# Research on the Optimal Leasing Strategy of Electric Vehicle Manufacturers

^{*}

## Abstract

**:**

_{1}and m

_{2}) were selected as research objects to construct two different leasing strategy models for electric vehicle manufacturers, namely, m

_{1}provided a unit rental electric vehicle strategy and m

_{2}provided a fixed rental electric vehicle strategy. We studied the optimal car rental strategy and pricing of the two manufacturers under the situation of m

_{2}providing and not providing rental service efforts, and the influence of relevant factors on the optimal decision are explored. It shows that the price of electric vehicles rented by consumers per unit increases with the combined effect of the coefficient of rental service effort and the marginal cost of the rental service effort, while the price of fixed rental electric vehicles decreases with the combined effect of both. When the unit rental preference coefficient is large, the unit rental of electric vehicles will give m

_{1}maximum profit. When the rental service effort coefficient is high, m

_{2}is the most profitable. The efforts to provide leasing services of m

_{2}increase their own interests to a certain extent. The greater the effort coefficient of leasing services, the smaller the marginal cost of leasing services, and the optimal social welfare reaches the maximum. The conclusion of the article can provide relevant leasing insights for electric vehicle manufacturers and also provide certain theoretical guidance for promoting electric vehicle leasing service strategies.

## 1. Introduction

_{2}providing and not providing rental service efforts, and to a certain extent, it enriches the research on electric vehicle leasing strategies and pricing decisions under the background of ‘double carbon’. In view of the above background, this paper mainly solves the following problems: First, the optimal equilibrium solution of electric vehicle manufacturers under different leasing strategies? Second, how do electric vehicle manufacturers m

_{1}and m

_{2}make decisions to maximize profits? Third, under which leasing strategy do consumers benefit the most? Therefore, the main academic contributions of this paper are as follows: Firstly, different from previous research, we study two different leasing strategies of electric vehicle manufacturers from reality, namely, the unit leasing electric vehicle strategy provided by m

_{1}and the fixed leasing electric vehicle strategy provided by m

_{2}. Secondly, the optimal decisions of two manufacturers under the scenario of m

_{2}with and without leasing service effort are studied. In addition, the insights provided by this paper can help electric vehicle manufacturers to determine the best electric vehicle rental strategy and pricing.

## 2. Literature Review

#### 2.1. Research on the Pricing Decision of Electric Vehicle Manufacturers

#### 2.2. Research on the Operation Strategy of Electric Vehicle Leasing

#### 2.3. The Discussion Section

_{2}providing and not providing rental service efforts. To a certain extent, it enriches the research on electric vehicle leasing strategies and pricing decisions under the background of double carbon. The main contributions of this paper are as follows: (1) Different from previous research, we study two different leasing strategies of electric vehicle manufacturers from reality, namely, the unit leasing electric vehicle strategy provided by m

_{1}and the fixed leasing electric vehicle strategy provided by m

_{2}. (2) The optimal decisions of two manufacturers under the scenario of m

_{2}with and without leasing service effort are studied. (3) In addition, the insights provided by this paper can help electric vehicle manufacturers determine the best electric vehicle rental strategy and pricing in the context of dual carbon.

## 3. Model Construction and Solution Analysis

_{1}and the best fixed leasing strategy of m

_{2}under the efforts of the manufacturer m

_{2}to provide and not to provide leasing services. At the same time, the optimal pricing and leasing service effort level of electric vehicle manufacturers are studied.

#### 3.1. Model Description and Basic Assumptions

_{1}and m

_{2}) in a duopoly market. They produce and lease electric vehicles with the battery-swapping mode and provide consumers with the strategy of unit leasing and fixed leasing of electric vehicles, respectively. Manufacturer m

_{1}provides consumers with the price of unit leasing electric vehicle of p

_{1}, and m

_{2}provides consumers with the price of fixed leasing electric vehicle of p

_{2}. Both of them will strive to maximize their own interests in order to increase sales and obtain greater benefits. Due to the prevalence of the unit rental mode in the current automobile market, m

_{2}will strive to improve the service level h for fixed rental consumers in order to enhance their competitiveness, that is, to provide consumers with post-rental battery upgrades, vehicle health management testing and consumer on-demand flexible battery swapping and other services. In view of this, this paper explores the optimal decision-making and pricing problems of m

_{1}that provides unit leasing electric vehicles and m

_{2}that provides fixed leasing electric vehicles under m

_{2}’s efforts to provide and not provide leasing services. Table 1 gives the specific model symbols and meanings involved in this paper.

_{1}and m

_{2}are only provided to consumers in the form of unit leases and fixed leases. (2) This article mainly studies the optimal leasing strategy and leasing service level, so it omits the manufacturer’s research on the wholesale price of electric vehicles. (3) According to the studies by Lu et al. [18] and Wu et al. [26], we assume that the utility obtained by heterogeneous consumers for electric vehicles provided by unit leasing m

_{1}and fixed leasing m

_{2}’s electric vehicles are u

_{1}and u

_{2}, where the random variable v is the consumers’ valuation of the electric vehicle, and v obeys the uniform distribution on [0, 1]. The rental price will have a negative impact on the utility u, but the consumers’ leasing preference coefficient θ (0 < θ < 1), the leasing service coefficient α and the rental service effort level h will have a positive impact on the utility. (4) We assume that the marginal cost of the manufacturer m

_{2}’s efforts to provide rental services is β (0 < β < 1).

#### 3.2. Basic Model: Do Not Provide Leasing Service Effort

_{1}and m

_{2}when m

_{2}does not provide leasing service efforts in the electric vehicle market. Through the behavior of consumers, it can be seen that there are three choices for consumers: (1) electric vehicles provided by unit leasing m

_{1}; (2) fixed leasing m

_{2}electric vehicles provided to consumers; (3) hold a conservative wait-and-see attitude. According to references [18,26], the utility functions that satisfy the above consumer choice are as follows:

_{1}is affected by the valuation v, the lease preference coefficient θ, the times of leases n and the lease price p

_{1}. In Equation (2), the consumer utility of the fixed leasing depends on the size of v and p

_{2}. Through the above utility function, it is easy to know that when u

_{1}− u

_{2}> 0 and both u

_{1}and u

_{2}are greater than 0, consumers are more willing to lease electric vehicles provided by m

_{1}; otherwise, when u

_{1}− u

_{2}< 0, consumers prefer fixed leasing electric vehicles. According to the different behavior choices of heterogeneous consumers and the non-negative demand, we will assume that $0<{p}_{2}<\frac{{np}_{1}-{p}_{2}}{\theta}<1$, and it is easy to derive the demand function of consumers’ unit and fixed leasing electric vehicles:

_{1}and m

_{2}are as follows:

_{1}and m

_{2}determine their optimal rental prices with the goal of maximizing their own profits. Because $\frac{{\partial}^{2}{\pi}_{1}}{\partial {{p}_{1}}^{2}}=\frac{-2{n}^{2}}{\theta}<0$, π

_{1}is a concave function of p

_{1}, and there is a unique maximum value. Similarly, $\frac{{\partial}^{2}{\pi}_{2}}{\partial {{p}_{2}}^{2}}=\frac{-2\left(1+\theta \right)}{\theta}<0$, so π

_{2}has a unique maximum. The Nash equilibrium simultaneous equation is used to obtain the optimal rental price:

**Theorem 1.**

_{2}does not provide leasing service efforts, only when the inequality $\frac{2{c}_{1}-{c}_{2}-2+\sqrt{{{c}_{2}}^{2}+4{{c}_{1}}^{2}-4{c}_{2}\left(1+{c}_{1}\right)+4}}{4}<\theta <\frac{{c}_{1}-{c}_{2}}{2{c}_{2}-1}$is established can the optimal solution that meets the interests of the two manufacturers and the requirements of the basic model be obtained, and the optimal equilibrium solution is given by Equations (8)–(10).

**The Proof of Theorem 1.**

_{1}and m

_{2}determine their optimal leasing prices with the goal of maximizing their own profits. Because $\frac{{\partial}^{2}{\pi}_{1}}{\partial {{p}_{1}}^{2}}=\frac{-2{n}^{2}}{\theta}<0$, π

_{1}is a concave function of p

_{1}, and there is a unique maximum value. Similarly, $\frac{{\partial}^{2}{\pi}_{2}}{\partial {{p}_{2}}^{2}}=\frac{-2\left(1+\theta \right)}{\theta}<0$, so π

_{2}has a unique maximum value. By using the Nash equilibrium simultaneous equation, the optimal rental price and the condition for obtaining the optimal equilibrium solution are obtained, and Theorem 1 is proved. □

_{1}and m

_{2}can obtain the optimal profit when m

_{2}does not provide leasing service efforts.

#### 3.3. Expanded Model: Rental Service Effort

_{2}will provide leasing service efforts to consumers of fixed-leasing electric vehicles. The decision variables in the model include the price of unit and fixed leasing electric vehicles and the level of leasing service efforts. At this time, the utility functions obtained by consumers from leasing are as follows:

_{2}providing leasing service effort is a quadratic function of leasing service effort level h, and it is $\frac{\beta {h}^{2}}{2}$, where β is the marginal cost of leasing service effort. Because $\frac{{\partial}^{2}{{\pi}_{1}}^{E}}{\partial {{{p}_{1}}^{E}}^{2}}=\frac{-2{n}^{2}}{\theta}<0$, ${{\pi}_{1}}^{E}$ is a concave function of ${{p}_{1}}^{E}$ with a unique maximum. The Hessian matrix of Equation (19) is $\left[\begin{array}{cc}\frac{-2\left(1+\theta \right)}{\theta}& \frac{\alpha \left(1+\theta \right)}{\theta}\\ \frac{\alpha \left(1+\theta \right)}{\theta}& -\beta \end{array}\right]$, its first-order order principal sub-formula is $\frac{-2\left(1+\theta \right)}{\theta}<0$, at the same time, the second-order order principal sub-formula is $\frac{-\left(1+\theta \right)\left(\theta {\alpha}^{2}+{\alpha}^{2}-2\beta \theta \right)}{{\theta}^{2}}>0$, and it has an optimal solution. Therefore, according to the non-negative consumer demand and Hessian negative definiteness, the conditions for obtaining the optimal solution under the extended model are as follows:$\beta >\frac{{\alpha}^{2}\left(1+\theta \right)}{2\theta}$ and $\frac{{c}_{1}{\alpha}^{2}-2{\alpha}^{2}-2\beta {c}_{1}+\beta {c}_{2}+2\beta +\sqrt{{{c}_{1}}^{2}{\alpha}^{4}-4{\alpha}^{2}\beta {{c}_{1}}^{2}+2{\alpha}^{2}\beta {c}_{1}{c}_{2}+4{\beta}^{2}{{c}_{1}}^{2}-4{\beta}^{2}{c}_{1}{c}_{2}+{\beta}^{2}{{c}_{2}}^{2}-4{\beta}^{2}{c}_{2}+4{\beta}^{2}}}{2\left({\alpha}^{2}-2\beta \right)}<\theta <\frac{{c}_{1}-{c}_{2}}{2{c}_{2}-1}$.

**Theorem 2.**

_{2}provides the leasing service effort, when the inequalities $\beta >\frac{{\alpha}^{2}\left(1+\theta \right)}{2\theta}$and$\frac{{c}_{1}{\alpha}^{2}-2{\alpha}^{2}-2\beta {c}_{1}+\beta {c}_{2}+2\beta +\sqrt{{{c}_{1}}^{2}{\alpha}^{4}-4{\alpha}^{2}\beta {{c}_{1}}^{2}+2{\alpha}^{2}\beta {c}_{1}{c}_{2}+4{\beta}^{2}{{c}_{1}}^{2}-4{\beta}^{2}{c}_{1}{c}_{2}+{\beta}^{2}{{c}_{2}}^{2}-4{\beta}^{2}{c}_{2}+4{\beta}^{2}}}{2\left({\alpha}^{2}-2\beta \right)}<\theta <\frac{{c}_{1}-{c}_{2}}{2{c}_{2}-1}$are satisfied at the same time, m

_{1}and m

_{2}can obtain the optimal Nash equilibrium solution. Otherwise, although the equilibrium results can be obtained, there is no optimal solution that meets the requirements. The optimal equilibrium solution is as follows:

**Proof of Theorem 2.**

_{2}provides the leasing service effort, the consumers’ leasing preference coefficient θ satisfies $\frac{{c}_{1}{\alpha}^{2}-2{\alpha}^{2}-2\beta {c}_{1}+\beta {c}_{2}+2\beta +\sqrt{{{c}_{1}}^{2}{\alpha}^{4}-4{\alpha}^{2}\beta {{c}_{1}}^{2}+2{\alpha}^{2}\beta {c}_{1}{c}_{2}+4{\beta}^{2}{{c}_{1}}^{2}-4{\beta}^{2}{c}_{1}{c}_{2}+{\beta}^{2}{{c}_{2}}^{2}-4{\beta}^{2}{c}_{2}+4{\beta}^{2}}}{2\left({\alpha}^{2}-2\beta \right)}<\theta <\frac{{c}_{1}-{c}_{2}}{2{c}_{2}-1}$, and when the marginal cost of the rental service effort paid by m

_{2}is $\beta >\frac{{\alpha}^{2}\left(1+\theta \right)}{2\theta}$, the electric vehicle manufacturers m

_{1}and m

_{2}can obtain the optimal Nash equilibrium solution.

## 4. Comparative Analysis of Different Leasing Strategies

_{1}= 0.8, c

_{2}= 0.65, α~[0.2, 0.4], β~[0.5, 0.7], θ~[0.4, 0.4], N~[24, 96]. At the same time, we use the Matlab 2023 calculation tool and Origin 2023b data drawing analysis tool for the example analysis. The comparative analysis of the best equilibrium solutions for electric vehicle manufacturers. is shown in Table 2.

#### 4.1. A Comparative Analysis of the Single Factor of **θ** on the Change of Optimal Decision Variables

**Corollary 1.**

- (a)
- The optimal price: ${{{p}_{2}}^{E}}^{\mathrm{*}}>{{p}_{2}}^{\mathrm{*}}>{{p}_{1}}^{\mathrm{*}}>{{{p}_{1}}^{E}}^{\mathrm{*}}$;
- (b)
- The optimal demand: when θ~(0, 0.21], ${{{D}_{2}}^{E}}^{\mathrm{*}}>{{D}_{1}}^{\mathrm{*}}>{{D}_{2}}^{\mathrm{*}}>{{{D}_{1}}^{E}}^{\mathrm{*}}$; when θ~(0.21, 0.225], ${{{D}_{2}}^{E}}^{\mathrm{*}}>{{D}_{1}}^{\mathrm{*}}>{{{D}_{1}}^{E}}^{\mathrm{*}}>{{D}_{2}}^{\mathrm{*}}$; when θ~(0.225, 0.25], ${{D}_{1}}^{\mathrm{*}}>{{{D}_{2}}^{E}}^{\mathrm{*}}>{{{D}_{1}}^{E}}^{\mathrm{*}}>{{D}_{2}}^{\mathrm{*}}$; when θ~(0.25, 0.4], ${{D}_{1}}^{\mathrm{*}}>{{{D}_{1}}^{E}}^{\mathrm{*}}>{{{D}_{2}}^{E}}^{\mathrm{*}}>{{D}_{2}}^{\mathrm{*}}$;
- (c)
- The optimal profit: when θ~(0, 0.225], ${{\pi}_{1}}^{\mathrm{*}}>{{{\pi}_{2}}^{E}}^{\mathrm{*}}>{{{\pi}_{1}}^{E}}^{\mathrm{*}}>{{\pi}_{2}}^{\mathrm{*}}$; when θ~(0.225, 0.4], ${{\pi}_{1}}^{\mathrm{*}}>{{{\pi}_{1}}^{E}}^{\mathrm{*}}>{{{\pi}_{2}}^{E}}^{\mathrm{*}}>{{\pi}_{2}}^{\mathrm{*}}$;
- (d)
- The optimal consumer surplus: when θ~(0, 0.275], ${{CS}_{1}}^{\mathrm{*}}>{{{CS}_{1}}^{E}}^{\mathrm{*}}>{{{CS}_{2}}^{E}}^{\mathrm{*}}>{{CS}_{2}}^{\mathrm{*}}$; when θ~(0.275, 0.4], ${{{CS}_{1}}^{E}}^{\mathrm{*}}>{{CS}_{1}}^{\mathrm{*}}>{{{CS}_{2}}^{E}}^{\mathrm{*}}>{{CS}_{2}}^{\mathrm{*}}$;
- (e)
- The optimal social welfare: when θ~(0, 0.275], ${{SW}^{E}}^{\mathrm{*}}>{SW}^{\mathrm{*}}$; when θ~(0.275, 0.4], ${SW}^{\mathrm{*}}>{{SW}^{E}}^{\mathrm{*}}$.

**Proof of Corollary 1.**

_{2}provides the rental service effort, when $\frac{{c}_{1}{\alpha}^{2}-2{\alpha}^{2}-2\beta {c}_{1}+\beta {c}_{2}+2\beta +\sqrt{{{c}_{1}}^{2}{\alpha}^{4}-4{\alpha}^{2}\beta {{c}_{1}}^{2}+2{\alpha}^{2}\beta {c}_{1}{c}_{2}+4{\beta}^{2}{{c}_{1}}^{2}-4{\beta}^{2}{c}_{1}{c}_{2}+{\beta}^{2}{{c}_{2}}^{2}-4{\beta}^{2}{c}_{2}+4{\beta}^{2}}}{2\left({\alpha}^{2}-2\beta \right)}<\theta <\frac{{c}_{1}-{c}_{2}}{2{c}_{2}-1}$ and $\beta >\frac{{\alpha}^{2}\left(1+\theta \right)}{2\theta}$ are satisfied at the same time, the manufacturers m

_{1}and m

_{2}can obtain the Nash equilibrium solution. Therefore, the optimal equilibrium solution in the above different models is analyzed, and the relationship between the optimal prices is ${{{p}_{2}}^{E}}^{\mathrm{*}}>{{p}_{2}}^{\mathrm{*}}>{{p}_{1}}^{\mathrm{*}}>{{{p}_{1}}^{E}}^{\mathrm{*}}$, and then, Corollary 1(a) is proved. The comparison of optimal demand: when θ~(0, 0.21], ${{{D}_{2}}^{E}}^{\mathrm{*}}>{{D}_{1}}^{\mathrm{*}}>{{D}_{2}}^{\mathrm{*}}>{{{D}_{1}}^{E}}^{\mathrm{*}}$; when θ~(0.21, 0.225], ${{{D}_{2}}^{E}}^{\mathrm{*}}>{{D}_{1}}^{\mathrm{*}}>{{{D}_{1}}^{E}}^{\mathrm{*}}>{{D}_{2}}^{\mathrm{*}}$; when θ~(0.225, 0.25], ${{D}_{1}}^{\mathrm{*}}>{{{D}_{2}}^{E}}^{\mathrm{*}}>{{{D}_{1}}^{E}}^{\mathrm{*}}>{{D}_{2}}^{\mathrm{*}}$; when θ~(0.25, 0.4], ${{D}_{1}}^{\mathrm{*}}>{{{D}_{1}}^{E}}^{\mathrm{*}}>{{{D}_{2}}^{E}}^{\mathrm{*}}>{{D}_{2}}^{\mathrm{*}}$; therefore, Corollary 1(b) can be proven. The optimal profit comparison analysis is as follows: when θ~(0, 0.225], ${{\pi}_{1}}^{\mathrm{*}}>{{{\pi}_{2}}^{E}}^{\mathrm{*}}>{{{\pi}_{1}}^{E}}^{\mathrm{*}}>{{\pi}_{2}}^{\mathrm{*}}$; when θ~(0.225, 0.4], ${{\pi}_{1}}^{\mathrm{*}}>{{{\pi}_{1}}^{E}}^{\mathrm{*}}>{{{\pi}_{2}}^{E}}^{\mathrm{*}}>{{\pi}_{2}}^{\mathrm{*}}$; Corollary 1(c) is proven. Comparing the optimal consumer surplus seen in the model: when θ~(0, 0.275], ${{CS}_{1}}^{\mathrm{*}}>{{{CS}_{1}}^{E}}^{\mathrm{*}}>{{{CS}_{2}}^{E}}^{\mathrm{*}}>{{CS}_{2}}^{\mathrm{*}}$; when θ~(0.275, 0.4], ${{{CS}_{1}}^{E}}^{\mathrm{*}}>{{CS}_{1}}^{\mathrm{*}}>{{{CS}_{2}}^{E}}^{\mathrm{*}}>{{CS}_{2}}^{\mathrm{*}}$; Corollary 1(d) is easy to prove. Similarly, the optimal social welfare in Corollary 1(e) can be proven. □

_{2}provides the leasing service effort or not, the price of the consumer unit leasing the electric vehicle is less than the price of the fixed leasing. When m

_{2}provides leasing service efforts, due to the cost, in order to balance the income, m

_{2}will increase the fixed leasing price, and at this time, the fixed leasing cost is the largest. In the two models, consumers can obtain greater benefits by choosing unit leasing. The optimal demand of consumers in the optimal demand of Corollary 1(b) is closely related to consumer’s unit leasing preference coefficient θ and the leasing price. When θ~(0, 0.225], the consumer unit leasing coefficient has less impact on consumer choice. When m

_{2}does not provide rental service efforts, consumers mainly choose unit leasing electric vehicles based on the leasing price. When m

_{2}provides leasing service efforts, more consumers will choose fixed leasing under higher leasing service levels, so more consumers will prefer fixed leasing electric vehicles under m

_{2}’s leasing service efforts. When the leasing coefficient θ~(0.25, 0.4], the consumer unit leasing coefficient has a greater impact on its choice. Regardless of whether m

_{2}provides leasing services or not, the larger unit leasing preference coefficient will encourage more consumers to choose unit leasing electric vehicles. Therefore, when the unit leasing coefficient is small, the demand of consumers is mainly determined by the leasing price and the level of leasing service effort; when the unit leasing coefficient is large, consumers will be more inclined to unit leasing electric vehicles. The optimal profit of Corollary 1(c) shows that when the unit leasing coefficient is small, the profit of consumers is determined by the leasing price, the level of leasing service effort and the demand; when the unit leasing coefficient is large, the unit leasing will bring the maximum profit to m

_{1}. The optimal consumer surplus in Corollary 1(d) explains that when the unit leasing coefficient is small, when consumers do not provide leasing service efforts in m

_{2}, the unit leasing electric vehicle obtains greater benefits. However, when the unit leasing coefficient is large, consumers will gain a lot in choosing unit leasing electric vehicles under any circumstances. The optimal social welfare in Corollary 1(e) explores that when θ is small and θ~(0, 0.275], m

_{2}’s efforts to provide leasing services will increase the welfare of the whole society; when θ~(0.275, 0.4], the larger unit leasing coefficient will encourage more consumers to choose independently. At this time, regardless of whether the manufacturer has improved the level of leasing service efforts, social welfare will increase with the increase in the unit leasing coefficient.

**Corollary 2.**

- (a)
- The optimal price: $\frac{\partial {{p}_{1}}^{\mathrm{*}}}{\partial \theta}>0$; $\frac{\partial {{p}_{2}}^{\mathrm{*}}}{\partial \theta}<0$; $\frac{\partial {{{p}_{1}}^{E}}^{\mathrm{*}}}{\partial \theta}>0$; $\frac{\partial {{{p}_{2}}^{E}}^{\mathrm{*}}}{\partial \theta}<0$;
- (b)
- The optimal demand: $\frac{\partial {{D}_{1}}^{\mathrm{*}}}{\partial \theta}>0$; $\frac{\partial {{D}_{2}}^{\mathrm{*}}}{\partial \theta}<0$; $\frac{\partial {{{D}_{1}}^{E}}^{\mathrm{*}}}{\partial \theta}>0$; $\frac{\partial {{{D}_{2}}^{E}}^{\mathrm{*}}}{\partial \theta}<0$;
- (c)
- The optimal profit: $\frac{\partial {{\pi}_{1}}^{\mathrm{*}}}{\partial \theta}>0$; $\frac{\partial {{\pi}_{2}}^{\mathrm{*}}}{\partial \theta}<0$; $\frac{\partial {{{\pi}_{1}}^{E}}^{\mathrm{*}}}{\partial \theta}>0$; $\frac{\partial {{{\pi}_{2}}^{E}}^{\mathrm{*}}}{\partial \theta}<0$.

**The Proof of Corollary 2.**

_{2}providing and not providing leasing service efforts, the price of the consumer unit leasing the electric vehicle will increase with the unit leasing coefficient, but the price of the consumer fixed leasing the electric vehicle will decrease with the increase in the unit leasing coefficient in Corollary 2(a). Combined with $\frac{\partial {{D}_{1}}^{\mathrm{*}}}{\partial \theta}=\frac{2{\theta}^{2}\left(4{c}_{1}-{2c}_{2}-1\right)+\left(8\theta +3\right)\left({c}_{1}-{c}_{2}\right)}{{\theta}^{2}{\left(4\theta +3\right)}^{2}}>0$ and $\frac{\partial {{D}_{2}}^{\mathrm{*}}}{\partial \theta}=\frac{{\theta}^{2}\left(6{c}_{2}-{4c}_{1}-1\right)+\left(8\theta +3\right)\left({c}_{2}-{c}_{1}\right)}{{\theta}^{2}{\left(4\theta +3\right)}^{2}}<0$ in Corollary 2(b), that is, the demand of consumers for unit leasing electric vehicles increases monotonically with the unit leasing coefficient, and the monotonicity of the price in Corollary 2(a) with respect to θ can be well understood. At this time, when the demand increases steadily, the manufacturer can obtain greater profits by real-time markup. Similarly, Corollary 2(c) shows that under a certain threshold, the optimal profit of m

_{1}increases with the unit leasing coefficient, while the optimal profit of m

_{2}decreases with the unit leasing coefficient.

#### 4.2. Comparative Analysis of Optimal Decision Variables under the Influence of α and β

_{2}reaches the maximum. At this time, consumers can better obtain services such as post-rent battery upgrades, vehicle health management inspections and consumers’ on-demand use of flexible battery swaps provided by m

_{2}.

_{2}reaches the maximum value. At this time, the profit of the combined manufacturer m

_{2}is higher, and the profitability is stronger than that of the manufacturer m

_{1}. Although the leasing price provided by the manufacturer m

_{2}is slightly higher at this time, based on the fact that consumers can better obtain services such as post-rent battery upgrades, vehicle health management inspections and consumers’ on-demand use of flexible battery swaps provided by m

_{2}, more and more consumers will lease the vehicles of m

_{2}. Thus, the demand on m

_{2}is larger, and m

_{2}will be more competitive in the electric vehicle market.

_{2}provides leasing service efforts, the price of electric vehicles leased by consumers per unit increases with the influence of α and β, while the price of fixed leasing electric vehicles decreases with the increase in α and β. In general, the price of fixed leasing electric vehicles in the extended model is always higher than the unit leasing price. This is consistent with the reality that since m

_{2}provides leasing services at a certain cost, it will correspondingly increase the fixed leasing price in order to increase its revenue.

_{2}provides leasing service efforts, consumers obtain greater benefits than unit leasing in the fixed leasing process. At this time, consumers’ demand for fixed leasing electric vehicles is higher than that of unit leasing.

_{1}and m

_{2}decrease with the increase in α and β. The profit of m

_{1}fluctuates greatly under the influence of α and β, and the profit of m

_{2}is less affected by α and β. When the leasing service effort coefficient of m

_{2}is small, that is α~(0.2, 0.3), the optimal profit of m

_{2}is less than m

_{1}regardless of the marginal cost of the input. When the leasing service effort coefficient takes the equilibrium value α = 0.3, the optimal profit of the two manufacturers is equal. When the leasing service effort coefficient is large, that is α~(0.3, 0.4), m

_{2}is the most profitable. It fully shows that m

_{2}’s efforts to provide leasing services have increased its own interests to a certain extent, and at the same time, they have a certain impact on the profits of their rival m

_{1}.

_{2}provides the leasing service effort, the consumer surplus of the unit leasing electric vehicle decreases with the influence of α and β. On the contrary, the consumer surplus of the fixed leasing electric vehicle increases under the combined influence of α and β, and the greater the leasing service effort coefficient, the greater the optimal consumer surplus. Therefore, the greater the coefficient of leasing effort provided by m

_{2}, the more consumers will choose fixed leasing electric vehicles.

_{2}provides the leasing service effort, the greater the leasing service effort coefficient, the smaller the marginal cost of leasing service effort, and the optimal social welfare reaches the maximum value.

## 5. Conclusions

_{1}and m

_{2}) as the research object and analyzes two different leasing strategies of electric vehicle manufacturers under the background of double carbon: the unit leasing electric vehicle strategy provided by m

_{1}and the fixed leasing electric vehicle strategy provided by m

_{2}. The main contributions of this paper are as follows: (1) Different from previous research, we study two different leasing strategies of electric vehicle manufacturers from reality, namely, the unit leasing electric vehicle strategy provided by m

_{1}and the fixed leasing electric vehicle strategy provided by m

_{2}. (2) The optimal decisions of two manufacturers under the scenario of m

_{2}with and without leasing service effort are studied. (3) In addition, the insights provided by this paper can help electric vehicle manufacturers determine the best electric vehicle rental strategy and pricing in the context of dual carbon. At the same time, the optimal car rental strategy and pricing problem of the two manufacturers under the situation of m

_{2}providing and not providing rental service efforts are explored. The research results provide the following insights.

_{2}provides rental service efforts, the price of electric vehicles leased by consumers increases with the influence of α and β, while the price of fixed rental electric vehicles decreases with the increase in α and β. In general, the price of fixed rental electric vehicles in the extended model is always higher than the unit rental price. Regardless of whether m

_{2}provides leasing services or not, the price of electric vehicles leased by consumers is less than the price of fixed leasing.

_{2}provides leasing service efforts, consumers obtain greater benefits than unit leasing in the fixed leasing process. At this time, consumers’ demand for fixed leasing electric vehicles is higher than that of unit leasing.

_{1}. The optimal profits of m

_{1}and m

_{2}decrease with the increase in α and β. The profit of m

_{1}fluctuates greatly under the influence of α and β, and the profit of m

_{2}is less affected by α and β. When the rental service effort coefficient of m

_{2}is small, the optimal profit of m

_{2}is less than m

_{1}regardless of the marginal cost of input. When the leasing service effort coefficient takes the equilibrium value, the optimal profits of the two manufacturers are evenly matched. When the rental service effort coefficient is large, m

_{2}is the most profitable. The efforts of m

_{2}to provide leasing services have increased its own interests to a certain extent, and at the same time, they have had a certain impact on the profits of competitor m

_{1}.

_{2}, the unit rental electric vehicle gains more benefits; when m

_{2}provides leasing service efforts, consumers’ fixed leasing electric vehicles are more profitable. However, when the unit leasing coefficient is large, consumers will gain a lot in choosing unit leasing electric vehicles under any circumstances. After m

_{2}provides leasing service efforts, the consumer surplus of selecting unit leasing electric vehicles decreases with the influence of α and β; on the contrary, the consumer surplus of selecting fixed leasing electric vehicles increases under the combined influence of α and β, and the greater the leasing service effort coefficient, the greater the optimal consumer surplus. Therefore, the greater the coefficient of leasing effort provided by m

_{2}, the more consumers will be attracted to choose leasing rental electric vehicles.

_{2}’s efforts to provide leasing services will increase the welfare of the whole society; when the unit leasing preference coefficient is large, more consumers will choose independently. At this time, regardless of whether the manufacturer has improved the level of leasing service efforts, social welfare will increase with the increase in the unit leasing coefficient. When m

_{2}provides the level of rental service effort, the greater the coefficient of rental service effort, the smaller the marginal cost of rental service effort, and the optimal social welfare reaches the maximum value.

_{1}: when the consumer unit leasing preference coefficient is large, m

_{1}will obtain greater profits. (2) m

_{2}: when the leasing service effort coefficient is large, m

_{2}is the most profitable. (3) Consumers: When m

_{2}does not provide rental service efforts, consumers mainly choose to lease electric vehicles according to the rental price. When m

_{2}provides leasing service efforts, more consumers will favor fixed leasing electric vehicles at a higher level of rental service. (4) Government: when the consumer unit rental preference coefficient is small, the government should encourage m

_{2}to actively provide rental services to increase the welfare of the whole society; when m

_{2}provides the level of leasing service effort, the government should call on m

_{2}to increase the coefficient of leasing service effort and reduce the marginal cost of the leasing service effort to achieve better social welfare.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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Variation | Meanings and Interpretation |
---|---|

p_{1} | Decision variable, m_{1} provides consumers with the price of unit leasing electric vehicle |

p_{2} | Decision variable, m_{2} provides consumers with the price of a fixed leasing electric vehicle |

h | Decision variables, leasing service effort level |

θ | Consumers’ preference coefficient for unit leasing electric vehicles provided by m_{1} |

v | Consumers’ valuation of electric vehicles |

n | The times of consumers’ leasing electric vehicles |

u_{1} | The utility obtained by the electric vehicle provided by the consumer unit leasing |

u_{2} | The utility obtained by the electric vehicle provided by the consumer fixed leasing |

α | Leasing service effort coefficient |

β | Marginal cost of leasing service effort |

c_{i} | The unit production cost of the manufacturers (i = 1 represents m_{1}, i = 2 represents m_{2}) |

D_{i} | Consumers’ demand for leasing electric vehicles (i = 1 represents m_{1}, i = 2 represents m_{2}) |

π_{i} | The total profit of the manufacturers (i = 1 represents m_{1}, i = 2 represents m_{2}) |

E | The superscript E indicates the extended model |

θ | α | β | N | ${{\mathit{p}}_{1}}^{\mathit{*}}$ | ${{\mathit{p}}_{2}}^{\mathit{*}}$ | ${{{\mathit{p}}_{1}}^{\mathit{E}}}^{\mathit{*}}$ | ${{{\mathit{p}}_{2}}^{\mathit{E}}}^{\mathit{*}}$ | ${{\mathit{D}}_{1}}^{\mathit{*}}$ | ${{\mathit{D}}_{2}}^{\mathit{*}}$ | ${{{\mathit{D}}_{1}}^{\mathit{E}}}^{\mathit{*}}$ | ${{{\mathit{D}}_{2}}^{\mathit{E}}}^{\mathit{*}}$ | ${{\mathit{\pi}}_{1}}^{\mathit{*}}$ |

0.2 | 0.2 | 0.5 | 24 | 0.385 | 0.674 | 0.329 | 0.739 | 1.842 | 1.421 | 1.171 | 2.36 | 0.679 |

0.25 | 0.2 | 0.5 | 24 | 0.394 | 0.669 | 0.341 | 0.728 | 2.375 | 0.938 | 2.057 | 1.415 | 1.41 |

0.3 | 0.2 | 0.5 | 24 | 0.403 | 0.664 | 0.351 | 0.72 | 2.738 | 0.619 | 2.575 | 0.881 | 2.249 |

0.35 | 0.2 | 0.5 | 24 | 0.412 | 0.66 | 0.361 | 0.714 | 3.003 | 0.395 | 2.918 | 0.539 | 3.157 |

0.4 | 0.2 | 0.5 | 24 | 0.421 | 0.657 | 0.421 | 0.709 | 3.207 | 0.228 | 3.165 | 0.303 | 4.113 |

θ | α | β | N | ${{\mathit{\pi}}_{2}}^{\mathit{*}}$ | ${{{\mathit{\pi}}_{1}}^{\mathit{E}}}^{\mathit{*}}$ | ${{{\mathit{\pi}}_{2}}^{\mathit{E}}}^{\mathit{*}}$ | ${{\mathit{C}\mathit{S}}_{1}}^{\mathit{*}}$ | ${{\mathit{C}\mathit{S}}_{2}}^{\mathit{*}}$ | ${{{\mathit{C}\mathit{S}}_{1}}^{\mathit{E}}}^{\mathit{*}}$ | ${{{\mathit{C}\mathit{S}}_{2}}^{\mathit{E}}}^{\mathit{*}}$ | ${\mathit{S}\mathit{W}}^{\mathit{*}}$ | ${{\mathit{S}\mathit{W}}^{\mathit{E}}}^{\mathit{*}}$ |

0.2 | 0.2 | 0.5 | 24 | 0.337 | 0.274 | 0.477 | 0.515 | 0.101 | 0.359 | 0.329 | 0.668 | 0.757 |

0.25 | 0.2 | 0.5 | 24 | 0.176 | 1.057 | 0.27 | 0.635 | 0.044 | 0.555 | 0.15 | 0.778 | 0.833 |

0.3 | 0.2 | 0.5 | 24 | 0.088 | 1.988 | 0.159 | 0.657 | 0.019 | 0.708 | 0.089 | 0.96 | 0.906 |

0.35 | 0.2 | 0.5 | 24 | 0.04 | 2.981 | 0.099 | 0.727 | 0.008 | 0.782 | 0.065 | 1.105 | 1.049 |

0.4 | 0.2 | 0.5 | 24 | 0.015 | 4.007 | 0.068 | 0.793 | 0.003 | 0.847 | 0.055 | 1.258 | 1.204 |

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**MDPI and ACS Style**

Wu, D.; Li, J.
Research on the Optimal Leasing Strategy of Electric Vehicle Manufacturers. *World Electr. Veh. J.* **2024**, *15*, 19.
https://doi.org/10.3390/wevj15010019

**AMA Style**

Wu D, Li J.
Research on the Optimal Leasing Strategy of Electric Vehicle Manufacturers. *World Electric Vehicle Journal*. 2024; 15(1):19.
https://doi.org/10.3390/wevj15010019

**Chicago/Turabian Style**

Wu, Doudou, and Jizi Li.
2024. "Research on the Optimal Leasing Strategy of Electric Vehicle Manufacturers" *World Electric Vehicle Journal* 15, no. 1: 19.
https://doi.org/10.3390/wevj15010019