📖 消费者剩余与生产者剩余

Economics

I. Lesson Focus

This lesson explains the concepts of consumer surplus, producer surplus, total surplus, and deadweight loss. Candidates must be able to calculate these values from linear demand and supply functions, discrete willingness-to-pay schedules, and after policy interventions such as per-unit taxes and price controls. The focus is on identifying efficiency losses and understanding tax incidence.

II. The Problem

Suppose a consumer is willing to pay $60 for the first cup of coffee, $45 for the second, and $30 for the third. The market price is $35 per cup and the consumer buys three cups, paying a total of $105. Yet the consumer values the three cups at $135. What is this $30 “extra value,” and how is it measured? If the government imposes a $10 per-cup tax, how much welfare is lost by consumers, producers, and society overall? CFA exams frequently require precise calculation of surplus changes, identification of deadweight loss triangles, and determination of whether price ceilings, floors, or taxes improve or reduce allocative efficiency. This lesson solves these core microeconomic problems.

III. Consumer Surplus (CS)

Consumer surplus is the difference between what consumers are willing to pay (WTP) and what they actually pay. It represents the net benefit consumers receive from market transactions.

On a demand curve, consumer surplus is the area above the equilibrium price line and below the demand curve. The downward-sloping demand curve reflects diminishing marginal willingness to pay.

Calculation Methods: - Discrete case: sum (WTP$_i$ – P) for all units purchased. - Continuous (linear) case: area of a triangle = $\frac{1}{2} \times$ base $\times$ height.

Formulas: $$ CS = \sum (WTP_i - P) \quad \text{or} \quad CS = \frac{1}{2} \times (P_{max} - P) \times Q $$

IV. Producer Surplus (PS)

Producer surplus is the difference between the price producers actually receive and their minimum acceptable price (marginal cost). It measures the net benefit received by producers.

On a supply curve, producer surplus is the area below the equilibrium price line and above the supply curve. The upward-sloping supply curve reflects increasing marginal cost.

Formulas: $$ PS = \sum (P - MC_i) \quad \text{or} \quad PS = \frac{1}{2} \times (P - P_{min}) \times Q $$

V. Total Surplus and Market Efficiency

Total surplus (TS) = Consumer Surplus + Producer Surplus = total social welfare.

In a perfectly competitive market without externalities, the equilibrium where supply equals demand maximizes total surplus. This point is Pareto efficient. Any deviation from equilibrium caused by price controls or taxes creates a deadweight loss (DWL) — the net reduction in total surplus.

VI. Effects of Price Controls

• Price Ceiling (below equilibrium): creates shortage. Only the quantity actually traded generates surplus; the untraded units create DWL.
• Price Floor (above equilibrium): creates surplus. Again generates DWL.

VII. Effects of Taxation

A per-unit tax shifts the supply curve upward (or demand downward) by the amount of the tax, creating a vertical wedge between the price paid by buyers and the price received by sellers. - Tax incidence depends on the relative elasticities of supply and demand. - Government tax revenue = tax rate × new equilibrium quantity traded. - Deadweight loss = area of the triangle = $\frac{1}{2} \times$ tax $\times$ reduction in quantity.

Taxes always reduce total surplus, although the government may use the revenue for other purposes.

Worked Cases

Case 1: Discrete Demand and Consumer Surplus

A consumer’s willingness to pay for successive units is $100, $80, $60, $40, and $20. The market price is $45. Calculate consumer surplus.

Solution:
The consumer buys the first three units (fourth unit WTP $40 < $45).
CS = (100−45) + (80−45) + (60−45) = 55 + 35 + 15 = $105.

Case 2: Linear Supply and Demand — Equilibrium Surpluses

Demand: $P = 120 - 2Q$; Supply: $P = 20 + 4Q$.
Find equilibrium price and quantity, then CS, PS, and total surplus.

Solution:
Set 120 − 2Q = 20 + 4Q → 100 = 6Q → Q ≈ 16.67, P ≈ $86.67.
CS = $\frac{1}{2} \times (120 - 86.67) \times 16.67 ≈ 277.78$.
PS = $\frac{1}{2} \times (86.67 - 20) \times 16.67 ≈ 555.56$.
Total surplus = 277.78 + 555.56 = $833.34.

Case 3: Tax Incidence, Surplus Changes, and Deadweight Loss

Using Case 2 data, impose a $12 per-unit tax on sellers. Find the new quantity, loss in CS, loss in PS, tax revenue, and DWL.

Solution:
New supply: P = 32 + 4Q.
120 − 2Q = 32 + 4Q → Q ≈ 14.67. Buyer price ≈ $90.67, seller net price ≈ $78.67.
New CS ≈ 214.22 → CS loss ≈ $63.56.
New PS ≈ 430.00 → PS loss ≈ $125.56.
Tax revenue = 12 × 14.67 ≈ $176.
DWL = $\frac{1}{2} \times 12 \times (16.67 - 14.67) ≈ 12$.
Note: CS loss + PS loss = tax revenue + DWL (rounding accounts for minor discrepancy).

Traps

Key Formulas

• Consumer Surplus: $CS = \int_{0}^{Q} (D(Q)-P)\,dQ$ or triangle formula
• Producer Surplus: $PS = \int_{0}^{Q} (P-S(Q))\,dQ$ or triangle formula
• Total Surplus: $TS = CS + PS$
• Deadweight Loss: $DWL = \frac{1}{2} \times tax \times \Delta Q$
• Tax incidence on consumers: $\frac{E_s}{E_s + |E_d|}$; on producers: $\frac{|E_d|}{E_s + |E_d|}$
• Maximum total surplus occurs where marginal benefit equals marginal cost (competitive equilibrium)

Practice Questions

Q1. A consumer’s willingness to pay for three units is $50, $40, and $25. Market price is $30. Consumer surplus equals:
A. $35 B. $45 C. $55 D. $65

Q2. In a linear model, equilibrium price is $40, demand vertical intercept is $100, equilibrium quantity is 30. Consumer surplus equals:
A. 600 B. 900 C. 1,200 D. 1,800

Q3. Which situation creates a positive deadweight loss?
A. Competitive market equilibrium B. Subsidy on a good with positive externality C. Effective price ceiling D. Per-unit tax on a good with no externality

Q4. If demand is perfectly inelastic and government levies a per-unit tax on sellers, then:
A. Consumers bear the entire tax B. Producers bear the entire tax C. Burden is split equally D. Large DWL is created

Q5. Original equilibrium Q = 100 at P = $20. After a $5 per-unit tax, Q falls to 90. Deadweight loss is approximately:
A. 12.5 B. 25 C. 50 D. 450

Q6. Producer surplus increases while consumer surplus decreases. Most likely situation:
A. Tax levied on buyers B. Price ceiling below equilibrium C. Price floor above equilibrium D. Demand curve shifts right

Q7. Total surplus is maximized when:
A. CS = PS B. Marginal willingness to pay equals marginal cost C. Tax revenue is maximized D. Price equals average cost

Q8. Government imposes a binding price ceiling; actual quantity traded is 70 % of the original equilibrium quantity. Then:
A. Total surplus increases B. Consumer surplus must increase C. Producer surplus must decrease D. Deadweight loss exists

Answers

Takeaways

• Consumer surplus lies above the price line under the demand curve; producer surplus lies below the price line above the supply curve.
• Competitive equilibrium without externalities maximizes total social surplus.
• Taxes and price controls create deadweight loss whose size depends on elasticities.
• Tax incidence is determined by relative supply and demand elasticities, independent of statutory incidence.
• Always calculate surpluses and tax revenue using the actual post-intervention quantity traded.
• Deadweight loss represents a net loss to society, not merely a transfer between parties.