A price on carbon is the Pigouvian prescription — a per-tonne charge equal to the marginal external cost, derived on the externalities page. Take that as settled and ask the next question: which instrument puts the price there? Carbon is a global commons, so no firm abates voluntarily (the free-rider logic sits on the public-goods page); the price must be imposed. Two instruments do it — a carbon tax and a cap-and-trade permit market. How they work, when they coincide, and when they part is the environmental-economics material that most often brings intermediate students to an economics tutor in Exeter.
1 · The marginal abatement cost curve
Start with one number: the marginal abatement cost, MAC — the cost of cutting one more tonne. A firm abates its cheapest tonnes first, so each further tonne is dearer: MAC rises with abatement A, an upward-sloping curve.
Now impose a carbon price t, a tax per tonne emitted. On every tonne the firm chooses: abate it at cost MAC, or emit it and pay t. It abates any tonne whose MAC is below t and stops where MAC = t.
A firm abates up to MAC = t, and pays the price on every tonne beyond.
The price never dictates how to cut — it lets each firm find its own cheapest tonnes.
2 · Cost-effectiveness: cut where it is cheapest
One firm is a warm-up; instrument design needs two. A cement plant and a power station must cut 40 tonnes between them — but cement abates dearly (its process emissions are hard to cut) while the power station abates cheaply (it can fuel-switch). Order each to cut the same 20 tonnes and you force expensive cuts at the cement plant while cheap cuts go spare at the power station: the same carbon at very different marginal costs.
Shift cutting from the dear source to the cheap one and society saves the gap, until none is left. That is the cost-effective rule — equalise marginal abatement cost across sources — and a uniform mandate cannot reach it, because it would need every firm’s MAC. A carbon price makes firms reveal it; the worked example puts numbers on the saving.
3 · Cap-and-trade: permits, allocation, and the clearing price
A market instrument reaches that split with no MAC known to the regulator. Fix the cap: issue permits for exactly the allowed emissions, one per tonne, auctioned or grandfathered free. Then let firms trade. A high-cost abater would rather buy a permit than cut an expensive tonne; a low-cost abater would rather cut more and sell. So permits flow from the cheap source to the dear one, and the price is bid up until each firm — abating to MAC = price — together just meets the cap. That clearing price is the carbon price, discovered rather than set, and it delivers the very split a tax would. Real systems — Europe’s, California’s — run exactly this logic at scale, with a market price that moves as the cap tightens.
4 · Prices versus quantities: the equivalence and where it breaks
Under certainty the two instruments coincide. Charge a per-tonne tax and each firm abates to MAC = the tax, so total abatement is exactly the cap, at exactly the cost a permit market reaches — fix the price and the quantity follows, fix the quantity and the price follows, two names for one point.
Uncertainty breaks it. The regulator rarely knows the MACs, and Weitzman’s rule then decides. Compare two slopes: the marginal benefit of abating (the marginal damage avoided) against the marginal cost (the MAC). If marginal benefit is flatter, a price is safer — a cost surprise then moves quantity a lot but its value little. If it is steeper — a threshold, a tipping point — the quantity matters most, and a cap is safer. Carbon’s damage builds slowly, a flat benefit curve, the textbook case for a price. Treat that as a compressed sketch, not the last word.
Worked example — two plants, a 40-tonne cap, and the gains from trade
Cement’s MACC = 6AC; the power station’s MACP = 2AP; cut 40 tonnes in total. A firm’s abatement cost is the triangular area under its MAC.
Step 1 — The mandate. Cut 20 each. Cement spends £1,200 and the power station £400 — £1,600 in all.
Step 2 — The waste. At 20 apiece, cement’s last tonne costs £120 and the power station’s £40. That £80 gap means carbon is being cut in the wrong place.
Step 3 — The efficient split. Equalise the MACs subject to the cap: cement cuts 10, the power station 30, at a common MAC of £60. Now cement spends £300 and the power station £900 — £1,200, a £400 saving for the same 40 tonnes.
Step 4 — Tighten the cap. Require 60 tonnes and the rule gives cement 15, the power station 45, at a common MAC of £90. The price rises as the cap tightens — it is what the cap costs at the margin.
Step 5 — Deliver with a price. A £60 tax makes each abate to MAC = £60; a 40-permit market is bid to £60. Both reach the 10-and-30 split, cement buying permits, the power station selling.
Step 6 — Interpret. The mandate fixed a quantity per firm — right total, wrong split, £1,600. A price fixes one number every firm faces and lets each pick its quantity, so the cheap abater does more: £1,200. Tax or cap, under certainty the two routes reach the same point.
Given two firms’ abatement-cost curves and a cap, could you find the split that minimises total cost — and show a tax and a permit market reach the very same point? Deriving the equal-MAC rule and the price–quantity equivalence, rather than naming the instruments, is exactly what environmental-economics questions reward. A one-on-one economics tutor builds the two-firm diagram with you until the cost-effective split falls out on its own. Book a trial session.
Practice
Q1. Two firms: MACA = 5AA and MACB = 3AB; a cap requires 32 tonnes cut. (a) The cost of an equal-cutback mandate, 16 each. (b) The cost-effective split, the carbon price, the cost, and the saving.
Q2. The cement plant and power station under the 40-tonne cap (price £60, split 10 and 30), but with permits grandfathered 25 to cement and 15 to the power station. (a) How much does each abate? (b) Who trades, and how many permits? (c) Does total abatement cost change?
Answers. Q1: (a) £1,024; (b) AA = 12, AB = 20, price £60, cost £960, saving £64. Q2: (a) the price is unchanged, so cement abates 10 and the power station 30; (b) cement buys 5 permits, the power station sells 5; (c) no — still £1,200, the allocation being a lump-sum transfer, not an efficiency lever.
Key takeaways
- A firm abates to MAC = the carbon price. The price sets the target and lets each firm find its own cheapest tonnes.
- Cost-effectiveness is equal MAC across sources. An equal-cutback mandate gets the total right and the split wrong — £1,600 against £1,200, a £400 penalty for the same 40 tonnes.
- Tax and cap-and-trade are equivalent under certainty. Both face every source with one carbon price; one fixes the price, the other the quantity.
- Uncertainty breaks the tie. Flat marginal benefit favours a price, a steep threshold a quantity; carbon’s slow damage tilts the case toward a price.
Why Exeter students choose our economics tutoring
- Models built, not memorised: sessions derive the equal-MAC rule and the tax–permit equivalence from the abatement-cost curves, so you can reproduce them under exam pressure rather than quote a slogan.
- The distinctions examiners test, drilled: cost-effectiveness versus efficiency, price versus quantity instruments, the permit allocation as pure distribution — the lines that separate a first from a 2:1.
- One-on-one and matched to your course: a tutor works from your own notation and past papers, whether your module follows Perman, Hanley, or Kolstad.
FAQ
Q: What is the marginal abatement cost curve?
A: It plots the cost of cutting one more tonne against how much a firm has already cut. It slopes up because firms use their cheapest options first.
Q: Why is a carbon price cheaper than a fixed rule?
A: It lets each firm cut only the tonnes worth less than the price, so cheap abaters do more. Marginal costs end up equal across firms — the cost-effective outcome a mandate misses.
Q: Are a carbon tax and cap-and-trade the same?
A: Under certainty, yes. A tax fixes the price and lets quantity adjust; a cap fixes the quantity and lets price adjust. Same abatement, same cost.
Q: Does it matter who gets the free permits?
A: Not for efficiency. The allocation is a lump-sum transfer — it sets who pays, not who abates. The cap fixes the environmental outcome.
Q: When is a cap better than a tax?
A: When the damage from missing the quantity rises steeply — a threshold. If marginal damage is flat, a price guards better against cost surprises. That is Weitzman’s rule.
Book an economics tutor in Exeter or online
Environmental economics rewards students who can derive the abatement rule and the tax–permit equivalence, not just name the instruments. One-on-one sessions build that fluency on your own past papers and notation. Tell us your course and exam date, and we will match you with the right tutor this week.
