Proof of Proposition 1. According to the Stackelberg differential game mode under the mode N-R, the reaction decision of the intermediary platform needs to be solved first with the help of the inverse induction method. According to Bellman’s theory of continuous type dynamic programming, the continuously differentiable value function
satisfies the Hamilton-Jacobi-Bellman (HJB) equation as:
where,
is the optimal value function of the platform under mode N-R and represents the profit of the platform in the whole operation planning period.
is the first order derivative of the platform optimal value function with respect to environmental goodwill, indicating the impact of a unit change in brand environmental goodwill on platform profits. For the sake of notation simplification, the time variable
is omitted in the following solving procedures. Based on the first-order optimality condition, the retail price
and the low-carbon publicity
of the platform are respectively:
and
. Substituting
p and
into the manufacturer’s objective functional, according to Bellman’s continuous dynamic programming method, the manufacturer’s HJB equation is obtained as follows.
where,
is the manufacturer’s optimal value function under mode N-R.
is the first-order derivative of the manufacturer’s optimal value function with respect to environmental goodwill, which reflects the impact of the unit change in environmental goodwill on the manufacturer’s profit. According to the first-order optimal conditions on the right side of Equation (A2), the optimal decisions of the manufacturer’s wholesale price and carbon emission reduction can be obtained as
and
. Substitute the wholesale price
of the product into the retail price of the product to obtain
.
Substituting
into the HJB equation of the manufacturer and platform, the following HJB equations can be obtained.
According to the relationship between the value functions at the left and right ends of Equation (A3) and environmental goodwill, it is assumed that the optimal value functions of the manufacturer and the platform meet the following relationship.
where
represent the undetermined coefficients of the optimal value function of the manufacturer and platform respectively, which are constants. Substituting Equation (A4) into HJB Equation (A3), the identity relation can be obtained as follows.
Based on the functional relationship of environmental goodwill on the left and right ends of Equation (A5), the specific expression of undetermined constant coefficient can be obtained as follows.
In order to further obtain the time trajectory of brand environmental goodwill, Equation (A6) was substituted into
to obtain the expression about relevant decisions, and further substituted into the dynamic equation of environmental goodwill
to obtain the first-order differential equation of environmental goodwill as follows.
By solving Equation (A7), the optimal time path of environmental goodwill under mode N-R can be obtained.
where,
represents the steady state of environmental goodwill in N-R mode. By substituting environmental goodwill into the optimal decision and optimal value function of the manufacturer and platform, relevant conclusions in Proposition 1 can be obtained accordingly. Furthermore, consumer surplus can be expressed as
, where
is the highest willingness of consumers to pay, then consumer surplus is
. Finally, the social welfare is expressed as the sum of total profit of supply chain and consumer surplus, then the social welfare under mode N-R is
.
Proposition 1 is proved. □
Proof of Proposition 2. According to the Stackelberg differential game mode under mode N-A, by backward induction, the optimal value function of the platform satisfies the HJB equation.
where
is the optimal value function of the platform under mode N-A, and
is the first partial derivative of the optimal value function with respect to the state variable.
According to the first-order optimality condition on the right end of Equation (A9), the low-carbon publicity on the platform can be obtained as
. Substituting it into the manufacturer’s objective functional, the HJB equation satisfied by the manufacturer’s optimal value function
is obtained as follows.
Based on the first-order optimality condition at the right end of Equation (A10), the manufacturer’s product retail price and carbon emission can be obtained as
,
. By substituting the optimal decisions of the manufacturer and platform into the HJB equation of the manufacturer and platform, the HJB equations of the optimal value function of the manufacturer and platform can be obtained as follows.
According to the functional relationship between the optimal value function and the state variable in (A11), it is assumed that the optimal value function of the manufacturer and platform satisfies the following form.
where
are the constant coefficient of the optimal value function to be determined. Substituting Equation (A12) into the system of HJB equations Equation (A11) yields the set of constant equations satisfied by the constant coefficient to be determined.
Based on the constancy relation in Equation (A13), the specific expressions for the constant coefficients to be determined can be obtained as follows.
Substituting Equation (A14) into
and
and further substituting the two into the environmental goodwill dynamics equation gives the first order differential equation for environmental goodwill
Solving this first order linear differential equation yields the optimal trajectory of environmental goodwill under mode N-A
where,
is the steady state of environmental goodwill under mode N-A. From this, the environmental goodwill time trajectory is substituted into the manufacturer and platform optimal decision and the optimal value function to obtain the optimal product retail price and carbon emission level for the manufacturer, the optimal low-carbon publicity level for the platform and the profit for both.
,
,
.
Similarly, consumer surplus can be calculated as
, Social welfare is
.
Proposition 2 is proved. □
Proof of Proposition 3. According to the Stackelberg differential game mode under the mode C-R, the reaction decision of the intermediary platform needs to be solved first with the help of the inverse induction method. According to the Bellman continuous dynamic programming theory, the continuously differentiable value function
satisfies the Hamilton-Jacobi-Bellman (HJB) equation as:
where
is the optimal value function of the platform under mode C-R and represents the profit of the platform over the entire operating plan period.
is the first-order derivative of the platform optimal value function with respect to environmental goodwill, indicating the impact of a unit change in environmental goodwill on the platform’s profits. From the first-order optimality condition, the retail price and the low-carbon publicity of the platform are:
and
. Substituting
and
into the manufacturer’s objective generalized function, the manufacturer’s HJB equation is obtained according to Bellman’s continuous dynamic programming method as follows.
where
is the manufacturer’s optimal value function under mode C-R and
is the first-order derivative of the manufacturer’s optimal value function with respect to environmental goodwill, reflecting the impact of a unit change in environmental goodwill on the manufacturer’s profit. According to the first-order optimality condition on the right side of Equation (A18), the optimal decision for the manufacturer’s wholesale price and carbon emission reduction can be obtained as
,
. Substituting
into the retail price of the product gives
.
Substituting
into the HJB equations in the manufacturer and platform yields the following set of HJB equations.
Based on the relationship between the value functions at the left and right ends of Equation (A19) and environmental goodwill, the optimal value functions for the manufacturer and platform, respectively, are assumed to satisfy the following relationship.
where
denote the coefficient to be undetermined for the manufacturer and platform optimal value functions, respectively, and are both constants. Substituting Equation (A20) into the system of HJB Equations (A19) yields the following constant relationship.
Based on the left and right ends of Equation (A21), as a function of environmental goodwill, the specific expressions for the constant coefficients to be determined can be obtained as follows.
To further obtain the temporal trajectory of environmental goodwill, substitute Equation (A22) into equation
to obtain an expression for the relevant decision and further substitute it into the environmental goodwill dynamics equation
to obtain the first order differential equation for environmental goodwill as follows:
Solving for Equation (A23) yields the optimal time path for environmental goodwill under mode C-R:
where
denotes the steady state of environmental goodwill in the C-R mode. Substituting the environmental goodwill into the optimal decision and optimal value functions of the manufacturer and the platform, the relevant conclusions in Proposition 3 can be obtained accordingly. Further, the consumer surplus can be expressed as
, where
is the consumer’s highest willingness to pay. Finally, by expressing social welfare as the sum of total supply chain profits and consumer surplus, social welfare under mode C-R is
.
Proposition 3 is proved. □
Proof of Proposition 4. According to the Stackelberg differential game mode under mode C-A, the HJB equation satisfied by the optimal value function of the platform is listed with the help of the inverse induction method.
where
is the optimal value function of the platform under mode C-A and
is the first order partial derivative of the optimal value function with respect to the state variables. From the first-order optimality condition on the right-hand end of Equation (A25), the low-carbon publicity on the platform can be obtained as
. Substituting this into the manufacturer’s objective generalization, the HJB equation for which the manufacturer’s optimal value function is satisfied is written as follows.
where
is the manufacturer’s optimal value function and
is the first order partial derivative of the optimal value function with respect to the state variables. According to the first-order optimality condition on the right-hand end of Equation (A26), the optimal decision on the manufacturer’s retail price of the product and the level of carbon reduction is obtained as
.
By substituting the optimal decision of manufacturer and platform into the HJB equation of manufacturer and platform, the HJB equations for the optimal value function of manufacturer and platform can be obtained as follows.
Based on the functional relationship between the optimal value function and the state variable in (A27), it is assumed that the optimal value functions of the manufacturer and the platform satisfy the following form.
where
are the constant coefficients of the optimal value function to be determined. Substituting Equation (A28) into the HJB equation system Equation (A27), we can obtain the set of constant equations satisfied by the constant coefficients to be determined.
According to the constancy relation in Equation (A29), the specific expressions for the constant coefficients to be determined can be obtained as follows.
Substituting Equation (A30) into
,
and the environmental goodwill dynamics equation gives the first order differential equation for environmental goodwill.
The optimal trajectory of environmental goodwill under mode C-A can be obtained by solving the first-order linear differential equation.
where,
is the steady state of environmental goodwill under mode C-A. Therefore, by substituting the time trajectory of environmental goodwill into the optimal decision and the optimal value function of the manufacturer and the platform, the optimal product retail price and carbon emission level of the manufacturer, the optimal online low-carbon publicity level of the platform, and the profits of the two can be obtained.
. The consumer surplus can be calculated as
, and social welfare is
.
Proposition 4 is proved. □
Second, we compare the platform profits in Scenario 2 and Scenario 1. .
It is found that Scenario 2 is better than Scenario 1 for the platform when the impact coefficient A of the manufacturer’s carbon emission reduction on environmental goodwill , where .
To make the profit comparison between the manufacturer and the platform clearer, we use the image to express it.
Figure A1.
Influence of reference effect on manufacturer and platform profit in reselling mode.
In the agency selling mode, the impact of the presence or absence of a reference effect of the consumer on the manufacturer’s profit is first compared. Scenario 2 was found to be better than Scenario 1 for the manufacturer when the level of carbon emission reduction by manufacturers had an impact factor on environmental goodwill , where . Then we analyze the impact of reference effect on its profit from the perspective of platform.
Again, we use the image to represent the combined profitability of the manufacturer and the platform in comparison to the agent selling mode.
Figure A2.
Influence of reference effect on manufacturer and platform profit in agency selling mode.
Proposition 6 is proved. □
Proof of Proposition 7. We start by comparing the manufacturer’s profits under the reselling and agent selling mode.
We discover that if , then , or , . So, when or , , else if , , where , .
If , that is , , so when or , else if , .
Therefore, if , or , , else if , . If , or , , else if , .
When , , and when , .
Secondly, we compare the profitability of the platform under the reselling and agent selling mode.
, We find that if , then , or . When or , . Else if , , where , .
If , then . , . It is clearly that , so , , else if , .
Therefore, if , or , , else if , . If , , , else if , .
When , , and when , .
Secondly, we compare the profitability of the platform under the reselling and Agent selling mode.
, We find that if , then , or . When or , , else if , , where , .
If , then . , . It is clearly that , so , , else if , .
Therefore, if , or , , else if , . If , , , else if , .
The interval in which the manufacturer is able to obtain a profit in the selling mode chosen by the platform, i.e., the interval in which the manufacturer and the platform reach a stable cooperation on the selling mode, is as follows. (1) When the platform has a high level of low-carbon publicity, select reselling mode when ; when , select agent selling mode. (2) When the platform has a low level of low-carbon publicity, select reselling mode when or ; when , select agent selling mode.
Proposition 7 is proved. □