Why the Allocation of Abatement Matters
A policy can fix the total amount of emissions while leaving firms flexibility over who reduces emissions. That flexibility matters because firms may have different costs of reducing pollution. Requiring every firm to make the same reduction can therefore meet the target at unnecessarily high cost. Trading permits lets firms with lower abatement costs reduce more and sell permits to firms for which reduction is more expensive.
The key comparison is not the firms’ total abatement costs or the amounts they each reduce. It is their marginal abatement cost: the extra cost of one more unit of abatement. For a cost-minimizing allocation, the marginal costs of firms undertaking abatement must be equal, except at a boundary where a firm does no abatement or has reached its maximum feasible reduction.
This analysis takes the aggregate emissions target as given. It answers how to meet that target at least cost, not what the target itself should be. The distinction is important: choosing an efficient target involves comparing the benefits and costs of emissions reductions, while permit trading minimizes the cost of achieving a specified target.
The Cost-Minimizing Condition
Suppose there are \(n\) firms, each with a differentiable, increasing, strictly convex abatement cost function \(C_i(a_i)\). Strict convexity means that marginal abatement cost rises as the firm abates more. Assume each firm can choose any nonnegative abatement level in the relevant range, and the regulator requires total abatement of \(\bar A\). The cost-minimization problem is:
For an interior allocation, where every firm abates a positive amount, the condition is:
Here \(\lambda\) is the common marginal cost of abatement at the optimum. The condition follows directly from the cost-minimization problem:
For an interior solution, let \(L=\sum_i C_i(a_i)-\lambda(\sum_i a_i-\bar A)\). The multiplier \(\lambda\) attaches a value to the total-abatement requirement.
The first-order condition for firm \(i\) is \(C_i'(a_i)-\lambda=0\), so \(C_i'(a_i)=\lambda\).
Because every active firm faces the same multiplier, every active firm has the same marginal abatement cost. If one firm had a higher marginal cost than another, shifting a small amount of abatement from the high-cost firm to the low-cost firm would reduce total cost without changing total abatement.
The last step also provides a direct justification. If firm A’s marginal cost is greater than firm B’s, transfer a small amount \(\epsilon\) of abatement from A to B. To first order, the change in total cost is approximately \(-MAC_A\epsilon+MAC_B\epsilon\), which is negative when \(MAC_A>MAC_B\). The original allocation therefore cannot minimize cost.
Strict convexity makes the marginal costs rise as firms take on more abatement. Under the stated assumptions, equalizing marginal costs identifies the unique cost-minimizing interior allocation. If a firm is at zero abatement, the condition instead is that its marginal cost at zero is at least as high as the common marginal cost of firms that do abate. Otherwise, assigning a small amount of abatement to that firm would lower total cost. Feasibility limits on a firm’s maximum abatement can also create boundary conditions.
How Permit Trading Implements the Condition
A permit authorizes its holder to emit a specified amount. With a fixed total number of permits, the total emissions allowed by the regulated firms are fixed. A firm can reduce its own abatement requirement by buying permits, or take on more abatement and sell permits it no longer needs. Trading reallocates the responsibility for meeting the aggregate emissions limit; it does not change the total number of permits.
Consider a competitive permit market with a common price \(p\) per permit. A firm compares the price of buying a permit with the cost of avoiding the corresponding emissions by abating. If its marginal abatement cost is below the permit price, it is less costly to abate the next unit and sell the permit. If its marginal abatement cost is above the permit price, buying a permit is less costly than undertaking that unit of abatement. For a firm choosing a positive interior amount, the resulting condition is:
Thus, if firms can trade freely and the market clears, active firms face the same permit price and have equal marginal abatement costs. A firm’s initial allocation of permits can affect its wealth, but under these assumptions it does not change the marginal condition that determines its abatement choice. This cost-minimizing result relies on the fixed aggregate cap, well-defined abatement costs, and trading that allows permits to move between firms; it does not by itself establish that the chosen cap is socially optimal.
Worked Examples
Worked Example: Comparing Trading With Equal Abatement
Two hypothetical firms must together abate 6 units. Their abatement costs, in cost units, are \(C_A(x)=5x+x^2\) and \(C_B(y)=y^2\). Find the cost-minimizing allocation and compare its total cost with the cost when each firm abates 3 units. Assume both firms can choose any nonnegative abatement level and their cost functions apply throughout the relevant range.
Method. Equalize marginal abatement costs and impose the aggregate requirement \(x+y=6\). The marginal costs are \(C_A'(x)=5+2x\) and \(C_B'(y)=2y\). At an interior minimum, \(5+2x=2y\).
Substitute \(y=6-x\): \(5+2x=2(6-x)=12-2x\). Therefore \(4x=7\), so \(x=1.75\), and \(y=6-1.75=4.25\). The common marginal cost is \(5+2(1.75)=8.5\); checking firm B gives \(2(4.25)=8.5\).
The cost for firm A is \(C_A(1.75)=5(1.75)+(1.75)^2=8.75+3.0625=11.8125\). The cost for firm B is \(C_B(4.25)=(4.25)^2=18.0625\). Total cost is \(11.8125+18.0625=29.875\) cost units.
With equal abatement, each firm reduces emissions by 3 units. Firm A’s cost is \(C_A(3)=15+9=24\), and firm B’s cost is \(C_B(3)=9\), for total cost \(24+9=33\). Trading lowers cost by \(33-29.875=3.125\) cost units. As a check, the efficient reductions still sum to \(1.75+4.25=6\).
Common mistake avoided: Equal reductions do not generally minimize cost. The calculation instead equalizes marginal costs; at the efficient allocation, both marginal costs are 8.5.
Worked Example: Interpreting a Permit Transfer
Two hypothetical firms must abate 12 units in total. Their costs are \(C_A(x)=2x^2\) and \(C_B(y)=y^2\), with both firms able to choose any nonnegative abatement level. Initially, each is assigned permits consistent with abating 6 units. Find the least-cost abatement allocation, the implied permit transfer, and the cost saving relative to the initial equal-abatement allocation.
Method. The marginal costs are \(4x\) for A and \(2y\) for B. Set them equal and use \(x+y=12\): \(4x=2y\), so \(y=2x\). Then \(x+2x=12\), giving \(x=4\) and \(y=8\). Both marginal costs equal \(16\).
At the equal-abatement allocation, each firm reduces emissions by 6. The cost is \(C_A(6)+C_B(6)=2(36)+36=108\). At the efficient allocation, the cost is \(C_A(4)+C_B(8)=2(16)+64=96\). The saving is \(108-96=12\) cost units.
Firm A abates 2 units less than under the initial allocation and therefore needs permits for those emissions. Firm B abates 2 units more and can sell 2 permits. The transfer preserves aggregate abatement: A’s 4 units plus B’s 8 units still equal 12. The calculation verifies the cost saving as well as the target.
Common mistake avoided: A permit transfer is measured in emissions units, not in dollars or in abatement-cost units. Here the transfer is 2 permits, while the cost saving is 12 cost units.
Worked Example: Checking a Corner Allocation
Two hypothetical firms must abate 2 units in total. Their costs are \(C_A(x)=10x+x^2\) and \(C_B(y)=2y^2\). Determine whether the least-cost allocation is interior, and compare its cost with equal abatement. Assume abatement cannot be negative.
Method. The marginal costs are \(10+2x\) for A and \(4y\) for B. If the solution were interior, the conditions \(10+2x=4y\) and \(x+y=2\) would both hold. Substituting \(y=2-x\) gives \(10+2x=8-4x\), or \(x=-\frac{1}{3}\). This violates \(x\geq0\), so the interior solution is infeasible.
Check the boundary \(x=0,\ y=2\). Firm B’s marginal cost at \(y=2\) is \(4(2)=8\), while firm A’s marginal cost at zero is \(10\). Moving a small amount of abatement from B to A would replace a unit with marginal cost near 8 by one with marginal cost 10, raising total cost. The boundary is therefore cost-minimizing. Its cost is \(C_A(0)+C_B(2)=0+2(4)=8\) cost units.
With equal abatement, \(x=y=1\), costs are \(C_A(1)=10+1=11\) and \(C_B(1)=2\), totaling 13 cost units. The boundary allocation saves \(13-8=5\) cost units. The check is that total abatement remains \(0+2=2\), and A’s marginal cost at zero exceeds B’s marginal cost at the chosen positive abatement.
Common mistake avoided: Do not accept a negative abatement amount just because it solves the equality equation. The nonnegativity constraint requires checking whether an interior solution is feasible and, if not, examining a boundary.
Common Traps and Practical Checks
A common exam error is to assume that every firm should abate the same amount. Equal reductions are efficient only in special cases, such as when the firms’ marginal-cost schedules make equal reductions satisfy the cost-minimizing condition. In the first example, the incorrect calculation would assign \(x=y=3\) and report cost \(33\) as the minimum. The correction is to compare marginal costs: at that allocation, A’s marginal cost is \(5+2(3)=11\), while B’s is \(2(3)=6\). Since A’s marginal cost is higher, shifting abatement from A to B reduces cost. The equal-marginal-cost solution is \(x=1.75,\ y=4.25\), with total cost \(29.875\).
Another trap is to equalize the firms’ total abatement costs. Total costs can differ at the optimum; what matters is whether a small feasible transfer of abatement could lower the sum. When a firm has zero abatement, do not force equality with the other firms. Check instead that its initial marginal cost is not below the common marginal cost of the firms that are abating.
For a calculation, a reliable sequence is: write each cost function, differentiate to obtain marginal costs, impose the total-abatement requirement, solve the equality conditions, check all feasibility constraints, and calculate total costs separately. Finally, verify that the allocations sum to the required target and that any claimed saving is the difference between the two total costs.
Check Your Understanding
For each problem, identify the relevant marginal-cost condition, check feasibility, and verify the total abatement requirement.
- Two firms must abate 6 units. Their costs are \(C_A(x)=x^2+4x\) and \(C_B(y)=2y^2\). Find the cost-minimizing allocation, common marginal cost, and total cost. (Answer: \(x=\frac{10}{3},\ y=\frac{8}{3}\); common marginal cost \(=\frac{32}{3}\); total cost \(=\frac{116}{3}\) cost units.)
- Two firms must abate 16 units and have costs \(C_A(a)=3a^2\) and \(C_B(b)=b^2\). Calculate the efficient reductions, total cost, and saving relative to equal abatement. (Answer: \(a=4,\ b=12\); efficient total cost \(=192\); equal-abatement cost \(=256\); saving \(=64\) cost units.)
- Two firms must abate 2 units, with costs \(C_A(x)=10x+x^2\) and \(C_B(y)=2y^2\). Check whether the solution is interior; then find the least-cost allocation and its saving relative to equal abatement. (Answer: the interior solution is infeasible; \(x=0,\ y=2\); efficient cost \(=8\); equal-abatement cost \(=13\); saving \(=5\) cost units.)
- A competitive permit price is 18 cost units per emissions unit. A firm’s abatement cost is \(C_A(x)=3x^2\), and another firm’s is \(C_B(y)=y^2\). Assuming both choose positive interior abatement, find each firm’s abatement and their combined abatement. (Answer: \(x=3,\ y=9\); combined abatement \(=12\) units.)
- Two firms must abate 10 units, with costs \(C_A(x)=2x^2\) and \(C_B(y)=4y^2\). Find the efficient allocation and compare its total cost with equal abatement. (Answer: \(x=\frac{20}{3},\ y=\frac{10}{3}\); common marginal cost \(=\frac{80}{3}\); efficient cost \(=\frac{400}{3}\); equal-abatement cost \(=150\); saving \(=\frac{50}{3}\) cost units.)