A factory that pollutes imposes a cost it never pays — the dirty air lands on everyone downwind, not on its accounts. Whenever a transaction spills costs or benefits onto people outside it, the market gets the quantity wrong, and the tool economists reach for to fix it is the Pigouvian tax. Externalities are where you first see why a free market can fail and how a well-designed tax repairs it — a staple of first-year micro that a private economics tutor in London will make sure you can both draw and defend.
1 · Private cost versus social cost
An externality is a cost or benefit that falls on someone who is not party to the transaction. A polluting factory creates a negative externality; a household that vaccinates, or a beekeeper whose bees pollinate a neighbour’s orchard, creates a positive one.
Focus on the negative case. The firm’s own marginal private cost (MPC) counts only what it pays — labour, materials, energy. Society also bears the pollution, the marginal external cost (MEC). Add them and you get the marginal social cost:
MSC = MPC + MEC.
With no externality on the buyers’ side, the demand curve already measures the marginal social benefit (MSB). The whole problem is the gap between MPC, which the firm acts on, and MSC, which society actually bears.
2 · Why the market overproduces
Left alone, the market settles where demand meets private cost — where MSB = MPC. That is privately optimal, because the firm ignores the external cost. But the socially efficient quantity is where demand meets the full social cost — where MSB = MSC.
Because MSC sits above MPC, the market’s equilibrium lies to the right of the social optimum: the market overproduces. For every unit between the efficient quantity and the market quantity, the social cost exceeds the social benefit — society would be better off if those units were never made.
Add up the excess cost over all those wasted units and you get the deadweight loss: a triangle between the MSC and MSB curves, running from the efficient quantity out to the market quantity. It is the money value of the harm the market inflicts by ignoring the externality.
3 · The Pigouvian tax
Arthur Pigou’s fix is elegant: charge the firm a per-unit tax equal to the external cost. The tax makes the firm feel society’s cost as its own — it internalises the externality. Private cost plus tax now equals social cost, so the firm’s profit-maximising quantity becomes the socially efficient one.
The size of the tax is the marginal external cost evaluated at the social optimum, not at the messy market quantity. When the external cost per unit is constant, the two coincide; when it rises with output, they differ, and using the market-quantity figure would overtax.
The mirror image works for positive externalities: there the market underproduces, and the fix is a Pigouvian subsidy equal to the external benefit, which nudges output up to the efficient level.
Worked example — a factory that pollutes
A factory’s output pollutes a river. Demand (the marginal social benefit) is P = 20 − Q; the firm’s marginal private cost is MPC = 2 + Q; each unit dumps a constant marginal external cost of MEC = £4. Prices are in pounds, Q in thousands of units.
Step 1 — The social cost. Marginal social cost is private plus external: MSC = (2 + Q) + 4 = 6 + Q, a curve running parallel to MPC, £4 above it everywhere.
Step 2 — The market equilibrium. The unregulated market sets demand equal to private cost: 20 − Q = 2 + Q, so 2Q = 18 and Q = 9. The price is P = 20 − 9 = £11. Call it E.
Step 3 — The social optimum. Efficiency sets demand equal to social cost: 20 − Q = 6 + Q, so 2Q = 14 and Q = 7, at a price of 20 − 7 = £13. Call it E*.
Step 4 — The overproduction. The market makes 9,000 units; society should want only 7,000. The extra 2,000 units are the problem — at each of them the social cost exceeds the social benefit. At the market quantity, MSC − MSB = 15 − 11 = £4, exactly the external cost.
Step 5 — The deadweight loss. The waste is the triangle between MSC and demand from Q = 7 to Q = 9: ½ × £4 × 2,000 = £4,000.
Step 6 — The Pigouvian tax. Set the tax equal to the external cost, £4 a unit. The firm’s cost becomes 2 + Q + 4 = 6 + Q — precisely MSC. It now maximises profit where 20 − Q = 6 + Q, giving Q = 7: the social optimum. The tax has closed the gap.
Step 7 — The revenue. The government collects £4 on each of 7,000 units — £28,000 — while the deadweight loss falls to zero. The externality is internalised, not merely priced.
Would you set the Pigouvian tax at the optimum, or at the market quantity? That single distinction separates a full-mark answer from a near miss, and drawing MPC, MSC and MSB cleanly under time pressure is exactly what a one-on-one economics tutor rehearses with you until it is automatic. Book a trial session.
Practice
Q1. Demand is P = 100 − Q, marginal private cost is MPC = 20 + Q, and the marginal external cost is a constant £10. Find the market quantity, the socially optimal quantity, the Pigouvian tax, and the deadweight loss.
Q2. A good has a positive consumption externality. Private benefit is MPB = 50 − Q, the marginal external benefit is £8, and marginal cost is MC = 2 + Q. Find the market and efficient quantities, and the correction needed.
Q3. Demand is P = 40 − Q, marginal private cost is MPC = 4 + Q, and the marginal external cost rises with output: MEC = Q. Find the market and efficient quantities and the correct Pigouvian tax.
Answers. Q1: the market sets 100 − Q = 20 + Q, so Q = 40; the optimum sets 100 − Q = 30 + Q, so Q = 35. The tax equals the external cost, £10, and the deadweight loss is ½ × 10 × 5 = £25. Q2: MSB = 58 − Q; the market sets 50 − Q = 2 + Q → Q = 24, while the optimum sets 58 − Q = 2 + Q → Q = 28. The market underproduces, so the fix is a £8 subsidy, with a deadweight loss of ½ × 8 × 4 = £16. Q3: MSC = 4 + 2Q; the market gives 40 − Q = 4 + Q → Q = 18, the optimum 40 − Q = 4 + 2Q → Q = 12. The tax is the external cost at the optimum, MEC(12) = £12 — not the £18 it reaches at the market quantity.
Key takeaways
- An externality is a cost or benefit landing on someone outside the transaction, so private and social values diverge: MSC = MPC + MEC.
- With a negative externality the market overproduces; the efficient quantity is where demand meets marginal social cost, not marginal private cost.
- The gap creates a deadweight loss — the triangle between the social-cost and benefit curves over the overproduced units.
- A Pigouvian tax equal to the marginal external cost at the optimum internalises the externality and restores efficiency. Positive externalities call for a subsidy instead.
Why London students choose our private economics tutoring
- Diagram, cold: whether you study at LSE, UCL, King’s or Warwick, sessions rehearse the MPC/MSC/MSB diagram and the deadweight-loss triangle until you can build it from a blank page under exam conditions.
- Both cases, and the traps: our tutors drill the negative and positive externality side by side, and the subtle point examiners love — that the tax equals the external cost at the optimum, not the market quantity.
- One-on-one pace: a private tutor works the numerical questions with you, catching the sign and quantity slips that quietly lose marks on this topic.
FAQ
Q: What is a negative externality?
A: It is a cost imposed on people outside a transaction, such as pollution from a factory. Because the producer does not pay it, the firm’s private cost is below the true social cost, and the market produces more than is socially efficient.
Q: Why does a market overproduce a good with a negative externality?
A: The market equates demand with marginal private cost, ignoring the external cost. The efficient quantity equates demand with marginal social cost, which is higher. Since private cost is understated, the market quantity is too large.
Q: What is a Pigouvian tax?
A: A per-unit tax set equal to the marginal external cost. It forces the producer to bear the cost it imposes on others, so private cost plus tax equals social cost and the firm chooses the socially efficient quantity. The externality is internalised.
Q: How large should the Pigouvian tax be?
A: Equal to the marginal external cost measured at the socially optimal quantity. If the external cost per unit is constant, that is simply the external cost. If it rises with output, you must evaluate it at the optimum, not at the larger market quantity.
Q: How are positive externalities corrected?
A: With a subsidy. A positive externality means the social benefit exceeds the private benefit, so the market underproduces. A per-unit subsidy equal to the marginal external benefit raises output to the efficient level — the mirror image of the tax.
Book a private economics tutor in London or online
Externalities reward the student who can draw MPC, MSC and MSB cleanly and read the overproduction, the deadweight loss and the correct tax straight off the diagram. A one-on-one session builds exactly that, on the questions your course sets — negative and positive cases, constant and rising external costs. Tell us your university and module, and we will match you with the right tutor this week, in London or fully online.
