摘要：This issue of tweets will introduce the Model of the intensive reading journal article "The Impact of Demand Uncertainty on Consum

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“喆学（108）：精读期刊论文

《需求不确定性对绿色技术采用的消费者补贴的影响》

模型（1）”

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"Zhexue (108): Intensive reading of journal articles

"The impact of demand uncertainty on consumer subsidies for green technology adoption"

The Model(1)"

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本期推文将从思维导图、精读内容、知识补充三个方面介绍精读期刊论文《需求不确定性对绿色技术采用的消费者补贴的影响》模型。

This issue of tweets will introduce the Model of the intensive reading journal article "The Impact of Demand Uncertainty on Consumer Subsidies for Green Technology Adoption" from three aspects: mind map, intensive reading content, and knowledge supplement.

一、思维导图（Mind Maps）

二、精读内容（Intensive reading content）

在政府补贴背景下，电动汽车等寡头市场产品的供应商需同时优化定价（p）与生产量（q），构成价格设定报童问题。研究通过分析加性和乘性需求不确定性模型（如线性与等弹性需求），对比确定性需求下的决策，量化忽略需求不确定性的成本。结果表明，明确考虑需求波动可优化政府补贴水平、价格及产量，为企业评估精准需求预测的价值提供依据，揭示仅依赖平均需求决策的潜在风险。

In the context of government subsidies, suppliers of oligopolistic market products such as electric vehicles need to optimize both pricing (p) and production volume (q), which constitutes a price-setting newsboy problem. This study analyzes additive and multiplicative demand uncertainty models (such as linear and isoelastic demand) and compares decisions under deterministic demand to quantify the cost of ignoring demand uncertainty. The results show that explicitly considering demand fluctuations can optimize government subsidy levels, prices, and output, provide a basis for companies to evaluate the value of accurate demand forecasts, and reveal the potential risks of relying solely on average demand decisions.

加性需求不确定性模型定义为D(z,ϵ)=y(z)+ϵ，其中z=p−r为有效价格，y(z)表示确定性需求（期望值）,ϵ为服从Fϵ分布的随机扰动项。

The additive demand uncertainty model is defined as D(z,ϵ)=y(z)+ϵ, where z=p−r is the effective price, y(z) represents the deterministic demand (expected value), and ϵ is a random disturbance term that follows the Fϵ distribution.

在政府补贴背景下，供应商需基于有效价格z = p - r优化生产量q^*(p,r)和定价p^*(r)，政府则通过非凸优化确定最优补贴。定理1证明目标约束在最优时紧致，确保解的有效性。通过对比确定性需求（以期望值y(z)替代随机需求）与随机需求模型，分析指标如补贴水平、价格、产量及利润、政府支出的差异，揭示忽略需求不确定性对决策的影响，为政策制定提供量化依据。

In the context of government subsidies, suppliers need to optimize production volume q^*(p,r) and pricing p^*(r) based on the effective price z = p - r, and the government determines the optimal subsidy through non-convex optimization. Theorem 1 proves that the target constraint is tight at the optimal time, ensuring the validity of the solution. By comparing deterministic demand (replacing random demand with expected value y(z)) with the random demand model, the differences in indicators such as subsidy level, price, output and profit, and government expenditure are analyzed to reveal the impact of ignoring demand uncertainty on decision-making, providing a quantitative basis for policy making.

在一般需求函数y(p−r)下，价格设定报童问题（如公式10）无法获得闭式解，需借助二分搜索等数值方法求解最优价格psto。假设条件（9）确保利润函数对价格的严格凹性，从而保证唯一最优解；若条件不满足，问题仍可通过数值方法处理。对于线性需求，关系式（15）提供了合理的充分条件，简化分析。

Under the general demand function y(p−r), the price setting newsboy problem (such as Equation 10) cannot obtain a closed-form solution, and it is necessary to use numerical methods such as binary search to solve the optimal price psto. Assumption condition (9) ensures the strict concavity of the profit function with respect to the price, thereby ensuring a unique optimal solution; if the condition is not met, the problem can still be handled by numerical methods. For linear demand, relation (15) provides a reasonable sufficient condition to simplify the analysis.

最优生产量q∗由预期需求y(z)与报童分位数(p−c)/p共同决定，后者反映需求不确定性下的库存策略。政府通过调控有效价格z=p−r，确保预期销量达到目标采用水平Γ。随机需求下，为避免缺货，政府需激励供应商生产超出Γ的数量，其差值由K(psto)量化，体现需求波动对生产决策的影响。

The optimal production quantity q∗ is determined by the expected demand y(z) and the newsboy quantile (p−c)/p, which reflects the inventory strategy under demand uncertainty. The government ensures that the expected sales volume reaches the target adoption level Γ by regulating the effective price z=p−r. Under random demand, in order to avoid stockouts, the government needs to incentivize suppliers to produce more than Γ, and the difference is quantified by K(psto), which reflects the impact of demand fluctuations on production decisions.

最优价格p满足边际成本等于边际收益的条件（即c=p(1−1/Ed(p))，其中Ed(p)为价格弹性。尽管无闭式解，可证明需求不确定性会降低最优价格及企业利润，凸显风险环境对定价策略的抑制作用。

The optimal price p satisfies the condition that marginal cost equals marginal revenue (i.e., c=p(1−1/Ed(p)), where Ed(p) is the price elasticity. Although there is no closed-form solution, it can be shown that demand uncertainty reduces the optimal price and corporate profits, highlighting the inhibitory effect of the risk environment on pricing strategies.

定理1揭示了需求不确定性对最优变量（有效价格z、产量q、定价p及利润Π）的动态影响。通过量化噪声幅度（如标准差σ）与关键指标K(psto)的关系K(psto)≤0且随噪声增强单调递减），可系统性比较随机与确定性场景的差异。例如，均匀分布下K(psto)与σ呈线性关系（公式13），正态、指数等单峰分布亦类似。

Theorem 1 reveals the dynamic impact of demand uncertainty on the optimal variables (effective price z, output q, pricing p, and profit Π). By quantifying the relationship between the noise amplitude (such as standard deviation σ) and the key indicator K(psto) (K(psto)≤0 and monotonically decreasing with the increase of noise), the difference between random and deterministic scenarios can be systematically compared. For example, under uniform distribution, K(psto) is linearly related to σ (Formula 13), and the same is true for unimodal distributions such as normal and exponential.

有效价格zsto=y−1(Γ−K(psto))随噪声幅度增加而降低，其确定性场景（无噪声）达到最大值zdet=y−1(Γ)。生产量需额外补偿∣K(psto)∣以应对缺货风险，导致随机需求下的产量与定价均低于确定性场景，且差距随噪声扩大而增大。

The effective price zsto=y−1(Γ−K(psto)) decreases as the noise amplitude increases, and its deterministic scenario (no noise) reaches the maximum value zdet=y−1(Γ). The production volume needs to be compensated additionally |K(psto)| to cope with the risk of out-of-stock, resulting in that the output and pricing under random demand are lower than those in the deterministic scenario, and the gap increases as the noise increases.

三、知识补充（Knowledge supplement）

加性模型（Additive Model）是一种统计学和机器学习中常用的建模方法，其核心思想是将多个变量的影响以相加的方式组合，从而描述目标变量（因变量）与多个预测变量（自变量）之间的关系。与传统的线性模型不同，加性模型允许每个变量通过非线性函数对目标变量产生影响，因此在处理复杂关系时更具灵活性。

The additive model is a commonly used modeling method in statistics and machine learning. Its core idea is to combine the effects of multiple variables in an additive manner to describe the relationship between the target variable (dependent variable) and multiple predictor variables (independent variables). Unlike traditional linear models, additive models allow each variable to affect the target variable through nonlinear functions, so they are more flexible in dealing with complex relationships.

加性模型的特点：

Characteristics of additive models:

1.灵活性：每个变量可以独立使用非线性函数（如样条函数、核平滑等），无需假设全局线性关系。

1. Flexibility: Each variable can use nonlinear functions (such as spline functions, kernel smoothing, etc.) independently without assuming a global linear relationship.

2.可解释性：每个变量的贡献可以单独可视化，便于理解变量如何影响目标变量。

2. Interpretability: The contribution of each variable can be visualized separately, making it easier to understand how the variable affects the target variable.

3.降维能力：通过避免变量间的高阶交互项，减少了模型复杂度。

3. Dimensionality reduction capability: By avoiding high-order interaction terms between variables, the model complexity is reduced.

4.适用性：广泛用于回归和分类问题（如广义加性模型）。

4. Applicability: It is widely used in regression and classification problems (such as generalized additive models).

常见的加性模型类型包括：

Common additive model types include:

1.广义加性模型（GAM）：通过链接函数（如logit、log）将加性项与目标变量连接，支持更广泛的分布类型。

1. Generalized additive model (GAM): connects additive terms to target variables through link functions (such as logit, log), supporting a wider range of distribution types,

2. 加性回归树：以梯度提升机（如XGBoost、LightGBM）为代表，通过叠加多个弱学习器（如决策树）逐步优化预测结果。

2. Additive regression tree: represented by gradient boosting machines (such as XGBoost, LightGBM), gradually optimizes prediction results by superimposing multiple weak learners (such as decision trees);

3. 结构化加性模型：针对时空数据等特殊结构，引入时间趋势或空间平滑项，增强对数据特性的捕捉能力。

3. Structured additive model: for special structures such as spatiotemporal data, introduces time trends or spatial smoothing terms to enhance the ability to capture data characteristics.

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翻译：谷歌翻译

参考资料：谷歌、Chat GPT

参考文献：Maxime C. Cohen, Ruben Lobel, Georgia Perakis. The Impact of Demand Uncertainty on Consumer Subsidies for Green Technology Adoption [J], Management Science, 2016, 62(5): 1235-1258.

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