# Coordinating a Two-Echelon Supply Chain under Carbon Tax

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Supply Chain Model Description

## 4. Decision Analysis

#### 4.1. The Integrated Supply Chain: A Benchmark

**Proposition**

**1.**

- (a)
- If $t\le \widehat{t}$, we have ${e}^{c\ast}=\frac{k+ty}{2h}$ , ${p}^{c\ast}=\frac{4ah+4ch+4hty+(3k-ty)(k+ty)}{8h}$ and ${\pi}^{c\ast}=\frac{{(4ah-4ch-4hty+{(k+ty)}^{2})}^{2}}{64{h}^{2}}$.
- (b)
- If $t>\widehat{t}$, we have ${e}^{c\ast}=1$, ${p}^{c\ast}=\frac{1}{2}(a+c+h+k)$ and ${\pi}^{c\ast}=\frac{1}{4}{(a-c-h+k)}^{2}$.

#### 4.2. Decentralized Case with a Whole Price Contract (WP)

**Proposition**

**2.**

- (a)
- If $t\le \widehat{t}$, ${e}^{w\ast}=\frac{k+ty}{2h}$, ${w}^{w\ast}=\frac{4ah+4ch+4hty+(3k-ty)(k+ty)}{8h}$, ${p}^{w\ast}=\frac{12ah+4ch+4hty+(7k-ty)(k+ty)}{16h}$, ${\pi}_{r}^{w\ast}=\frac{{(4ah-4ch-4hty+{(k+ty)}^{2})}^{2}}{256{h}^{2}}$ and ${\pi}_{m}^{w\ast}=\frac{{(4ah-4ch-4hty+{(k+ty)}^{2})}^{2}}{128{h}^{2}}$.
- (b)
- If $t>\widehat{t}$, we have ${e}^{w\ast}=1$, ${w}^{w\ast}=\frac{1}{2}(a+c+h+k)$, ${p}^{w\ast}=\frac{1}{4}(3a+c+h+3k)$, ${\pi}_{r}^{w\ast}=\frac{1}{16}{(a-c-h+k)}^{2}$ and ${\pi}_{m}^{w\ast}=\frac{1}{8}{(a-c-h+k)}^{2}$.

**Theorem**

**1.**

#### 4.3. Decentralized Case with a Cost-Sharing Contract (CS)

**Proposition**

**3.**

- (a)
- If $t\le \widehat{t}$, ${e}^{cs\ast}=\frac{k+ty}{2h}$, ${w}^{cs\ast}=\frac{4ah+4ch+4hty+(k+ty)(k-3ty+2(k+ty)\rho )}{8h}$, ${p}^{cs\ast}=\frac{12ah+4ch+4hty+(7k-ty)(k+ty)}{16h}$, ${\pi}_{r}^{cs\ast}=\frac{{(4ah-4ch-4hty+{(k+ty)}^{2})}^{2}}{256{h}^{2}}$ and ${\pi}_{m}^{cs}=\frac{{(4ah-4ch-4hty+{(k+ty)}^{2})}^{2}}{128{h}^{2}}$.
- (b)
- If $t>\widehat{t}$, ${e}^{cs\ast}=1$, ${w}^{cs\ast}=\frac{1}{2}(a+c-h+k+2h\rho )$, ${p}^{cs\ast}=\frac{1}{4}(3a+c+h+3k)$, ${\pi}_{r}^{cs\ast}=\frac{1}{16}{(a-c-h+k)}^{2}$ and ${\pi}_{m}^{cs\ast}=\frac{1}{8}{(a-c-h+k)}^{2}$.

## 5. Coordination with Two Contract Mechanisms

#### 5.1. Coordination by MW Contract

**Proposition**

**4.**

- (a)
- If $t\le \widehat{t}$, we have ${e}^{mw\ast}=\frac{k+ty}{2h}$, ${w}^{mw\ast}=\frac{4ch+{k}^{2}+ty(4h-ty)}{4h}$, ${p}^{mw\ast}=\frac{4ah+4ch+4hty+(3k-ty)(k+ty)}{8h}$, ${\pi}_{r}^{mw\ast}=\frac{{(4ah-4ch-4hty+{(k+ty)}^{2})}^{2}}{64{h}^{2}}$ and ${\pi}_{m}^{mw\ast}=0$.
- (b)
- if $t>\widehat{t}$, we have ${e}^{mw\ast}=1$, ${w}^{mw\ast}=c+h$, ${p}^{mw\ast}=\frac{1}{2}(a+c+h+k)$, ${\pi}_{r}^{mw\ast}=\frac{1}{4}{(a-c-h+k)}^{2}$ and ${\pi}_{m}^{mw\ast}=0$.

#### 5.2. Coordination by MS Contract

**Proposition**

**5.**

- (a)
- If $t\le \widehat{t}$, we have ${e}^{ms\ast}=\frac{k+ty}{2h}$, ${w}^{ms\ast}=\frac{4ch+4hty+(k+ty)(ty(-2+\rho )+k\rho )}{4h}$, ${p}^{ms\ast}=\frac{4ah+4ch+4hty+(3k-ty)(k+ty)}{8h}$, ${\pi}_{r}^{ms\ast}=\frac{{(4ah-4ch-4hty+{(k+ty)}^{2})}^{2}}{64{h}^{2}}$, ${\pi}_{m}^{ms\ast}=0$.
- (b)
- if $t>\widehat{t}$, we have ${e}^{ms\ast}=1$, ${w}^{ms\ast}=c+h\rho $, ${p}^{ms\ast}=\frac{1}{2}(a+c+h+k)$, ${\pi}_{r}^{ms\ast}=\frac{1}{4}{(a-c-h+k)}^{2}$ and ${\pi}_{m}^{ms\ast}=0$.

## 6. Implementing the Contracts (${\mathit{w}}^{\mathit{m}\mathit{w}\mathbf{\ast}}$, ${\mathit{e}}^{\mathit{m}\mathit{w}\mathbf{\ast}}$) and (${\mathit{w}}^{\mathit{m}\mathit{s}\mathbf{\ast}}$, ${\mathit{\rho}}^{\mathit{m}\mathit{s}\mathbf{\ast}}$, ${\mathit{e}}^{\mathit{m}\mathit{s}\mathbf{\ast}}$) with TPT

**Proposition**

**6.**

## 7. Numerical Analysis

#### 7.1. Impact of $t$ on ${p}^{w\ast}$ and ${e}^{w\ast}$ under the WP Contract

#### 7.2. Impact of $t$ on ${w}^{\ast}$ under the MW and MS Contract

#### 7.3. Win–Win Outcome with the TPT Contract

## 8. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

**Proof of Proposition**

**1.**

**Case**

**1.**

**Case**

**2.**

**Proof of Proposition**

**2.**

**Proof of Proposition**

**3.**

**Proof of Theorem**

**1.**

**Proof of Proposition**

**4.**

**Proof of Proposition**

**5.**

**Proof of Proposition**

**6.**

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Parameters | Description |
---|---|

$a$ | Initial market potential demand |

${p}^{i}$ | Retail price |

${w}^{i}$ | Wholesale price |

${q}^{i}$ | Product quantity |

$k$ | Consumer environmental awareness (CEA) |

$h$ | Cost coefficient of unit product abatement investment |

$c$ | Unit production cost |

$y$ | Initial unit amount of carbon emissions from production process |

${e}^{i}$ | Unit product emission reduction ratio (abatement effort level) |

$t$ | Unit product carbon tax |

${\pi}_{j}^{i}$ | The supply chain member’s profit |

Superscript $i$ | $i\in \{c,w,n,cs,mw,ms\}$ refer to the integrated model, wholesale price contract, without reduce emissions, cost sharing contract, modified wholesale price and modified cost sharing contracts, respectively |

Subscript $j$ | $j\in \{m,r\}$ refer to the manufacturer and the retailer, respectively |

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**MDPI and ACS Style**

Yu, W.; Han, R.
Coordinating a Two-Echelon Supply Chain under Carbon Tax. *Sustainability* **2017**, *9*, 2360.
https://doi.org/10.3390/su9122360

**AMA Style**

Yu W, Han R.
Coordinating a Two-Echelon Supply Chain under Carbon Tax. *Sustainability*. 2017; 9(12):2360.
https://doi.org/10.3390/su9122360

**Chicago/Turabian Style**

Yu, Wei, and Ruizhu Han.
2017. "Coordinating a Two-Echelon Supply Chain under Carbon Tax" *Sustainability* 9, no. 12: 2360.
https://doi.org/10.3390/su9122360