Signature
Q_p = (a - c) / b; Q_s = (a + m - c) / (b + g); s = m - g * (a + m - c) / (b + g); DWL = 0.5 * ((a + m - c) / (b + g) - (a - c) / b) * (m - g * (a - c) / b)
| Inputs | Definition | Unit |
|---|---|---|
a | Marginal private benefit of the first vaccination, the intercept of the demand curve | currency per vaccination |
c | Constant marginal cost of a vaccination, borne in full by the person vaccinated and equal to marginal social cost | currency per vaccination |
b | Fall in marginal private benefit for each additional thousand vaccinations, greater than zero | currency per vaccination per thousand vaccinations |
m | Marginal external benefit of the first vaccination | currency per vaccination |
g | Fall in marginal external benefit for each additional thousand vaccinations, zero when the external benefit is constant | currency per vaccination per thousand vaccinations |
Q_p | Vaccinations at which marginal private benefit equals marginal cost | thousand vaccinations |
|---|---|---|
Q_s | Vaccinations at which marginal social benefit equals marginal cost | thousand vaccinations |
s | Subsidy equal to the marginal external benefit at the efficient quantity | currency per vaccination |
DWL | Net social benefit forgone by stopping at Q_p instead of Q_s | thousand units of currency |
Function
Social marginal cost and benefit function
Adds the marginal effect that an activity has on third parties to the marginal cost or benefit that the decision-maker weighs, giving the marginal social cost or benefit. The efficient quantity is where marginal social benefit equals marginal social cost; a private decision that ignores the external term stops at a different quantity, and a Pigouvian tax or subsidy set at the marginal external effect at the efficient quantity closes the gap.
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Implementations
Excel
Linear Pigouvian subsidy and deadweight loss
With the curve parameters in named cells DemandIntercept, DemandSlope, MargCost, ExtIntercept and ExtSlope, the four formulas return the private quantity, the efficient quantity, the subsidy and the deadweight loss.
=(DemandIntercept-MargCost)/DemandSlope; =(DemandIntercept+ExtIntercept-MargCost)/(DemandSlope+ExtSlope); =ExtIntercept-ExtSlope*EfficientQty; =0.5*(EfficientQty-PrivateQty)*(ExtIntercept-ExtSlope*PrivateQty)
Assumptions
Linear curves over the relevant range
Marginal private and external benefit fall linearly and marginal cost is constant between Q_p and Q_s. Transmission models usually give a non-linear external benefit, so the linear form is an approximation around the current coverage.
Interior solution below full coverage
a is greater than c, so some people vaccinate privately, and Q_s lies within the population that can be vaccinated. At very high coverage the marginal external benefit can reach zero and the formulas no longer apply.
Subsidy passed through to the price faced
The subsidy lowers the price faced by each person by s, so private choice moves along the same demand curve to the point where marginal private benefit equals c minus s.
Worked examples
Vaccination subsidy consistent with the article's margin
Marginal private benefit is 100 minus Q per vaccination, the marginal cost 60 and the marginal external benefit 29 minus 0.2 × Q, with Q in thousands; at 45 thousand these give the article's margin of 55, 20 and 60. People choose 40 thousand vaccinations, the efficient number is 57.5 thousand, the Pigouvian subsidy is 17.5 per vaccination and the deadweight loss is 183.75 thousand. The figures are illustrative.
a = 100; b = 1; c = 60; m = 29; g = 0.2; Q_p = 40; Q_s = 57.5; s = 17.5; DWL = 183.75
Constant external benefit
If the external benefit stays at 20 per vaccination whatever the coverage, the subsidy equals that constant benefit, the efficient number rises to 60 thousand and the deadweight loss is 200 thousand.
a = 100; b = 1; c = 60; m = 20; g = 0; Q_p = 40; Q_s = 60; s = 20; DWL = 200
No external benefit leaves take-up unchanged
With no external benefit (m and g both zero), the private and efficient quantities coincide at 40 thousand, so no subsidy is needed and there is no deadweight loss. The case checks that the formula reduces to the private market when the externality disappears.
a = 100; b = 1; c = 60; m = 0; g = 0; Q_p = 40; Q_s = 40; s = 0; DWL = 0
Common errors
Setting the subsidy at the external benefit at the private quantity
In the first example the marginal external benefit at 40 thousand is 21. A subsidy of 21 moves take-up to 61 thousand, beyond the efficient 57.5 thousand, because the external benefit has fallen to 17.5 by then. The Pigouvian rate is the marginal external benefit at the efficient quantity.
Treating the deadweight triangle as the value of the whole programme
The 183.75 thousand is the welfare gain from moving from private to efficient take-up. It does not represent the total net benefit of the vaccination programme and excludes the benefits already generated by vaccinations undertaken privately. The calculation also abstracts from any distortionary cost of financing the subsidy, since the subsidy payment itself is a transfer rather than a social cost.
Sources
Pigou on bounties and taxes to correct a divergence
Pigou AC. The Economics of Welfare. 4th ed. London: Macmillan; 1932. Part II, Chapter IX, section 13: bounties for activities whose social net product exceeds the private, and taxes where it falls short.
Textbook derivation of the Pigouvian correction
Mas-Colell A, Whinston MD, Green JR. Microeconomic Theory. New York: Oxford University Press; 1995. Chapter 11, section 11.B: the optimal Pigouvian tax equals the marginal externality at the optimum, and the welfare loss of the unregulated equilibrium.
Canonical Identity
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