# Optimal Pricing and Greening Strategy in a Competitive Green Supply Chain: Impact of Government Subsidy and Tax Policy

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## Abstract

**:**

## 1. Introduction

- (a)
- How do the constructions of the supply chain influence the benefits of manufacturers, retailers furthermore, governments, and the green level of green items?
- (b)
- In the two models with and without government interference, what are the distinctions in the green items’ evaluating decision-making procedure?
- (c)
- What are the effects of government subsidy and imposing a tax on the manufacturer on the greening level, the retail cost, and the expected benefits of the players?
- (d)
- How does manufacturing greener items influence the final cost of green and non-green items?

## 2. Literature Review

## 3. Notations and Problem Definition

#### 3.1. Notations

Decision Variables | |

${p}_{g}$ | retail price of green product per unit item (${p}_{g}$ ∈ ${R}^{+}$) |

${p}_{n}$ | retail price of non-green product per unit item (${p}_{n}$ ∈ ${R}^{+}$) |

${w}_{g}$ | wholesale price of green product per unit item (${w}_{g}$ ∈ ${R}^{+}$) |

${w}_{n}$ | wholesale price of non-green product per unit item (${w}_{n}$ ∈ ${R}^{+}$) |

g | greening level of green product (g ∈ ${R}^{+}\cup \left\{0\right\}$) |

Parameters | |

$\alpha $ | degree of customers loyalty in green product ($\alpha \in (0,1)$) |

$\left(1-\alpha \right)$ | degree of customers loyalty in non-green product |

a | market based-demand ($0<a\in {R}^{+}$) |

b | price elasticity ($0<b\in {R}^{+}$) |

c | cross price elasticity ($0<c\in {R}^{+}$) |

$\gamma $ | sensitivity coefficient of greening level per unit green item |

${M}_{g}$ | manufacturing cost of the per unit green item (${M}_{g}$ ∈ ${R}^{+}$) |

${M}_{n}$ | manufacturing cost of the per unit non-green item (${M}_{n}$ ∈ ${R}^{+}$) |

e | cost factor of enhancing greening level (e ∈ ${R}^{+}$) |

${G}_{s}$ | subsidy rate coefficient for green products (${G}_{s}$ ∈ ${R}^{+}$) |

${G}_{t}$ | tax rate for non-green products (${G}_{t}$ ∈ ${R}^{+}$) |

${D}_{g}$ | demand of the green product (${D}_{g}$ ∈ ${R}^{+}$) |

${D}_{n}$ | demand of the non-green product (${D}_{n}$ ∈ ${R}^{+}$) |

$T{P}_{{M}_{g}}$ | profit of green manufacturer ($T{P}_{{M}_{g}}$ ∈ ${R}^{+}$) |

$T{P}_{{M}_{n}}$ | profit of non-green manufacturer ($T{P}_{{M}_{n}}$ ∈ ${R}^{+}$) |

$T{P}_{R}$ | profit of the retailer ($T{P}_{R}$ ∈ ${R}^{+}$) |

$T{P}_{SC}$ | profit of the supply chain ($T{P}_{SC}$ ∈ ${R}^{+}$) |

#### 3.2. Problem Definition

- Market demand is a linear function of the products’ sales price and greening level. The demand faced by the non-green manufacturer together with green manufacturer varied for without government interference model and with government interference model.
- The manufacturing cost of the non-green item is less than that of the green item (${M}_{n}<{M}_{g}$).

## 4. Mathematical Model Construction and Solution

#### 4.1. Without Government Interference Model (Model 1)

#### 4.1.1. Centralized Model (Model 1-A)

**Theorem**

**1.**

**Proof.**

#### 4.1.2. Decentralized Model (Model 1-B)

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

#### 4.2. With Government Interference Model (Model 2)

#### 4.2.1. Centralized Model (Model 2-A)

**Theorem**

**5.**

**Proof.**

#### 4.2.2. Decentralized Model (Model 2-B)

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

#### 4.2.3. Government Profit Function

## 5. Numerical Experiment

#### Comparative Study of Optimal Solutions of the Model

## 6. Sensitivity Analysis

#### 6.1. Effect of Demand Elasticity Parameter $a,b,c$

#### 6.2. Effect of Manufacturing Cost ${M}_{g}$ and ${M}_{n}$ in Profit Function

#### 6.3. Effect of Customer Green Preference $\gamma $

#### 6.4. Effect of Government Subsidy Parameter ${G}_{s}$

#### 6.5. Effect of Tax Rate Parameter ${G}_{t}$

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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parameters | $\alpha $ | a | b | c | ${M}_{g}$ | ${M}_{n}$ | $\gamma $ | e | ${G}_{s}$ | ${G}_{t}$ |

values | 0.5 | 1200 | 5 | 1 | 30 | 15 | 1.2 | 5 | 0.5 | 4 |

Different Scenario of Model | Without Govt. Interference; Model 1 (When in Centralized Case ${\mathit{w}}_{\mathit{g}}=75;{\mathit{w}}_{\mathit{n}}=50$) | With Govt. Interference; Model 2 (When in Centralized Case ${\mathit{w}}_{\mathit{g}}=75;{\mathit{w}}_{\mathit{n}}=50$) | With Govt. Interference; Model 2 (When in Centralized Case ${\mathit{w}}_{\mathit{g}}=85;{\mathit{w}}_{\mathit{n}}=60$) |
---|---|---|---|

Centralized Model (A) | |||

${p}_{g}^{\ast}$ | 91.85 | 109.58 | 109.58 |

${p}_{n}^{\ast}$ | 82.87 | 81.80 | 81.80 |

${g}^{\ast}$ | 14.84 | 52.20 | 52.20 |

$T{P}_{{M}_{g}}^{\ast}$ | 10,312 | 8086 | 11,395 |

$T{P}_{{M}_{n}}^{\ast}$ | 9713 | 8905 | 11,450 |

$T{P}_{R}^{\ast}$ | 13,190 | 19,540 | 13,686 |

$T{P}_{SC}^{\ast}$ | 33,215 | 36,531 | 36,531 |

$T{P}_{Govt.}$ | − | −7624 | −7624 |

Decentralized Model (B) | |||

${p}_{g}^{\ast}$ | 117.48 | 127.88 | 127.88 |

${w}_{g}^{\ast}$ | 83.35 | 89.32 | 89.32 |

${p}_{n}^{\ast}$ | 113.08 | 102.53 | 102.53 |

${w}_{n}^{\ast}$ | 75.83 | 57.97 | 57.97 |

${g}^{\ast}$ | 6.40 | 21.95 | 21.95 |

$T{P}_{{M}_{g}}^{\ast}$ | 7014 | 7592 | 7592 |

$T{P}_{{M}_{n}}^{\ast}$ | 9252 | 7917 | 7917 |

$T{P}_{R}^{\ast}$ | 10,215 | 13,930 | 13,930 |

$T{P}_{SC}^{\ast}$ | 26,481 | 29,439 | 29,439 |

$T{P}_{Govt.}$ | − | −890 | −890 |

Centralized Model | Decentralized Model | |||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Parameters | Policy | ${\mathit{p}}_{\mathit{g}}^{\ast}$ | ${\mathit{p}}_{\mathit{n}}^{\ast}$ | ${\mathit{g}}^{\ast}$ | ${\mathit{TP}}_{{\mathit{M}}_{\mathit{g}}}^{\ast}$ | ${\mathit{TP}}_{{\mathit{M}}_{\mathit{n}}}^{\ast}$ | ${\mathit{TP}}_{\mathit{R}}^{\ast}$ | ${\mathit{TP}}_{\mathit{SC}}^{\ast}$ | ${\mathit{TP}}_{\mathit{Govt}.}$ | ${\mathit{p}}_{\mathit{g}}^{\ast}$ | ${\mathit{w}}_{\mathit{g}}^{\ast}$ | ${\mathit{p}}_{\mathit{n}}^{\ast}$ | ${\mathit{w}}_{\mathit{n}}^{\ast}$ | ${\mathit{g}}^{\ast}$ | ${\mathit{TP}}_{{\mathit{M}}_{\mathit{g}}}^{\ast}$ | ${\mathit{TP}}_{{\mathit{M}}_{\mathit{n}}}^{\ast}$ | ${\mathit{TP}}_{\mathit{R}}^{\ast}$ | ${\mathit{TP}}_{\mathit{SC}}^{\ast}$ | ${\mathit{TP}}_{\mathit{Govt}.}$ | |

$a=800$ | $b=4$, | Without Govt. Interference | 79.35 | 70.37 | 11.84 | 6539 | 6370 | 4373 | 17,282 | − | 99.40 | 72.23 | 95.02 | 64.72 | 5.07 | 3502 | 4944 | 5308 | 13,756 | − |

$c=0.8$ | With Govt. Interference | 91.81 | 69.10 | 35.23 | 6075 | 5843 | 6618 | 18,537 | −2925 | 106.83 | 76.55 | 86.95 | 51.98 | 14.90 | 3780 | 4277 | 6864 | 14,923 | −230 | |

$b=5$, | Without Govt. Interference | 66.08 | 57.72 | 8.66 | 6008 | 6212 | 141 | 12,363 | − | 80.86 | 60.80 | 76.88 | 53.58 | 3.69 | 2338 | 3720 | 3792 | 9851 | − | |

$c=1$ | With Govt. Interference | 76.30 | 56.25 | 30.14 | 6290 | 5598 | 1247 | 13,137 | −2227 | 86.93 | 64.34 | 69.52 | 42.40 | 12.70 | 2544 | 3096 | 5004 | 10,644 | −93 | |

$b=6$ | Without Govt. Interference | 57.35 | 49.30 | 6.56 | 5424 | 6055 | −2290 | 9189 | − | 68.57 | 53.23 | 64.080 | 46.15 | 2.78 | 1599 | 2912 | 2811 | 7323 | − | |

$c=1.2$ | With Govt. Interference | 65.98 | 47.71 | 26.29 | 6219 | 5359 | −1943 | 9635 | −1709 | 73.66 | 56.20 | 58.00 | 36.20 | 11.00 | 1757 | 2328 | 3766 | 7852 | 6.63 | |

$a=1200$ | $b=4$, | Without Govt. Interference | 111.82 | 101.86 | 19.64 | 10,636 | 9869 | 24,116 | 44,622 | − | 143.33 | 100.52 | 140.28 | 95.55 | 8.46 | 9768 | 12,028 | 13,725 | 35,522 | − |

$c=0.8$ | With Govt. Interference | 132.64 | 101.08 | 58.80 | 6731 | 9178 | 33,091 | 49,001 | −8996 | 157.63 | 107.60 | 128.85 | 72.64 | 24.83 | 10,501 | 10,651 | 18,153 | 39,305 | −1187 | |

$b=5$, | Without Govt. Interference | 91.85 | 82.87 | 14.84 | 10,312 | 9713 | 13,190 | 33,215 | − | 117.48 | 83.35 | 113.08 | 75.83 | 6.40 | 7013 | 9252 | 10,215 | 26,481 | − | |

$c=1$ | With Govt. Interference | 109.58 | 81.80 | 52.20 | 8086 | 8905 | 19,540 | 36,531 | −7624 | 127.88 | 89.32 | 102.53 | 57.97 | 21.95 | 7592 | 7917 | 13,930 | 29,439 | −890 | |

$b=6$ | Without Govt. Interference | 78.72 | 78.24 | 11.69 | 9828 | 9555 | 6366 | 25,750 | − | 99.01 | 71.97 | 94.95 | 64.69 | 5.03 | 5221 | 7409 | 7915 | 20,547 | − | |

$c=1.2$ | With Govt. Interference | 94.35 | 69.00 | 47.57 | 8800 | 8635 | 10,902 | 28,338 | −6656 | 108.10 | 77.16 | 85.09 | 48.35 | 19.81 | 5693 | 6114 | 11,109 | 22,917 | −688 | |

$a=1600$ | $b=4$, | Without Govt. Interference | 144.28 | 133.35 | 27.42 | 14,430 | 13,370 | 56,953 | 84,755 | − | 191.26 | 128.82 | 185.56 | 120.38 | 11.85 | 19,177 | 22,210 | 26,078 | 67,467 | − |

$c=0.8$ | With Govt. Interference | 173.46 | 133.07 | 82.37 | 4610 | 12,513 | 76,902 | 94,027 | −18,313 | 208.44 | 138.63 | 170.73 | 93.29 | 34.76 | 20,584 | 19,879 | 34,831 | 75,295 | −2761 | |

$b=5$, | Without Govt. Interference | 117.63 | 108.03 | 21.03 | 14,424 | 13,212 | 36,617 | 64,253 | − | 154.09 | 105.90 | 149.27 | 98.09 | 9.11 | 14,195 | 17,259 | 19,775 | 51,230 | − | |

$c=1$ | With Govt. Interference | 1435 | 107.35 | 74.27 | 7444 | 12,212 | 52,033 | 71,690 | −16,130 | 168.84 | 114.29 | 135.55 | 73.54 | 31.18 | 15,331 | 14,956 | 27,339 | 57,627 | −2264 | |

$b=6$ | Without Govt. Interference | 100.08 | 91.18 | 16.82 | 14,102 | 13,055 | 23,616 | 50,773 | − | 129.45 | 90.72 | 125.10 | 83.24 | 7.28 | 10,928 | 13,969 | 15,626 | 40,524 | − | |

$c=1.2$ | With Govt. Interference | 122.72 | 90.26 | 68.85 | 9118 | 11,912 | 35,940 | 56,971 | −14,680 | 142.53 | 98.13 | 112.18 | 60.52 | 28.62 | 11,879 | 11,686 | 22,341 | 45,907 | −1897 |

With Govt. Intf. | Centralized Model | Decentralized Model | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Parameters | ${\mathit{p}}_{\mathit{g}}^{\ast}$ | ${\mathit{p}}_{\mathit{n}}^{\ast}$ | ${\mathit{g}}^{\ast}$ | ${\mathit{TP}}_{{\mathit{M}}_{\mathit{g}}}^{\ast}$ | ${\mathit{TP}}_{{\mathit{M}}_{\mathit{n}}}^{\ast}$ | ${\mathit{TP}}_{\mathit{R}}^{\ast}$ | ${\mathit{TP}}_{\mathit{SC}}^{\ast}$ | ${\mathit{TP}}_{\mathit{Govt}.}$ | ${\mathit{p}}_{\mathit{g}}^{\ast}$ | ${\mathit{w}}_{\mathit{g}}^{\ast}$ | ${\mathit{p}}_{\mathit{n}}^{\ast}$ | ${\mathit{w}}_{\mathit{n}}^{\ast}$ | ${\mathit{g}}^{\ast}$ | ${\mathit{TP}}_{{\mathit{M}}_{\mathit{g}}}^{\ast}$ | ${\mathit{TP}}_{{\mathit{M}}_{\mathit{n}}}^{\ast}$ | ${\mathit{TP}}_{\mathit{R}}^{\ast}$ | ${\mathit{TP}}_{\mathit{SC}}^{\ast}$ | ${\mathit{TP}}_{\mathit{Govt}.}$ |

${G}_{s}=0.6$, ${G}_{t}=10$ | 118.22 | 79.16 | 66.40 | 5937 | 8140 | 23,075 | 37,153 | −12,694 | 129.83 | 88.70 | 82.38 | 23.53 | 24.65 | 7096 | 2159 | 20,933 | 30,189 | 359 |

${G}_{s}=0.4$, ${G}_{t}=8$ | 103.64 | 79.55 | 41.96 | 9261 | 8718 | 16,056 | 34,036 | −3104 | 123.78 | 85.94 | 93.53 | 44.16 | 17.90 | 7023 | 6095 | 15,609 | 28,728 | 670 |

${G}_{s}=0.2$, ${G}_{t}=8$ | 95.95 | 79.16 | 26.45 | 10,103 | 8920 | 12,996 | 32,016 | 633 | 119.52 | 83.72 | 97.89 | 53.20 | 11.82 | 6865 | 7169 | 13,196 | 27,231 | 1184 |

${G}_{s}=0.2$, ${G}_{t}=6$ | 95.94 | 80.16 | 26.40 | 10,161 | 9095 | 13,377 | 32,633 | 161 | 119.73 | 84.10 | 101.73 | 58.86 | 11.90 | 6964 | 7838 | 12,478 | 27,281 | 750 |

${G}_{s}=0$, ${G}_{t}=0$ | 91.85 | 82.87 | 14.84 | 10,312 | 9711 | 13,190 | 33,215 | 0 | 117.48 | 83.35 | 113.08 | 75.83 | 6.40 | 7013 | 9252 | 10,215 | 26,481 | 0 |

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**MDPI and ACS Style**

Barman, A.; Das, R.; De, P.K.; Sana, S.S.
Optimal Pricing and Greening Strategy in a Competitive Green Supply Chain: Impact of Government Subsidy and Tax Policy. *Sustainability* **2021**, *13*, 9178.
https://doi.org/10.3390/su13169178

**AMA Style**

Barman A, Das R, De PK, Sana SS.
Optimal Pricing and Greening Strategy in a Competitive Green Supply Chain: Impact of Government Subsidy and Tax Policy. *Sustainability*. 2021; 13(16):9178.
https://doi.org/10.3390/su13169178

**Chicago/Turabian Style**

Barman, Abhijit, Rubi Das, Pijus Kanti De, and Shib Sankar Sana.
2021. "Optimal Pricing and Greening Strategy in a Competitive Green Supply Chain: Impact of Government Subsidy and Tax Policy" *Sustainability* 13, no. 16: 9178.
https://doi.org/10.3390/su13169178