Optimal Green Operation and Information Leakage Decisions under Government Subsidy and Supply Uncertainty

Abstract

This study investigated optimal green operation and information leakage decisions in a green supply chain system. The system consists of one supplier, one leader retailer 1, one follower retailer 2, and the government. In this system, the government subsidizes each retailer based on the selling price of the product. The supplier is subject to a yield uncertainty process. The suppler decides whether to leak leader retailer 1′s order quantity to follower retailer 2 or not. In this study, we first built a Stackelberg game to address the equilibrium green operation decisions, when the supplier has and has not information leakage behavior, respectively. Subsequently, we identify the supplier’s information leakage equilibrium and how such behavior affects retailers’ ex ante profits, consumer surplus, and social welfare through a numerical study. Interestingly, we obtained the following results: (1) Supplier leaks are the unique equilibrium of the supplier. The product’s green degree and wholesale price at supplier’s equilibrium are higher under information leakage than under no information leakage. (2) The supplier’s information leakage behavior is good for leader retailer 1 and bad for follower retailer 2. (3) Information leakage behavior increases both consumer surplus and social welfare under certain conditions. (4) In general, key system parameters (e.g., the subsidy rate, supply uncertainty, supply correlation, and forecast accuracies) positively correlate with consumer surplus and social welfare in the same direction, while they affect retailer 1′s and retailer 2′s ex ante profit in the opposite direction. These findings provide useful insights for businesses to manage demand forecast information and make decisions on the green level of the product in green supply chain management.

1. Introduction

In the context of economic globalization, environmental issues have attracted the attention of both industry and the government. The concept of a green supply chain (GSC) refers to a supply chain (SC) that adds environmental components to its management practices. GSC’s actions include the R&D of green products, the recycling of used products, and green procurement. Green products have attracted the attention of GSC managers since they cause less environmental pollution during their total life cycle (http://www.thegreensupplychain.com/, accessed on 1 November 2021). For example, to incentivize green production, the Chinese government has subsidized the development and production of electric cars made by BYD Co., Ltd. (https://www.byd.com/cn/Investor/InvestorAnnals.html, accessed on November 2021). Yang et al. [1] and Bian et al. [2] examined how government subsidies can affect an industry’s green technology investment decisions.

The increasing economic integration and interdependence has intensified economic globalization and exacerbated supply and demand uncertainty. Demand variability, caused by volatility, ambiguity, and complexity, is a major challenge affecting the effectiveness of supply chain management. Firms can forecast future demand, using sales information. For example, BYD and BM can forecast the future demand for electric cars from their sales data. In this research, based on the market share, we denote retailer 1 (R1) as the leader and retailer 2 (R2) as the follower. After receiving R1′s order quantity, the supplier may leak R1′s order quantity to R2. This phenomenon is referred to as information leakage (Kong et al. [3]). Alternatively, supply variability may be caused by equipment failure, human error, as often seen in the agriculture, automobile, and vaccine industries (Zhou et al. [4]; Dai et al. [5]; Arifoğlu et al. [6]; Jung [7]). Uncertainties in the yield processes of a common industry are often positively correlated (Jung [7]). In their works, Liu et al. [8], Chen and Ozer [9], and Kong et al. [3] examined the issue of information leakage under different types of contract. However, these authors did not consider the impact of suppliers’ green investment behaviors on their own information leakage decisions. Thus, our research focus is different from the existing works and contribute to the practice and theory of the literature.

Government subsidy strategies can effectively encourage firms to invest in green products. For example, BYD Co., Ltd. (BYD) and BAIC Motor (BM) invested in electric cars, whose market share has greatly increased under the government’s sales subsidy strategy. This is because, given the incentive of a sales subsidy policy, electric car manufacturers can increase their sales volume by setting a lower selling price. In their studies, Huang et al. [10] and Ma et al. [11] developed game frameworks to explore the impact of subsidy strategies on green product investment decisions in supply chains. In this research, we further discuss the impact of sales subsidies on the information leakage decisions of upstream firms.

Motivated by the above business practices, this study explores the optimal green operation and information leakage decisions that can be made in a GSC system. This system consists of one supplier (referred to as “she”), two retailers (both referred to as “he”), and the government. The two retailers purchase products from a common supplier. Without a loss of generality, we consider retailer 1′s (R1) order first. After receiving R1′s order quantity, the supplier must decide whether to leak R1′s order quantity to R2 (referred to as strategy L) or not (referred to as strategy N). The supplier’s yield process is uncertain (supply uncertainty). The government subsidizes retailers to sell green products at the end of the sales season.

In particular, we aim to address the following three questions:

(1)

What are the optimal green operational decisions of the supplier and the two retailers with/without information leakage?

(2)

What is the supplier’s information leakage equilibrium?

(3)

How does a supplier’s information leakage behavior affect the two retailers’ ex ante payoffs, consumer surplus, and social welfare when the subsidy rate, supply uncertainty, supply correlation, and the two retailers’ forecast accuracies change?

To answer these questions, we formulate a Stackelberg game model to examine the optimal green operation and information leakage decisions that can be made under the conditions of government subsidies and supply uncertainty. We obtain the optimal green operation decisions and profits of GSC members and the corresponding total subsidies of the government, consumer surplus, and social welfare through backward induction.

The main findings of this work and the corresponding management insights are summarized as follows.

(1)

We provide the GSC members’ equilibrium green operational decisions to help suppliers make green product investment decisions as well as information leakage decisions. We find that compared with the suppliers without information leakage behavior, the supplier can set a higher green product level and a higher wholesale price, leading to higher profit when information leakage behavior exists.

(2)

We point out that information leak (strategy L) is the unique equilibrium state of the supplier. In general, information leakage benefits R1 but hurts R2. The key parameters (e.g., the subsidy rate, supply uncertainty, supply correlation, and forecast accuracies) affect retailer 1 and retailer 2′ ex ante profit in the opposite direction.

(3)

However, the supplier’s information leakage behavior increases both consumer surplus and social welfare levels. Under certain conditions, the subsidy rate, supply uncertainty, supply correlation, and forecast accuracies, respectively, affect consumer surplus and social welfare in the same direction.

The remainder of this paper is organized as follows: Section 2 summarizes the related literature. The proposed model is discussed in Section 3. The optimal green operational decisions are identified in Section 4. Section 5 presents the supplier’s information leakage equilibrium and describes how information leakage affects two retailers’ ex ante payoffs, consumer surplus, and social welfare. Finally, Section 6 contains the concluding remarks. Appendix A shows all our proofs.

2. Literature Review

This study is related to two streams of research: information leakage in an SC and government subsidies to GSCs.

2.1. Information Leakage in an SC

We focus on an SC with one upstream supplier and two downstream retailers. Li [12] and Zhang [13] built a three-stage game model to investigate the incentive of vertical information sharing and pointed out that vertical information sharing cannot occur freely when an upstream enterprise exhibits information leakage behavior. They also stated that vertical information sharing benefits upstream suppliers and hurts downstream retailers. Subsequently, Jain et al. [14], Qian et al. [15], and Wang et al. [16] built a Stackelberg game to investigate the information sharing equilibrium of downstream enterprises when upstream enterprises exhibit information leakage behavior. Anand and Goyal [17], Kong et al. [3], Liu et al. [8], and Chen and Ozer [9] explored upstream enterprises’ equilibrium information leakage decisions by considering the two retailers from the Stackelberg Cournot competition, who have asymmetric information. Shamir [18] pointed out that upstream enterprises leaking information can cause downstream competing enterprises to form a cartel. Lu et al. [19] and Huang et al. [20] built a Stackelberg game model to identify the equilibrium of information sharing from the perspectives of information errors and channel structures, respectively. In contrast, we build a Stackelberg game model to investigate information leakage in a GSC system with government subsidies and supply uncertainty.

2.2. Government Subsidies in a GSC

Researchers have studied government subsidy choices made in GSC management. As consumers’ passion for the environment increases, researchers believe a high degree of product greenness could lead to a high market demand. For example, Bigerna et al. [21], Zhang et al. [22], Song et al. [23], Chen et al. [24], Zhou and Yuen [25], and Shao et al. [26] developed a multi-stage game model framework to investigate government subsidy decisions under different operating environments and market structures. Additionally, Yu et al. [27] discussed the impact of green subsidies on firms’ production decisions, while Bian et al. [2] built a game model to compare two subsidies: subsidized manufacturers and subsidized consumers in a GSC. Others have focused on government subsidies under the condition of competition. Huang et al. [28] and Yang et al. [1] explored government subsidies under the condition of competition at both price and green service levels. Xu et al. [29] and Yu et al. [30] investigated government subsidies from a consumer welfare perspective through developing a game model with horizontal competition. Huang et al. [10] built a multi-stage game model to investigate government subsidy choices and GSC members’ pricing decisions in a capital-constrained GSC. Moreover, Yang et al. [1] and Meng et al. [31] explored the choice of subsidy strategy used in GSC management. However, they did not consider how the government’s subsidy policy would affect the supplier’s information leakage decisions. This study aims to fill this gap by examining the impact of sales subsidies on upstream firms’ information leakage decisions.

Table 1. Related literature vs. this paper.

Authors	Supply Uncertainty	Supply Correlation	Product Green Level	Government Subsidy	Information Leakage	Competition
This paper	√	√	√	√	√	√
Yang et al. [1]			√	√
Bian et al. [2]			√	√
Li [12]					√	√
Zhang [13]					√	√
Jain et al. [14]	√				√	√
Qian et al. [15]					√	√
Wang et al. [16]					√	√
Anand and Goyal [17]					√	√
Kong et al. [3]					√	√
Liu et al. [8]					√	√
Chen and Ozer [9]					√	√
Shamir [18]					√	√
Lu et al. [19]			√	√
Huang et al. [20]			√	√
Bigerna et al. [21]			√	√
Zhang et al. [22]			√	√
Song et al. [23]			√	√
Chen et al. [24]			√	√
Zhou and Yuen [25]			√	√
Shao et al. [26]			√	√
Yu et al. [27]			√	√
Li et al. [32]			√	√
Huang et al. [28]			√	√		√
Xu et al. [29]			√	√		√
Yu et al. [30]			√	√		√
Huang et al. [10]			√	√		√

shows how this research contributes to the literature by comparing it with other studies.

3. Model Descriptions

Our model considers a supplier, two retailers, and the government, with the two retailers purchasing from the same supplier. The supplier is subject to supply uncertainty caused by yield uncertainty. The market demand in the system is uncertain. The detailed GSC system structure and corresponding transaction processes are displayed in Figure 1.

3.1. Supply Uncertainty

The supplier is subject to an uncertain yield process. That is, if retailer i purchases $q_i$ from the supplier, they will only receive $y_i{q}_i$. Similar to Jung [7], we assume a random variable $y_i\in \left(0,1\right]$ and $y_i~N\left(\mu ,{\sigma }_y^2\right)$, where $i=1,2$. $y_1$ and $y_2$ are positively correlated, and we assume that $\rho =\frac{Cov\left(y_1{y}_2\right)}{\sqrt{VaR\left(y_1\right)VaR\left(y_2\right)}}$. The supplier decides on the wholesale price ($w$) and the green level of the product ($g$) simultaneously. The supplier’s cost structure consists of three parts: (1) per unit manufacturer cost ($c_1$), (2) per unit distribution cost ($c_2$), and (3) the investment cost of green technology ($\frac{1}{2}k{g}^2$). Retailers pay $w$ per unit of product received.

3.2. Demand Uncertainty

Similar to Li [12], Zhang [13], and Huang et al. [10], we model the inverse demand function as:

$$P=a+\theta +\lambda g-\left(Q_i+Q_j\right)$$

(1)

where parameter $P$ denotes the selling price of a green product. Parameters $a$ and $\lambda$ represent the potential market size and green preferences of consumers, respectively. $Q_i$ and $Q_j$ represent retailers i and j’s total sales quantities, respectively. The random variable $\theta$ represents the demand variability, and $\theta ~N\left(0,{\sigma }_{\theta }^2\right)$. Each retailer i has access to a private demand forecast signal (information) $X_i$, which is an unbiased estimator of $\theta$. In this case, $E\left[X_i|\theta \right]=\theta$. $X_i$ and $X_j$, which are conditionally dependent on $\theta$, are identically distributed.

Thus, we have:

$$E\left[\theta |X_i\right]=E\left[X_j|X_i\right]=\frac{t_i{\sigma }_{\theta }^2}{1+t_i{\sigma }_{\theta }^2}{X}_i;E\left[\theta |X_i,X_j\right]=\frac{t_i{\sigma }_{\theta }^2{X}_i+t_j{\sigma }_{\theta }^2{X}_j}{1+t_i{\sigma }_{\theta }^2+t_j{\sigma }_{\theta }^2};$$

$$E\left[X_i^2\right]=\frac{1}{t_i}+{\sigma }_{\theta }^2=\frac{1+t_i{\sigma }_{\theta }^2}{t_i}; E\left[X_i{X}_j\right]={\sigma }_{\theta }^2.$$

where $t_i=1/E\left[VaR\left[X_i|\theta \right]\right]$ describes retailer i’s demand forecasting ability.

3.3. Government Subsidy

The government offers a subsidy rate ($\tau$) per unit of product sold by the retailers. To gain more subsidies, each retailer must disclose its forecast information to the government. This means that given the available to sell quantity of $y_1{q}_1+y_2{q}_2$, the total subsidy of the government is:

$$G=E\left[P\left(y_1{q}_1+y_2{q}_2\right)|X_1,X_2\right]$$

(2)

3.4. Sequence of Events

The decision process of the supplier and two retailers is illustrated as follows:

• The supplier simultaneously decides on $w$ and $g$.
• R1 has access to forecast information $X_1$ and decides on $q_1$.
• The supplier receives $q_1$ and decides whether to leak $q_1$ to R2 (L) or not (N).
• R2 has access to forecast information $X_2$ and decides on $q_2$ based on $q_1$ (if the supplier leaks).
• The supplier receives retailer $q_2$. The supplier manufactures and distributes products.
• The two retailers sell all the available products from the supplier and gain a subsidy rate ($\tau$) per unit of products sold from the government.
• Both supply and demand variabilities are realized.

The game sequence of the GSC system is shown in Figure 2.

From Figure 2, we can see that the supplier is the leader in the vertical Stackelberg game, and the corresponding two retailers are the followers. In addition, when the supplier leaks retailer 1′s order quantity to retailer 2, the two retailers form a horizontal Stackelberg game with retailer 1 as the leader and retailer 2 as the follower.

The parameters and variables used in this research are summarized in

Table 2. Parameters and variables.

Variables	Descriptions
$\theta$	Demand variability and $\theta ~N\left(0,{\sigma }_{\theta }^2\right)$ (a random variable)
$X_1$,$X_2$	Demand forecast information (signals)
$q_i^Z$	Retailer i’s order quantity under strategy$Z$; $Z=L, N$(a decision variable)
$w^Z$, $g^Z$	Supplier’s wholesale price and green level of product under strategy $Z$ (decision variables)
$y_i$	Yield variability factor, $y_i~N\left(\mu ,{\sigma }_y^2\right)$, i = 1,2 (a random variable)
${\Pi }_S^Z$, ${\Pi }_{R_i}^Z$	The ex ante payoffs of supplier and retailer i under strategy$Z$
$C{S}^Z$, $S{W}^Z$	Consumer surplus and social welfare under strategy$Z$
Parameters	Descriptions
${\delta }_y$	Supply uncertainty, and ${\delta }_y\in \left(0,\infty \right)$
$\rho$	Supply correlation, and $\rho \in \left(0,1\right)$
$\tau$	Subsidy rate
$\lambda$	Green preference of the consumer
$k$	The investment cost coefficient of green technology
$a$	Potential market size
$t_1$,$t_2$	Forecast accuracy
Indices	Descriptions
Subscript	$S$, $R_i$, and $R_j$ represent supplier, retailer i$,$ and retailer j, respectively
Superscript	L (or N) means supplier leaks (or does not leak) R1′s order quantity to R2; * denotes that the value is “optimal”

.

4. Optimal Green Operation Decisions

We now show the optimal green operation decisions and profits of the GSC members and the corresponding total subsidy of the government, consumer surplus, and social welfare.

4.1. Optimal Green Operation Decisions under No Information Leakage

In this research, we assume that retailers only have to pay for the inventory they ordered and received and that the retailers’ revenues are composed of two parts: product sales revenue and government subsidies. In strategy N, there is no information leakage, and each retailer $i$ makes an inference regarding $\theta$ using $E\left[\theta |X_i\right]$. Each retailer $i$ makes his own order quantity decision based on his own forecast information $X_i$.

Thus, the two retailer 1′ optimization problems can be expressed as follows:

$$\underset{q_1^N}{\max }{\pi }_{R_1}^N=E\left\{P^N{y}_1{q}_1^N+\tau P^N{y}_1{q}_1^N-w^N{y}_1{q}_1^N|X_1\right\}$$

(3)

where $P^N=a+\theta +\lambda g^N-\left(y_1{q}_1^N+y_2{q}_2^N\right)$ denotes the green product’s unit selling price. $P^N{y}_1{q}_1^N$ denotes retailer 1′s total sales revenue. $\tau P^N{y}_1{q}_1^N$ denotes the subsidies that retailer 1 receives from government. $w^N{y}_1{q}_1^N$ denotes retailer 1′s total purchase cost that retailer 1 pays to his supplier.

The two retailer 2′ optimization problems can be expressed as follows:

$$\underset{q_2^N}{\max }{\pi }_{R_2}^N=E\left\{P^N{y}_2{q}_2^N+\tau P^N{y}_2{q}_2^N-w^N{y}_2{q}_2^N|X_2\right\}$$

(4)

where $P^N{y}_2{q}_2^N$ denotes retailer 2′s total sales revenue. $\tau P^N{y}_2{q}_2^N$ denotes the subsidies that retailer 2 obtains from government. $w^N{y}_2{q}_2^N$ denotes retailer 2′s total purchase cost that retailer 2 pays to his supplier.

Accordingly, the supplier’s optimization problem is:

$$\underset{w^N, g^N }{\max }{\pi }_S^N=E\left\{w^N\left(y_1{q}_1^N+y_2{q}_2^N\right)-c_1\left(q_1^N+q_2^N\right)-c_2\left(y_1{q}_1^N+y_2{q}_2^N\right)-\frac{1}{2}k{\left(g^N\right)}^2\right\}$$

(5)

where $w^N\left(y_1{q}_1^N+y_2{q}_2^N\right)$ denotes the amount that the supplier receives from retailer 1 and retailer 2. $c_1\left(q_1^N+q_2^N\right)$ denotes the supplier’s total manufacture cost. $c_2\left(y_1{q}_1^N+y_2{q}_2^N\right)$ denotes the supplier’s distribution cost, while $\frac{1}{2}k{\left(g^N\right)}^2$ denotes the total investment cost of green technology.

Solving problems (3)–(5) by backward induction, we obtain the GSC members’ equilibrium decisions under strategy N in Proposition 1 and summarize them below.

Proposition 1.

Given$k>\frac{{\lambda }^2\left(1+\tau \right)}{3+\left(2+\rho \right){\delta }_y^2}$, for strategy N, we have:

(1)

The supplier’s equilibrium decisions are:

$$w^{N\ast }=\frac{k\left[3+\left(2+\rho \right){\delta }_y^2\right]\left[\left(1+\tau \right)a+\left(\frac{\overline{c}}{{\mu }_y}\right)\right]-2\left(1+\tau \right){\lambda }^2\left(\frac{\overline{c}}{{\mu }_y}\right)}{2\left\{k\left[3+\left(2+\rho \right){\delta }_y^2\right]-\left(1+\tau \right){\lambda }^2\right\}}$$

(6)

$$g^{N\ast }=\frac{\lambda \left[\left(1+\tau \right)a-\left(\frac{\overline{c}}{{\mu }_y}\right)\right]}{k\left[3+\left(2+\rho \right){\delta }_y^2\right]-{\lambda }^2\left(1+\tau \right)}$$

(7)

(2)

The two retailers’ equilibrium decisions are:

$$q_1^{N\ast }={\overline{q}}_1^{N\ast }+f_{11}^{N\ast }{X}_1$$

(8)

$$q_2^{N\ast }={\overline{q}}_2^{N\ast }+f_{22}^{N\ast }{X}_2$$

(9)

$${\overline{q}}_1^{N\ast }={\overline{q}}_2^{N\ast }=\frac{k\left[\left(1+\tau \right)a-\left(\frac{\overline{c}}{{\mu }_y}\right)\right]}{2{\mu }_y\left(1+\tau \right)\left\{k\left[3+\left(2+\rho \right){\delta }_y^2\right]-{\lambda }^2\left(1+\tau \right)\right\}}$$

(10)

$$f_{11}^{N\ast }=\frac{\left\{2\left(1+{\delta }_y^2\right)+\left[1+\left(2-\rho \right){\delta }_y^2\right]t_2{\sigma }_{\theta }^2\right\}{t}_1{\sigma }_{\theta }^2}{{\mu }_y\left[4{\left(1+{\delta }_y^2\right)}^2\left(1+t_1{\sigma }_{\theta }^2\right)\left(1+t_2{\sigma }_{\theta }^2\right)-{\left(1+\rho {\delta }_y^2\right)}^2{t}_1{t}_2{\sigma }_{\theta }^4\right]}$$

(11)

$$f_{22}^{N\ast }=\frac{\left\{2\left(1+{\delta }_y^2\right)+\left[1+\left(2-\rho \right){\delta }_y^2\right]t_1{\sigma }_{\theta }^2\right\}{t}_2{\sigma }_{\theta }^2}{{\mu }_y\left[4{\left(1+{\delta }_y^2\right)}^2\left(1+t_1{\sigma }_{\theta }^2\right)\left(1+t_2{\sigma }_{\theta }^2\right)-{\left(1+\rho {\delta }_y^2\right)}^2{t}_1{t}_2{\sigma }_{\theta }^4\right]}$$

(12)

Part (1) of Proposition 1 shows that the supplier’s optimal wholesale price ($w^{N\ast }$) and level of product greenness ($g^{N\ast }$) are dependent on the subsidy rate ($\tau$), supply uncertainty (${\delta }_y$), supply correlation ($\rho$), green preference of the consumer ($\lambda$), potential market size ($a$), and their own cost parameters ($k$ and $\overline{c}$). Meanwhile, $w^{N\ast }$ and $g^{N\ast }$ are independent of the retailers’ forecast accuracies ($t_1$ and $t_2$). This suggests that the supplier makes optimal decisions concerning the wholesale price and the level of product greenness and does not need to consider retailers’ forecast accuracies ($t_1$ and $t_2$).

Part (2) of Proposition 1 shows that retailer 1′s (retailer 2′s) optimal order quantity is linearly related to their demand forecast information $X_1$ ($X_2$). This is because each retailer has their own demand forecast information for the demand uncertainty factor $\theta$. The retailers make order quantity decisions based on their own forecast information. By observing $f_{11}^{N\ast }$ in Equation (11) and $f_{22}^{N\ast }$ in Equation (12), we find that $f_{11}^{N\ast }>0$ and $f_{22}^{N\ast }>0$. $f_{11}^{N\ast }>0$ ($f_{22}^{N\ast }>0$) indicates that retailer 1′s (retailer 2′s) optimal order quantity $q_1^{N\ast }$ ($q_2^{N\ast }$) increases with $X_1$ ($X_2$). This suggests that each retailer should provide a positive response to their own forecast information.

Proposition 1 states the equilibrium decisions of the GSC members. This can help to guide the supplier in making investment decisions concerning the R&D of green products. It also helps retailers make suitable order quantity decisions, given government subsidies and supply uncertainty.

We can obtain the next proposition based on Proposition 1.

Proposition 2.

Given $k>\frac{{\lambda }^2\left(1+\tau \right)}{3+\left(2+\rho \right){\delta }_y^2}$, for strategy N, we have:

(1)

The supplier’s ex ante payoff is:

$${\Pi }_S^{N\ast }=\frac{k{\left[\left(1+\tau \right)a-\left(\frac{\overline{c}}{{\mu }_y}\right)\right]}^2}{2\left(1+\tau \right)\left\{k\left[3+\left(2+\rho \right){\delta }_y^2\right]-{\lambda }^2\left(1+\tau \right)\right\}}$$

(13)

(2)

The retailers’ ex ante payoffs are:

$${\Pi }_{R_1}^{N\ast } = \frac{k^2\left(1+{\delta }_y^2\right){\left[\left(1+\tau \right)a-\left(\frac{\overline{c}}{{\mu }_y}\right)\right]}^2}{4\left(1+\tau \right){\left\{k\left[3+\left(2+\rho \right){\delta }_y^2\right]-{\lambda }^2\left(1+\tau \right)\right\}}^2} + \frac{\left(1+{\delta }_y^2\right)\left(1+\tau \right){\left[2\left(1+{\delta }_y^2\right)\left(1+t_2{\sigma }_{\theta }^2\right)-\left(1+\rho {\delta }_y^2\right)t_2{\sigma }_{\theta }^2\right]}^2{t}_1\left(1+t_1{\sigma }_{\theta }^2\right){\sigma }_{\theta }^4}{{\left[4{\left(1+{\delta }_y^2\right)}^2\left(1+t_1{\sigma }_{\theta }^2\right)\left(1+t_2{\sigma }_{\theta }^2\right)-{\left(1+\rho {\delta }_y^2\right)}^2{t}_1{t}_2{\sigma }_{\theta }^4\right]}^2}$$

(14)

$${\Pi }_{R_2}^{N\ast } = \frac{k^2\left(1+{\delta }_y^2\right){\left[\left(1+\tau \right)a-\left(\frac{\overline{c}}{{\mu }_y}\right)\right]}^2}{4\left(1+\tau \right){\left\{k\left[3+\left(2+\rho \right){\delta }_y^2\right]-{\lambda }^2\left(1+\tau \right)\right\}}^2} + \frac{\left(1+{\delta }_y^2\right)\left(1+\tau \right){\left[2\left(1+{\delta }_y^2\right)\left(1+t_1{\sigma }_{\theta }^2\right)-\left(1+\rho {\delta }_y^2\right)t_1{\sigma }_{\theta }^2\right]}^2{t}_2\left(1+t_2{\sigma }_{\theta }^2\right){\sigma }_{\theta }^4}{{\left[4{\left(1+{\delta }_y^2\right)}^2\left(1+t_1{\sigma }_{\theta }^2\right)\left(1+t_2{\sigma }_{\theta }^2\right)-{\left(1+\rho {\delta }_y^2\right)}^2{t}_1{t}_2{\sigma }_{\theta }^4\right]}^2}$$

(15)

(3)

Total government subsidies are:

$$G^{N\ast }=\left\{\begin{array}{c}\left(a+\lambda g^N\right){\mu }_y\left({\overline{q}}_1^{N\ast }+{\overline{q}}_2^{N\ast }\right)\tau +\frac{{\mu }_y{f}_{11}^{N\ast }\left(1+t_1{\sigma }_{\theta }^2\right){\sigma }_{\theta }^2\tau }{1+t_1{\sigma }_{\theta }^2+t_2{\sigma }_{\theta }^2}+\frac{{\mu }_y{f}_{22}^{N\ast }{t}_1{\sigma }_{\theta }^4\tau }{1+t_1{\sigma }_{\theta }^2+t_2{\sigma }_{\theta }^2}\hfill \\ +\frac{{\mu }_y{f}_{11}^{N\ast }{t}_2{\sigma }_{\theta }^4\tau }{1+t_1{\sigma }_{\theta }^2+t_2{\sigma }_{\theta }^2}+\frac{{\mu }_y{f}_{22}^{N\ast }\left(1+t_2{\sigma }_{\theta }^2\right){\sigma }_{\theta }^2\tau }{1+t_1{\sigma }_{\theta }^2+t_2{\sigma }_{\theta }^2}-{\mu }_y^2\left(1+{\delta }_y^2\right)\left({\left({\overline{q}}_1^{N\ast }\right)}^2+{\left(f_{11}^{N\ast }\right)}^2\frac{1+t_1{\sigma }_{\theta }^2}{t_1}\right)\tau \hfill \\ -2{\mu }_y^2\left(1+\rho {\delta }_y^2\right)\left({\overline{q}}_2^{N\ast }{\overline{q}}_1^{N\ast }+f_{22}^{N\ast }{f}_{11}^{N\ast }{\sigma }_{\theta }^2\right)\tau -{\mu }_y^2\left(1+{\delta }_y^2\right)\left({\left({\overline{q}}_2^{N\ast }\right)}^2+{\left(f_{22}^{N\ast }\right)}^2\frac{1+t_2{\sigma }_{\theta }^2}{t_2}\right)\tau \hfill \end{array}\right\}$$

(16)

(4)

Consumer surplus and social welfare are:

$$C{S}^{N\ast }=\frac{1}{2}E\left[{\left(y_1{q}_1^{N\ast }+y_2{q}_1^{N\ast }\right)}^2\right]$$

$$=\frac{{\mu }_y^2}{2}\left\{\begin{array}{c}\left(1+{\delta }_y^2\right){\left({\overline{q}}_1^{N\ast }\right)}^2+\left(1+{\delta }_y^2\right){\left({\overline{q}}_2^{N\ast }\right)}^2+2\left(1+\rho {\delta }_y^2\right){\overline{q}}_2^{N\ast }{\overline{q}}_1^{N\ast }\\ +2\left(1+\rho {\delta }_y^2\right)f_{22}^{N\ast }{f}_{11}^{N\ast }{\sigma }_{\theta }^2+\left(1+{\delta }_y^2\right){\left(f_{11}^{N\ast }\right)}^2\frac{1+t_1{\sigma }_{\theta }^2}{t_1}\\ +\left(1+{\delta }_y^2\right){\left(f_{22}^{N\ast }\right)}^2\frac{1+t_2{\sigma }_{\theta }^2}{t_2}\end{array}\right\}$$

(17)

$$S{W}^{N\ast }={\Pi }_S^{N\ast }+{\Pi }_{R_1}^{N\ast }+{\Pi }_{R_2}^{N\ast }+C{S}^{N\ast }-G^{N\ast }$$

(18)

Proposition 2 describes the ex ante payoffs of GSC members, the total subsidy of the government, consumer surplus, and social welfare under equilibrium. Part (1) indicates that the supplier’s ex ante payoff (${\Pi }_S^{N\ast }$) is independent of the retailers’ forecast accuracies ($t_1$ and $t_2$). Part (2) indicates that the retailers’ ex ante payoffs (${\Pi }_{R_1}^{N\ast }$ and ${\Pi }_{R_2}^{N\ast }$) depend on the forecast accuracies ($t_1$ and $t_2$). Parts (3) and (4) show that the total government subsidies ($G^{N\ast }$), consumer surplus ($C{S}^{N\ast }$), and social welfare ($S{W}^{N\ast }$) also depend on $t_1$ and $t_2$.

4.2. Optimal Green Operation Decisions under Information Leakage

Because R1 is the leader, R1 makes order quantity decisions after gaining access to their own forecast information $X_1$. Similar to Section 4.1, R1′s optimization problem is:

$$\underset{q_1^L}{\max }{\pi }_{R_1}^L=E\left\{P^L{y}_1{q}_1^L+\tau P^L{y}_1{q}_1^L-w^L{y}_1{q}_1^L|X_1\right\}$$

(19)

where $P^L=a+\theta +\lambda g^L-\left(y_1{q}_1^L+y_2{q}_2^L\right)$ denotes the green product’s unit selling price under strategy L (i.e., the supplier without information leakage behavior). Correspondingly, retailer 1′s total sales revenue is $P^L{y}_1{q}_1^L$. The subsidies that retailer 1 obtains from the government are $\tau P^L{y}_1{q}_1^L$. Retailer 1′s total purchase cost that is paid to his supplier is $w^L{y}_1{q}_1^L$.

For Strategy L, R2 can obtain R1′s actual order quantity ($q_1^{L\ast }$) from the supplier and infer the value of $X_1$ from $q_1^{L\ast }$. Thus, R2′s optimization problem is expressed as follows:

$$\underset{q_2^L}{\max }{\pi }_{R_2}^L=E\left\{\left[P^L{y}_2{q}_2^L+\tau P^L{y}_2{q}_2^L-w^L{y}_2{q}_2^L\right]|X_1,X_2\right\}$$

(20)

where $P^L{y}_2{q}_2^L$, $\tau P^L{y}_2{q}_2^L$, and $w^L{y}_2{q}_2^L$ respectively denote retailer 2′s total sales revenue, the subsidies revenue, and the total purchase cost.

Accordingly, the supplier’s optimization problem is:

$$\underset{w^L, g^L }{\max }{\pi }_S^L=E\left\{w^L\left(y_1{q}_1^L+y_2{q}_2^L\right)-c_1\left(q_1^L+q_2^L\right)-c_2\left(y_1{q}_1^L+y_2{q}_2^L\right)-\frac{1}{2}k{\left(g^L\right)}^2\right\}$$

(21)

where $w^L\left(y_1{q}_1^L+y_2{q}_2^L\right)$ denotes the supplier’s total revenue. $c_1\left(q_1^L+q_2^L\right)$, $c_2\left(y_1{q}_1^L+y_2{q}_2^L\right)$, and $\frac{1}{2}k{\left(g^L\right)}^2$ respectively denote the supplier’s total manufacture cost, total distribute cost, and the total investment cost of green technology.

Solving problems (19)–(21) through backward induction, we can obtain the GSC members’ equilibrium decisions and summarize them in Proposition 3 below:

Proposition 3.

Given $k>\frac{\left[2H+\left(1+\rho {\delta }_y^2\right)\right]{\lambda }^2\left(1+\tau \right)}{8\left(1+{\delta }_y^2\right)F}$, for strategy L, we have:

(1)

The supplier’s equilibrium decisions are:

$$w^{L\ast }=\frac{4k\left(1+{\delta }_y^2\right)F\left[\left(1+\tau \right)a+\left(\frac{\overline{c}}{{\mu }_y}\right)\right]-\left(1+\tau \right){\lambda }^2\left[2H+\left(1+\rho {\delta }_y^2\right)\right]\left(\frac{\overline{c}}{{\mu }_y}\right)}{8k\left(1+{\delta }_y^2\right)F-\left(1+\tau \right){\lambda }^2\left[2H+\left(1+\rho {\delta }_y^2\right)\right]}$$

(22)

$$g^{L\ast }=\frac{\lambda \left[2H+\left(1+\rho {\delta }_y^2\right)\right]\left[\left(1+\tau \right)a-\left(\frac{\overline{c}}{{\mu }_y}\right)\right]}{8k\left(1+{\delta }_y^2\right)F-\left(1+\tau \right){\lambda }^2\left[2H+\left(1+\rho {\delta }_y^2\right)\right]}$$

(23)

where $F=2{\left(1+{\delta }_y^2\right)}^2-{\left(1+\rho {\delta }_y^2\right)}^2$; $H=4{\left(1+{\delta }_y^2\right)}^2-2\left(1+{\delta }_y^2\right)\left(1+\rho {\delta }_y^2\right)-{\left(1+\rho {\delta }_y^2\right)}^2$.

(2)

The equilibrium decisions of the two retailers are:

$$q_1^{L\ast }={\overline{q}}_1^{L\ast }+f_{11}^{L\ast }{X}_1$$

(24)

$$q_2^{L\ast }={\overline{q}}_2^{L\ast }+f_{21}^{L\ast }{X}_1+f_{22}^{L\ast }{X}_2$$

(25)

where:

$${\overline{q}}_1^{L\ast }=\frac{2k\left(1+{\sigma }_y^2\right)\left[1+\left(2-\rho \right){\delta }_y^2\right]\left[\left(1+\tau \right)a-\left(\frac{\overline{c}}{{\mu }_y}\right)\right]}{{\mu }_y\left(1+\tau \right)\left\{8k\left(1+{\delta }_y^2\right)F-\left(1+\tau \right)\left[2H+\left(1+\rho {\delta }_y^2\right)\right]{\lambda }^2\right\}}$$

(26)

$${\overline{q}}_2^{L\ast }=\frac{kH\left[\left(1+\tau \right)a-\left(\frac{\overline{c}}{{\mu }_y}\right)\right]}{{\mu }_y\left(1+\tau \right)\left\{8k\left(1+{\delta }_y^2\right)F-\left(1+\tau \right)\left[2H+\left(1+\rho {\delta }_y^2\right)\right]{\lambda }^2\right\}}$$

(27)

$$f_{11}^{L\ast }=\frac{\left[1+\left(2-\rho \right){\delta }_y^2\right]t_1{\sigma }_{\theta }^2}{2{\mu }_y\left(1+t_1{\sigma }_{\theta }^2\right)F}$$

(28)

$$f_{21}^{L\ast }=\frac{\left\{2F\left(1+t_1{\sigma }_{\theta }^2\right)-\left[2\left(1+{\delta }_y^2\right)\left(1+\rho {\delta }_y^2\right)-{\left(1+\rho {\delta }_y^2\right)}^2\right]\left(1+t_1{\sigma }_{\theta }^2+t_2{\sigma }_{\theta }^2\right)\right\}{t}_1{\sigma }_{\theta }^2}{4{\mu }_y\left(1+{\delta }_y^2\right)F\left(1+t_1{\sigma }_{\theta }^2\right)\left(1+t_1{\sigma }_{\theta }^2+t_2{\sigma }_{\theta }^2\right)}$$

(29)

$$f_{22}^{L\ast }=\frac{t_2{\sigma }_{\theta }^2}{2{\mu }_y\left(1+{\delta }_y^2\right)\left(1+t_1{\sigma }_{\theta }^2+t_2{\sigma }_{\theta }^2\right)}$$

(30)

Part (1) of Proposition 3 shows the supplier’s optimal wholesale price and the level of product greenness. It is similar to Part (1) of Proposition 1, hence will not be discussed here.

Part (2) of Proposition 3 shows that retailer 1′s optimal order quantity ($q_1^{L\ast }$) is linearly related to their own forecast information ($X_1$). Retailer 2′s optimal order quantity ($q_2^{L\ast }$) is linearly related to their own forecast information ($X_2$) and linearly related to competitor retailer 1′s forecast information ($X_1$). It can be seen that $f_{11}^{L\ast }>0$ in Equation (28) and $f_{22}^{L\ast }>0$ in Equation (30), indicating that each retailer responds positively to their own forecast information. The inequality $f_{21}^{L\ast }>0$ in Equation (29) holds if $\frac{2F}{2\left(1+{\delta }_y^2\right)\left(1+\rho {\delta }_y^2\right)-{\left(1+\rho {\delta }_y^2\right)}^2}>\frac{1+t_1{\sigma }_{\theta }^2+t_2{\sigma }_{\theta }^2}{1+t_1{\sigma }_{\theta }^2}$. This means that R2 responds positively to their competitor R1′s forecast information only under certain conditions.

We can obtain the next proposition based on Proposition 3:

Proposition 4.

Given $k>\frac{\left[2H+\left(1+\rho {\delta }_y^2\right)\right]{\lambda }^2\left(1+\tau \right)}{8\left(1+{\delta }_y^2\right)F}$, for strategy L, we have:

(1)

The supplier’s ex ante profit is:

$${\Pi }_S^{L\ast }=\frac{k\left[2H+\left(1+\rho {\delta }_y^2\right)\right]{\left[\left(1+\tau \right)a-\left(\frac{\overline{c}}{{\mu }_y}\right)\right]}^2}{2\left(1+\tau \right)\left\{8k\left(1+{\delta }_y^2\right)F-\left(1+\tau \right){\lambda }^2\left[2H+\left(1+\rho {\delta }_y^2\right)\right]\right\}}$$

(31)

(2)

The retailers’ ex ante profits are:

$${\Pi }_{R_1}^L=\frac{2\left(1+\tau \right)k^2\left(1+{\delta }_y^2\right)F{\left[1+\left(2-\rho \right){\delta }_y^2\right]}^2{\left[\left(1+\tau \right)a-\left(\frac{\overline{c}}{{\mu }_y}\right)\right]}^2}{{\left\{8k\left(1+\tau \right)\left(1+{\delta }_y^2\right)F-{\left(1+\tau \right)}^2{\lambda }^2\left[2H+\left(1+\rho {\delta }_y^2\right)\right]\right\}}^2}+\frac{\left(1+\tau \right){\left[1+\left(2-\rho \right){\delta }_y^2\right]}^2{t}_1{\sigma }_{\theta }^4}{8\left(1+{\delta }_y^2\right)F\left(1+t_1{\sigma }_{\theta }^2\right)}$$

(32)

$${\Pi }_{R_2}^{L\ast }=\left\{\begin{array}{c}\frac{\left(1+\tau \right)k^2\left(1+{\delta }_y^2\right)H^2{\left[\left(1+\tau \right)a-\left(\frac{\overline{c}}{{\mu }_y}\right)\right]}^2}{{\left\{8k\left(1+\tau \right)\left(1+{\delta }_y^2\right)F-{\left(1+\tau \right)}^2\left(2H+\left(1+\rho {\delta }_y^2\right)\right){\lambda }^2\right\}}^2}+\frac{\left(1+\tau \right)\left(t_1+t_2\right){\sigma }_{\theta }^4}{4\left(1+{\sigma }_y^2\right)\left(1+t_1{\sigma }_{\theta }^2+t_2{\sigma }_{\theta }^2\right)}\hfill \\ -\frac{\left(1+\tau \right)\left(H+2F\right)\left[1+\left(2-\rho \right){\delta }_y^2\right]\left(1+\rho {\delta }_y^2\right)t_1{\sigma }_{\theta }^4}{16\left(1+{\delta }_y^2\right)F^2\left(1+t_1{\sigma }_{\theta }^2\right)}\hfill \end{array}\right\}$$

(33)

(3)

The total subsidies from the government are:

$$G^L=\left\{\begin{array}{c}{\mu }_y\left(a+\lambda g^{L\ast }\right)\left({\overline{q}}_1^{L\ast }+{\overline{q}}_2^{L\ast }\right)\tau +\frac{{\mu }_y{t}_1{\sigma }_{\theta }^2\tau }{1+t_1{\sigma }_{\theta }^2+t_2{\sigma }_{\theta }^2}\left[\left(f_{11}^{L\ast }+f_{21}^{L\ast }\right)\frac{1+t_1{\sigma }_{\theta }^2}{t_1}+f_{22}^{L\ast }{\sigma }_{\theta }^2\right]\hfill \\ +\frac{{\mu }_y{t}_2{\sigma }_{\theta }^2\tau }{1+t_1{\sigma }_{\theta }^2+t_2{\sigma }_{\theta }^2}\left[\left(f_{11}^{L\ast }+f_{21}^{L\ast }\right){\sigma }_{\theta }^2+f_{22}^{L\ast }\frac{1+t_2{\sigma }_{\theta }^2}{t_2}\right]-{\mu }_y^2\tau \left(1+{\delta }_y^2\right)\left({\left({\overline{q}}_1^{L\ast }\right)}^2+{\left(f_{11}^{L\ast }\right)}^2\frac{1+t_1{\sigma }_{\theta }^2}{t_1}\right)\hfill \\ -\tau {\mu }_y^2\left(1+{\delta }_y^2\right)\left({\left({\overline{q}}_2^{L\ast }\right)}^2+{\left(f_{21}^{L\ast }\right)}^2\frac{1+t_1{\sigma }_{\theta }^2}{t_1}+2f_{21}^{L\ast }{f}_{22}^{L\ast }{\sigma }_{\theta }^2+{\left(f_{22}^{L\ast }\right)}^2\frac{1+t_2{\sigma }_{\theta }^2}{t_2}\right)\hfill \\ -2\tau {\mu }_y^2\left(1+\rho {\delta }_y^2\right)\left({\overline{q}}_1^{L\ast }{\overline{q}}_2^{L\ast }+f_{11}^{L\ast }{f}_{21}^{L\ast }\frac{1+t_1{\sigma }_{\theta }^2}{t_1}+f_{11}^{L\ast }{f}_{22}^{L\ast }{\sigma }_{\theta }^2\right)\hfill \end{array}\right\}$$

(34)

(4)

The consumer surplus and social welfare are:

$$C{S}^{L\ast }=\frac{1}{2}E\left[{\left(y_1{q}_1^{L\ast }+y_2{q}_1^{L\ast }\right)}^2\right]$$

$$=\frac{{\mu }_y^2}{2}\left\{\begin{array}{c}\left(1+{\delta }_y^2\right){\left({\overline{q}}_1^{L\ast }\right)}^2+2\left(1+\rho {\delta }_y^2\right){\overline{q}}_2^{L\ast }{\overline{q}}_1^{L\ast }+\left(1+{\delta }_y^2\right){\left({\overline{q}}_2^{L\ast }\right)}^2\\ +2\left(\left(1+\rho {\delta }_y^2\right)f_{22}^{L\ast }{f}_{11}^{L\ast }+\left(1+{\delta }_y^2\right)f_{22}^{L\ast }{f}_{21}^{L\ast }\right){\sigma }_{\theta }^2+\left(1+{\delta }_y^2\right){\left(f_{22}^{L\ast }\right)}^2\frac{1+t_2{\sigma }_{\theta }^2}{t_2}\\ +\left[\left(1+{\delta }_y^2\right){\left(f_{11}^{L\ast }\right)}^2+2\left(1+\rho {\delta }_y^2\right)f_{21}^{L\ast }{f}_{11}^{L\ast }+\left(1+{\delta }_y^2\right){\left(f_{21}^{L\ast }\right)}^2\right]\frac{1+t_1{\sigma }_{\theta }^2}{t_1}\end{array}\right\}$$

(35)

$$S{W}^{L\ast }={\Pi }_S^{L\ast }+{\Pi }_{R_1}^{L\ast }+{\Pi }_{R_2}^{L\ast }+C{S}^{L\ast }-G^{L\ast }$$

(36)

Proposition 4 describes the ex ante payoffs of GSC members, the total subsidy of the government, consumer surplus, and social welfare under equilibrium. It is similar to Proposition 2, and thus, will not be repeated here.

5. Information Leakage Decisions

In this section, we first study the supplier’s optimal information leakage decision in Section 5.1. Then, we examine how information leakage affects the two retailers’ ex ante payoffs, consumer surplus, and social welfare in Section 5.2.

5.1. The Supplier’s Information Leakage Decision

In Propositions 1 and 3, we examine the supplier’s optimal information leakage decision by comparing the supplier’s equilibrium operation decisions and ex ante payoff under strategies N and L. The main findings are given in the proposition below.

Proposition 5.

Compared with strategy N, the supplier sets a higher greenness level of the product and a higher wholesale price and gains more profit under strategy L (i.e.,$g^{L\ast }>g^{N\ast },$$w^{L\ast }>w^{N\ast }$, and ${\Pi }_S^{L\ast }>{\Pi }_S^{N\ast }$).

Proposition 5 indicates that the supplier’s information leakage behavior leads to higher level of product greenness and higher wholesale price. It implies that a higher level of product greenness results in a higher market demand. A higher level of product greenness leads to a higher cost for the supplier. Thus, the supplier can set a high wholesale price and may gain a high level of profit. This suggests that the supplier would choose to leak information, adopt a higher product greenness level and a higher wholesale price so as to gain a higher level of profit.

5.2. Information Leakage Preference from the Perspectives of Retailers, Consumer Surplus, and Social Welfare

We set the values of the parameters at moderate levels (i.e., $a=20$, ${\mu }_y=1$, $k=20$, ${\mathsf{\sigma }}_{\theta }=5$, $\overline{c}=1$, and $\lambda =0.5$) throughout this subsection. Note that Table 2 gives the range of the parameters. We define:

$$\Delta {\Pi }_{R_1}={\Pi }_{R_1}^{L\ast }-{\Pi }_{R_1}^{N\ast }$$

(37)

$$\Delta {\Pi }_{R_2}={\Pi }_{R_2}^{L\ast }-{\Pi }_{R_2}^{N\ast }$$

(38)

$$\Delta CS=C{S}^{L\ast }-C{S}^{N\ast }$$

(39)

$$\Delta SW=S{W}^{L\ast }-S{W}^{N\ast }$$

(40)

Next, we investigate the impact of the subsidy rate ($\tau$), supply correlation ($\rho$), supply uncertainty (${\delta }_y$), and forecast accuracy ($t_1$ and $t_2$) on $\Delta {\Pi }_{R_1}$, $\Delta {\Pi }_{R_2}$, $\Delta CS$, and $\Delta SW$, respectively.

5.2.1. Impact of Subsidy Rate ($\tau$)

Figure 3 shows how the subsidy rate ($\tau$) affects $\Delta {\Pi }_{R_1}$, $\Delta {\Pi }_{R_2}$, $\Delta CS,$ and $\Delta SW$ when the supply correlation ($\rho$), supply uncertainty (${\delta }_y$), and forecast accuracy ($t_1$ and $t_2$) are at moderate levels (i.e., $\rho =0.5$, ${\delta }_y=5$, $t_1=5$, and $t_2=5$).

From Figure 3a,c,d, we can see that $\Delta {\Pi }_{R_1}$, $\Delta CS,$ and $\Delta SW$ are always positive. Figure 3b shows that $\Delta {\Pi }_{R_2}$ is negative. This indicates that retailer 1 benefits from the supplier’s information leakage behavior, while retailer 2 suffers from the supplier’s information leakage behavior. In addition, the supplier’s information leakage behavior enhances both the consumer surplus and social welfare.

5.2.2. Impact of Supply Correlation ($\rho$)

Figure 4 shows how supply correlation ($\rho$) affects $\Delta {\Pi }_{R_1}$, $\Delta {\Pi }_{R_2}$, $\Delta CS,$ and $\Delta SW$ when the subsidy rate ($\tau$), supply uncertainty (${\delta }_y$), and forecast accuracy ($t_1$ and $t_2$) are at moderate levels (i.e., $\tau =0.05$, ${\delta }_y=5$, $t_1=5$, and $t_2=5$).

As shown in Figure 4a, when $\rho$ increases from 0 to 1, $\Delta {\Pi }_{R_1}$ gradually changes from negative to positive. This means that when $\rho$ is high (i.e., $\rho >0.2$), R1′s ex ante profit under strategy L is higher than it is under strategy N. This demonstrates that the value of $\Delta {\Pi }_{R_1}$ (i.e., negative or positive) depends on $\rho$, suggesting that R1 prefers supplier leaks only when $\rho$ is not too low (i.e., $\rho >0.2$).

Figure 4b shows that $\Delta {\Pi }_{R_2}$ gradually changes from positive to negative as $\rho$ increases from 0 to 1. This indicates that, for a high $\rho$ (i.e., $\rho >0.2$), R2′s ex ante payoff under strategy L is higher than that under strategy N.

From Figure 4c, we can see that $\Delta CS$ is always positive, indicating that the supplier’s information behavior enhances the consumer surplus.

Figure 4d shows that $\Delta SW>0$ only when the supply correlation is not too large (i.e., $\rho <0.9$).

5.2.3. Impact of Supply Uncertainty (${\delta }_{\mathrm{y}}$)

Figure 5 shows how supply uncertainty (${\delta }_y$) affects $\Delta {\Pi }_{R_1}$, $\Delta {\Pi }_{R_2}$, $\Delta CS,$ and $\Delta SW$ when the subsidy rate ($\tau$), supply correlation ($\rho$), and forecast accuracy ($t_1$ and $t_2$) are at moderate levels (i.e., $\tau =0.05$, $\rho =0.5$, $t_1=5$, and $t_2=5$).

From Figure 5a,c,d, we can see the impacts of supply uncertainty (${\delta }_y$) on $\Delta {\Pi }_{R_1}$, $\Delta CS,$ and $\Delta SW$ are in the same direction. From Figure 5a,b, we can find the impacts of supply uncertainty (${\delta }_y$) on $\Delta {\Pi }_{R_1}$ and $\Delta {\Pi }_{R_2}$ are in the opposite direction. $\Delta {\Pi }_{R_1}$ is positive in Figure 5a, while $\Delta {\Pi }_{R_2}$ is negative in Figure 5b. This indicates that the supplier’s information leakage behavior benefits retailer 1 while hindering retailer 2. Both consumer surplus and social welfare are enhanced by the supplier’s information leakage behavior.

5.2.4. Impact of Forecast Accuracies ($t_1$, and $t_2$)

Figure 6 shows how forecast accuracy ($t_1$ and $t_2$) affects $\Delta {\Pi }_{R_1}$, $\Delta {\Pi }_{R_2}$, $\Delta CS,$ and $\Delta SW$ when the subsidy rate ($\tau$), supply uncertainty (${\delta }_y$), and supply correlation ($\rho$) are at a moderate level (i.e., $\tau =0.05$, ${\delta }_y=5$, and $\rho =0.5$).

Figure 6a,b show that $\Delta {\Pi }_{R_1}>0$ and $\Delta {\Pi }_{R_2}<0$. This indicates that the supplier’s information leakage behavior benefits retailer 1 while hurting retailer 2. Figure 5c,d shows that the supplier’s information leakage behavior enhances both consumer surplus and social welfare when the forecast accuracy meets certain conditions (e.g., $t_1<5$, $t_2>5$).

The main findings from Figure 3, Figure 4, Figure 5 and Figure 6 and proposition 5 suggest that the supplier always prefers to leak information, as this enhances their profit through a higher product greenness level and wholesale price. Under certain conditions (e.g., $t_1<5$ or $t_2>5$ and $\rho <0.9$), the supplier’s information leakage behavior can increase both consumer surplus and social welfare.

6. Conclusions

This study explores optimal green operation decisions (including the greenness level of products) and information leakage in a GSC system with one supplier, one leader (retailer 1), one follower (retailer 2), and the government. The GSC faces variations in supply and demand. The supplier decides on whether to leak information (strategy L) or not (strategy N).

Our contributions are threefold. First, this study involves green product investment decisions. The analysis of the equilibrium product greenness degree under strategy L (i.e., the supplier engages in information leakage behavior) and strategy N (i.e., the supplier does not engage in information leakage behavior) provides valuable insights for upstream suppliers in terms of their investment in green product R&D. The supplier’s information leakage behavior motivates them to set a high wholesale price as well as a high product greenness level in order to gain more profit. This suggests that if the supplier decides to leak demand information, they will invest more in the R&D of green products.

Second, our findings enrich the literature on information leakage from suppliers in a new GSC setting with government subsidies and supply uncertainty. Interestingly, we found that when the supply correlation is high, the dominant retailer (i.e., retailer 1) benefits from the supplier’s information leakage behavior. Under certain conditions, the supplier’s information leakage behavior improves retailer 1′s profit and worsens retailer 2′s profits. This is because retailer 1′s gain (retailer 2′s loss) on the first-move advantage exceeds the loss (gain) of the supplier’s information leakage.

Finally, this study contributes to the extant knowledge on GSC management with government subsidies and supply uncertainty. Our findings demonstrate that when the government’s subsidy rate, supplier correlation, and forecast accuracy satisfy certain conditions, the supplier’s information leakage behavior increases both consumer surplus and social welfare. All the findings of our research will be useful for the management of green supply chains with government subsidies and supply uncertainty.

Two limitations of this work exist, leading to our suggested future research directions. One is that this research only considers government-subsidized retailers; it would be interesting to further examine other subsidy modes (e.g., subsidizing upstream suppliers and manufacturers) and compare different subsidy modes with each other. The other is that this research assumes that the upstream and downstream SC members are well-funded. In the future, we will examine the interplay between subsidy mode choices and financing decisions when SC members are cash strapped.