How to Calculate the Equilibrium Permit Price
The equilibrium permit price equals the marginal abatement cost of the last unit of abatement the cap requires, the level where every firm's marginal abatement cost is equal.
The Permit Price formula
Calculator
Enter baseline emissions, the cap and each firm's abatement cost slope to get the equilibrium permit price.
What the two firms emit together when pollution is free.
The number of permits the regulator issues. Every ton above this has to be abated.
Firm A's marginal abatement cost is this figure times the tons it cuts, so at 10 tons its last ton costs 10 times this.
The same rule for the second firm. A steeper slope means abatement gets expensive faster.
Trading settles at $60 a ton, the marginal abatement cost of the last ton the cap forces out of the industry.
- Abatement the cap requires
- 40
- Tons cut by the low-cost firm
- 30
- Tons cut by the high-cost firm
- 10
- Total abatement cost under trading
- $1,200
- Extra cost of forcing equal cuts
- $400
Baseline emissions minus the cap leaves 40 tons that somebody has to cut.
The firm with the flatter cost curve does 30 tons of the cutting and sells the permits it no longer needs.
The firm with the steeper cost curve cuts only 10 tons and buys permits to cover the rest, which is cheaper for it than abating.
Adding both firms' cost of cutting comes to $1,200, the cheapest way to reach this cap.
Ordering both firms to cut the same amount would cost $1,600, which is $400 more for exactly the same tons of pollution removed.
How to calculate Permit Price, step by step
- 1Find how much abatement the cap forces. Subtract the cap from what the industry would emit if pollution were free. Permits are scarce by exactly that amount, and that gap is what gives them a price.
- 2Write down each firm's marginal abatement cost. Marginal abatement cost rises as a firm cuts more, because the cheap fixes get used first. Two firms with different technology have different slopes, which is the whole reason there is anything to trade.
- 3Split the cuts so marginal costs are equal. A firm whose marginal abatement cost sits below the permit price cuts more and sells its spare permits, a firm above the price buys instead. Trading stops only when both firms sit at the same marginal cost.
- 4Read the price off that common level. The permit price is the marginal abatement cost of the last ton the cap requires. The regulator never learns either firm's costs, since the trading does that work.
- 5Compare against a uniform order. Price the same total cut when every firm is told to cut the same amount. The difference is what trading saves, and it is the standard case for permits over a mandate.
Worked example: Permit Price
Two firms emit 120 tons between them and the regulator issues permits for 80, so 40 tons of cuts have to happen. Firm A's marginal abatement cost is $2 for each ton it has already cut, so at 30 tons its last ton costs 2 × 30 = $60. Firm B is steeper at $6, so at 10 tons its last ton also costs 6 × 10 = $60. That split adds to the 40 tons required and leaves both firms at the same marginal cost, so the permit price settles at $60. Total abatement cost is ½ × 2 × 30² + ½ × 6 × 10² = 900 + 300 = $1,200. Ordering each firm to cut 20 tons instead would cost ½ × 2 × 20² + ½ × 6 × 20² = 400 + 1,200 = $1,600, so trading removes the same 40 tons for $400 less.
Permit Price questions
Who sets the price of a permit?
The market does. The regulator fixes the number of permits, which makes supply vertical, and demand comes from firms comparing the price with their own cost of abating. If the cap is loose or a downturn cuts output, permit demand falls and the price can collapse without any policy change at all.
Why does trading equalize marginal abatement cost across firms?
Because any firm whose marginal abatement cost is below the permit price gains by cutting one more ton and selling the permit, and any firm above the price gains by buying a permit instead of cutting. Those trades keep happening while a gap exists, so the gap closes. Equal marginal abatement cost across sources is the condition for reaching a given total cut at the lowest possible cost.
What happens to the permit price if the cap is tightened?
A tighter cap forces more abatement, and firms move up their rising marginal abatement cost curves, so the price rises. In the example above, cutting the cap from 80 tons to 60 raises required abatement from 40 tons to 60 and lifts the price from $60 to $90. This is the trade-off students are asked to name: permits fix the quantity of pollution and let the price move, while a tax fixes the price and lets the quantity move.
Does it matter whether permits are auctioned or handed out free?
It changes who captures the value of the permits, not how much pollution there is or what a permit trades for. The cap sets emissions either way, and a permit used is a permit not sold, so holding one carries an opportunity cost equal to its market price. Free allocation transfers wealth to the firms that receive the permits and an auction sends that money to the government instead.
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