Signature
Q_0 = (a - c) / b; Q_star = (a - c - m) / (b + g); t_star = m + g * Q_star
| Inputs | Definition | Unit |
|---|---|---|
a | Marginal private benefit at zero consumption | pounds per unit |
c | Cost of one more unit borne by the consumer before any tax | 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 when the external cost per unit is constant | 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 |
t_star | Tax per unit equal to the marginal external cost at Q_star | pounds per unit |
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
Untaxed quantity, efficient quantity and Pigouvian tax in three cells
With the curve parameters in DemandIntercept, DemandSlope, PrivCost, ExtIntercept and ExtSlope, the three formulas return the untaxed quantity in PrivateQty, the efficient quantity in EfficientQty and the Pigouvian tax in PigouTax.
=(DemandIntercept-PrivCost)/DemandSlope; =(DemandIntercept-PrivCost-ExtIntercept)/(DemandSlope+ExtSlope); =ExtIntercept+ExtSlope*EfficientQty
Assumptions
Demand measures the true benefit to consumers
The demand curve is the marginal private benefit, so there are no internalities, and the market is otherwise undistorted. A tax that also targets costs people impose on themselves follows a different rule.
Linear curves and an interior efficient quantity
Marginal private benefit falls and marginal external cost rises linearly over the relevant range, marginal private cost is constant, and a is greater than c plus m, so the efficient quantity is above zero.
Tax passed through in full to the price consumers face
The tax raises the private cost of each unit by t_star, so consumers move along the same demand curve to the point where marginal private benefit equals c plus t_star.
Worked examples
Tax of 3.20 pounds with external cost rising at 0.025 per thousand units
In the article's illustrative product, marginal private benefit is 20 minus 0.1Q pounds, marginal private cost is 4 pounds and marginal external cost is 0.025Q pounds, with Q in thousands of units a year. Consumers choose 160 thousand units, the efficient quantity is 128 thousand and the Pigouvian tax is 3.20 pounds per unit. The tax also raises 3.2 × 128 = 409.6, or 409,600 pounds a year, which is a transfer from consumers to the government and not a cost to society.
a = 20; b = 0.1; c = 4; m = 0; g = 0.025; Q_0 = 160; Q_star = 128; t_star = 3.2
Constant marginal external cost of 3.20 pounds per unit
If each unit imposed a constant external cost of 3.20 pounds, the efficient quantity and the tax would be the same as in the article's example, and the external cost measured before the tax would give the right rate. The case is computed here for illustration and is not printed in the article.
a = 20; b = 0.1; c = 4; m = 3.2; g = 0; Q_0 = 160; Q_star = 128; t_star = 3.2
No external cost leaves consumption untaxed
With m and g both zero the private and efficient quantities coincide at 160 thousand units and the Pigouvian tax is zero, a limiting case that checks the formula.
a = 20; b = 0.1; c = 4; m = 0; g = 0; Q_0 = 160; Q_star = 160; t_star = 0
Common errors
Average external cost used instead of marginal
Dividing the total external cost at the efficient quantity by the number of units gives an average of 1.60 pounds per unit in the article's example, half the marginal 3.20 pounds, because the external cost rises with consumption. A tax of 1.60 pounds leaves consumption at 144 thousand units, well above the efficient 128 thousand. These figures are computed here for illustration.
Pigouvian tax revenue counted as a cost to society
The 409,600 pounds raised in the article's example is a transfer from consumers to the government and cancels out at the level of society. Counting it as a cost, or as part of the welfare gain, misstates the case for the tax, which comes from the change in consumption.
Sources
Pigou on the rate of tax with the optimum effect
Pigou AC. The Economics of Welfare. 4th ed. London: Macmillan; 1932. First published 1920. Part II, Chapter IX, section 13, on bounties and taxes as the most obvious ways for the State to encourage or restrain an activity whose social and private net products diverge, and Chapter XI, section 11, which states that there is one rate of tax that would have the optimum effect.
External cost priced at the socially optimal level
Fullerton D, Leicester A, Smith S. Environmental taxes. NBER Working Paper 14197. Cambridge, MA: National Bureau of Economic Research; 2008. Section 5.2, which states that the carbon price should reflect the damage from the marginal tonne of carbon dioxide at the socially optimal level of abatement, as with any other externality tax.
Average and marginal damage in the calculation of a Pigouvian tax
Coase RH. The problem of social cost. Journal of Law and Economics. 1960;3:1-44. Section IX, which notes that a tax proposal of this kind faces the problem of calculation and the difference between average and marginal damage.
Canonical Identity
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