In this section, we first introduce the scenario in which the retailer is concerned about fairness but the manufacturer does not care about the fairness. Regarding the retailer’s fairness concern, we consider the effect of government subsidies on manufacturer’s and retailer’s strategies. Especially, we consider that there are two types of subsidies for the consumer [

13]: fixed subsidy and discount subsidy for unit product. Finally, we compare the optimal carbon emission reduction rate, wholesale price, and retail price in above different scenarios, and analyze the influence of fairness concern, CEA, and government subsidies on the manufacturer’s, retailer’s, and the whole supply chain’s profits.

#### 4.1. Fairness Concern

In this subsection, we consider that the retailer is concerned about fairness but the manufacturer does not care about fairness; in other words, the retailer maximizes her utility depending on the profits of both members. Similar to Du et al. (2010) [

12], we capture the fairness in the retailer’s objective function through the following utility function:

in which

$\lambda (0<\lambda <1)$ represents the retailer’s fairness concern coefficient.

The manufacturer’s utility function is still the profit function, because the manufacturer is not concerned about fairness. Then, the manufacturer’s utility function is as follows:

From Equation (8), we can see that the retailer’s utility function is determined by his profit and the profit difference between the manufacturer and the retailer. If the manufacturer makes much more money than that of the retailer in the supply chain, the retailer’s utility will be lower, even if he has earned much less money. Additionally, the larger the retailer’s fairness concern, the higher the retailer’s utility.

Similar to

Section 3.2, we assume the manufacturer is the leader and the retailer is the follower; then, we derive optimal solutions using the backward method.

**Theorem** **3.** The optimal carbon emission reduction rate is Compared with Theorem 1, the optimal carbon emission reduction rate decreases when the retailer cares about fairness; in other words, retailer’s fairness concern will result in decreasing environmental quality of the green products.

**Theorem** **4.** When the retailer is concern about fairness, then the optimal wholesale price and the optimal retail price are as follows: Above, Theorems 3 and 4 present the optimal strategies when the retailer cares about fairness. Let $\lambda =0$; this means the retailer is not concern about fairness, and the Formulas (9)–(11) are same as Formulas (4)–(6).

**Proposition** **3.** When the retailer is concerned about fairness, then the optimal wholesale price, retail price, manufacturer’s profit, retailer's profit, retailer’s utility, and supply chain’s utility increase with CEA.

From Proposition 3, we can see that the change trends of the optimal solutions with CEA when the retailer is concerned about the fairness are the same as those without fairness;

**Proposition** **4.** When the retailer cares about fairness, then the manufacturer's optimal carbon emission reduction rate of green products’ optimal wholesale price and retail price decrease with $\lambda $.

Proposition 4 shows that the retailer's fairness concern decreases the optimal strategies of both the manufacturer and the retailer, especially the carbon emission reduction rate. Therefore, when the retailer cares about the fairness, the manufacturer will lower carbon emission reduction rate. This is not good news for the public, because the lower carbon emission reduction rate, the lower environment quality. Next, we will discuss the profit change with regard to the fairness concern.

**Proposition** **5.** When the retailer cares about fairness, then

- (i)
When $\lambda <\frac{4b{c}_{e}-2{\tau}^{2}}{5{\tau}^{2}-8b{c}_{e}}$, then the retailer's profit increases with $\lambda $; and when $\frac{4b{c}_{e}-2{\tau}^{2}}{5{\tau}^{2}-8b{c}_{e}}<\lambda <\frac{4b{c}_{e}-{\tau}^{2}}{{\tau}^{2}-8b{c}_{e}}$, then the retailer's profit decreases with $\lambda $;

- (ii)
The optimal manufacturer’s profit decreases with $\lambda $.

When the retailer cares about fairness, Proposition 5 shows that the manufacturer’s profit decreases all the time; the retailer’s profit only increases in a certain zone. Once the level of the fairness concern exceeds a certain zone, the retailer’s profit decreases. In other words, the retailer’s profit first increases then decreases with $\lambda $.

Propositions 4 and 5 show that when the retailer cares about fairness, the retailer will remove the unfairness and increase his profit and utility in a certain zone. However, when the retailer pays excessive attention to fairness, both manufacturer’s and retailer’s profits decrease; in other words, the retailer’s behavior makes both members suffer loss, and the retailer not only cannot remove unfairness but also makes own their situation worse.

#### 4.2. Fairness Concern and Government Subsidy

With retailer’s fairness concern, in this subsection, we will discuss the effects of two government subsidies on the supply chain. Here we will consider two types subsidy: F-type subsidy (in which the consumer obtains a fixed subsidy $s$ when the consumer purchases one unit product) and D-type subsidy (in which the consumer obtains a discount subsidy that is related to the green level of the product when the consumer purchases one unit product). Firstly, we will discuss how the F-type subsidy impacts the optimal solutions such as the optimal wholesale price, retail price, each party’s profit, and so on. Then, we will discuss how the D-type subsidy affects supply chain members’ strategies. Finally, we’ll compare the optimal solutions with two subsidies and give the differences between them.

#### 4.2.1. F-Type Subsidy

Assume that the government provides fixed unit product subsidy

$s$ to the consumer when the consumer purchases green products. Then, the demand function is as follows:

Then, the retailer’s and manufacturer’s profit functions are as follows:

Similar to

Section 4.1, the retailer’s utility function with fairness concern is

The manufacturer’s utility function is as follows:

**Theorem** **5.** With F-type subsidy and retailer’s fairness concern, the optimal carbon emission reduction rate of the green product is Compared with Theorem 3, we can obtain that the optimal carbon emission reduction rate increases when it is subsidized.

**Theorem** **6.** With F-type subsidy and retailer’s fairness concern, the optimal wholesale price and retail price are as follows: Theorems 5 and 6 present the changes of optimal carbon emission reduction rate, the wholesale price and price with CEA, the fairness concern, and F-type subsidy.

**Proposition 6. **

- (i)
The optimal carbon emission reduction rate of the green product, wholesale price, and retail price increase with CEA;

- (ii)
The manufacturer’s and retailer’s profits increase with CEA.

Proposition 6 gives the change trends with CEA; the results are similar to Proposition 3, so the introduction of government subsidies does not influence the effects of CEA on each party’s strategies.

**Proposition 7. **

- (i)
The optimal carbon emission reduction rate, optimal wholesale price, and retail price decrease with $\lambda $;

- (ii)
The manufacturer’s profit decreases with $\lambda $;

- (iii)
the retailer’s profit increases with $\lambda $ when $\lambda <\frac{4b{c}_{e}-2{\tau}^{2}}{5{\tau}^{2}-8b{c}_{e}}$ and decreases when $\frac{4b{c}_{e}-2{\tau}^{2}}{5{\tau}^{2}-8b{c}_{e}}<\lambda <\frac{4b{c}_{e}-{\tau}^{2}}{{\tau}^{2}-8b{c}_{e}}$.

Notice that Propositions 5 and 7 have same results even there is F-type subsidy in Proposition 7. In other words, the F-type subsidy and the retailer’s fairness concern do not interact with each other.

**Proposition 8. **

- (i)
The optimal carbon emission reduction rate, wholesale price, and retail price increase with the fixed subsidy $s$;

- (ii)
The manufacturer’s and retailer’s profits increase with the fixed subsidy $s$.

Proposition 8 shows the impact of government subsidies on the optimal carbon emission reduction rate, wholesale price, retail price, and manufacturer’s and retailer’s profits. Obviously, the fixed subsidy $s$ makes all members of the supply chain better-off.

#### 4.2.2. D-Type Subsidy

In this subsection, we consider the scenario that the consumer will obtain a discount subsidy when he purchases green products. Assume that the subsidy is related to the environmental quality of green products, that is,

$s=\beta {e}_{4}$; the higher environmental quality of the green product, the higher subsidy, in which

$\beta $ and

${e}_{4}$ represent subsidy coefficient and carbon emission reduction rate of per unit of green product, respectively. Then, the demand function is as follows:

Then, the retailer’s and manufacturer’s profit functions are as follows:

The retailer’s and manufacturer’s utility functions with fairness concern are as follows:

**Assumption** **2.** Consumer environmental awareness satisfies
$\tau <2\sqrt{\frac{b{c}_{e}(2\lambda +1)}{\lambda +1}}-b\beta $.

Similar to Assumption 1, with the D-type subsidy, CEA is not larger than a threshold, and the threshold is determined not only by the cost coefficient and the demand sensitivity coefficient to price but also by the level of fairness concern and the subsidy coefficient.

**Theorem** **7.** With D-type subsidy and retailer’s fairness concern, the optimal carbon emission reduction rate of green products is as follows: Compared with Theorem 5, we can derive that ${e}_{4}>{e}_{3}$; that is, the optimal carbon emission reduction rate with D-type subsidy is much larger than that with F-type subsidy. In other words, the D-type subsidy will stimulate the manufacturer to produce much more environmentally-friendly products.

**Theorem** **8.** With D-type subsidy and retailer’s fairness concern, the optimal wholesale price and retail price are as follows: **Proposition 9. **

- (i)
The optimal carbon emission reduction rate of green products, wholesale price, and retail price increase with CEA;

- (ii)
The manufacturer’s and retailer’s profits increase with CEA.

**Proposition 10. **

- (i)
The optimal carbon emission reduction rate, optimal wholesale price, and retail price decrease with $\lambda $;

- (ii)
The manufacturer’s profit decreases with $\lambda $;

- (iii)
the retailer’s profit increases with $\lambda $ when $\lambda <\frac{4b{c}_{e}-2{(\tau +b\beta )}^{2}}{5{(\tau +b\beta )}^{2}-8b{c}_{e}}$ and decreases when $\frac{4b{c}_{e}-2{(\tau +b\beta )}^{2}}{5{(\tau +b\beta )}^{2}-8b{c}_{e}}<\lambda <\frac{4b{c}_{e}-{(\tau +b\beta )}^{2}}{{(\tau +b\beta )}^{2}-8b{c}_{e}}$.

Similar to Propositions 5 and 7, the retailer’s profit only increases with fairness concern when the level of the fairness concern is not larger than a threshold; however, differently from Propositions 5 and 7, the threshold in Proposition 10 is less than that in Propositions 5 and 7. In other words, with the D-type subsidy, the retailer’s profit will be more likely to decrease when the retailer is concerned about fairness. In other words, the D-type subsidy decreases the possibility that the retailer increases his profit by paying attention to fairness.

**Proposition 11. **

- (i)
The optimal carbon emission reduction rate, wholesale price, and retail price increase with the subsidy coefficient $\beta $;

- (ii)
The manufacturer’s and retailer’s profits increase with government subsidy coefficient $\beta $.

Proposition 11 shows the impact of D-type subsidy on the optimal carbon emission reduction rate, wholesale price, retail price, and manufacturer’s and retailer’s profits.

In summary, we present the equilibrium carbon emission reduction rates, wholesale prices, and prices of above four scenarios in

Table 3; in

Table 4, we give the retailer’s, manufacturer’s profits and retailer’s utility with the optimal solutions in

Table 3. We use N, F, F-F, and F-D to represent the benchmark model, the model with fairness concern, the model with fairness concern and F-type subsidy, and the model with fairness concern and D-type subsidy, respectively.