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“小宇分享（八）：

精读《上下游联合减排与低碳宣传的微分博弈模型》成本分担下的分散式决策模型求解过程

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Xiaoyu's Sharing (Ⅷ):

In-depth reading of the "Differential Game Model of Upstream and Downstream Joint Emission Reduction and Low-Carbon Promotion" and the solution process of the decentralized decision-making model under cost sharing

一、思维导图（Mind mapping）

二、精读内容（Intensive reading content）

求解制造商与零售商的HJB方程

Solving the HJB equation for manufacturers and retailers

首先求解成本分担契约下的分散式决策零售商的最优控制问题，由求解命题（1）中的证明过程，同理可得出分散式决策下零售商的最优控制函数，t时刻零售商的最优利润值函数，如下图所示：

First, we solve the optimal control problem of the retailer with decentralized decision-making under the cost-sharing contract. By solving the proof process in Proposition (1), we can also derive the optimal control function of the retailer under decentralized decision-making and the optimal profit value function of the retailer at time t, as shown in the following figure:

此时零售商的最优控制问题满足如下的HJB方程,根据零售商的HJB方程形式可知此方程为关于E_R的凹函数(与命题一证明一致，原理海塞矩阵正负定判断最值)，对E_R一阶求导可得t时刻E_R的结果，如下图所示：

At this point, the retailer's optimal control problem satisfies the following HJB equation. Based on the retailer's HJB equation, this equation is a concave function about ER (consistent with the proof of Proposition 1, based on the principle that the positive and negative definiteness of the Hessian matrix determines the maximum value). Taking the first-order derivative of ER yields the result of ER at time t, as shown in the figure below:

根据以上计算结果可知，零售商的低碳宣传努力程度E_Ｒ随制造商为其承担的低碳宣传努力成本比例X、自身边际利润π_Ｒ和对需求影响程度系数β的增加而提高，随其自身的宣传努力成本系数η_Ｒ的增加而降低.

According to the above calculation results, the retailer's low-carbon publicity effort E_R increases with the increase of the manufacturer's low-carbon publicity effort cost ratio X, its own marginal profit π_R, and the coefficient of influence on demand β, and decreases with the increase of its own publicity effort cost coefficient η_R.

根据上述计算零售商的计算过程，同理可得t时刻制造商的最优值利润函数，如下图所示：

Based on the above calculation process for retailers, the optimal profit function of manufacturers at time t can be obtained by analogy, as shown in the following figure:

此时制造商的最优控制问题满足如下的HJB方程,根据制造商的HJB方程形式可知此方程为关于E_M的凹函数(与命题一证明一致，原理海塞矩阵正负定判断最值)，将零售商的反应式函数E_R带入HJB方程中，且对E_M一阶求导可得t时刻E_M和X的结果，如下图所示：

At this point, the manufacturer's optimal control problem satisfies the following HJB equation. Based on the form of the manufacturer's HJB equation, we know that this equation is a concave function with respect to E_M (consistent with the proof of Proposition 1, based on the principle that the Hessian matrix is positive or negative and determines the maximum value). Substituting the retailer's reaction function E_R into the HJB equation and taking the first-order derivative of E_M yields the results for E_M and X at time t, as shown in the figure below:

根据以上计算结果可知，制造商的减排努力程度随自身减排努力对产品减排量的影响程度 γ的增加而增加，即，单位减排努力带来的减排量的提高越多制造商减排越努力;随自身减排成本系数 η_M的增加而降低，制造商为零售商提供低碳宣传成本补贴比例随制造商边际收益的增加而增加，随零售商边际收益的增加而减少，且当 2πM＞ πＲ时制造商才为零售商提供补贴，否则制造商不但不给零售商提供补贴，而且会以零售商的促销成本为基础收取一定的费用．

Based on the above calculations, we can see that a manufacturer's emission reduction efforts increase with the impact of its own efforts on product emission reductions, γ. That is, the greater the increase in emissions reductions per unit of emission reduction effort, the more diligent the manufacturer is in reducing emissions. Furthermore, the manufacturer's emission reduction efforts decrease with an increase in its own emission reduction cost coefficient, η_M. The proportion of low-carbon promotion cost subsidies provided by manufacturers to retailers increases with the manufacturer's marginal revenue and decreases with the retailer's marginal revenue. Manufacturers only provide subsidies to retailers when 2πM > πR. Otherwise, not only do they not provide subsidies to retailers, but they charge a fee based on the retailer's promotional costs.

求解过程

Solution process

求解系数

Solve for the coefficients

将E_M，X，E_Ｒ分别代入制造商与零售商的HJB方程中，整理得如下结果：

Substituting E_M, X, and E_R into the HJB equations of the manufacturer and the retailer respectively, we obtain the following results:

根据式(47) 和式(48)微分方程的特点，推测关于τ的线形最优值函数是 HJB 方程的解，设函数VN_M(τ)和 VN_Ｒ(τ)的表达式为VN_M(τ)=a_Y1 * τ+b_Y1,VN_R(τ)=a_Y2 * τ+ b_Y2,如下图示：

Based on the characteristics of the differential equations (47) and (48), it is inferred that the linear optimal value function about τ is the solution of the HJB equation. Assume that the expressions of the functions VN_M(τ) and VN_R(τ) are VN_M(τ)=a_Y1 * τ+b_Y1, and VN_R(τ)=a_Y2 * τ+b_Y2, as shown in the figure below:

其中a1、b1、a2、b2均为未知常数，将式图中两式分别代入式制造商和零售商的HJB方程中求解系数，求解过程如下：

Where a1, b1, a2, and b2 are all unknown constants, substitute the two equations in the figure into the HJB equations of the manufacturer and retailer respectively to solve for the coefficients. The solution process is as follows:

上述系数求解过程可证明命题（3）成本契约分担下分散式决策的制造商和零售商各自的利润最优值函数分别为:

The above coefficient solution process can prove Proposition (3) that the optimal profit functions of the manufacturer and retailer under decentralized decision-making under cost contract sharing are:

求解均衡结果

Solving the equilibrium result

将式VN_M(τ)=a_Y1 * τ+b_Y1,VN_R(τ)=a_Y2 * τ+ b_Y2和E_M、E_Ｒ中可以得到制造商和零售商的均衡解，如下图所示：

By combining VN_M(τ)=a_Y1 * τ+b_Y1, VN_R(τ)=a_Y2 * τ+ b_Y2 with E_M and E_R, we can obtain the equilibrium solutions for manufacturers and retailers, as shown in the figure below:

求解轨迹方程

Solving the trajectory equation

最后将EY*_M带入tau_dot(t)=gamma * E_M(t)- delta *tau(t)，再求解微分方程，即可求出无成本分担的分散式决策下的减排量轨迹方程，结构如下图所示：

Finally, substitute EY*_M into tau_dot(t)=gamma * E_M(t)-delta *tau(t) and solve the differential equation to obtain the emission reduction trajectory equation under decentralized decision-making without cost sharing. The structure is shown in the figure below:

综上所述，命题（三）成本分担契约下的分散式决策证明完毕。

To sum up, Proposition (3) decentralized decision-making under cost-sharing contracts is proven.

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翻译：谷歌翻译

资料来源：ChatGPT、百度百科

参考文献：[1]徐春秋,赵道致,原白云,等.上下游联合减排与低碳宣传的微分博弈模型[J].管理科学学报,2016,19(02):53-65.

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来源：LearningYard学苑