Signature
Q_0 = (a - c) / b; Q_star = (a - c - m) / (b + g); DWL = 0.5 * (Q_0 - Q_star) * (m + g * Q_0)
| Inputs | Definition | Unit |
|---|---|---|
a | Marginal private benefit at zero consumption | pounds per unit |
c | Constant cost of one more unit borne by the consumer | pounds per unit |
b | Fall in marginal private benefit per thousand units, above zero | pounds per unit per thousand units |
m | Marginal external cost of the first unit, zero or above | pounds per unit |
g | Rise in marginal external cost per thousand units, zero or above | pounds per unit per thousand units |
Q_0 | Quantity at which marginal private benefit equals marginal private cost | thousand units a year |
|---|---|---|
Q_star | Quantity at which marginal private benefit equals marginal social cost | thousand units a year |
DWL | Net social loss from consuming Q_0 rather than Q_star | thousand pounds a year |
Function
Pigouvian tax on a negative externality set at marginal external cost at the efficient quantity
Finds the quantity at which the marginal private benefit of an activity equals its marginal social cost, the marginal private cost plus the marginal external cost, and sets a tax per unit equal to the marginal external cost at that quantity, so that the decision-maker faces the full social cost of each unit. It is the negative-externality mirror of the vaccination subsidy HE-FM-EXT-002 within the social marginal cost and benefit framework HE-FM-EXT-001 on the Externality page. The formulae below apply the rule with a linear demand curve, a constant marginal private cost and a marginal external cost that rises with the quantity consumed.
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Implementations
Excel
Deadweight loss triangle of an untaxed external cost
With the quantities in PrivateQty and EfficientQty and the external cost parameters in ExtIntercept and ExtSlope, the cell returns the deadweight loss in DeadweightLoss.
=0.5*(PrivateQty-EfficientQty)*(ExtIntercept+ExtSlope*PrivateQty)
Assumptions
Linear curves between the efficient and untaxed quantities
The gap between marginal social cost and marginal private benefit changes linearly from zero at Q_star to the marginal external cost at Q_0, so the loss is a triangle. With curved schedules the area has to be integrated.
Welfare measured as consumer benefit less social cost
The loss counts the benefit to consumers less private and external costs, valued in the same currency and year. Tax revenue is a transfer and plays no part, and any cost of administering the tax is left out.
Worked examples
Deadweight loss of 64 thousand pounds a year in the article's example
Between 128 and 160 thousand units the gap rises to 4 pounds per unit at 160 thousand, so the loss is 0.5 × 32 × 4 = 64, or 64,000 pounds a year, as in the article.
a = 20; b = 0.1; c = 4; m = 0; g = 0.025; Q_0 = 160; Q_star = 128; DWL = 64
Deadweight loss with a constant external cost of 3.20 pounds
With a constant external cost of 3.20 pounds per unit the quantities are the same, but the gap at 160 thousand is 3.20 pounds, so the loss is 51.2 thousand pounds a year. The case is computed here for illustration.
a = 20; b = 0.1; c = 4; m = 3.2; g = 0; Q_0 = 160; Q_star = 128; DWL = 51.2
Common errors
Triangle height taken from the external cost at the efficient quantity
Using the marginal external cost at Q_star, 3.20 pounds, as the height of the triangle gives 51.2 instead of 64 thousand pounds in the article's example. When the external cost rises with consumption, the height is the gap at Q_0. The 51.2 is computed here for illustration.
Deadweight loss read as the total external cost
The triangle is the loss from the units consumed beyond Q_star. The external cost imposed by all 160 thousand units is much larger, but most of it comes from units whose benefit exceeds their social cost, which the tax keeps.
Sources
Welfare gain and revenue of an ideal Pigouvian tax
Fullerton D, Leicester A, Smith S. Environmental taxes. NBER Working Paper 14197. Cambridge, MA: National Bureau of Economic Research; 2008. Section 4, Figure 4.1, in which an ideal Pigouvian tax reduces purchases to the efficient quantity, the tax revenue is one area and the welfare gain is the area by which social costs exceed the marginal benefits to consumers for the purchases removed.
Canonical Identity
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