How to Calculate a Pigouvian Tax
A Pigouvian tax equals the marginal external cost at the socially optimal quantity, the vertical distance between MSC and MPC at the optimal quantity.
The Pigouvian Tax formula
Calculator
Enter demand, private cost and social cost to get the optimal quantity and the corrective tax that closes the gap.
Demand is P = 40 − 0.5Q, so the intercept is 40.
How far price falls per extra unit. Enter 0.5 for P = 40 − 0.5Q.
MPC = 10 + 0.5Q, the cost the firm itself pays.
How far private cost rises per extra unit.
MSC is MPC plus the external damage, so 10 + 6 = 16.
Normally the same slope as MPC, since the spillover is a constant per unit.
Set the tax at $6 per unit, the gap between MSC and MPC at 24 units, and private cost lands on social cost.
- Free-market quantity
- 30
- Socially optimal quantity
- 24
- Price buyers pay after the tax
- $28
- Tax revenue
- $144
- Market verdict
- Overproduction
Where MPC meets demand, so the market ignores the spillover and trades 30 units.
Where MSB meets MSC. The market overshoots it by 6 units.
Buyers pay $28 and sellers keep $22 once the tax is handed over.
Tax per unit times the quantity traded after the tax, which is the optimal quantity.
How to calculate Pigouvian Tax, step by step
- 1Find the socially optimal quantity. Set marginal social benefit equal to marginal social cost; Q optimal is where MSB = MSC, not where the private market lands.
- 2Measure the external cost there. At Q optimal, subtract marginal private cost from marginal social cost; that vertical gap is the marginal external cost.
- 3Set the tax equal to that gap. The per-unit tax equals the marginal external cost at Q optimal, which raises private cost until firms face the full social cost.
- 4Check the new market outcome. With the tax in place the firm's private cost curve sits on marginal social cost, so it produces Q optimal and the deadweight loss disappears.
- 5Find revenue if asked. Tax revenue = per-unit tax × the quantity traded after the tax, which is Q optimal.
Worked example: Pigouvian Tax
A steel mill has marginal private cost MPC = 10 + 0.5Q and faces demand (MSB) P = 40 − 0.5Q. Each ton dumps $6 of pollution damage on neighbors, so MSC = 16 + 0.5Q. Free market: 10 + 0.5Q = 40 − 0.5Q gives Q = 30 at a price of 40 − 15 = $25. Social optimum: 16 + 0.5Q = 40 − 0.5Q gives Q = 24, where MSC = 16 + 12 = $28 and MSB = 40 − 12 = $28. The marginal external cost at Q = 24 is 28 − 22 = $6, so the Pigouvian tax is $6 per ton. After the tax buyers pay $28, sellers keep 28 − 6 = $22, output falls from 30 to 24 tons, and revenue = $6 × 24 = $144.
Pigouvian Tax questions
Do you measure the external cost at the market quantity or the optimal quantity?
At the socially optimal quantity, since that is where the tax has to close the gap between private and social cost. When the external cost per unit is constant the two readings give the same number, but on a free-response question read it at Q optimal.
What does a Pigouvian tax do to the supply curve?
It shifts marginal private cost up by exactly the tax amount, so the firm's new supply curve lies on top of marginal social cost. Output then falls to the socially optimal quantity.
Does a Pigouvian tax create deadweight loss?
No, set equal to the marginal external cost it removes the deadweight loss from overproduction rather than creating any. A tax larger than the external cost overshoots and causes underproduction, which is a new deadweight loss.
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