晴读（38）：精读复刻论文

摘要：Today, the editor will present the Recycling Channel Competitive Relationship Incentive Model (Model SN) from the paper "Design of

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“晴读（38）：精读复刻论文《基于委托代理理论的逆向供应链激励机制设计与回收模式选择》的回收渠道竞争关系激励模型（模型SN）”

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"Qing Du (38): Intensive Reading and Replication of the Recycling Channel Competitive Relationship Incentive Model (Model SN) in the Paper "Design of Incentive Mechanism and Selection of Recovery Mode for Reverse Supply Chain Based on Principal-Agent Theory"

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今天小编将从思维导图、精读内容、知识补充三个板块为大家带来《基于委托代理理论的逆向供应链激励机制设计与回收模式选择》的回收渠道竞争关系激励模型（模型SN）。

Today, the editor will present the Recycling Channel Competitive Relationship Incentive Model (Model SN) from the paper "Design of Incentive Mechanism and Selection of Recovery Mode for Reverse Supply Chain Based on Principal-Agent Theory" to everyone through three sections: mind map, intensive reading content, and knowledge supplement.

一、思维导图（Mind Mapping）

二、精读内容（Conduct in-depth reading of the material）

1.模型背景。首先考虑回收商能力信息对称的情况。此时，制造商知道回收商i的能力禀赋θi，但无法观测到回收商的努力行为。为此，制造商针对回收商i设计一个契约(αi，βi）（为书写简洁，将能力信息对称下的α(θi)和β(θi)分别记为αi和βi）激励回收商最优努力。“回收渠道竞争关系激励模型（模型SN）”是分析对称信息下双回收商竞争行为的关键框架，其核心是解决制造商如何设计契约，在回收商努力不可观测但能力已知的竞争场景下，实现自身与回收商的效用平衡。

1. Model Background. First, consider the scenario where recyclers' capability information is symmetric. In this case, the manufacturer knows the capability endowment of recycler i but cannot observe the recycler's effort behavior. To address this, the manufacturer designs a contract (αᵢ, βᵢ) for recycler i (for the sake of concise writing, α(θᵢ) and β(θᵢ) under symmetric capability information are denoted as αᵢ and βᵢ, respectively) to incentivize the recycler to exert optimal effort. The "Recycling Channel Competitive Relationship Incentive Model (Model SN)" is a key framework for analyzing the competitive behavior of dual recyclers under symmetric information, whose core is to solve how the manufacturer designs contracts to achieve the utility balance between itself and recyclers in a competitive scenario where recyclers' efforts are unobservable but their capabilities are known.

2.模型假设

（1）信息结构：回收商能力禀赋（θᵢ）为对称信息，制造商完全知晓；但回收商的努力行为（eᵢ）不可观测，存在道德风险。

（2）渠道关系：双回收商为竞争关系，一方努力水平提升会降低另一方的回收业务表现，具体表现为回收绩效函数中，竞争强度系数k（k>0）越大，两回收商在回收价格、地域、产品类目等方面的竞争越激烈。

（3）主体特征：制造商为风险中性，目标是最大化自身期望效用；回收商为风险规避，效用函数满足不变绝对风险厌恶（U(Πᵢ)=-e-ρΠᵢ，ρ为绝对风险规避系数），且努力成本为二次型（cᵢ=aeᵢ²/2，a为努力成本系数）。

2. Model Assumptions

(1) Information Structure: The recyclers' capability endowments (θᵢ) are symmetric information and fully known to the manufacturer; however, the recyclers' effort behaviors (eᵢ) are unobservable, leading to moral hazard.

(2) Channel Relationship: The two recyclers are in a competitive relationship—an increase in one recycler's effort level will reduce the other's recycling business performance. Specifically, in the recycling performance function, the larger the competition intensity coefficient k (k>0), the fiercer the competition between the two recyclers in terms of recycling price, region, product category, and other aspects.

(3) Subject Characteristics: The manufacturer is risk-neutral, to maximize its own expected utility; the recyclers are risk-averse, whose utility functions satisfy constant absolute risk aversion (U(Πᵢ) = -e^(-ρΠᵢ), where ρ is the absolute risk aversion coefficient), and their effort cost follows a quadratic form (cᵢ = aeᵢ²/2, where a is the effort cost coefficient).

3.核心变量与函数

（1）核心变量定义

θᵢ：回收商i的能力禀赋，反映其在回收产能、技术、物流等方面的基础能力。

eᵢ：回收商i的努力水平，为决策变量，影响回收绩效。

βᵢ^SN：回收绩效分享比例，制造商设计的激励强度，β越大意味着回收商从回收绩效中获得的收益越多。

αᵢ^SN：固定支付，与回收绩效无关，用于保障回收商的基础参与意愿。

Qᵢ^SN：回收商i的回收绩效，公式为Qᵢ^SN=θᵢ+beᵢ-k(eⱼ-eᵢ)+ε（b为努力产出系数，ε为市场不确定性，服从N(0,σ²)）。

3. Core Variables and Functions

(1) Definition of Core Variables

θᵢ: The capability endowment of recycler i, reflecting its basic capabilities in recycling capacity, technology, logistics, and other aspects.

eᵢ: The effort level of recycler i, which is a decision variable and affects recycling performance.

βᵢ^SN: The recycling performance sharing ratio, an incentive intensity designed by the manufacturer. A larger β indicates that the recycler obtains more benefits from the recycling performance.

αᵢ^SN: Fixed payment, which is irrelevant to recycling performance and used to ensure the recycler's basic willingness to participate.

Qᵢ^SN: The recycling performance of recycler i, with the formula Qᵢ^SN = θᵢ + beᵢ - k(eⱼ - eᵢ) + ε (where b is the effort output coefficient, ε represents market uncertainty, and follows the normal distribution N(0, σ²)).

（2）目标函数与约束条件

制造商目标：最大化期望效用，需扣除给双回收商的固定支付和绩效分成，公式为：

(2) Objective Function and Constraints

Manufacturer's Objective: Maximize expected utility, which needs to deduct the fixed payments and performance shares given to the two recyclers. The formula is:

参与约束（IR）：回收商接受契约的最低条件，其期望效用不低于保留效用(ω̄)，即：

Participation Constraint (IR): The minimum condition for recyclers to accept the contract, requiring their expected utility to be no less than the reservation utility (ω̄), i.e.:

激励相容约束（IC）：回收商选择的努力水平能最大化自身期望效用，即：

Incentive Compatibility Constraint (IC): The effort level chosen by the recycler maximizes its own expected utility, i.e.:

4.最优解与关键结论

通过求解制造商的规划问题（P₁），得到模型SN的最优契约与行为结果，核心结论如下：

（1）最优回收绩效分享比例（βᵢSN*）公式为：

4. Optimal Solutions and Key Conclusions

By solving the manufacturer's programming problem (P₁), the optimal contract and behavioral results of Model SN are obtained, with the core conclusions as follows:

(1) The formula for the optimal recycling performance sharing ratio (βᵢ^SN*) is:

影响规律：β随努力产出系数b、竞争强度k的增大而增大（但k超过临界值K₀后会下降，避免恶性竞争）；随努力成本系数a、绝对风险规避系数ρ、市场不确定性σ²的增大而减小。逻辑：b越大，努力的边际收益越高，需提高激励；ρ或σ²越大，回收商风险承受能力越弱，需降低激励以减少风险成本。

Influence Law: β increases with the increase of the effort output coefficient b and competition intensity k (but decreases when k exceeds the critical value K₀ to avoid vicious competition); it decreases with the increase of the effort cost coefficient a, absolute risk aversion coefficient ρ, and market uncertainty σ². Logic: A larger b means higher marginal returns from effort, so incentives need to be increased; a larger ρ or σ² indicates weaker risk-bearing capacity of recyclers, so incentives need to be reduced to lower risk costs.

（2）最优努力水平（eᵢSN*）

Optimal effort level（eᵢSN*）

影响规律：e与b、k正相关，与a、ρ、σ²负相关。逻辑：k增大时，竞争压力促使回收商提升努力以抢占市场；a增大时，努力的边际成本上升，回收商会减少努力投入。

Influence Law: e is positively correlated with b and k, and negatively correlated with a, ρ, and σ². Logic: When k increases, competitive pressure drives recyclers to increase their efforts to seize the market; when a increases, the marginal cost of effort rises, so recyclers will reduce their effort input.

（3）参与主体期望效用

回收商：仅获得保留效用（UᵢSN*=ω̄），因能力信息对称，制造商无需支付额外信息租金。制造商：期望效用随k的增大而增大。公式为：

(3) Expected Utility of Participating Entities

Recyclers: Only obtain the reservation utility (Uᵢ^SN* = ω̄). Due to the symmetry of capability information, manufacturers do not need to pay additional information rent. Manufacturer: The expected utility increases with the growth of k. The formula is:

5.模型意义：

（1）激励强度调整：当回收市场竞争较弱（k

（2）回收商选择：优先选择努力产出系数b 高、努力成本系数a 低的回收商，这类回收商在相同激励下能付出更高努力，提升回收效率。

风险控制：若市场不确定性（σ²）高或回收商风险规避程度（ρ）强，需降低激励强度，同时通过固定支付（α）保障回收商参与，平衡风险与激励。

5. Model Significance:

(1) Adjustment of Incentive Intensity: When competition in the recycling market is weak (k

(2) Recycler Selection: Prioritize recyclers with a high effort output coefficient b and a low effort cost coefficient a. Such recyclers can exert higher efforts under the same incentives, thereby improving recycling efficiency.

(3) Risk Control: If market uncertainty (σ²) is high or recyclers' risk aversion degree (ρ) is strong, the incentive intensity needs to be reduced. At the same time, fixed payments (α) should be used to ensure recyclers' participation and balance risks and incentives.

三、知识补充（Supplementary knowledge）

1.IR：在经济学和博弈论相关领域，IR通常是“Individual Rationality”的缩写，即个体理性约束，也常被称为参与约束。它在契约设计、委托代理关系等情境中意义重大，是确保经济主体参与某项活动的关键条件。在委托代理关系里，委托人设计契约时，要保证代理人参与契约的收益不低于其不参与时的收益，也就是代理人的保留收益，这样代理人才会愿意参与契约，让契约得以成立并执行。

1. IR: In the fields related to economics and game theory, IR is usually the abbreviation of "Individual Rationality", i.e., the Individual Rationality Constraint, which is also often referred to as the Participation Constraint. It is of great significance in scenarios such as contract design and principal-agent relationships, and serves as a key condition to ensure that economic entities participate in a certain activity. In a principal-agent relationship, when designing a contract, the principal must ensure that the agent's benefit from participating in the contract is not lower than the benefit when not participating, which is the agent's reservation benefit. Only in this way will the agent be willing to participate in the contract, enabling the contract to be established and implemented.

2.IC：在委托代理理论及相关的供应链研究中，IC通常代表“Incentive Compatibility”，即激励相容约束。它是指在委托代理关系里，委托人设计的契约要让代理人追求自身利益最大化的行为，恰好与委托人的目标达成一致，以此解决信息不对称带来的逆向选择和道德风险问题。

2. IC: In principal-agent theory and related supply chain research, IC usually stands for "Incentive Compatibility", i.e., the Incentive Compatibility Constraint. It means that in a principal-agent relationship, the contract designed by the principal should make the agent's behavior of pursuing the maximization of its own interests exactly consistent with the principal's goals, so as to solve the problems of adverse selection and moral hazard caused by information asymmetry.

3.绝对风险规避：是经济学和金融学中用于衡量个体在面对风险时规避倾向的重要概念，反映了个体对风险的厌恶程度以及在投资、消费等决策中对风险的态度，与个体的效用函数密切相关。

3. Absolute Risk Aversion: It is an important concept in economics and finance used to measure an individual's tendency to avoid risk when facing it. It reflects the individual's degree of risk aversion and attitude towards risk in decisions such as investment and consumption, and is closely related to the individual's utility function.

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