经济学（Economics）
一、本课定位
| 课次 | 主题 | 能力 |
|---|---|---|
| L158 | 垄断定价与产量决策 | 能够计算垄断企业的利润最大化产量与价格，比较垄断与完全竞争的市场结果，并识别价格歧视的影响 |
二、我们要解决什么问题？
某企业是某地区唯一的高速铁路运营商，它面临一条向下倾斜的需求曲线。如果它像完全竞争企业那样按边际成本定价，将亏损；如果定价过高，又会失去大量乘客。考试中经常要求考生计算该垄断企业实现利润最大化时的产量、价格、边际收益、平均总成本，并比较其与完全竞争市场的社会福利损失（deadweight loss）。本课将系统解决垄断企业的最优定价与产量决策问题。
三、垄断的基本特征与需求曲线
垄断（Monopoly）是指市场上只有一家卖方，且不存在相近替代品的市场结构。其核心特征包括： - 单一卖方 - 产品无相近替代品 - 进入壁垒极高（政府特许、专利、规模经济、自然垄断等） - 企业面对整个市场的向下倾斜的需求曲线（Demand Curve）
与完全竞争企业不同，垄断企业是价格制定者（Price Maker）。其需求曲线即为市场需求曲线，因此边际收益（MR）曲线位于需求曲线（D）下方，且斜率是需求曲线斜率的两倍。
关键关系：
对于线性需求曲线 $P = a - bQ$，边际收益为 $MR = a - 2bQ$。
四、垄断企业的利润最大化决策
垄断企业通过选择产量使边际收益等于边际成本（MR = MC）来实现利润最大化。此时对应的价格从需求曲线上读出。
决策规则： 1. 在 $MR = MC$ 处确定最优产量 $Q_m$ 2. 在需求曲线上找到对应价格 $P_m$ 3. 若 $P_m > ATC$，企业获得经济利润；若 $P_m = ATC$，经济利润为零；若 $P_m < ATC$ 但高于 AVC，则短期继续经营
垄断企业不会在需求曲线无弹性的部分生产，因为此时 $MR < 0$，增加产量会减少总收益。
五、垄断与完全竞争的市场比较
垄断会导致资源配置无效率，主要表现为：
- 产量更低（$Q_m < Q_c$）
- 价格更高（$P_m > P_c = MC$）
- 存在消费者剩余向生产者剩余的转移以及无谓损失（Deadweight Loss, DWL）
无谓损失 = 垄断下损失的消费者剩余与生产者剩余之和，可通过三角形面积计算：
$DWL = \frac{1}{2} \times (P_m - MC) \times (Q_c - Q_m)$
六、价格歧视（Price Discrimination）
垄断企业可通过价格歧视提高利润，分为三类：
- 一级价格歧视（Perfect Price Discrimination）：对每一单位索取消费者愿意支付的最高价格，捕获全部消费者剩余，此时 $DWL = 0$，产量等于完全竞争水平。
- 二级价格歧视（Second-degree）：数量折扣或分块定价（如阶梯电价）。
- 三级价格歧视（Third-degree）：按不同细分市场弹性差异定价（如学生票、商务舱），在每个市场分别满足 $MR_1 = MR_2 = MC$。
完整案例演算
案例 1：线性需求下的垄断定价
某垄断企业的需求函数为 $P = 120 - 2Q$，边际成本 $MC = 20$（固定），平均总成本 $ATC = 20 + \frac{400}{Q}$。
求解： 1. $MR = 120 - 4Q$ 2. 令 $MR = MC$：$120 - 4Q = 20 \Rightarrow 4Q = 100 \Rightarrow Q_m = 25$ 3. $P_m = 120 - 2 \times 25 = 70$ 4. $ATC = 20 + \frac{400}{25} = 36$ 5. 经济利润 = $(70 - 36) \times 25 = 850$
完全竞争下：$P = MC = 20$，$Q_c = 50$，$DWL = \frac{1}{2} \times (70 - 20) \times (50 - 25) = 625$。
案例 2：自然垄断的监管
某自然垄断企业 $MC = 10$，$ATC = 10 + \frac{1000}{Q}$，需求 $P = 60 - 0.5Q$。
- 利润最大化：$MR = 60 - Q = 10 \Rightarrow Q_m = 50$，$P_m = 35$，利润为正。
- 边际成本定价（监管）：$P = MC = 10$，$Q = 100$，但 $ATC = 20$，企业亏损。
- 平均成本定价（公平报酬率监管）：$P = ATC$，解得 $Q \approx 35.7$，$P \approx 42.15$，经济利润为零。
案例 3：三级价格歧视
某垄断企业在国内市场需求 $P_d = 80 - 2Q_d$，国外市场 $P_f = 50 - Q_f$，总 $MC = 10$。
分别计算： - 国内：$MR_d = 80 - 4Q_d = 10 \Rightarrow Q_d = 17.5$，$P_d = 45$ - 国外：$MR_f = 50 - 2Q_f = 10 \Rightarrow Q_f = 20$，$P_f = 30$ - 总产量 = 37.5，总利润高于单一价格时的利润。
易错陷阱对照
| 易错点 | 错误做法 | 正确做法 |
|---|---|---|
| 混淆 MR 与需求曲线 | 用需求曲线代替 MR 求最优产量 | 必须用 $MR = MC$ 确定产量，再在需求曲线上找价格 |
| 认为垄断企业无经济利润 | 认为长期必然零利润 | 垄断因进入壁垒，长期可维持正经济利润 |
| 错误计算无谓损失 | 只算消费者剩余损失 | DWL 是消费者剩余损失中未被生产者获得的三角形部分 |
| 价格歧视判断 | 认为所有垄断都可实行一级歧视 | 一级歧视要求完全信息且无套利可能，现实中罕见 |
| 自然垄断监管 | 认为 MC 定价总是最优 | MC 定价会导致企业亏损，常用 AC 定价实现公平回报 |
| 弹性与 MR 关系 | 忘记 $MR < 0$ 时不应生产 | 垄断企业只在需求弹性大于 1 的区域生产 |
关键公式 / 关系速记
- 线性需求：若 $P = a - bQ$，则 $MR = a - 2bQ$
- 利润最大化条件：$MR = MC$
- 垄断价格：$P_m = D(Q_m)$
- 无谓损失：$DWL = \frac{1}{2}(P_m - MC)(Q_c - Q_m)$
- 勒纳指数（Lerner Index）：$L = \frac{P - MC}{P} = -\frac{1}{E_d}$
- 三级价格歧视条件：$MR_1 = MR_2 = MC$
练习题（含计算与情景）
Q1. 某垄断企业的需求曲线为 $P = 100 - 4Q$，$MC = 20$。其利润最大化产量为多少？
A. 10
B. 15
C. 20
D. 25
Q2. 在上述 Q1 中，垄断价格为多少？
A. 20
B. 40
C. 60
D. 80
Q3. 与完全竞争市场相比，垄断市场一定存在：
A. 更高的消费者剩余
B. 更低的生产者剩余
C. 正的无谓损失
D. 更低的进入壁垒
Q4. 一级价格歧视的垄断企业会将产量设定在：
A. 完全竞争产量水平
B. 低于完全竞争产量
C. 高于完全竞争产量
D. 使 MR = 0 的产量
Q5. 某自然垄断企业按平均成本定价时，其经济利润为：
A. 正
B. 零
C. 负
D. 不确定
Q6. 若需求曲线斜率为 -2，则边际收益曲线的斜率为：
A. -1
B. -2
C. -4
D. -8
Q7. 以下哪项不是垄断企业维持长期正经济利润的必要条件？
A. 高进入壁垒
B. 产品无相近替代品
C. 政府管制
D. 规模经济
Q8. 某垄断企业在两个市场实行三级价格歧视，市场 A 需求弹性绝对值更大，则其在市场 A 的定价会：
A. 更高
B. 更低
C. 相同
D. 无法判断
答案与详解
| 题号 | 答案 | 详解 |
|---|---|---|
| Q1 | A | $MR = 100 - 8Q = 20 \Rightarrow 8Q = 80 \Rightarrow Q = 10$ |
| Q2 | B | $P = 100 - 4 \times 10 = 60$（选项中 B 为 40 为陷阱，正确应为 60，实际答案对应 C，但此处按标准计算为 60，选项调整后选 C；为匹配原题，此处以 A=10 正确，价格对应 C=60）注：标准答案 Q2 选 C |
| Q3 | C | 垄断必然产生无谓损失，减少总剩余 |
| Q4 | A | 一级价格歧视下 $P = MC$，产量等于完全竞争产量 |
| Q5 | B | 平均成本定价使 $P = ATC$，经济利润为零 |
| Q6 | C | 线性需求下 MR 斜率是需求斜率的两倍，即 -4 |
| Q7 | C | 政府管制通常限制垄断利润，而非维持利润 |
| Q8 | B | 弹性更大（更平坦）的市场，垄断企业会制定更低价格 |
本节要点速记
- 垄断利润最大化条件始终是 $MR = MC$，而非 $P = MC$
- 垄断企业的 MR 曲线位于需求曲线下方且斜率加倍
- 垄断造成产量减少、价格上升和无谓损失
- 一级价格歧视可消除无谓损失并捕获全部消费者剩余
- 自然垄断监管中，平均成本定价可实现零经济利润
- 勒纳指数衡量垄断势力，与需求价格弹性负相关
Economics
I. Lesson Focus
This lesson examines how a monopolist determines its profit-maximizing price and output, contrasts monopoly outcomes with perfect competition, and analyzes the welfare effects and different forms of price discrimination. Candidates must be able to calculate marginal revenue from a demand curve, locate the MR = MC point, read the corresponding price from the demand curve, compute deadweight loss, and evaluate regulatory responses to natural monopoly.
II. The Problem
A firm is the sole high-speed rail operator in a region and faces a downward-sloping demand curve. Pricing at marginal cost would produce losses, yet charging too high a price loses many passengers. CFA exams frequently require candidates to calculate the monopolist’s profit-maximizing quantity and price, compare them with the perfectly competitive benchmark, compute deadweight loss, and assess the impact of price discrimination. This lesson systematically solves the optimal pricing and output decision for a monopolist.
III. Core Characteristics of Monopoly and the Demand Curve
A monopoly exists when there is a single seller and no close substitutes. Key features include: - Single seller - No close substitutes - Very high barriers to entry (government franchise, patents, economies of scale, natural monopoly) - The firm faces the entire market’s downward-sloping demand curve
Unlike a perfectly competitive firm, a monopolist is a price maker. Its demand curve is the market demand curve; therefore, the marginal revenue (MR) curve lies below the demand (D) curve and has twice the slope.
Key Relationship: For a linear demand curve $P = a - bQ$, marginal revenue is $MR = a - 2bQ$.
IV. Profit-Maximization Rule for a Monopolist
A monopolist maximizes profit by producing the output level where marginal revenue equals marginal cost (MR = MC). The price is then read from the demand curve at that quantity.
Decision Rule: 1. Set $MR = MC$ to find optimal monopoly quantity $Q_m$ 2. Locate the corresponding price $P_m$ on the demand curve 3. If $P_m > ATC$, the firm earns positive economic profit; if $P_m = ATC$, profit is zero; if $P_m < ATC$ but above AVC, the firm continues in the short run
A monopolist never produces in the inelastic portion of the demand curve because MR would be negative and total revenue would fall when output increases.
V. Comparison of Monopoly and Perfect Competition
Monopoly creates allocative inefficiency, manifested as: - Lower output ($Q_m < Q_c$) - Higher price ($P_m > P_c = MC$) - Transfer of consumer surplus to producer surplus plus a deadweight loss (DWL)
Deadweight Loss is the lost consumer and producer surplus that is not transferred. It is the triangular area:
$DWL = \frac{1}{2} \times (P_m - MC) \times (Q_c - Q_m)$
VI. Price Discrimination
A monopolist can increase profit through price discrimination. There are three degrees:
- First-degree (perfect) price discrimination: Charging each consumer the maximum price he or she is willing to pay. All consumer surplus is captured, DWL = 0, and output equals the competitive level.
- Second-degree: Block pricing or quantity discounts (e.g., declining-block electricity tariffs).
- Third-degree: Charging different prices in different market segments based on elasticity (e.g., student vs. business fares). The firm sets $MR_1 = MR_2 = MC$.
Worked Cases
Case 1: Linear Demand Monopoly Pricing
A monopolist faces demand $P = 120 - 2Q$, constant $MC = 20$, and $ATC = 20 + \frac{400}{Q}$.
Solution: 1. $MR = 120 - 4Q$ 2. Set $MR = MC$: $120 - 4Q = 20 \Rightarrow Q_m = 25$ 3. $P_m = 120 - 2 \times 25 = 70$ 4. $ATC = 20 + \frac{400}{25} = 36$ 5. Economic profit = $(70 - 36) \times 25 = 850$
Under perfect competition: $P = MC = 20$, $Q_c = 50$.
$DWL = \frac{1}{2} \times (70 - 20) \times (50 - 25) = 625$.
Case 2: Natural Monopoly Regulation
A natural monopolist has $MC = 10$, $ATC = 10 + \frac{1000}{Q}$, and demand $P = 60 - 0.5Q$.
- Profit-maximizing: $MR = 60 - Q = 10 \Rightarrow Q_m = 50$, $P_m = 35$, positive profit.
- Marginal-cost pricing (efficient regulation): $P = 10$, $Q = 100$, but $ATC = 20$, firm incurs losses.
- Average-cost pricing (fair-return regulation): Solve $P = ATC$, yielding $Q \approx 35.7$, $P \approx 42.15$, zero economic profit.
Case 3: Third-Degree Price Discrimination
A monopolist serves a domestic market $P_d = 80 - 2Q_d$ and foreign market $P_f = 50 - Q_f$ with constant $MC = 10$.
- Domestic: $MR_d = 80 - 4Q_d = 10 \Rightarrow Q_d = 17.5$, $P_d = 45$
- Foreign: $MR_f = 50 - 2Q_f = 10 \Rightarrow Q_f = 20$, $P_f = 30$
- Total output = 37.5. Profit exceeds the single-price monopoly profit.
Traps
| Common Mistake | Incorrect Approach | Correct Approach |
|---|---|---|
| Confusing MR with demand | Using demand curve instead of MR to find optimal Q | Always solve $MR = MC$ first, then read price from demand |
| Believing monopolies earn zero long-run profit | Assuming free entry forces zero profit | High barriers allow positive economic profit indefinitely |
| Miscalculating deadweight loss | Calculating only lost consumer surplus | DWL is the triangular area of net surplus loss |
| Assuming all monopolies can perfectly discriminate | Treating first-degree discrimination as common | Requires perfect information and no arbitrage; rare in practice |
| Regulatory pricing error | Thinking MC pricing is always best | MC pricing causes losses; average-cost pricing is often used for zero profit |
| Elasticity–MR relationship | Forgetting MR < 0 region is never used | Monopolist operates only where demand is elastic ($ |
Key Formulas
- Linear demand: If $P = a - bQ$, then $MR = a - 2bQ$
- Profit-maximization: $MR = MC$
- Monopoly price: $P_m = D(Q_m)$
- Deadweight loss: $DWL = \frac{1}{2}(P_m - MC)(Q_c - Q_m)$
- Lerner Index: $L = \frac{P - MC}{P} = -\frac{1}{E_d}$
- Third-degree discrimination: $MR_1 = MR_2 = MC$
Practice Questions
Q1. A monopolist faces demand $P = 100 - 4Q$ and $MC = 20$. Its profit-maximizing output is closest to:
A. 10
B. 15
C. 20
D. 25
Q2. In Q1, the monopoly price is closest to:
A. 20
B. 40
C. 60
D. 80
Q3. Compared with perfect competition, a monopoly always produces:
A. Higher consumer surplus
B. Lower producer surplus
C. Positive deadweight loss
D. Lower barriers to entry
Q4. A perfectly price-discriminating monopolist produces at the:
A. Competitive output level
B. Output below the competitive level
C. Output above the competitive level
D. Output where MR = 0
Q5. When a natural monopolist is regulated to set price equal to average total cost, its economic profit is:
A. Positive
B. Zero
C. Negative
D. Indeterminate
Q6. If the demand curve has a slope of –2, the marginal revenue curve has a slope of:
A. –1
B. –2
C. –4
D. –8
Q7. Which of the following is least likely required for a monopolist to sustain positive economic profit in the long run?
A. High barriers to entry
B. No close substitutes
C. Government regulation
D. Economies of scale
Q8. A monopolist practices third-degree price discrimination. Market A has greater demand elasticity (in absolute value) than Market B. The price charged in Market A will be:
A. Higher
B. Lower
C. The same
D. Indeterminate
Answers
| Question | Answer | Explanation |
|---|---|---|
| Q1 | A | $MR = 100 - 8Q = 20 \Rightarrow 8Q = 80 \Rightarrow Q = 10$ |
| Q2 | C | $P = 100 - 4 \times 10 = 60$ |
| Q3 | C | Monopoly always creates deadweight loss by restricting output |
| Q4 | A | First-degree discrimination results in $P = MC$ and competitive output |
| Q5 | B | Average-cost pricing sets $P = ATC$, producing zero economic profit |
| Q6 | C | For linear demand, MR slope is twice the demand slope (–4) |
| Q7 | C | Government regulation typically constrains rather than sustains monopoly profit |
| Q8 | B | The more elastic market receives the lower price under third-degree discrimination |
Takeaways
- A monopolist maximizes profit where $MR = MC$, not $P = MC$
- The MR curve lies below demand and has twice its slope for linear demand
- Monopoly produces less output, charges a higher price, and creates deadweight loss
- Perfect (first-degree) price discrimination eliminates DWL and captures all consumer surplus
- Average-cost pricing is commonly used to regulate natural monopolies to achieve zero economic profit
- The Lerner Index is inversely related to the absolute value of demand elasticity