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9708 · 7.4

Private costs and benefits, externalities and social costs and benefits — practice questions

Practice and worked examples for 9708 Private costs and benefits, externalities and social costs and benefits. Short previews only — attempt the full question in MarkScheme against the official scheme.

Worked example 1

The market for pesticide use is shown below (linear curves):

  • MPC = 10 + 0.5Q
  • External cost = 4 per unit (constant)
  • MPB = MSB = 30 − 0.5Q

(a) Find Q_market and Q_social. (b) Calculate the Pigouvian tax per unit. (c) Shade and describe the deadweight loss.

Show solution outline

(a) Q_market: MPC = MPB 10 + 0.5Q = 30 − 0.5Q Q = 20 units, P = £20

Q_social: MSC = MSB, where MSC = MPC + 4 = 14 + 0.5Q 14 + 0.5Q = 30 − 0.5Q Q = 16 units, P = £22

Market overproduces by 4 units (20 − 16).

(b) Pigouvian tax = marginal external cost = £4 per unit (constant in this case).

With tax, effective MPC = 10 + 0.5Q + 4 = 14 + 0.5Q = MSC → Q falls to 16.

(c) DWL: Triangle between MSC and MSB from Q_social (16) to Q_market (20).

At Q = 20: MSC = 14 + 10 = £24, MSB = £20 → gap = £4. At Q = 16: MSC = MSB = £22 → gap = £0.

DWL = ½ × (20 − 16) × £4 = £8 — welfare lost from the 4 extra units.

Worked example 2

The market for university education exhibits positive externalities. Assume the market can be described by the following equations, where Q is the number of students in thousands and P is the annual tuition fee in thousands of pounds (£'000).

  • Marginal Private Benefit (MPB): P = 60 - Q
  • Marginal Private Cost (MPC): P = 15 + 0.5Q
  • Marginal External Benefit (MEB) is constant at £15,000 per student.

(a) Calculate the free market equilibrium quantity of students and the tuition fee. (b) Determine the socially optimal quantity of students. (c) What is the value of the per-unit subsidy required to achieve the social optimum? (d) Calculate the value of the deadweight welfare loss at the market equilibrium.

Show solution outline

(a) Market Equilibrium (MPB = MPC): Set the private benefit equal to the private cost: 60 - Q = 15 + 0.5Q 45 = 1.5Q Q = 45 / 1.5 = 30 So, Q_market = 30,000 students. Substitute Q=30 into either equation to find the price: P = 60 - 30 = 30 So, P_market = £30,000.

(b) Socially Optimal Equilibrium (MSB = MSC): First, find the Marginal Social Benefit (MSB). Assume no external costs, so MPC = MSC. MSB = MPB + MEB MSB = (60 - Q) + 15 = 75 - Q Now, set MSB = MSC: 75 - Q = 15 + 0.5Q 60 = 1.5Q Q = 60 / 1.5 = 40 So, Q_social = 40,000 students. The market under-provides education by 10,000 students.

(c) Optimal Subsidy: The optimal per-unit subsidy is equal to the Marginal External Benefit (MEB) at the socially optimal quantity. Since MEB is constant: Subsidy = MEB = £15,000 per student.

(d) Deadweight Welfare Loss (DWL): The DWL is the area of the triangle between the MSB and MSC curves, from Q_market to Q_social. Area = 0.5 * (Q_social - Q_market) * (Vertical gap at Q_market) The vertical gap at Q_market (30) is MSB(30) - MSC(30). MSB(30) = 75 - 30 = 45 (£45,000) MSC(30) = 15 + 0.5(30) = 15 + 15 = 30 (£30,000) Gap = 45 - 30 = £15,000. DWL = 0.5 * (40 - 30) * 15 DWL = 0.5 * 10 * 15 = 75 Since Q is in thousands and P is in thousands, the DWL is in millions of pounds. DWL = £75 million.

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