.
• Region IV (Low $\gamma$, High $k$): CR Express faces a dual deficit of low service appeal and high operational costs, rendering direct subsidies inefficient—they offer only temporary relief without establishing a viable long-term position. Large-scale direct subsidies should be avoided, and resources are better allocated to supporting maritime alternatives or addressing the root causes of low $\gamma$. This scenario typically arises in price-sensitive emerging markets with underdeveloped inland logistics, where speed is undervalued and operational hurdles inflate costs. Subsidies cannot overcome this unfavorable condition, and efforts should focus on maritime efficiency.

6. Results

To delve deeper into the quantitative impact of key parameters and policies on decision-making within the EV cross-border supply chain and to make the theoretical conclusions more intuitive, this section employs numerical simulation for analysis. By assigning reasonable benchmark values to the parameters, we can visualize the changing trends of various equilibrium solutions, verifying the propositions from the previous sections while revealing interactive effects and critical thresholds not easily observed directly in the theoretical model. Based on the general constraints that parameters must satisfy during the equilibrium solution process, combined with practical reasonableness, the parameters are set as follows: $c_m=0.3$, $c_r=0.1$ and $\delta =0.2$ [43,44].

6.1. Impact of Key Parameters on Decisions Under Different Transportation Modes

This subsection uses numerical simulation to compare and analyze how the logistics cost coefficient $k$ and the service premium coefficient $\gamma$ differentially affect market demand and logistics service levels under the two transportation modes, thereby intuitively revealing the underlying driving logic. Varying the parameter $k$ from 0.1 to 0.5 while holding other parameters constant yields the results shown in Figure 3. And Figure 4 presents results obtained by jointly varying $k$ over the same range and $\gamma$ from 0 to 0.1.

Figure 3a shows that market demand $D^{*}$ decreases as $k$ increases in both the no-subsidy (Model NM) and with-subsidy (Model GM) scenarios, verifying the dampening effect of rising costs stated in Proposition 1. Notably, the demand curve under Model GM exhibits an inflection point at approximately $k\approx 0.18$ beyond which the rate of decline slows significantly. The logistics cost coefficient $k$ serves as a composite indicator that captures variable logistics expenditures such as fuel costs, port handling fees, and canal tolls. The value $k\approx 0.18$ defines a critical cost threshold: when the actual integrated cost falls below this threshold (for instance, during periods of low fuel prices), a given government subsidy can effectively translate into market-demand growth; Conversely, if the actual cost exceeds this threshold due to factors such as a surge in oil prices or increases in port tariffs, the cost-substitution effect of the subsidy shifts its focus from stimulating growth to sustaining basic market demand. Figure 3b indicates that the optimal logistics level $t^{*}$ is highly sensitive to changes in $k$, and the curves for Models GM and NM decline nearly in parallel. This confirms that under the maritime shipping mode, while the subsidy can raise the absolute baseline of logistics investment, it cannot alter its fundamental elasticity constrained by the cost coefficient. The system thus exhibits a distinct cost-linear-driven characteristic.

Figure 4, through three-dimensional surfaces, intuitively reveals the joint influence of the logistics cost coefficient $k$ and the service premium coefficient $\gamma$ on the equilibrium solutions under the CR Express mode. The market demand surface in Figure 4a shows that an increase in $k$ similarly leads to demand contraction, but an increase in the service premium $\gamma$ can effectively buffer this trend, manifested as an elevation of the surface along the $\gamma$-axis. Furthermore, comparing the surfaces for Model GR (with subsidy) and Model NR (without subsidy) indicates that the subsidy provides an additional boost to demand across any combination of $k$ and $\gamma$. The logistics level surface in Figure 4b exhibits a steeper morphology, indicating that $t^{*}$ is far more sensitive to changes in $k$ than $D^{*}$ is. Although increasing $\gamma$ can marginally alleviate the decline in $t^{*}$, its effect is quite limited and insufficient to counteract the dominant negative influence of $k$. This implies that when fundamental logistics costs are prohibitively high, the demand pull from the service premium is insufficient to incentivize further logistics investment. Enhancing core logistics efficiency is thus a prerequisite for realizing the full-service value of the CR Express.

6.2. Impact of Tariff Reduction Degree and Government Subsidy Efficiency

China has established tariff preferential mechanisms with Belt and Road partners like Uzbekistan, Belarus, and Ethiopia. Notably, the tariff reduction ratio for products such as EVs can reach up to 100%. To leverage this policy context, this subsection employs numerical simulations to analyze how the degree of tariff reduction $\lambda$ affects profit trends for the manufacturer and retailer, and further benchmarks the efficiency of government production subsidies. Figure 5 shows the results of varying $\lambda$ from 0 to 1 while holding other parameters constant, and Figure 6 details the outcomes for $\mu$ in the range of 0 to 0.2.

Figure 5 illustrates the effect of the tariff reduction degree $\lambda$ on the profits of the manufacturer and the retailer, respectively. Figure 5a indicates that, although the direct policy target is the retailer, aiming at lowering its import tariff costs, the manufacturer’s profit also increases significantly with a higher tariff reduction degree $\lambda$. The underlying mechanism is that the tariff reduction lowers the retailer’s import costs ($\delta \cdot (1-\lambda )\cdot (w+c_r)$), thereby stimulating market demand. As the upstream of the supply chain, the manufacturer benefits from the expansion of wholesale quantities, sharing in the gains from downstream demand growth. For the retailer, as shown in Figure 5b, although its profit is positively correlated with the tariff reduction degree $\lambda$, the rate of profit growth is notably lower than that of the manufacturer. This disparity stems from the power structure of the supply chain—in the Stackelberg game where the manufacturer acts as the leader, it can leverage its first-mover advantage to strategically increase the wholesale price $w$, thereby partially capturing the benefits of the tariff reduction that would otherwise fully accrue to the retailer, ultimately amplifying its own profit.

Next, an indicator $\mathrm{\Psi }=(\partial {\pi }_M/\partial \mu )/(\partial D\cdot \mu /\partial \mu )$ is constructed to analyze the efficiency of government subsidies. The numerator $\partial {\pi }_M/\partial \mu$ represents the marginal impact of the subsidy on the manufacturer’s profit, reflecting the policy’s incentive effect, while the denominator $\partial D\cdot \mu /\partial \mu$ denotes the marginal fiscal cost of the subsidy. The economic interpretation of $\mathrm{\Psi }$ is the change in the manufacturer’s profit driven per unit of fiscal expenditure, which can be used to assess the policy effectiveness of converting subsidy funds into corporate profits.

Figure 6 illustrates the variation trend of government subsidy efficiency $\mathrm{\Psi }$ with respect to the unit subsidy $\mu$ under Models GM and GR. The results show that the subsidy efficiency under both transport modes exhibits a diminishing marginal trend as $\mu$ increases, a pattern consistent with the economic principle of diminishing marginal returns. Notably, at the same subsidy level, the subsidy efficiency of CR Express remains consistently and significantly higher than that of the maritime shipping. This is because the service premium $\gamma$ associated with CR Express enhances the product’s demand elasticity and profit margin, enabling government subsidies to be transmitted more effectively along the supply chain and incentivizing the manufacturer to improve its logistics level.

6.3. Change Trends in Profits and Logistics Mode Selection

This subsection focuses on the distribution of supply chain profits, analyzing how logistics costs, fixed costs, and subsidy policies affect the profitability of manufacturers and retailers, and clarifying the boundary conditions for the profit advantage of the CR Express mode. In this experiment, $k$ varies from 0.1 to 0.5 and $F$ from 0 to 0.1, while other parameters are held constant.

Figure 7 illustrates the trend of retailer’s profit with respect to the logistics cost coefficient $k$. As shown in Figure 7a, in the scenario without subsidies, the profits for retailer using the CR Express mode (Model NR) are consistently higher than those using maritime shipping mode (Model NM), and both decline as $k$ increases. This trend is highly synchronized with market demand, confirming that retailer profits are directly derived from sales volume. Figure 7b indicates that after introducing government subsidies (Models GM and GR), the baseline of retailer profits shifts upward overall, and the declining trend of the profit curve with increasing $k$ becomes more gradual. This demonstrates that subsidies, transmitted through the supply chain, indirectly enhance the stability of retailer profitability, helping them better cope with fluctuations in logistics costs. Notably, the flattening effect on the profit curve is most pronounced under Model GR, which further reveals a complementary relationship between the subsidy policy and the service premium.

Figure 8 employs three-dimensional surfaces to further examine how manufacturer profit is jointly constrained by the logistics cost coefficient $k$ and the fixed cost $F$, and how subsidy policy fundamentally reshapes the logic of mode selection. Figure 8a clearly illustrates the manufacturer’s dilemma in the absence of subsidies. Under maritime shipping (Model NM), its profit is influenced only by $k$ and remains insensitive to $F$; in contrast, under the CR Express (Model NR), profit is eroded by both $k$ and $F$. The pink critical line ($F=F_0(k)$) highlighted in the figure plays a decisive role: when $F<F_0$, the profit in Model NR exceeds that in Model NM, whereas when $F>F_0$, the relationship reverses. This visually quantifies the conditional advantage outlined in Proposition 3, showing that without policy support, the CR Express is economically viable only when fixed cost is sufficiently controlled. Figure 8b demonstrates how subsidies radically alter this dynamic—with subsidies (Models GM and GR), the profit surface for the CR Express (Model GR) lies entirely above that for maritime shipping (Model GM) across all combinations of $k$ and $F$. Thus, the subsidy mitigates the negative effect of the fixed cost $F$, thereby raising the critical $F$-value for manufacturer profitability ($F_0<{F_0}^{\prime }$).

7. Discussion

This section focuses on analyzing the impact of capacity constraints and power structure on supply chain equilibrium, as well as testing the robustness of the model’s core findings by altering the distribution of consumer valuations and the functional form of demand.

7.1. Effects of Capacity Constraints

When market demand is high, both maritime shipping and CR Express are subject to capacity constraints due to the indivisibility of transport resources (e.g., vessels, train sets) and the lead time required for capacity expansion, making it difficult to rapidly scale up transport volume in the short term. This subsection examines the performance of the two transport modes under such capacity constraints.

By introducing a binding capacity upper limit $\overline{D}$ into Models GM and GR, extended Models $\overline{GM}$ and $\overline{GR}$ are constructed respectively. The optimal solutions for decision variables and profits are derived using backward induction, listed as follows:

$$t^{\overline{GM}*}={\displaystyle \frac{\overline{D}}{k\cdot \phi }}, t^{\overline{GR}*}={\displaystyle \frac{\beta \cdot \overline{D}}{k\cdot \phi }},$$

$$w^{\overline{GM}*}={\displaystyle \frac{k\cdot \phi \cdot \left(1-\overline{D}-\epsilon \right)+\overline{D}}{k\cdot {\phi }^2}}, w^{\overline{GR}*}={\displaystyle \frac{{\beta }^2\cdot \overline{D}-\phi \cdot k\cdot (\epsilon -(1+\gamma )\cdot (1-\overline{D}))}{k\cdot {\phi }^2}},$$

$$p^{\overline{GM}*}={\displaystyle \frac{k\cdot \phi \cdot \left(1-\overline{D}\right)+\overline{D}}{k\cdot \phi }}, p^{\overline{GR}*}=(1+\gamma )\cdot (1-\overline{D})+{\displaystyle \frac{{\beta }^2\cdot \overline{D}}{k\cdot \phi }},$$

$${\pi }_M^{\overline{GM}*}={\displaystyle \frac{\overline{D}\cdot \left(2\cdot \phi \cdot ((\mu -c_m-c_r)\cdot \delta \cdot (1-\lambda )+(\mu -c_m)+1-\overline{D})\cdot k+\overline{D}\right)}{2\cdot k\cdot {\phi }^2}}$$

$${\pi }_M^{\overline{GR}*}={\displaystyle \frac{2\cdot k^2\cdot \phi \cdot \left(((1+\gamma +\mu -c_m)-F\cdot \phi -\delta \cdot (\mu -c_m-c_r)\cdot (1-\lambda ))\cdot \overline{D}-(1+\gamma )\cdot {\overline{D}}^2\right)+{\beta }^2\cdot {\overline{D}}^2}{2\cdot k^2\cdot {\phi }^2}}$$

By comparing the equilibrium outcomes of Models $\overline{GM}$ and $\overline{GR}$, the following main conclusions can be drawn:

Proposition 5 (Comparison of different transportation modes under capacity constraints).

The differences in equilibrium solutions between the maritime shipping model ($\overline{GM}$) and the CR Express model ($\overline{GR}$) are as follows:

(1)  $t^{\overline{GR}*}<t^{\overline{GM}*}$;

(2)  $w^{\overline{GR}*}>w^{\overline{GM}*}$  if  $\overline{D}>{\overline{D}}_0$, otherwise  $w^{\overline{GR}*}<w^{\overline{GM}*}$;

(3)  $p^{\overline{GR}*}>p^{\overline{GM}*}$  if  $\overline{D}>{\overline{D}}_0$, otherwise  $p^{\overline{GR}*}<p^{\overline{GM}*}$;

(4)  ${\pi }_M^{\overline{GR}*}>{\pi }_M^{\overline{GM}*}$  if  $\gamma >{\overline{\gamma }}_0$, otherwise  ${\pi }_M^{\overline{GR}*}<{\pi }_M^{\overline{GM}*}$;

• where ${\overline{D}}_0={\displaystyle \frac{k\cdot \gamma \cdot (1+\delta \cdot (1-\lambda ))}{k\cdot \gamma \cdot ((1+\delta \cdot (1-\lambda ))+1-{\delta }^2}}$ and ${\overline{\gamma }}_0={\displaystyle \frac{2\cdot F\cdot k\cdot {(1+\delta \cdot (1-\lambda ))}^2+(1-{\beta }^2)\cdot {\overline{D}}^2}{2\cdot k\cdot (1+\delta \cdot (1-\lambda ))\cdot \overline{D}\cdot (1-\overline{D})}}$.

As shown in Proposition 5, under capacity constraints, the equilibrium outcomes of the two transport modes exhibit systematic differences. The logistics level under the CR Express (Model $\overline{GR}$) is consistently lower than that under maritime shipping (Model $\overline{GM}$). This is primarily because the higher fixed cost $F$ of the CR Express, coupled with the sales limit $\overline{D}$, incentivizes the manufacturer to reduce logistics investment in order to control total costs. When the capacity constraint is relatively relaxed ($\overline{D}>{\overline{D}}_0$), the CR Express (Model $\overline{GR}$) commands higher FOB and retail prices than maritime shipping (Model $\overline{GM}$), owing to the pricing advantage conferred by its service premium $\gamma$, reflecting consumers’ willingness to pay for higher logistics reliability. Meanwhile, in such a capacity-limited setting, the manufacturer, as the Stackelberg leader, can fully extract the surplus from the retail segment through wholesale pricing, leaving the retailer with zero profit. The manufacturer’s own profit depends on the market’s perceived intensity of the service premium $\gamma$: only when $\gamma$ is sufficiently high ($\gamma >{\overline{\gamma }}_0$) can the premium revenue generated by the CR Express cover its higher fixed cost F, thereby delivering greater profit for the manufacturer than the maritime shipping.

7.2. Effects of Shifting Channel Leadership

This subsection analyzes the retailer-led power structure, in which overseas retailers dominate via local market access and domestic manufacturers operate at a disadvantage due to lower brand recognition and information asymmetry. The focus is on the resulting equilibrium and profit distribution.

In Model $G{R}^{\prime }$, the retailer acts as the Stackelberg leader and determines a unit markup $m$ on the manufacturer’s FOB price $w$ to maximize its own profit. The manufacturer, as the follower, observes $m$ and then decides the FOB price $w$ and the logistics level $t$. The profit functions of both parties are as follows:

$${\pi }_M^{G{R}^{\prime }}(w^{G{R}^{\prime }},t^{G{R}^{\prime }})=(w^{G{R}^{\prime }}-c_m+\mu )\cdot (1-{\displaystyle \frac{w^{G{R}^{\prime }}+m^{G{R}^{\prime }}+\beta \cdot t^{G{R}^{\prime }}}{1+\gamma }})-{\displaystyle \frac{1}{2}}k\cdot {(t^{G{R}^{\prime }})}^2-F$$

(13)

$${\pi }_R^{G{R}^{\prime }}(m^{G{R}^{\prime }})=(m^{G{R}^{\prime }}-\delta \cdot (1-\lambda )\cdot (w^{G{R}^{\prime }}+c_r))\cdot (1-{\displaystyle \frac{w^{G{R}^{\prime }}+m^{G{R}^{\prime }}+\beta \cdot t^{G{R}^{\prime }}}{1+\gamma }})$$

(14)

The game is solved by backward induction. First, we derive the manufacturer’s reaction functions for $w$ and $t$. These are then substituted into the retailer’s profit function to solve the optimal markup $m^{*}$. Finally, we back-substitute to obtain the equilibrium values $w^{*}$, $t^{*}$, $D^{*}$ and the corresponding profits.

$$t^{G{R}^{\prime }*}={\displaystyle \frac{(\mu -c_m-(1+\gamma )+(1-\phi )\cdot c_r)\cdot \beta }{2\cdot ((1+\gamma )\cdot (3-\phi )\cdot k-{\beta }^2)}}$$

$$w^{G{R}^{\prime }*}={\displaystyle \frac{(1+\gamma )\cdot ((\mu -c_m-1-\gamma )+(1-\phi )\cdot c_r)\cdot k+2\cdot (\mu -c_m-1-\gamma )\cdot {\beta }^2}{2\cdot ((1+\gamma )\cdot (3-\phi )\cdot k-{\beta }^2)}}$$

$$m^{G{R}^{\prime }*}={\displaystyle \frac{2(1+\gamma )\cdot (\mu -2\cdot (1+\gamma )-c_m-c_r)\cdot k-(\mu -c_m-(1+\gamma )+(1-\phi )\cdot c_r)\cdot {\beta }^2}{2\cdot ((1+\gamma )\cdot (3-\phi )\cdot k-{\beta }^2)}}$$

$$D^{G{R}^{\prime }*}={\displaystyle \frac{(\mu -(1+\gamma )-c_m+(1-\phi )\cdot c_r)\cdot k}{2\cdot ((1+\gamma )\cdot (3-\phi )\cdot k-{\beta }^2)}}$$

$${\pi }_M^{G{R}^{\prime }*}={\displaystyle \frac{{\beta }^2\cdot (H_4-(1-\phi )\cdot H_3)\cdot k+F\cdot {\beta }^4-(1+\gamma )\cdot ({(1-\phi )}^2\cdot H_1+4\cdot (1-\phi )\cdot H_2-H_0)\cdot k^2}{{((1+\gamma )\cdot (3-\phi )\cdot k-{\beta }^2)}^2}}-F$$

$${\pi }_R^{G{R}^{\prime }*}={\displaystyle \frac{{(1+\gamma -\mu +c_m-(1-\phi )\cdot c_r)}^2\cdot k}{2\cdot ((1+\gamma )\cdot (3-\phi )\cdot k-{\beta }^2)}}$$

where $H_0={(1+\gamma )}^2+4\cdot (1+\gamma )\cdot (\mu -c_m)+4\cdot {(\mu -c_m)}^2$, $H_1={\displaystyle \frac{{(\mu -c_m-c_r)}^2}{4}}+F$, $H_2={\displaystyle \frac{(\mu -c_m-c_r)\cdot (1+\gamma )}{8}}+F$, $H_3={\displaystyle \frac{(\mu -c_m-c_r)\cdot (1+\gamma )}{2}}+4\cdot F$, $H_4={(1+\gamma )}^2+2\cdot (1+\gamma )\cdot (\mu -c_m)+{(\mu -c_m)}^2-4\cdot F\cdot (1+\gamma )$.

To clearly compare the effect of power structure, Figure 9 presents a numerical comparison of the manufacturer’s profits under Model GR (manufacturer-led) and Model $G{R}^{\prime }$ (retailer-led).

A comparison of the equilibrium outcomes between Model GR (manufacturer-led) and Model $G{R}^{\prime }$ (retailer-led) reveals that the manufacturer’s profit is significantly higher under Model GR. This disparity stems from fundamental changes in economic mechanisms induced by the shift in supply-chain power structure. When the retailer acts as the leader, it can use the markup $m$ to compress the manufacturer’s profit down to its participation-constraint boundary, while making decisions based solely on its own profit objective. In doing so, the retailer often fails to fully internalize the positive contribution of logistics investment to overall channel performance. Consequently, although the government subsidy $\mu$ is formally allocated to the manufacturer, a portion of its benefit may be absorbed by the retailer through adjustment of $m$, thereby diluting the incentive effect of the subsidy on the manufacturer’s logistics improvement.

This demonstrates that the power structure within the supply chain is a critical factor in evaluating subsidy effectiveness. If the policy goal is to encourage the manufacturer to enhance logistics performance, direct subsidies to the manufacturer may have limited impact in retailer-dominated supply chains. Instead, more sophisticated contractual designs—such as performance-linked mechanisms—are required to ensure that incentives are effectively transmitted.

7.3. Robustness Tests with Relaxed Assumptions

This subsection tests the robustness of the model’s core findings by introducing two critical generalizations: a non-uniform distribution of consumer valuation and a nonlinear demand function. To examine whether the conclusions depend on the original simplified form, the following numerical analysis takes the most comprehensive scenario, Model GR, as the test case.

It is assumed that the consumer base valuation $v$ follows a symmetric triangular distribution on the interval [0,1] with a mode of 0.5. This specification aims to capture a more realistic market scenario in which consumer value perceptions are more concentrated around the market center rather than uniformly dispersed across the interval. Under this assumption, the probability density function and the cumulative distribution function of $v$ are given as follows:

$$f(v)=\left\{\begin{array}{ll}4\cdot v,& 0\le v\le 0.5\\ 4\cdot (1-v),& 0.5<v\le 1\end{array}\right.$$

(15)

$$F(v)=\left\{\begin{array}{ll}2\cdot v^2,& 0\le v\le 0.5\\ 4\cdot v-2\cdot v^2-1,& 0.5<v\le 1\end{array}\right.$$

(16)

Furthermore, a quadratic term in the logistics level is introduced into the consumer utility function to reflect diminishing marginal utility, i.e., $U=v\cdot (1+\gamma )-p+\beta \cdot t-\xi \cdot t^2$, where $\xi >0$ denotes the coefficient of diminishing marginal utility with respect to the logistics level $t$. Accordingly, the market demand function $D(v)$ is re-derived based on the new distribution and utility relationship, as shown below:

$$D(v)=\left\{\begin{array}{ll}1-2\cdot {(\frac{p-\beta \cdot t+\xi \cdot t^2}{1+\gamma })}^2,& 0\le v\le 0.5\\ 2-4\cdot ({\displaystyle \frac{p-\beta \cdot t+\xi \cdot t^2}{1+\gamma }})+2\cdot {(\frac{p-\beta \cdot t+\xi \cdot t^2}{1+\gamma })}^2,& 0.5<v\le 1\end{array}\right.$$

(17)

Under the relaxed assumptions, a numerical simulation is employed to analyze the impact of the logistics cost coefficient $k$ on the manufacturer’s profit in Model GR, with the results presented in Figure 10.

The results indicate that, despite changes in the consumer distribution form and the introduction of demand nonlinearity, the core dynamic relationship of the model remains robust: the manufacturer’s profit declines continuously as the logistics cost coefficient $k$ increases. This robustness stems from the fundamental and directional impact of rising logistics costs on profitability, which persists across different functional forms and distributional assumptions. Different distribution and nonlinear assumptions affect only the specific slope of the profit curve but do not alter its negative trend. This outcome supports the robustness of the study’s main conclusions with respect to the model assumptions.

8. Conclusions

8.1. Conclusions

This study focuses on a cross-border EV supply chain consisting of a domestic manufacturer and an overseas retailer. It investigates the interactive effects between transportation mode choice (maritime shipping and CR Express) and government production subsidy policies in the typical context of exports to countries along the Belt and Road. By constructing Stackelberg game models for four scenarios integrating the subsidy policy with the transportation mode (Models NM, NR, GM and GR), this paper systematically analyzes supply chain equilibrium decisions and further assesses the effects of capacity constraints and power structures, while verifying the robustness of the core findings. These insights not only provide actionable guidance for low-carbon transport policy and strategy but also yield an analytical framework replicable for other cross-border industries facing similar service–cost trade-offs.

Through rigorous theoretical derivation and numerical simulation, the following key conclusions are drawn:

(1)

The driving logics of transportation modes are fundamentally different. The maritime shipping mode exhibits a cost-linear-driven pattern, where its performance is in a simple proportional relationship with cost changes. In contrast, the CR Express mode shows a nonlinear-driven characteristic integrating cost and service level.

(2)

The advantages of the CR Express mode are conditional and policy-dependent. In a market scenario without government subsidies, the CR Express can lower retail prices, expand market share, and benefit retailers through its service advantage. However, its profitability for the manufacturer is strictly constrained by the fixed cost $F$ (threshold $F_0$). The higher fixed costs further incentivize the manufacturer to reduce logistics investment, especially under capacity constraints. The introduction of government subsidies can exert a positive influence on this landscape, making it a more viable strategic option.

(3)

The efficacy of government subsidies unfolds along an optimal, context-dependent path. The effectiveness of subsidy policies is not universal; its efficiency highly depends on key market and operational parameters. The synergy between the subsidy and the CR Express is strongest when the service premium $\gamma$ is high ($\gamma >{\gamma }_0$) and the logistics cost coefficient $k$ is low ($k<k_0$). Conversely, in environments where $\gamma$ is insufficient or $k$ is excessively high, the transmission efficiency of subsidies is significantly diminished. Furthermore, the study reveals that the internal power structure of the supply chain mediates the final allocation of subsidy benefits, thereby influencing the actual effectiveness of policy incentives.

8.2. Implications

Based on these conclusions, this study offers the following implications for supply chain managers and policymakers:

(1)

For supply chain managers: If the target market exhibits a high service-premium sensitivity ($\gamma >{\gamma }_0$) and the fixed cost of using CR Express is manageable ($F<F_0$ or $F<{F^{\prime }}_0$), then adopting the CR Express mode is optimal to build a differentiated advantage and capture the service-premium benefit. When facing broad-based logistics-cost inflation ($k>k_0$), managers should recognize that the CR Express mode offers stronger risk-buffering capacity. Furthermore, companies should proactively seek and align with government subsidy programs, especially those tied to the CR Express, to achieve the dual goals of cost reduction and value enhancement.

(2)

For policymakers: The government should implement a differentiated subsidy scheme guided by the key parameter thresholds. In particular, increasing subsidy intensity for enterprises adopting the CR Express in timeliness-sensitive markets ($\gamma >{\gamma }_0$). In regions with weak logistics infrastructure and high systemic costs ($k>k_0$), policy focus should prioritize improving the logistics ecosystem and reducing these systemic costs ($k$). Additionally, assisting enterprises in identifying the service premium level ($\gamma$) of different markets can enhance the overall resource allocation efficiency of the supply chain.

8.3. Limitations and Future Research

Despite offering the aforementioned theoretical and managerial insights, this study is subject to several limitations that suggest productive avenues for future work. First, future research could develop a dynamic extension to examine how the gradual evolution of infrastructure and market conditions along the Belt and Road shapes long-term supply chain decisions. Second, incorporating uncertainties—such as fluctuations in service reliability, infrastructure accessibility, demand variability, or broader disruptions—would enhance the model’s relevance to practical complexities. Finally, extending the framework to more complex networks—for instance, moving beyond the bilateral monopoly to an oligopolistic market, or incorporating elements such as multi-tier suppliers, competing firms, or diversified retail channels—would enable a more systematic investigation of real-world supply chain dynamics and policy synergies. Beyond these theoretical extensions, empirical validation using firm-level or customs data would be a valuable next step.