# Pricing and Coordinating the Lease-Oriented Closed-Loop Supply Chain for Construction Machinery in the Era of Carbon Tax

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## Abstract

**:**

## 1. Introduction

## 2. Research Framework

## 3. Model Framework

#### 3.1. Model Description

#### 3.2. Definition of Symbols

_{n}, and the unit cost of remanufacturing a product is represented by c

_{r}. The manufacturer sells the new and remanufactured products to the lessor at prices w

_{n}and w

_{r}, respectively. The lessor then leases these products to consumers at prices p

_{n}and p

_{r}, respectively. Consumers have the flexibility to choose between new and remanufactured products based on their preferences.

_{n}and h

_{r}, respectively. The government imposes a carbon tax rate of t CNY/kg based on the total carbon emissions K, generated by the manufacturer’s production activities.

#### 3.3. Related Assumptions

**Assumption**

**1.**

_{n}> w

_{n}and p

_{r}> w

_{r}.

**Assumption**

**2.**

**Assumption**

**3.**

**Assumption**

**4.**

_{n}> c

_{r}and h

_{n}> h

_{r}, respectively.

**Assumption**

**5.**

_{n}> w

_{r}and p

_{n}> p

_{r}, respectively.

**Assumption**

**6.**

## 4. Model Solutions and Contract Coordination Design

#### 4.1. Demand Functions and Profit Functions

#### 4.2. Centralized Decision-Making Scenario Analysis

_{n}, p

_{r}, and e.

_{n}, p

_{r}, and e. By solving the equations $\frac{\partial {\pi}_{t}^{c}}{\partial {p}_{n}}=0$, $\frac{\partial {\pi}_{t}^{c}}{\partial {p}_{r}}=0$, and $\frac{\partial {\pi}_{t}^{c}}{\partial e}=0$, we can derive the following equations:

**Proposition**

**1.**

**Proof**

**of**

**Proposition**

**1.**

_{3}< 0, there is $\frac{\partial {\pi}_{t}^{c\ast}}{\partial t}<0$. □

**Proposition**

**2.**

**Proof**

**of**

**Proposition**

**2.**

_{4}> 0, there is $\frac{\partial {Q}_{r}^{c\ast}}{\partial \beta}>0$.

_{5}> 0, there is $\frac{\partial {K}_{c}^{\ast}}{\partial \beta}<0$; otherwise, there is $\frac{\partial {K}_{c}^{\ast}}{\partial \beta}>0$.

_{6}> 0, there is $\frac{\partial {\pi}_{t}^{c\ast}}{\partial \beta}>0$; otherwise, there is $\frac{\partial {\pi}_{t}^{c\ast}}{\partial \beta}<0$. □

#### 4.3. Decentralized Decision-Making Scenario Analysis

_{n}and p

_{r}.

_{n}and p

_{r}. By solving $\frac{\partial {\pi}_{t}^{c\ast}}{\partial {p}_{n}}=0$ and $\frac{\partial {\pi}_{t}^{c\ast}}{\partial {p}_{r}}=0$, we can obtain the following equations:

_{n}, e, and w

_{r}on ${\pi}_{m}^{s\ast}$, the Hessian matrix is constructed.

_{n}, w

_{r}, and e. By solving $\frac{\partial {\pi}_{m}^{s\ast}}{\partial {w}_{n}}=0$, $\frac{\partial {\pi}_{m}^{s\ast}}{\partial {w}_{r}}=0$, and $\frac{\partial {\pi}_{m}^{s\ast}}{\partial e}=0$, we can obtain the following equations:

**Proposition**

**3.**

**Proof**

**of**

**Proposition**

**3.**

_{7}< 0, there is $\frac{\partial {\pi}_{m}^{s\ast \ast}}{\partial t}<0$.

_{8}< 0, there is $\frac{\partial {\pi}_{r}^{s\ast \ast}}{\partial t}<0$. □

**Proposition**

**4.**

**Proof**

**of**

**Proposition**

**4.**

_{9}> 0, there is $\frac{\partial {Q}_{r}^{s\ast}}{\partial \beta}>0$.

_{10}> 0, there is $\frac{\partial {K}_{s}^{\ast}}{\partial \beta}>0$; otherwise, there is $\frac{\partial {K}_{s}^{\ast}}{\partial \beta}<0$.

_{11}> 0, there is $\frac{\partial {\pi}_{m}^{s\ast \ast}}{\partial \beta}>0$; otherwise, there is $\frac{\partial {\pi}_{m}^{s\ast \ast}}{\partial \beta}<0$.

**Proposition**

**5.**

**Proof**

**of**

**Proposition**

**5.**

#### 4.4. Leasing Compensation–Cost Apportioning Combined Coordination Contract Model

_{n}and p

_{r}in order to construct the Hessian matrix.

_{n}and p

_{r}. By solving $\frac{\partial {\pi}_{r}^{v}}{\partial {p}_{n}}=0$ and $\frac{\partial {\pi}_{r}^{v}}{\partial {p}_{r}}=0$, we can obtain the following equations:

_{n}, w

_{r}, and e, the Hessian matrix is constructed.

_{n}, w

_{r}, and e. By solving $\frac{\partial {\pi}_{m}^{v\ast}}{\partial {w}_{n}}=0$, $\frac{\partial {\pi}_{m}^{v\ast}}{\partial {w}_{r}}=0$, and $\frac{\partial {\pi}_{m}^{v\ast}}{\partial e}=0$, we can obtain the following equations:

## 5. Numerical Analysis

_{n}= 100, c

_{r}= 50, h

_{n}= 500, h

_{n}= 250, α = 15,000, k = 10, v = 50, β = 0.8, and t < 0.2, which align with the current carbon tax policy in China [47].

#### 5.1. Determining the Value Range of λ and f

_{1}< f < F

_{2}

_{t}) between the total supply chain profits achieved under the combined contract and those obtained under decentralized decision making.

_{t}> 0, indicating that the implementation of the combined contract leads to an increase in the total supply chain profit and represents a Pareto improvement. As the cost apportioning proportional coefficient (λ) increases, the leasing compensation proportional coefficient (f) also increases. This suggests that the lessor assumes a larger proportion of the industrial Internet platform cost, resulting in the need for a higher leasing compensation fee from the manufacturer. Furthermore, the manufacturer, with increased financial resources, can actively engage in recycling activities and enhance the recovery rate.

#### 5.2. The Impact of the Carbon Tax Rate t

#### 5.3. The Impact of the Remanufactured Product Preference Coefficient of Consumer β

## 6. Conclusions, Discussion, Limitations, and Implications

#### 6.1. Discussion

- (1)
- In the context of a leasing compensation–cost apportioning combined contract, as the cost apportioning proportional coefficient (λ) increases, the corresponding leasing compensation proportional coefficient (f) also experiences an increment. This increase in the allocation of platform construction costs to the lessor necessitates higher compensation fees to ensure the viability of operations. Nevertheless, the most substantial enhancement in total closed-loop supply chain profits is observed when λ is set at 0.35;
- (2)
- As the carbon tax rate (t) increases, the total profits of the supply chain exhibit a decline across all decision-making scenarios, including both the centralized and decentralized decision-making approaches as well as the combined contract. Notably, when considering the combined contract, both manufacturers and lessors achieve comparatively higher profits compared with the decentralized decision-making strategy. Furthermore, the increase in t is found to result in higher selling and leasing prices for both new and remanufactured products, spanning all scenarios. Specifically, under the combined contract, the leasing prices of new products remain consistently lower than those under decentralized decision making. Additionally, the increase in t leads to a decrease in the demand for new products across all scenarios, whereas the demand for remanufactured products and total carbon emissions also exhibit a downward trend;
- (3)
- As consumer awareness and acceptance of remanufactured products grow, there is an increase in demand for these products. This results in a decline in profits for all parties in the supply chain initially, followed by an eventual increase. However, the final profit level is higher than the initial profit level. Additionally, the production of remanufactured products contributes to lower production carbon emissions, leading to an overall reduction in carbon emissions within the closed-loop supply chain.

#### 6.2. Conclusions

- (1)
- The leasing compensation–cost apportioning combined contract serves as an effective solution to mitigate the challenges posed by dual marginal effects in a decentralized decision-making environment. This approach significantly enhances the overall profitability of the supply chain. Moreover, it leads to reduced leasing prices, thereby enabling consumers to enjoy improved services and benefits;
- (2)
- Although the carbon tax policy does exert a certain influence on enterprise profitability, it concurrently generates positive societal outcomes by effectively curbing carbon emissions within the lease-oriented closed-loop supply chain of construction machinery. This policy also incentivizes manufacturers to prioritize the production of remanufactured products, leading to higher utilization rates of machinery equipment;
- (3)
- As the carbon tax rate increases, the initial decline in profits for various stakeholders in the supply chain, owing to the enhanced consumer acceptance of remanufactured products, is eventually offset by the subsequent increase in overall supply chain profits. This shift is accompanied by a reduction in carbon emissions within the closed-loop supply chain. Furthermore, the trend fosters a higher substitution rate for remanufactured products over new ones, reflecting the positive impact of consumer preference changes on the sustainability of the supply chain.

#### 6.3. Limitations

#### 6.4. Implications

- (1)
- The notable effectiveness of the combined contract underscores its substantial managerial significance. This highlights the importance of collaborative efforts between manufacturers and lessors in boosting the overall efficiency of the supply chain. Moreover, this collaboration not only fosters environmental sustainability but also aligns with the industry’s trajectory towards a low-carbon transformation and progress;
- (2)
- The carbon tax policy, recognized as a potent instrument for emissions reduction, has showcased considerable effectiveness in curtailing carbon emissions. Nonetheless, it is crucial to recognize that setting carbon tax rates at excessively low levels might prompt manufacturers to curtail their recycling efforts. Conversely, excessively high rates could potentially impede corporate profitability and dampen the incentive for pursuing energy conservation and emissions reduction. Consequently, in establishing carbon tax rates, governments must undertake comprehensive research and formulate a well-calibrated carbon tax policy. This policy would play a pivotal role in steering environmentally conscious development within enterprises;
- (3)
- The enhancement of consumer environmental awareness and the execution of governmental low-carbon policies play equally pivotal roles. Governments ought to bolster consumer eco-consciousness via low-carbon campaigns, whereas manufacturing enterprises should remain committed to advancing remanufacturing technologies. This endeavor ensures the maintenance of product quality and fosters increased consumer acceptance of remanufactured goods.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Flow chart of lease-oriented closed-loop supply chain for construction machinery under the carbon tax policy.

**Figure 3.**The impact of the cost apportioning proportional coefficient (f) on the profit increment (Δπ

_{t}).

**Figure 6.**The impact of the remanufactured product preference coefficient of consumers (β) on the product demand (Q).

**Figure 7.**The impact of the remanufactured product preference coefficient of consumers (β) on the total production carbon emissions (K).

Notation | Definition |
---|---|

${Q}_{n}^{i}$ | Demand for the new product under different decision-making situations, i ∈ {c, s, v}. c, s, and v represent centralized decision making, decentralized decision making, and combined contract, respectively. |

${Q}_{n}^{i\ast}$ | The optimal value after solving for partial optimal decision variables |

${Q}_{r}^{i}$ | Demand for the remanufactured product |

${\pi}_{m}^{i}$ | Profit of the manufacturer |

${\pi}_{m}^{i\ast \ast}$ | The optimal value after further solving for all optimal decision variables |

${\pi}_{r}^{i}$ | Profit of the lessor |

${\pi}_{t}^{i}$ | Profit of the total supply chain |

p_{n} | New product leasing price |

p_{r} | Remanufactured product leasing price |

w_{n} | New product selling price |

w_{r} | Remanufactured product selling price |

c_{n} | Unit new product cost |

c_{r} | Unit remanufactured product cost |

v | Unit recovery cost |

β | Remanufactured product preference coefficient of consumer |

t | Carbon tax rate |

e | Product recovery rate |

φ | Industrial Internet platform cost coefficient |

h_{n} | Production carbon emissions of unit new product |

h_{r} | Remanufactured carbon emissions of unit new product |

λ | Cost apportioning proportional coefficient |

f | Leasing compensation proportional coefficient |

k | Unit revenue of parts that cannot be remanufactured |

K | Total production carbon emission |

$\mathit{\lambda}$ | $\mathit{f}$ | ${\mathit{p}}_{\mathit{n}}^{\mathit{v}\ast \ast}$ | ${\mathit{w}}_{\mathit{n}}^{\mathit{v}\ast \ast}$ | ${\mathit{e}}_{\mathit{v}}$ | $\mathsf{\Delta}{\mathit{\pi}}_{\mathit{t}}$ |
---|---|---|---|---|---|

0.05 | (0.03, 15.3) | 2286.3 | 1596.3 | 0.47 | 15.3 |

0.15 | (45.3, 267.9) | 2286.2 | 1598.8 | 0.53 | 222.6 |

0.25 | (165.8, 506.7) | 2286.0 | 1602.1 | 0.60 | 340.9 |

0.35 | (438.5, 820.0) | 2285.8 | 1606.3 | 0.69 | 381.5 |

0.45 | (1022.6, 1248.9) | 2285.4 | 1612.0 | 0.82 | 226.2 |

$\mathit{t}$ | ${\mathit{\pi}}_{\mathit{t}}^{\mathit{c}\ast}$ | ${\mathit{\pi}}_{\mathit{t}}^{\mathit{s}\ast}$ | ${\mathit{\pi}}_{\mathit{t}}^{\mathit{v}\ast}$ | ${\mathit{\pi}}_{\mathit{m}}^{\mathit{s}\ast \ast}$ | ${\mathit{\pi}}_{\mathit{r}}^{\mathit{s}\ast \ast}$ | ${\mathit{\pi}}_{\mathit{m}}^{\mathit{v}\ast \ast}$ | ${\mathit{\pi}}_{\mathit{r}}^{\mathit{v}\ast \ast}$ |
---|---|---|---|---|---|---|---|

0.02 | 2,095,400 | 1,569,900 | 1,570,300 | 1,046,000 | 523,840 | 1,046,300 | 523,970 |

0.05 | 2,073,900 | 1,553,800 | 1,554,200 | 1,035,400 | 518,480 | 1,035,600 | 518,620 |

0.08 | 2,052,700 | 1,537,900 | 1,538,300 | 1,024,800 | 513,160 | 1,025,000 | 513,320 |

0.11 | 2,031,600 | 1,522,200 | 1,522,600 | 1,014,300 | 507,890 | 1,014,500 | 508,060 |

0.14 | 2,010,700 | 1,506,500 | 1,506,900 | 1,003,900 | 202,670 | 1,004,100 | 502,840 |

0.17 | 1,990,000 | 1,491,000 | 1,491,400 | 993,600 | 497,490 | 993,700 | 497,670 |

0.20 | 1,969,400 | 1,475,700 | 1,476,000 | 983,300 | 492,350 | 983,500 | 492,550 |

$\mathit{t}$ | ${\mathit{p}}_{\mathit{n}}^{\mathit{c}\ast}$ | ${\mathit{p}}_{\mathit{r}}^{\mathit{c}\ast}$ | ${\mathit{p}}_{\mathit{n}}^{\mathit{s}\ast \ast}$ | ${\mathit{p}}_{\mathit{r}}^{\mathit{s}\ast \ast}$ | ${\mathit{p}}_{\mathit{n}}^{\mathit{v}\ast \ast}$ | ${\mathit{p}}_{\mathit{r}}^{\mathit{v}\ast \ast}$ |
---|---|---|---|---|---|---|

0.02 | 1550.3 | 1232.5 | 2276.3 | 1816.2 | 2275.7 | 1816.2 |

0.05 | 1557.9 | 1236.3 | 2280.1 | 1818.1 | 2279.5 | 1818.1 |

0.08 | 1565.4 | 1240.0 | 2283.9 | 1820.0 | 2283.2 | 1820.0 |

0.11 | 1573.0 | 1243.7 | 2287.6 | 1821.9 | 2287.0 | 1821.9 |

0.14 | 1580.6 | 1247.5 | 2291.4 | 1823.7 | 2290.8 | 1823.7 |

0.17 | 1588.1 | 1251.2 | 2295.2 | 1825.6 | 2294.6 | 1825.6 |

0.20 | 1595.7 | 1255.0 | 2298.9 | 1827.5 | 2298.3 | 1827.5 |

$\mathit{t}$ | ${\mathit{Q}}_{\mathit{n}}^{\mathit{c}\ast}$ | ${\mathit{Q}}_{\mathit{r}}^{\mathit{c}\ast}$ | ${\mathit{Q}}_{\mathit{n}}^{\mathit{s}\ast}$ | ${\mathit{Q}}_{\mathit{r}}^{\mathit{s}\ast}$ | ${\mathit{Q}}_{\mathit{n}}^{\mathit{v}\ast}$ | ${\mathit{Q}}_{\mathit{r}}^{\mathit{v}\ast}$ |
---|---|---|---|---|---|---|

0.02 | 1410.0 | 48.4 | 699.6 | 30.1 | 702.8 | 26.9 |

0.05 | 1391.9 | 62.7 | 690.1 | 37.2 | 693.3 | 34.1 |

0.08 | 1372.9 | 77.1 | 680.7 | 44.3 | 683.8 | 41.2 |

0.11 | 1353.8 | 91.5 | 671.2 | 51.4 | 674.3 | 48.4 |

0.14 | 1334.7 | 105.9 | 661.8 | 58.5 | 664.8 | 55.5 |

0.17 | 1315.7 | 120.3 | 652.3 | 65.7 | 655.3 | 62.7 |

0.20 | 1296.6 | 134.6 | 642.8 | 72.8 | 645.8 | 69.8 |

**Table 6.**The supply chain profits under different remanufactured product preference coefficients of consumers (β).

$\mathit{\beta}$ | ${\mathit{\pi}}_{\mathit{t}}^{\mathit{c}\ast}$ | ${\mathit{\pi}}_{\mathit{t}}^{\mathit{s}\ast}$ | ${\mathit{\pi}}_{\mathit{t}}^{\mathit{v}\ast}$ | ${\mathit{\pi}}_{\mathit{m}}^{\mathit{s}\ast \ast}$ | ${\mathit{\pi}}_{\mathit{r}}^{\mathit{s}\ast \ast}$ | ${\mathit{\pi}}_{\mathit{m}}^{\mathit{v}\ast \ast}$ | ${\mathit{\pi}}_{\mathit{r}}^{\mathit{v}\ast \ast}$ |
---|---|---|---|---|---|---|---|

0.05 | 2,069,400 | 1,550,300 | 1,550,700 | 1,032,900 | 517,350 | 1,033,900 | 517,430 |

0.2 | 2,042,500 | 1,530,100 | 1,530,500 | 1,019,500 | 516,020 | 1,020,400 | 510,700 |

0.35 | 2,038,800 | 1,527,400 | 1,527,800 | 1,017,600 | 509,710 | 1,018,600 | 509,800 |

0.5 | 2,037,600 | 1,526,500 | 1,526,900 | 1,017,000 | 509,410 | 1,018,000 | 509,510 |

0.65 | 2,037,500 | 1,526,400 | 1,526,800 | 1,017,100 | 509,360 | 1,018,000 | 906,480 |

0.8 | 2,038,600 | 1,527,400 | 1,527,800 | 1,017,800 | 509,640 | 1,018,600 | 509,810 |

0.95 | 2,050,600 | 1,527,300 | 1,537,400 | 1,024,600 | 512,640 | 1,025,000 | 513,060 |

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## Share and Cite

**MDPI and ACS Style**

Yin, J.; Fang, Y.; Zhang, H.; Wang, T.; Cai, S.
Pricing and Coordinating the Lease-Oriented Closed-Loop Supply Chain for Construction Machinery in the Era of Carbon Tax. *Buildings* **2023**, *13*, 2145.
https://doi.org/10.3390/buildings13092145

**AMA Style**

Yin J, Fang Y, Zhang H, Wang T, Cai S.
Pricing and Coordinating the Lease-Oriented Closed-Loop Supply Chain for Construction Machinery in the Era of Carbon Tax. *Buildings*. 2023; 13(9):2145.
https://doi.org/10.3390/buildings13092145

**Chicago/Turabian Style**

Yin, Jing, Yifan Fang, Hengxi Zhang, Tingting Wang, and Shunyao Cai.
2023. "Pricing and Coordinating the Lease-Oriented Closed-Loop Supply Chain for Construction Machinery in the Era of Carbon Tax" *Buildings* 13, no. 9: 2145.
https://doi.org/10.3390/buildings13092145