A limit with a market inside it
Two lectures ago, two firms polluted a river, and the government wanted 20 units of pollution cut. Firm A could cut at $2 a unit, and Firm B only at $8. A rule that each firm cut 10 units cost $100. A $5 tax per unit of pollution got the same cut for $40, because A did all the cutting. But the tax had a weakness. The government had to guess the right tax. If it guessed too low, the cut would fall short, and for a dangerous pollutant, falling short can be serious.
That lecture ended with a promise: a third approach that combines the certainty of a rule with the flexibility of a tax. It is called cap and trade, or tradeable permits. It sounds strange at first: using a market to fix a market failure. This lecture shows how it works, why it finds the cheapest cuts, and where it can go wrong.
How cap and trade works
The idea has three steps.
- Set a cap. The government decides the total amount of pollution allowed in a year, for example a number of tonnes of carbon dioxide.
- Issue permits. It creates exactly that many permits. Each permit allows its owner to emit one unit, such as one tonne. The permits can be sold at auction, or given free to firms, usually in proportion to what they emitted in the past. Free allocation is often called grandfathering.
- Let firms trade. At the end of each year, every firm must hand over one permit for each unit it emitted. A firm that cuts its pollution can sell the permits it no longer needs. A firm that wants to pollute more must buy permits from someone else. Firms that emit without permits pay heavy fines.
The government fixes the total quantity of pollution with a cap and issues that many permits. Firms buy and sell the permits, and the price is set by the permit market.
Because the permits add up to the cap, total pollution cannot exceed the cap, however the permits are traded.
Firms A and B each emit 30 units. The government wants a total of 40, a cut of 20. It issues 40 permits and gives each firm 20 free. A can cut up to 20 units at $2 each; B can cut at $8 each.
Without trading, each firm must cut 10 units to match its permits. Cost: , the same as the rule.
With trading, look at it from each side. For B, every permit it can buy for less than $8 saves it an $8 cut. For A, every permit it can sell for more than $2 is worth an extra $2 cut. So there is a deal at any price between $2 and $8. Suppose they agree on $5. A cuts all 20 units (cost $40) and sells its 10 spare permits to B for $50. B cuts nothing and uses the permits it bought.
Total pollution: A emits 10, B emits 30, so 40, exactly the cap. Total cost of the cuts: $40, as with the tax. And both firms are better off than without trading. A spends $40 and receives $50, a gain of $10, instead of paying $20. B pays $50 instead of $80.
Look at what happened. Nobody told A to do the cutting. The permit price made it profitable. And the quantity was certain from the start, because there were only ever 40 permits.
What sets the permit price
Real markets have many firms, and each firm's cuts get more expensive as it cuts more. The first tonne is cheap: switch off a wasteful machine. The fiftieth tonne is expensive: rebuild a factory. Economists call the cost of cutting one more unit the marginal abatement cost ("abate" means reduce).
Suppose permits trade at a price . A firm compares two choices for each tonne: cut it, or keep emitting it and use a permit. If its marginal abatement cost is below , cutting is cheaper, so it cuts, and sells or saves the permit. If its marginal abatement cost is above , it keeps emitting and pays for a permit. So every firm cuts until its marginal abatement cost reaches the permit price, and then stops.
This is the key to why cap and trade is cheap. Every firm ends up with the same marginal abatement cost, equal to the permit price. If two firms had different marginal costs, the total cost could be lowered by moving a tonne of cutting from the expensive one to the cheap one. When the marginal costs are equal, there is no such move left. The total cut is being made at the lowest possible cost. That is it. That is the whole idea. A rule that tells every firm to cut the same amount leaves the marginal costs unequal, as the grey circles show, so it cannot be the cheapest.
Change the cap below and watch the permit market respond.
Three invented firms each start with 40 t of emissions and are given permits for an equal share of the cap. Choose the total cut the cap requires. Watch the permit price, who cuts how much, who sells permits, and the cost compared with a rule that every firm cuts the same amount. Try 21, 42 and 63 tonnes.
Three invented firms, with marginal abatement costs of $1, $2 and $4 per tonne for each tonne already cut. Each is given permits for an equal share of the cap. The permit price, the cuts and the costs:
| Total cut | Permit price | Cuts by A, B, C | Cost with trading | Cost of equal cuts |
|---|---|---|---|---|
| 21 t | $12 | 12, 6, 3 t | $126 | $171.50 |
| 42 t | $24 | 24, 12, 6 t | $504 | $686 |
| 63 t | $36 | 36, 18, 9 t | $1 134 | $1 543.50 |
In every case, trading cuts the total cost by about 27%. Firm A, the cheapest cutter, always cuts more than its share and sells permits; B and C buy them.
As the cap tightens, the price rises, and every firm cuts more. But the cheapest cutter, Firm A, always does the most, and the saving over equal cuts stays at about a quarter of the cost. In this model the permit price rises in step with the cut: double the cut, double the price.
Does it matter who gets the permits?
Go back to Worked Example 1, and give all 40 permits to B, none to A. B emits 30, so it has 10 spare. A emits 30 and holds nothing. A can cut 20 units at $2 each, which is cheaper than buying permits at any price above $2. So A cuts 20 units and buys B's 10 spare permits for the rest. Now give all 40 permits to A instead. A cuts 20 units, because each unit cut frees a permit that B will buy for anything up to $8. A uses 10 permits and sells 30 to B, who cuts nothing. Either way, A does all 20 units of cutting.
The allocation changes who pays and who receives money. It does not change who does the cutting, or the total cost of the cuts. This surprising idea goes back to the economist Ronald Coase. In 1960 he argued that once rights are clearly defined and can be traded cheaply, they flow to whoever values them most, whoever held them first. A permit system creates exactly such a right: a right to emit one tonne.
A firm is given its permits free. A student says: "Then the firm has no reason to cut pollution, because polluting costs it nothing." Is that right?
Show the answer
No. A free permit is still worth its market price, because the firm could sell it. If permits sell for $24 and the firm can cut a tonne for $15, it gains $9 by cutting and selling the spare permit. Using a permit to pollute means giving up $24. Economists call this an opportunity cost. So free permits keep the incentive to cut. What free allocation changes is who keeps the money: the firm, instead of the government.
Certain quantity or certain price?
A tax and a permit system can both reach the cheapest cuts. Both give firms a reason to keep cutting, because every tonne cut saves a tax payment or frees a permit to sell. Both can raise money for the government, if permits are auctioned. The real difference is what each one fixes.
- A tax fixes the price of polluting. The quantity depends on how firms respond, which the government cannot know exactly.
- A cap fixes the quantity. The price depends on how costly the cuts turn out to be, which the government cannot know exactly either.
So the choice depends on which uncertainty is more harmful. If the damage rises sharply past a threshold, as in the chemical question two lectures ago, a certain quantity matters most, and a cap is safer. If firms need a steady price to plan long investments, a tax is more predictable. Some permit systems combine the two, with a minimum price, a maximum price, or both.
Sort the features below.
For each feature, decide whether it belongs to a carbon tax, to cap and trade, or to both.
Eight features sorted, each with its reason.
- Carbon tax: the price of a tonne is known in advance; in a boom, total emissions can rise.
- Cap and trade: total emissions are known in advance; the price can collapse if the cap is too generous; the price can jump if cuts prove costly.
- Both: the cheapest cuts are made first; every tonne cut saves the firm money; can raise revenue for the government.
A real case: the EU Emissions Trading System
The policy. In 2005, the European Union started a cap-and-trade system for carbon dioxide from power stations and energy-intensive industry, later extended to aviation and shipping. Firms must hand in one allowance (permit) for each tonne they emit. Prices are in euros. In the first phase, 2005–2007, nearly all allowances were given free.
The prediction. If the cap is below what firms would otherwise emit, allowances become scarce and valuable. Firms then cut where cutting is cheaper than the allowance price, and emissions fall to the cap.
What happened. In the first phase, there was little reliable data on emissions, so the cap was set from estimates. According to the European Commission, more allowances were issued than were needed, and the allowance price fell to zero in 2007. The cap was tightened in later phases. The Commission's 2024 carbon market report states that in 2023, emissions from the installations covered were about 47.6% below their 2005 level, and that auctions raised €43.6 billion that year.
The verdict. The first phase shows the weakness of a cap: if it is too generous, the price collapses and nothing changes. Once the cap was tight, the system put a real price on carbon. How much of the fall in emissions the system caused, and how much came from other changes, such as cheaper renewable electricity, is still debated. The EU ETS was the first major carbon market. China's national system, announced in 2017, covers more emissions: about 3 billion tonnes from its power sector at launch, according to the UN climate secretariat.
Three classic mistakes
A firm can buy extra permits, but only from another firm that has cut its own pollution. The total is fixed by the cap. Trading moves the cuts; it does not add pollution.
A free permit can still be sold, so using it has an opportunity cost equal to the permit price. The incentive is the same as with auctioned permits. Only the money ends up in different hands.
Both find the cheapest cuts, both keep the incentive to cut, and both can raise revenue. The real difference is certainty: permits fix the quantity, a tax fixes the price.
Work each one on paper first. Open a hint only when you are stuck.
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Firms A and B each emit 25 units. The cap is 30, and each firm is given 15 permits. A can cut up to 25 units at $3 a unit; B can cut at $9 a unit. Compare the cost of the cut without trading and with trading. Between which prices could a permit trade?
Hint 1
Without trading, each firm must cut 10 units.
Hint 2
With trading, the cheaper cutter does all 20 units of cutting.
Full answer
Without trading: . With trading: A cuts 20 units for and sells 10 permits to B. Total cost $60, a saving of $60, with total emissions still . A will sell for more than $3, and B will buy for less than $9, so the price lies between $3 and $9.
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Use the three firms from the explorer (marginal abatement costs of $1, $2 and $4 per tonne for each tonne already cut). The cap requires a total cut of 21 t. Show that a permit price of $12 gives exactly 21 t of cuts, and that every firm's last tonne then costs the same.
Hint 1
Each firm cuts until its marginal cost reaches $12: Firm B's cost is $2 times the tonnes cut.
Hint 2
A cuts 12 t, B cuts , C cuts .
Full answer
A cuts 12 t, B cuts 6 t and C cuts 3 t: t. A's last tonne costs , B's and C's . All equal to the permit price, so no cheaper way to share the 21 t exists. (A higher price would give more than 21 t of cuts, and a lower one less, so $12 is the price at which the permit market clears.)
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An economic slowdown means factories produce much less. What happens to emissions and to the price of pollution (a) under a carbon tax and (b) under a cap-and-trade system?
Hint 1
Less production means firms would emit less, so they need fewer permits.
Hint 2
Which tool holds the price still, and which holds the quantity?
Full answer
(a) Under a tax, the price stays the same, and emissions fall because firms produce less. (b) Under cap and trade, firms need fewer permits, so demand for permits falls and the permit price drops. Emissions stay at the cap, or below it if permits are left unused. So the slowdown cuts emissions under a tax, but mostly cuts the permit price under a cap. This is one reason the EU's system had a large surplus of allowances after the 2008 economic crisis.
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A power company is given 1000 permits free. The permit price is $20 a tonne. It can cut 300 t of its emissions at $15 a tonne. (a) Should it cut those 300 t? (b) How much better off is it? (c) A politician says: "Giving permits free is a gift to polluters." Explain what is right and what is wrong in this claim.
Hint 1
Each tonne cut frees a permit that can be sold for $20.
Hint 2
Think about who receives the value of the permits under free allocation, compared with an auction.
Full answer
(a) Yes. Each tonne costs $15 to cut and frees a permit worth $20. (b) It gains . (c) Right: the permits are worth , and under free allocation the firm keeps that value; under an auction, the government would receive it. Wrong: free permits do not remove the incentive to cut, because every permit used to pollute could have been sold. So free allocation changes who gains the money, not how much pollution is cut.
The original problem was that the air and the rivers had no price: anyone could dump into them for free. A tax puts a price on pollution directly. Cap and trade does something different. It creates a new kind of property, the right to emit, and lets a market set the price. This is why Coase's idea matters so much: many market failures come from missing property rights, and some can be fixed by creating them.
But every one of these tools depends on a government choice: the size of the tax, or the size of the cap. Set too generously, as in the EU's first phase, the tool does nothing; set badly in other ways, it can make things worse. That risk, government failure, is part of this unit too.
- Under cap and trade, the government fixes total pollution and issues that many permits, which firms can buy and sell.
- Each firm cuts until its marginal abatement cost equals the permit price, so all firms' marginal costs are equal and the total cut is made at the lowest cost.
- Who receives the permits changes who gains money, not who does the cutting; free permits still carry an opportunity cost.
- A tax fixes the price of pollution and leaves the quantity uncertain; a cap fixes the quantity and leaves the price uncertain.
- A cap works only if it is tight enough: the EU system's first-phase price fell to zero because too many permits were issued.