Why the Initial Allocation Matters
A tradable-permit program sets a total limit on emissions and requires each unit of emissions to be covered by a permit. The regulator must also decide how permits first enter the market. It can give them to firms, often using a historical emissions measure, or sell them, usually through an auction. This initial decision can change who receives permit value and whether the government collects revenue. It need not change the total emissions limit or the least-cost pattern of abatement.
The distinction is important: the cap determines total permitted emissions, while the initial allocation determines who initially owns the permits. When permits are transferable and trading works competitively, firms respond to the market price whether their permits were received free or purchased. The earlier material on tradable emissions permits and cost-minimizing abatement provides the setting for this comparison.
Why Trading Equalizes Marginal Abatement Costs
Let firm \(i\) have baseline emissions \(E_i^0\), abate \(a_i\), and emit \(e_i=E_i^0-a_i\). Let \(C_i(a_i)\) be its abatement cost and \(MAC_i(a_i)=C_i'(a_i)\) its marginal abatement cost. With a fixed total cap, the required aggregate abatement is fixed. If firms can trade permits at a common price \(p\), each firm compares the cost of one more unit of abatement with the permit revenue it could earn by not emitting that unit.
For an interior choice, a firm abates until its marginal abatement cost equals the permit price. If one firm’s marginal cost were higher than another’s, the firms could reduce total cost by shifting some abatement from the high-cost firm to the low-cost firm, while keeping total emissions unchanged. This is why trading can deliver cost-effective abatement even when the initial permits are distributed unevenly.
This equality is a condition for an interior optimum, not a claim that every firm must have the same emissions or the same abatement. At a boundary—for example, a firm unable to abate more—the firm’s marginal cost need not equal the permit price. The argument also depends on a common, enforceable emissions unit and permits that firms can actually trade. Transaction costs, trading restrictions, or weak enforcement can prevent the predicted result.
If two firms have different marginal abatement costs, the firm with the lower cost can abate more at lower expense.
The low-cost firm abates an additional unit and sells the permit it no longer needs. The high-cost firm buys that permit and abates one less unit.
One firm’s emissions fall by one unit and the other’s rise by one unit, so aggregate emissions are unchanged. Total abatement cost falls whenever the original marginal costs differed.
Worked Example: Trading From an Unequal Free Allocation
Two firms each have baseline emissions of 10 units. The regulator sets a cap of 8 units, so the firms must abate 12 units in total. Firm A has abatement cost \(C_A(a_A)=\tfrac12a_A^2\), and firm B has \(C_B(a_B)=a_B^2\). The regulator initially gives A 6 permits and B 2. Find the cost-minimizing abatement, permit price, trade, and total abatement cost. Compare this with the outcome if the firms do not trade.
Method and assumptions. Assume each firm is a price taker, permits are freely transferable, and the cap is enforced. The marginal abatement costs are \(MAC_A=a_A\) and \(MAC_B=2a_B\). At an interior cost-minimizing allocation, these marginal costs are equal, and aggregate abatement must equal 12.
Substitute \(a_A=2a_B\) into the total-abatement equation: \(2a_B+a_B=12\), so \(a_B=4\) and \(a_A=8\). The common marginal cost, and hence the permit price, is \(p=MAC_A=8\); checking firm B gives \(MAC_B=2(4)=8\).
Firm A then emits \(10-8=2\) units, while firm B emits \(10-4=6\). These emissions sum to the cap: \(2+6=8\). A has 6 permits but needs only 2, so it sells 4. B has 2 permits but needs 6, so it buys 4. At a price of 8, the payment is \(4(8)=32\).
The abatement costs are \(C_A(8)=\tfrac12(8^2)=32\) and \(C_B(4)=4^2=16\), for a total of 48. To check why trading matters, without trade A’s 6 permits would require it to abate 4, while B’s 2 permits would require it to abate 8. Their costs would be \(\tfrac12(4^2)=8\) and \(8^2=64\), totaling 72. Trading reduces total abatement cost by \(72-48=24\) and preserves the cap.
Common mistake avoided: treating the initial permit holdings as permanent emissions quotas. They are starting endowments, not a barrier to trade. The relevant cost-minimizing condition is equal marginal abatement costs, not equal permit holdings or equal abatement.
Free Permits, Auctions, and the Value of an Endowment
A free permit still has a market value if it can be sold. A firm that uses a permit gives up the opportunity to sell it. Thus, when deciding whether to abate one more unit, a firm compares the cost of that abatement with the permit revenue it could otherwise receive. Giving a firm a fixed number of free permits changes its initial wealth, but does not eliminate the marginal incentive created by the permit price.
For a given emissions outcome, the firm’s net permit-related cash outlay can be written as the permit price times emissions minus the value of its initial free allocation. A negative outlay is a net receipt from selling permits. This accounting isolates permit transactions; it does not include abatement costs, changes in product prices, or other effects on the firm.
Here \(q_i^0\) is firm \(i\)’s initial free allocation. If permits are auctioned instead, the firm pays \(pe_i\) for the permits it needs, and the government receives the auction proceeds. The value \(pq_i^0\) is therefore a useful measure of the cash-flow difference between receiving that allocation free and having to buy the same quantity. It is not a measure of total social cost.
Worked Example: Comparing Free Allocation With Auctioning
A regulator issues 100 permits at a market price of $25 per permit. Firm A emits 40 units and firm B emits 60. Under free allocation, each firm receives 50 permits. Their abatement costs are $600 for A and $900 for B. Calculate permit cash flows, firm cash burdens including abatement costs, and government revenue under free allocation and auctioning. Assume emissions and the permit price are unchanged between the two allocation methods.
Method and assumptions. The total permits issued equal emissions: \(40+60=100\), so the cap is met. Under free allocation, use \(p(e_i-q_i^0)\) for each firm’s net permit cash outlay. Under auctioning, each firm pays for all permits it uses.
With free allocation, A has 50 permits and emits 40, leaving 10 to sell. Its net permit cash outlay is \(25(40-50)=-\$250\), so A receives $250. B emits 60 but holds 50 permits, so it buys 10. Its net permit cash outlay is \(25(60-50)=\$250\). The trade balances: A sells 10 and B buys 10, with a payment of \(10(25)=\$250\).
Including abatement costs, A’s cash burden is \(\$600-\$250=\$350\), and B’s is \(\$900+\$250=\$1{,}150\). Their combined burden is $1,500, equal to their combined abatement costs because their permit payments between one another cancel. The government receives no auction revenue under this free-allocation arrangement.
Under auctioning, A pays \(40(25)=\$1{,}000\) and B pays \(60(25)=\$1{,}500\). Government revenue is \(\$1{,}000+\$1{,}500=\$2{,}500\). Including abatement costs, A’s cash burden is $1,600 and B’s is $2,400, totaling $4,000. The $2,500 difference from the free-allocation total is exactly the government’s auction revenue.
The emissions and abatement costs in the question are held fixed, so auctioning changes who receives the value of the permits, not the real abatement cost. This comparison is about firms’ direct cash flows and government receipts. The ultimate distribution can also depend on how firms adjust wages, output, or prices, and on how the government uses auction revenue.
Common mistake avoided: counting auction payments as though they were destroyed resources. The $2,500 is a payment from firms to the government. In this example, the real abatement cost is $1,500 under either allocation method.
Fixed Allocation Can Change Wealth Without Changing Abatement
An allocation fixed in advance—such as one based on a historical emissions measure—can give some firms a more valuable endowment than others. If that allocation does not change when the firm changes current emissions or output, it acts like a wealth transfer while leaving the marginal permit price intact. Firms still compare the marginal cost of abatement with the value of a permit they could sell or would need to buy.
The qualification matters: if future free allocations are increased when a firm produces more or emits more, the allocation is no longer independent of the firm’s current choice. That design can change the marginal incentive. The conclusion that allocation affects wealth but not abatement applies to a fixed endowment, not to every rule described as free allocation.
Worked Example: Reassigning Fixed Permits Between Firms
Two firms each have baseline emissions of 20 units, and the cap is 32. Both have abatement cost \(C_i(a_i)=\tfrac32a_i^2\), so their marginal abatement cost is \(3a_i\). The permit price is $12. First, find the firms’ abatement and check the cap. Then compare their permit cash flows when the free allocations are 18 permits for A and 14 for B, and when the allocations are reversed.
Method and assumptions. Assume the firms are price takers and their allocations are fixed regardless of their current abatement. Each firm chooses abatement where \(3a_i=12\), giving \(a_A=a_B=4\). Each emits \(20-4=16\), so total emissions are \(16+16=32\), exactly the cap.
With allocations of 18 to A and 14 to B, A has 2 permits to sell and receives \(2(12)=\$24\). B needs 2 more permits and pays $24. Their net permit cash outlays are therefore \(-\$24\) for A and \(+\$24\) for B. Each firm’s abatement cost is \(\tfrac32(4^2)=\$24\), so their respective abatement-cost-plus-permit-cash totals are $0 for A and $48 for B.
Now reverse the allocations: A receives 14 and B receives 18. Emissions and abatement do not change, because each firm still faces the same $12 permit price. A buys 2 permits and pays $24; B sells 2 and receives $24. The corresponding totals are $48 for A and $0 for B. The allocation change shifts $48 of endowment value between the firms: the permit difference is 4 permits, each worth $12. Total abatement cost remains \(\$24+\$24=\$48\), and total net permit cash flow remains zero.
This cash-flow calculation does not mean that A’s use of a free permit has no economic cost. Using a permit rather than selling it gives up $12 in possible revenue. That opportunity cost is why the permit price still affects abatement decisions.
Common Traps and Why the Distinction Matters
- Assuming free permits are costless at the margin. A firm that uses a tradable permit gives up the opportunity to sell it. Its relevant marginal permit cost is the market price, even when the permit was initially free.
- Equating auction revenue with social cost. Auction revenue is a transfer to the government. Abatement uses resources; permit payments transfer purchasing power between firms and the government.
- Assuming free allocation changes the cap. The cap is the total number of permits issued. Changing which firms receive those permits does not change that total.
- Claiming every free-allocation rule leaves incentives unchanged. A fixed endowment independent of current decisions generally changes wealth rather than the marginal incentive. An allocation that varies with current emissions or output can alter that incentive.
These distinctions matter when comparing policy designs. An auction can raise public revenue that may be returned to households, used to reduce other taxes, or spent on public programs. Free allocation can instead direct permit value to firms. Neither statement alone determines who ultimately bears the policy’s burden: that depends in part on firms’ responses and on the use of public revenue. The key analytical separation is between cost-effective abatement, governed by marginal incentives, and distribution, influenced by initial ownership and revenue policy.
Check Your Understanding
Work through each problem by separating the cap and marginal incentives from the initial distribution of permit value.
- Two firms must abate 10 units in total. Their marginal abatement costs are \(MAC_X=a_X\) and \(MAC_Y=3a_Y\). Find each firm’s cost-minimizing abatement and the common permit price. (Answer: \(a_X=7.5\), \(a_Y=2.5\), and \(p=7.5\).)
- A permit price is $9. Firm A has 40 free permits and emits 25 units; firm B has 20 free permits and emits 35. How many permits are traded, in which direction, and what is the payment? (Answer: A sells 15 permits to B; the payment is $135.)
- A regulator auctions 80 permits for $14 each. Firms A and B use 30 and 50 permits, respectively. Find each firm’s payment and the government’s total revenue. (Answer: A pays $420, B pays $700, and government revenue is $1,120.)
- Two firms must abate 4 units in total. Their marginal abatement costs are \(MAC_A=2a_A\) and \(MAC_B=a_B\). Find cost-minimizing abatement and the permit price. Would fixed free allocations change this result if firms can trade? (Answer: \(a_A=\tfrac43\), \(a_B=\tfrac83\), and \(p=\tfrac83\); fixed allocations do not change the result under the stated trading assumptions.)
- A cap issues 50 permits at $20 each. Firms A and B emit 20 and 30 units and incur abatement costs of $300 and $500. With free allocations of 25 permits each, calculate the firms’ net permit cash flows and their combined cash burden including abatement costs. Then find auction revenue and combined firm cash burden under auctioning. (Answer: Under free allocation, A pays $100 and B receives $100, so the combined burden is $800. Under auctioning, revenue is $1,000 and the combined firm cash burden is $1,800.)