Consider a steel industry where the marginal private cost (MPC) of producing Q thousand tonnes is MPC = 50 + 2Q (Rs per tonne). The marginal external damage (MED) from pollution is MED = Q (Rs per tonne). The market price of steel is fixed at Rs.150 per tonne. (a) Find the equilibrium output without any environmental regulation. (b) Determine the optimal Pigouvian tax per tonne. (c) Compute the new equilibrium output after imposing this tax. (d) Calculate the reduction in total social cost due to the tax.

Step-by-step solution

1. (a) Set price equal to MPC: 150 = 50 + 2Q ⇒ 2Q = 100 ⇒ Q₀ = 50 thousand tonnes (unregulated equilibrium).
2. (b) Optimal Pigouvian tax equals MED evaluated at Q₀: t = MED|_{Q=50} = 50 Rs per tonne.
3. (c) With tax, effective marginal cost = MPC + t = (50 + 2Q) + 50 = 100 + 2Q. Set equal to price: 150 = 100 + 2Q ⇒ 2Q = 50 ⇒ Q₁ = 25 thousand tonnes (regulated equilibrium).
4. (d) Compute total social cost (TSC) before tax: • Private total cost = average MPC × Q₀. MPC at 0 = 50, at Q₀ = 150, average = (50+150)/2 = 100 Rs. Private cost = 100 × 50 = Rs.5,000. • External total cost = ∫₀^{50} MED dQ = ½ × 50² = 1,250 Rs. TSC₀ = 5,000 + 1,250 = Rs.6,250. Compute TSC after tax: • Private cost = average MPC at Q₁ = (50 + (50+2·25))/2 = (50+100)/2 = 75 Rs. Private cost = 75 × 25 = Rs.1,875. • External cost = ½ × 25² = 312.5 Rs. TSC₁ = 1,875 + 312.5 = Rs.2,187.5. • Reduction = TSC₀ – TSC₁ = 6,250 – 2,187.5 = Rs.4,062.5.

Concept tested: Pigouvian tax and welfare analysis

Published 4 October 2026 · How this solution was prepared and checked