Calculating Pigouvian Taxes with Linear Curves

From Linear Curves to a Corrective Tax

A negative externality makes the private cost of producing a good smaller than the full cost to society. A Pigouvian tax can bring those costs into alignment: producers respond to a higher private cost, and the market quantity can move toward the socially efficient level. With linear demand, private marginal cost, and marginal damage curves, the calculations are direct—but the tax must be evaluated at the right quantity.

The method here assumes a competitive market with price-taking buyers and sellers, an interior equilibrium, and no external benefits or other market distortions. The tax is a constant amount per unit, chosen using the marginal damage associated with efficient output. All prices and costs below are in dollars per unit, and quantities use the units specified in each example.

Definition: Let inverse demand, private marginal cost, and marginal damage be $P_D(Q)=a-bQ$, $MPC(Q)=c+dQ$, and $MD(Q)=e+fQ$, respectively. Here $Q$ is output, $P_D(Q)$ is the price buyers are willing to pay, $MPC(Q)$ is the producer’s marginal cost, and $MD(Q)$ is the marginal external damage. Social marginal cost is $SMC(Q)=MPC(Q)+MD(Q)$. The unregulated competitive quantity equates inverse demand and private marginal cost. The efficient quantity equates inverse demand and social marginal cost. A constant per-unit Pigouvian tax set at the efficient quantity is $t^*=MD(Q^*)$, where $Q^*$ is efficient output.

Three Equations, Three Different Roles

The unregulated market does not include external damage in the seller’s private cost. Its equilibrium is therefore found from $P_D(Q_0)=MPC(Q_0)$, where $Q_0$ is unregulated output. To find the socially efficient quantity, include the marginal damage caused by each additional unit: $P_D(Q^*)=MPC(Q^*)+MD(Q^*)$. Finally, the corrective tax is the marginal damage evaluated at $Q^*$, not at $Q_0$.

After a per-unit tax is imposed, the price paid by buyers and the amount received by sellers are different. At output $Q$, buyers pay $P_D(Q)$; sellers receive that price less the tax. In a competitive equilibrium with tax $t$, the buyers’ price must satisfy $P_D(Q)=MPC(Q)+t$. If $t=MD(Q^*)$, that market equation matches the efficient-output condition at $Q^*$.

Why the Efficient-Output Condition Holds

The condition for efficient output follows from the change in total surplus when output increases slightly. Define total surplus as the benefit buyers receive, less the private production costs and the external damage. A marginal increase in output adds the value buyers place on the last unit and subtracts both kinds of marginal cost.

W(Q)=\int_0^Q \bigl[P_D(q)-MPC(q)-MD(q)\bigr]\,dq

For an interior maximum, the derivative of total surplus with respect to output must equal zero. Differentiating the integral gives the marginal contribution of the last unit. Thus the efficient quantity satisfies inverse demand equal to social marginal cost. This argument assumes the curves are continuous and that the maximum occurs at a positive quantity rather than at a boundary.

\frac{dW}{dQ}=P_D(Q)-MPC(Q)-MD(Q)=0 \quad\Longleftrightarrow\quad P_D(Q^*)=SMC(Q^*)

With linear curves, the output calculations can be written explicitly. The unregulated quantity is $(a-c)/(b+d)$, and the efficient quantity is $(a-c-e)/(b+d+f)$, provided the resulting quantities are positive and the relevant total marginal-cost curve rises relative to inverse demand. These expressions come from solving the two equilibrium conditions; they do not replace checking that the assumed interior solution is appropriate.

Worked Calculations

Worked Example: A Tax That Matches Marginal Damage

A competitive market has inverse demand $P_D(Q)=100-2Q$, private marginal cost $MPC(Q)=20+Q$, and marginal damage $MD(Q)=10+Q$. Find the unregulated equilibrium, efficient output, Pigouvian tax, and post-tax equilibrium. Quantities are measured in units per period.

Method. First solve the private market equilibrium, then include marginal damage to find efficient output. Evaluate the tax at that efficient quantity and verify the taxed market equilibrium.

Unregulated equilibrium. Set inverse demand equal to private marginal cost:

100-2Q_0=20+Q_0 \quad\Longrightarrow\quad 80=3Q_0 \quad\Longrightarrow\quad Q_0=\frac{80}{3}=26.6667

The unregulated price is $100-2(80/3)=140/3=\$46.67$ per unit, rounded to the nearest cent.

Efficient output. Social marginal cost is $20+Q+10+Q=30+2Q$. Equate this with inverse demand:

100-2Q^*=30+2Q^* \quad\Longrightarrow\quad 70=4Q^* \quad\Longrightarrow\quad Q^*=17.5

Tax. Evaluate marginal damage at $Q^*=17.5$:

t^*=MD(17.5)=10+17.5=\$27.50\text{ per unit}

Post-tax check. The taxed supply condition is $P_D(Q)=MPC(Q)+t^*$, or $100-2Q=20+Q+27.5=47.5+Q$. Hence $52.5=3Q$, so $Q=17.5$. Buyers pay $100-2(17.5)=\$65.00$; sellers receive $65-27.50=\$37.50$, which equals $MPC(17.5)=20+17.5=\$37.50$. The quantity and both prices are consistent. The common mistake is to set the tax using damage at $Q_0$; that would use $10+80/3=\$36.67$, not the damage at efficient output.

Worked Example: Keeping the Two Market Prices Straight

Suppose $P_D(Q)=150-3Q$, $MPC(Q)=30+2Q$, and $MD(Q)=6+Q$. Find the unregulated quantity and price, then calculate the efficient quantity, tax, price paid by buyers, and net price received by sellers.

Method. Solve the unregulated and efficient conditions separately. In the taxed market, compare the buyer price with marginal cost plus the tax.

Unregulated equilibrium.

150-3Q_0=30+2Q_0 \quad\Longrightarrow\quad 120=5Q_0 \quad\Longrightarrow\quad Q_0=24

The unregulated price is $150-3(24)=\$78.00$ per unit.

Efficient output. Social marginal cost is $30+2Q+6+Q=36+3Q$. Therefore:

150-3Q^*=36+3Q^* \quad\Longrightarrow\quad 114=6Q^* \quad\Longrightarrow\quad Q^*=19

Tax and post-tax prices. The tax is $MD(19)=6+19=\$25.00$ per unit. The taxed market condition is $150-3Q=30+2Q+25=55+2Q$, so $95=5Q$ and $Q=19$. Buyers pay $150-3(19)=\$93.00$. Sellers receive $93-25=\$68.00$, equal to their marginal cost $30+2(19)=\$68.00$. The check confirms the efficient quantity. Do not report the buyer price as the seller’s net receipt: the tax creates a \$25.00 difference.

Worked Example: A Sloped Marginal-Damage Curve

Consider inverse demand $P_D(Q)=120-Q$, private marginal cost $MPC(Q)=24+2Q$, and marginal damage $MD(Q)=12+0.5Q$. Find the unregulated outcome, the efficient output, the corrective tax, and the post-tax buyer and seller prices.

Method. Add the damage curve to private marginal cost before solving for efficient output. Then use the damage at that output as the per-unit tax.

Unregulated equilibrium.

120-Q_0=24+2Q_0 \quad\Longrightarrow\quad 96=3Q_0 \quad\Longrightarrow\quad Q_0=32

The unregulated price is $120-32=\$88.00$ per unit.

Efficient output. Social marginal cost is $(24+2Q)+(12+0.5Q)=36+2.5Q$. Thus:

120-Q^*=36+2.5Q^* \quad\Longrightarrow\quad 84=3.5Q^* \quad\Longrightarrow\quad Q^*=24

Tax and post-tax check. Marginal damage at 24 units is $12+0.5(24)=\$24.00$ per unit. The taxed condition is $120-Q=24+2Q+24=48+2Q$, which gives $72=3Q$, or $Q=24$. Buyers pay $120-24=\$96.00$. Sellers receive $96-24=\$72.00$, exactly $MPC(24)=24+2(24)=\$72.00$. The tax has shifted the market condition to the efficient one.

Common Exam Trap: Taxing at the Unregulated Quantity

Marginal damage generally changes with output. The efficient tax in this model is $MD(Q^*)$, because the goal is to make the taxed market condition hold at efficient output. Using $MD(Q_0)$ instead can push output below the efficient quantity.

In the first worked example, $Q_0=80/3$, so the incorrect tax would be $MD(Q_0)=10+80/3=110/3$, or about \$36.67 per unit. The resulting market equation would be $100-2Q=20+Q+110/3$, giving $Q=130/9$, or about 14.44 units. That is below the efficient quantity of 17.5. The correct tax is $MD(17.5)=\$27.50$, which gives $Q=17.5$.

The calculation relies on the specified marginal-damage function and the model’s assumptions. A constant per-unit tax equal to $MD(Q^*)$ implements the efficient quantity in this linear-curve setting. It does not mean that total damage equals the tax, nor that the tax payment is the same thing as the external harm. The tax is a per-unit incentive; total tax revenue would be the tax per unit multiplied by the number of units sold.

Key takeaway: Solve $P_D(Q_0)=MPC(Q_0)$ for the unregulated outcome. Solve $P_D(Q^*)=MPC(Q^*)+MD(Q^*)$ for efficient output. Then set $t^*=MD(Q^*)$ and verify $P_D(Q^*)=MPC(Q^*)+t^*$. This method is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Check Your Understanding

For each problem, calculate the unregulated quantity and price, then the efficient quantity and corrective tax. Where requested, check the taxed prices or revenue.

A market has $P_D(Q)=90-Q$, $MPC(Q)=18+2Q$, and $MD(Q)=6+Q$. Find the unregulated quantity and price, efficient quantity, tax, buyer price, and seller net price. (Answer: $Q_0=24$, unregulated price $\$66.00$; $Q^*=16.5$, tax $\$22.50$, buyer price $\$73.50$, seller net price $\$51.00$.)
Let $P_D(Q)=200-4Q$, $MPC(Q)=40+Q$, and $MD(Q)=8+2Q$. Find $Q_0$, $Q^*$, and the tax per unit. (Answer: $Q_0=32$; $Q^*=152/7\approx21.7143$; tax $=360/7\approx\$51.43$ per unit.)
Suppose $P_D(Q)=140-2Q$, $MPC(Q)=20+Q$, and $MD(Q)=4+0.5Q$. Find the unregulated price, efficient output, tax, and buyer price after the tax. (Answer: $Q_0=40$, unregulated price $\$60.00$; $Q^*=232/7\approx33.1429$; tax $=144/7\approx\$20.57$ per unit; buyer price $=516/7\approx\$73.71$.)
A market has $P_D(Q)=110-Q$, $MPC(Q)=10+Q$, and $MD(Q)=10+0.5Q$. Find the unregulated quantity, efficient quantity, tax, and tax revenue at the efficient quantity. (Answer: $Q_0=50$; $Q^*=36$; tax $=\$28.00$ per unit; revenue $=\$1{,}008.00$.)
For $P_D(Q)=180-2Q$, $MPC(Q)=30+Q$, and $MD(Q)=6+0.5Q$, find the unregulated quantity and price, efficient quantity, tax, and the buyer and seller prices after the tax. (Answer: $Q_0=50$, unregulated price $\$80.00$; $Q^*=288/7\approx41.1429$; tax $=186/7\approx\$26.57$ per unit; buyer price $=684/7\approx\$97.71$; seller net price $=498/7\approx\$71.14$.)

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