📚 Mastering Economics: Worked Examples & Model Answers | 经济学典型例题详解与满分答案
Economics exams at IB and CIE level require not only a solid understanding of theory but also the ability to apply concepts to quantitative and qualitative questions. This article presents ten carefully selected worked examples spanning microeconomics and macroeconomics. Each example is broken down step by step, demonstrating the analytical approach expected in high-scoring answers. Use these models to sharpen your technique and deepen your understanding.
IB 和 CIE 的经济学考试不仅要求学生牢固掌握理论，还需要能够将概念应用于定量和定性分析题目。本文精选了十个典型例题，涵盖微观与宏观经济，并逐步拆解解答思路，展示高分答案的分析方法。通过这些示范，你可以提升解题技巧并加深对知识的理解。
1. Demand and Supply Equilibrium | 供需均衡计算
Question: The demand for a good is given by Qd = 500 – 8P and the supply is given by Qs = -100 + 12P, where P is price in dollars. Determine the equilibrium price and quantity. Show all steps.
问题： 某商品的需求函数为 Qd = 500 – 8P，供给函数为 Qs = -100 + 12P，其中 P 为美元价格。求均衡价格和均衡数量，并写出全部步骤。
Answer: At equilibrium, quantity demanded equals quantity supplied: Qd = Qs. Set the two equations equal: 500 – 8P = -100 + 12P. Collect like terms: 500 + 100 = 12P + 8P, leading to 600 = 20P. Hence P = 600/20 = 30. Substitute P = 30 into either equation. Using the demand equation: Qd = 500 – 8(30) = 500 – 240 = 260. Therefore, equilibrium price is $30 and equilibrium quantity is 260 units.
解答： 在市场均衡时，需求量等于供给量：Qd = Qs。令两式相等：500 – 8P = -100 + 12P。合并同类项：500 + 100 = 12P + 8P，得出 600 = 20P，即 P = 30。将 P = 30 代入需求方程：Qd = 500 – 8(30) = 500 – 240 = 260。因此，均衡价格为 30 美元，均衡数量为 260 单位。
Examiner tip: Always state the equilibrium condition first. Then show algebraic manipulation clearly. Double-check by substituting P into both equations to ensure they yield the same Q.
考官提示： 务必先写明均衡条件，然后清晰地展示代数运算。最后将求得的 P 代入两个方程验证，以确保得到相同的 Q 值。
2. Price Elasticity of Demand | 需求价格弹性计算
Question: When the price of a cinema ticket rises from $8 to $12, the quantity demanded falls from 900 tickets per week to 600 tickets. Using the midpoint formula, calculate the price elasticity of demand (PED). Interpret your result.
问题： 电影票价从 8 美元上涨到 12 美元，每周需求量从 900 张下降到 600 张。请使用中点公式计算需求价格弹性（PED），并解释结果。
Answer: Midpoint formula: Ed = (ΔQ / Qavg) / (ΔP / Pavg). ΔQ = 600 – 900 = -300, so absolute change is 300. Qavg = (900+600)/2 = 750. ΔP = 12 – 8 = 4. Pavg = (8+12)/2 = 10. Therefore Ed = (-300/750) / (4/10) = (-0.4) / 0.4 = -1.0. Taking absolute value, PED = 1.0. This indicates unit elastic demand; the percentage change in quantity demanded equals the percentage change in price. Total revenue remains unchanged.
解答： 中点公式：Ed = (ΔQ / Qavg) / (ΔP / Pavg)。ΔQ = 600 – 900 = -300，取绝对值变化量为 300。Qavg = (900+600)/2 = 750。ΔP = 12 – 8 = 4。Pavg = (8+12)/2 = 10。因此 Ed = (-300/750) / (4/10) = (-0.4) / 0.4 = -1.0。取绝对值，PED = 1.0。这表明需求具有单位弹性，需求量变动的百分比等于价格变动的百分比，总收益保持不变。
Note: Always use the midpoint formula when asked to avoid the inconsistency caused by the direction of change. Interpret the magnitude (elastic, inelastic, unit elastic) and the resulting revenue effect.
注意： 题目要求用时，必须使用中点公式，以避免因变化方向不同而导致结果不一致。解释弹性的大小（富有弹性、缺乏弹性、单位弹性）以及对总收益的影响。
3. Consumer and Producer Surplus | 消费者剩余与生产者剩余
Question: The demand curve is P = 40 – 0.5Q, and the supply curve is P = 5 + 1.5Q. Find the equilibrium, then calculate consumer surplus and producer surplus. Illustrate the areas.
问题： 需求曲线为 P = 40 – 0.5Q，供给曲线为 P = 5 + 1.5Q。求均衡点，并计算消费者剩余和生产者剩余，说明其代表的区域。
Answer: Set demand equal to supply: 40 – 0.5Q = 5 + 1.5Q. Solving: 40 – 5 = 2Q, so 35 = 2Q, Q = 17.5. Substitute to find P: P = 40 – 0.5(17.5) = 40 – 8.75 = 31.25. Consumer surplus is the area of the triangle above the price and below the demand curve. The demand curve’s vertical intercept is 40. Height = 40 – 31.25 = 8.75. Area = 0.5 × base × height = 0.5 × 17.5 × 8.75 = 76.5625. Producer surplus is the area below the price and above the supply curve. The supply curve’s vertical intercept is 5. Height = 31.25 – 5 = 26.25. Area = 0.5 × 17.5 × 26.25 = 229.6875. Total surplus = 76.56 + 229.69 = 306.25 (approx.).
解答： 令需求等于供给：40 – 0.5Q = 5 + 1.5Q。解方程：40 – 5 = 2Q，故 Q = 17.5。代入求 P：P = 40 – 0.5(17.5) = 31.25。消费者剩余是均衡价格之上、需求曲线之下的三角形面积。需求曲线的纵截距为 40，高度 = 40 – 31.25 = 8.75。面积 = 0.5 × 底 × 高 = 0.5 × 17.5 × 8.75 = 76.5625。生产者剩余是均衡价格之下、供给曲线之上的三角形面积。供给曲线的纵截距为 5，高度 = 31.25 – 5 = 26.25。面积 = 0.5 × 17.5 × 26.25 = 229.6875。总剩余约 306.25。
Exam technique: Draw the diagram even if not asked; it helps to identify intercepts and heights. Be precise with arithmetic and show the formula for triangle area.
考试技巧： 即使题目没有要求，也建议画出图形，这有助于确认截距和高度。计算要精确，并写出三角形面积公式。
4. Government Intervention: Indirect Tax | 政府干预：间接税效应
Question: The original supply equation is Qs = -80 + 20P. The government imposes a specific tax of $5 per unit. Derive the new supply equation and find the new equilibrium if demand is Qd = 400 – 10P. Calculate the tax revenue and the deadweight loss.
问题： 原供给方程为 Qs = -80 + 20P。政府对每单位征收 5 美元的从量税。推导新的供给方程，并求当需求为 Qd = 400 – 10P 时的新均衡。计算税收收入和福利净损失。
Answer: A per-unit tax shifts the supply curve upward by the tax amount. The new price received by producers is P – 5. So the new supply function is Qs_new = -80 + 20(P – 5) = -80 + 20P – 100 = -180 + 20P. Set Qd = Qs_new: 400 – 10P = -180 + 20P. Solve: 400 + 180 = 30P, 580 = 30P, P = 19.33 (approx.). This is the price paid by consumers. Producers receive P – 5 = 14.33. Equilibrium quantity Q = 400 – 10(19.33) = 206.7. Tax revenue = tax per unit × Q = 5 × 206.7 = 1033.5. Original equilibrium (no tax): 400 – 10P = -80 + 20P gives 480 = 30P, P = 16, Q = 240. Deadweight loss = 0.5 × tax × (Q_old – Q_new) = 0.5 × 5 × (240 – 206.7) = 0.5 × 5 × 33.3 = 83.25.
解答： 从量税使供给曲线向上平移税额。生产者得到的价格变为 P – 5，故新供给函数为 Qs_new = -80 + 20(P – 5) = -180 + 20P。令需求等于新供给：400 – 10P = -180 + 20P。解得 P = 19.33（消费者支付价格），生产者实际收到 14.33。均衡数量 Q = 400 – 10(19.33) = 206.7。税收 = 税额 × 数量 = 1033.5。原均衡（无税）：P = 16，Q = 240。福利净损失 = 0.5 × 税额 × 数量减少量 = 0.5 × 5 × 33.3 = 83.25。
Key point: The legal incidence falls on producers, but the economic burden is shared. Embolden the welfare analysis with a discussion of consumer and producer incidence.
关键点： 法定税负由生产者承担，但经济负担由双方分担。福利分析中要讨论消费者和生产者的税负分摊比例。
5. Price Ceiling and Shortages | 价格上限与短缺分析
Question: The market for rental apartments has demand Qd = 1000 – 10P and supply Qs = 200 + 10P. The government sets a rent ceiling of $30. Calculate the shortage and the change in consumer surplus after the price ceiling, compared to the free-market equilibrium.
问题： 租房市场需求为 Qd = 1000 – 10P，供给为 Qs = 200 + 10P。政府设定租金上限为 30 美元。计算短缺量，以及相较于自由市场均衡，实施价格上限后消费者剩余的变化。
Answer: Free-market equilibrium: 1000 – 10P = 200 + 10P → 800 = 20P, P* = 40, Q* = 1000 – 10(40) = 600. At ceiling P = 30: Qd = 1000 – 10(30) = 700, Qs = 200 + 10(30) = 500. Shortage = 700 – 500 = 200 units. Consumer surplus change: originally CS = 0.5 × (100 – 40) × 600 = 0.5 × 60 × 600 = 18000. After ceiling, consumers who rent get a lower price. We calculate the new CS area under the demand curve and above the actual price $30, but only up to quantity supplied 500. It is a trapezoid: CS = 0.5 × (100 – 30 + (100 – (30 + (500/10)?)) Actually more straightforward: height from 30 to demand intercept 100 is 70. CS after = 0.5 × (100 – 30) × 500 = 0.5 × 70 × 500 = 17500. So consumer surplus decreases by 500. Some consumers gain from lower price but lose from reduced quantity available.
解答： 自由市场均衡：1000 – 10P = 200 + 10P → P* = 40, Q* = 600。价格上限 P = 30 时：Qd = 700, Qs = 500，短缺 = 200 单位。消费者剩余变化：原 CS = 0.5 × (100 – 40) × 600 = 18000。实行上限后，由于实际成交量为 500，消费者剩余为需求曲线以下、价格 30 以上的三角形（或梯形），即 0.5 × (100 – 30) × 500 = 17500。消费者剩余减少了 500。部分消费者虽享受低租金，但可供出租的房屋减少导致总剩余下降。
Note: Always separate those who remain in the market from those who are rationed out. Producer surplus falls, and deadweight loss emerges. Use a diagram to visualize the welfare effects.
注意： 要把仍留在市场中的消费者和被挤出市场的消费者区分开。生产者剩余也会下降，并产生无谓损失。用图形直观展示福利效应。
6. Profit Maximization under Perfect Competition | 完全竞争下的利润最大化
Question: A perfectly competitive firm has total cost TC = 50 + 4Q + 2Q². The market price is $44. Find the profit-maximizing output and the profit level. Determine the firm’s shutdown price.
问题： 某完全竞争厂商的总成本函数为 TC = 50 + 4Q + 2Q²，市场价格为 44 美元。求利润最大化产量和利润水平，并确定该厂商的停业价格。
Answer: In perfect competition, P = MC. MC = d(TC)/dQ = 4 + 4Q. Set P = MC: 44 = 4 + 4Q → 40 = 4Q, Q = 10. Profit = TR – TC = (44 × 10) – [50 + 4(10) + 2(10)²] = 440 – (50 + 40 + 200) = 440 – 290 = 150. To find the shutdown price, we need minimum AVC. AVC = (4Q + 2Q²)/Q = 4 + 2Q. AVC is linear and increasing, minimum occurs at the smallest Q, but in the short run firm will shut down if P < minimum AVC. Minimum AVC occurs at output where derivative = 0? Since AVC = 4 + 2Q, it increases with Q, so theoretical minimum is at Q=0, AVC=4. However the firm only operates if P >= AVC. Set P = AVC: 4 + 2Q, at Q where MC = AVC? Not necessary; just note that shutdown point is where P = minimum AVC = 4. So if price falls below $4, the firm shuts down.
解答： 在完全竞争市场中，利润最大化条件为 P = MC。MC = 4 + 4Q。令 44 = 4 + 4Q，得 Q = 10。利润 = 总收益 – 总成本 = 440 – (50 + 40 + 200) = 150。停业价格对应平均可变成本 AVC 的最低点。AVC = (4Q + 2Q²)/Q = 4 + 2Q，该函数随 Q 递增，因此从理论上讲最低 AVC 出现在 Q 趋近于 0 时，为 4。若价格低于 4 美元，厂商应停止生产。
Exam tip: Clearly show the MC curve and its equality to price. For shutdown, distinguish between short-run (P < AVC) and long-run (P < ATC) conditions.
考试提示： 要清晰地写出 MC 以及它与价格相等的条件。对于停业决策，要区分短期（P < AVC）和长期（P < ATC）的条件。
7. Negative Externalities and Pigouvian Tax | 负外部性与庇古税
Question: The marginal private cost (MPC) of producing a chemical is MPC = 10 + 2Q. The marginal external cost (MEC) is constant at $8 per unit. Demand is given by P = 50 – Q. Find the socially optimal quantity and calculate the Pigouvian tax that corrects the externality. Illustrate the welfare gain.
问题： 生产某种化学品的边际私人成本为 MPC = 10 + 2Q，边际外部成本为常数 8 美元。需求为 P = 50 – Q。求社会最优产量，并计算能纠正外部性的庇古税，说明福利改善。
Answer: Marginal social cost MSC = MPC + MEC = 10 + 2Q + 8 = 18 + 2Q. Socially optimal output where P = MSC: 50 – Q = 18 + 2Q → 50 – 18 = 3Q, 32 = 3Q, Q = 10.67. Private equilibrium (no tax): 50 – Q = 10 + 2Q → 40 = 3Q, Q = 13.33. Overproduction of 2.67 units. Pigouvian tax should equal marginal external cost at the optimal output, which is constant $8. Imposing a tax of $8 shifts MPC to MPC + tax = 18 + 2Q, aligning with MSC. Deadweight loss without tax is the triangle between MSC and demand from Q=10.67 to 13.33. Height of DWL at Q_private: MSC(13.33) – demand price? MSC at 13.33 = 18 + 2(13.33) = 44.66, demand price = 50 – 13.33 = 36.67. The DWL = 0.5 × (13.33-10.67) × (44.66 – 36.67) ≈ 0.5 × 2.66 × 7.99 ≈ 10.63. Tax revenue = 8 × 10.67 = 85.36, used to compensate or reduce other taxes.
解答： 边际社会成本 MSC = MPC + MEC = 18 + 2Q。社会最优产量满足 P = MSC：50 – Q = 18 + 2Q，解得 Q = 10.67。私人市场均衡（无税）：50 – Q = 10 + 2Q，Q = 13.33，过度生产 2.67 单位。庇古税应等于社会最优产量下的边际外部成本，这里为常数 8 美元。征税 8 美元后，MPC 上移为 18 + 2Q，与 MSC 重叠。无税时无谓损失是 MSC 和需求曲线之间从 Q=10.67 到 13.33 的三角形面积（DWL ≈ 10.63）。税收收入 85.36 可用于补偿或减税。
Comment: This example demonstrates the need to compare private and social optima. Emphasize that the tax ‘internalizes the externality’, making polluters pay.
评注： 本例展示了对比私人最优与社会最优的必要性。强调税收“将外部性内部化”，使污染者承担成本。
8. AD-AS Model and Macroeconomic Equilibrium | AD-AS 模型与宏观均衡
Question: Suppose an economy’s aggregate demand is given by AD: Y = 1200 – 20P, and short-run aggregate supply is SRAS: Y = -400 + 40P. Full-employment output Yf = 800. Find the short-run equilibrium price level and output. Calculate the output gap. Propose a fiscal policy to close the gap.
问题： 假设某经济体的总需求为 Y = 1200 – 20P，短期总供给为 Y = -400 + 40P，充分就业产出 Yf = 800。求短期均衡价格水平和产出，计算产出缺口，并提出能够消除缺口的财政政策。
Answer: Set AD = SRAS: 1200 – 20P = -400 + 40P → 1200 + 400 = 60P, 1600 = 60P, P = 26.67. Equilibrium Y = 1200 – 20(26.67) = 1200 – 533.4 = 666.6. Since Y < Yf (800), there is a recessionary gap of 800 – 666.6 = 133.4. To close the gap, expansionary fiscal policy is needed. An increase in government spending or a cut in taxes shifts AD to the right. The spending multiplier effect can be used to calculate the required injection. If MPC = 0.8, multiplier = 1/(1-0.8) = 5. Needed ΔAD = 133.4, so ΔG = 133.4/5 = 26.68. New AD: Y = 1200 + (26.68 × 5) – 20P = 1200 + 133.4 – 20P = 1333.4 – 20P. New equilibrium: 1333.4 – 20P = -400 + 40P → 1733.4 = 60P, P = 28.89, Y ≈ 1333.4 – 20(28.89) = 755.6 (with this linear model it does not fully reach Yf due to price effect; in reality further adjustments follow). In practice, government would adjust spending iteratively.
解答： 令 AD = SRAS：1200 – 20P = -400 + 40P，得 P = 26.67，均衡产出 Y = 666.6。因 Y < Yf(800)，存在 133.4 的衰退缺口。为消除缺口，需实施扩张性财政政策。增加政府支出或减税可使 AD 右移。已知 MPC = 0.8，乘数 = 5。需要增加的总需求为 133.4，故政府支出应增加 ΔG = 133.4/5 = 26.68。新 AD 为 Y = 1200 + 133.4 – 20P。求解新均衡得 P = 28.89，Y ≈ 755.6（由于价格上升的挤出效应，未完全达到充分就业，实际中需多次调整）。
Note: Recognise the limitations of a simple fixed-price multiplier. In an AD-AS context, the upward-sloping SRAS reduces the multiplier effect.
注意： 要认识到固定价格乘数模型的局限性。在 AD-AS 框架下，向上倾斜的 SRAS 会削弱乘数效应。
9. The Multiplier and National Income | 乘数效应与国民收入
Question: In a closed economy, the consumption function is C = 120 + 0.75Yd, where Yd = Y – T. Investment I = 80, government spending G = 100, and lump-sum taxes T = 100. Calculate the equilibrium national income. What is the value of the government expenditure multiplier? If the government aims to increase income by 200, by how much must it increase G?
问题： 在一个封闭经济中，消费函数为 C = 120 + 0.75Yd，其中 Yd = Y – T。投资 I = 80，政府支出 G = 100，一次性总付税 T = 100。求均衡国民收入，政府支出乘数是多少？如果政府希望收入增加 200，G 需要增加多少？
Answer: Aggregate expenditure AE = C + I + G = 120 + 0.75(Y – 100) + 80 + 100 = 120 + 0.75Y – 75 + 180 = 225 + 0.75Y. Equilibrium condition Y = AE: Y = 225 + 0.75Y → 0.25Y = 225, Y = 900. The government spending multiplier k = 1/(1 – MPC) = 1/0.25 = 4. To increase Y by 200, the required ΔG = ΔY / k = 200/4 = 50. Thus, an increase of 50 in government spending will raise national income by 200.
解答： 总支出 AE = C + I + G = 120 + 0.75(Y – 100) + 80 + 100 = 225 + 0.75Y。由均衡条件 Y = AE 得 Y = 225 + 0.75Y，解得 Y = 900。政府支出乘数 k = 1/(1 – MPC) = 4。要使收入增加 200，需增加政府支出 ΔG = 200/4 = 50。
Extension: The tax multiplier is -MPC/(1-MPC) = -3. A tax cut of 66.67 would also achieve the same income increase. Compare the two for policy evaluation.
拓展： 税收乘数为 -MPC/(1-MPC) = -3，减税 66.67 也能实现同样的收入增长。可在政策评估中对比两者的差异。
10. International Trade and Comparative Advantage | 国际贸易与比较优势
Question: Two countries, Alpha and Beta, produce cloth and wine. In Alpha, 1 unit of cloth requires 10 hours of labour, 1 unit of wine requires 5 hours. In Beta, 1 cloth requires 6 hours, 1 wine requires 4 hours. Which country has absolute advantage in each good? Calculate opportunity costs and determine comparative advantage. Propose a mutually beneficial trade.
问题： 假设有 Alpha 和 Beta 两国，均生产布和酒。在 Alpha，生产 1 单位布需 10 小时劳动，1 单位酒需 5 小时。在 Beta，生产 1 单位布需 6 小时，1 单位酒需 4 小时。哪国在哪种产品上具有绝对优势？计算机会成本并确定比较优势，提出一个互利的贸易方案。
Answer: Beta has absolute advantage in both goods: it uses fewer labour hours per unit (6<10 for cloth, 4<5 for wine). Opportunity cost of cloth in Alpha: 10/5 = 2 wine per cloth. In Beta: 6/4 = 1.5 wine per cloth. Alpha has lower opportunity cost in wine? Wait: cloth opportunity cost in Alpha is 2 wine, Beta 1.5 wine, so Beta has lower opportunity cost for cloth. For wine: Alpha gives up 5/10 = 0.5 cloth per wine; Beta gives up 4/6 ≈ 0.67 cloth per wine. Thus Alpha has lower opportunity cost in wine (0.5 < 0.67), so Alpha has comparative advantage in wine, Beta in cloth. Trade pattern: Alpha specialises in wine, Beta in cloth. Terms of trade between the two opportunity cost ratios: between 0.5 and 0.67 cloth per wine. For example, exchange 1 wine for 0.6 cloth benefits both.
解答： Beta 在两种产品上都具有绝对优势，因为其生产单位产品所需的劳动时间均少于 Alpha（布 6<10，酒 4<5）。Alpha 生产 1 布的机会成本为 2 酒，Beta 为 1.5 酒，故 Beta 生产布的机会成本更低。Alpha 生产 1 酒的机会成本为 0.5 布，Beta 为 0.67 布，因此 Alpha 在酒上具有比较优势（0.5 < 0.67），Beta 在布上具有比较优势。贸易格局：Alpha 专业化生产酒，Beta 专业化生产布。贸易条件应介于两国机会成本比率之间，即 1 酒交换 0.5 至 0.67 单位布。例如，1 酒换 0.6 布，可使双方获益。
Essential point: Comparative advantage is based on opportunity cost, not absolute productivity. Show gains from trade by comparing consumption possibilities before and after trade.
要点： 比较优势基于机会成本，而非绝对生产率。通过对比贸易前后消费可能性边界来展示贸易收益。
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