相关例题 （应用题）

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Question 1 — Hiring With a Stepwise MRP Schedule (利润最大化雇佣)

A bakery faces the following value of marginal product of labor (MRP_L) per worker per day: Worker 1–2: $260each;Worker3–4:$220 each; Worker 5–6: $170each;Worker7–8:$120 each. The market wage is $180/day. How many workers should the bakery hire?

一家面包店的劳动边际产品价值（MRP_L）如下： 第1–2名：各 $260；第3–4名：各$220；第5–6名：各 $170；第7–8名：各$120。 市场工资为 $180/天。应雇佣多少名工人？

📖 点击查看答案

Hire 4 workers.
应雇佣 4 名工人。

📝 点击查看解析

（若含公式示例：$\mathrm{MRPL}\ge W$ 继续雇佣）
Hire as long as MRP_L ≥ W. For workers 1–4, $260and$220 ≥ $180(profitable).Forworkers5–6,$170 < $180(unprofitable).Sotheprofit-maximizingemploymentis4.规则：只要MRP\_L\ge 工资就继续雇佣。前四名的$260 和 $220\ge$180；第5–6名 $170<$180，不应雇佣，故最优雇佣量为 4。

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Question 2 — Product Price Shock and Labor Demand Shift (产品价格冲击与劳动需求)

An electronics firm sells a device at $500.EachaddedworkerhasMP=3devices/day.Afterasuccessfulmarketingcampaign,thedevicepricerisesto$600 with MP unchanged. If the wage is $1,400/day, how does the optimal employment decision change?

一家电子公司产品单价 $500，每名新增工人的MP=3台/天。营销成功后价格升至$600，MP 不变。工资 $1,400/天。最优雇佣决策如何变化？

📖 点击查看答案

Before: MRP = $1,500>$1,400 → hire the worker; After: MRP = $1,800>$1,400 → even stronger hiring incentive.
变动前：MRP=$1,500>$1,400，应雇；变动后：MRP=$1,800>$1,400，更应雇佣。

📝 点击查看解析

$\mathrm{MRP}=P\times \mathrm{MP}$
MRP = P × MP. Price increase shifts labor demand (MRP curve) right. With higher MRP, the firm hires at least as much labor as before, possibly more until the marginal worker’s MRP falls to $1,400.MRP=价格\times MP。提价使劳动需求右移。MRP更高，企业会至少维持原雇佣量，并继续加雇直到边际工人的MRP下降至$1,400。

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Question 3 — Backward-Bending Supply Threshold (后弯供给拐点)

Suppose a consultant’s labor supply is upward sloping up to $80/hour and backward-bending beyond that:

• At $60/h\to 6hours;$80/h → 8 hours; $100/h\to 7hours;$120/h → 6 hours. If the market rate rises from $80to$100, what happens to total labor supplied?

某顾问的劳动供给：$80/h 之前向上倾斜，之后后弯：

• $60/h\to 6小时；$80/h → 8 小时；$100/h\to 7小时；$120/h → 6 小时。 工资由 $80提至$100，供给的劳动时长如何变化？

📖 点击查看答案

It decreases from 8 to 7 hours.
从 8 小时降至 7 小时。

📝 点击查看解析

Beyond $80/h,theincomeeffectdominatesthesubstitutioneffect;higherwagebuysmoreleisure,reducinghours.在$80/h 以上，收入效应强于替代效应；工资提高带来更多闲暇消费，工时反而下降。

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Question 4 — Immigration and Minimum Wage (移民与最低工资)

In a city, labor demand for dishwashers is D: W = 20 − 0.002L and labor supply is S: W = 8 + 0.001L (W in $/hour,Linworkers).a)Findcompetitiveequilibrium(W\ast ,L\ast ).b)AsuddenimmigrationshiftssupplytoS\prime :W=6+0.001L.Computethenewcompetitiveequilibrium.c)Ifabindingminimumwageof$14 is imposed after the shift, what are employment and unemployment (excess supply)?

某市洗碗工的需求与供给为：D：W = 20 − 0.002L；S：W = 8 + 0.001L（W 为美元/小时，L 为人数）。
a) 求竞争均衡 (W*, L*)。
b) 移民使供给变为 S′：W = 6 + 0.001L。求新均衡。
c) 若在变动后设置 $14 的约束性最低工资，就业与失业（超额供给）分别是多少？

📖 点击查看答案

a) W*=$12,L\ast =4,000.b)W\prime \approx$10.67, L′≈4,666.67.
c) At W=$14: demand Ld = 3,000; supply Ls = 8,000 → employment 3,000; unemployment 5,000.

📝 点击查看解析

Solve a) $20-0.002L=8+0.001L\Rightarrow 12=0.003L\Rightarrow L=4,000$; then $W=20-0.002\times 4,000=12$。
b) $20-0.002L=6+0.001L\Rightarrow 14=0.003L\Rightarrow L\approx 4,666.67$，$W\approx 10.67$。
c) 最低工资 $14：需求$14=20-0.002L \Rightarrow L_d=3{,}000$；供给$14=6+0.001L \Rightarrow L_s=8{,}000$。就业 3,000；失业 5,000。

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Question 5 — Education Investment NPV (教育投资净现值)

A 4-year degree costs $25,000peryear(paidatyear-end)anddelaysworkthatwouldpay$30,000/year otherwise. After graduation, expected wage premium is $18,000 per year for 35 years. Discount rate is 5%. Is the NPV positive?

四年本科学费每年 $25,000（年末支付），同时放弃每年$30,000 的工作收入。毕业后预计每年多赚 $18,000，持续 35 年。折现率 5%。NPV 是否为正？

📖 点击查看答案

Yes, NPV > 0 (about $78,000).是，净现值为正（约$78,000）。

📝 点击查看解析

年金现值：$\mathrm{PV}=A\times \frac{1-(1+r{)}^{-n}}{r}$
Cost PV (tuition + foregone earnings) for 4 years is an annuity of $55,000:$\mathrm{PV_{cost}} = 55{,}000 \times \frac{1-1/1.05^{4}}{0.05} \approx 55{,}000 \times 3.54595 \approx $195{,}027$。BenefitPV:$18{,}000 \times \frac{1-1/1.05^{35}}{0.05} \approx 18{,}000 \times 15.3725 \approx $276{,}705$。NPV\approx 276,705-195,027\approx$81,678（四舍五入约 $78–82k）。为正。
（中文同上要点）

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Question 6 — Capital Rental Rule (资本租用规则)

A factory can rent an additional robot arm at $900/day.Itincreasesoutputby60units;theproductsellsat$20 each, and marginal cost excluding the robot is unchanged. Should the firm rent it?

某工厂可每天 $900租一台机械臂，可多产出60件产品；产品售价$20/件，除机械臂外的边际成本不变。是否应租？

📖 点击查看答案

Yes. MRPK = 60 × $20=$1,200 > $900.应租。MRPK=60\times$20 = $1,200＞$900。

📝 点击查看解析

Rent until MRPK = rental rate r. Here MRPK−r = $300>0,soaddingtherobotarmincreasesprofit.最优条件为$\mathrm{MRPK} = r$。此处$\mathrm{MRPK}-r = 300>0$，租用提高利润。

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Question 7 — Natural Resource Quota (资源配额)

Oil demand: $P=120-0.1Q$; short-run supply: $P=40+0.05Q$ (P in $/barrel,Qinmillionbarrels/day).a)Competitiveequilibrium?b)Governmentsetsanextractioncap$\bar{Q} = 300$. What is the new price?

原油需求：$P=120-0.1Q$；供给：$P=40+0.05Q$。
a) 竞争均衡？
b) 若政府限产 $\bar{Q}=300$，价格变为多少？

📖 点击查看答案

a) Q^* = 533.33,\; P^* = \66.67$。b)With$\bar{Q} = 300$,pricebecomes$P = 120 - 0.1 \times 300 = $90$。

📝 点击查看解析

a) 解 $120-0.1Q=40+0.05Q\Rightarrow 80=0.15Q\Rightarrow Q=533.33$；代回 $P=66.67$。
b) 配额使供给在 $\bar{Q}$ 处垂直，市场在需求侧出清：$P=90$。这反映了稀缺地租的上升。

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Question 8 — Monopsony ME Curve (买方垄断的边际支出)

A hospital faces labor supply $W=10+0.05L$ (W in $/hour).Totalexpenditureonlaboris$E(L) = W \times L$。a)Derivethemarginalexpenditure$ME(L)$。b)Ifthehospital’slabordemand(MR{P}_L)is$W = 40 - 0.05L$，find the monopsony employment and wage.
c) Compare with the competitive outcome.

一家医院面临的劳动供给：$W=10+0.05L$。总工资支出 $E(L)=W\times L$。
a) 推导边际支出 $ME(L)$。
b) 若医院的劳动需求（MRP_L）为 $W=40-0.05L$，求买方垄断下的工资与雇佣量。
c) 与竞争均衡比较。

📖 点击查看答案

a) $ME=10+0.10L$。
b) Set $ME={\mathrm{MRP}}_L\Rightarrow 10+0.10L=40-0.05L\Rightarrow L=200$；wage from supply W = 10 + 0.05\times 200 = \20$。c)Competitive:$10 + 0.05L = 40 - 0.05L \Rightarrow L = 300$；$W = $25$。Monopsony hires fewer workers at lower wage.

📝 点击查看解析

Because $W(L)=10+0.05L$, $E(L)=(10+0.05L)L=10L+0.05L^2\Rightarrow ME=\frac{dE}{dL}=10+0.10L$。
买方垄断选择 $ME={\mathrm{MRP}}_L$ 的 L，再到供给曲线读出工资 W。竞争均衡直接令供给=需求。由于工资随 L 上升，$ME$ 曲线在供给曲线之上；买方垄断雇佣更少、工资更低。

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Question 9 — Union Wage Floor in Monopsony (工会工资底线)

Use Question 8’s market. Suppose a union negotiates a wage floor W_u = \24$。What is employment under the wage floor and how does total employment compare to the monopsony case?

沿用题8的市场。若工会谈判到工资底线 W_u = \24$，就业如何？与买方垄断无工会时相比怎样？

📖 点击查看答案

At $W_u=24$，the hospital hires where ${\mathrm{MRP}}_L=24\Rightarrow 24=40-0.05L\Rightarrow L=320$。Employment rises from 200 to 320.

📝 点击查看解析

A binding but not excessive wage floor in a monopsony effectively makes the supply curve horizontal at $W_u$ up to the original supply quantity; the firm hires where ${\mathrm{MRP}}_L$ intersects that floor. This can increase both wage and employment, reducing deadweight loss.
在买方垄断下，适度的工资底线把供给变成水平线，企业在 ${\mathrm{MRP}}_L$ 与该线的交点雇佣，工资与就业均可上升，降低无谓损失。

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Question 10 — Human Capital Shock vs. Product Demand (人力资本与产品需求对比)

Two identical firms face the same wage. Firm A doubles worker productivity via training (MP ×2). Firm B faces a 20% increase in product price with MP unchanged. Which firm’s labor demand shifts more, and by how much in MRP terms?

两家相同企业工资一致。A 通过培训使 MP 翻倍；B 的产品价格上涨 20%，MP 不变。哪家劳动需求右移更多？MRP 变化幅度分别是多少？

📖 点击查看答案

Firm A shifts more.
${\mathrm{MRP}}_A$ doubles (×2); ${\mathrm{MRP}}_B$ rises by 20% (×1.2).

📝 点击查看解析

Labor demand is the MRP curve: $\mathrm{MRP}=P\times \mathrm{MP}$。Increasing MP by 100% doubles MRP; raising P by 20% raises MRP by 20%. Hence A’s rightward shift is larger.
劳动需求即 MRP 曲线；$\mathrm{MRP}=价格\times \mathrm{MP}$。MP 翻倍使 MRP 翻倍；价格涨 20% 仅使 MRP 增 20%，故 A 的右移更大。

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