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9708 · 7.3

Efficiency and market failure — practice questions

Practice and worked examples for 9708 Efficiency and market failure. Short previews only — attempt the full question in MarkScheme against the official scheme.

Worked example 1

A chemical factory produces at Q_market = 100 units where MPC = MPB = £20. The external cost is £5 per unit (constant). MSC = MPC + £5.

(a) Find Q_social where MSB = MSC. (b) Explain the deadweight loss from overproduction.

Show solution outline

(a) At Q_social, MSB = MSC. MSB = MPB = £20 (no external benefit). MSC = MPC + external cost = £20 + £5 = £25 at the market quantity.

Since MSC > MSB at Q = 100, the market overproduces. Q_social is where MSC = MSB = £20, meaning MPC = £15 (since MSC = MPC + 5).

On a diagram, Q_social < 100 — society wants fewer than 100 units.

(b) Deadweight loss: For each unit from Q_social to 100, MSC > MSB — society loses the excess of MSC over MSB.

DWL = area of triangle between MSC and MSB curves from Q_social to 100.

At the margin near Q_market, welfare loss per unit ≈ £5 (external cost). Total DWL = ½ × (100 − Q_social) × £5.

Policy: A Pigouvian tax of £5 per unit would shift supply to MSC and restore Q_social.

Worked example 2

A monopolist faces a market demand curve of P = 100 - 2Q and has a constant marginal cost (MC) of £20. There are no externalities.

(a) Calculate the monopolist's profit-maximizing price and quantity. (b) Calculate the allocatively efficient price and quantity. (c) Calculate the deadweight loss due to the monopoly.

Show solution outline

(a) Monopolist's Equilibrium (Profit Maximisation)

A monopolist maximises profit where Marginal Revenue (MR) equals Marginal Cost (MC).

  1. Find MR: For a linear demand curve P = a - bQ, the MR curve is MR = a - 2bQ. Here, P = 100 - 2Q, so MR = 100 - 4Q.
  2. Set MR = MC: 100−4Q=20100 - 4Q = 20 80=4Q80 = 4Q Qmonopoly=20 unitsQ_{monopoly} = 20 \text{ units}
  3. Find Price: Substitute Q=20 back into the demand curve: P=100−2(20)=100−40=£60P = 100 - 2(20) = 100 - 40 = £60

Answer (a): The profit-maximizing quantity is 20 units and the price is £60.

(b) Allocatively Efficient Equilibrium

Allocative efficiency occurs where Price (representing Marginal Social Benefit, MSB) equals Marginal Cost (representing Marginal Social Cost, MSC). The condition is P = MC.

  1. Set P = MC: 100−2Q=20100 - 2Q = 20 80=2Q80 = 2Q Qefficient=40 unitsQ_{efficient} = 40 \text{ units}
  2. Find Price: At this quantity, P = MC = £20.

Answer (b): The allocatively efficient quantity is 40 units and the price is £20.

(c) Deadweight Loss (DWL)

The deadweight loss is the welfare lost due to the monopolist under-producing compared to the social optimum. It's the area of the triangle between the demand curve and the MC curve, over the range of underproduction (from Q=20 to Q=40).

  1. Formula: DWL = 0.5 × (change in Quantity) × (change in Price/Cost)
  2. Calculate:
    • Base of triangle (change in Q) = Qefficient−Qmonopoly=40−20=20Q_{efficient} - Q_{monopoly} = 40 - 20 = 20
    • Height of triangle (price gap at monopoly Q) = Pmonopoly−MC=£60−£20=£40P_{monopoly} - MC = £60 - £20 = £40 DWL=0.5×20×40=£400DWL = 0.5 \times 20 \times 40 = £400

Answer (c): The deadweight loss resulting from the monopoly is £400.

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