📖 政府干预：价格上限/下限、税收

Economics

I. Lesson Focus

This lesson examines how government interventions in competitive markets—price ceilings, price floors, and per-unit taxes—alter equilibrium price and quantity, redistribute consumer surplus (CS) and producer surplus (PS), generate tax revenue or fiscal costs, and create deadweight loss (DWL). Candidates must be able to calculate changes in surplus, identify the incidence of a tax according to elasticities, determine whether a price control is binding, and quantify DWL using geometric areas under supply and demand curves.

II. The Problem

A government imposes a price ceiling of 4 yuan per kilogram on rice when the free-market equilibrium price is 6 yuan to protect low-income consumers. At the same time, it levies a 2-yuan per-unit tax on each bottle of liquor to raise revenue. The rice market experiences severe shortages, while the liquor market sees buyers paying more, sellers receiving less, reduced total surplus, and positive deadweight loss. CFA questions frequently require candidates to compute the exact changes in CS, PS, government revenue, and DWL before and after intervention and to evaluate whether the policy achieves its stated objective. This lesson provides the analytical framework and calculation techniques needed to solve such problems accurately.

III. The Economic Rationale for Government Intervention

In an unregulated competitive market, equilibrium occurs where the supply and demand curves intersect. At this point, total surplus (CS + PS) is maximized and resources are allocated efficiently (Pareto optimal). Governments intervene for reasons of equity, price stability, or fiscal needs using three primary tools: price ceilings, price floors, and taxes. Each tool disrupts the equilibrium, reallocates surplus between consumers, producers, and government, and typically creates a deadweight loss—the net loss in total surplus that is not transferred to any party.

IV. Price Ceilings

A price ceiling is a legally mandated maximum price. When set below the equilibrium price (a binding ceiling), it creates a shortage.
- The quantity transacted is determined by the short side of the market (quantity supplied).
- Consumer surplus usually rises for those who can buy at the lower price but falls overall because of reduced quantity.
- Producer surplus declines sharply.
- The net decline in total surplus appears as a triangular deadweight loss.
- Secondary effects include black markets, queuing, deterioration of product quality, and rent-seeking behavior.

Calculation
Shortage = Qd(Pc) − Qs(Pc)
Changes in surplus are found by comparing the triangular and trapezoidal areas before and after the ceiling.

V. Price Floors

A price floor is a legally mandated minimum price. When set above equilibrium (binding), it creates a surplus. Common examples are agricultural support prices and minimum wages.
- Quantity transacted equals quantity demanded.
- Producers who sell receive a higher price, but total producer surplus may rise or fall depending on elasticities and the size of the quantity reduction.
- Consumer surplus falls significantly.
- Deadweight loss again emerges as the triangle between the supply and demand curves from the new quantity to the original equilibrium quantity.
- Policy costs may include government purchases of the surplus or increased unemployment (in the labor market).

VI. Per-unit Taxes

A per-unit tax is a fixed monetary amount levied on each unit sold. The tax can be legally imposed on buyers or sellers, but the economic incidence depends on relative price elasticities of supply (Es) and demand (Ed).
- The supply curve shifts upward (if levied on sellers) or the demand curve shifts downward (if levied on buyers) by the amount of the tax; the effect on equilibrium is identical.
- Equilibrium quantity falls.
- Buyers pay a higher price (Pb), sellers receive a lower net price (Ps), and Pb − Ps equals the tax.
- Tax revenue = tax per unit × new equilibrium quantity.
- Both CS and PS decline; the portion of the loss not captured as tax revenue is deadweight loss.
- The side with the less elastic curve bears a larger share of the tax burden.

Key Incidence Formulas
Buyer burden proportion = Es / (Es + |Ed|)
Seller burden proportion = |Ed| / (Es + |Ed|)
Pb = Pe + t × (Es / (Es + |Ed|))
Ps = Pe − t × (|Ed| / (Es + |Ed|))
DWL = ½ × t × (Qe − Qtax)

Worked Cases

Case 1: Binding Price Ceiling, Shortage, and DWL

Demand: Qd = 120 − 10P; Supply: Qs = 20 + 10P.
Equilibrium: 120 − 10P = 20 + 10P → Pe = 5, Qe = 70.
Government sets Pc = 3 (binding).

At Pc = 3: Qs = 50, Qd = 90 → shortage = 40 units. Actual quantity traded = 50.

Surplus calculations (area method):
Original CS = ½ × (12 − 5) × 70 = 245
New CS = ½ × (12 − 3) × 50 + (5 − 3) × 50 = 325 (gain of 80)
Original PS = ½ × (5 − 2) × 70 = 105
New PS = ½ × (3 − 2) × 50 = 25 (loss of 80)
DWL = ½ × (5 − 3) × (70 − 50) = 20

Although consumers who obtain the good appear to gain, total surplus falls by 20, confirming inefficiency.

Case 2: Minimum Wage (Price Floor) Analysis

Labor demand: Qd = 400 − 20W; Labor supply: Qs = 40 + 10W.
Equilibrium wage = 12, employment = 160.
Government sets floor Wf = 16.

At Wf = 16: Qd = 80, Qs = 200 → surplus (unemployment) = 120. Actual employment = 80.
DWL = ½ × (16 − 12) × (160 − 80) = 160.
While employed workers earn more, the reduction in employment and resulting DWL reduce total social welfare.

Case 3: Tax Incidence and DWL with Equal Elasticities

Demand: Qd = 200 − 10P; Supply: Qs = 20 + 10P.
Equilibrium: Pe = 9, Qe = 110.
Government imposes a 4-unit tax on sellers.

New supply: Qs = −20 + 10P.
New equilibrium: 200 − 10P = −20 + 10P → Pb = 11, Ps = 7, Qtax = 90.
Tax revenue = 4 × 90 = 360.
DWL = ½ × 4 × (110 − 90) = 40.
Because supply and demand have identical slopes (equal elasticities at equilibrium), the tax burden is split 50/50.

Traps

Key Formulas

• Shortage (ceiling) = Qd(Pc) − Qs(Pc)
• Surplus (floor) = Qs(Pf) − Qd(Pf)
• Buyer price after tax: Pb = Pe + t × (Es / (Es + |Ed|))
• Seller net price after tax: Ps = Pe − t × (|Ed| / (Es + |Ed|))
• Tax revenue = t × Qtax
• DWL (tax) = ½ × t × (Qe − Qtax)
• DWL (price control) = ½ × |Pcontrol − Pe| × |Qe − Qcontrol|
• Total surplus after intervention = CS + PS + Government revenue − DWL

Practice Questions

Q1. When a government imposes a binding price ceiling below the equilibrium price, which of the following must occur?
A. Consumer surplus always increases
B. Producer surplus always increases
C. A shortage occurs
D. Total surplus increases

Q2. In a market with an equilibrium price of 10, a price floor is set at 12. If demand is more elastic than supply, then:
A. Consumers bear most of the welfare loss
B. Producers bear most of the welfare loss
C. Producer surplus must increase
D. Deadweight loss is zero

Q3. Imposing a per-unit tax on sellers causes the supply curve to:
A. Shift rightward
B. Shift upward by the amount of the tax
C. Shift downward by the amount of the tax
D. Remain unchanged

Q4. Deadweight loss is largest when:
A. Demand is perfectly inelastic
B. Supply is perfectly elastic
C. Both supply and demand have moderate elasticities
D. Both supply and demand are perfectly inelastic

Q5. Demand is Qd = 100 − 5P and supply is Qs = 20 + 5P. Equilibrium price is 8. After a 4 per-unit tax, the price paid by buyers is closest to:
A. 10
B. 9
C. 11
D. 12

Q6. Black markets are most likely to emerge when:
A. A price ceiling is set above equilibrium
B. Enforcement is perfect
C. A binding ceiling creates shortage and some buyers are willing to pay more
D. Producer surplus increases

Q7. A minimum wage law typically leads to:
A. A shortage of labor
B. A surplus of labor (higher unemployment)
C. An increase in total surplus
D. Higher firm profits

Q8. If demand is perfectly inelastic, a tax imposed on producers will result in:
A. Consumers bearing the entire tax burden
B. Producers bearing the entire tax burden
C. The largest possible reduction in quantity
D. The largest possible deadweight loss

Answers

Takeaways

• Binding price ceilings create shortages and are resolved by supply; binding floors create surpluses resolved by demand.
• Tax incidence depends only on relative supply and demand elasticities, not on whom the tax is legally levied.
• Deadweight loss equals one-half times the price distortion times the reduction in quantity for both controls and taxes.
• Government intervention typically trades efficiency for perceived equity; candidates must master surplus-area calculations.
• The less elastic side of the market bears more of the economic burden of taxes or price controls.
• Always compute the new equilibrium quantity first, then the resulting CS, PS, revenue, and DWL triangles or trapezoids.