# Carbon Reduction Incentives under Multi-Market Interactions: Supply Chain Vertical Cooperation Perspective

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Carbon Emission Reduction Incentives

#### 2.2. Low-Carbon Supply Chain Cooperation

#### 2.3. Cross-Interface between Finance and Green Operations

## 3. Problem Description

## 4. Model Building and Analysis

#### 4.1. Decision Analysis under No Cooperation Model

**Proposition**

**1.**

**Corollary**

**1.**

#### 4.2. Decision Analysis under Cost Cooperation Model

**Proposition**

**2.**

**Corollary**

**2.**

- (1)
- ${e}^{CC*}$decrease with $m$ and $r$ and increase with $\beta $ and $k$.
- (2)
- ${e}^{CC*}$, ${q}^{CC*}$, ${\pi}_{S}^{CC*}$, and ${\pi}_{R}^{CC*}$increase first and then decrease with $\theta $.

**Corollary**

**3.**

#### 4.3. Decision Analysis under Alliance Cooperation Model

**Proposition**

**3.**

**Corollary**

**4.**

## 5. The Impact of Vertical Cooperation

#### 5.1. The Impact of Cooperation on Carbon Reduction Level

**Proposition**

**4.**

- (1)
- ${e}^{AC*}>{e}^{NC*}$.
- (2)
- ${e}^{CC*}>{e}^{NC*}$.
- (3)
- When $m\ge \frac{{(bk+\beta )}^{2}[-1+(2+r)\theta ]}{2b{(1+r)}^{2}\theta (-1+2\theta )}$(or $\beta \le \sqrt{\frac{2bm{(1+r)}^{2}\theta (-1+2\theta )}{-1+(2+r)\theta}}-bk$), ${e}^{AC*}\ge {e}^{CC*}$; Otherwise, ${e}^{CC*}>{e}^{AC*}$.

#### 5.2. The Impact of Cooperation on Retail Price

**Proposition**

**5.**

- (1)
- When $m\ge \frac{\beta (bk+\beta )}{b(1+r)}$(or $\beta \le \frac{1}{2}[\sqrt{{b}^{2}{k}^{2}+4bm(1+r)}-bk]$), ${p}^{AC*}\le {p}^{NC*}$; Otherwise, ${p}^{AC*}>{p}^{NC*}$.
- (2)
- ${p}^{CC*}>{p}^{NC*}$.
- (3)
- ${p}^{AC*}\le {p}^{CC*}$.

#### 5.3. The Impact of Cooperation on Market Demand

**Proposition**

**6.**

- (1)
- ${q}^{AC*}>{q}^{NC*}$.
- (2)
- ${q}^{CC*}\ge {q}^{NC*}$.
- (3)
- When $m\ge \frac{(5+8r+4{r}^{2}){(bk+\beta )}^{2}}{8b{(1+r)}^{2}}$(or $\beta \le \sqrt{\frac{8bm{(1+r)}^{2}}{5+8r+4{r}^{2}}}-bk$), ${q}^{AC*}\ge {q}^{CC*}$; Otherwise, ${q}^{AC*}<{q}^{CC*}$.

#### 5.4. The Impact of Cooperation on Supply Chain Profit

**Proposition**

**7.**

- (1)
- ${\pi}_{T}^{AC*}>{\pi}_{T}^{NC*}$.
- (2)
- When $m\ge \frac{{(bk+\beta )}^{2}(3+2r)(1+16r+12{r}^{2})}{64br{(1+r)}^{2}}$(or $\beta \le 8(1+r)\sqrt{\frac{bmr}{(3+2r)(1+16r+12{r}^{2})}}-bk$), ${\pi}_{T}^{CC*}\ge {\pi}_{T}^{NC*}$; Otherwise ${\pi}_{T}^{AC*}>{\pi}_{T}^{CC*}$.
- (3)
- ${\pi}_{T}^{AC*}>{\pi}_{T}^{CC*}$.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Proof of Proposition**

**1.**

**Proof of Corollary**

**1.**

**Proof of Corollary**

**2.**

- (1)
- The impact of $m$, $r$, $\beta $ and $k$ on ${e}^{CC*}$:$\frac{\partial {e}^{CC*}}{\partial m}=\frac{-4b{(1+r)}^{3}{\theta}^{3}(bk+\beta )(a-bc-bk{e}_{0})}{{\{4bm{(1+r)}^{2}{\theta}^{2}-{(bk+\beta )}^{2}[(3+2r)\theta -1]\}}^{2}}<0$; $\frac{\partial {e}^{CC*}}{\partial r}=\frac{-(bk+\beta )\theta [4bm{(1+r)}^{2}{\theta}^{2}-(1-\theta ){(bk+\beta )}^{2}](a-bc-bk{e}_{0})}{{\{4bm{(1+r)}^{2}{\theta}^{2}-{(bk+\beta )}^{2}[(3+2r)\theta -1]\}}^{2}}<0$; $\frac{\partial {e}^{CC*}}{\partial \beta}=\frac{(1+r)\theta [4bm{(1+r)}^{2}{\theta}^{2}+{(bk+\beta )}^{2}((3+2r)\theta -1)](a-bc-bk{e}_{0})}{{\{4bm{(1+r)}^{2}{\theta}^{2}-{(bk+\beta )}^{2}[(3+2r)\theta -1]\}}^{2}}>0$; $\begin{array}{l}\frac{\partial {e}^{CC*}}{\partial k}=\frac{b(1+r)\theta (a-bc)[4bm{(1+r)}^{2}{\theta}^{2}+{(bk+\beta )}^{2}((3+2r)\theta -1)]}{{\{4bm{(1+r)}^{2}{\theta}^{2}-{(bk+\beta )}^{2}[(3+2r)\theta -1]\}}^{2}}+\\ \frac{b(1+r)\theta {e}_{0}[4bm{(1+r)}^{2}(2bk+\beta ){\theta}^{2}+\beta {(bk+\beta )}^{2}((3+2r)\theta -1)]}{{\{4bm{(1+r)}^{2}{\theta}^{2}-{(bk+\beta )}^{2}[(3+2r)\theta -1]\}}^{2}}>0\end{array}$
- (2)
- The impact of $\theta $ on ${e}^{CC*}$, ${q}^{CC*}$, ${\pi}_{R}^{CC*}$ and ${\pi}_{S}^{CC*}$:

**Proof of Corollary**

**3.**

**Proof of Corollary**

**4.**

**Proof of Proposition**

**4.**

- (1)
- Comparing ${e}^{AC*}$ and ${e}^{NC*}$, we can achieve$${e}^{AC*}-{e}^{NC*}=\frac{(bk+\beta )(a-bc-bk{e}_{0})}{2[2bm(1+r)-{(bk+\beta )}^{2}]}>0.$$
- (2)
- Comparing ${e}^{CC*}$ and ${e}^{NC*}$:

- (3)
- Comparing ${e}^{AC*}$ and ${e}^{CC*}$:

**Proof of Proposition**

**5.**

- (1)
- Comparing ${p}^{AC*}$ and ${p}^{NC*}$:

- (2)
- Comparing ${p}^{CC*}$ and ${p}^{NC*}$:

- (3)
- Comparing ${p}^{AC*}$ and ${p}^{CC*}$:

**Proof of Proposition**

**6.**

- (1)
- Comparing ${q}^{AC*}$ and ${q}^{NC*}$, we have$${q}^{AC*}-{q}^{NC*}=\frac{bm(1+r)(a-bc-bk{e}_{0})}{2[2bm(1+r)-{(bk+\beta )}^{2}]}>0.$$
- (2)
- Comparing ${q}^{CC*}$ and ${q}^{NC*}$, we have$${q}^{CC*}-{q}^{NC*}=\frac{bm(1+r){(1+2r)}^{2}{(bk+\beta )}^{2}(a-bc-bk{e}_{o})}{2[2bm(1+r)-{(bk+\beta )}^{2}][16bm{(1+r)}^{2}-{(3+2r)}^{2}{(bk+\beta )}^{2}]}>0.$$
- (3)
- Comparing ${q}^{AC*}$ and ${q}^{CC*}$, we have

**Proof of Proposition**

**7.**

- (1)
- Comparing ${\pi}_{T}^{AC*}$ and ${\pi}_{T}^{NC*}$, we have$${\pi}_{T}^{AC*}-{\pi}_{T}^{NC*}=\frac{m(1+r){(a-bc-bk{e}_{0})}^{2}}{8[2bm(1+r)-{(bk+\beta )}^{2}]}>0.$$
- (2)
- Comparing ${\pi}_{T}^{CC*}$ and ${\pi}_{T}^{NC*}$, we have

- (3)
- Comparing ${\pi}_{T}^{AC*}$ and ${\pi}_{T}^{CC*}$, we have

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**Figure 1.**Carbon reduction level under different cooperation models. (

**a**) Impact of the difficulty of carbon reduction on carbon reduction level. (

**b**) Impact of the carbon sensitivity of consumer market on carbon reduction level.

**Figure 2.**Impact of financial market and carbon market on the advantages of cooperation for carbon reduction. (

**a**) Impact of interest rates. (

**b**) Impact of carbon price.

**Figure 3.**Retail price under different cooperation models. (

**a**) Impact of the difficulty of carbon reduction on retail price. (

**b**) Impact of the carbon sensitivity of consumer market on retail price.

**Figure 4.**Impact of financial market and carbon market on the advantages of cooperation for retail price. (

**a**) Impact of interest rates. (

**b**) Impact of carbon price.

**Figure 5.**Market demand under different cooperation models. (

**a**) Impact of the difficulty of carbon reduction on market demand. (

**b**) Impact of the carbon sensitivity of consumer market on market demand.

**Figure 6.**Impact of financial market and carbon market on the advantages of cooperation for market demand. (

**a**) Impact of interest rates. (

**b**) Impact of carbon price.

**Figure 7.**The profit of the whole supply chain under different cooperation models. (

**a**) Impact of the difficulty of carbon reduction on supply chain profit. (

**b**) Impact of carbon sensitivity of the consumer market on supply chain profit.

Symbols | Description |
---|---|

$a$ | initial market size |

$b$ | the price sensitivity of consumer market, $b>0$ |

$\beta $ | the carbon sensitivity of consumer market, $\beta >0$ |

$c$ | production cost per unit product |

$w$ | the wholesale price |

$l$ | the retailer’s unit sales profit |

$p$ | the retail price, $p=w+l$ |

$q$ | the market demand |

$m$ | the difficulty of carbon reduction |

$E$ | the carbon allowance |

${e}_{0}$ | initial carbon emission per unit product |

$e$ | carbon reduction level |

$k$ | the carbon price |

$r$ | financing rate |

$\theta $ | the cost-sharing ratio |

$\pi $ | the enterprise’s profit |

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## Share and Cite

**MDPI and ACS Style**

Huang, X.; He, J.; Mao, L.
Carbon Reduction Incentives under Multi-Market Interactions: Supply Chain Vertical Cooperation Perspective. *Mathematics* **2024**, *12*, 599.
https://doi.org/10.3390/math12040599

**AMA Style**

Huang X, He J, Mao L.
Carbon Reduction Incentives under Multi-Market Interactions: Supply Chain Vertical Cooperation Perspective. *Mathematics*. 2024; 12(4):599.
https://doi.org/10.3390/math12040599

**Chicago/Turabian Style**

Huang, Xiaohui, Juan He, and Lin Mao.
2024. "Carbon Reduction Incentives under Multi-Market Interactions: Supply Chain Vertical Cooperation Perspective" *Mathematics* 12, no. 4: 599.
https://doi.org/10.3390/math12040599