Signature
Q_0 = (a - c) / b; Q_star = (a - c - m) / (b + g); t_0 = m + g * Q_0; Q_t = (a - c - t_0) / b; L_t = 0.5 * (Q_star - Q_t) * ((a - b * Q_t) - (c + m + g * Q_t))
| Inputs | Definition | Unit |
|---|---|---|
a | Marginal private benefit at zero consumption | pounds per unit |
c | Constant cost of one more unit borne by the consumer before 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 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 |
t_0 | Marginal external cost at the untaxed quantity Q_0, used as the tax per unit | pounds per unit |
Q_t | Quantity at which marginal private benefit equals c plus t_0 | thousand units a year |
L_t | Net social benefit forgone by consuming Q_t 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
Overshooting tax, taxed quantity and remaining loss
With the curve parameters in the named cells used for the Pigouvian tax and the quantities in PrivateQty and EfficientQty, the formulas return the tax set at pre-tax damage in PreTaxMEC, the taxed quantity in TaxedQty and the remaining welfare loss.
=ExtIntercept+ExtSlope*PrivateQty; =(DemandIntercept-PrivCost-PreTaxMEC)/DemandSlope; =0.5*(EfficientQty-TaxedQty)*((DemandIntercept-DemandSlope*TaxedQty)-(PrivCost+ExtIntercept+ExtSlope*TaxedQty))
Assumptions
Marginal external cost rising or constant
g is zero or above, so the tax set at pre-tax damage is at least the Pigouvian rate and Q_t is no higher than Q_star. With g equal to zero the two taxes coincide and the remaining loss is zero.
Positive quantity under the overshooting tax
a is greater than c plus t_0, so some consumption remains under the tax. Otherwise consumption stops altogether and the triangle formula no longer applies.
Worked examples
Tax of 4 pounds overshoots to 120 thousand units
In the article's example the marginal external cost at 160 thousand units is 4 pounds. A tax of 4 pounds cuts consumption to 120 thousand units, where marginal private benefit is 8 and marginal social cost 7, so the remaining loss is 0.5 × 8 × 1 = 4, or 4,000 pounds a year, against 64,000 pounds with no tax.
a = 20; b = 0.1; c = 4; m = 0; g = 0.025; Q_0 = 160; Q_star = 128; t_0 = 4; Q_t = 120; L_t = 4
Constant external cost leaves no overshoot
With a constant external cost of 3.20 pounds per unit, the damage measured before the tax equals the Pigouvian rate, consumption falls to the efficient 128 thousand units and no loss remains. 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; t_0 = 3.2; Q_t = 128; L_t = 0
Common errors
Damage observed before the tax taken as the Pigouvian rate
Setting the tax at the marginal external cost observed at current consumption, 4 pounds in the article's example, overshoots the Pigouvian 3.20 pounds whenever the external cost rises with consumption. The overshoot cuts consumption 8 thousand units below the efficient level, and each of those units was worth up to 1 pound more than its social cost.
Overshooting tax judged only against the untaxed market
A loss of 4,000 pounds against 64,000 pounds with no tax shows that the overshooting tax is better than none, not that it is efficient. The comparison that identifies the error is with the Pigouvian tax, which leaves no loss.
Sources
Externality tax at the damage of the marginal unit at the optimum
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 price should reflect the damage of the marginal tonne at the socially optimal level of abatement, and identifies the Pigouvian carbon tax by the point where marginal damage equals the marginal cost of abatement.
Pigou on a single optimal rate of tax
Pigou AC. The Economics of Welfare. 4th ed. London: Macmillan; 1932. First published 1920. Part II, Chapter XI, section 11, which states that where marginal social net product falls short of marginal private net product there is one rate of tax that would have the optimum effect.
Canonical Identity
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