algo-price-elasticity

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Price Elasticity of Demand

需求价格弹性

Overview

概述

Price elasticity measures the percentage change in quantity demanded for a 1% change in price. Ed = %ΔQ / %ΔP. |Ed| > 1 = elastic (price-sensitive), |Ed| < 1 = inelastic (price-insensitive). Critical for pricing decisions and revenue optimization.

价格弹性衡量的是价格每变化1%时，需求量的百分比变化。公式为Ed = %ΔQ / %ΔP。|Ed| > 1表示需求富有弹性（对价格敏感），|Ed| < 1表示需求缺乏弹性（对价格不敏感）。这一指标对于定价决策和收入优化至关重要。

When to Use

适用场景

Trigger conditions:

Estimating how a price change will affect unit sales and revenue
Determining if demand is elastic or inelastic for a product
Optimizing price for maximum revenue or profit

When NOT to use:

When you need consumer willingness-to-pay distribution (use Van Westendorp or conjoint)
When pricing multiple products together (use bundle pricing)

触发条件：

估算价格变化对单位销量和收入的影响
判断某产品的需求是富有弹性还是缺乏弹性
优化价格以实现收入或利润最大化

不适用场景：

需要了解消费者支付意愿分布时（应使用Van Westendorp模型或联合分析法）
对多产品组合定价时（应使用捆绑定价策略）

Algorithm

算法

IRON LAW: Elasticity Is NOT Constant Along a Linear Demand Curve
It varies at every price point. At high prices, demand is elastic
(small price increase → big volume drop). At low prices, demand is
inelastic. Always calculate at the SPECIFIC price point of interest.
Revenue-maximizing price is where Ed = -1 (unit elastic).

IRON LAW: Elasticity Is NOT Constant Along a Linear Demand Curve
It varies at every price point. At high prices, demand is elastic
(small price increase → big volume drop). At low prices, demand is
inelastic. Always calculate at the SPECIFIC price point of interest.
Revenue-maximizing price is where Ed = -1 (unit elastic).

Phase 1: Input Validation

阶段1：输入验证

Collect: price-quantity pairs over time (or across markets). Control for: seasonality, promotions, competitor actions, other confounders. Gate: Minimum 10 price-quantity observations, confounders identified.

收集：不同时间（或不同市场）的价格-销量数据对。需控制以下变量：季节性、促销活动、竞品动作及其他干扰因素。 准入条件： 至少10组价格-销量观测数据，且已识别所有干扰因素。

Phase 2: Core Algorithm

阶段2：核心算法

Point elasticity: Ed = (dQ/dP) × (P/Q) at a specific price point Arc elasticity: Ed = ((Q₂-Q₁)/((Q₂+Q₁)/2)) / ((P₂-P₁)/((P₂+P₁)/2)) between two points Regression method: log(Q) = α + β×log(P) + controls → β is the elasticity (constant elasticity model)

点弹性： Ed = (dQ/dP) × (P/Q)，针对特定价格点计算 弧弹性： Ed = ((Q₂-Q₁)/((Q₂+Q₁)/2)) / ((P₂-P₁)/((P₂+P₁)/2))，计算两个价格点之间的弹性 回归法： log(Q) = α + β×log(P) + 控制变量 → β即为弹性值（恒定弹性模型）

Phase 3: Verification

阶段3：验证

Check: sign should be negative (price up → quantity down). Cross-validate with holdout periods. Gate: Elasticity is negative, confidence interval is reasonable.

检查：弹性值应为负数（价格上升→需求量下降）。使用留存期数据进行交叉验证。 准入条件： 弹性值为负，且置信区间合理。

Phase 4: Output

阶段4：输出

Return elasticity estimate with revenue impact projection.

返回弹性估算值及收入影响预测结果。

Output Format

输出格式

json

{
  "elasticity": -1.5,
  "interpretation": "elastic — 1% price increase → 1.5% quantity decrease",
  "revenue_impact": {"price_change_pct": 10, "quantity_change_pct": -15, "revenue_change_pct": -6.5},
  "metadata": {"method": "log-log regression", "r_squared": 0.82, "observations": 52}
}

json

{
  "elasticity": -1.5,
  "interpretation": "elastic — 1% price increase → 1.5% quantity decrease",
  "revenue_impact": {"price_change_pct": 10, "quantity_change_pct": -15, "revenue_change_pct": -6.5},
  "metadata": {"method": "log-log regression", "r_squared": 0.82, "observations": 52}
}

Examples

示例

Sample I/O

输入输出样例

Input: Price increased 10% from $100 to $110, quantity dropped from 1000 to 850 Expected: Arc elasticity = ((-150/925) / (10/105)) = -1.70 (elastic)

输入： 价格从100美元上涨10%至110美元，销量从1000降至850 预期结果： 弧弹性 = ((-150/925) / (10/105)) = -1.70（富有弹性）

Edge Cases

边缘情况

Input	Expected	Why
Luxury good	May be positive (Veblen)	Higher price → higher perceived value
Necessity (insulin)	Near zero	Demand barely responds to price
Perfect substitute available	Very elastic (< -3)	Customers switch immediately

输入	预期结果	原因
奢侈品	弹性可能为正（凡勃伦商品）	价格越高，感知价值越高
必需品（如胰岛素）	弹性接近0	需求几乎不随价格变化
存在完美替代品	弹性极高（< -3）	客户会立即转向替代产品

Gotchas

注意事项

Omitted variable bias: Without controlling for advertising, seasonality, and competitor prices, elasticity estimates are biased.
Short-run vs long-run: Short-run elasticity is typically lower (customers are locked in). Long-run gives them time to find substitutes.
Cross-price elasticity: Demand for product A may depend on product B's price. Ignoring this in a portfolio context leads to suboptimal pricing.
Asymmetric elasticity: Consumers may react differently to price increases vs decreases. Don't assume symmetry.
Small sample noise: With few observations, elasticity estimates have wide confidence intervals. Report intervals, not just point estimates.

遗漏变量偏差： 若未控制广告、季节性和竞品价格等因素，弹性估算结果会存在偏差。
短期vs长期： 短期弹性通常较低（客户被锁定），长期弹性会让客户有时间寻找替代品。
交叉价格弹性： 产品A的需求可能受产品B价格影响。在组合定价场景中忽略这一点会导致次优定价。
非对称弹性： 消费者对涨价和降价的反应可能不同，切勿假设弹性对称。
小样本噪声： 观测数据过少时，弹性估算的置信区间会很宽。应报告区间而非仅点估算值。

Scripts

脚本

Script	Description	Usage
`scripts/arc_elasticity.py`	Compute arc elasticity and revenue impact	`python scripts/arc_elasticity.py --help`

Run

python scripts/arc_elasticity.py --verify

to execute built-in sanity tests.

脚本	描述	使用方式
`scripts/arc_elasticity.py`	计算弧弹性及收入影响	`python scripts/arc_elasticity.py --help`

运行

python scripts/arc_elasticity.py --verify

可执行内置的合理性测试。

References

参考资料

For regression-based elasticity estimation, see
```
references/regression-estimation.md
```
For cross-price elasticity analysis, see
```
references/cross-price.md
```

基于回归的弹性估算方法，请参阅
```
references/regression-estimation.md
```
交叉价格弹性分析，请参阅
```
references/cross-price.md
```