Taxes and Permits Under Cost Uncertainty

Why Cost Uncertainty Matters

A regulator may know the expected cost of reducing emissions but not the actual cost firms will face when they make their decisions. A tax fixes the price firms pay for emissions, while a permit cap fixes the quantity of emissions allowed. When costs turn out differently than expected, those instruments generally produce different abatement quantities. The policy choice is therefore about which type of error is less costly: allowing abatement to vary, or holding it fixed.

The comparison depends on the slopes of two curves. The marginal benefit of abatement measures the additional social benefit from reducing emissions by one more unit. The marginal abatement cost measures the additional private cost of doing so. A steep marginal benefit curve means that a small change in abatement can substantially change social benefits; a steep marginal cost curve means that a small change can substantially change abatement costs.

Definition: Let \(A\) denote total abatement, measured as emissions avoided relative to a baseline. Let \(MB(A)\) be the marginal social benefit of abatement, and let \(MAC(A)\) be the marginal cost to firms of abatement. A regulator’s quantity policy specifies \(A\), equivalently setting an emissions cap consistent with that abatement. A price policy sets a tax per unit of emissions; a firm then abates until the marginal cost of another unit of abatement equals the tax it avoids by abating.

A Linear Model of Uncertain Costs

To isolate the instrument comparison, assume there is one aggregate abatement decision, firms respond to the same marginal price, and the regulator evaluates total social surplus. Suppose the marginal benefit curve is known and slopes downward, while the marginal abatement cost curve slopes upward but has an uncertain vertical shift:

MB(A)=m-bA,\qquad MAC(A;\theta)=c+\theta+dA,

where \(b>0\) and \(d>0\) are the absolute slopes, and \(\theta\) is an uncertain cost shock. A positive \(\theta\) makes abatement more expensive at every quantity; a negative \(\theta\) makes it less expensive. Assume \(E[\theta]=0\), and assume all policy and optimal abatement quantities in the examples are interior to the feasible range. The regulator knows the shock’s distribution but not its realized value when setting policy.

With known costs, efficient abatement equates marginal benefit and marginal cost. Before the shock is observed, the expected-cost target \(\bar A\) therefore satisfies \(MB(\bar A)=E[MAC(\bar A;\theta)]\). A tax calibrated to this target is \(\tau=E[MAC(\bar A;\theta)]\). A quantity policy instead sets abatement at \(\bar A\). When the shock is realized, the tax still equals \(\tau\), but the quantity cap still requires \(\bar A\).

Worked Example: Costlier Abatement Makes a Fixed Target Safer

A hypothetical regulator estimates \(MB(A)=40-4A\) and \(MAC(A;\theta)=4+\theta+2A\), with abatement measured in thousands of tons and money measured in thousands of dollars. The cost shock is equally likely to be \(-6\) or \(+6\). Compare a tax calibrated to expected costs with a permit policy that fixes the expected efficient abatement quantity. Use the method of calculating the efficient target, the policy outcomes in each state, and the resulting surplus losses.

Expected target and tax. Since the two shocks are equally likely, \(E[\theta]=0\), so expected marginal cost is \(4+2A\). Equating it to marginal benefit gives \(40-4\bar A=4+2\bar A\), hence \(36=6\bar A\) and \(\bar A=6\). The calibrated tax is \(\tau=4+2(6)=16\) thousand dollars per thousand tons.

Efficient abatement in each state. At any realized shock, set \(40-4A=4+\theta+2A\). This gives \(A^*(\theta)=(36-\theta)/6\). When \(\theta=-6\), efficient abatement is \(42/6=7\); when \(\theta=+6\), it is \(30/6=5\).

Policy outcomes. Under permits, abatement remains 6 in either state. Under the tax, firms choose abatement where \(MAC=\tau\): \(16=4+\theta+2A\), so \(A_{\text{tax}}=6-\theta/2\). The tax produces \(9\) when \(\theta=-6\) and \(3\) when \(\theta=+6\).

The total marginal surplus \(MB-MAC\) has slope \(-(4+2)=-6\). As justified below, the loss from being a distance \(e\) from the realized efficient quantity is \(\frac12(6)e^2\). The permit policy differs from the efficient quantity by 1 in either state, so its loss is \(\frac12(6)(1)^2=3\) thousand dollars in each state. The tax differs by 2 in either state, so its loss is \(\frac12(6)(2)^2=12\) thousand dollars in each state. Expected losses are therefore 3 and 12 thousand dollars, respectively. The fixed quantity performs better here because the marginal benefit slope, 4, is steeper than the marginal cost slope, 2. The calculation assumes the stated linear curves and interior outcomes; it is not a general claim that permits always dominate taxes.

Why the Slopes Determine the Comparison

The result follows by comparing each policy’s abatement to the efficient quantity after the shock is known. The realized efficient quantity satisfies \(m-bA^*=c+\theta+dA^*\), so

A^*(\theta)=\bar A-\frac{\theta}{b+d}.

The permit policy holds abatement at \(\bar A\). The tax holds the marginal price at \(\tau=c+d\bar A\); firms choose \(A_{\text{tax}}\) such that \(c+\theta+dA_{\text{tax}}=\tau\). Solving gives \(A_{\text{tax}}=\bar A-\theta/d\). Thus, a cost increase reduces abatement under either policy, but the amount of that reduction differs.

For a complete welfare justification, define total net surplus at abatement \(A\) and shock \(\theta\) as the area under marginal benefit minus the area under marginal cost:

W(A;\theta)=\int_0^A \bigl[MB(q)-MAC(q;\theta)\bigr]\,dq.

The derivative \(W_A=MB(A)-MAC(A;\theta)\) is zero at \(A^*\). Because the marginal curves are linear, \(W_A=-(b+d)(A-A^*)\). Integrating this difference between \(A^*\) and a policy quantity \(A\) gives an exact loss, not just a local approximation:

W(A^*;\theta)-W(A;\theta)=\frac12(b+d)(A-A^*)^2.

Under permits, the quantity error is \(\bar A-A^*=\theta/(b+d)\). Under the tax, it is \(A_{\text{tax}}-A^*=-\theta b/[d(b+d)]\). Substituting these errors into the loss expression and taking expectations yields:

E[L_{\text{permits}}]=\frac{E[\theta^2]}{2(b+d)},\qquad E[L_{\text{tax}}]=\frac{b^2E[\theta^2]}{2d^2(b+d)}.

For a nonzero amount of cost uncertainty, comparing these losses gives the standard slope rule:

Result: In this linear model with a mean-zero additive shift in marginal abatement cost, a fixed abatement quantity has lower expected welfare loss when \(b>d\), meaning marginal benefit is steeper than marginal cost. A tax has lower expected loss when \(d>b\). They have equal expected loss when \(b=d\). The result compares the slopes’ absolute values and relies on the stated model assumptions.

Worked Example: A Steep Cost Curve Favors the Tax

Suppose a hypothetical water authority estimates \(MB(A)=30-A\) and \(MAC(A;\theta)=5+\theta+4A\), where \(A\) is abatement in thousands of units and dollar amounts are in thousands. The cost shock is equally likely to be \(-4\) or \(+4\). Find the expected efficient target and calibrated tax, then compare expected losses under the two policies.

Target and tax. Expected marginal cost is \(5+4A\). Equating \(30-A=5+4A\) gives \(25=5A\), so \(\bar A=5\). The calibrated tax is \(\tau=5+4(5)=25\) thousand dollars per thousand units.

Efficient and policy quantities. With the realized shock, \(30-A=5+\theta+4A\), so \(A^*(\theta)=(25-\theta)/5\). Efficient abatement is \(5.8\) for \(\theta=-4\) and \(4.2\) for \(\theta=+4\). Permits hold abatement at 5. Under the tax, \(25=5+\theta+4A\), so \(A_{\text{tax}}=5-\theta/4\): the tax quantities are 6 and 4 in those two states.

Welfare losses. The total marginal surplus has slope \(-(1+4)=-5\), so loss is \(\frac12(5)e^2\), where \(e\) is the deviation from \(A^*\). Permits have an error of \(0.8\) in either state, giving \(\frac12(5)(0.8)^2=1.6\) thousand dollars. The tax has an error of \(0.2\) in either state, giving \(\frac12(5)(0.2)^2=0.1\) thousand dollars. These are also the expected losses because each policy has the same loss in both states. The tax is preferred: the marginal cost slope, 4, is steeper than the marginal benefit slope, 1. The tempting mistake is to assume that a cap must be safer because it fixes emissions; the calculation shows that fixing quantity can be costly when the marginal cost curve is steep.

Unequal Probabilities and Expected Loss

The formula depends on the second moment \(E[\theta^2]\), not on whether the cost shock has just two outcomes. The examples so far used symmetric shocks, but the same comparison applies to any distribution with a finite second moment and mean zero. For a discrete distribution, calculate each state’s loss and weight it by that state’s probability. Do not compare only the most likely state or average the state-specific quantities before calculating losses.

Worked Example: Weighting Several Cost States

A hypothetical regulator faces \(MB(A)=30-2A\) and \(MAC(A;\theta)=6+\theta+A\). The cost shock is \(-3\) with probability \(0.25\), zero with probability \(0.50\), and \(+3\) with probability \(0.25\). Calculate the expected loss under a fixed quantity and under a tax calibrated to expected costs.

Expected target and tax. The expected shock is \((-3)(0.25)+(0)(0.50)+(3)(0.25)=0\). Equating expected marginal cost to marginal benefit gives \(30-2\bar A=6+\bar A\), so \(24=3\bar A\) and \(\bar A=8\). The calibrated tax is \(\tau=6+8=14\).

State-specific choices. The realized optimum solves \(30-2A=6+\theta+A\), so \(A^*(\theta)=8-\theta/3\). It is 9, 8, and 7 for shocks \(-3,0,+3\), respectively. Permits hold abatement at 8. Under the tax, firms solve \(14=6+\theta+A\), giving \(A_{\text{tax}}=8-\theta\), or 11, 8, and 5 across those states.

Expected losses. The total marginal surplus has slope \(-(2+1)=-3\), so loss is \(\frac12(3)e^2\). Permit errors are \(-1,0,1\), with losses \(1.5,0,1.5\) thousand dollars. The expected permit loss is \((0.25)(1.5)+(0.50)(0)+(0.25)(1.5)=0.75\) thousand dollars. Tax errors are \(2,0,-2\), with losses \(6,0,6\) thousand dollars. The expected tax loss is \((0.25)(6)+(0.50)(0)+(0.25)(6)=3\) thousand dollars. The fixed quantity has the lower expected loss because \(b=2>d=1\). As a check, \(E[\theta^2]=(9)(0.25)+(0)(0.50)+(9)(0.25)=4.5\), and the permit formula gives \(4.5/[2(3)]=0.75\), matching the state-by-state calculation.

Assumptions, Interpretation, and Exam Traps

The slope rule is useful because it converts a policy comparison into a comparison of marginal-curve slopes. It does not say that one instrument is always superior. Its clean form assumes the regulator can estimate the expected target and the relevant slopes; uncertainty is an additive shift in marginal cost; marginal benefits are known; firms respond to the tax or permit price as modeled; and outcomes remain within the feasible range. If costs are uncertain in ways that change their slope, if marginal benefits are also uncertain, or if quantities hit a boundary, the simple loss formulas need not describe the comparison. In particular, uncertainty about cost levels is not the same as uncertainty about how costs change with abatement.

A common calculation error is to substitute the policy quantity into the realized marginal curves and treat the resulting marginal surplus as the total welfare loss. Marginal surplus is a rate at one quantity; total loss is the area between the curves across the quantity error. For example, in the first worked example, when \(\theta=6\), permits choose \(A=6\) while the optimum is 5. At \(A=6\), \(MB-MAC=16-(4+6+12)=-6\). The loss is not 6 thousand dollars: the marginal surplus gap grows linearly from zero at \(A=5\) to 6 at \(A=6\), so the loss is the triangle \(\frac12(1)(6)=3\) thousand dollars. Use the integrated loss formula, and keep the units of the quantity and money amounts consistent.

Key takeaway: Under additive abatement-cost uncertainty, a tax fixes the marginal price and lets abatement vary; a quantity limit fixes abatement and lets the implied price vary. Compare the absolute slopes of marginal benefit and marginal cost: the steeper curve generally identifies which policy makes the less costly adjustment.

Check Your Understanding

For each problem, calculate the expected target before comparing policy losses. Unless stated otherwise, assume an additive, mean-zero cost shock and interior quantities.

A regulator estimates \(MB(A)=26-2A\) and \(MAC(A;\theta)=2+\theta+A\), with \(\theta=-3\) or \(+3\), each equally likely. Find the expected efficient abatement target and determine which instrument has lower expected loss. (Answer: \(\bar A=8\); the fixed quantity has lower expected loss because \(b=2>d=1\).)
Suppose \(MB(A)=24-A\) and \(MAC(A;\theta)=4+\theta+3A\), with \(\theta=-2\) or \(+2\), each equally likely. Find the calibrated tax and identify the instrument with lower expected loss. (Answer: \(\bar A=5\), \(\tau=19\); the tax has lower expected loss because \(d=3>b=1\).)
Let \(MB(A)=32-3A\) and \(MAC(A;\theta)=2+\theta+A\), with \(\theta=-4\) or \(+4\), equally likely. Calculate the expected target and choose the lower-loss instrument. (Answer: \(\bar A=7.5\); fixed quantity has lower expected loss because \(b=3>d=1\).)
For \(MB(A)=28-2A\) and \(MAC(A;\theta)=4+\theta+2A\), the shock is \(-4\) with probability \(0.25\), zero with probability \(0.50\), and \(+4\) with probability \(0.25\). Calculate the expected loss under each instrument. (Answer: \(\bar A=6\); expected permit loss is \(1\) thousand dollars and expected tax loss is \(1\) thousand dollars.)
A model has \(MB(A)=35-A\) and \(MAC(A;\theta)=5+\theta+2A\). The shock is \(-3\) with probability \(0.50\) and \(+3\) with probability \(0.50\). Find expected efficient abatement and compare expected losses. (Answer: \(\bar A=10\); the tax has lower expected loss because \(d=2>b=1\).)

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