How Tradable Permits Work
A regulator can limit pollution by setting a maximum number of emissions permits and requiring firms to surrender one permit for each unit they emit. The total number of permits sets an overall emissions limit; trading lets firms change how many permits each holds. The cap controls the permitted total, while a firm’s final emissions depend on its permit holdings and compliance.
This approach is useful when firms differ in the cost of reducing emissions. A firm that can cut emissions at relatively low cost may be willing to abate more and sell permits. A firm facing higher abatement costs may prefer to buy permits. The trades change the distribution of emissions across firms, not the cap itself. This topic is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.
The regulator may initially allocate permits to firms, for example by a free allocation or an auction. Whatever the initial allocation, permitted trading transfers existing permits between firms: a sale reduces the seller’s holdings and increases the buyer’s by the same amount. It does not create additional permits. If the initial allocations sum to the cap, then trades leave that sum unchanged.
A permit allocation is not the same thing as a required abatement amount. Given a firm’s baseline emissions, the number of units it must abate to emit no more than its permit holdings is \(E_i^0-q_i\), if that quantity is nonnegative. When a firm holds more permits than its baseline emissions, it can comply without using all its permits; holding an unused permit does not force it to emit.
Why the Cap Limits Aggregate Emissions
The cap’s accounting logic follows directly from the firm-level compliance requirement. Add \(e_i\leq q_i\) across all firms. If the regulator issues \(Q\) permits and permits cannot be duplicated, the sum of holdings is \(Q\). Therefore aggregate emissions cannot exceed \(Q\). If every permit is surrendered for an emission unit, the inequality is an equality.
The same relationship can be written in terms of abatement. Since \(e_i=E_i^0-a_i\), using all permits gives \(\sum_i(E_i^0-a_i)=Q\). Rearranging shows that total abatement must equal total baseline emissions minus the cap.
This is a short derivation of the required aggregate reduction: the cap fixes total emissions, and emissions are baseline emissions less abatement. The equality assumes all issued permits are used and firms’ baseline emissions are measured consistently. If some permits are left unused, actual emissions are below the cap and aggregate abatement is greater than the amount implied by equality.
Worked Example: Tracking a Permit Sale
Two firms have baseline emissions of 80 units for Firm A and 60 units for Firm B. The regulator issues 100 permits, initially allocating 55 to A and 45 to B. Both firms use all permits. A then sells 10 permits to B, and both continue to use all permits. Find initial and final abatement and emissions, and verify the cap after the trade.
Method. For each firm, calculate abatement as baseline emissions minus permit holdings. Then check that the holdings and emissions add to the cap. The stated assumption that all permits are used means emissions equal holdings for each firm.
Initially: A abates \(80-55=25\) units and emits 55. B abates \(60-45=15\) units and emits 45. Total emissions are \(55+45=100\), equal to the cap.
After the trade: A sells 10 permits, so A’s holdings fall to \(55-10=45\). B buys those 10, so B’s holdings rise to \(45+10=55\). A now abates \(80-45=35\) units and emits 45. B now abates \(60-55=5\) units and emits 55.
Check: A’s 10-permit sale and B’s 10-permit purchase balance. Final total emissions are \(45+55=100\), and final total abatement is \(35+5=40\), which equals total baseline emissions \(140\) minus the cap \(100\). A’s abatement rises by 10 while B’s falls by 10; the trade has not changed total emissions.
Common mistake avoided: A’s holdings decrease, so A sells permits; it does not buy them. The direction of a trade is determined by the change in holdings.
How Marginal Abatement Costs Guide Trading
A firm’s marginal abatement cost (MAC) is the additional cost of abating one more unit, measured in dollars per unit. Write the firm’s total abatement cost as \(C_i(a_i)\); where differentiable, its marginal abatement cost is \(C_i'(a_i)\). A firm compares that additional cost with the permit price \(p\).
If the firm can abate one more unit for less than \(p\), abating saves it the cost of buying a permit, or allows it to sell a permit for more than the cost of the extra abatement. If the additional abatement costs more than \(p\), it is less costly at the margin to obtain a permit instead. Under a competitive permit market, a firm with an interior choice—neither zero abatement nor its maximum—chooses abatement until its MAC equals the permit price:
This condition applies to a price-taking firm choosing among feasible abatement levels, with a known permit price and a smooth cost function. At a boundary, equality need not hold: a firm may abate zero if its initial MAC is already above the price, or abate as much as technically possible if its MAC remains below the price. The MAC comparison explains the direction of incentives; it does not by itself establish that every observed trade is profitable, since the firms’ costs and the price must also be considered.
Worked Example: Finding Abatement at a Permit Price
Two firms have baseline emissions of 60 and 50 units, and a cap of 70 permits. Firm A’s abatement cost is \(C_A(a_A)=\tfrac12a_A^2\); Firm B’s is \(C_B(a_B)=\tfrac32a_B^2\). A competitive permit price is \(30\) dollars per unit. Initially, A holds 35 permits and B holds 35. Find each firm’s abatement at the stated price, the permit trade from the initial allocation, and total emissions.
Method. Differentiate each cost function to obtain MAC, set MAC equal to the permit price for an interior choice, and then convert abatement into emissions and permit holdings. Verify that the resulting holdings sum to the cap.
For A, \(C_A'(a_A)=a_A\). Thus \(a_A=30\) when \(C_A'(a_A)=30\). For B, \(C_B'(a_B)=3a_B\), so \(3a_B=30\) and \(a_B=10\). Both choices are feasible because they are between zero and the firms’ respective baseline emissions.
A’s emissions are \(60-30=30\); B’s are \(50-10=40\). To cover those emissions, A needs 30 permits and B needs 40. Relative to their initial holdings of 35 each, A sells \(35-30=5\) permits and B buys \(40-35=5\).
Cost and cap check: A’s abatement cost is \(\tfrac12(30)^2=450\) dollars. B’s is \(\tfrac32(10)^2=150\) dollars. Total emissions are \(30+40=70\), and final holdings are \(30+40=70\). The permit payment is \(5\times30=150\) dollars from B to A; it is a transfer between firms, not an additional abatement cost for the economy as a whole.
Common mistake avoided: The permit price is set equal to marginal cost, not total abatement cost. For B, solving \(3a_B=30\) gives 10 units; dividing the cost function’s total value by the price would not give the firm’s chosen abatement.
Following Trades Across Several Firms
With several firms, keep a separate ledger for each: baseline emissions, abatement, emissions, and permit holdings. A consistent calculation satisfies \(e_i=E_i^0-a_i\) and, if all permits are used, \(e_i=q_i\). For trades, the total quantity sold must equal the total quantity bought. These checks catch sign errors that can otherwise reverse the economic interpretation.
Worked Example: Three Firms and Balanced Trades
Firms A, B, and C have baseline emissions of 50, 40, and 30 units. The cap is 92 permits. Their initial permit allocations are 40, 30, and 22. Their abatement cost functions are \(C_A(a_A)=\tfrac12a_A^2\), \(C_B(a_B)=a_B^2\), and \(C_C(a_C)=2a_C^2\). At a permit price of 16 dollars, find their abatement choices, resulting permit holdings, and trades relative to the initial allocation. Assume all permits are used.
Method. Calculate each MAC and equate it to the stated price. Then derive emissions and required holdings, compare those holdings with the initial allocation, and balance buyers against sellers.
The marginal costs are \(C_A'(a_A)=a_A\), \(C_B'(a_B)=2a_B\), and \(C_C'(a_C)=4a_C\). At a price of 16, the firms choose \(a_A=16\), \(2a_B=16\), or \(a_B=8\), and \(4a_C=16\), or \(a_C=4\).
A’s emissions and required permits are \(50-16=34\). B’s are \(40-8=32\), and C’s are \(30-4=26\). Compared with their initial holdings, A sells \(40-34=6\) permits, B buys \(32-30=2\), and C buys \(26-22=4\). Total purchases are \(2+4=6\), equal to A’s sale.
Checks: Final permits total \(34+32+26=92\), the cap. Emissions also total 92, and total abatement is \(16+8+4=28\), equal to baseline emissions \(120\) minus the cap \(92\). Each firm’s MAC at its chosen abatement is 16 dollars per unit.
The abatement costs are \(C_A(16)=128\), \(C_B(8)=64\), and \(C_C(4)=32\), for a total of 224 dollars. A receives \(6\times16=96\) dollars; B pays \(2\times16=32\), and C pays \(4\times16=64\). The buyers’ total payment equals the seller’s revenue.
Common mistake avoided: A’s holdings fall from 40 to 34, so A sells 6 permits. B’s and C’s holdings rise, so they buy 2 and 4. Treating a decrease in holdings as a purchase would reverse the trade directions even if the quantities happened to balance.
Common Compliance and Calculation Traps
- Confusing the cap with an individual allocation. The cap applies to total permits. A firm’s allocation can rise or fall through trading while the aggregate total remains fixed.
- Reversing the trade direction. If holdings fall from 45 to 30, the firm sells \(45-30=15\) permits. If another firm’s holdings rise from 20 to 30, it buys \(30-20=10\).
- Confusing permits with abatement. Abatement is baseline emissions minus actual emissions. With full permit use, it is also baseline emissions minus holdings. A fall in holdings therefore increases, not decreases, required abatement.
- Forgetting the full-use assumption. Compliance requires emissions no greater than holdings. A firm need not emit every unit its permits authorize; if it leaves permits unused, aggregate emissions can be below the cap.
- Comparing total cost with the permit price. The relevant comparison is between the permit price and the marginal cost of one additional unit of abatement, not total abatement cost.
Check Your Understanding
For each problem, track abatement and permit holdings separately. Assume all permits are used unless the question says otherwise.
- Firms A and B have baseline emissions of 42 and 38 units. The cap is 60 permits, initially allocated as 30 to each firm. A sells 3 permits to B. Find each firm’s final holdings and abatement, and verify total emissions. (Answer: A holds 27 permits and abates 15; B holds 33 and abates 5; total emissions are 60.)
- Two firms must abate 30 units in total. Their marginal abatement costs are \(MAC_A=2a_A\) and \(MAC_B=a_B\). If both choose interior abatement at a common permit price, find each firm’s abatement and the price. (Answer: \(a_A=10\), \(a_B=20\), and the permit price is 20 dollars per unit.)
- Firms A and B have baseline emissions of 40 and 50 units, a cap of 75, and initial allocations of 30 and 45 permits. B sells 4 permits to A. Find final holdings, abatement, and total emissions. (Answer: A holds 34 and abates 6; B holds 41 and abates 9; total emissions are 75.)
- Two firms must abate 25 units in total, with \(MAC_A=a_A\) and \(MAC_B=4a_B\). If both have interior choices at the same permit price, solve for their abatement levels and the price. (Answer: \(a_A=20\), \(a_B=5\), and the permit price is 20 dollars per unit.)
- Firms A, B, and C have baseline emissions of 35, 30, and 25 units, and a cap of 70. Their marginal abatement costs are \(a_A\), \(2a_B\), and \(3a_C\), respectively. At a permit price of 12 dollars per unit, find each firm’s abatement, verify the aggregate reduction, and find total emissions. (Answer: Abatement is 12, 6, and 4 units; aggregate abatement is 22, equal to \(90-70\); total emissions are 70.)