# Profit Distribution Model for Construction Supply Chain with Cap-and-Trade Policy

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

_{2}, which has become one of the three largest greenhouse gas emission sources in the world [3]. Since the construction industry plays such a major role in global environment degradation, controlling and reducing GHG emissions has become one of its major tasks [4]. Apart from the development of GHG emission reduction technologies and new energy technologies, more and more researchers follow with interest the optimization of the supply chain operation strategies to reduce GHG emissions [2].

- (1)
- A complete profit distribution model of construction supply chain in pure competition, co-opetition, and pure cooperation under cap-and-trade is constructed, and the optimal emission reduction decisions and profit distribution of construction enterprises are obtained.
- (2)
- Systematic comparison of the differences of emission reduction amount and profits of enterprises under three different cooperation modes, probing the reasons for the differences.
- (3)
- The impacts of emission reduction cost coefficient, emission reduction efficiency coefficient, and cap-and-trade policy’s constraints on the profits of construction supply chain.

## 2. Literature Review

#### 2.1. Supply Chain Operation Decisions under Cap-and-Trade Policy

#### 2.2. Profit Distribution of Supply Chain

#### 2.3. Construction Supply Chain Operation Decisions for Multi-Echelon Supply Chains

## 3. Model Description and Assumption

- (1)
- In the general contract signed by the proprietor and the general contractor, we assumed that the proprietor signs a contract with fixed total price and emission reduction bonuses with the general contractor to encourage contractor to achieve emission reduction, which is given by ${P}_{1}={R}_{1}+t$ [44]. Among them, ${P}_{1}$ is the total contract price, and ${R}_{1}$ is the fixed total contract price of the general contractor given by the proprietor. In addition, $t$ is the final emission reduction bonuses of the general contractor given by proprietor.
- (2)
- Because $t$ is the emission reduction bonuses, it must be related to the amount of carbon emissions reduction. Similar to the simple linear incentive contract mentioned by Ward, we assume a linear relationship between the emission bonuses and the amount of emission reductions [45]. Parameter $\phi $($\phi \ge 0$) is the bonuses coefficient of unit emission reduction negotiated by proprietor and general contractor in advance, which is given by $t=\phi s\left({e}_{1}+{e}_{2}\right)$. The ${e}_{1}$, ${e}_{2}$ (${e}_{1},\text{}{e}_{2}\ge 0$) offered above are the emission reduction amount of the unit construction area of general contractor and subcontractor. In addition, $s$ is construction areas.
- (3)
- In addition, ${e}_{1},\text{}{e}_{2}$ are affected by the emission reduction efforts ${\theta}_{1},\text{}{\theta}_{2}$. With the increase of the investment in the reduction emission, the marginal cost of the carbon emission reduction will increase gradually, so the increase of the emission reduction will be slower. That is, carbon emission reduction amount ${e}_{1}$ is a continuous increasing convex function about emission reduction efforts ${\theta}_{i}$, and the relationship of the two can be assumed as ${e}_{i}={\beta}_{i}{\theta}_{i}^{\frac{1}{2}},\text{}i=1,2$. ${\beta}_{i}$ is the emission reduction efficiency coefficient of enterprise $i$, which is the indication about emission reduction technology, management ability and so on of enterprise $i$. In addition, the bigger ${\beta}_{i}$ is, the higher the emission reduction efficiency of the enterprise $i$ is.
- (4)
- To achieve the goal of emission reduction, enterprises will increase the investment of emission reduction, upgrade the technology, adopt new equipment, and so on, to reduce the emission reduction of the unit construction area. The costs of emission reduction accelerated increase with the increase of the amount of emission reduction. Similar to the model of research and development cost adopted by many scholars in the field of product research and development [46], the cost of emission reduction of enterprise $i$ is assumed as ${G}_{i}=\frac{1}{2}{\epsilon}_{i}{e}_{i}^{2}$. Among them ${\epsilon}_{i}$ is cost coefficient of emission reduction of enterprise $i$. In addition, the bigger ${\epsilon}_{i}$ is, the more the cost of emission reduction is.
- (5)
- Under the constraints of cap-and-trade policy, we assumed that the government uses the benchmark-based emission method to allocate the quotas for the construction products, which means that the government establishes the carbon emission quotas according to the advanced carbon emission level of construction area [47]. When enterprises use up the free quotas allocated by the government, they need to purchase additional quotas from the carbon trading market to meet demand. Conversely, if the enterprises still have remainders after the completion of the construction products, the quotas can be sold for additional benefits. We assumed that the benchmark carbon emission of unit construction area in the construction industry (the advanced level in industry) is ${e}_{s}$. the initial carbon emission of unit construction area is ${e}_{0}$. If the construction area is s, and the transaction price of the carbon trading market is m, then carbon trading price of the supply chain after production is given as $f=ms\left({e}_{0}-{e}_{1}-{e}_{2}-{e}_{s}\right)$.
- (6)
- In the subcontract signed by the general contractor and the subcontractor, the general contractor will also give the subcontractor a fixed contract price ${R}_{2}$ at first. In addition, to encourage the subcontractor to increase the emission reduction efforts, the general contractor will give the subcontractor a certain bonus. At this time, the amount of bonus that the general contractor can dominate consists of two parts, which are the final reduction bonuses of general contractor given by proprietor t and f in the carbon trading market, and the total amount of the governable bonuses is ($t-f$). When carbon trading price $f$ is positive, it is indicated that the carbon emissions from the supply chain exceed the free quotas from the government, and they need to buy quotas from the carbon trading market, and on the contrary to indicate that the enterprises still have surplus quotas at the end of production, which can be sold in the carbon trading market to gain extra income. The general contractor assigns part of its domination to the subcontractor for reward. Assuming the profit distribution proportion of subcontractor given by general contractor is $\lambda \text{}\left(0\le \lambda \le 1\right)$, the total contract price ${P}_{2}$ of the subcontractor can be expressed as ${P}_{2}={R}_{2}+\lambda \left(t-f\right)$.
- (7)
- Supposing the fixed cost of general contractor and subcontractor are ${C}_{1}$ and ${C}_{2}$, the profits of general contractor and subcontractor are ${\pi}_{1}$ and ${\pi}_{2}$ while the profits of supply chain is $\pi $. Then we can draw their expression as$${\pi}_{1}={P}_{1}-f-{G}_{1}-{C}_{1}-{P}_{2}={R}_{1}-{R}_{2}-{C}_{1}-\frac{1}{2}{\epsilon}_{1}{\theta}_{1}{\beta}_{1}^{2}+\left(1-\lambda \right)\phantom{\rule{0ex}{0ex}}\left[\phi s\left({\beta}_{1}{\theta}_{1}^{\frac{1}{2}}+{\beta}_{2}{\theta}_{2}^{\frac{1}{2}}\right)-ms\left({e}_{0}-{\beta}_{1}{\theta}_{1}^{\frac{1}{2}}-{\beta}_{2}{\theta}_{2}^{\frac{1}{2}}-{e}_{s}\right)\right]$$$${\pi}_{2}={P}_{2}-{G}_{2}-{C}_{2}={R}_{2}-{C}_{2}-\frac{1}{2}{\epsilon}_{2}{\theta}_{2}{\beta}_{2}^{2}+\lambda \left[\phi s\left({\beta}_{1}{\theta}_{1}^{\frac{1}{2}}+{\beta}_{2}{\theta}_{2}^{\frac{1}{2}}\right)\phantom{\rule{0ex}{0ex}}-ms\left({e}_{0}-{\beta}_{1}{\theta}_{1}^{\frac{1}{2}}-{\beta}_{2}{\theta}_{2}^{\frac{1}{2}}-{e}_{s}\right)\right]$$$$\pi ={\pi}_{1}+{\pi}_{2}={P}_{1}-f-{C}_{1}-{C}_{2}-{G}_{1}-{G}_{2}={R}_{1}-{C}_{1}-{C}_{2}-\frac{1}{2}{\epsilon}_{1}{\theta}_{1}{\beta}_{1}^{2}-\frac{1}{2}{\epsilon}_{2}{\theta}_{2}{\beta}_{2}^{2}\phantom{\rule{0ex}{0ex}}+\phi s\left({\beta}_{1}{\theta}_{1}^{\frac{1}{2}}+{\beta}_{2}{\theta}_{2}^{\frac{1}{2}}\right)-ms\left({e}_{0}-{\beta}_{1}{\theta}_{1}^{\frac{1}{2}}-{\beta}_{2}{\theta}_{2}^{\frac{1}{2}}-{e}_{s}\right)$$
- (8)
- Supposing that the general contractor and the subcontractor are both risk neutral enterprises.

## 4. Model Analysis and Solution

#### 4.1. Pure Competition Model

**Proposition**

**1.**

**Proposition**

**2.**

**Proof.**

#### 4.2. Co-Opetition Model

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

#### 4.3. Pure Cooperation Model

**Proposition**

**6.**

**Proof.**

**Proposition**

**7.**

**Proof.**

**Proposition**

**8.**

**Proof.**

**Proposition**

**9.**

**Proof.**

## 5. Model Comparison

**Proposition**

**10.**

**Proof.**

**Proposition**

**11.**

**Proof.**

**Proposition**

**12.**

**Proof.**

## 6. Numerical Analysis

#### 6.1. The Influence of Emission Reduction Cost Coefficient and Efficiency Coefficient of Emission Reduction on Supply Chain Decision

**Observation**

**1.**

**Observation**

**2.**

#### 6.2. The Impact of the Constraints of Cap-and-Trade Policy on Supply Chain’s Decision-Making

## 7. Discussions and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Relationship between cost coefficient ${\epsilon}_{i}$ and emission reduction efforts ${\theta}_{i}$.

Nomenclature | Definition |
---|---|

${P}_{i}$ | the total contract price of enterprise i (i = 1, 2; 1 represents general contractor, 2 represents subcontractor) |

${R}_{i}$ | the fixed contract price of enterprise i (i = 1, 2) |

$t$ | the final reduction bonus of general contractor given by proprietor |

$\phi $ | bonus coefficient of unit emission reduction |

$s$ | construction area |

${e}_{i}$ | the emission reduction of the unit construction area of enterprise i (i = 1, 2) |

${\beta}_{i}$ | the emission reduction efficiency coefficient of enterprise i (i = 1, 2) |

${\theta}_{i}$ | emission reduction efforts of enterprise i (i = 1, 2) |

${G}_{i}$ | cost of emission reduction of enterprise i (i = 1, 2) |

${\epsilon}_{i}$ | cost coefficient of emission reduction of enterprise i (i = 1, 2) |

m | carbon trading price of each unit |

f | the amount of carbon trading |

${e}_{0}$ | the initial carbon emission of unit construction area |

${e}_{s}$ | the benchmark carbon emission of unit construction area in construction industry (the advanced level in industry) |

$\lambda $ | the profit distribution proportion of subcontractor given by general contractor |

**Table 2.**The impact of emission reduction cost coefficient ${\epsilon}_{1}$ on the optimal decision of the general contractor under pure competition decision-making.

${\mathit{\epsilon}}_{1}$ | In the Case of Pure Competition Decision-Making | |||||
---|---|---|---|---|---|---|

λ | ${\mathit{\theta}}_{1}^{\mathit{s}}$ | ${\mathit{\theta}}_{2}^{\mathit{s}}$ | ${\mathit{\pi}}_{1}^{\mathit{s}}$ | ${\mathit{\pi}}_{2}^{\mathit{s}}$ | ${\mathit{\pi}}_{\mathit{S}}$ | |

72,000 | 0 | 1.6 | 0 | 209,520 | 120,000 | 329,520 |

81,000 | 0 | 1.23 | 0 | 205,520 | 120,000 | 325,520 |

90,000 | 0 | 1 | 0 | 202,320 | 120,000 | 322,320 |

99,000 | 0 | 0.83 | 0 | 199,701.8 | 120,000 | 319,701.8 |

108,000 | 0 | 0.69 | 0 | 197,520 | 120,000 | 317,520 |

**Table 3.**The impact of emission reduction cost coefficient ${\epsilon}_{2}$ on the optimal decision of the subcontractor under pure competition decision-making.

${\mathit{\epsilon}}_{2}$ | In the Case of Pure Competition Decision-Making | |||||
---|---|---|---|---|---|---|

λ | ${\mathit{\theta}}_{1}^{\mathit{s}}$ | ${\mathit{\theta}}_{2}^{\mathit{s}}$ | ${\mathit{\pi}}_{1}^{\mathit{s}}$ | ${\mathit{\pi}}_{2}^{\mathit{s}}$ | ${\mathit{\pi}}_{\mathit{S}}$ | |

76,000 | 0 | 1.5625 | 0 | 209,520 | 120,000 | 329,520 |

85,500 | 0 | 1.5625 | 0 | 209,520 | 120,000 | 329,520 |

95,000 | 0 | 1.5625 | 0 | 209,520 | 120,000 | 329,520 |

104,500 | 0 | 1.5625 | 0 | 209,520 | 120,000 | 329,520 |

114,000 | 0 | 1.5625 | 0 | 209,520 | 120,000 | 329,520 |

**Table 4.**The impact of emission reduction cost coefficient ${\epsilon}_{1}$ on the optimal decision of the general contractor under co-opetition decision-making.

${\mathit{\epsilon}}_{1}$ | In the Case of Co-Opetition Decision-Making | |||||
---|---|---|---|---|---|---|

λ | ${\mathit{\theta}}_{1}^{\mathit{d}}$ | ${\mathit{\theta}}_{2}^{\mathit{d}}$ | ${\mathit{\pi}}_{1}^{\mathit{d}}$ | ${\mathit{\pi}}_{2}^{\mathit{d}}$ | ${\mathit{\pi}}_{\mathit{D}}$ | |

72,000 | 0.43 | 0.51 | 0.30 | 196,554.77 | 136,304.48 | 332,859.25 |

81,000 | 0.46 | 0.36 | 0.34 | 194,834.36 | 134,818.58 | 329,652.94 |

90,000 | 0.49 | 0.26 | 0.38 | 193,573.18 | 133,596.22 | 327,169.40 |

99,000 | 0.51 | 0.20 | 0.42 | 192,616.28 | 132,584.93 | 325,201.20 |

108,000 | 0.53 | 0.15 | 0.45 | 191,867.51 | 131,744.22 | 323,611.74 |

**Table 5.**The impact of emission reduction cost coefficient ${\epsilon}_{2}$ on the optimal decision of the subcontractor under co-opetition decision-making.

${\mathit{\epsilon}}_{2}$ | In the Case of Co-Opetition Decision-Making | |||||
---|---|---|---|---|---|---|

λ | ${\mathit{\theta}}_{1}^{\mathit{d}}$ | ${\mathit{\theta}}_{2}^{\mathit{d}}$ | ${\mathit{\pi}}_{1}^{\mathit{d}}$ | ${\mathit{\pi}}_{2}^{\mathit{d}}$ | ${\mathit{\pi}}_{\mathit{D}}$ | |

76,000 | 0.54 | 0.21 | 0.73 | 196,144.58 | 136,242.23 | 332,386.81 |

85,500 | 0.51 | 0.24 | 0.52 | 194,722.51 | 134,720.05 | 329,442.56 |

95,000 | 0.49 | 0.26 | 0.38 | 193,573.18 | 133,596.22 | 327,169.40 |

104,500 | 0.46 | 0.29 | 0.28 | 192,638.98 | 132,734.37 | 325,373.35 |

114,000 | 0.44 | 0.31 | 0.22 | 191,876.14 | 132,050.82 | 323,926.96 |

**Table 6.**The impact of emission reduction cost coefficient ${\epsilon}_{1}$ on the optimal decision of the general contractor under pure cooperation decision-making.

${\mathit{\epsilon}}_{1}$ | In the Case of Pure Cooperation Decision-Making | ||||
---|---|---|---|---|---|

${\mathit{\theta}}_{1}^{\mathit{c}}$ | ${\mathit{\theta}}_{2}^{\mathit{c}}$ | ${\mathit{\psi}}_{1}\left(\mathit{v}\right)$ | ${\mathit{\psi}}_{2}\left(\mathit{v}\right)$ | ${\mathit{\pi}}_{\mathit{C}}$ | |

72,000 | 1.56 | 1.60 | 204,315.25 | 144,064.96 | 348,380.21 |

81,000 | 1.23 | 1.60 | 202,198.00 | 142,182.21 | 344,380.21 |

90,000 | 1.00 | 1.60 | 200,578.59 | 140,601.62 | 341,180.21 |

99,000 | 0.83 | 1.60 | 199,296.69 | 139,265.34 | 338,562.03 |

108,000 | 0.69 | 1.60 | 198,251.75 | 138,128.46 | 336,380.21 |

**Table 7.**The impact of emission reduction cost coefficient ${\epsilon}_{2}$ on the optimal decision of the subcontractor under cooperative decision-making.

${\mathit{\epsilon}}_{2}$ | In the Case of Cooperation Decision-Making | ||||
---|---|---|---|---|---|

${\mathit{\theta}}_{1}^{\mathit{c}}$ | ${\mathit{\theta}}_{2}^{\mathit{c}}$ | ${\mathit{\psi}}_{1}\left(\mathit{v}\right)$ | ${\mathit{\psi}}_{2}\left(\mathit{v}\right)$ | ${\mathit{\pi}}_{\mathit{C}}$ | |

76,000 | 1 | 2.49 | 203,951.80 | 144,049.46 | 348,001.26 |

85,500 | 1 | 1.97 | 202,107.12 | 142,104.67 | 344,211.79 |

95,000 | 1 | 1.60 | 200,578.59 | 140,601.62 | 341,180.21 |

104,500 | 1 | 1.32 | 199,302.22 | 139,397.61 | 338,699.83 |

114,000 | 1 | 1.11 | 198,229.08 | 138,403.76 | 336,632.84 |

**Table 8.**The impact of emission reduction efficiency coefficient ${\beta}_{1}$ on the optimal decision under pure competition decision-making.

β_{1} | In the Case of Pure Competition Decision-Making | |||||
---|---|---|---|---|---|---|

λ | ${\mathit{\theta}}_{1}^{\mathit{s}}$ | ${\mathit{\theta}}_{2}^{\mathit{s}}$ | ${\mathit{\pi}}_{1}^{\mathit{s}}$ | ${\mathit{\pi}}_{2}^{\mathit{s}}$ | ${\mathit{\pi}}_{\mathit{S}}$ | |

0.64 | 0 | 2.44 | 0 | 209,520 | 120,000 | 329,520 |

0.72 | 0 | 1.93 | 0 | 209,520 | 120,000 | 329,520 |

0.80 | 0 | 1.56 | 0 | 209,520 | 120,000 | 329,520 |

0.88 | 0 | 1.29 | 0 | 209,520 | 120,000 | 329,520 |

0.96 | 0 | 1.09 | 0 | 209,520 | 120,000 | 329,520 |

**Table 9.**The impact of emission reduction efficiency coefficient ${\beta}_{1}$ on the optimal decision under co-opetition decision-making.

β_{1} | In the Case of Co-Opetition Decision-Making | |||||
---|---|---|---|---|---|---|

λ | ${\mathit{\theta}}_{1}^{\mathit{d}}$ | ${\mathit{\theta}}_{2}^{\mathit{d}}$ | ${\mathit{\pi}}_{1}^{\mathit{d}}$ | ${\mathit{\pi}}_{2}^{\mathit{d}}$ | ${\mathit{\pi}}_{\mathit{D}}$ | |

0.64 | 0.49 | 0.41 | 0.38 | 197,899.02 | 137,694.38 | 335,593.40 |

0.72 | 0.49 | 0.33 | 0.38 | 197,899.02 | 137,694.38 | 335,593.40 |

0.80 | 0.49 | 0.26 | 0.38 | 197,899.02 | 137,694.38 | 335,593.40 |

0.88 | 0.49 | 0.22 | 0.38 | 197,899.02 | 137,694.38 | 335,593.40 |

0.96 | 0.49 | 0.18 | 0.38 | 197,899.02 | 137,694.38 | 335,593.40 |

**Table 10.**The impact of emission reduction efficiency coefficient ${\beta}_{1}$ on the optimal decision under pure cooperation decision-making.

β_{1} | In the Case of Pure Cooperation Decision-Making | ||||
---|---|---|---|---|---|

${\mathit{\theta}}_{1}^{\mathit{c}}$ | ${\mathit{\theta}}_{2}^{\mathit{c}}$ | ${\mathit{\psi}}_{1}\left(\mathit{v}\right)$ | ${\mathit{\psi}}_{2}\left(\mathit{v}\right)$ | ${\mathit{\pi}}_{\mathit{C}}$ | |

0.64 | 1.56 | 1.60 | 204,904.43 | 144,699.78 | 349,604.21 |

0.72 | 1.23 | 1.60 | 204,904.43 | 140,601.62 | 349,604.21 |

0.80 | 1.00 | 1.60 | 204,904.43 | 144,699.78 | 349,604.21 |

0.88 | 0.83 | 1.60 | 204,904.43 | 144,699.78 | 349,604.21 |

0.96 | 0.69 | 1.60 | 204,904.43 | 144,699.78 | 349,604.21 |

**Table 11.**The impact of the emission reduction efficiency coefficient ${\beta}_{2}$ on the optimal decision under pure competition decision-making.

β_{2} | In the Case of Pure Competition Decision-Making | |||||
---|---|---|---|---|---|---|

λ | ${\mathit{\theta}}_{\mathbf{1}}^{\mathit{s}}$ | ${\mathit{\theta}}_{\mathbf{2}}^{\mathit{s}}$ | ${\mathit{\pi}}_{\mathbf{1}}^{\mathit{s}}$ | ${\mathit{\pi}}_{\mathbf{2}}^{\mathit{s}}$ | ${\mathit{\pi}}_{\mathit{S}}$ | |

0.48 | 0 | 1.56 | 0 | 209,520 | 120,000 | 329,520 |

0.54 | 0 | 1.56 | 0 | 209,520 | 120,000 | 325,520 |

0.60 | 0 | 1.56 | 0 | 209,520 | 120,000 | 329,520 |

0.66 | 0 | 1.56 | 0 | 209,520 | 120,000 | 329,520 |

0.72 | 0 | 1.56 | 0 | 209,520 | 120,000 | 329,520 |

**Table 12.**The impact of the emission reduction efficiency coefficient ${\beta}_{2}$ on the optimal decision under co-opetition decision-making.

β_{2} | In the Case of Co-Opetition Decision-Making | |||||
---|---|---|---|---|---|---|

λ | ${\mathit{\theta}}_{1}^{\mathit{d}}$ | ${\mathit{\theta}}_{2}^{\mathit{d}}$ | ${\mathit{\pi}}_{1}^{\mathit{d}}$ | ${\mathit{\pi}}_{2}^{\mathit{d}}$ | ${\mathit{\pi}}_{\mathit{D}}$ | |

0.48 | 0.49 | 0.26 | 0.59 | 197,899.02 | 137,694.38 | 335,593.40 |

0.54 | 0.49 | 0.26 | 0.47 | 197,899.02 | 137,694.38 | 335,593.40 |

0.60 | 0.49 | 0.26 | 0.38 | 197,899.02 | 137,694.38 | 335,593.40 |

0.66 | 0.49 | 0.26 | 0.31 | 197,899.02 | 137,694.38 | 335,593.40 |

0.72 | 0.49 | 0.26 | 0.26 | 197,899.02 | 137,694.38 | 335,593.40 |

**Table 13.**The impact of the emission reduction efficiency coefficient ${\beta}_{2}$ on the optimal decision under pure cooperative decision-making.

β_{2} | In the Case of Pure Cooperation Decision-Making | ||||
---|---|---|---|---|---|

${\mathit{\theta}}_{1}^{\mathit{c}}$ | ${\mathit{\theta}}_{2}^{\mathit{c}}$ | ${\mathit{\psi}}_{1}\left(\mathit{v}\right)$ | ${\mathit{\psi}}_{2}\left(\mathit{v}\right)$ | ${\mathit{\pi}}_{\mathit{C}}$ | |

0.48 | 1.00 | 2.49 | 204,904.43 | 144,699.78 | 349,604.21 |

0.54 | 1.00 | 1.97 | 204,904.43 | 144,699.78 | 349,604.21 |

0.60 | 1.00 | 1.60 | 204,904.43 | 144,699.78 | 349,604.21 |

0.66 | 1.00 | 1.32 | 204,904.43 | 144,699.78 | 349,604.21 |

0.72 | 1.00 | 1.11 | 204,904.43 | 144,699.78 | 349,604.21 |

**Table 14.**The impact of different carbon trading prices $m$ on the optimal decision under pure competition decision-making.

m | In the Case of Pure Competition Decision-Making | |||||
---|---|---|---|---|---|---|

λ | ${\mathit{\theta}}_{1}^{\mathit{s}}$ | ${\mathit{\theta}}_{2}^{\mathit{s}}$ | ${\mathit{\pi}}_{1}^{\mathit{s}}$ | ${\mathit{\pi}}_{2}^{\mathit{s}}$ | ${\mathit{\pi}}_{\mathit{S}}$ | |

24 | 0 | 1.13 | 0 | 200,826 | 120,000 | 320,826 |

27 | 0 | 1.34 | 0 | 204,970.5 | 120,000 | 324,970.5 |

30 | 0 | 1.56 | 0 | 209,520 | 120,000 | 329,520 |

33 | 0 | 1.81 | 0 | 214,474.5 | 120,000 | 334,474.5 |

36 | 0 | 2.07 | 0 | 219,834 | 120,000 | 339,834 |

**Table 15.**The impact of different carbon trading prices $m$ on the optimal decision under co-opetition decision-making.

m | In the Case of Co-Opetition Decision-Making | |||||
---|---|---|---|---|---|---|

λ | ${\mathit{\theta}}_{1}^{\mathit{d}}$ | ${\mathit{\theta}}_{2}^{\mathit{d}}$ | ${\mathit{\pi}}_{1}^{\mathit{d}}$ | ${\mathit{\pi}}_{2}^{\mathit{d}}$ | ${\mathit{\pi}}_{\mathit{D}}$ | |

24 | 0.49 | 0.19 | 0.27 | 192,674.17 | 132,539.87 | 325,214.03 |

27 | 0.49 | 0.23 | 0.32 | 195,167.19 | 134,999.83 | 330,167.05 |

30 | 0.49 | 0.26 | 0.38 | 197,899.02 | 137,694.38 | 335,593.40 |

33 | 0.49 | 0.30 | 0.44 | 200,869.65 | 140,623.42 | 341,493.07 |

36 | 0.49 | 0.35 | 0.50 | 204,079.08 | 143,786.98 | 347,866.07 |

**Table 16.**The impact of different carbon trading prices $m$ on the optimal decision under pure cooperation decision-making.

m | In the Case of Pure Cooperation Decision-Making | ||||
---|---|---|---|---|---|

${\mathit{\theta}}_{1}^{\mathit{c}}$ | ${\mathit{\theta}}_{2}^{\mathit{c}}$ | ${\mathit{\psi}}_{1}\left(\mathit{v}\right)$ | ${\mathit{\psi}}_{2}\left(\mathit{v}\right)$ | ${\mathit{\pi}}_{\mathit{C}}$ | |

24 | 0.72 | 1.15 | 197,735.56 | 137,601.28 | 335,336.8 |

27 | 0.86 | 1.37 | 201,161.19 | 140,993.86 | 342,155.1 |

30 | 1.00 | 1.60 | 204,904.43 | 144,699.78 | 349,604.2 |

33 | 1.16 | 1.84 | 208,965.27 | 148,719.04 | 357,684.3 |

36 | 1.32 | 2.11 | 213,343.73 | 153,051.64 | 366,395.4 |

**Table 17.**The impact of the benchmark carbon emission of unit construction area ${e}_{s}$ on the optimal decision under pure competition decision-making.

e_{s} | In the Case of Pure Competition Decision-Making | |||||
---|---|---|---|---|---|---|

λ | ${\mathit{\theta}}_{1}^{\mathit{s}}$ | ${\mathit{\theta}}_{2}^{\mathit{s}}$ | ${\mathit{\pi}}_{1}^{\mathit{s}}$ | ${\mathit{\pi}}_{2}^{\mathit{s}}$ | ${\mathit{\pi}}_{\mathit{S}}$ | |

0.62 | 0 | 1.56 | 0 | 201,096 | 120,000 | 321,096 |

0.70 | 0 | 1.56 | 0 | 205,308 | 120,000 | 325,308 |

0.78 | 0 | 1.56 | 0 | 209,520 | 120,000 | 329,520 |

0.86 | 0 | 1.56 | 0 | 213,732 | 120,000 | 333,732 |

0.94 | 0 | 1.56 | 0 | 217,944 | 120,000 | 337,944 |

**Table 18.**The impact of the benchmark carbon emission of unit construction area ${e}_{s}$ on the optimal decision under co-opetition decision-making.

e_{s} | In the Case of Co-Opetition Decision-Making | |||||
---|---|---|---|---|---|---|

λ | ${\mathit{\theta}}_{1}^{\mathit{d}}$ | ${\mathit{\theta}}_{2}^{\mathit{d}}$ | ${\mathit{\pi}}_{1}$ | ${\mathit{\pi}}_{2}$ | ${\mathit{\pi}}_{\mathit{D}}$ | |

0.62 | 0.49 | 0.26 | 0.26 | 193,573.18 | 133,596.22 | 327,169.40 |

0.70 | 0.49 | 0.26 | 0.26 | 195,736.10 | 135,645.30 | 331,381.40 |

0.78 | 0.49 | 0.26 | 0.26 | 197,899.02 | 137,694.38 | 335,593.40 |

0.86 | 0.49 | 0.26 | 0.26 | 200,061.94 | 139,743.46 | 339,805.40 |

0.94 | 0.49 | 0.26 | 0.26 | 202,224.86 | 141,792.54 | 344,017.40 |

**Table 19.**The impact of the benchmark carbon emission of unit construction area ${e}_{s}$ on the optimal decision under pure cooperation decision-making.

e_{s} | In the Case of Pure Cooperation Decision-Making | ||||
---|---|---|---|---|---|

${\mathit{\theta}}_{1}^{\mathit{c}}$ | ${\mathit{\theta}}_{2}^{\mathit{c}}$ | ${\mathit{\psi}}_{1}\left(\mathit{v}\right)$ | ${\mathit{\psi}}_{2}\left(\mathit{v}\right)$ | ${\mathit{\pi}}_{\mathit{C}}$ | |

0.62 | 1.00 | 1.11 | 200,578.59 | 140,601.62 | 341,180.21 |

0.70 | 1.00 | 1.11 | 202,741.51 | 142,650.70 | 345,392.21 |

0.78 | 1.00 | 1.11 | 204,904.43 | 144,699.78 | 349,604.21 |

0.86 | 1.00 | 1.11 | 207,067.35 | 146,748.87 | 353,816.21 |

0.94 | 1.00 | 1.11 | 209,230.26 | 148,797.95 | 358,028.21 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Jiang, W.; Lu, W.; Xu, Q.
Profit Distribution Model for Construction Supply Chain with Cap-and-Trade Policy. *Sustainability* **2019**, *11*, 1215.
https://doi.org/10.3390/su11041215

**AMA Style**

Jiang W, Lu W, Xu Q.
Profit Distribution Model for Construction Supply Chain with Cap-and-Trade Policy. *Sustainability*. 2019; 11(4):1215.
https://doi.org/10.3390/su11041215

**Chicago/Turabian Style**

Jiang, Wen, Wenfei Lu, and Qianwen Xu.
2019. "Profit Distribution Model for Construction Supply Chain with Cap-and-Trade Policy" *Sustainability* 11, no. 4: 1215.
https://doi.org/10.3390/su11041215