Negative externality graph explained
The negative externality graph has one line more than the supply and demand graph you know, and that line is where the marks go. Students draw it under the supply curve instead of above it, shade a triangle that points the wrong way, or decide that the "efficient" amount of pollution is zero. Read the three lines in order and the graph answers every question on its own.
The graph in one minute
Take the mill of Lesson 10. The vertical axis is dollars per ton, the horizontal axis tons of paper a day. Demand is what buyers pay for one more ton: P = 100 − Q. MPC is what the mill itself pays to make one more ton, wood, workers and electricity: P = Q. That is the supply curve you already know, under a new name. Every ton also costs the crews downstream $20 in lost fish. The mill never sees that bill, so it is not in MPC. Add it and you get MSC, the full cost to society of one more ton: P = Q + 20. Same slope as MPC, $20 higher at every quantity.
Where the market stops, and where it should
The mill weighs only its own cost, so it keeps making paper while the price covers MPC. Society would want it to stop while the price covers MSC. Slide the quantity and compare the three numbers at each ton.
At ton 25 the price is $75 and the full cost $45: making it is a gain for everyone. At ton 40 price and MSC are both $60, and the gain is gone. The mill does not stop there, because its own cost is $40 and the price is still $60. It stops at ton 50, where the price has fallen to $50 and finally matches its own cost. Those last ten tons cost society more than anyone pays for them.
The deadweight triangle
Each ton between 40 and 50 is a small loss: $0 at ton 40, growing to $20 at ton 50, where buyers pay $50 for a ton that costs society $70. Add the losses up and you get the triangle on the graph below, between the efficient and the market quantity, with demand as its floor and MSC as its roof. Its area is ½ × 10 tons × $20 = $100 a day. This is the deadweight loss of the externality, exactly as a tax's deadweight loss in Lesson 4 was a triangle of trades that should have happened. Here it is a triangle of tons that should not.
| Ton | Buyers pay | Mill's cost | Full cost | Worth it? |
|---|---|---|---|---|
| 30 | $70 | $30 | $50 | Yes, +$20 |
| 40 | $60 | $40 | $60 | Just. Efficient |
| 45 | $55 | $45 | $65 | No, −$10 |
| 50 | $50 | $50 | $70 | No, −$20. Market stops |
The fix: a tax equal to the harm
The mill ignores the $20 because nobody charges it. So charge it. A Pigouvian tax (after Arthur Pigou, the British economist who proposed it in 1920) adds $20 to every ton the mill makes. Its private cost becomes Q + 20: MPC slides up onto MSC, and the mill, still looking only at its own bill, chooses 40 tons by itself. Move the tax and watch the quantity, the loss and the money raised.
Three readings. With no tax, 50 tons and a $100 loss. At $20 the mill's cost line is MSC, it chooses 40 tons, the loss is gone and the tax raises $20 × 40 = $800 a day, on every ton still made, not only the ten it gave up. At $40 the mill cuts to 30 tons, and now ten tons that were worth more than their full cost go unmade: the loss is back, $100, from the other side. The right tax is the harm per ton, no more and no less. The tax is not the only fix: Ronald Coase (British economist, Nobel Prize 1991) argued that if it is clear who owns the river and bargaining is cheap, the mill and the crews could strike their own deal; with many people affected, a tax is usually simpler.
The trap on tests
- MSC goes above MPC. A negative externality adds a cost, so the social line sits higher. If your social cost line is below the private one, you have drawn a positive externality.
- The market is where demand meets MPC. The efficient quantity is where demand meets MSC, to its left. Label both, and say the market makes too much.
- The efficient quantity is not zero. At 40 tons the harm is still $20 a ton; it is worth bearing because buyers pay $60 for a ton that costs society $60. Only tons past that point are a loss.
- The triangle points left. Its corners are the efficient point, the market point and the point on MSC straight above the market quantity. Area = ½ × extra tons × harm per ton.
- The tax equals the external cost per unit, charged on every unit. Revenue is tax × the quantity still made, not × the quantity cut.
- Consumption externalities move the other line. If the harm comes from using the good (cigarette smoke, not paper making), the gap is on the demand side: social benefit below private benefit, same answer, too much.
Quick check
1. A cement plant faces demand P = 80 − Q and has MPC P = Q (dollars per ton, tons a day). Each ton costs the neighbors $10 in dust. Find the market quantity, the efficient quantity and the deadweight loss.
Market: 80 − Q = Q, so 40 tons at $40. MSC = Q + 10. Efficient: 80 − Q = Q + 10, so 2Q = 70 and Q = 35 at $45. Deadweight loss: ½ × 5 extra tons × $10 harm = $25 a day.
2. Same plant. The city sets a tax of $10 a ton. How much does it raise, and what happens to the loss?
The tax equals the harm, so the plant's cost becomes Q + 10 = MSC and it chooses 35 tons on its own. Revenue: $10 × 35 = $350 a day, charged on all 35 tons still made. The deadweight loss falls to $0.
3. Same plant, but the dust is found to cost $20 a ton, not $10. What changes?
MSC = Q + 20. Efficient: 80 − Q = Q + 20, so Q = 30 at $50. The market still makes 40, now 10 tons too many instead of 5, and the loss is ½ × 10 × $20 = $100: double the harm per ton, four times the triangle. The right tax is now $20 a ton.