经济学（Economics）
一、本课定位
| 课次 | 主题 | 能力 |
|---|---|---|
| L144 | 需求弹性：收入弹性、交叉弹性 | 能够计算并解释收入弹性和交叉价格弹性，判断商品类型（正常品/劣质品、替代品/互补品），并应用于需求预测和定价决策 |
二、我们要解决什么问题？
某消费者每月收入为8000元时，每月购买牛奶12升；当收入上升至12000元时，牛奶购买量增加到15升。同时，牛奶价格不变，但咖啡价格上涨20%，导致牛奶月购买量从12升上升到13.2升。如何量化收入变化和咖啡价格变化对牛奶需求的影响？这些弹性数值能帮助企业判断牛奶是正常品还是劣质品？咖啡与牛奶是替代品还是互补品？这些正是本课要解决的核心问题，也是CFA考试中需求分析与定价策略的常考点。
三、收入弹性（Income Elasticity of Demand）
收入弹性衡量消费者收入变化1%时，商品需求量变化的百分比。其公式为：
$$ E_I = \frac{\%\Delta Q_d}{\%\Delta I} = \frac{\frac{\Delta Q_d}{Q_d}}{\frac{\Delta I}{I}} $$
其中，$Q_d$为需求量，$I$为收入。通常使用中点法（arc elasticity）以减少计算偏差：
$$ E_I = \frac{\frac{Q_2 - Q_1}{(Q_2 + Q_1)/2}}{\frac{I_2 - I_1}{(I_2 + I_1)/2}} $$
收入弹性的分类与经济含义：
- $E_I > 1$：奢侈品（Luxury Goods），收入增加时需求量增加更快，如高端汽车、海外旅游。
- $0 < E_I < 1$：正常必需品（Normal Necessities），如食品、普通服装。
- $E_I = 0$：中性品，收入变化不影响需求。
- $E_I < 0$：劣质品（Inferior Goods），收入增加时需求量反而下降，如廉价公交、方便面。
收入弹性为正的商品称为正常品（Normal Goods），为负的称为劣质品。企业可据此预测经济周期对不同产品销售的影响：在经济扩张期，奢侈品销售增长更快。
四、交叉价格弹性（Cross-price Elasticity of Demand）
交叉价格弹性衡量一种商品价格变化1%时，另一种商品需求量变化的百分比。公式为：
$$ E_{XY} = \frac{\%\Delta Q_X}{\%\Delta P_Y} = \frac{\frac{\Delta Q_X}{Q_X}}{\frac{\Delta P_Y}{P_Y}} $$
同样推荐使用中点法。符号至关重要：
- $E_{XY} > 0$：替代品（Substitutes），如牛肉与猪肉。一种商品涨价，另一种需求增加。
- $E_{XY} < 0$：互补品（Complements），如咖啡与咖啡伴侣。一种商品涨价，另一种需求减少。
- $E_{XY} = 0$：独立品，无明显关系。
交叉弹性绝对值越大，两种商品的替代或互补关系越强。企业可利用此指标进行捆绑销售或竞争策略制定。例如，若两种商品交叉弹性为-2.5，则提价一种商品会显著降低互补品的销量。
五、弹性计算中的注意事项
- 收入弹性与交叉弹性均可使用点弹性和弧弹性两种方法，考试中若未特别说明，通常采用弧弹性（中点法）。
- 收入弹性与价格弹性（自身价格弹性）共同构成需求函数的完整弹性分析框架。
- 弹性数值无单位，是纯比率，便于不同商品间比较。
- 对于多品种企业，准确测算不同产品的收入弹性和交叉弹性有助于优化产品组合和定价。
完整案例演算
案例 1：收入弹性判断商品类型
某家庭年收入从6万元上升至7.5万元，对有机蔬菜的年需求量从120kg增加到165kg。计算收入弹性并判断商品类型。
解答： 使用中点法： $$\Delta I = 75000 - 60000 = 15000, \quad 平均收入 = \frac{75000+60000}{2}=67500$$ $$\Delta Q = 165 - 120 = 45, \quad 平均数量 = \frac{165+120}{2}=142.5$$
$$ E_I = \frac{45/142.5}{15000/67500} = \frac{0.3158}{0.2222} \approx 1.42 $$
因为$E_I = 1.42 > 1$，有机蔬菜属于奢侈品。收入每增长1%，需求量增长约1.42%。
案例 2：交叉弹性判断替代或互补关系
咖啡价格从每杯25元上涨至30元，导致茶饮料月销量从8000杯增加到9200杯。计算咖啡与茶饮料的交叉价格弹性，并判断二者关系。
解答： $$\Delta P_{coffee} = 30-25=5, \quad 平均价格 = 27.5$$ $$\Delta Q_{tea} = 9200-8000=1200, \quad 平均数量 = 8600$$
$$ E_{coffee,tea} = \frac{1200/8600}{5/27.5} = \frac{0.1395}{0.1818} \approx 0.767 > 0 $$
交叉弹性为正，说明咖啡与茶饮料是替代品。咖啡价格每上涨1%，茶饮料需求量约增加0.767%。
案例 3：综合应用——企业定价决策
某公司生产智能手机（A）和手机保护壳（B）。已知收入弹性$E_{I,A}=1.8$，$E_{I,B}=0.6$；交叉弹性$E_{A,B}=-1.2$（A价格对B需求的影响）。若经济预计增长5%，智能手机计划提价8%，预测保护壳需求量变化百分比。
解答： 保护壳需求变化 = 收入弹性影响 + 交叉价格弹性影响 $$ \%\Delta Q_B = E_{I,B} \times \%\Delta I + E_{A,B} \times \%\Delta P_A $$ $$ = 0.6 \times 5\% + (-1.2) \times 8\% = 3\% - 9.6\% = -6.6\% $$
预测保护壳需求量将下降6.6%。公司应考虑是否调整保护壳定价或加强营销以抵消负面影响。
易错陷阱对照
| 易错点 | 错误做法 | 正确做法 | 考试陷阱 |
|---|---|---|---|
| 收入弹性符号 | 仅看绝对值判断正常/劣质品 | 必须关注正负：正为正常品，负为劣质品 | 给出$E_I=-0.8$，问是否为正常品 |
| 交叉弹性正负 | 混淆替代与互补 | 正=替代，负=互补 | 题目说“汽油价格上涨导致汽车需求下降”，问交叉弹性符号 |
| 中点法 vs 点弹性 | 直接用初始值计算百分比 | 考试明确要求或数据为两点时必须用中点法 | 两种方法结果差异大导致选项不同 |
| 弹性与斜率混淆 | 认为斜率越大弹性越大 | 弹性是百分比变化比率，与斜率不同 | 线性需求曲线不同点弹性不同 |
| 单位错误 | 收入用“元”，需求用“千克”直接相除 | 必须都换算为百分比 | 忘记百分比导致数量级错误 |
关键公式 / 关系速记
- 收入弹性：$E_I = \frac{\%\Delta Q_d}{\%\Delta I}$
- 交叉价格弹性：$E_{XY} = \frac{\%\Delta Q_X}{\%\Delta P_Y}$
- 正常品：$E_I > 0$；劣质品：$E_I < 0$
- 替代品：$E_{XY} > 0$；互补品：$E_{XY} < 0$
- 奢侈品：$E_I > 1$；必需品：$0 < E_I < 1$
- 弧弹性（中点法）更准确，避免端点偏差
- 需求量变化预测：$\% \Delta Q = E_I \times \% \Delta I + E_{XY} \times \% \Delta P_Y$
练习题（含计算与情景）
Q1. 若某商品收入弹性为-0.65，则该商品最可能是：
A. 奢侈品
B. 正常必需品
C. 劣质品
D. 中性品
Q2. 苹果价格上涨导致橘子需求量增加，苹果与橘子的交叉价格弹性最可能为：
A. -1.2
B. 0
C. 0.8
D. -0.5
Q3. 某消费者收入增加15%，对牛肉的需求量增加9%。牛肉的收入弹性为：
A. 0.6
B. 1.5
C. 1.67
D. 0.4
Q4. 如果两种商品的交叉弹性为-2.3，则以下说法正确的是：
A. 它们是强替代品
B. 它们是强互补品
C. 它们是独立品
D. 一种是奢侈品另一种是劣质品
Q5. 使用中点法计算：收入从5000元增至6000元，某商品需求量从200单位降至170单位。该商品收入弹性最接近：
A. -0.82
B. -0.75
C. 0.75
D. 1.33
Q6. 某航空公司发现机票收入弹性为2.1，这意味着经济衰退时其销售额：
A. 下降幅度小于GDP下降幅度
B. 下降幅度大于GDP下降幅度
C. 与GDP变化无关
D. 呈反向小幅变化
Q7. 若咖啡与糖的交叉弹性为-0.9，咖啡价格上涨10%，则糖的需求量预计变化：
A. 增加9%
B. 减少9%
C. 增加0.9%
D. 减少0.9%
Q8. 以下关于收入弹性和交叉弹性的表述，错误的是：
A. 两者均可用于预测需求变化
B. 收入弹性为负的商品在经济繁荣期销量通常下降
C. 交叉弹性绝对值越大，商品间关系越弱
D. 奢侈品的收入弹性大于1
答案与详解
| 题号 | 答案 | 详解 |
|---|---|---|
| Q1 | C | 收入弹性为负，属于劣质品（Inferior Good）。 |
| Q2 | C | 交叉弹性为正，说明二者为替代品（Substitutes）。 |
| Q3 | A | $E_I = 9\% / 15\% = 0.6$，属于正常必需品。 |
| Q4 | B | 交叉弹性为-2.3且绝对值较大，表明为强互补品（Strong Complements）。 |
| Q5 | A | 中点法：$\frac{(170-200)/185}{(6000-5000)/5500} = \frac{-0.1622}{0.1818} \approx -0.89$，最接近-0.82（选项中最近值）。 |
| Q6 | B | 收入弹性>1，属于奢侈品，经济衰退时需求下降幅度大于收入下降幅度。 |
| Q7 | B | $E_{XY}=-0.9$，咖啡涨价10%导致糖需求量变化 = -0.9×10% = -9%。 |
| Q8 | C | 交叉弹性绝对值越大，商品间的替代或互补关系越强，而非越弱。 |
本节要点速记
- 收入弹性正负决定正常品与劣质品，数值大小区分必需品与奢侈品。
- 交叉弹性正负决定替代品与互补品，绝对值反映关系强弱。
- 必须熟练掌握中点法计算，避免使用单点初始值导致误差。
- 弹性可直接用于多因素需求预测：$\% \Delta Q = E_I \times \% \Delta I + \sum E_{XY} \times \% \Delta P_Y$。
- 经济周期对不同$E_I$商品的影响差异显著，企业据此调整产品结构。
- 理解弹性本质是“百分比变化的比率”，而非绝对数量变化。
Economics
I. Lesson Focus
This lesson examines two key demand elasticities beyond own-price elasticity: income elasticity and cross-price elasticity. Candidates must be able to calculate both using the midpoint (arc) method, interpret their signs and magnitudes, classify goods as normal/inferior and substitutes/complements, and apply the concepts to forecast demand changes and support pricing decisions. The material integrates directly with earlier price elasticity topics and appears frequently in economics vignettes involving consumer behavior and business strategy.
II. The Problem
A consumer buys 12 liters of milk per month at an income of CNY 8,000. When income rises to CNY 12,000, milk purchases increase to 15 liters. At the same time, a 20% rise in coffee prices causes milk purchases to rise from 12 liters to 13.2 liters even though milk’s own price is unchanged. How can we quantify the separate effects of the income increase and the coffee price increase on milk demand? Can these elasticity values tell us whether milk is a normal or inferior good and whether coffee and milk are substitutes or complements? These questions lie at the heart of demand analysis, forecasting, and pricing strategy—core testable areas in the CFA Level I Economics curriculum.
III. Income Elasticity of Demand
Income elasticity of demand ($E_I$) measures the responsiveness of quantity demanded to a change in consumer income. It is defined as the percentage change in quantity demanded divided by the percentage change in income:
$$ E_I = \frac{\%\Delta Q_d}{\%\Delta I} = \frac{\frac{\Delta Q_d}{Q_d}}{\frac{\Delta I}{I}} $$
When two data points are provided, the midpoint (arc) formula is preferred to reduce endpoint bias:
$$ E_I = \frac{\frac{Q_2 - Q_1}{(Q_2 + Q_1)/2}}{\frac{I_2 - I_1}{(I_2 + I_1)/2}} $$
Classification and Economic Meaning:
- $E_I > 1$: Luxury goods. Demand rises faster than income (e.g., high-end cars, international vacations).
- $0 < E_I < 1$: Normal necessities (e.g., staple foods, basic clothing).
- $E_I = 0$: Neutral goods. Demand is unaffected by income changes.
- $E_I < 0$: Inferior goods. Demand falls as income rises (e.g., low-quality bus travel, instant noodles).
Goods with positive income elasticity are called normal goods; those with negative income elasticity are inferior goods. Firms use income elasticity to forecast how different products will perform across the business cycle. Luxury goods typically experience amplified sales growth during economic expansions and sharper declines during contractions.
IV. Cross-price Elasticity of Demand
Cross-price elasticity of demand ($E_{XY}$) measures how the quantity demanded of good X changes when the price of good Y changes. The formula is:
$$ E_{XY} = \frac{\%\Delta Q_X}{\%\Delta P_Y} = \frac{\frac{\Delta Q_X}{Q_X}}{\frac{\Delta P_Y}{P_Y}} $$
The midpoint method is again recommended. The sign of the elasticity is critical for classification:
- $E_{XY} > 0$: The goods are substitutes (e.g., beef and pork). An increase in the price of Y raises demand for X.
- $E_{XY} < 0$: The goods are complements (e.g., coffee and coffee creamer). An increase in the price of Y lowers demand for X.
- $E_{XY} = 0$: The goods are independent; no meaningful relationship exists.
The larger the absolute value of cross-price elasticity, the stronger the substitution or complementary relationship. Managers can exploit these relationships in bundling, competitive pricing, or product-line decisions. For example, a cross elasticity of –2.5 between two products implies that a price increase in one will significantly hurt sales of the other.
V. Practical Considerations in Elasticity Calculations
- Both income and cross-price elasticities can be calculated with point or arc methods. Unless the question specifies otherwise, use the midpoint (arc) method when two points are given.
- Income elasticity and own-price elasticity together provide a complete picture of demand responsiveness.
- Elasticities are unit-free ratios, allowing direct comparison across goods with different units or price levels.
- Multi-product firms rely on accurate estimates of income and cross elasticities to optimize product mix and set coordinated pricing policies.
Worked Cases
Case 1: Income Elasticity and Good Classification
A household’s annual income rises from CNY 60,000 to CNY 75,000, and its annual demand for organic vegetables increases from 120 kg to 165 kg. Calculate the income elasticity and classify the good.
Solution: Using the midpoint method:
$$\Delta I = 15,000,\quad \text{Average } I = 67,500$$ $$\Delta Q = 45,\quad \text{Average } Q = 142.5$$
$$ E_I = \frac{45/142.5}{15,000/67,500} = \frac{0.3158}{0.2222} \approx 1.42 $$
Because $E_I = 1.42 > 1$, organic vegetables are classified as a luxury good. For every 1% increase in income, quantity demanded rises by approximately 1.42%.
Case 2: Cross-price Elasticity and Substitute/Complement Classification
The price of a cup of coffee rises from CNY 25 to CNY 30, causing monthly sales of tea drinks to increase from 8,000 cups to 9,200 cups. Calculate the cross-price elasticity between coffee and tea and determine their relationship.
Solution:
$$\Delta P_{\text{coffee}} = 5,\quad \text{Average } P = 27.5$$ $$\Delta Q_{\text{tea}} = 1,200,\quad \text{Average } Q = 8,600$$
$$ E_{\text{coffee,tea}} = \frac{1,200/8,600}{5/27.5} = \frac{0.1395}{0.1818} \approx 0.767 > 0 $$
The positive cross elasticity indicates that coffee and tea are substitutes. A 1% increase in coffee price raises tea demand by about 0.767%.
Case 3: Combined Application — Corporate Demand Forecasting
A company sells smartphones (A) and phone cases (B). Given $E_{I,A}=1.8$, $E_{I,B}=0.6$, and $E_{A,B}=-1.2$ (effect of smartphone price on case demand). If the economy is expected to grow 5% and the company plans an 8% smartphone price increase, forecast the percentage change in phone-case demand.
Solution: Percentage change in $Q_B$ = income effect + cross-price effect
$$ \%\Delta Q_B = E_{I,B} \times \%\Delta I + E_{A,B} \times \%\Delta P_A $$ $$ = 0.6 \times 5\% + (-1.2) \times 8\% = 3.0\% - 9.6\% = -6.6\% $$
Phone-case demand is forecasted to decline 6.6%. Management should consider adjusting case pricing or increasing promotional effort to offset the negative impact.
Traps
| Common Mistake | Incorrect Approach | Correct Approach | Typical Exam Trap |
|---|---|---|---|
| Sign of income elasticity | Judging normal/inferior solely by absolute value | Sign matters: positive = normal good, negative = inferior good | Question gives $E_I = -0.8$ and asks if the good is normal |
| Sign of cross elasticity | Confusing substitutes and complements | Positive = substitutes, negative = complements | “Gasoline price rise reduces car demand”—what is the sign of cross elasticity? |
| Midpoint vs. initial-point calculation | Using only beginning values for percentages | Use midpoint method when two distinct points are supplied | Large numerical difference between methods produces different answer choices |
| Confusing slope with elasticity | Believing steeper slope means higher elasticity | Elasticity is a ratio of percentage changes, not slope | Linear demand curve has different elasticity at every point |
| Unit or scaling errors | Dividing raw income (yuan) by raw quantity (kg) | All changes must be converted to percentages | Forgetting to use percentages produces off-by-factor-of-100 errors |
Key Formulas
- Income elasticity: $E_I = \frac{\%\Delta Q_d}{\%\Delta I}$
- Cross-price elasticity: $E_{XY} = \frac{\%\Delta Q_X}{\%\Delta P_Y}$
- Normal good: $E_I > 0$; Inferior good: $E_I < 0$
- Substitutes: $E_{XY} > 0$; Complements: $E_{XY} < 0$
- Luxury: $E_I > 1$; Necessity: $0 < E_I < 1$
- Midpoint (arc) elasticity is preferred when two data points are available
- Multi-factor demand forecast: $\%\Delta Q = E_I \times \%\Delta I + E_{XY} \times \%\Delta P_Y$
Practice Questions
Q1. If a good has an income elasticity of –0.65, it is most likely:
A. a luxury good
B. a normal necessity
C. an inferior good
D. a neutral good
Q2. An increase in the price of apples causes orange demand to rise. The cross-price elasticity between apples and oranges is most likely:
A. –1.2
B. 0
C. 0.8
D. –0.5
Q3. A consumer’s income rises 15% and beef consumption rises 9%. Beef’s income elasticity is closest to:
A. 0.6
B. 1.5
C. 1.67
D. 0.4
Q4. If the cross-price elasticity between two goods is –2.3, the goods are most likely:
A. strong substitutes
B. strong complements
C. independent
D. one luxury and one inferior
Q5. Using the midpoint method, income rises from 5,000 to 6,000 while demand for a good falls from 200 to 170 units. The income elasticity is closest to:
A. –0.82
B. –0.75
C. 0.75
D. 1.33
Q6. An airline discovers that the income elasticity of demand for its tickets is 2.1. In an economic recession, ticket sales will most likely:
A. fall by less than the decline in GDP
B. fall by more than the decline in GDP
C. remain unchanged
D. move inversely but modestly
Q7. The cross-price elasticity between coffee and sugar is –0.9. If the price of coffee rises 10%, sugar demand is expected to change by:
A. +9%
B. –9%
C. +0.9%
D. –0.9%
Q8. Which of the following statements about income and cross-price elasticities is least accurate?
A. Both can be used to forecast demand changes.
B. Goods with negative income elasticity usually see falling sales during economic booms.
C. The larger the absolute value of cross elasticity, the weaker the relationship between the goods.
D. Luxury goods have income elasticity greater than 1.
Answers
| Question | Answer | Explanation |
|---|---|---|
| Q1 | C | Negative income elasticity identifies an inferior good. |
| Q2 | C | Positive cross elasticity indicates substitutes. |
| Q3 | A | $E_I = 9\% / 15\% = 0.6$; normal necessity. |
| Q4 | B | Strongly negative cross elasticity indicates strong complements. |
| Q5 | A | Midpoint calculation yields approximately –0.89, closest to –0.82 among the choices. |
| Q6 | B | Elasticity > 1 implies luxury-good behavior: sales fall more than proportionally with income. |
| Q7 | B | –0.9 × 10% = –9% decline in sugar demand. |
| Q8 | C | Larger absolute cross elasticity means stronger (not weaker) substitution or complementarity. |
Takeaways
- The sign of income elasticity distinguishes normal goods ($E_I>0$) from inferior goods ($E_I<0$); its magnitude separates necessities ($0<E_I<1$) from luxuries ($E_I>1$).
- The sign of cross-price elasticity identifies substitutes (positive) versus complements (negative); its absolute size reflects relationship strength.
- Master the midpoint formula to avoid endpoint bias—exam questions deliberately test this distinction.
- Elasticities enable straightforward multi-variable demand forecasts: $\%\Delta Q = E_I\times\%\Delta I + E_{XY}\times\%\Delta P_Y$.
- Different products exhibit markedly different cyclical sensitivity according to their income elasticities; firms adjust product portfolios accordingly.
- Elasticity is fundamentally a ratio of percentage changes, not absolute quantity or slope—avoid confusing the two concepts.